diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 79ec0bf..f9f1ef6 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-04T17:22:47","documenter_version":"1.5.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-05T07:19:23","documenter_version":"1.5.0"}} \ No newline at end of file diff --git a/dev/api-ctbase.html b/dev/api-ctbase.html index 6e69bde..4cce40b 100644 --- a/dev/api-ctbase.html +++ b/dev/api-ctbase.html @@ -1,5 +1,5 @@ -CTBase API · CTDirect.jl
GitHub

CTBase API

This is just a dump of CTBase API documentation. For more details about CTBase.jl package, see the documentation.

Index

Documentation

CTBase.CTBase — Module

CTBase module.

Lists all the imported modules and packages:

  • Base
  • Core
  • DataStructures
  • DocStringExtensions
  • LinearAlgebra
  • MLStyle
  • Parameters
  • PrettyTables
  • Printf
  • ReplMaker
  • Unicode

List of all the exported names:

source
CTBase.TimesDisc — Type

Type alias for a grid of times. This is used to define a discretization of time interval given to solvers.

julia> const TimesDisc = Union{Times, StepRangeLen}

See also: Time, Times.

source
CTBase.AmbiguousDescription — Type
struct AmbiguousDescription <: CTException

Exception thrown when the description is ambiguous / incorrect.

Fields

  • var::Tuple{Vararg{Symbol}}
source
CTBase.BoundaryConstraint — Type
struct BoundaryConstraint{variable_dependence}

Fields

  • f::Function

The default value for variable_dependence is Fixed.

Constructor

The constructor BoundaryConstraint returns a BoundaryConstraint of a function. The function must take 2 or 3 arguments (x0, xf) or (x0, xf, v), if the function is variable, it must be specified. Dependencies are specified with a boolean, variable, false by default or with a DataType, NonFixed/Fixed, Fixed by default.

Examples

julia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])
+CTBase API · CTDirect.jl
GitHub
CTBase API
This is just a dump of CTBase API documentation. For more details about CTBase.jl package, see the documentation.
Index
Documentation
CTBase.CTBase — Module
CTBase module.
Lists all the imported modules and packages:
  • Base
  • Core
  • DataStructures
  • DocStringExtensions
  • LinearAlgebra
  • MLStyle
  • Parameters
  • PrettyTables
  • Printf
  • ReplMaker
  • Unicode
List of all the exported names:
source
CTBase.TimesDisc — Type
Type alias for a grid of times. This is used to define a discretization of time interval given to solvers.
julia> const TimesDisc = Union{Times, StepRangeLen}
See also: Time, Times.
source
CTBase.AmbiguousDescription — Type
struct AmbiguousDescription <: CTException
Exception thrown when the description is ambiguous / incorrect.
Fields
  • var::Tuple{Vararg{Symbol}}
source
CTBase.BoundaryConstraint — Type
struct BoundaryConstraint{variable_dependence}
Fields
  • f::Function
The default value for variable_dependence is Fixed.
Constructor
The constructor BoundaryConstraint returns a BoundaryConstraint of a function. The function must take 2 or 3 arguments (x0, xf) or (x0, xf, v), if the function is variable, it must be specified.  Dependencies are specified with a boolean, variable, false by default or with a DataType, NonFixed/Fixed, Fixed by default.
Examples
julia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])
 julia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], variable=true)
 julia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], NonFixed)
Warning
When the state is of dimension 1, consider x0 and xf as a scalar. When the constraint is dimension 1, return a scalar.
Call
The call returns the evaluation of the BoundaryConstraint for given values. If a variable is given for a non variable dependent boundary constraint, it will be ignored.
Examples
julia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])
 julia> B([0, 0], [1, 1])
@@ -1695,4 +1695,4 @@
     startOnLoad: true,
     theme: "neutral"
 });
-

+
diff --git a/dev/api.html b/dev/api.html index 07f86db..2b23d2c 100644 --- a/dev/api.html +++ b/dev/api.html @@ -1,20 +1,20 @@ -API · CTDirect.jl
GitHub

CTDirect API

Index

Documentation

CTDirect.DOCP — Type

Struct for discretized optimal control problem DOCP

Contains:

  • a copy of the original OCP
  • a NLP formulation of the DOCP
  • data required to link the two problems
source
CTDirect.OCPSolutionFromDOCP — Method
OCPSolutionFromDOCP(
+API · CTDirect.jl
GitHub
CTDirect API
Index
Documentation
CTDirect.DOCP — Type
Struct for discretized optimal control problem DOCP
Contains:
  • a copy of the original OCP
  • a NLP formulation of the DOCP
  • data required to link the two problems
source
CTDirect.OCPSolutionFromDOCP — Method
OCPSolutionFromDOCP(
     docp,
     docp_solution_ipopt
 ) -> OptimalControlSolution
-
Build OCP functional solution from DOCP vector solution (given as a GenericExecutionStats)
source
CTDirect.available_methods — Method
available_methods() -> Tuple{Tuple{Vararg{Symbol}}}
-
Return the list of available methods to solve the optimal control problem.
source
CTDirect.available_methods — Method
available_methods() -> Tuple{Tuple{Vararg{Symbol}}}
+
Return the list of available methods to solve the optimal control problem.
source
CTDirect.directTranscription — Method
directTranscription(
     ocp::OptimalControlModel,
     description...;
     init,
     grid_size,
     time_grid
 ) -> DOCP
-
Discretize an optimal control problem into a nonlinear optimization problem (ie direct transcription)
source
CommonSolve.solve — Function
CommonSolve.solve(args...; kwargs...)
Solves an equation or other mathematical problem using the algorithm specified in the arguments. Generally, the interface is:
CommonSolve.solve(prob::ProblemType,alg::SolverType; kwargs...)::SolutionType
where the keyword arguments are uniform across all choices of algorithms.
By default, solve defaults to using solve! on the iterator form, i.e.:
solve(args...; kwargs...) = solve!(init(args...; kwargs...))
source
solve(
+
Discretize an optimal control problem into a nonlinear optimization problem (ie direct transcription)
source
CommonSolve.solve — Function
CommonSolve.solve(args...; kwargs...)
Solves an equation or other mathematical problem using the algorithm specified in the arguments. Generally, the interface is:
CommonSolve.solve(prob::ProblemType,alg::SolverType; kwargs...)::SolutionType
where the keyword arguments are uniform across all choices of algorithms.
By default, solve defaults to using solve! on the iterator form, i.e.:
solve(args...; kwargs...) = solve!(init(args...; kwargs...))
source
solve(
     docp::DOCP;
     init,
     display,
@@ -25,7 +25,7 @@
     linear_solver,
     kwargs...
 ) -> SolverCore.GenericExecutionStats
-
Solve a discretized optimal control problem DOCP
source
solve(
+
Solve a discretized optimal control problem DOCP
source
solve(
     ocp::OptimalControlModel,
     description...;
     init,
@@ -39,21 +39,21 @@
     linear_solver,
     kwargs...
 ) -> OptimalControlSolution
-
Solve an optimal control problem OCP by direct method
source

+
diff --git a/dev/continuation-17b6381d.svg b/dev/continuation-0fd0ad39.svg similarity index 84% rename from dev/continuation-17b6381d.svg rename to dev/continuation-0fd0ad39.svg index 9901366..b468b69 100644 --- a/dev/continuation-17b6381d.svg +++ b/dev/continuation-0fd0ad39.svg @@ -1,287 +1,287 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/continuation.html b/dev/continuation.html index 4a5e3d5..e2c1cb5 100644 --- a/dev/continuation.html +++ b/dev/continuation.html @@ -83,9 +83,9 @@ ylabel!("Maximal altitude r(tf)") plot(sol0) p = plot!(sol) -plot(pobj, p, layout=2)Example block output + diff --git a/dev/dev-api.html b/dev/dev-api.html index 2d17c90..f093b8f 100644 --- a/dev/dev-api.html +++ b/dev/dev-api.html @@ -1,19 +1,19 @@ -Developers · CTDirect.jl
GitHub

Internal functions

CTDirect.DOCP_initial_guess — Function
DOCP_initial_guess(docp) -> Any
+Developers · CTDirect.jl
GitHub
Internal functions
CTDirect.DOCP_initial_guess — Function
DOCP_initial_guess(docp) -> Any
 DOCP_initial_guess(docp, init::OptimalControlInit) -> Any
-
Build initial guess for discretized problem
source
CTDirect.get_lagrange_cost_at_time_step — Method
get_lagrange_cost_at_time_step(xu, docp, i) -> Any
-
Retrieve the additional state variable corresponding to the lagrange (running) cost at given time step from the NLP variables
source
CTDirect.get_lagrange_cost_at_time_step — Method
get_lagrange_cost_at_time_step(xu, docp, i) -> Any
+
Retrieve the additional state variable corresponding to the lagrange (running) cost at given time step from the NLP variables
source
CTDirect.parse_DOCP_solution — Method
parse_DOCP_solution(
     docp,
     solution,
     multipliers_constraints,
@@ -21,7 +21,7 @@
     multipliers_UB,
     constraints
 ) -> NTuple{18, Any}
-
Parse DOCP solution into OCP variables, constraints and multipliers
source
CTDirect.setPathConstraintsAtTimeStep! — Method
setPathConstraintsAtTimeStep!(
     docp,
     index,
     target;
@@ -30,7 +30,7 @@
     lb,
     ub
 ) -> Any
-
Set the path constraints / bounds for given time step target = :constraints | :bounds
source
CTDirect.setPunctualConditions! — Method
setPunctualConditions!(
     docp,
     index,
     target;
@@ -40,20 +40,20 @@
     lb,
     ub
 ) -> Any
-
Set the boundary and variable constraints / their bounds target = :constraints | :bounds
source
CTDirect.set_variable! — Method
set_variable!(xu, v_init, docp) -> Any
-
Set optimization variables in the NLP variables (for initial guess)
source
CTDirect.variables_bounds — Method
variables_bounds(docp) -> Tuple{Any, Any}
-
Build upper and lower bounds vectors for the DOCP variable box constraints.
source

+
diff --git a/dev/index.html b/dev/index.html index 4e28ac2..c948768 100644 --- a/dev/index.html +++ b/dev/index.html @@ -7,7 +7,7 @@ P --> B F --> B D --> B -style D fill:#FFF0F5
Install

To install a package from the control-toolbox ecosystem, please visit the installation page.

An optimal control problem with fixed initial and final times, denoted (OCP), can be described as minimising the cost functional

\[g(x(t_0), x(t_f)) + \int_{t_0}^{t_f} f^{0}(t, x(t), u(t))~\mathrm{d}t\]

where the state $x$ and the control $u$ are functions subject, for $t \in [t_0, t_f]$, to the differential constraint

\[ \dot{x}(t) = f(t, x(t), u(t))\]

and other constraints such as

\[\begin{array}{llcll} +style D fill:#FBF275

Install

To install a package from the control-toolbox ecosystem, please visit the installation page.

An optimal control problem with fixed initial and final times, denoted (OCP), can be described as minimising the cost functional

\[g(x(t_0), x(t_f)) + \int_{t_0}^{t_f} f^{0}(t, x(t), u(t))~\mathrm{d}t\]

where the state $x$ and the control $u$ are functions subject, for $t \in [t_0, t_f]$, to the differential constraint

\[ \dot{x}(t) = f(t, x(t), u(t))\]

and other constraints such as

\[\begin{array}{llcll} ~\xi_l &\le& \xi(t, u(t)) &\le& \xi_u, \\ \eta_l &\le& \eta(t, x(t)) &\le& \eta_u, \\ \psi_l &\le& \psi(t, x(t), u(t)) &\le& \psi_u, \\ @@ -34,4 +34,4 @@ startOnLoad: true, theme: "neutral" }); -

  • 1J. T. Betts. Practical methods for optimal control using nonlinear programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.
  • 2J. Nocedal and S.J. Wright. Numerical optimization. Springer-Verlag, New York, 1999.****
+
  • 1J. T. Betts. Practical methods for optimal control using nonlinear programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.
  • 2J. Nocedal and S.J. Wright. Numerical optimization. Springer-Verlag, New York, 1999.****
diff --git a/dev/search_index.js b/dev/search_index.js index 858a06a..eda9d92 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"api.html#CTDirect-API","page":"API","title":"CTDirect API","text":"","category":"section"},{"location":"api.html#Index","page":"API","title":"Index","text":"","category":"section"},{"location":"api.html","page":"API","title":"API","text":"Pages = [\"api.md\"]\nModules = [CTDirect, CommonSolve]\nOrder = [:module, :constant, :type, :function, :macro]","category":"page"},{"location":"api.html#Documentation","page":"API","title":"Documentation","text":"","category":"section"},{"location":"api.html","page":"API","title":"API","text":"Modules = [CTDirect]\nOrder = [:module, :constant, :type, :function, :macro]\nPrivate = false","category":"page"},{"location":"api.html#CTDirect.DOCP","page":"API","title":"CTDirect.DOCP","text":"Struct for discretized optimal control problem DOCP\n\nContains:\n\na copy of the original OCP\na NLP formulation of the DOCP\ndata required to link the two problems\n\n\n\n\n\n","category":"type"},{"location":"api.html#CTDirect.OCPSolutionFromDOCP-Tuple{Any, Any}","page":"API","title":"CTDirect.OCPSolutionFromDOCP","text":"OCPSolutionFromDOCP(\n docp,\n docp_solution_ipopt\n) -> OptimalControlSolution\n\n\nBuild OCP functional solution from DOCP vector solution (given as a GenericExecutionStats)\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.available_methods-Tuple{}","page":"API","title":"CTDirect.available_methods","text":"available_methods() -> Tuple{Tuple{Vararg{Symbol}}}\n\n\nReturn the list of available methods to solve the optimal control problem.\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.directTranscription-Tuple{OptimalControlModel, Vararg{Any}}","page":"API","title":"CTDirect.directTranscription","text":"directTranscription(\n ocp::OptimalControlModel,\n description...;\n init,\n grid_size,\n time_grid\n) -> DOCP\n\n\nDiscretize an optimal control problem into a nonlinear optimization problem (ie direct transcription)\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.getNLP-Tuple{DOCP}","page":"API","title":"CTDirect.getNLP","text":"getNLP(docp::DOCP) -> Any\n\n\nExtract the NLP problem from the DOCP\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.is_solvable-Tuple{Any}","page":"API","title":"CTDirect.is_solvable","text":"is_solvable(ocp) -> Bool\n\n\nCheck if an OCP is solvable by the method solve.\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.setInitialGuess-Tuple{DOCP, Any}","page":"API","title":"CTDirect.setInitialGuess","text":"setInitialGuess(docp::DOCP, init) -> Any\n\n\nSet initial guess in the DOCP\n\n\n\n\n\n","category":"method"},{"location":"api.html","page":"API","title":"API","text":"solve\nsave_OCP_solution\nload_OCP_solution\nexport_OCP_solution\nread_OCP_solution","category":"page"},{"location":"api.html#CommonSolve.solve","page":"API","title":"CommonSolve.solve","text":"CommonSolve.solve(args...; kwargs...)\n\nSolves an equation or other mathematical problem using the algorithm specified in the arguments. 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sol::OptimalControlSolution;\n filename_prefix\n)\n\n\nSave OCP solution in JLD2 format\n\n\n\n\n\n","category":"function"},{"location":"api.html#CTDirect.load_OCP_solution","page":"API","title":"CTDirect.load_OCP_solution","text":"load_OCP_solution() -> Any\nload_OCP_solution(filename_prefix) -> Any\n\n\nLoad OCP solution in JLD2 format\n\n\n\n\n\n","category":"function"},{"location":"api.html#CTDirect.export_OCP_solution","page":"API","title":"CTDirect.export_OCP_solution","text":"export_OCP_solution(\n sol::OptimalControlSolution;\n filename_prefix\n)\n\n\nExport OCP solution in JSON format\n\n\n\n\n\n","category":"function"},{"location":"api.html#CTDirect.read_OCP_solution","page":"API","title":"CTDirect.read_OCP_solution","text":"read_OCP_solution() -> Any\nread_OCP_solution(filename_prefix) -> Any\n\n\nRead OCP solution in JSON 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\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.constraints_bounds-Tuple{Any}","page":"Developers","title":"CTDirect.constraints_bounds","text":"constraints_bounds(docp) -> Tuple{Any, Any}\n\n\nBuild upper and lower bounds vectors for the DOCP nonlinear constraints.\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_control_at_time_step-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.get_control_at_time_step","text":"get_control_at_time_step(xu, docp, i) -> Any\n\n\nRetrieve control variables at given time step from the NLP variables\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_final_time-Tuple{Any, Any}","page":"Developers","title":"CTDirect.get_final_time","text":"get_final_time(xu, docp) -> Any\n\n\nRetrieve final time for OCP (may be fixed or variable)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_initial_time-Tuple{Any, Any}","page":"Developers","title":"CTDirect.get_initial_time","text":"get_initial_time(xu, docp) -> Any\n\n\nRetrieve initial time for OCP (may be fixed or variable)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_lagrange_cost_at_time_step-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.get_lagrange_cost_at_time_step","text":"get_lagrange_cost_at_time_step(xu, docp, i) -> Any\n\n\nRetrieve the additional state variable corresponding to the lagrange (running) cost at given time step from the NLP variables\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_state_at_time_step-Tuple{Any, Any, Int64}","page":"Developers","title":"CTDirect.get_state_at_time_step","text":"get_state_at_time_step(xu, docp, i::Int64) -> Any\n\n\nRetrieve state variables at given time step from the NLP variables\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_time_at_time_step-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.get_time_at_time_step","text":"get_time_at_time_step(xu, docp, i) -> Any\n\n\nGet actual (un-normalized) time at give time step\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_unnormalized_time-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.get_unnormalized_time","text":"get_unnormalized_time(xu, docp, t_normalized) -> Any\n\n\nGet actual (un-normalized) time value\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_variable-Tuple{Any, Any}","page":"Developers","title":"CTDirect.get_variable","text":"get_variable(xu, docp) -> Any\n\n\nRetrieve optimization variables from the NLP variables\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.parse_DOCP_solution-NTuple{6, Any}","page":"Developers","title":"CTDirect.parse_DOCP_solution","text":"parse_DOCP_solution(\n docp,\n solution,\n multipliers_constraints,\n multipliers_LB,\n multipliers_UB,\n constraints\n) -> NTuple{18, Any}\n\n\nParse DOCP solution into OCP variables, constraints and multipliers\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.setPathConstraintsAtTimeStep!-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.setPathConstraintsAtTimeStep!","text":"setPathConstraintsAtTimeStep!(\n docp,\n index,\n target;\n c,\n args,\n lb,\n ub\n) -> Any\n\n\nSet the path constraints / bounds for given time step target = :constraints | :bounds\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.setPunctualConditions!-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.setPunctualConditions!","text":"setPunctualConditions!(\n docp,\n index,\n target;\n c,\n args_0,\n args_f,\n lb,\n ub\n) -> Any\n\n\nSet the boundary and variable constraints / their bounds target = :constraints | :bounds\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.setStateEquationAtTimeStep!-NTuple{5, Any}","page":"Developers","title":"CTDirect.setStateEquationAtTimeStep!","text":"setStateEquationAtTimeStep!(\n docp,\n c,\n index,\n args_i,\n args_ip1\n) -> Any\n\n\nSet the constraints corresponding to the state equation\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.set_control_at_time_step!-NTuple{4, Any}","page":"Developers","title":"CTDirect.set_control_at_time_step!","text":"set_control_at_time_step!(xu, u_init, docp, i) -> Any\n\n\nSet control variables at given time step in the NLP variables (for initial guess)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.set_state_at_time_step!-NTuple{4, Any}","page":"Developers","title":"CTDirect.set_state_at_time_step!","text":"set_state_at_time_step!(xu, x_init, docp, i) -> Any\n\n\nSet state variables at given time step in the NLP variables (for initial guess)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.set_variable!-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.set_variable!","text":"set_variable!(xu, v_init, docp) -> Any\n\n\nSet optimization variables in the NLP variables (for initial guess)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.variables_bounds-Tuple{Any}","page":"Developers","title":"CTDirect.variables_bounds","text":"variables_bounds(docp) -> Tuple{Any, Any}\n\n\nBuild upper and lower bounds vectors for the DOCP variable box constraints.\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html","page":"Developers","title":"Developers","text":"","category":"page"},{"location":"continuation.html#Discrete-continuation","page":"Discrete continuation","title":"Discrete continuation","text":"","category":"section"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Using the warm start option, it is easy to implement a basic discrete continuation method, where a sequence of problems is solved using each solution as initial guess for the next problem. This usually gives better and faster convergence than solving each problem with the same initial guess, and is a way to handle problems that require a good initial guess.","category":"page"},{"location":"continuation.html#Continuation-on-parametric-OCP","page":"Discrete continuation","title":"Continuation on parametric OCP","text":"","category":"section"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"The most compact syntax to perform a discrete continuation is to use a function that returns the OCP for a given value of the continuation parameter, and solve a sequence of these problems. We illustrate this on a very basic double integrator with increasing fixed final time.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"First we load the required packages","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"using CTDirect\nusing CTBase\nusing NLPModelsIpopt\nusing Printf\nusing Plots","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"and write a function that returns the OCP for a given final time","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"function ocp_T(T)\n @def ocp begin\n t ∈ [ 0, T ], time\n x ∈ R², state\n u ∈ R, control\n q = x₁\n v = x₂\n q(0) == 0\n v(0) == 0\n q(T) == 1\n v(T) == 0\n ẋ(t) == [ v(t), u(t) ]\n ∫(u(t)^2) → min\n end\n return ocp\nend\nnothing # hide","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Then we perform the continuation with a simple for loop, using each solution to initialize the next problem.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"init1 = ()\niter_list = []\nfor T=1:5\n ocp1 = ocp_T(T) \n sol1 = solve(ocp1, print_level=0, init=init1)\n global init1 = sol1\n @printf(\"T %.2f objective %.6f iterations %d\\n\", T, sol1.objective, sol1.iterations)\n push!(iter_list, sol1.iterations)\nend","category":"page"},{"location":"continuation.html#Continuation-on-global-variable","page":"Discrete continuation","title":"Continuation on global variable","text":"","category":"section"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"As a second example, we show how to avoid redefining a new OCP each time, and modify the original one instead. More precisely we now solve a Goddard problem for a decreasing maximal thrust. If we store the value for Tmax in a global variable, we can simply modify this variable and keep the same OCP problem during the continuation.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Let us first define the Goddard problem (note that the formulation below illustrates all the possible constraints types, and the problem could be defined in a more compact way).","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Cd = 310\nTmax = 3.5\nβ = 500\nb = 2\nfunction F0(x)\n r, v, m = x\n D = Cd * v^2 * exp(-β*(r - 1))\n return [ v, -D/m - 1/r^2, 0 ]\nend\nfunction F1(x)\n r, v, m = x\n return [ 0, Tmax/m, -b*Tmax ]\nend\n\nocp = Model(variable=true)\nr0 = 1\nv0 = 0\nm0 = 1\nmf = 0.6\nx0=[r0,v0,m0]\nvmax = 0.1\nstate!(ocp, 3)\ncontrol!(ocp, 1)\nvariable!(ocp, 1)\ntime!(ocp, t0=0, indf=1)\nconstraint!(ocp, :initial, lb=x0, ub=x0)\nconstraint!(ocp, :final, rg=3, lb=mf, ub=Inf)\nconstraint!(ocp, :state, lb=[r0,v0,mf], ub=[r0+0.2,vmax,m0])\nconstraint!(ocp, :control, lb=0, ub=1)\nconstraint!(ocp, :variable, lb=0.01, ub=Inf)\nobjective!(ocp, :mayer, (x0, xf, v) -> xf[1], :max)\ndynamics!(ocp, (x, u, v) -> F0(x) + u*F1(x) )\n\nsol0 = solve(ocp, print_level=0)\nsol = sol0\n@printf(\"Objective for reference solution %.6f\\n\", sol0.objective)","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Then we perform the continuation on the maximal thrust.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Tmax_list = []\nobj_list = []\nfor Tmax_local=3.5:-0.5:1\n global Tmax = Tmax_local \n global sol = solve(ocp, print_level=0, init=sol)\n @printf(\"Tmax %.2f objective %.6f iterations %d\\n\", Tmax, sol.objective, sol.iterations)\n push!(Tmax_list, Tmax)\n push!(obj_list, sol.objective)\nend ","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"We plot now the objective w.r.t the maximal thrust, as well as both solutions for Tmax=3.5 and Tmax=1.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"pobj = plot(Tmax_list, obj_list, label=\"r(tf)\",seriestype=:scatter)\nxlabel!(\"Maximal thrust (Tmax)\")\nylabel!(\"Maximal altitude r(tf)\")\nplot(sol0)\np = plot!(sol)\nplot(pobj, p, layout=2)","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"","category":"page"},{"location":"index.html#CTDirect.jl","page":"Introduction","title":"CTDirect.jl","text":"","category":"section"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"CurrentModule = CTDirect","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"The CTDirect.jl package is part of the control-toolbox ecosystem.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"flowchart TD\nO(OptimalControl) --> B(CTBase)\nO --> D(CTDirect)\nO --> F(CTFlows)\nP(CTProblems) --> F\nP --> B\nF --> B\nD --> B\nstyle D fill:#FFF0F5","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"note: Install\nTo install a package from the control-toolbox ecosystem, please visit the installation page.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"An optimal control problem with fixed initial and final times, denoted (OCP), can be described as minimising the cost functional","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"g(x(t_0) x(t_f)) + int_t_0^t_f f^0(t x(t) u(t))mathrmdt","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"where the state x and the control u are functions subject, for t in t_0 t_f, to the differential constraint","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":" dotx(t) = f(t x(t) u(t))","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"and other constraints such as","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"beginarrayllcll\nxi_l le xi(t u(t)) le xi_u \neta_l le eta(t x(t)) le eta_u \npsi_l le psi(t x(t) u(t)) le psi_u \nphi_l le phi(t_0 x(t_0) t_f x(t_f)) le phi_u\nendarray","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"The so-called direct approach transforms the infinite dimensional optimal control problem (OCP) into a finite dimensional optimization problem (NLP). This is done by a discretization in time by Runge-Kutta methods applied to the state and control variables, as well as the dynamics equation. These methods are usually less precise than indirect methods based on Pontryagin’s Maximum Principle, but more robust with respect to the initialization. Also, they are more straightforward to apply, hence their wide use in industrial applications. We refer the reader to for instance[1] and [2] for more details on direct transcription methods and NLP algorithms.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"[1]: J. T. Betts. Practical methods for optimal control using nonlinear programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"[2]: J. Nocedal and S.J. Wright. Numerical optimization. Springer-Verlag, New York, 1999.****","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"Example of the time discretization by the trapezoidal rule:","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"beginarraylcl\nt in t_0t_f to t_0 ldots t_N=t_f02em\nx(cdot) u(cdot) to X=x_0 ldots x_N u_0 ldots u_N 1em\nhline\n\ntextstep to h = (t_f-t_0)N02em\ntextcriterion to min g(x_0 x_N) 02em\ntextdynamics to x_i+i = x_i + (h2) (f(t_i x_i u_i) + f(t_i+1 x_i+1 u_i+1)) 02em\ntextcontrol constraints to xi_l le xi(t_i u_i) le xi_u 02em\ntextpath constraints to eta_l le eta(t_i x_i) le eta_u 02em\ntextmixed constraints to psi_l le psi(t_i x_i u_i) le psi_u 02em\ntextlimit conditions to phi_l le phi(x_0 x_N) le phi_u\nendarray","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"We therefore obtain a nonlinear programming problem on the discretized state and control variables of the general form:","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"(NLP)quad left\nbeginarraylr\nmin F(X) \nLB le C(X) le UB\nendarray\nright","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"We use packages from JuliaSmoothOptimizers to solve the (NLP) problem.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"As input of this package we use an OptimalControlModel structure from CTBase.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"note: Current limitations\nThe current implemented is limited totrapezoidal rule for the ODE discretization\nIpopt for the optimization software","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"note: Related packages\nThis package is equivalent to the bocop software.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"","category":"page"},{"location":"api-ctbase.html#CTBase-API","page":"CTBase API","title":"CTBase API","text":"","category":"section"},{"location":"api-ctbase.html","page":"CTBase API","title":"CTBase API","text":"This is just a dump of CTBase API documentation. For more details about CTBase.jl package, see the documentation.","category":"page"},{"location":"api-ctbase.html#Index","page":"CTBase API","title":"Index","text":"","category":"section"},{"location":"api-ctbase.html","page":"CTBase API","title":"CTBase API","text":"Pages = [\"api-ctbase.md\"]\nModules = [CTBase]\nOrder = [:module, :constant, :type, :function, :macro]","category":"page"},{"location":"api-ctbase.html#Documentation","page":"CTBase API","title":"Documentation","text":"","category":"section"},{"location":"api-ctbase.html","page":"CTBase API","title":"CTBase API","text":"Modules = [CTBase]\nOrder = [:module, :constant, :type, :function, :macro]\nPrivate = false","category":"page"},{"location":"api-ctbase.html#CTBase.CTBase","page":"CTBase API","title":"CTBase.CTBase","text":"CTBase module.\n\nLists all the imported modules and packages:\n\nBase\nCore\nDataStructures\nDocStringExtensions\nLinearAlgebra\nMLStyle\nParameters\nPrettyTables\nPrintf\nReplMaker\nUnicode\n\nList of all the exported names:\n\nAbstractHamiltonian\nAmbiguousDescription\nAutonomous\nBoundaryConstraint\nCTCallback\nCTCallbacks\nCTException\nControl\nControlConstraint\nControlLaw\nControls\nCostate\nCostates\nDCostate\nDState\nDescription\nDimension\nDynamics\nFeedbackControl\nFixed\nHamiltonian\nHamiltonianLift\nHamiltonianVectorField\nIncorrectArgument\nIncorrectMethod\nIncorrectOutput\nIndex\nLagrange\n@Lie\nLie\nLift\nMayer\nMixedConstraint\nModel\nMultiplier\nNonAutonomous\nNonFixed\nNotImplemented\nOptimalControlInit\nOptimalControlModel\nOptimalControlSolution\nParsingError\nPoisson\nPrintCallback\nState\nStateConstraint\nStates\nStopCallback\nTime\nTimeDependence\nTimes\nTimesDisc\nUnauthorizedCall\nVariable\nVariableConstraint\nVariableDependence\nVectorField\n__OCPModel\nadd\nconstraint\nconstraint!\nconstraint_type\nconstraints_labels\ncontrol!\nctNumber\nctVector\nct_repl\nct_repl_update_model\nctgradient\nctindices\nctinterpolate\nctjacobian\nctupperscripts\n@def\ndim_boundary_constraints\ndim_control_constraints\ndim_control_range\ndim_mixed_constraints\ndim_path_constraints\ndim_state_constraints\ndim_state_range\ndim_variable_constraints\ndim_variable_range\ndynamics!\ngetFullDescription\nget_priority_print_callbacks\nget_priority_stop_callbacks\nhas_free_final_time\nhas_free_initial_time\nhas_lagrange_cost\nhas_mayer_cost\nis_autonomous\nis_fixed\nis_max\nis_min\nis_time_dependent\nis_time_independent\nis_variable_dependent\nis_variable_independent\nnlp_constraints!\nobjective!\nremove_constraint!\nreplace_call\nset_AD_backend\nstate!\ntime!\nvariable!\n∂ₜ\n⋅\n\n\n\n\n\n","category":"module"},{"location":"api-ctbase.html#CTBase.Control","page":"CTBase API","title":"CTBase.Control","text":"Type alias for a control in Rᵐ.\n\njulia> const Control = ctVector\n\nSee also: ctVector, State, Costate, Variable.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Costate","page":"CTBase API","title":"CTBase.Costate","text":"Type alias for a costate in Rⁿ.\n\njulia> const Costate = ctVector\n\nSee also: ctVector, State, Control, Variable.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.DCostate","page":"CTBase API","title":"CTBase.DCostate","text":"Type alias for a tangent vector to the costate space.\n\njulia> const DCostate = ctVector\n\nSee also: ctVector, DState.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.DState","page":"CTBase API","title":"CTBase.DState","text":"Type alias for a tangent vector to the state space.\n\njulia> const DState = ctVector\n\nSee also: ctVector, DCostate.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.State","page":"CTBase API","title":"CTBase.State","text":"Type alias for a state in Rⁿ.\n\njulia> const State = ctVector\n\nSee also: ctVector, Costate, Control, Variable.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.TimesDisc","page":"CTBase API","title":"CTBase.TimesDisc","text":"Type alias for a grid of times. This is used to define a discretization of time interval given to solvers.\n\njulia> const TimesDisc = Union{Times, StepRangeLen}\n\nSee also: Time, Times.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Variable","page":"CTBase API","title":"CTBase.Variable","text":"Type alias for a variable in Rᵏ.\n\njulia> const Variable = ctVector\n\nSee also: ctVector, State, Costate, Control.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ctVector","page":"CTBase API","title":"CTBase.ctVector","text":"Type alias for a vector of real numbers.\n\njulia> const ctVector = Union{ctNumber, AbstractVector{<:ctNumber}}\n\nSee also: ctNumber, State, Costate, Control, Variable.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.AbstractHamiltonian","page":"CTBase API","title":"CTBase.AbstractHamiltonian","text":"abstract type AbstractHamiltonian{time_dependence, variable_dependence}\n\nAbstract type for hamiltonians.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.AmbiguousDescription","page":"CTBase API","title":"CTBase.AmbiguousDescription","text":"struct AmbiguousDescription <: CTException\n\nException thrown when the description is ambiguous / incorrect.\n\nFields\n\nvar::Tuple{Vararg{Symbol}}\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Autonomous","page":"CTBase API","title":"CTBase.Autonomous","text":"abstract type Autonomous <: TimeDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.BoundaryConstraint","page":"CTBase API","title":"CTBase.BoundaryConstraint","text":"struct BoundaryConstraint{variable_dependence}\n\nFields\n\nf::Function\n\nThe default value for variable_dependence is Fixed.\n\nConstructor\n\nThe constructor BoundaryConstraint returns a BoundaryConstraint of a function. The function must take 2 or 3 arguments (x0, xf) or (x0, xf, v), if the function is variable, it must be specified. Dependencies are specified with a boolean, variable, false by default or with a DataType, NonFixed/Fixed, Fixed by default.\n\nExamples\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], variable=true)\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], NonFixed)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x0 and xf as a scalar. When the constraint is dimension 1, return a scalar.\n\nCall\n\nThe call returns the evaluation of the BoundaryConstraint for given values. If a variable is given for a non variable dependent boundary constraint, it will be ignored.\n\nExamples\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B([0, 0], [1, 1])\n[1, 2]\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B([0, 0], [1, 1],Real[])\n[1, 2]\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], variable=true)\njulia> B([0, 0], [1, 1], [1, 2, 3])\n[4, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.BoundaryConstraint-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.BoundaryConstraint","text":"BoundaryConstraint(\n f::Function,\n dependencies::DataType...\n) -> BoundaryConstraint{Fixed}\n\n\nReturn a BoundaryConstraint of a function. Dependencies are specified with a DataType, NonFixed/Fixed, Fixed by default.\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.BoundaryConstraint-Tuple{Function}","page":"CTBase API","title":"CTBase.BoundaryConstraint","text":"BoundaryConstraint(\n f::Function;\n variable\n) -> BoundaryConstraint{Fixed}\n\n\nReturn a BoundaryConstraint of a function. Dependencies are specified with a boolean, variable, false by default.\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.BoundaryConstraint-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.BoundaryConstraint","text":"Return the evaluation of the BoundaryConstraint.\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B([0, 0], [1, 1])\n[1, 2]\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B([0, 0], [1, 1],Real[])\n[1, 2]\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], variable=true)\njulia> B([0, 0], [1, 1], [1, 2, 3])\n[4, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.CTCallback","page":"CTBase API","title":"CTBase.CTCallback","text":"abstract type CTCallback\n\nAbstract type for callbacks.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.CTCallbacks","page":"CTBase API","title":"CTBase.CTCallbacks","text":"Tuple of callbacks\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.CTException","page":"CTBase API","title":"CTBase.CTException","text":"abstract type CTException <: Exception\n\nAbstract type for exceptions.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ControlConstraint","page":"CTBase API","title":"CTBase.ControlConstraint","text":"struct ControlConstraint{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to VectorField in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor ControlConstraint returns a ControlConstraint of a function. The function must take 1 to 3 arguments, u to (t, u, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], Autonomous, Fixed)\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], Autonomous, NonFixed)\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], NonAutonomous, Fixed)\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], NonAutonomous, NonFixed)\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], autonomous=true, variable=false)\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], autonomous=true, variable=true)\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], autonomous=false, variable=false)\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the control is of dimension 1, consider u as a scalar.\n\nCall\n\nThe call returns the evaluation of the ControlConstraint for given values.\n\nExamples\n\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], autonomous=true, variable=false)\njulia> C([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> C(t, [1, -1], v)\n[1, -2]\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], autonomous=true, variable=true)\njulia> C([1, -1], [1, 2, 3])\n[1, 1]\njulia> C(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], autonomous=false, variable=false)\njulia> C(1, [1, -1])\n[2, -2]\njulia> C(1, [1, -1], v)\n[2, -2]\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], autonomous=false, variable=true)\njulia> C(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ControlConstraint-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.ControlConstraint","text":"ControlConstraint(\n f::Function,\n dependencies::DataType...\n) -> ControlConstraint{Autonomous, Fixed}\n\n\nReturn the StateConstraint of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], Autonomous, Fixed)\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], Autonomous, NonFixed)\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], NonAutonomous, Fixed)\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlConstraint-Tuple{Function}","page":"CTBase API","title":"CTBase.ControlConstraint","text":"ControlConstraint(\n f::Function;\n autonomous,\n variable\n) -> ControlConstraint{Autonomous, Fixed}\n\n\nReturn the ControlConstraint of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], autonomous=true, variable=false)\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], autonomous=true, variable=true)\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], autonomous=false, variable=false)\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlConstraint-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.ControlConstraint","text":"Return the value of the ControlConstraint function.\n\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], autonomous=true, variable=false)\njulia> C([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> C(t, [1, -1], v)\n[1, -2]\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], autonomous=true, variable=true)\njulia> C([1, -1], [1, 2, 3])\n[1, 1]\njulia> C(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], autonomous=false, variable=false)\njulia> C(1, [1, -1])\n[2, -2]\njulia> C(1, [1, -1], v)\n[2, -2]\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], autonomous=false, variable=true)\njulia> C(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlLaw","page":"CTBase API","title":"CTBase.ControlLaw","text":"struct ControlLaw{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to Hamiltonian in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor ControlLaw returns a ControlLaw of a function. The function must take 2 to 4 arguments, (x, p) to (t, x, p, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], Autonomous, Fixed)\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], Autonomous, NonFixed)\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], NonAutonomous, Fixed)\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], NonAutonomous, NonFixed)\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the ControlLaw for given values.\n\nExamples\n\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> u([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> u(t, [1, 0], [0, 1])\nMethodError\njulia> u([1, 0], [0, 1], v)\nMethodError\njulia> u(t, [1, 0], [0, 1], v)\n3\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> u([1, 0], [0, 1], [1, 2, 3])\n6\njulia> u(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> u(1, [1, 0], [0, 1])\n4\njulia> u(1, [1, 0], [0, 1], v)\n4\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> u(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ControlLaw-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.ControlLaw","text":"ControlLaw(\n f::Function,\n dependencies::DataType...\n) -> ControlLaw{Autonomous, Fixed}\n\n\nReturn the ControlLaw of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], Autonomous, Fixed)\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], Autonomous, NonFixed)\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], NonAutonomous, Fixed)\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlLaw-Tuple{Function}","page":"CTBase API","title":"CTBase.ControlLaw","text":"ControlLaw(\n f::Function;\n autonomous,\n variable\n) -> ControlLaw{Autonomous, Fixed}\n\n\nReturn the ControlLaw of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlLaw-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.ControlLaw","text":"Return the value of the ControlLaw function.\n\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> u([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> u(t, [1, 0], [0, 1])\nMethodError\njulia> u([1, 0], [0, 1], v)\nMethodError\njulia> u(t, [1, 0], [0, 1], v)\n3\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> u([1, 0], [0, 1], [1, 2, 3])\n6\njulia> u(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> u(1, [1, 0], [0, 1])\n4\njulia> u(1, [1, 0], [0, 1], v)\n4\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> u(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Controls","page":"CTBase API","title":"CTBase.Controls","text":"Type alias for a vector of controls.\n\njulia> const Controls = AbstractVector{<:Control}\n\nSee also: Control, States, Costates.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Costates","page":"CTBase API","title":"CTBase.Costates","text":"Type alias for a vector of costates.\n\njulia> const Costates = AbstractVector{<:Costate}\n\nSee also: Costate, States, Controls.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Description","page":"CTBase API","title":"CTBase.Description","text":"A description is a tuple of symbols. Description is a type alias for a tuple of symbols.\n\njulia> const Description = Tuple{DescVarArg}\n\nSee also: DescVarArg.\n\nExample\n\nBase.show is overloaded for descriptions, that is tuple of descriptions are printed as follows:\n\njulia> display( ( (:a, :b), (:b, :c) ) )\n(:a, :b)\n(:b, :c)\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Dimension","page":"CTBase API","title":"CTBase.Dimension","text":"Type alias for a dimension. This is used to define the dimension of the state space, the costate space, the control space, etc.\n\njulia> const Dimension = Integer\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Dynamics","page":"CTBase API","title":"CTBase.Dynamics","text":"struct Dynamics{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default value for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor Dynamics returns a Dynamics of a function. The function must take 2 to 4 arguments, (x, u) to (t, x, u, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> Dynamics((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> Dynamics((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], Autonomous, Fixed)\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], Autonomous, NonFixed)\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], NonAutonomous, Fixed)\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], NonAutonomous, NonFixed)\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar. Same for the control.\n\nCall\n\nThe call returns the evaluation of the Dynamics for given values.\n\nExamples\n\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> D([1, 0], 1)\n[-1, 1]\njulia> t = 1\njulia> v = Real[]\njulia> D(t, [1, 0], 1, v)\n[-1, 1]\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> D([1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> D(t, [1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> D(1, [1, 0], 1)\n[0, 1]\njulia> D(1, [1, 0], 1, v)\n[0, 1]\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\njulia> D(1, [1, 0], 1, [1, 2, 3])\n[3, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Dynamics-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Dynamics","text":"Dynamics(\n f::Function,\n dependencies::DataType...\n) -> Dynamics{Autonomous, Fixed}\n\n\nReturn the Dynamics of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> Dynamics((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> Dynamics((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], Autonomous, Fixed)\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], Autonomous, NonFixed)\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], NonAutonomous, Fixed)\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Dynamics-Tuple{Function}","page":"CTBase API","title":"CTBase.Dynamics","text":"Dynamics(\n f::Function;\n autonomous,\n variable\n) -> Dynamics{Autonomous, Fixed}\n\n\nReturn the Dynamics of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Dynamics-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Dynamics","text":"Return the value of the Dynamics function.\n\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> D([1, 0], 1)\n[-1, 1]\njulia> t = 1\njulia> v = Real[]\njulia> D(t, [1, 0], 1, v)\n[-1, 1]\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> D([1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> D(t, [1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> D(1, [1, 0], 1)\n[0, 1]\njulia> D(1, [1, 0], 1, v)\n[0, 1]\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\njulia> D(1, [1, 0], 1, [1, 2, 3])\n[3, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.FeedbackControl","page":"CTBase API","title":"CTBase.FeedbackControl","text":"struct FeedbackControl{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to VectorField in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor FeedbackControl returns a FeedbackControl of a function. The function must take 1 to 3 arguments, x to (t, x, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], Autonomous, Fixed)\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], Autonomous, NonFixed)\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], NonAutonomous, Fixed)\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], NonAutonomous, NonFixed)\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], autonomous=true, variable=false)\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], autonomous=true, variable=true)\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], autonomous=false, variable=false)\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar.\n\nCall\n\nThe call returns the evaluation of the FeedbackControl for given values.\n\nExamples\n\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], autonomous=true, variable=false)\njulia> u([1, 0])\n1\njulia> t = 1\njulia> v = Real[]\njulia> u(t, [1, 0])\nMethodError\njulia> u([1, 0], v)\nMethodError\njulia> u(t, [1, 0], v)\n1\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], autonomous=true, variable=true)\njulia> u([1, 0], [1, 2, 3])\n4\njulia> u(t, [1, 0], [1, 2, 3])\n4\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], autonomous=false, variable=false)\njulia> u(1, [1, 0])\n2\njulia> u(1, [1, 0], v)\n2\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], autonomous=false, variable=true)\njulia> u(1, [1, 0], [1, 2, 3])\n5\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.FeedbackControl-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.FeedbackControl","text":"FeedbackControl(\n f::Function,\n dependencies::DataType...\n) -> FeedbackControl{Autonomous, Fixed}\n\n\nReturn the FeedbackControl of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], Autonomous, Fixed)\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], Autonomous, NonFixed)\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], NonAutonomous, Fixed)\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.FeedbackControl-Tuple{Function}","page":"CTBase API","title":"CTBase.FeedbackControl","text":"FeedbackControl(\n f::Function;\n autonomous,\n variable\n) -> FeedbackControl{Autonomous, Fixed}\n\n\nReturn the FeedbackControl of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], autonomous=true, variable=false)\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], autonomous=true, variable=true)\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], autonomous=false, variable=false)\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.FeedbackControl-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.FeedbackControl","text":"Return the value of the FeedbackControl function.\n\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], autonomous=true, variable=false)\njulia> u([1, 0])\n1\njulia> t = 1\njulia> v = Real[]\njulia> u(t, [1, 0])\nMethodError\njulia> u([1, 0], v)\nMethodError\njulia> u(t, [1, 0], v)\n1\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], autonomous=true, variable=true)\njulia> u([1, 0], [1, 2, 3])\n4\njulia> u(t, [1, 0], [1, 2, 3])\n4\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], autonomous=false, variable=false)\njulia> u(1, [1, 0])\n2\njulia> u(1, [1, 0], v)\n2\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], autonomous=false, variable=true)\njulia> u(1, [1, 0], [1, 2, 3])\n5\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Fixed","page":"CTBase API","title":"CTBase.Fixed","text":"abstract type Fixed <: VariableDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Hamiltonian","page":"CTBase API","title":"CTBase.Hamiltonian","text":"struct Hamiltonian{time_dependence, variable_dependence} <: AbstractHamiltonian{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor Hamiltonian returns a Hamiltonian of a function. The function must take 2 to 4 arguments, (x, p) to (t, x, p, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> Hamiltonian((x, p) -> x + p, Int64)\nIncorrectArgument \njulia> Hamiltonian((x, p) -> x + p, Int64)\nIncorrectArgument\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H = Hamiltonian((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3]], autonomous=false, variable=true)\njulia> H = Hamiltonian((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3]], NonAutonomous, NonFixed)\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the Hamiltonian for given values.\n\nExamples\n\njulia> H = Hamiltonian((x, p) -> [x[1]^2+2p[2]]) # autonomous=true, variable=false\njulia> H([1, 0], [0, 1])\nMethodError # H must return a scalar\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> H(t, [1, 0], [0, 1])\nMethodError\njulia> H([1, 0], [0, 1], v)\nMethodError \njulia> H(t, [1, 0], [0, 1], v)\n3\njulia> H = Hamiltonian((x, p, v) -> x[1]^2+2p[2]+v[3], variable=true)\njulia> H([1, 0], [0, 1], [1, 2, 3])\n6\njulia> H(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> H = Hamiltonian((t, x, p) -> t+x[1]^2+2p[2], autonomous=false)\njulia> H(1, [1, 0], [0, 1])\n4\njulia> H(1, [1, 0], [0, 1], v)\n4\njulia> H = Hamiltonian((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> H(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Hamiltonian-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Hamiltonian","text":"Hamiltonian(\n f::Function,\n dependencies::DataType...\n) -> Hamiltonian{Autonomous, Fixed}\n\n\nReturn an Hamiltonian of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H = Hamiltonian((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Hamiltonian-Tuple{Function}","page":"CTBase API","title":"CTBase.Hamiltonian","text":"Hamiltonian(\n f::Function;\n autonomous,\n variable\n) -> Hamiltonian{Autonomous, Fixed}\n\n\nReturn an Hamiltonian of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H = Hamiltonian((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Hamiltonian-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Hamiltonian","text":"Return the value of the Hamiltonian.\n\njulia> Hamiltonian((x, p) -> x + p, Int64)\nIncorrectArgument \njulia> Hamiltonian((x, p) -> x + p, Int64)\nIncorrectArgument\njulia> H = Hamiltonian((x, p) -> [x[1]^2+2p[2]]) # autonomous=true, variable=false\njulia> H([1, 0], [0, 1])\nMethodError # H must return a scalar\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> H(t, [1, 0], [0, 1])\nMethodError\njulia> H([1, 0], [0, 1], v)\nMethodError \njulia> H(t, [1, 0], [0, 1], v)\n3\njulia> H = Hamiltonian((x, p, v) -> x[1]^2+2p[2]+v[3], variable=true)\njulia> H([1, 0], [0, 1], [1, 2, 3])\n6\njulia> H(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> H = Hamiltonian((t, x, p) -> t+x[1]^2+2p[2], autonomous=false)\njulia> H(1, [1, 0], [0, 1])\n4\njulia> H(1, [1, 0], [0, 1], v)\n4\njulia> H = Hamiltonian((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> H(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianLift","page":"CTBase API","title":"CTBase.HamiltonianLift","text":"struct HamiltonianLift{time_dependence, variable_dependence} <: AbstractHamiltonian{time_dependence, variable_dependence}\n\nLifts\n\nX::VectorField\n\nThe values for time_dependence and variable_dependence are deternimed by the values of those for the VectorField.\n\nConstructor\n\nThe constructor HamiltonianLift returns a HamiltonianLift of a VectorField.\n\nExamples\n\njulia> H = HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]]))\njulia> H = HamiltonianLift(VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true))\njulia> H = HamiltonianLift(VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false))\njulia> H = HamiltonianLift(VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true))\njulia> H = HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]]))\njulia> H = HamiltonianLift(VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], NonFixed))\njulia> H = HamiltonianLift(VectorField((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous))\njulia> H = HamiltonianLift(VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed))\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the HamiltonianLift for given values.\n\nExamples\n\njulia> H = HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]]))\njulia> H([1, 2], [1, 1])\n5\njulia> t = 1\njulia> v = Real[]\njulia> H(t, [1, 0], [0, 1])\nMethodError\njulia> H([1, 0], [0, 1], v)\nMethodError \njulia> H(t, [1, 0], [0, 1], v)\n5\njulia> H = HamiltonianLift(VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true))\njulia> H([1, 0], [0, 1], [1, 2, 3])\n3\njulia> H(t, [1, 0], [0, 1], [1, 2, 3])\n3\njulia> H = HamiltonianLift(VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false))\njulia> H(1, [1, 2], [1, 1])\n6\njulia> H(1, [1, 0], [0, 1], v)\n6\njulia> H = HamiltonianLift(VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true))\njulia> H(1, [1, 0], [0, 1], [1, 2, 3])\n3\n\nAlternatively, it is possible to construct the HamiltonianLift from a Function being the VectorField.\n\njulia> HL1 = HamiltonianLift((x, v) -> [x[1]^2,x[2]^2+v], autonomous=true, variable=true)\njulia> HL2 = HamiltonianLift(VectorField((x, v) -> [x[1]^2,x[2]^2+v], autonomous=true, variable=true))\njulia> HL1([1, 0], [0, 1], 1) == HL2([1, 0], [0, 1], 1)\ntrue\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.HamiltonianLift-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.HamiltonianLift","text":"HamiltonianLift(\n f::Function,\n dependences::DataType...\n) -> HamiltonianLift\n\n\nReturn an HamiltonianLift of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> HamiltonianLift(HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]], Int64))\nIncorrectArgument \njulia> HL = HamiltonianLift(x -> [x[1]^2,x[2]^2], Autonomous, Fixed)\njulia> HL = HamiltonianLift((x, v) -> [x[1]^2,x[2]^2+v], Autonomous, NonFixed)\njulia> HL = HamiltonianLift((t, x) -> [t+x[1]^2,x[2]^2], NonAutonomous, Fixed)\njulia> HL = HamiltonianLift((t, x, v) -> [t+x[1]^2,x[2]^2+v], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianLift-Tuple{Function}","page":"CTBase API","title":"CTBase.HamiltonianLift","text":"HamiltonianLift(\n f::Function;\n autonomous,\n variable\n) -> HamiltonianLift\n\n\nReturn an HamiltonianLift of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> HL = HamiltonianLift(x -> [x[1]^2,x[2]^2], autonomous=true, variable=false)\njulia> HL = HamiltonianLift((x, v) -> [x[1]^2,x[2]^2+v], autonomous=true, variable=true)\njulia> HL = HamiltonianLift((t, x) -> [t+x[1]^2,x[2]^2], autonomous=false, variable=false)\njulia> HL = HamiltonianLift((t, x, v) -> [t+x[1]^2,x[2]^2+v], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianLift-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.HamiltonianLift","text":"Return the value of the HamiltonianLift.\n\nExamples\n\njulia> HamiltonianLift(HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]], Int64))\nIncorrectArgument \njulia> H = HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]]))\njulia> H([1, 2], [1, 1])\n5\njulia> t = 1\njulia> v = Real[]\njulia> H(t, [1, 0], [0, 1])\nMethodError\njulia> H([1, 0], [0, 1], v)\nMethodError \njulia> H(t, [1, 0], [0, 1], v)\n5\njulia> H = HamiltonianLift(VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true))\njulia> H([1, 0], [0, 1], [1, 2, 3])\n3\njulia> H(t, [1, 0], [0, 1], [1, 2, 3])\n3\njulia> H = HamiltonianLift(VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false))\njulia> H(1, [1, 2], [1, 1])\n6\njulia> H(1, [1, 0], [0, 1], v)\n6\njulia> H = HamiltonianLift(VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true))\njulia> H(1, [1, 0], [0, 1], [1, 2, 3])\n3\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianVectorField","page":"CTBase API","title":"CTBase.HamiltonianVectorField","text":"struct HamiltonianVectorField{time_dependence, variable_dependence} <: CTBase.AbstractVectorField{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor HamiltonianVectorField returns a HamiltonianVectorField of a function. The function must take 2 to 4 arguments, (x, p) to (t, x, p, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], variable=true)\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], autonomous=false)\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], autonomous=false, variable=true)\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], NonFixed)\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], NonAutonomous)\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], NonAutonomous, NonFixed)\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the HamiltonianVectorField for given values.\n\nExamples\n\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv([1, 0], [0, 1])\n[3, -3]\njulia> t = 1\njulia> v = Real[]\njulia> Hv(t, [1, 0], [0, 1])\nMethodError\njulia> Hv([1, 0], [0, 1], v)\nMethodError\njulia> Hv(t, [1, 0], [0, 1], v)\n[3, -3]\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], variable=true)\njulia> Hv([1, 0], [0, 1], [1, 2, 3, 4])\n[6, -3]\njulia> Hv(t, [1, 0], [0, 1], [1, 2, 3, 4])\n[6, -3]\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], autonomous=false)\njulia> Hv(1, [1, 0], [0, 1])\n[4, -3]\njulia> Hv(1, [1, 0], [0, 1], v)\n[4, -3]\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], autonomous=false, variable=true)\njulia> Hv(1, [1, 0], [0, 1], [1, 2, 3, 4])\n[7, -3]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.HamiltonianVectorField-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.HamiltonianVectorField","text":"HamiltonianVectorField(\n f::Function,\n dependencies::DataType...\n) -> HamiltonianVectorField{Autonomous, Fixed}\n\n\nReturn an HamiltonianVectorField of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], NonFixed)\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], NonAutonomous)\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianVectorField-Tuple{Function}","page":"CTBase API","title":"CTBase.HamiltonianVectorField","text":"HamiltonianVectorField(\n f::Function;\n autonomous,\n variable\n) -> HamiltonianVectorField{Autonomous, Fixed}\n\n\nReturn an HamiltonianVectorField of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], variable=true)\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], autonomous=false)\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianVectorField-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.HamiltonianVectorField","text":"Return the value of the HamiltonianVectorField.\n\nExamples\n\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv([1, 0], [0, 1])\n[3, -3]\njulia> t = 1\njulia> v = Real[]\njulia> Hv(t, [1, 0], [0, 1])\nMethodError\njulia> Hv([1, 0], [0, 1], v)\nMethodError\njulia> Hv(t, [1, 0], [0, 1], v)\n[3, -3]\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], variable=true)\njulia> Hv([1, 0], [0, 1], [1, 2, 3, 4])\n[6, -3]\njulia> Hv(t, [1, 0], [0, 1], [1, 2, 3, 4])\n[6, -3]\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], autonomous=false)\njulia> Hv(1, [1, 0], [0, 1])\n[4, -3]\njulia> Hv(1, [1, 0], [0, 1], v)\n[4, -3]\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], autonomous=false, variable=true)\njulia> Hv(1, [1, 0], [0, 1], [1, 2, 3, 4])\n[7, -3]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.IncorrectArgument","page":"CTBase API","title":"CTBase.IncorrectArgument","text":"struct IncorrectArgument <: CTException\n\nException thrown when an argument is inconsistent.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.IncorrectMethod","page":"CTBase API","title":"CTBase.IncorrectMethod","text":"struct IncorrectMethod <: CTException\n\nException thrown when a method is incorrect.\n\nFields\n\nvar::Symbol\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.IncorrectOutput","page":"CTBase API","title":"CTBase.IncorrectOutput","text":"struct IncorrectOutput <: CTException\n\nException thrown when the output is incorrect.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Index","page":"CTBase API","title":"CTBase.Index","text":"mutable struct Index\n\nFields\n\nval::Integer\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Lagrange","page":"CTBase API","title":"CTBase.Lagrange","text":"struct Lagrange{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default value for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor Lagrange returns a Lagrange cost of a function. The function must take 2 to 4 arguments, (x, u) to (t, x, u, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], Autonomous, Fixed)\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, Autonomous, Fixed)\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], Autonomous, NonFixed)\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, Fixed)\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, NonFixed)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar. Same for the control.\n\nCall\n\nThe call returns the evaluation of the Lagrange cost for given values.\n\nExamples\n\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L([1, 0], [1])\nMethodError\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L([1, 0], [1])\n-1\njulia> t = 1\njulia> v = Real[]\njulia> L(t, [1, 0], [1])\nMethodError\njulia> L([1, 0], [1], v)\nMethodError\njulia> L(t, [1, 0], [1], v)\n-1\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L([1, 0], [1], [1, 2, 3])\n2\njulia> L(t, [1, 0], [1], [1, 2, 3])\n2\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L(1, [1, 0], [1])\n0\njulia> L(1, [1, 0], [1], v)\n0\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\njulia> L(1, [1, 0], [1], [1, 2, 3])\n3\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Lagrange-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Lagrange","text":"Lagrange(\n f::Function,\n dependencies::DataType...\n) -> Lagrange{Autonomous, Fixed}\n\n\nReturn a Lagrange cost of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\n\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lagrange-Tuple{Function}","page":"CTBase API","title":"CTBase.Lagrange","text":"Lagrange(\n f::Function;\n autonomous,\n variable\n) -> Lagrange{Autonomous, Fixed}\n\n\nReturn a Lagrange cost of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\n\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lagrange-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Lagrange","text":"Return the value of the Lagrange function.\n\nExamples\n\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L([1, 0], [1])\nMethodError\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L([1, 0], [1])\n-1\njulia> t = 1\njulia> v = Real[]\njulia> L(t, [1, 0], [1])\nMethodError\njulia> L([1, 0], [1], v)\nMethodError\njulia> L(t, [1, 0], [1], v)\n-1\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L([1, 0], [1], [1, 2, 3])\n2\njulia> L(t, [1, 0], [1], [1, 2, 3])\n2\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L(1, [1, 0], [1])\n0\njulia> L(1, [1, 0], [1], v)\n0\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\njulia> L(1, [1, 0], [1], [1, 2, 3])\n3\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Mayer","page":"CTBase API","title":"CTBase.Mayer","text":"struct Mayer{variable_dependence}\n\nFields\n\nf::Function\n\nThe default value for variable_dependence is Fixed.\n\nConstructor\n\nThe constructor Mayer returns a Mayer cost of a function. The function must take 2 or 3 arguments (x0, xf) or (x0, xf, v), if the function is variable, it must be specified. Dependencies are specified with a boolean, variable, false by default or with a DataType, NonFixed/Fixed, Fixed by default.\n\nExamples\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], variable=true)\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], NonFixed)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x0 and xf as a scalar.\n\nCall\n\nThe call returns the evaluation of the Mayer cost for given values. If a variable is given for a non variable dependent Mayer cost, it will be ignored.\n\nExamples\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G([0, 0], [1, 1])\n1\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G([0, 0], [1, 1],Real[])\n1\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], variable=true)\njulia> G([0, 0], [1, 1], [1, 2, 3])\n4\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Mayer-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Mayer","text":"Mayer(\n f::Function,\n dependencies::DataType...\n) -> Mayer{Fixed}\n\n\nReturn a Mayer cost of a function. Dependencies are specified with a DataType, NonFixed/Fixed, Fixed by default.\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Mayer-Tuple{Function}","page":"CTBase API","title":"CTBase.Mayer","text":"Mayer(f::Function; variable) -> Mayer{Fixed}\n\n\nReturn a Mayer cost of a function. Dependencies are specified with a boolean, variable, false by default.\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Mayer-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Mayer","text":"Return the evaluation of the Mayer cost.\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G([0, 0], [1, 1])\n1\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G([0, 0], [1, 1], Real[])\n1\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], variable=true)\njulia> G([0, 0], [1, 1], [1, 2, 3])\n4\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.MixedConstraint","page":"CTBase API","title":"CTBase.MixedConstraint","text":"struct MixedConstraint{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to Lagrange in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default value for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor MixedConstraint returns a MixedConstraint of a function. The function must take 2 to 4 arguments, (x, u) to (t, x, u, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Autonomous, Fixed)\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], Autonomous, NonFixed)\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], NonAutonomous, Fixed)\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], NonAutonomous, NonFixed)\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar. Same for the control.\n\nCall\n\nThe call returns the evaluation of the MixedConstraint for given values.\n\nExamples\n\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> M([1, 0], 1)\n[-1, 1]\njulia> t = 1\njulia> v = Real[]\njulia> MethodError M(t, [1, 0], 1)\njulia> MethodError M([1, 0], 1, v)\njulia> M(t, [1, 0], 1, v)\n[-1, 1]\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> M([1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> M(t, [1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> M(1, [1, 0], 1)\n[0, 1]\njulia> M(1, [1, 0], 1, v)\n[0, 1]\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\njulia> M(1, [1, 0], 1, [1, 2, 3])\n[3, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.MixedConstraint-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.MixedConstraint","text":"MixedConstraint(\n f::Function,\n dependencies::DataType...\n) -> MixedConstraint{Autonomous, Fixed}\n\n\nReturn the MixedConstraint of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Autonomous, Fixed)\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], Autonomous, NonFixed)\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], NonAutonomous, Fixed)\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.MixedConstraint-Tuple{Function}","page":"CTBase API","title":"CTBase.MixedConstraint","text":"MixedConstraint(\n f::Function;\n autonomous,\n variable\n) -> MixedConstraint{Autonomous, Fixed}\n\n\nReturn the MixedConstraint of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.MixedConstraint-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.MixedConstraint","text":"Return the value of the MixedConstraint function.\n\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> M([1, 0], 1)\n[-1, 1]\njulia> t = 1\njulia> v = Real[]\njulia> MethodError M(t, [1, 0], 1)\njulia> MethodError M([1, 0], 1, v)\njulia> M(t, [1, 0], 1, v)\n[-1, 1]\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> M([1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> M(t, [1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> M(1, [1, 0], 1)\n[0, 1]\njulia> M(1, [1, 0], 1, v)\n[0, 1]\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\njulia> M(1, [1, 0], 1, [1, 2, 3])\n[3, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Multiplier","page":"CTBase API","title":"CTBase.Multiplier","text":"struct Multiplier{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to ControlLaw in the usage.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor Multiplier returns a Multiplier of a function. The function must take 2 to 4 arguments, (x, p) to (t, x, p, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], Autonomous, Fixed)\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], Autonomous, NonFixed)\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], NonAutonomous, Fixed)\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], NonAutonomous, NonFixed)\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the Multiplier for given values.\n\nExamples\n\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> μ([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> μ(t, [1, 0], [0, 1])\nMethodError\njulia> μ([1, 0], [0, 1], v)\nMethodError\njulia> μ(t, [1, 0], [0, 1], v)\n3\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> μ([1, 0], [0, 1], [1, 2, 3])\n6\njulia> μ(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> μ(1, [1, 0], [0, 1])\n4\njulia> μ(1, [1, 0], [0, 1], v)\n4\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> μ(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Multiplier-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Multiplier","text":"Multiplier(\n f::Function,\n dependencies::DataType...\n) -> Multiplier{Autonomous, Fixed}\n\n\nReturn the Multiplier of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], Autonomous, Fixed)\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], Autonomous, NonFixed)\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], NonAutonomous, Fixed)\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Multiplier-Tuple{Function}","page":"CTBase API","title":"CTBase.Multiplier","text":"Multiplier(\n f::Function;\n autonomous,\n variable\n) -> Multiplier{Autonomous, Fixed}\n\n\nReturn the Multiplier of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Multiplier-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Multiplier","text":"Return the value of the Multiplier function.\n\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> μ([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> μ(t, [1, 0], [0, 1])\nMethodError\njulia> μ([1, 0], [0, 1], v)\nMethodError\njulia> μ(t, [1, 0], [0, 1], v)\n3\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> μ([1, 0], [0, 1], [1, 2, 3])\n6\njulia> μ(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> μ(1, [1, 0], [0, 1])\n4\njulia> μ(1, [1, 0], [0, 1], v)\n4\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> μ(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.NonAutonomous","page":"CTBase API","title":"CTBase.NonAutonomous","text":"abstract type NonAutonomous <: TimeDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.NonFixed","page":"CTBase API","title":"CTBase.NonFixed","text":"abstract type NonFixed <: VariableDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.NotImplemented","page":"CTBase API","title":"CTBase.NotImplemented","text":"struct NotImplemented <: CTException\n\nException thrown when a method is not implemented.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.OptimalControlInit","page":"CTBase API","title":"CTBase.OptimalControlInit","text":"Initial guess for OCP, contains\n\nfunctions of time for the state and control variables\nvector for optimization variables\n\nInitialization data for each field can be left to default or: \n\nvector for optimization variables\nconstant / vector / function for state and control \nexisting solution ('warm start') for all fields\n\nConstructors:\n\nOptimalControlInit(): default initialization\nOptimalControlInit(state, control, variable, time): constant vector, function handles and / or matrices / vectors interpolated along given time grid\nOptimalControlInit(sol): from existing solution\n\nExamples\n\njulia> init = OptimalControlInit()\njulia> init = OptimalControlInit(state=[0.1, 0.2], control=0.3)\njulia> init = OptimalControlInit(state=[0.1, 0.2], control=0.3, variable=0.5)\njulia> init = OptimalControlInit(state=[0.1, 0.2], controlt=t->sin(t), variable=0.5)\njulia> init = OptimalControlInit(state=[[0, 0], [1, 2], [5, -1]], time=[0, .3, 1.], controlt=t->sin(t))\njulia> init = OptimalControlInit(sol)\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.OptimalControlModel","page":"CTBase API","title":"CTBase.OptimalControlModel","text":"mutable struct OptimalControlModel{time_dependence<:TimeDependence, variable_dependence<:VariableDependence} <: CTBase.AbstractOptimalControlModel\n\nFields\n\nmodel_expression::Union{Nothing, Expr}: Default: nothing\ninitial_time::Union{Nothing, Index, Real}: Default: nothing\ninitial_time_name::Union{Nothing, String}: Default: nothing\nfinal_time::Union{Nothing, Index, Real}: Default: nothing\nfinal_time_name::Union{Nothing, String}: Default: nothing\ntime_name::Union{Nothing, String}: Default: nothing\ncontrol_dimension::Union{Nothing, Integer}: Default: nothing\ncontrol_components_names::Union{Nothing, Vector{String}}: Default: nothing\ncontrol_name::Union{Nothing, String}: Default: nothing\nstate_dimension::Union{Nothing, Integer}: Default: nothing\nstate_components_names::Union{Nothing, Vector{String}}: Default: nothing\nstate_name::Union{Nothing, String}: Default: nothing\nvariable_dimension::Union{Nothing, Integer}: Default: nothing\nvariable_components_names::Union{Nothing, Vector{String}}: Default: nothing\nvariable_name::Union{Nothing, String}: Default: nothing\nlagrange::Union{Nothing, Lagrange}: Default: nothing\nmayer::Union{Nothing, Mayer}: Default: nothing\ncriterion::Union{Nothing, Symbol}: Default: nothing\ndynamics::Union{Nothing, Dynamics}: Default: nothing\nconstraints::Dict{Symbol, Tuple}: Default: Dict{Symbol, Tuple{Vararg{Any}}}()\ndim_control_constraints::Union{Nothing, Integer}: Default: nothing\ndim_state_constraints::Union{Nothing, Integer}: Default: nothing\ndim_mixed_constraints::Union{Nothing, Integer}: Default: nothing\ndim_boundary_constraints::Union{Nothing, Integer}: Default: nothing\ndim_variable_constraints::Union{Nothing, Integer}: Default: nothing\ndim_control_range::Union{Nothing, Integer}: Default: nothing\ndim_state_range::Union{Nothing, Integer}: Default: nothing\ndim_variable_range::Union{Nothing, Integer}: Default: nothing\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.OptimalControlSolution","page":"CTBase API","title":"CTBase.OptimalControlSolution","text":"mutable struct OptimalControlSolution <: CTBase.AbstractOptimalControlSolution\n\nType of an optimal control solution.\n\nFields\n\ntimes::Union{Nothing, StepRangeLen, AbstractVector{<:Real}}: Default: nothing\ninitial_time_name::Union{Nothing, String}: Default: nothing\nfinal_time_name::Union{Nothing, String}: Default: nothing\ntime_name::Union{Nothing, String}: Default: nothing\ncontrol_dimension::Union{Nothing, Integer}: Default: nothing\ncontrol_components_names::Union{Nothing, Vector{String}}: Default: nothing\ncontrol_name::Union{Nothing, String}: Default: nothing\ncontrol::Union{Nothing, Function}: Default: nothing\nstate_dimension::Union{Nothing, Integer}: Default: nothing\nstate_components_names::Union{Nothing, Vector{String}}: Default: nothing\nstate_name::Union{Nothing, String}: Default: nothing\nstate::Union{Nothing, Function}: Default: nothing\nvariable_dimension::Union{Nothing, Integer}: Default: nothing\nvariable_components_names::Union{Nothing, Vector{String}}: Default: nothing\nvariable_name::Union{Nothing, String}: Default: nothing\nvariable::Union{Nothing, Real, AbstractVector{<:Real}}: Default: nothing\ncostate::Union{Nothing, Function}: Default: nothing\nobjective::Union{Nothing, Real}: Default: nothing\niterations::Union{Nothing, Integer}: Default: nothing\nstopping::Union{Nothing, Symbol}: Default: nothing\nmessage::Union{Nothing, String}: Default: nothing\nsuccess::Union{Nothing, Bool}: Default: nothing\ninfos::Dict{Symbol, Any}: Default: Dict{Symbol, Any}()\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ParsingError","page":"CTBase API","title":"CTBase.ParsingError","text":"struct ParsingError <: CTException\n\nException thrown for syntax error during abstract parsing.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.PrintCallback","page":"CTBase API","title":"CTBase.PrintCallback","text":"mutable struct PrintCallback <: CTCallback\n\nCallback for printing.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.PrintCallback-Tuple","page":"CTBase API","title":"CTBase.PrintCallback","text":"Call the callback.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.StateConstraint","page":"CTBase API","title":"CTBase.StateConstraint","text":"struct StateConstraint{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to VectorField in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor StateConstraint returns a StateConstraint of a function. The function must take 1 to 3 arguments, x to (t, x, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], Autonomous, Fixed)\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], Autonomous, NonFixed)\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous, Fixed)\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed)\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], autonomous=true, variable=false)\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], autonomous=true, variable=true)\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false, variable=false)\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar.\n\nCall\n\nThe call returns the evaluation of the StateConstraint for given values.\n\nExamples\n\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], autonomous=true, variable=false)\njulia> S([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> S(t, [1, -1], v)\n[1, -2]\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], autonomous=true, variable=true)\njulia> S([1, -1], [1, 2, 3])\n[1, 1]\njulia> S(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false, variable=false)\njulia> S(1, [1, -1])\n[2, -2]\njulia> S(1, [1, -1], v)\n[2, -2]\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> S(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.StateConstraint-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.StateConstraint","text":"StateConstraint(\n f::Function,\n dependencies::DataType...\n) -> StateConstraint{Autonomous, Fixed}\n\n\nReturn the StateConstraint of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], Autonomous, Fixed)\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], Autonomous, NonFixed)\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous, Fixed)\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.StateConstraint-Tuple{Function}","page":"CTBase API","title":"CTBase.StateConstraint","text":"StateConstraint(\n f::Function;\n autonomous,\n variable\n) -> StateConstraint{Autonomous, Fixed}\n\n\nReturn the StateConstraint of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], autonomous=true, variable=false)\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], autonomous=true, variable=true)\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false, variable=false)\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.StateConstraint-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.StateConstraint","text":"Return the value of the StateConstraint function.\n\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], autonomous=true, variable=false)\njulia> S([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> S(t, [1, -1], v)\n[1, -2]\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], autonomous=true, variable=true)\njulia> S([1, -1], [1, 2, 3])\n[1, 1]\njulia> S(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false, variable=false)\njulia> S(1, [1, -1])\n[2, -2]\njulia> S(1, [1, -1], v)\n[2, -2]\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> S(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.States","page":"CTBase API","title":"CTBase.States","text":"Type alias for a vector of states.\n\njulia> const States = AbstractVector{<:State}\n\nSee also: State, Costates, Controls.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.StopCallback","page":"CTBase API","title":"CTBase.StopCallback","text":"Stopping callback.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.StopCallback-Tuple","page":"CTBase API","title":"CTBase.StopCallback","text":"Call the callback.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Time","page":"CTBase API","title":"CTBase.Time","text":"Type alias for a time.\n\njulia> const Time = ctNumber\n\nSee also: ctNumber, Times, TimesDisc.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.TimeDependence","page":"CTBase API","title":"CTBase.TimeDependence","text":"abstract type TimeDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Times","page":"CTBase API","title":"CTBase.Times","text":"Type alias for a vector of times.\n\njulia> const Times = AbstractVector{<:Time}\n\nSee also: Time, TimesDisc.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.UnauthorizedCall","page":"CTBase API","title":"CTBase.UnauthorizedCall","text":"struct UnauthorizedCall <: CTException\n\nException thrown when a call to a function is not authorized.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.VariableConstraint","page":"CTBase API","title":"CTBase.VariableConstraint","text":"struct VariableConstraint\n\nFields\n\nf::Function\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor VariableConstraint returns a VariableConstraint of a function. The function must take 1 argument, v.\n\nExamples\n\njulia> V = VariableConstraint(v -> [v[1]^2, 2v[2]])\n\nwarning: Warning\nWhen the variable is of dimension 1, consider v as a scalar.\n\nCall\n\nThe call returns the evaluation of the VariableConstraint for given values.\n\nExamples\n\njulia> V = VariableConstraint(v -> [v[1]^2, 2v[2]])\njulia> V([1, -1])\n[1, -2]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.VariableConstraint-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.VariableConstraint","text":"Return the value of the VariableConstraint function.\n\njulia> V = VariableConstraint(v -> [v[1]^2, 2v[2]])\njulia> V([1, -1])\n[1, -2]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.VariableDependence","page":"CTBase API","title":"CTBase.VariableDependence","text":"abstract type VariableDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.VectorField","page":"CTBase API","title":"CTBase.VectorField","text":"struct VectorField{time_dependence, variable_dependence} <: CTBase.AbstractVectorField{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor VectorField returns a VectorField of a function. The function must take 1 to 3 arguments, x to (t, x, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true)\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false)\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], NonFixed)\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous)\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar.\n\nCall\n\nThe call returns the evaluation of the VectorField for given values.\n\nExamples\n\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> V(t, [1, -1])\nMethodError\njulia> V([1, -1], v)\nMethodError\njulia> V(t, [1, -1], v)\n[1, -2]\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true)\njulia> V([1, -1], [1, 2, 3])\n[1, 1]\njulia> V(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false)\njulia> V(1, [1, -1])\n[2, -2]\njulia> V(1, [1, -1], v)\n[2, -2]\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> V(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.VectorField-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.VectorField","text":"VectorField(\n f::Function,\n dependencies::DataType...\n) -> VectorField{Autonomous, Fixed}\n\n\nReturn a VectorField of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], NonFixed)\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous)\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.VectorField-Tuple{Function}","page":"CTBase API","title":"CTBase.VectorField","text":"VectorField(\n f::Function;\n autonomous,\n variable\n) -> VectorField{Autonomous, Fixed}\n\n\nReturn a VectorField of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true)\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false)\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.VectorField-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.VectorField","text":"Return the value of the VectorField.\n\nExamples\n\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> V(t, [1, -1])\nMethodError\njulia> V([1, -1], v)\nMethodError\njulia> V(t, [1, -1], v)\n[1, -2]\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true)\njulia> V([1, -1], [1, 2, 3])\n[1, 1]\njulia> V(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false)\njulia> V(1, [1, -1])\n[2, -2]\njulia> V(1, [1, -1], v)\n[2, -2]\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> V(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctNumber","page":"CTBase API","title":"CTBase.ctNumber","text":"Type alias for a real number.\n\njulia> const ctNumber = Real\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.:⋅-Tuple{Function, Function}","page":"CTBase API","title":"CTBase.:⋅","text":"⋅(X::Function, f::Function) -> Function\n\n\nLie derivative of a scalar function along a function. In this case both functions will be considered autonomous and non-variable.\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> f = x -> x[1]^2 + x[2]^2\njulia> (φ⋅f)([1, 2])\n0\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> (φ⋅f)(1, [1, 2], [2, 1])\nMethodError\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.:⋅-Tuple{VectorField{Autonomous, <:VariableDependence}, Function}","page":"CTBase API","title":"CTBase.:⋅","text":"⋅(\n X::VectorField{Autonomous, <:VariableDependence},\n f::Function\n) -> CTBase.var\"#97#99\"{VectorField{Autonomous, var\"#s98\"}, <:Function} where var\"#s98\"<:VariableDependence\n\n\nLie derivative of a scalar function along a vector field : L_X(f) = X⋅f, in autonomous case\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> X = VectorField(φ)\njulia> f = x -> x[1]^2 + x[2]^2\njulia> (X⋅f)([1, 2])\n0\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.:⋅-Tuple{VectorField{NonAutonomous, <:VariableDependence}, Function}","page":"CTBase API","title":"CTBase.:⋅","text":"⋅(\n X::VectorField{NonAutonomous, <:VariableDependence},\n f::Function\n) -> CTBase.var\"#101#103\"{VectorField{NonAutonomous, var\"#s98\"}, <:Function} where var\"#s98\"<:VariableDependence\n\n\nLie derivative of a scalar function along a vector field : L_X(f) = X⋅f, in nonautonomous case\n\nExample\n\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> X = VectorField(φ, NonAutonomous, NonFixed)\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> (X⋅f)(1, [1, 2], [2, 1])\n10\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Tuple{Function, Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Lie","text":"Lie(\n X::Function,\n f::Function,\n dependences::DataType...\n) -> Function\n\n\nLie derivative of a scalar function along a vector field or a function. Dependencies are specified with DataType : Autonomous, NonAutonomous and Fixed, NonFixed.\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> f = x -> x[1]^2 + x[2]^2\njulia> Lie(φ,f)([1, 2])\n0\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> Lie(φ, f, NonAutonomous, NonFixed)(1, [1, 2], [2, 1])\n10\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Tuple{Function, Function}","page":"CTBase API","title":"CTBase.Lie","text":"Lie(\n X::Function,\n f::Function;\n autonomous,\n variable\n) -> Function\n\n\nLie derivative of a scalar function along a function. Dependencies are specified with boolean : autonomous and variable.\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> f = x -> x[1]^2 + x[2]^2\njulia> Lie(φ,f)([1, 2])\n0\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> Lie(φ, f, autonomous=false, variable=true)(1, [1, 2], [2, 1])\n10\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Tuple{VectorField, Function}","page":"CTBase API","title":"CTBase.Lie","text":"Lie(X::VectorField, f::Function) -> Function\n\n\nLie derivative of a scalar function along a vector field.\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> X = VectorField(φ)\njulia> f = x -> x[1]^2 + x[2]^2\njulia> Lie(X,f)([1, 2])\n0\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> X = VectorField(φ, NonAutonomous, NonFixed)\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> Lie(X, f)(1, [1, 2], [2, 1])\n10\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Union{Tuple{V}, Tuple{VectorField{Autonomous, V}, VectorField{Autonomous, V}}} where V<:VariableDependence","page":"CTBase API","title":"CTBase.Lie","text":"Lie(\n X::VectorField{Autonomous, V<:VariableDependence},\n Y::VectorField{Autonomous, V<:VariableDependence}\n) -> VectorField\n\n\nLie bracket of two vector fields: [X, Y] = Lie(X, Y), autonomous case\n\nExample\n\njulia> f = x -> [x[2], 2x[1]]\njulia> g = x -> [3x[2], -x[1]]\njulia> X = VectorField(f)\njulia> Y = VectorField(g)\njulia> Lie(X, Y)([1, 2])\n[7, -14]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Union{Tuple{V}, Tuple{VectorField{NonAutonomous, V}, VectorField{NonAutonomous, V}}} where V<:VariableDependence","page":"CTBase API","title":"CTBase.Lie","text":"Lie(\n X::VectorField{NonAutonomous, V<:VariableDependence},\n Y::VectorField{NonAutonomous, V<:VariableDependence}\n) -> VectorField{NonAutonomous}\n\n\nLie bracket of two vector fields: [X, Y] = Lie(X, Y), nonautonomous case\n\nExample\n\njulia> f = (t, x, v) -> [t + x[2] + v, -2x[1] - v]\njulia> g = (t, x, v) -> [t + 3x[2] + v, -x[1] - v]\njulia> X = VectorField(f, NonAutonomous, NonFixed)\njulia> Y = VectorField(g, NonAutonomous, NonFixed)\njulia> Lie(X, Y)(1, [1, 2], 1)\n[-7,12]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lift-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Lift","text":"Lift(\n X::Function,\n dependences::DataType...\n) -> HamiltonianLift\n\n\nReturn the HamiltonianLift of a VectorField or a function. Dependencies are specified with DataType : Autonomous, NonAutonomous and Fixed, NonFixed.\n\nExample\n\njulia> H = Lift(x -> 2x)\njulia> H(1, 1)\n2\njulia> H = Lift((t, x, v) -> 2x + t - v, NonAutonomous, NonFixed)\njulia> H(1, 1, 1, 1)\n2\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lift-Tuple{Function}","page":"CTBase API","title":"CTBase.Lift","text":"Lift(X::Function; autonomous, variable) -> HamiltonianLift\n\n\nReturn the HamiltonianLift of a function. Dependencies are specified with boolean : autonomous and variable.\n\nExample\n\njulia> H = Lift(x -> 2x)\njulia> H(1, 1)\n2\njulia> H = Lift((t, x, v) -> 2x + t - v, autonomous=false, variable=true)\njulia> H(1, 1, 1, 1)\n2\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lift-Tuple{VectorField}","page":"CTBase API","title":"CTBase.Lift","text":"Lift(X::VectorField) -> HamiltonianLift\n\n\nReturn the HamiltonianLift of a VectorField.\n\nExample\n\njulia> HL = Lift(VectorField(x -> [x[1]^2,x[2]^2], autonomous=true, variable=false))\njulia> HL([1, 0], [0, 1])\n0\njulia> HL = Lift(VectorField((t, x, v) -> [t+x[1]^2,x[2]^2+v], autonomous=false, variable=true))\njulia> HL(1, [1, 0], [0, 1], 1)\n1\njulia> H = Lift(x -> 2x)\njulia> H(1, 1)\n2\njulia> H = Lift((t, x, v) -> 2x + t - v, autonomous=false, variable=true)\njulia> H(1, 1, 1, 1)\n2\njulia> H = Lift((t, x, v) -> 2x + t - v, NonAutonomous, NonFixed)\njulia> H(1, 1, 1, 1)\n2\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Model-Tuple{Vararg{DataType}}","page":"CTBase API","title":"CTBase.Model","text":"Model(\n dependencies::DataType...\n) -> OptimalControlModel{Autonomous, Fixed}\n\n\nReturn a new OptimalControlModel instance, that is a model of an optimal control problem.\n\nThe model is defined by the following argument:\n\ndependencies: either Autonomous or NonAutonomous. Default is Autonomous. And either NonFixed or Fixed. Default is Fixed.\n\nExamples\n\njulia> ocp = Model()\njulia> ocp = Model(NonAutonomous)\njulia> ocp = Model(Autonomous, NonFixed)\n\nnote: Note\nIf the time dependence of the model is defined as nonautonomous, then, the dynamics function, the lagrange cost and the path constraints must be defined as functions of time and state, and possibly control. If the model is defined as autonomous, then, the dynamics function, the lagrange cost and the path constraints must be defined as functions of state, and possibly control.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Model-Tuple{}","page":"CTBase API","title":"CTBase.Model","text":"Model(\n;\n autonomous,\n variable\n) -> OptimalControlModel{Autonomous, Fixed}\n\n\nReturn a new OptimalControlModel instance, that is a model of an optimal control problem.\n\nThe model is defined by the following optional keyword argument:\n\nautonomous: either true or false. Default is true.\nvariable: either true or false. Default is false.\n\nExamples\n\njulia> ocp = Model()\njulia> ocp = Model(autonomous=false)\njulia> ocp = Model(autonomous=false, variable=true)\n\nnote: Note\nIf the time dependence of the model is defined as nonautonomous, then, the dynamics function, the lagrange cost and the path constraints must be defined as functions of time and state, and possibly control. If the model is defined as autonomous, then, the dynamics function, the lagrange cost and the path constraints must be defined as functions of state, and possibly control.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Tuple{Function, Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::Function,\n g::Function,\n dependences::DataType...\n) -> Hamiltonian\n\n\nPoisson bracket of two functions : {f, g} = Poisson(f, g) Dependencies are specified with DataType : Autonomous, NonAutonomous and Fixed, NonFixed.\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> Poisson(f, g)([1, 2], [2, 1])\n-20 \njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> Poisson(f, g, NonAutonomous, NonFixed)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Tuple{Function, Function}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::Function,\n g::Function;\n autonomous,\n variable\n) -> Hamiltonian\n\n\nPoisson bracket of two functions : {f, g} = Poisson(f, g) Dependencies are specified with boolean : autonomous and variable.\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> Poisson(f, g)([1, 2], [2, 1])\n-20 \njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> Poisson(f, g, autonomous=false, variable=true)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{AbstractHamiltonian{Autonomous, V}, AbstractHamiltonian{Autonomous, V}}} where V<:VariableDependence","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::AbstractHamiltonian{Autonomous, V<:VariableDependence},\n g::AbstractHamiltonian{Autonomous, V<:VariableDependence}\n) -> HamiltonianLift\n\n\nPoisson bracket of two Hamiltonian functions (subtype of AbstractHamiltonian) : {f, g} = Poisson(f, g), autonomous case\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> F = Hamiltonian(f)\njulia> G = Hamiltonian(g)\njulia> Poisson(f, g)([1, 2], [2, 1])\n-20 \njulia> Poisson(f, G)([1, 2], [2, 1])\n-20\njulia> Poisson(F, g)([1, 2], [2, 1])\n-20\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{AbstractHamiltonian{NonAutonomous, V}, AbstractHamiltonian{NonAutonomous, V}}} where V<:VariableDependence","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::AbstractHamiltonian{NonAutonomous, V<:VariableDependence},\n g::AbstractHamiltonian{NonAutonomous, V<:VariableDependence}\n) -> HamiltonianLift\n\n\nPoisson bracket of two Hamiltonian functions (subtype of AbstractHamiltonian) : {f, g} = Poisson(f, g), non autonomous case\n\nExample\n\njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> F = Hamiltonian(f, autonomous=false, variable=true)\njulia> G = Hamiltonian(g, autonomous=false, variable=true)\njulia> Poisson(F, G)(2, [1, 2], [2, 1], [4, 4])\n-76\njulia> Poisson(f, g, NonAutonomous, NonFixed)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{T}, Tuple{AbstractHamiltonian{T, V}, Function}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::AbstractHamiltonian{T<:TimeDependence, V<:VariableDependence},\n g::Function\n) -> Hamiltonian\n\n\nPoisson bracket of an Hamiltonian function (subtype of AbstractHamiltonian) and a function : {f, g} = Poisson(f, g), autonomous case\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> F = Hamiltonian(f)\njulia> Poisson(F, g)([1, 2], [2, 1])\n-20\njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> F = Hamiltonian(f, autonomous=false, variable=true)\njulia> Poisson(F, g)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{T}, Tuple{Function, AbstractHamiltonian{T, V}}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::Function,\n g::AbstractHamiltonian{T<:TimeDependence, V<:VariableDependence}\n) -> Hamiltonian\n\n\nPoisson bracket of a function and an Hamiltonian function (subtype of AbstractHamiltonian) : {f, g} = Poisson(f, g)\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> G = Hamiltonian(g) \njulia> Poisson(f, G)([1, 2], [2, 1])\n-20\njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> G = Hamiltonian(g, autonomous=false, variable=true)\njulia> Poisson(f, G)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{T}, Tuple{HamiltonianLift{T, V}, HamiltonianLift{T, V}}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::HamiltonianLift{T<:TimeDependence, V<:VariableDependence},\n g::HamiltonianLift{T<:TimeDependence, V<:VariableDependence}\n) -> HamiltonianLift\n\n\nPoisson bracket of two HamiltonianLift functions : {f, g} = Poisson(f, g)\n\nExample\n\njulia> f = x -> [x[1]^2+x[2]^2, 2x[1]^2]\njulia> g = x -> [3x[2]^2, x[2]-x[1]^2]\njulia> F = Lift(f)\njulia> G = Lift(g)\njulia> Poisson(F, G)([1, 2], [2, 1])\n-64\njulia> f = (t, x, v) -> [t*v[1]*x[2]^2, 2x[1]^2 + + v[2]]\njulia> g = (t, x, v) -> [3x[2]^2 + -x[1]^2, t - v[2]]\njulia> F = Lift(f, NonAutonomous, NonFixed)\njulia> G = Lift(g, NonAutonomous, NonFixed)\njulia> Poisson(F, G)(2, [1, 2], [2, 1], [4, 4])\n100\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.__OCPModel-Tuple","page":"CTBase API","title":"CTBase.__OCPModel","text":"__OCPModel(args...; kwargs...) -> OptimalControlModel\n\n\nRedirection to Model to avoid confusion with other functions Model from other packages if imported. This function is used by @def.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.add-Tuple{Tuple{Vararg{Tuple{Vararg{Symbol}}}}, Tuple{Vararg{Symbol}}}","page":"CTBase API","title":"CTBase.add","text":"add(\n x::Tuple{Vararg{Tuple{Vararg{Symbol}}}},\n y::Tuple{Vararg{Symbol}}\n) -> Tuple{Tuple{Vararg{Symbol}}}\n\n\nConcatenate the description y to the tuple of descriptions x if x does not contain y and return the new tuple of descriptions. Throw an error if the description y is already contained in x.\n\nExample\n\njulia> descriptions = ()\njulia> descriptions = add(descriptions, (:a,))\n(:a,)\njulia> descriptions = add(descriptions, (:b,))\n(:a,)\n(:b,)\njulia> descriptions = add(descriptions, (:b,))\nERROR: IncorrectArgument: the description (:b,) is already in ((:a,), (:b,))\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.add-Tuple{Tuple{}, Tuple{Vararg{Symbol}}}","page":"CTBase API","title":"CTBase.add","text":"add(\n x::Tuple{},\n y::Tuple{Vararg{Symbol}}\n) -> Tuple{Tuple{Vararg{Symbol}}}\n\n\nReturn a tuple containing only the description y.\n\nExample\n\njulia> descriptions = ()\njulia> descriptions = add(descriptions, (:a,))\n(:a,)\njulia> print(descriptions)\n((:a,),)\njulia> descriptions[1]\n(:a,)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.constraint!-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Symbol}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.constraint!","text":"constraint!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol;\n rg,\n f,\n lb,\n ub,\n label\n)\n\n\nAdd a constraint to an optimal control problem, denoted ocp.\n\nnote: Note\nThe state, control and variable dimensions must be set before. Use state!, control! and variable!.\nThe initial and final times must be set before. Use time!.\nWhen an element is of dimension 1, consider it as a scalar.\n\nYou can add an :initial, :final, :control, :state or :variable box constraint (whole range). \n\nRange constraint on the state, control or variable\n\nYou can add an :initial, :final, :control, :state or :variable box constraint on a range of it, that is only on some components. If not range is specified, then the constraint is on the whole range. We denote by x, u and v respectively the state, control and variable. We denote by n, m and q respectively the dimension of the state, control and variable. The range of the constraint must be contained in 1:n if the constraint is on the state, or 1:m if the constraint is on the control, or 1:q if the constraint is on the variable.\n\nExamples\n\njulia> constraint!(ocp, :initial; rg=1:2:5, lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :initial; rg=2:3, lb=[ 0, 0 ], ub=[ 1, 2 ])\njulia> constraint!(ocp, :final; rg=1, lb=0, ub=2)\njulia> constraint!(ocp, :control; rg=1, lb=0, ub=2)\njulia> constraint!(ocp, :state; rg=2:3, lb=[ 0, 0 ], ub=[ 1, 2 ])\njulia> constraint!(ocp, :variable; rg=1:2, lb=[ 0, 0 ], ub=[ 1, 2 ])\njulia> constraint!(ocp, :initial; lb=[ 0, 0, 0 ]) # [ 0, 0, 0 ] ≤ x(t0), dim(x) = 3\njulia> constraint!(ocp, :initial; lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ]) # [ 0, 0, 0 ] ≤ x(t0) ≤ [ 1, 2, 1 ], dim(x) = 3\njulia> constraint!(ocp, :final; lb=-1, ub=1) # -1 ≤ x(tf) ≤ 1, dim(x) = 1\njulia> constraint!(ocp, :control; lb=0, ub=2) # 0 ≤ u(t) ≤ 2, t ∈ [t0, tf], dim(u) = 1\njulia> constraint!(ocp, :state; lb=[ 0, 0 ], ub=[ 1, 2 ]) # [ 0, 0 ] ≤ x(t) ≤ [ 1, 2 ], t ∈ [t0, tf], dim(x) = 2\njulia> constraint!(ocp, :variable; lb=[ 0, 0 ], ub=[ 1, 2 ]) # [ 0, 0 ] ≤ v ≤ [ 1, 2 ], dim(v) = 2\n\nFunctional constraint\n\nYou can add a :boundary, :control, :state, :mixed or :variable box functional constraint.\n\nExamples\n\n# variable independent ocp\njulia> constraint!(ocp, :boundary; f = (x0, xf) -> x0[3]+xf[2], lb=0, ub=1)\n\n# variable dependent ocp\njulia> constraint!(ocp, :boundary; f = (x0, xf, v) -> x0[3]+xf[2]*v[1], lb=0, ub=1)\n\n# time independent and variable independent ocp\njulia> constraint!(ocp, :control; f = u -> 2u, lb=0, ub=1)\njulia> constraint!(ocp, :state; f = x -> x-1, lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :mixed; f = (x, u) -> x[1]-u, lb=0, ub=1)\n\n# time dependent and variable independent ocp\njulia> constraint!(ocp, :control; f = (t, u) -> 2u, lb=0, ub=1)\njulia> constraint!(ocp, :state; f = (t, x) -> t * x, lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :mixed; f = (t, x, u) -> x[1]-u, lb=0, ub=1)\n\n# time independent and variable dependent ocp\njulia> constraint!(ocp, :control; f = (u, v) -> 2u * v[1], lb=0, ub=1)\njulia> constraint!(ocp, :state; f = (x, v) -> x * v[1], lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :mixed; f = (x, u, v) -> x[1]-v[2]*u, lb=0, ub=1)\n\n# time dependent and variable dependent ocp\njulia> constraint!(ocp, :control; f = (t, u, v) -> 2u+v[2], lb=0, ub=1)\njulia> constraint!(ocp, :state; f = (t, x, v) -> x-t*v[1], lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :mixed; f = (t, x, u, v) -> x[1]*v[2]-u, lb=0, ub=1)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.constraint-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Symbol}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.constraint","text":"constraint(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n label::Symbol\n) -> Any\n\n\nRetrieve a labeled constraint. The result is a function associated with the constraint computation (not taking into account provided value / bounds).\n\nExample\n\njulia> constraint!(ocp, :initial, 0, :c0)\njulia> c = constraint(ocp, :c0)\njulia> c(1)\n1\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.constraint_type-NTuple{7, Any}","page":"CTBase API","title":"CTBase.constraint_type","text":"constraint_type(\n e,\n t,\n t0,\n tf,\n x,\n u,\n v\n) -> Union{Symbol, Tuple{Symbol, Any}}\n\n\nReturn the type constraint among :initial, :final, :boundary, :control_range, :control_fun, :state_range, :state_fun, :mixed, :variable_range, :variable_fun (:other otherwise), together with the appropriate value (range, updated expression...) Expressions like u(t0) where u is the control and t0 the initial time return :other.\n\nExample\n\njulia> t = :t; t0 = 0; tf = :tf; x = :x; u = :u; v = :v\n\njulia> constraint_type(:( ẏ(t) ), t, t0, tf, x, u, v)\n:other\n\njulia> constraint_type(:( ẋ(s) ), t, t0, tf, x, u, v)\n:other\n\njulia> constraint_type(:( x(0)' ), t, t0, tf, x, u, v)\n:boundary\n\njulia> constraint_type(:( x(t)' ), t, t0, tf, x, u, v)\n:state_fun\n\njulia> constraint_type(:( x(0) ), t, t0, tf, x, u, v)\n(:initial, nothing)\n\njulia> constraint_type(:( x[1:2:5](0) ), t, t0, tf, x, u, v)\n(:initial, 1:2:5)\n\njulia> constraint_type(:( x[1:2](0) ), t, t0, tf, x, u, v)\n(:initial, 1:2)\n\njulia> constraint_type(:( x[1](0) ), t, t0, tf, x, u, v)\n(:initial, 1)\n\njulia> constraint_type(:( 2x[1](0)^2 ), t, t0, tf, x, u, v)\n:boundary\n\njulia> constraint_type(:( x(tf) ), t, t0, tf, x, u, v)\n(:final, nothing)\nj\njulia> constraint_type(:( x[1:2:5](tf) ), t, t0, tf, x, u, v)\n(:final, 1:2:5)\n\njulia> constraint_type(:( x[1:2](tf) ), t, t0, tf, x, u, v)\n(:final, 1:2)\n\njulia> constraint_type(:( x[1](tf) ), t, t0, tf, x, u, v)\n(:final, 1)\n\njulia> constraint_type(:( 2x[1](tf)^2 ), t, t0, tf, x, u, v)\n:boundary\n\njulia> constraint_type(:( x[1](tf) - x[2](0) ), t, t0, tf, x, u, v)\n:boundary\n\njulia> constraint_type(:( u[1:2:5](t) ), t, t0, tf, x, u, v)\n(:control_range, 1:2:5)\n\njulia> constraint_type(:( u[1:2](t) ), t, t0, tf, x, u, v)\n(:control_range, 1:2)\n\njulia> constraint_type(:( u[1](t) ), t, t0, tf, x, u, v)\n(:control_range, 1)\n\njulia> constraint_type(:( u(t) ), t, t0, tf, x, u, v)\n(:control_range, nothing)\n\njulia> constraint_type(:( 2u[1](t)^2 ), t, t0, tf, x, u, v)\n:control_fun\n\njulia> constraint_type(:( x[1:2:5](t) ), t, t0, tf, x, u, v)\n(:state_range, 1:2:5)\n\njulia> constraint_type(:( x[1:2](t) ), t, t0, tf, x, u, v)\n(:state_range, 1:2)\n\njulia> constraint_type(:( x[1](t) ), t, t0, tf, x, u, v)\n(:state_range, 1)\n\njulia> constraint_type(:( x(t) ), t, t0, tf, x, u, v)\n(:state_range, nothing)\n\njulia> constraint_type(:( 2x[1](t)^2 ), t, t0, tf, x, u, v)\n:state_fun\n\njulia> constraint_type(:( 2u[1](t)^2 * x(t) ), t, t0, tf, x, u, v)\n:mixed\n\njulia> constraint_type(:( 2u[1](0)^2 * x(t) ), t, t0, tf, x, u, v)\n:other\n\njulia> constraint_type(:( 2u[1](0)^2 * x(t) ), t, t0, tf, x, u, v)\n:other\n\njulia> constraint_type(:( 2u[1](t)^2 * x(t) + v ), t, t0, tf, x, u, v)\n:mixed\n\njulia> constraint_type(:( v[1:2:10] ), t, t0, tf, x, u, v)\n(:variable_range, 1:2:9)\n\njulia> constraint_type(:( v[1:10] ), t, t0, tf, x, u, v)\n(:variable_range, 1:10)\n\njulia> constraint_type(:( v[2] ), t, t0, tf, x, u, v)\n(:variable_range, 2)\n\njulia> constraint_type(:( v ), t, t0, tf, x, u, v)\n(:variable_range, nothing)\n\njulia> constraint_type(:( v^2 + 1 ), t, t0, tf, x, u, v)\n:variable_fun\njulia> constraint_type(:( v[2]^2 + 1 ), t, t0, tf, x, u, v)\n:variable_fun\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.constraints_labels-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.constraints_labels","text":"constraints_labels(\n ocp::OptimalControlModel\n) -> Base.KeySet{Symbol, Dict{Symbol, Tuple}}\n\n\nReturn the labels of the constraints as a Base.keys.\n\nExample\n\njulia> constraints_labels(ocp)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.control!","page":"CTBase API","title":"CTBase.control!","text":"control!(ocp::OptimalControlModel, m::Integer)\ncontrol!(ocp::OptimalControlModel, m::Integer, name::String)\ncontrol!(\n ocp::OptimalControlModel,\n m::Integer,\n name::String,\n components_names::Vector{String}\n)\n\n\nDefine the control dimension and possibly the names of each coordinate.\n\nnote: Note\nYou must use control! only once to set the control dimension.\n\nExamples\n\njulia> control!(ocp, 1)\njulia> ocp.control_dimension\n1\njulia> ocp.control_components_names\n[\"u\"]\n\njulia> control!(ocp, 1, \"v\")\njulia> ocp.control_dimension\n1\njulia> ocp.control_components_names\n[\"v\"]\n\njulia> control!(ocp, 2)\njulia> ocp.control_dimension\n2\njulia> ocp.control_components_names\n[\"u₁\", \"u₂\"]\n\njulia> control!(ocp, 2, :v)\njulia> ocp.control_dimension\n2\njulia> ocp.control_components_names\n[\"v₁\", \"v₂\"]\n\njulia> control!(ocp, 2, \"v\")\njulia> ocp.control_dimension\n2\njulia> ocp.control_components_names\n[\"v₁\", \"v₂\"]\n\n\n\n\n\n","category":"function"},{"location":"api-ctbase.html#CTBase.ct_repl-Tuple{}","page":"CTBase API","title":"CTBase.ct_repl","text":"ct_repl(; debug, verbose)\n\n\nCreate a ct REPL.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ct_repl_update_model-Tuple{Expr}","page":"CTBase API","title":"CTBase.ct_repl_update_model","text":"ct_repl_update_model(e::Expr)\n\n\nUpdate the model adding the expression e. It must be public since in the ct repl, this function is quoted each time an expression is parsed and is valid. \n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctgradient-Tuple{Function, Any}","page":"CTBase API","title":"CTBase.ctgradient","text":"ctgradient(f::Function, x; backend) -> Any\n\n\nReturn the gradient of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctgradient-Tuple{Function, Real}","page":"CTBase API","title":"CTBase.ctgradient","text":"ctgradient(f::Function, x::Real; backend) -> Any\n\n\nReturn the gradient of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctgradient-Tuple{VectorField, Any}","page":"CTBase API","title":"CTBase.ctgradient","text":"ctgradient(X::VectorField, x) -> Any\n\n\nReturn the gradient of X at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctindices-Tuple{Integer}","page":"CTBase API","title":"CTBase.ctindices","text":"ctindices(i::Integer) -> String\n\n\nReturn i > 0 as a subscript.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctinterpolate-Tuple{Any, Any}","page":"CTBase API","title":"CTBase.ctinterpolate","text":"ctinterpolate(x, f) -> Any\n\n\nReturn the interpolation of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctjacobian-Tuple{Function, Any}","page":"CTBase API","title":"CTBase.ctjacobian","text":"ctjacobian(f::Function, x; backend) -> Any\n\n\nReturn the Jacobian of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctjacobian-Tuple{Function, Real}","page":"CTBase API","title":"CTBase.ctjacobian","text":"ctjacobian(f::Function, x::Real; backend) -> Any\n\n\nReturn the Jacobian of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctjacobian-Tuple{VectorField, Any}","page":"CTBase API","title":"CTBase.ctjacobian","text":"ctjacobian(X::VectorField, x) -> Any\n\n\nReturn the Jacobian of X at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctupperscripts-Tuple{Integer}","page":"CTBase API","title":"CTBase.ctupperscripts","text":"ctupperscripts(i::Integer) -> String\n\n\nReturn i > 0 as an upperscript.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_boundary_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_boundary_constraints","text":"dim_boundary_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of the boundary constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_control_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_control_constraints","text":"dim_control_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of nonlinear control constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_control_range-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_control_range","text":"dim_control_range(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of range constraints on control (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_mixed_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_mixed_constraints","text":"dim_mixed_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of nonlinear mixed constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_path_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_path_constraints","text":"dim_path_constraints(ocp::OptimalControlModel) -> Any\n\n\nReturn the dimension of nonlinear path (state + control + mixed) constraints (nothing if one of them is not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_state_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_state_constraints","text":"dim_state_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of nonlinear state constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_state_range-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_state_range","text":"dim_state_range(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of range constraints on state (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_variable_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_variable_constraints","text":"dim_variable_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of nonlinear variable constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_variable_range-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_variable_range","text":"dim_variable_range(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of range constraints on variable (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dynamics!-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Function}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.dynamics!","text":"dynamics!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n f::Function\n)\n\n\nSet the dynamics.\n\nnote: Note\nYou can use dynamics! only once to define the dynamics.The state, control and variable dimensions must be set before. Use state!, control! and variable!.\nThe times must be set before. Use time!.\nWhen an element is of dimension 1, consider it as a scalar.\n\nExample\n\njulia> dynamics!(ocp, f)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.getFullDescription-Tuple{Tuple{Vararg{Symbol}}, Tuple{Vararg{Tuple{Vararg{Symbol}}}}}","page":"CTBase API","title":"CTBase.getFullDescription","text":"getFullDescription(\n desc::Tuple{Vararg{Symbol}},\n desc_list::Tuple{Vararg{Tuple{Vararg{Symbol}}}}\n) -> Tuple{Vararg{Symbol}}\n\n\nReturn a complete description from an incomplete description desc and a list of complete descriptions desc_list. If several complete descriptions are possible, then the first one is returned.\n\nExample\n\njulia> desc_list = ((:a, :b), (:b, :c), (:a, :c))\n(:a, :b)\n(:b, :c)\n(:a, :c)\njulia> getFullDescription((:a,), desc_list)\n(:a, :b)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.get_priority_print_callbacks-Tuple{Tuple{Vararg{CTCallback}}}","page":"CTBase API","title":"CTBase.get_priority_print_callbacks","text":"get_priority_print_callbacks(\n cbs::Tuple{Vararg{CTCallback}}\n) -> Tuple{Vararg{CTCallback}}\n\n\nGet the highest priority print callbacks.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.get_priority_stop_callbacks-Tuple{Tuple{Vararg{CTCallback}}}","page":"CTBase API","title":"CTBase.get_priority_stop_callbacks","text":"get_priority_stop_callbacks(\n cbs::Tuple{Vararg{CTCallback}}\n) -> Tuple{Vararg{CTCallback}}\n\n\nGet the highest priority stop callbacks.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.has_free_final_time-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.has_free_final_time","text":"has_free_final_time(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined with free final time.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.has_free_initial_time-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.has_free_initial_time","text":"has_free_initial_time(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined with free initial time.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.has_lagrange_cost-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.has_lagrange_cost","text":"has_lagrange_cost(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined with lagrange cost.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.has_mayer_cost-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.has_mayer_cost","text":"has_mayer_cost(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined with mayer cost.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_autonomous-Tuple{OptimalControlModel{Autonomous}}","page":"CTBase API","title":"CTBase.is_autonomous","text":"is_autonomous(ocp::OptimalControlModel{Autonomous}) -> Bool\n\n\nReturn true if the model is autonomous.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_fixed-Tuple{OptimalControlModel{<:TimeDependence, Fixed}}","page":"CTBase API","title":"CTBase.is_fixed","text":"is_fixed(\n ocp::OptimalControlModel{<:TimeDependence, Fixed}\n) -> Bool\n\n\nReturn true if the model is fixed (= has no variable).\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_max-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_max","text":"is_max(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the criterion type of ocp is :max.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_min-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_min","text":"is_min(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the criterion type of ocp is :min.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_time_dependent-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_time_dependent","text":"is_time_dependent(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined as time dependent.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_time_independent-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_time_independent","text":"is_time_independent(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined as time independent.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_variable_dependent-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_variable_dependent","text":"is_variable_dependent(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined as variable dependent.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_variable_independent-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_variable_independent","text":"is_variable_independent(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined as variable independent.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.nlp_constraints!-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.nlp_constraints!","text":"nlp_constraints!(\n ocp::OptimalControlModel\n) -> Tuple{Tuple{Any, CTBase.var\"#ξ#91\", Vector{Real}}, Tuple{Any, CTBase.var\"#η#92\", Vector{Real}}, Tuple{Any, CTBase.var\"#ψ#93\", Vector{Real}}, Tuple{Any, CTBase.var\"#ϕ#94\", Vector{Real}}, Tuple{Any, CTBase.var\"#θ#95\", Vector{Real}}, Tuple{Vector{Real}, Vector{Int64}, Vector{Real}}, Tuple{Vector{Real}, Vector{Int64}, Vector{Real}}, Tuple{Vector{Real}, Vector{Int64}, Vector{Real}}}\n\n\nReturn a 6-tuple of tuples:\n\n(ξl, ξ, ξu) are control constraints\n(ηl, η, ηu) are state constraints\n(ψl, ψ, ψu) are mixed constraints\n(ϕl, ϕ, ϕu) are boundary constraints\n(θl, θ, θu) are variable constraints\n(ul, uind, uu) are control linear constraints of a subset of indices\n(xl, xind, xu) are state linear constraints of a subset of indices\n(vl, vind, vu) are variable linear constraints of a subset of indices\n\nand update information about constraints dimensions of ocp.\n\nnote: Note\nThe dimensions of the state and control must be set before calling nlp_constraints!.\nFor a Fixed problem, dimensions associated with constraints on the variable are set to zero.\n\nExample\n\njulia> (ξl, ξ, ξu), (ηl, η, ηu), (ψl, ψ, ψu), (ϕl, ϕ, ϕu), (θl, θ, θu),\n (ul, uind, uu), (xl, xind, xu), (vl, vind, vu) = nlp_constraints!(ocp)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.objective!-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Symbol, Function, Function}, Tuple{OptimalControlModel{T, V}, Symbol, Function, Function, Symbol}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.objective!","text":"objective!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol,\n g::Function,\n f⁰::Function\n)\nobjective!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol,\n g::Function,\n f⁰::Function,\n criterion::Symbol\n)\n\n\nSet the criterion to the function g and f⁰. Type can be :bolza. Criterion is :min or :max.\n\nnote: Note\nYou can use objective! only once to define the objective.The state, control and variable dimensions must be set before. Use state!, control! and variable!.\nThe times must be set before. Use time!.\nWhen an element is of dimension 1, consider it as a scalar.\n\nExample\n\njulia> objective!(ocp, :bolza, (x0, xf) -> x0[1] + xf[2], (x, u) -> x[1]^2 + u^2) # the control is of dimension 1\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.objective!-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Symbol, Function}, Tuple{OptimalControlModel{T, V}, Symbol, Function, Symbol}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.objective!","text":"objective!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol,\n f::Function\n)\nobjective!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol,\n f::Function,\n criterion::Symbol\n)\n\n\nSet the criterion to the function f. Type can be :mayer or :lagrange. Criterion is :min or :max.\n\nnote: Note\nYou can use objective! only once to define the objective.The state, control and variable dimensions must be set before. Use state!, control! and variable!.\nThe times must be set before. Use time!.\nWhen an element is of dimension 1, consider it as a scalar.\n\nExamples\n\njulia> objective!(ocp, :mayer, (x0, xf) -> x0[1] + xf[2])\njulia> objective!(ocp, :lagrange, (x, u) -> x[1]^2 + u^2) # the control is of dimension 1\n\nwarning: Warning\nIf you set twice the objective, only the last one will be taken into account.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.remove_constraint!-Tuple{OptimalControlModel, Symbol}","page":"CTBase API","title":"CTBase.remove_constraint!","text":"remove_constraint!(ocp::OptimalControlModel, label::Symbol)\n\n\nRemove a labeled constraint.\n\nExample\n\njulia> remove_constraint!(ocp, :con)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.replace_call-Tuple{Any, Symbol, Any, Any}","page":"CTBase API","title":"CTBase.replace_call","text":"replace_call(e, x::Symbol, t, y) -> Any\n\n\nReplace calls in e of the form (...x...)(t) by (...y...).\n\nExample\n\n\njulia> t = :t; t0 = 0; tf = :tf; x = :x; u = :u;\n\njulia> e = :( x[1](0) * 2x(tf) - x[2](tf) * 2x(0) )\n:((x[1])(0) * (2 * x(tf)) - (x[2])(tf) * (2 * x(0)))\n\njulia> x0 = Symbol(x, 0); e = replace_call(e, x, t0, x0)\n:(x0[1] * (2 * x(tf)) - (x[2])(tf) * (2x0))\n\njulia> xf = Symbol(x, \"f\"); replace_call(ans, x, tf, xf)\n:(x0[1] * (2xf) - xf[2] * (2x0))\n\njulia> e = :( A*x(t) + B*u(t) ); replace_call(replace_call(e, x, t, x), u, t, u)\n:(A * x + B * u)\n\njulia> e = :( F0(x(t)) + u(t)*F1(x(t)) ); replace_call(replace_call(e, x, t, x), u, t, u)\n:(F0(x) + u * F1(x))\n\njulia> e = :( 0.5u(t)^2 ); replace_call(e, u, t, u)\n:(0.5 * u ^ 2)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.replace_call-Tuple{Any, Vector{Symbol}, Any, Any}","page":"CTBase API","title":"CTBase.replace_call","text":"replace_call(e, x::Vector{Symbol}, t, y) -> Any\n\n\nReplace calls in e of the form (...x1...x2...)(t) by (...y1...y2...) for all symbols x1, x2... in the vector x.\n\nExample\n\n\njulia> t = :t; t0 = 0; tf = :tf; x = :x; u = :u;\n\njulia> e = :( (x^2 + u[1])(t) ); replace_call(e, [ x, u ], t , [ :xx, :uu ])\n:(xx ^ 2 + uu[1])\n\njulia> e = :( ((x^2)(t) + u[1])(t) ); replace_call(e, [ x, u ], t , [ :xx, :uu ])\n:(xx ^ 2 + uu[1])\n\njulia> e = :( ((x^2)(t0) + u[1])(t) ); replace_call(e, [ x, u ], t , [ :xx, :uu ])\n:((xx ^ 2)(t0) + uu[1])\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.state!","page":"CTBase API","title":"CTBase.state!","text":"state!(ocp::OptimalControlModel, n::Integer)\nstate!(ocp::OptimalControlModel, n::Integer, name::String)\nstate!(\n ocp::OptimalControlModel,\n n::Integer,\n name::String,\n components_names::Vector{String}\n)\n\n\nDefine the state dimension and possibly the names of each component.\n\nnote: Note\nYou must use state! only once to set the state dimension.\n\nExamples\n\njulia> state!(ocp, 1)\njulia> ocp.state_dimension\n1\njulia> ocp.state_components_names\n[\"x\"]\n\njulia> state!(ocp, 1, \"y\")\njulia> ocp.state_dimension\n1\njulia> ocp.state_components_names\n[\"y\"]\n\njulia> state!(ocp, 2)\njulia> ocp.state_dimension\n2\njulia> ocp.state_components_names\n[\"x₁\", \"x₂\"]\n\njulia> state!(ocp, 2, :y)\njulia> ocp.state_dimension\n2\njulia> ocp.state_components_names\n[\"y₁\", \"y₂\"]\n\njulia> state!(ocp, 2, \"y\")\njulia> ocp.state_dimension\n2\njulia> ocp.state_components_names\n[\"y₁\", \"y₂\"]\n\n\n\n\n\n","category":"function"},{"location":"api-ctbase.html#CTBase.time!-Union{Tuple{OptimalControlModel{<:TimeDependence, VT}}, Tuple{VT}} where VT","page":"CTBase API","title":"CTBase.time!","text":"time!(\n ocp::OptimalControlModel{<:TimeDependence, VT};\n t0,\n tf,\n ind0,\n indf,\n name\n)\n\n\nSet the initial and final times. We denote by t0 the initial time and tf the final time. The optimal control problem is denoted ocp. When a time is free, then one must provide the corresponding index of the ocp variable.\n\nnote: Note\nYou must use time! only once to set either the initial or the final time, or both.\n\nExamples\n\njulia> time!(ocp, t0=0, tf=1 ) # Fixed t0 and fixed tf\njulia> time!(ocp, t0=0, indf=2) # Fixed t0 and free tf\njulia> time!(ocp, ind0=2, tf=1 ) # Free t0 and fixed tf\njulia> time!(ocp, ind0=2, indf=3) # Free t0 and free tf\n\nWhen you plot a solution of an optimal control problem, the name of the time variable appears. By default, the name is \"t\". Consider you want to set the name of the time variable to \"s\".\n\njulia> time!(ocp, t0=0, tf=1, name=\"s\") # name is a String\n# or\njulia> time!(ocp, t0=0, tf=1, name=:s ) # name is a Symbol \n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.variable!","page":"CTBase API","title":"CTBase.variable!","text":"variable!(ocp::OptimalControlModel, q::Integer)\nvariable!(\n ocp::OptimalControlModel,\n q::Integer,\n name::String\n)\nvariable!(\n ocp::OptimalControlModel,\n q::Integer,\n name::String,\n components_names::Vector{String}\n)\n\n\nDefine the variable dimension and possibly the names of each component.\n\nnote: Note\nYou can use variable! once to set the variable dimension when the model is NonFixed.\n\nExamples\n\njulia> variable!(ocp, 1, \"v\")\njulia> variable!(ocp, 2, \"v\", [ \"v₁\", \"v₂\" ])\n\n\n\n\n\n","category":"function"},{"location":"api-ctbase.html#CTBase.∂ₜ-Tuple{Any}","page":"CTBase API","title":"CTBase.∂ₜ","text":"∂ₜ(f) -> CTBase.var\"#106#108\"\n\n\nPartial derivative wrt time of a function.\n\nExample\n\njulia> ∂ₜ((t,x) -> t*x)(0,8)\n8\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.@Lie-Tuple{Expr, Any, Any}","page":"CTBase API","title":"CTBase.@Lie","text":"Macros for Poisson brackets\n\nExample\n\njulia> H0 = (x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2)\njulia> H1 = (x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2)\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7]) autonomous=true variable=false\n#\njulia> H0 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2)\njulia> H1 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7]) autonomous=false variable=false\n#\njulia> H0 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v)\njulia> H1 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v)\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7], 2) autonomous=true variable=true\n#\njulia> H0 = (t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v)\njulia> H1 = (t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7], 2) autonomous=false variable=true\n\n\n\n\n\n","category":"macro"},{"location":"api-ctbase.html#CTBase.@Lie-Tuple{Expr, Any}","page":"CTBase API","title":"CTBase.@Lie","text":"Macros for Lie and Poisson brackets\n\nExample\n\njulia> H0 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2)\njulia> H1 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7]) autonomous=false\n#\njulia> H0 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v)\njulia> H1 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v)\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7], 2) variable=true\n#\n\n\n\n\n\n","category":"macro"},{"location":"api-ctbase.html#CTBase.@Lie-Tuple{Expr}","page":"CTBase API","title":"CTBase.@Lie","text":"Macros for Lie and Poisson brackets\n\nExample\n\njulia> F0 = VectorField(x -> [x[1], x[2], (1-x[3])])\njulia> F1 = VectorField(x -> [0, -x[3], x[2]])\njulia> @Lie [F0, F1]([1, 2, 3])\n[0, 5, 4]\n#\njulia> F0 = VectorField((t, x) -> [t+x[1], x[2], (1-x[3])], autonomous=false)\njulia> F1 = VectorField((t, x) -> [t, -x[3], x[2]], autonomous=false)\njulia> @Lie [F0, F1](1, [1, 2, 3])\n#\njulia> F0 = VectorField((x, v) -> [x[1]+v, x[2], (1-x[3])], variable=true)\njulia> F1 = VectorField((x, v) -> [0, -x[3]-v, x[2]], variable=true)\njulia> @Lie [F0, F1]([1, 2, 3], 2)\n#\njulia> F0 = VectorField((t, x, v) -> [t+x[1]+v, x[2], (1-x[3])], autonomous=false, variable=true)\njulia> F1 = VectorField((t, x, v) -> [t, -x[3]-v, x[2]], autonomous=false, variable=true)\njulia> @Lie [F0, F1](1, [1, 2, 3], 2)\n#\njulia> H0 = Hamiltonian((x, p) -> 0.5*(2x[1]^2+x[2]^2+p[1]^2))\njulia> H1 = Hamiltonian((x, p) -> 0.5*(3x[1]^2+x[2]^2+p[2]^2))\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7])\n3.0\n#\njulia> H0 = Hamiltonian((t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2), autonomous=false)\njulia> H1 = Hamiltonian((t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2), autonomous=false)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7])\n#\njulia> H0 = Hamiltonian((x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v), variable=true)\njulia> H1 = Hamiltonian((x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v), variable=true)\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7], 2)\n#\njulia> H0 = Hamiltonian((t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v), autonomous=false, variable=true)\njulia> H1 = Hamiltonian((t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v), autonomous=false, variable=true)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7], 2)\n#\n\n\n\n\n\n","category":"macro"},{"location":"api-ctbase.html#CTBase.@def","page":"CTBase API","title":"CTBase.@def","text":"Define an optimal control problem. One pass parsing of the definition.\n\nExample\n\n@def ocp begin\n tf ∈ R, variable\n t ∈ [ 0, tf ], time\n x ∈ R², state\n u ∈ R, control\n tf ≥ 0\n -1 ≤ u(t) ≤ 1\n q = x₁\n v = x₂\n q(0) == 1\n v(0) == 2\n q(tf) == 0\n v(tf) == 0\n 0 ≤ q(t) ≤ 5, (1)\n -2 ≤ v(t) ≤ 3, (2)\n ẋ(t) == [ v(t), u(t) ]\n tf → min\nend\n\n\n\n\n\n","category":"macro"},{"location":"api-ctbase.html","page":"CTBase API","title":"CTBase API","text":"","category":"page"},{"location":"tutorial.html#First-example:-Goddard-problem","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"(Image: R.H. Goddard)","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"We start with the well-known so-called Goddard problem, which models the ascent of a rocket through the atmosphere (see for instance [1],[2]). We restrict here ourselves to vertical (monodimensional) trajectories, and the state variables are the altitude, speed and mass of the rocket during the flight, for a total dimension of 3. The rocket is subject to gravity, thrust and drag forces. The final time is free, and the objective is here to reach a maximal altitude with a given fuel consumption, i.e a fixed final mass. We also impose a speed limit. All units are renormalized.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"[1]: R.H. Goddard. A Method of Reaching Extreme Altitudes, volume 71(2) of Smithsonian Miscellaneous Collections. Smithsonian institution, City of Washington, 1919.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"[2]: H. Seywald and E.M. Cliff. Goddard problem in presence of a dynamic pressure limit. Journal of Guidance, Control, and Dynamics, 16(4):776–781, 1993.","category":"page"},{"location":"tutorial.html#Problem-definition","page":"First example: Goddard problem","title":"Problem definition","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"First import the necessary modules","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"using CTDirect\nusing CTBase\nusing NLPModelsIpopt\nusing Plots","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Then define the OCP for the Goddard problem. Note that the free final time is modeled as an optimization variable, and has both a lower bound to prevent the optimization getting stuck at tf=0. In this particular case an upper bound is not needed for the final time since the final mass is prescribed.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Cd = 310\nβ = 500\nTmax = 3.5\nb = 2\nvmax = 0.1\nfunction F0(x)\n r, v, m = x\n D = Cd * v^2 * exp(-β*(r - 1))\n return [ v, -D/m - 1/r^2, 0 ]\nend\nfunction F1(x)\n r, v, m = x\n return [ 0, Tmax/m, -b*Tmax ]\nend\n@def ocp begin\n tf ∈ R, variable\n t ∈ [ 0, tf ], time\n x ∈ R^3, state\n u ∈ R, control\n 0.1 ≤ tf ≤ Inf\n r = x[1]\n v = x[2]\n m = x[3]\n x(0) == [1, 0, 1]\n m(tf) == 0.6\n 1 ≤ r(t) ≤ 1.1\n 0 ≤ v(t) ≤ vmax\n 0 ≤ u(t) ≤ 1\n ẋ(t) == F0(x(t)) + u(t)*F1(x(t))\n r(tf) → max\nend\nnothing # hide","category":"page"},{"location":"tutorial.html#Basic-solve","page":"First example: Goddard problem","title":"Basic solve","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"We can solve the problem directly using the default options.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"sol1 = solve(ocp, print_level=5)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Then plot the solution with the state and control variables, as well as the costate recovered from the lagrange multipliers of the discretized problem. ","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"plot(sol1)","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"The most common option for solve is the number of time steps for the discretized problem (default 100), that can be set with the argument grid_size. A larger grid size will increase the computational cost, while a smaller value may lead to a very coarse solution.","category":"page"},{"location":"tutorial.html#Initial-guess-options","page":"First example: Goddard problem","title":"Initial guess options","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"The function solve uses a default constant initialisation of 0.1 for all variables. More advanced options include constant and/or functional initialisation for each individual state or control component, as well as reusing an existing solution, also known as warm start[3].","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"[3]: Currently only the primal variables are reused for the warm start, not the lagrange multipliers. It should be noted that previous experiments with the Bocop software seemed to indicate that initializing also the multipliers gave little benefit.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Let us start with the simplest case, constant initialisation.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"x_const = [1.05, 0.2, 0.8]\nu_const = 0.5\nv_const = 0.15\ninit1 = (state=x_const, control=u_const, variable=v_const)\nsol2 = solve(ocp, print_level=0, init=init1)\nprintln(\"Objective \", sol2.objective, \" after \", sol2.iterations, \" iterations\")","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Now we illustrate the functional initialisation, with some random functions. Note that we only consider the state and control variables, since the optimization variables are scalar and therefore a functional initialisation is not relevant. In the example notice that the initialization does not provide an argument for the optimization variables, therefore the default initial guess will be used. ","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"x_func = t->[1+t^2, sqrt(t), 1-t]\nu_func = t->(cos(t)+1)*0.5\ninit2 = (state=x_func, control=u_func)\nsol3 = solve(ocp, print_level=0, init=init2)\nprintln(\"Objective \", sol3.objective, \" after \", sol3.iterations, \" iterations\")","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"More generally, the default, constant and functional initialisations can be mixed, as shown in the example below that uses a functional initial guess for the state, a constant initial guess for the control, and the default initial guess for the optimization variables. ","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"init3 = (state=x_func, control=u_const)\nsol4 = solve(ocp, print_level=0, init=init3)\nprintln(\"Objective \", sol4.objective, \" after \", sol4.iterations, \" iterations\")","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Finally, we can also use a so-called warmstart strategy and use an existing solution as initial guess (note that the OCP solution returned by the solve call is functional, thus it is not necessary to use the same time grid). Notice that the objective and constraint violation values start much closer to the solution than with the previous initialisations.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"sol4 = solve(ocp, grid_size=200, print_level=5, init=sol1)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"plot(sol4)","category":"page"},{"location":"tutorial.html#The-discretized-problem","page":"First example: Goddard problem","title":"The discretized problem","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Instead of calling solve directly on the OCP problem, you can first obtain the discretized problem (DOCP) by calling directTranscription, then call solve on the DOCP. The resulting solution of the discretized problem can be used to generate the corresponding OCP solution with OCPSolutionFromDOCP.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"docp = directTranscription(ocp, grid_size=100)\ndsol = solve(docp, print_level=5)\nsol5 = OCPSolutionFromDOCP(docp, dsol)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"The initial guess can be passed to solve same as before.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"dsol = solve(docp, print_level=0, init=sol1)\nsol6 = OCPSolutionFromDOCP(docp, dsol)\nprintln(\"Objective \", sol6.objective, \" after \", sol6.iterations, \" iterations\")","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Another possibility is to set the initial guess associated to the DOCP, using the function setInitialGuess.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"setInitialGuess(docp, sol1)\ndsol = solve(docp, print_level=5)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Finally, the direct transcription also accept an initial guess.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"docp = directTranscription(ocp, grid_size=100, init=sol1)\ndsol = solve(docp, print_level=5)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"","category":"page"}] +[{"location":"api.html#CTDirect-API","page":"API","title":"CTDirect API","text":"","category":"section"},{"location":"api.html#Index","page":"API","title":"Index","text":"","category":"section"},{"location":"api.html","page":"API","title":"API","text":"Pages = [\"api.md\"]\nModules = [CTDirect, CommonSolve]\nOrder = [:module, :constant, :type, :function, :macro]","category":"page"},{"location":"api.html#Documentation","page":"API","title":"Documentation","text":"","category":"section"},{"location":"api.html","page":"API","title":"API","text":"Modules = [CTDirect]\nOrder = [:module, :constant, :type, :function, :macro]\nPrivate = false","category":"page"},{"location":"api.html#CTDirect.DOCP","page":"API","title":"CTDirect.DOCP","text":"Struct for discretized optimal control problem DOCP\n\nContains:\n\na copy of the original OCP\na NLP formulation of the DOCP\ndata required to link the two problems\n\n\n\n\n\n","category":"type"},{"location":"api.html#CTDirect.OCPSolutionFromDOCP-Tuple{Any, Any}","page":"API","title":"CTDirect.OCPSolutionFromDOCP","text":"OCPSolutionFromDOCP(\n docp,\n docp_solution_ipopt\n) -> OptimalControlSolution\n\n\nBuild OCP functional solution from DOCP vector solution (given as a GenericExecutionStats)\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.available_methods-Tuple{}","page":"API","title":"CTDirect.available_methods","text":"available_methods() -> Tuple{Tuple{Vararg{Symbol}}}\n\n\nReturn the list of available methods to solve the optimal control problem.\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.directTranscription-Tuple{OptimalControlModel, Vararg{Any}}","page":"API","title":"CTDirect.directTranscription","text":"directTranscription(\n ocp::OptimalControlModel,\n description...;\n init,\n grid_size,\n time_grid\n) -> DOCP\n\n\nDiscretize an optimal control problem into a nonlinear optimization problem (ie direct transcription)\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.getNLP-Tuple{DOCP}","page":"API","title":"CTDirect.getNLP","text":"getNLP(docp::DOCP) -> Any\n\n\nExtract the NLP problem from the DOCP\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.is_solvable-Tuple{Any}","page":"API","title":"CTDirect.is_solvable","text":"is_solvable(ocp) -> Bool\n\n\nCheck if an OCP is solvable by the method solve.\n\n\n\n\n\n","category":"method"},{"location":"api.html#CTDirect.setInitialGuess-Tuple{DOCP, Any}","page":"API","title":"CTDirect.setInitialGuess","text":"setInitialGuess(docp::DOCP, init) -> Any\n\n\nSet initial guess in the DOCP\n\n\n\n\n\n","category":"method"},{"location":"api.html","page":"API","title":"API","text":"solve\nsave_OCP_solution\nload_OCP_solution\nexport_OCP_solution\nread_OCP_solution","category":"page"},{"location":"api.html#CommonSolve.solve","page":"API","title":"CommonSolve.solve","text":"CommonSolve.solve(args...; kwargs...)\n\nSolves an equation or other mathematical problem using the algorithm specified in the arguments. Generally, the interface is:\n\nCommonSolve.solve(prob::ProblemType,alg::SolverType; kwargs...)::SolutionType\n\nwhere the keyword arguments are uniform across all choices of algorithms.\n\nBy default, solve defaults to using solve! on the iterator form, i.e.:\n\nsolve(args...; kwargs...) = solve!(init(args...; kwargs...))\n\n\n\n\n\nsolve(\n docp::DOCP;\n init,\n display,\n print_level,\n mu_strategy,\n max_iter,\n tol,\n linear_solver,\n kwargs...\n) -> SolverCore.GenericExecutionStats\n\n\nSolve a discretized optimal control problem DOCP\n\n\n\n\n\nsolve(\n ocp::OptimalControlModel,\n description...;\n init,\n grid_size,\n time_grid,\n display,\n print_level,\n mu_strategy,\n max_iter,\n tol,\n linear_solver,\n kwargs...\n) -> OptimalControlSolution\n\n\nSolve an optimal control problem OCP by direct method\n\n\n\n\n\n","category":"function"},{"location":"api.html#CTDirect.save_OCP_solution","page":"API","title":"CTDirect.save_OCP_solution","text":"save_OCP_solution(\n sol::OptimalControlSolution;\n filename_prefix\n)\n\n\nSave OCP solution in JLD2 format\n\n\n\n\n\n","category":"function"},{"location":"api.html#CTDirect.load_OCP_solution","page":"API","title":"CTDirect.load_OCP_solution","text":"load_OCP_solution() -> Any\nload_OCP_solution(filename_prefix) -> Any\n\n\nLoad OCP solution in JLD2 format\n\n\n\n\n\n","category":"function"},{"location":"api.html#CTDirect.export_OCP_solution","page":"API","title":"CTDirect.export_OCP_solution","text":"export_OCP_solution(\n sol::OptimalControlSolution;\n filename_prefix\n)\n\n\nExport OCP solution in JSON format\n\n\n\n\n\n","category":"function"},{"location":"api.html#CTDirect.read_OCP_solution","page":"API","title":"CTDirect.read_OCP_solution","text":"read_OCP_solution() -> Any\nread_OCP_solution(filename_prefix) -> Any\n\n\nRead OCP solution in JSON format\n\n\n\n\n\n","category":"function"},{"location":"api.html","page":"API","title":"API","text":"","category":"page"},{"location":"dev-api.html#Internal-functions","page":"Developers","title":"Internal functions","text":"","category":"section"},{"location":"dev-api.html","page":"Developers","title":"Developers","text":"CurrentModule = CTDirect ","category":"page"},{"location":"dev-api.html","page":"Developers","title":"Developers","text":"Modules = [CTDirect]\nOrder = [:module, :type, :function, :macro]\nPublic = false","category":"page"},{"location":"dev-api.html#CTDirect.ArgsAtTimeStep","page":"Developers","title":"CTDirect.ArgsAtTimeStep","text":"Useful values at a time step: time, state, control, dynamics...\n\n\n\n\n\n","category":"type"},{"location":"dev-api.html#CTDirect.DOCP_constraints!-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.DOCP_constraints!","text":"DOCP_constraints!(c, xu, docp) -> Any\n\n\nCompute the constraints C for the DOCP problem (modeled as LB <= C(X) <= UB).\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.DOCP_constraints_check!-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.DOCP_constraints_check!","text":"DOCP_constraints_check!(cb, constraints, docp)\n\n\nCheck the nonlinear constraints violation for the DOCP problem. \n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.DOCP_initial_guess","page":"Developers","title":"CTDirect.DOCP_initial_guess","text":"DOCP_initial_guess(docp) -> Any\nDOCP_initial_guess(docp, init::OptimalControlInit) -> Any\n\n\nBuild initial guess for discretized problem\n\n\n\n\n\n","category":"function"},{"location":"dev-api.html#CTDirect.DOCP_variables_check!-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.DOCP_variables_check!","text":"DOCP_variables_check!(vb, variables, docp)\n\n\nCheck the variables box constraints violation for the DOCP problem. \n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.constraints_bounds-Tuple{Any}","page":"Developers","title":"CTDirect.constraints_bounds","text":"constraints_bounds(docp) -> Tuple{Any, Any}\n\n\nBuild upper and lower bounds vectors for the DOCP nonlinear constraints.\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_control_at_time_step-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.get_control_at_time_step","text":"get_control_at_time_step(xu, docp, i) -> Any\n\n\nRetrieve control variables at given time step from the NLP variables\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_final_time-Tuple{Any, Any}","page":"Developers","title":"CTDirect.get_final_time","text":"get_final_time(xu, docp) -> Any\n\n\nRetrieve final time for OCP (may be fixed or variable)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_initial_time-Tuple{Any, Any}","page":"Developers","title":"CTDirect.get_initial_time","text":"get_initial_time(xu, docp) -> Any\n\n\nRetrieve initial time for OCP (may be fixed or variable)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_lagrange_cost_at_time_step-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.get_lagrange_cost_at_time_step","text":"get_lagrange_cost_at_time_step(xu, docp, i) -> Any\n\n\nRetrieve the additional state variable corresponding to the lagrange (running) cost at given time step from the NLP variables\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_state_at_time_step-Tuple{Any, Any, Int64}","page":"Developers","title":"CTDirect.get_state_at_time_step","text":"get_state_at_time_step(xu, docp, i::Int64) -> Any\n\n\nRetrieve state variables at given time step from the NLP variables\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_time_at_time_step-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.get_time_at_time_step","text":"get_time_at_time_step(xu, docp, i) -> Any\n\n\nGet actual (un-normalized) time at give time step\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_unnormalized_time-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.get_unnormalized_time","text":"get_unnormalized_time(xu, docp, t_normalized) -> Any\n\n\nGet actual (un-normalized) time value\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.get_variable-Tuple{Any, Any}","page":"Developers","title":"CTDirect.get_variable","text":"get_variable(xu, docp) -> Any\n\n\nRetrieve optimization variables from the NLP variables\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.parse_DOCP_solution-NTuple{6, Any}","page":"Developers","title":"CTDirect.parse_DOCP_solution","text":"parse_DOCP_solution(\n docp,\n solution,\n multipliers_constraints,\n multipliers_LB,\n multipliers_UB,\n constraints\n) -> NTuple{18, Any}\n\n\nParse DOCP solution into OCP variables, constraints and multipliers\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.setPathConstraintsAtTimeStep!-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.setPathConstraintsAtTimeStep!","text":"setPathConstraintsAtTimeStep!(\n docp,\n index,\n target;\n c,\n args,\n lb,\n ub\n) -> Any\n\n\nSet the path constraints / bounds for given time step target = :constraints | :bounds\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.setPunctualConditions!-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.setPunctualConditions!","text":"setPunctualConditions!(\n docp,\n index,\n target;\n c,\n args_0,\n args_f,\n lb,\n ub\n) -> Any\n\n\nSet the boundary and variable constraints / their bounds target = :constraints | :bounds\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.setStateEquationAtTimeStep!-NTuple{5, Any}","page":"Developers","title":"CTDirect.setStateEquationAtTimeStep!","text":"setStateEquationAtTimeStep!(\n docp,\n c,\n index,\n args_i,\n args_ip1\n) -> Any\n\n\nSet the constraints corresponding to the state equation\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.set_control_at_time_step!-NTuple{4, Any}","page":"Developers","title":"CTDirect.set_control_at_time_step!","text":"set_control_at_time_step!(xu, u_init, docp, i) -> Any\n\n\nSet control variables at given time step in the NLP variables (for initial guess)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.set_state_at_time_step!-NTuple{4, Any}","page":"Developers","title":"CTDirect.set_state_at_time_step!","text":"set_state_at_time_step!(xu, x_init, docp, i) -> Any\n\n\nSet state variables at given time step in the NLP variables (for initial guess)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.set_variable!-Tuple{Any, Any, Any}","page":"Developers","title":"CTDirect.set_variable!","text":"set_variable!(xu, v_init, docp) -> Any\n\n\nSet optimization variables in the NLP variables (for initial guess)\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html#CTDirect.variables_bounds-Tuple{Any}","page":"Developers","title":"CTDirect.variables_bounds","text":"variables_bounds(docp) -> Tuple{Any, Any}\n\n\nBuild upper and lower bounds vectors for the DOCP variable box constraints.\n\n\n\n\n\n","category":"method"},{"location":"dev-api.html","page":"Developers","title":"Developers","text":"","category":"page"},{"location":"continuation.html#Discrete-continuation","page":"Discrete continuation","title":"Discrete continuation","text":"","category":"section"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Using the warm start option, it is easy to implement a basic discrete continuation method, where a sequence of problems is solved using each solution as initial guess for the next problem. This usually gives better and faster convergence than solving each problem with the same initial guess, and is a way to handle problems that require a good initial guess.","category":"page"},{"location":"continuation.html#Continuation-on-parametric-OCP","page":"Discrete continuation","title":"Continuation on parametric OCP","text":"","category":"section"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"The most compact syntax to perform a discrete continuation is to use a function that returns the OCP for a given value of the continuation parameter, and solve a sequence of these problems. We illustrate this on a very basic double integrator with increasing fixed final time.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"First we load the required packages","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"using CTDirect\nusing CTBase\nusing NLPModelsIpopt\nusing Printf\nusing Plots","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"and write a function that returns the OCP for a given final time","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"function ocp_T(T)\n @def ocp begin\n t ∈ [ 0, T ], time\n x ∈ R², state\n u ∈ R, control\n q = x₁\n v = x₂\n q(0) == 0\n v(0) == 0\n q(T) == 1\n v(T) == 0\n ẋ(t) == [ v(t), u(t) ]\n ∫(u(t)^2) → min\n end\n return ocp\nend\nnothing # hide","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Then we perform the continuation with a simple for loop, using each solution to initialize the next problem.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"init1 = ()\niter_list = []\nfor T=1:5\n ocp1 = ocp_T(T) \n sol1 = solve(ocp1, print_level=0, init=init1)\n global init1 = sol1\n @printf(\"T %.2f objective %.6f iterations %d\\n\", T, sol1.objective, sol1.iterations)\n push!(iter_list, sol1.iterations)\nend","category":"page"},{"location":"continuation.html#Continuation-on-global-variable","page":"Discrete continuation","title":"Continuation on global variable","text":"","category":"section"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"As a second example, we show how to avoid redefining a new OCP each time, and modify the original one instead. More precisely we now solve a Goddard problem for a decreasing maximal thrust. If we store the value for Tmax in a global variable, we can simply modify this variable and keep the same OCP problem during the continuation.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Let us first define the Goddard problem (note that the formulation below illustrates all the possible constraints types, and the problem could be defined in a more compact way).","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Cd = 310\nTmax = 3.5\nβ = 500\nb = 2\nfunction F0(x)\n r, v, m = x\n D = Cd * v^2 * exp(-β*(r - 1))\n return [ v, -D/m - 1/r^2, 0 ]\nend\nfunction F1(x)\n r, v, m = x\n return [ 0, Tmax/m, -b*Tmax ]\nend\n\nocp = Model(variable=true)\nr0 = 1\nv0 = 0\nm0 = 1\nmf = 0.6\nx0=[r0,v0,m0]\nvmax = 0.1\nstate!(ocp, 3)\ncontrol!(ocp, 1)\nvariable!(ocp, 1)\ntime!(ocp, t0=0, indf=1)\nconstraint!(ocp, :initial, lb=x0, ub=x0)\nconstraint!(ocp, :final, rg=3, lb=mf, ub=Inf)\nconstraint!(ocp, :state, lb=[r0,v0,mf], ub=[r0+0.2,vmax,m0])\nconstraint!(ocp, :control, lb=0, ub=1)\nconstraint!(ocp, :variable, lb=0.01, ub=Inf)\nobjective!(ocp, :mayer, (x0, xf, v) -> xf[1], :max)\ndynamics!(ocp, (x, u, v) -> F0(x) + u*F1(x) )\n\nsol0 = solve(ocp, print_level=0)\nsol = sol0\n@printf(\"Objective for reference solution %.6f\\n\", sol0.objective)","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Then we perform the continuation on the maximal thrust.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"Tmax_list = []\nobj_list = []\nfor Tmax_local=3.5:-0.5:1\n global Tmax = Tmax_local \n global sol = solve(ocp, print_level=0, init=sol)\n @printf(\"Tmax %.2f objective %.6f iterations %d\\n\", Tmax, sol.objective, sol.iterations)\n push!(Tmax_list, Tmax)\n push!(obj_list, sol.objective)\nend ","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"We plot now the objective w.r.t the maximal thrust, as well as both solutions for Tmax=3.5 and Tmax=1.","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"pobj = plot(Tmax_list, obj_list, label=\"r(tf)\",seriestype=:scatter)\nxlabel!(\"Maximal thrust (Tmax)\")\nylabel!(\"Maximal altitude r(tf)\")\nplot(sol0)\np = plot!(sol)\nplot(pobj, p, layout=2)","category":"page"},{"location":"continuation.html","page":"Discrete continuation","title":"Discrete continuation","text":"","category":"page"},{"location":"index.html#CTDirect.jl","page":"Introduction","title":"CTDirect.jl","text":"","category":"section"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"CurrentModule = CTDirect","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"The CTDirect.jl package is part of the control-toolbox ecosystem.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"flowchart TD\nO(OptimalControl) --> B(CTBase)\nO --> D(CTDirect)\nO --> F(CTFlows)\nP(CTProblems) --> F\nP --> B\nF --> B\nD --> B\nstyle D fill:#FBF275","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"note: Install\nTo install a package from the control-toolbox ecosystem, please visit the installation page.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"An optimal control problem with fixed initial and final times, denoted (OCP), can be described as minimising the cost functional","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"g(x(t_0) x(t_f)) + int_t_0^t_f f^0(t x(t) u(t))mathrmdt","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"where the state x and the control u are functions subject, for t in t_0 t_f, to the differential constraint","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":" dotx(t) = f(t x(t) u(t))","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"and other constraints such as","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"beginarrayllcll\nxi_l le xi(t u(t)) le xi_u \neta_l le eta(t x(t)) le eta_u \npsi_l le psi(t x(t) u(t)) le psi_u \nphi_l le phi(t_0 x(t_0) t_f x(t_f)) le phi_u\nendarray","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"The so-called direct approach transforms the infinite dimensional optimal control problem (OCP) into a finite dimensional optimization problem (NLP). This is done by a discretization in time by Runge-Kutta methods applied to the state and control variables, as well as the dynamics equation. These methods are usually less precise than indirect methods based on Pontryagin’s Maximum Principle, but more robust with respect to the initialization. Also, they are more straightforward to apply, hence their wide use in industrial applications. We refer the reader to for instance[1] and [2] for more details on direct transcription methods and NLP algorithms.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"[1]: J. T. Betts. Practical methods for optimal control using nonlinear programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"[2]: J. Nocedal and S.J. Wright. Numerical optimization. Springer-Verlag, New York, 1999.****","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"Example of the time discretization by the trapezoidal rule:","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"beginarraylcl\nt in t_0t_f to t_0 ldots t_N=t_f02em\nx(cdot) u(cdot) to X=x_0 ldots x_N u_0 ldots u_N 1em\nhline\n\ntextstep to h = (t_f-t_0)N02em\ntextcriterion to min g(x_0 x_N) 02em\ntextdynamics to x_i+i = x_i + (h2) (f(t_i x_i u_i) + f(t_i+1 x_i+1 u_i+1)) 02em\ntextcontrol constraints to xi_l le xi(t_i u_i) le xi_u 02em\ntextpath constraints to eta_l le eta(t_i x_i) le eta_u 02em\ntextmixed constraints to psi_l le psi(t_i x_i u_i) le psi_u 02em\ntextlimit conditions to phi_l le phi(x_0 x_N) le phi_u\nendarray","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"We therefore obtain a nonlinear programming problem on the discretized state and control variables of the general form:","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"(NLP)quad left\nbeginarraylr\nmin F(X) \nLB le C(X) le UB\nendarray\nright","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"We use packages from JuliaSmoothOptimizers to solve the (NLP) problem.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"As input of this package we use an OptimalControlModel structure from CTBase.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"note: Current limitations\nThe current implemented is limited totrapezoidal rule for the ODE discretization\nIpopt for the optimization software","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"note: Related packages\nThis package is equivalent to the bocop software.","category":"page"},{"location":"index.html","page":"Introduction","title":"Introduction","text":"","category":"page"},{"location":"api-ctbase.html#CTBase-API","page":"CTBase API","title":"CTBase API","text":"","category":"section"},{"location":"api-ctbase.html","page":"CTBase API","title":"CTBase API","text":"This is just a dump of CTBase API documentation. For more details about CTBase.jl package, see the documentation.","category":"page"},{"location":"api-ctbase.html#Index","page":"CTBase API","title":"Index","text":"","category":"section"},{"location":"api-ctbase.html","page":"CTBase API","title":"CTBase API","text":"Pages = [\"api-ctbase.md\"]\nModules = [CTBase]\nOrder = [:module, :constant, :type, :function, :macro]","category":"page"},{"location":"api-ctbase.html#Documentation","page":"CTBase API","title":"Documentation","text":"","category":"section"},{"location":"api-ctbase.html","page":"CTBase API","title":"CTBase API","text":"Modules = [CTBase]\nOrder = [:module, :constant, :type, :function, :macro]\nPrivate = false","category":"page"},{"location":"api-ctbase.html#CTBase.CTBase","page":"CTBase API","title":"CTBase.CTBase","text":"CTBase module.\n\nLists all the imported modules and packages:\n\nBase\nCore\nDataStructures\nDocStringExtensions\nLinearAlgebra\nMLStyle\nParameters\nPrettyTables\nPrintf\nReplMaker\nUnicode\n\nList of all the exported names:\n\nAbstractHamiltonian\nAmbiguousDescription\nAutonomous\nBoundaryConstraint\nCTCallback\nCTCallbacks\nCTException\nControl\nControlConstraint\nControlLaw\nControls\nCostate\nCostates\nDCostate\nDState\nDescription\nDimension\nDynamics\nFeedbackControl\nFixed\nHamiltonian\nHamiltonianLift\nHamiltonianVectorField\nIncorrectArgument\nIncorrectMethod\nIncorrectOutput\nIndex\nLagrange\n@Lie\nLie\nLift\nMayer\nMixedConstraint\nModel\nMultiplier\nNonAutonomous\nNonFixed\nNotImplemented\nOptimalControlInit\nOptimalControlModel\nOptimalControlSolution\nParsingError\nPoisson\nPrintCallback\nState\nStateConstraint\nStates\nStopCallback\nTime\nTimeDependence\nTimes\nTimesDisc\nUnauthorizedCall\nVariable\nVariableConstraint\nVariableDependence\nVectorField\n__OCPModel\nadd\nconstraint\nconstraint!\nconstraint_type\nconstraints_labels\ncontrol!\nctNumber\nctVector\nct_repl\nct_repl_update_model\nctgradient\nctindices\nctinterpolate\nctjacobian\nctupperscripts\n@def\ndim_boundary_constraints\ndim_control_constraints\ndim_control_range\ndim_mixed_constraints\ndim_path_constraints\ndim_state_constraints\ndim_state_range\ndim_variable_constraints\ndim_variable_range\ndynamics!\ngetFullDescription\nget_priority_print_callbacks\nget_priority_stop_callbacks\nhas_free_final_time\nhas_free_initial_time\nhas_lagrange_cost\nhas_mayer_cost\nis_autonomous\nis_fixed\nis_max\nis_min\nis_time_dependent\nis_time_independent\nis_variable_dependent\nis_variable_independent\nnlp_constraints!\nobjective!\nremove_constraint!\nreplace_call\nset_AD_backend\nstate!\ntime!\nvariable!\n∂ₜ\n⋅\n\n\n\n\n\n","category":"module"},{"location":"api-ctbase.html#CTBase.Control","page":"CTBase API","title":"CTBase.Control","text":"Type alias for a control in Rᵐ.\n\njulia> const Control = ctVector\n\nSee also: ctVector, State, Costate, Variable.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Costate","page":"CTBase API","title":"CTBase.Costate","text":"Type alias for a costate in Rⁿ.\n\njulia> const Costate = ctVector\n\nSee also: ctVector, State, Control, Variable.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.DCostate","page":"CTBase API","title":"CTBase.DCostate","text":"Type alias for a tangent vector to the costate space.\n\njulia> const DCostate = ctVector\n\nSee also: ctVector, DState.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.DState","page":"CTBase API","title":"CTBase.DState","text":"Type alias for a tangent vector to the state space.\n\njulia> const DState = ctVector\n\nSee also: ctVector, DCostate.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.State","page":"CTBase API","title":"CTBase.State","text":"Type alias for a state in Rⁿ.\n\njulia> const State = ctVector\n\nSee also: ctVector, Costate, Control, Variable.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.TimesDisc","page":"CTBase API","title":"CTBase.TimesDisc","text":"Type alias for a grid of times. This is used to define a discretization of time interval given to solvers.\n\njulia> const TimesDisc = Union{Times, StepRangeLen}\n\nSee also: Time, Times.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Variable","page":"CTBase API","title":"CTBase.Variable","text":"Type alias for a variable in Rᵏ.\n\njulia> const Variable = ctVector\n\nSee also: ctVector, State, Costate, Control.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ctVector","page":"CTBase API","title":"CTBase.ctVector","text":"Type alias for a vector of real numbers.\n\njulia> const ctVector = Union{ctNumber, AbstractVector{<:ctNumber}}\n\nSee also: ctNumber, State, Costate, Control, Variable.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.AbstractHamiltonian","page":"CTBase API","title":"CTBase.AbstractHamiltonian","text":"abstract type AbstractHamiltonian{time_dependence, variable_dependence}\n\nAbstract type for hamiltonians.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.AmbiguousDescription","page":"CTBase API","title":"CTBase.AmbiguousDescription","text":"struct AmbiguousDescription <: CTException\n\nException thrown when the description is ambiguous / incorrect.\n\nFields\n\nvar::Tuple{Vararg{Symbol}}\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Autonomous","page":"CTBase API","title":"CTBase.Autonomous","text":"abstract type Autonomous <: TimeDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.BoundaryConstraint","page":"CTBase API","title":"CTBase.BoundaryConstraint","text":"struct BoundaryConstraint{variable_dependence}\n\nFields\n\nf::Function\n\nThe default value for variable_dependence is Fixed.\n\nConstructor\n\nThe constructor BoundaryConstraint returns a BoundaryConstraint of a function. The function must take 2 or 3 arguments (x0, xf) or (x0, xf, v), if the function is variable, it must be specified. Dependencies are specified with a boolean, variable, false by default or with a DataType, NonFixed/Fixed, Fixed by default.\n\nExamples\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], variable=true)\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], NonFixed)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x0 and xf as a scalar. When the constraint is dimension 1, return a scalar.\n\nCall\n\nThe call returns the evaluation of the BoundaryConstraint for given values. If a variable is given for a non variable dependent boundary constraint, it will be ignored.\n\nExamples\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B([0, 0], [1, 1])\n[1, 2]\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B([0, 0], [1, 1],Real[])\n[1, 2]\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], variable=true)\njulia> B([0, 0], [1, 1], [1, 2, 3])\n[4, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.BoundaryConstraint-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.BoundaryConstraint","text":"BoundaryConstraint(\n f::Function,\n dependencies::DataType...\n) -> BoundaryConstraint{Fixed}\n\n\nReturn a BoundaryConstraint of a function. Dependencies are specified with a DataType, NonFixed/Fixed, Fixed by default.\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.BoundaryConstraint-Tuple{Function}","page":"CTBase API","title":"CTBase.BoundaryConstraint","text":"BoundaryConstraint(\n f::Function;\n variable\n) -> BoundaryConstraint{Fixed}\n\n\nReturn a BoundaryConstraint of a function. Dependencies are specified with a boolean, variable, false by default.\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.BoundaryConstraint-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.BoundaryConstraint","text":"Return the evaluation of the BoundaryConstraint.\n\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B([0, 0], [1, 1])\n[1, 2]\njulia> B = BoundaryConstraint((x0, xf) -> [xf[2]-x0[1], 2xf[1]+x0[2]^2])\njulia> B([0, 0], [1, 1],Real[])\n[1, 2]\njulia> B = BoundaryConstraint((x0, xf, v) -> [v[3]+xf[2]-x0[1], v[1]-v[2]+2xf[1]+x0[2]^2], variable=true)\njulia> B([0, 0], [1, 1], [1, 2, 3])\n[4, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.CTCallback","page":"CTBase API","title":"CTBase.CTCallback","text":"abstract type CTCallback\n\nAbstract type for callbacks.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.CTCallbacks","page":"CTBase API","title":"CTBase.CTCallbacks","text":"Tuple of callbacks\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.CTException","page":"CTBase API","title":"CTBase.CTException","text":"abstract type CTException <: Exception\n\nAbstract type for exceptions.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ControlConstraint","page":"CTBase API","title":"CTBase.ControlConstraint","text":"struct ControlConstraint{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to VectorField in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor ControlConstraint returns a ControlConstraint of a function. The function must take 1 to 3 arguments, u to (t, u, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], Autonomous, Fixed)\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], Autonomous, NonFixed)\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], NonAutonomous, Fixed)\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], NonAutonomous, NonFixed)\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], autonomous=true, variable=false)\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], autonomous=true, variable=true)\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], autonomous=false, variable=false)\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the control is of dimension 1, consider u as a scalar.\n\nCall\n\nThe call returns the evaluation of the ControlConstraint for given values.\n\nExamples\n\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], autonomous=true, variable=false)\njulia> C([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> C(t, [1, -1], v)\n[1, -2]\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], autonomous=true, variable=true)\njulia> C([1, -1], [1, 2, 3])\n[1, 1]\njulia> C(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], autonomous=false, variable=false)\njulia> C(1, [1, -1])\n[2, -2]\njulia> C(1, [1, -1], v)\n[2, -2]\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], autonomous=false, variable=true)\njulia> C(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ControlConstraint-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.ControlConstraint","text":"ControlConstraint(\n f::Function,\n dependencies::DataType...\n) -> ControlConstraint{Autonomous, Fixed}\n\n\nReturn the StateConstraint of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], Autonomous, Fixed)\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], Autonomous, NonFixed)\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], NonAutonomous, Fixed)\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlConstraint-Tuple{Function}","page":"CTBase API","title":"CTBase.ControlConstraint","text":"ControlConstraint(\n f::Function;\n autonomous,\n variable\n) -> ControlConstraint{Autonomous, Fixed}\n\n\nReturn the ControlConstraint of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], autonomous=true, variable=false)\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], autonomous=true, variable=true)\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], autonomous=false, variable=false)\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlConstraint-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.ControlConstraint","text":"Return the value of the ControlConstraint function.\n\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> IncorrectArgument ControlConstraint(u -> [u[1]^2, 2u[2]], Int64)\njulia> C = ControlConstraint(u -> [u[1]^2, 2u[2]], autonomous=true, variable=false)\njulia> C([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> C(t, [1, -1], v)\n[1, -2]\njulia> C = ControlConstraint((u, v) -> [u[1]^2, 2u[2]+v[3]], autonomous=true, variable=true)\njulia> C([1, -1], [1, 2, 3])\n[1, 1]\njulia> C(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> C = ControlConstraint((t, u) -> [t+u[1]^2, 2u[2]], autonomous=false, variable=false)\njulia> C(1, [1, -1])\n[2, -2]\njulia> C(1, [1, -1], v)\n[2, -2]\njulia> C = ControlConstraint((t, u, v) -> [t+u[1]^2, 2u[2]+v[3]], autonomous=false, variable=true)\njulia> C(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlLaw","page":"CTBase API","title":"CTBase.ControlLaw","text":"struct ControlLaw{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to Hamiltonian in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor ControlLaw returns a ControlLaw of a function. The function must take 2 to 4 arguments, (x, p) to (t, x, p, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], Autonomous, Fixed)\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], Autonomous, NonFixed)\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], NonAutonomous, Fixed)\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], NonAutonomous, NonFixed)\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the ControlLaw for given values.\n\nExamples\n\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> u([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> u(t, [1, 0], [0, 1])\nMethodError\njulia> u([1, 0], [0, 1], v)\nMethodError\njulia> u(t, [1, 0], [0, 1], v)\n3\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> u([1, 0], [0, 1], [1, 2, 3])\n6\njulia> u(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> u(1, [1, 0], [0, 1])\n4\njulia> u(1, [1, 0], [0, 1], v)\n4\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> u(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ControlLaw-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.ControlLaw","text":"ControlLaw(\n f::Function,\n dependencies::DataType...\n) -> ControlLaw{Autonomous, Fixed}\n\n\nReturn the ControlLaw of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], Autonomous, Fixed)\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], Autonomous, NonFixed)\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], NonAutonomous, Fixed)\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlLaw-Tuple{Function}","page":"CTBase API","title":"CTBase.ControlLaw","text":"ControlLaw(\n f::Function;\n autonomous,\n variable\n) -> ControlLaw{Autonomous, Fixed}\n\n\nReturn the ControlLaw of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ControlLaw-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.ControlLaw","text":"Return the value of the ControlLaw function.\n\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> ControlLaw((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> u = ControlLaw((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> u([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> u(t, [1, 0], [0, 1])\nMethodError\njulia> u([1, 0], [0, 1], v)\nMethodError\njulia> u(t, [1, 0], [0, 1], v)\n3\njulia> u = ControlLaw((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> u([1, 0], [0, 1], [1, 2, 3])\n6\njulia> u(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> u = ControlLaw((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> u(1, [1, 0], [0, 1])\n4\njulia> u(1, [1, 0], [0, 1], v)\n4\njulia> u = ControlLaw((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> u(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Controls","page":"CTBase API","title":"CTBase.Controls","text":"Type alias for a vector of controls.\n\njulia> const Controls = AbstractVector{<:Control}\n\nSee also: Control, States, Costates.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Costates","page":"CTBase API","title":"CTBase.Costates","text":"Type alias for a vector of costates.\n\njulia> const Costates = AbstractVector{<:Costate}\n\nSee also: Costate, States, Controls.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Description","page":"CTBase API","title":"CTBase.Description","text":"A description is a tuple of symbols. Description is a type alias for a tuple of symbols.\n\njulia> const Description = Tuple{DescVarArg}\n\nSee also: DescVarArg.\n\nExample\n\nBase.show is overloaded for descriptions, that is tuple of descriptions are printed as follows:\n\njulia> display( ( (:a, :b), (:b, :c) ) )\n(:a, :b)\n(:b, :c)\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Dimension","page":"CTBase API","title":"CTBase.Dimension","text":"Type alias for a dimension. This is used to define the dimension of the state space, the costate space, the control space, etc.\n\njulia> const Dimension = Integer\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Dynamics","page":"CTBase API","title":"CTBase.Dynamics","text":"struct Dynamics{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default value for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor Dynamics returns a Dynamics of a function. The function must take 2 to 4 arguments, (x, u) to (t, x, u, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> Dynamics((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> Dynamics((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], Autonomous, Fixed)\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], Autonomous, NonFixed)\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], NonAutonomous, Fixed)\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], NonAutonomous, NonFixed)\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar. Same for the control.\n\nCall\n\nThe call returns the evaluation of the Dynamics for given values.\n\nExamples\n\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> D([1, 0], 1)\n[-1, 1]\njulia> t = 1\njulia> v = Real[]\njulia> D(t, [1, 0], 1, v)\n[-1, 1]\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> D([1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> D(t, [1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> D(1, [1, 0], 1)\n[0, 1]\njulia> D(1, [1, 0], 1, v)\n[0, 1]\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\njulia> D(1, [1, 0], 1, [1, 2, 3])\n[3, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Dynamics-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Dynamics","text":"Dynamics(\n f::Function,\n dependencies::DataType...\n) -> Dynamics{Autonomous, Fixed}\n\n\nReturn the Dynamics of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> Dynamics((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> Dynamics((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], Autonomous, Fixed)\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], Autonomous, NonFixed)\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], NonAutonomous, Fixed)\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Dynamics-Tuple{Function}","page":"CTBase API","title":"CTBase.Dynamics","text":"Dynamics(\n f::Function;\n autonomous,\n variable\n) -> Dynamics{Autonomous, Fixed}\n\n\nReturn the Dynamics of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Dynamics-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Dynamics","text":"Return the value of the Dynamics function.\n\njulia> D = Dynamics((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> D([1, 0], 1)\n[-1, 1]\njulia> t = 1\njulia> v = Real[]\njulia> D(t, [1, 0], 1, v)\n[-1, 1]\njulia> D = Dynamics((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> D([1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> D(t, [1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> D = Dynamics((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> D(1, [1, 0], 1)\n[0, 1]\njulia> D(1, [1, 0], 1, v)\n[0, 1]\njulia> D = Dynamics((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\njulia> D(1, [1, 0], 1, [1, 2, 3])\n[3, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.FeedbackControl","page":"CTBase API","title":"CTBase.FeedbackControl","text":"struct FeedbackControl{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to VectorField in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor FeedbackControl returns a FeedbackControl of a function. The function must take 1 to 3 arguments, x to (t, x, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], Autonomous, Fixed)\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], Autonomous, NonFixed)\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], NonAutonomous, Fixed)\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], NonAutonomous, NonFixed)\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], autonomous=true, variable=false)\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], autonomous=true, variable=true)\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], autonomous=false, variable=false)\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar.\n\nCall\n\nThe call returns the evaluation of the FeedbackControl for given values.\n\nExamples\n\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], autonomous=true, variable=false)\njulia> u([1, 0])\n1\njulia> t = 1\njulia> v = Real[]\njulia> u(t, [1, 0])\nMethodError\njulia> u([1, 0], v)\nMethodError\njulia> u(t, [1, 0], v)\n1\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], autonomous=true, variable=true)\njulia> u([1, 0], [1, 2, 3])\n4\njulia> u(t, [1, 0], [1, 2, 3])\n4\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], autonomous=false, variable=false)\njulia> u(1, [1, 0])\n2\njulia> u(1, [1, 0], v)\n2\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], autonomous=false, variable=true)\njulia> u(1, [1, 0], [1, 2, 3])\n5\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.FeedbackControl-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.FeedbackControl","text":"FeedbackControl(\n f::Function,\n dependencies::DataType...\n) -> FeedbackControl{Autonomous, Fixed}\n\n\nReturn the FeedbackControl of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], Autonomous, Fixed)\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], Autonomous, NonFixed)\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], NonAutonomous, Fixed)\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.FeedbackControl-Tuple{Function}","page":"CTBase API","title":"CTBase.FeedbackControl","text":"FeedbackControl(\n f::Function;\n autonomous,\n variable\n) -> FeedbackControl{Autonomous, Fixed}\n\n\nReturn the FeedbackControl of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], autonomous=true, variable=false)\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], autonomous=true, variable=true)\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], autonomous=false, variable=false)\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.FeedbackControl-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.FeedbackControl","text":"Return the value of the FeedbackControl function.\n\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> FeedbackControl(x -> x[1]^2+2x[2], Int64)\nIncorrectArgument\njulia> u = FeedbackControl(x -> x[1]^2+2x[2], autonomous=true, variable=false)\njulia> u([1, 0])\n1\njulia> t = 1\njulia> v = Real[]\njulia> u(t, [1, 0])\nMethodError\njulia> u([1, 0], v)\nMethodError\njulia> u(t, [1, 0], v)\n1\njulia> u = FeedbackControl((x, v) -> x[1]^2+2x[2]+v[3], autonomous=true, variable=true)\njulia> u([1, 0], [1, 2, 3])\n4\njulia> u(t, [1, 0], [1, 2, 3])\n4\njulia> u = FeedbackControl((t, x) -> t+x[1]^2+2x[2], autonomous=false, variable=false)\njulia> u(1, [1, 0])\n2\njulia> u(1, [1, 0], v)\n2\njulia> u = FeedbackControl((t, x, v) -> t+x[1]^2+2x[2]+v[3], autonomous=false, variable=true)\njulia> u(1, [1, 0], [1, 2, 3])\n5\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Fixed","page":"CTBase API","title":"CTBase.Fixed","text":"abstract type Fixed <: VariableDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Hamiltonian","page":"CTBase API","title":"CTBase.Hamiltonian","text":"struct Hamiltonian{time_dependence, variable_dependence} <: AbstractHamiltonian{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor Hamiltonian returns a Hamiltonian of a function. The function must take 2 to 4 arguments, (x, p) to (t, x, p, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> Hamiltonian((x, p) -> x + p, Int64)\nIncorrectArgument \njulia> Hamiltonian((x, p) -> x + p, Int64)\nIncorrectArgument\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H = Hamiltonian((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3]], autonomous=false, variable=true)\njulia> H = Hamiltonian((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3]], NonAutonomous, NonFixed)\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the Hamiltonian for given values.\n\nExamples\n\njulia> H = Hamiltonian((x, p) -> [x[1]^2+2p[2]]) # autonomous=true, variable=false\njulia> H([1, 0], [0, 1])\nMethodError # H must return a scalar\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> H(t, [1, 0], [0, 1])\nMethodError\njulia> H([1, 0], [0, 1], v)\nMethodError \njulia> H(t, [1, 0], [0, 1], v)\n3\njulia> H = Hamiltonian((x, p, v) -> x[1]^2+2p[2]+v[3], variable=true)\njulia> H([1, 0], [0, 1], [1, 2, 3])\n6\njulia> H(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> H = Hamiltonian((t, x, p) -> t+x[1]^2+2p[2], autonomous=false)\njulia> H(1, [1, 0], [0, 1])\n4\njulia> H(1, [1, 0], [0, 1], v)\n4\njulia> H = Hamiltonian((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> H(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Hamiltonian-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Hamiltonian","text":"Hamiltonian(\n f::Function,\n dependencies::DataType...\n) -> Hamiltonian{Autonomous, Fixed}\n\n\nReturn an Hamiltonian of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H = Hamiltonian((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Hamiltonian-Tuple{Function}","page":"CTBase API","title":"CTBase.Hamiltonian","text":"Hamiltonian(\n f::Function;\n autonomous,\n variable\n) -> Hamiltonian{Autonomous, Fixed}\n\n\nReturn an Hamiltonian of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H = Hamiltonian((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Hamiltonian-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Hamiltonian","text":"Return the value of the Hamiltonian.\n\njulia> Hamiltonian((x, p) -> x + p, Int64)\nIncorrectArgument \njulia> Hamiltonian((x, p) -> x + p, Int64)\nIncorrectArgument\njulia> H = Hamiltonian((x, p) -> [x[1]^2+2p[2]]) # autonomous=true, variable=false\njulia> H([1, 0], [0, 1])\nMethodError # H must return a scalar\njulia> H = Hamiltonian((x, p) -> x[1]^2+2p[2])\njulia> H([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> H(t, [1, 0], [0, 1])\nMethodError\njulia> H([1, 0], [0, 1], v)\nMethodError \njulia> H(t, [1, 0], [0, 1], v)\n3\njulia> H = Hamiltonian((x, p, v) -> x[1]^2+2p[2]+v[3], variable=true)\njulia> H([1, 0], [0, 1], [1, 2, 3])\n6\njulia> H(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> H = Hamiltonian((t, x, p) -> t+x[1]^2+2p[2], autonomous=false)\njulia> H(1, [1, 0], [0, 1])\n4\njulia> H(1, [1, 0], [0, 1], v)\n4\njulia> H = Hamiltonian((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> H(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianLift","page":"CTBase API","title":"CTBase.HamiltonianLift","text":"struct HamiltonianLift{time_dependence, variable_dependence} <: AbstractHamiltonian{time_dependence, variable_dependence}\n\nLifts\n\nX::VectorField\n\nThe values for time_dependence and variable_dependence are deternimed by the values of those for the VectorField.\n\nConstructor\n\nThe constructor HamiltonianLift returns a HamiltonianLift of a VectorField.\n\nExamples\n\njulia> H = HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]]))\njulia> H = HamiltonianLift(VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true))\njulia> H = HamiltonianLift(VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false))\njulia> H = HamiltonianLift(VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true))\njulia> H = HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]]))\njulia> H = HamiltonianLift(VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], NonFixed))\njulia> H = HamiltonianLift(VectorField((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous))\njulia> H = HamiltonianLift(VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed))\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the HamiltonianLift for given values.\n\nExamples\n\njulia> H = HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]]))\njulia> H([1, 2], [1, 1])\n5\njulia> t = 1\njulia> v = Real[]\njulia> H(t, [1, 0], [0, 1])\nMethodError\njulia> H([1, 0], [0, 1], v)\nMethodError \njulia> H(t, [1, 0], [0, 1], v)\n5\njulia> H = HamiltonianLift(VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true))\njulia> H([1, 0], [0, 1], [1, 2, 3])\n3\njulia> H(t, [1, 0], [0, 1], [1, 2, 3])\n3\njulia> H = HamiltonianLift(VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false))\njulia> H(1, [1, 2], [1, 1])\n6\njulia> H(1, [1, 0], [0, 1], v)\n6\njulia> H = HamiltonianLift(VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true))\njulia> H(1, [1, 0], [0, 1], [1, 2, 3])\n3\n\nAlternatively, it is possible to construct the HamiltonianLift from a Function being the VectorField.\n\njulia> HL1 = HamiltonianLift((x, v) -> [x[1]^2,x[2]^2+v], autonomous=true, variable=true)\njulia> HL2 = HamiltonianLift(VectorField((x, v) -> [x[1]^2,x[2]^2+v], autonomous=true, variable=true))\njulia> HL1([1, 0], [0, 1], 1) == HL2([1, 0], [0, 1], 1)\ntrue\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.HamiltonianLift-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.HamiltonianLift","text":"HamiltonianLift(\n f::Function,\n dependences::DataType...\n) -> HamiltonianLift\n\n\nReturn an HamiltonianLift of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> HamiltonianLift(HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]], Int64))\nIncorrectArgument \njulia> HL = HamiltonianLift(x -> [x[1]^2,x[2]^2], Autonomous, Fixed)\njulia> HL = HamiltonianLift((x, v) -> [x[1]^2,x[2]^2+v], Autonomous, NonFixed)\njulia> HL = HamiltonianLift((t, x) -> [t+x[1]^2,x[2]^2], NonAutonomous, Fixed)\njulia> HL = HamiltonianLift((t, x, v) -> [t+x[1]^2,x[2]^2+v], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianLift-Tuple{Function}","page":"CTBase API","title":"CTBase.HamiltonianLift","text":"HamiltonianLift(\n f::Function;\n autonomous,\n variable\n) -> HamiltonianLift\n\n\nReturn an HamiltonianLift of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> HL = HamiltonianLift(x -> [x[1]^2,x[2]^2], autonomous=true, variable=false)\njulia> HL = HamiltonianLift((x, v) -> [x[1]^2,x[2]^2+v], autonomous=true, variable=true)\njulia> HL = HamiltonianLift((t, x) -> [t+x[1]^2,x[2]^2], autonomous=false, variable=false)\njulia> HL = HamiltonianLift((t, x, v) -> [t+x[1]^2,x[2]^2+v], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianLift-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.HamiltonianLift","text":"Return the value of the HamiltonianLift.\n\nExamples\n\njulia> HamiltonianLift(HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]], Int64))\nIncorrectArgument \njulia> H = HamiltonianLift(VectorField(x -> [x[1]^2, 2x[2]]))\njulia> H([1, 2], [1, 1])\n5\njulia> t = 1\njulia> v = Real[]\njulia> H(t, [1, 0], [0, 1])\nMethodError\njulia> H([1, 0], [0, 1], v)\nMethodError \njulia> H(t, [1, 0], [0, 1], v)\n5\njulia> H = HamiltonianLift(VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true))\njulia> H([1, 0], [0, 1], [1, 2, 3])\n3\njulia> H(t, [1, 0], [0, 1], [1, 2, 3])\n3\njulia> H = HamiltonianLift(VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false))\njulia> H(1, [1, 2], [1, 1])\n6\njulia> H(1, [1, 0], [0, 1], v)\n6\njulia> H = HamiltonianLift(VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true))\njulia> H(1, [1, 0], [0, 1], [1, 2, 3])\n3\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianVectorField","page":"CTBase API","title":"CTBase.HamiltonianVectorField","text":"struct HamiltonianVectorField{time_dependence, variable_dependence} <: CTBase.AbstractVectorField{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor HamiltonianVectorField returns a HamiltonianVectorField of a function. The function must take 2 to 4 arguments, (x, p) to (t, x, p, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], variable=true)\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], autonomous=false)\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], autonomous=false, variable=true)\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], NonFixed)\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], NonAutonomous)\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], NonAutonomous, NonFixed)\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the HamiltonianVectorField for given values.\n\nExamples\n\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv([1, 0], [0, 1])\n[3, -3]\njulia> t = 1\njulia> v = Real[]\njulia> Hv(t, [1, 0], [0, 1])\nMethodError\njulia> Hv([1, 0], [0, 1], v)\nMethodError\njulia> Hv(t, [1, 0], [0, 1], v)\n[3, -3]\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], variable=true)\njulia> Hv([1, 0], [0, 1], [1, 2, 3, 4])\n[6, -3]\njulia> Hv(t, [1, 0], [0, 1], [1, 2, 3, 4])\n[6, -3]\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], autonomous=false)\njulia> Hv(1, [1, 0], [0, 1])\n[4, -3]\njulia> Hv(1, [1, 0], [0, 1], v)\n[4, -3]\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], autonomous=false, variable=true)\njulia> Hv(1, [1, 0], [0, 1], [1, 2, 3, 4])\n[7, -3]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.HamiltonianVectorField-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.HamiltonianVectorField","text":"HamiltonianVectorField(\n f::Function,\n dependencies::DataType...\n) -> HamiltonianVectorField{Autonomous, Fixed}\n\n\nReturn an HamiltonianVectorField of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], NonFixed)\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], NonAutonomous)\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianVectorField-Tuple{Function}","page":"CTBase API","title":"CTBase.HamiltonianVectorField","text":"HamiltonianVectorField(\n f::Function;\n autonomous,\n variable\n) -> HamiltonianVectorField{Autonomous, Fixed}\n\n\nReturn an HamiltonianVectorField of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], variable=true)\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], autonomous=false)\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.HamiltonianVectorField-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.HamiltonianVectorField","text":"Return the value of the HamiltonianVectorField.\n\nExamples\n\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2], Int64)\nIncorrectArgument\njulia> Hv = HamiltonianVectorField((x, p) -> [x[1]^2+2p[2], x[2]-3p[2]^2]) # autonomous=true, variable=false\njulia> Hv([1, 0], [0, 1])\n[3, -3]\njulia> t = 1\njulia> v = Real[]\njulia> Hv(t, [1, 0], [0, 1])\nMethodError\njulia> Hv([1, 0], [0, 1], v)\nMethodError\njulia> Hv(t, [1, 0], [0, 1], v)\n[3, -3]\njulia> Hv = HamiltonianVectorField((x, p, v) -> [x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], variable=true)\njulia> Hv([1, 0], [0, 1], [1, 2, 3, 4])\n[6, -3]\njulia> Hv(t, [1, 0], [0, 1], [1, 2, 3, 4])\n[6, -3]\njulia> Hv = HamiltonianVectorField((t, x, p) -> [t+x[1]^2+2p[2], x[2]-3p[2]^2], autonomous=false)\njulia> Hv(1, [1, 0], [0, 1])\n[4, -3]\njulia> Hv(1, [1, 0], [0, 1], v)\n[4, -3]\njulia> Hv = HamiltonianVectorField((t, x, p, v) -> [t+x[1]^2+2p[2]+v[3], x[2]-3p[2]^2+v[4]], autonomous=false, variable=true)\njulia> Hv(1, [1, 0], [0, 1], [1, 2, 3, 4])\n[7, -3]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.IncorrectArgument","page":"CTBase API","title":"CTBase.IncorrectArgument","text":"struct IncorrectArgument <: CTException\n\nException thrown when an argument is inconsistent.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.IncorrectMethod","page":"CTBase API","title":"CTBase.IncorrectMethod","text":"struct IncorrectMethod <: CTException\n\nException thrown when a method is incorrect.\n\nFields\n\nvar::Symbol\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.IncorrectOutput","page":"CTBase API","title":"CTBase.IncorrectOutput","text":"struct IncorrectOutput <: CTException\n\nException thrown when the output is incorrect.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Index","page":"CTBase API","title":"CTBase.Index","text":"mutable struct Index\n\nFields\n\nval::Integer\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Lagrange","page":"CTBase API","title":"CTBase.Lagrange","text":"struct Lagrange{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default value for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor Lagrange returns a Lagrange cost of a function. The function must take 2 to 4 arguments, (x, u) to (t, x, u, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], Autonomous, Fixed)\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, Autonomous, Fixed)\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], Autonomous, NonFixed)\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, Fixed)\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, NonFixed)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar. Same for the control.\n\nCall\n\nThe call returns the evaluation of the Lagrange cost for given values.\n\nExamples\n\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L([1, 0], [1])\nMethodError\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L([1, 0], [1])\n-1\njulia> t = 1\njulia> v = Real[]\njulia> L(t, [1, 0], [1])\nMethodError\njulia> L([1, 0], [1], v)\nMethodError\njulia> L(t, [1, 0], [1], v)\n-1\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L([1, 0], [1], [1, 2, 3])\n2\njulia> L(t, [1, 0], [1], [1, 2, 3])\n2\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L(1, [1, 0], [1])\n0\njulia> L(1, [1, 0], [1], v)\n0\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\njulia> L(1, [1, 0], [1], [1, 2, 3])\n3\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Lagrange-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Lagrange","text":"Lagrange(\n f::Function,\n dependencies::DataType...\n) -> Lagrange{Autonomous, Fixed}\n\n\nReturn a Lagrange cost of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\n\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lagrange-Tuple{Function}","page":"CTBase API","title":"CTBase.Lagrange","text":"Lagrange(\n f::Function;\n autonomous,\n variable\n) -> Lagrange{Autonomous, Fixed}\n\n\nReturn a Lagrange cost of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\n\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lagrange-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Lagrange","text":"Return the value of the Lagrange function.\n\nExamples\n\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> Lagrange((x, u) -> 2x[2]-u[1]^2, Int64)\nIncorrectArgument\njulia> L = Lagrange((x, u) -> [2x[2]-u[1]^2], autonomous=true, variable=false)\njulia> L([1, 0], [1])\nMethodError\njulia> L = Lagrange((x, u) -> 2x[2]-u[1]^2, autonomous=true, variable=false)\njulia> L([1, 0], [1])\n-1\njulia> t = 1\njulia> v = Real[]\njulia> L(t, [1, 0], [1])\nMethodError\njulia> L([1, 0], [1], v)\nMethodError\njulia> L(t, [1, 0], [1], v)\n-1\njulia> L = Lagrange((x, u, v) -> 2x[2]-u[1]^2+v[3], autonomous=true, variable=true)\njulia> L([1, 0], [1], [1, 2, 3])\n2\njulia> L(t, [1, 0], [1], [1, 2, 3])\n2\njulia> L = Lagrange((t, x, u) -> t+2x[2]-u[1]^2, autonomous=false, variable=false)\njulia> L(1, [1, 0], [1])\n0\njulia> L(1, [1, 0], [1], v)\n0\njulia> L = Lagrange((t, x, u, v) -> t+2x[2]-u[1]^2+v[3], autonomous=false, variable=true)\njulia> L(1, [1, 0], [1], [1, 2, 3])\n3\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Mayer","page":"CTBase API","title":"CTBase.Mayer","text":"struct Mayer{variable_dependence}\n\nFields\n\nf::Function\n\nThe default value for variable_dependence is Fixed.\n\nConstructor\n\nThe constructor Mayer returns a Mayer cost of a function. The function must take 2 or 3 arguments (x0, xf) or (x0, xf, v), if the function is variable, it must be specified. Dependencies are specified with a boolean, variable, false by default or with a DataType, NonFixed/Fixed, Fixed by default.\n\nExamples\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], variable=true)\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], NonFixed)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x0 and xf as a scalar.\n\nCall\n\nThe call returns the evaluation of the Mayer cost for given values. If a variable is given for a non variable dependent Mayer cost, it will be ignored.\n\nExamples\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G([0, 0], [1, 1])\n1\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G([0, 0], [1, 1],Real[])\n1\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], variable=true)\njulia> G([0, 0], [1, 1], [1, 2, 3])\n4\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Mayer-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Mayer","text":"Mayer(\n f::Function,\n dependencies::DataType...\n) -> Mayer{Fixed}\n\n\nReturn a Mayer cost of a function. Dependencies are specified with a DataType, NonFixed/Fixed, Fixed by default.\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Mayer-Tuple{Function}","page":"CTBase API","title":"CTBase.Mayer","text":"Mayer(f::Function; variable) -> Mayer{Fixed}\n\n\nReturn a Mayer cost of a function. Dependencies are specified with a boolean, variable, false by default.\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Mayer-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Mayer","text":"Return the evaluation of the Mayer cost.\n\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G([0, 0], [1, 1])\n1\njulia> G = Mayer((x0, xf) -> xf[2]-x0[1])\njulia> G([0, 0], [1, 1], Real[])\n1\njulia> G = Mayer((x0, xf, v) -> v[3]+xf[2]-x0[1], variable=true)\njulia> G([0, 0], [1, 1], [1, 2, 3])\n4\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.MixedConstraint","page":"CTBase API","title":"CTBase.MixedConstraint","text":"struct MixedConstraint{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to Lagrange in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default value for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor MixedConstraint returns a MixedConstraint of a function. The function must take 2 to 4 arguments, (x, u) to (t, x, u, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Autonomous, Fixed)\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], Autonomous, NonFixed)\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], NonAutonomous, Fixed)\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], NonAutonomous, NonFixed)\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar. Same for the control.\n\nCall\n\nThe call returns the evaluation of the MixedConstraint for given values.\n\nExamples\n\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> M([1, 0], 1)\n[-1, 1]\njulia> t = 1\njulia> v = Real[]\njulia> MethodError M(t, [1, 0], 1)\njulia> MethodError M([1, 0], 1, v)\njulia> M(t, [1, 0], 1, v)\n[-1, 1]\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> M([1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> M(t, [1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> M(1, [1, 0], 1)\n[0, 1]\njulia> M(1, [1, 0], 1, v)\n[0, 1]\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\njulia> M(1, [1, 0], 1, [1, 2, 3])\n[3, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.MixedConstraint-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.MixedConstraint","text":"MixedConstraint(\n f::Function,\n dependencies::DataType...\n) -> MixedConstraint{Autonomous, Fixed}\n\n\nReturn the MixedConstraint of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Autonomous, Fixed)\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], Autonomous, NonFixed)\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], NonAutonomous, Fixed)\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.MixedConstraint-Tuple{Function}","page":"CTBase API","title":"CTBase.MixedConstraint","text":"MixedConstraint(\n f::Function;\n autonomous,\n variable\n) -> MixedConstraint{Autonomous, Fixed}\n\n\nReturn the MixedConstraint of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.MixedConstraint-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.MixedConstraint","text":"Return the value of the MixedConstraint function.\n\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], Int64)\nIncorrectArgument\njulia> M = MixedConstraint((x, u) -> [2x[2]-u^2, x[1]], autonomous=true, variable=false)\njulia> M([1, 0], 1)\n[-1, 1]\njulia> t = 1\njulia> v = Real[]\njulia> MethodError M(t, [1, 0], 1)\njulia> MethodError M([1, 0], 1, v)\njulia> M(t, [1, 0], 1, v)\n[-1, 1]\njulia> M = MixedConstraint((x, u, v) -> [2x[2]-u^2+v[3], x[1]], autonomous=true, variable=true)\njulia> M([1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> M(t, [1, 0], 1, [1, 2, 3])\n[2, 1]\njulia> M = MixedConstraint((t, x, u) -> [t+2x[2]-u^2, x[1]], autonomous=false, variable=false)\njulia> M(1, [1, 0], 1)\n[0, 1]\njulia> M(1, [1, 0], 1, v)\n[0, 1]\njulia> M = MixedConstraint((t, x, u, v) -> [t+2x[2]-u^2+v[3], x[1]], autonomous=false, variable=true)\njulia> M(1, [1, 0], 1, [1, 2, 3])\n[3, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Multiplier","page":"CTBase API","title":"CTBase.Multiplier","text":"struct Multiplier{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to ControlLaw in the usage.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor Multiplier returns a Multiplier of a function. The function must take 2 to 4 arguments, (x, p) to (t, x, p, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], Autonomous, Fixed)\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], Autonomous, NonFixed)\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], NonAutonomous, Fixed)\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], NonAutonomous, NonFixed)\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state and costate are of dimension 1, consider x and p as scalars.\n\nCall\n\nThe call returns the evaluation of the Multiplier for given values.\n\nExamples\n\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> μ([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> μ(t, [1, 0], [0, 1])\nMethodError\njulia> μ([1, 0], [0, 1], v)\nMethodError\njulia> μ(t, [1, 0], [0, 1], v)\n3\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> μ([1, 0], [0, 1], [1, 2, 3])\n6\njulia> μ(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> μ(1, [1, 0], [0, 1])\n4\njulia> μ(1, [1, 0], [0, 1], v)\n4\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> μ(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Multiplier-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Multiplier","text":"Multiplier(\n f::Function,\n dependencies::DataType...\n) -> Multiplier{Autonomous, Fixed}\n\n\nReturn the Multiplier of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], Autonomous, Fixed)\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], Autonomous, NonFixed)\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], NonAutonomous, Fixed)\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Multiplier-Tuple{Function}","page":"CTBase API","title":"CTBase.Multiplier","text":"Multiplier(\n f::Function;\n autonomous,\n variable\n) -> Multiplier{Autonomous, Fixed}\n\n\nReturn the Multiplier of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Multiplier-Tuple{Union{Real, AbstractVector{<:Real}}, Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.Multiplier","text":"Return the value of the Multiplier function.\n\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> Multiplier((x, p) -> x[1]^2+2p[2], Int64)\nIncorrectArgument\njulia> μ = Multiplier((x, p) -> x[1]^2+2p[2], autonomous=true, variable=false)\njulia> μ([1, 0], [0, 1])\n3\njulia> t = 1\njulia> v = Real[]\njulia> μ(t, [1, 0], [0, 1])\nMethodError\njulia> μ([1, 0], [0, 1], v)\nMethodError\njulia> μ(t, [1, 0], [0, 1], v)\n3\njulia> μ = Multiplier((x, p, v) -> x[1]^2+2p[2]+v[3], autonomous=true, variable=true)\njulia> μ([1, 0], [0, 1], [1, 2, 3])\n6\njulia> μ(t, [1, 0], [0, 1], [1, 2, 3])\n6\njulia> μ = Multiplier((t, x, p) -> t+x[1]^2+2p[2], autonomous=false, variable=false)\njulia> μ(1, [1, 0], [0, 1])\n4\njulia> μ(1, [1, 0], [0, 1], v)\n4\njulia> μ = Multiplier((t, x, p, v) -> t+x[1]^2+2p[2]+v[3], autonomous=false, variable=true)\njulia> μ(1, [1, 0], [0, 1], [1, 2, 3])\n7\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.NonAutonomous","page":"CTBase API","title":"CTBase.NonAutonomous","text":"abstract type NonAutonomous <: TimeDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.NonFixed","page":"CTBase API","title":"CTBase.NonFixed","text":"abstract type NonFixed <: VariableDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.NotImplemented","page":"CTBase API","title":"CTBase.NotImplemented","text":"struct NotImplemented <: CTException\n\nException thrown when a method is not implemented.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.OptimalControlInit","page":"CTBase API","title":"CTBase.OptimalControlInit","text":"Initial guess for OCP, contains\n\nfunctions of time for the state and control variables\nvector for optimization variables\n\nInitialization data for each field can be left to default or: \n\nvector for optimization variables\nconstant / vector / function for state and control \nexisting solution ('warm start') for all fields\n\nConstructors:\n\nOptimalControlInit(): default initialization\nOptimalControlInit(state, control, variable, time): constant vector, function handles and / or matrices / vectors interpolated along given time grid\nOptimalControlInit(sol): from existing solution\n\nExamples\n\njulia> init = OptimalControlInit()\njulia> init = OptimalControlInit(state=[0.1, 0.2], control=0.3)\njulia> init = OptimalControlInit(state=[0.1, 0.2], control=0.3, variable=0.5)\njulia> init = OptimalControlInit(state=[0.1, 0.2], controlt=t->sin(t), variable=0.5)\njulia> init = OptimalControlInit(state=[[0, 0], [1, 2], [5, -1]], time=[0, .3, 1.], controlt=t->sin(t))\njulia> init = OptimalControlInit(sol)\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.OptimalControlModel","page":"CTBase API","title":"CTBase.OptimalControlModel","text":"mutable struct OptimalControlModel{time_dependence<:TimeDependence, variable_dependence<:VariableDependence} <: CTBase.AbstractOptimalControlModel\n\nFields\n\nmodel_expression::Union{Nothing, Expr}: Default: nothing\ninitial_time::Union{Nothing, Index, Real}: Default: nothing\ninitial_time_name::Union{Nothing, String}: Default: nothing\nfinal_time::Union{Nothing, Index, Real}: Default: nothing\nfinal_time_name::Union{Nothing, String}: Default: nothing\ntime_name::Union{Nothing, String}: Default: nothing\ncontrol_dimension::Union{Nothing, Integer}: Default: nothing\ncontrol_components_names::Union{Nothing, Vector{String}}: Default: nothing\ncontrol_name::Union{Nothing, String}: Default: nothing\nstate_dimension::Union{Nothing, Integer}: Default: nothing\nstate_components_names::Union{Nothing, Vector{String}}: Default: nothing\nstate_name::Union{Nothing, String}: Default: nothing\nvariable_dimension::Union{Nothing, Integer}: Default: nothing\nvariable_components_names::Union{Nothing, Vector{String}}: Default: nothing\nvariable_name::Union{Nothing, String}: Default: nothing\nlagrange::Union{Nothing, Lagrange}: Default: nothing\nmayer::Union{Nothing, Mayer}: Default: nothing\ncriterion::Union{Nothing, Symbol}: Default: nothing\ndynamics::Union{Nothing, Dynamics}: Default: nothing\nconstraints::Dict{Symbol, Tuple}: Default: Dict{Symbol, Tuple{Vararg{Any}}}()\ndim_control_constraints::Union{Nothing, Integer}: Default: nothing\ndim_state_constraints::Union{Nothing, Integer}: Default: nothing\ndim_mixed_constraints::Union{Nothing, Integer}: Default: nothing\ndim_boundary_constraints::Union{Nothing, Integer}: Default: nothing\ndim_variable_constraints::Union{Nothing, Integer}: Default: nothing\ndim_control_range::Union{Nothing, Integer}: Default: nothing\ndim_state_range::Union{Nothing, Integer}: Default: nothing\ndim_variable_range::Union{Nothing, Integer}: Default: nothing\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.OptimalControlSolution","page":"CTBase API","title":"CTBase.OptimalControlSolution","text":"mutable struct OptimalControlSolution <: CTBase.AbstractOptimalControlSolution\n\nType of an optimal control solution.\n\nFields\n\ntimes::Union{Nothing, StepRangeLen, AbstractVector{<:Real}}: Default: nothing\ninitial_time_name::Union{Nothing, String}: Default: nothing\nfinal_time_name::Union{Nothing, String}: Default: nothing\ntime_name::Union{Nothing, String}: Default: nothing\ncontrol_dimension::Union{Nothing, Integer}: Default: nothing\ncontrol_components_names::Union{Nothing, Vector{String}}: Default: nothing\ncontrol_name::Union{Nothing, String}: Default: nothing\ncontrol::Union{Nothing, Function}: Default: nothing\nstate_dimension::Union{Nothing, Integer}: Default: nothing\nstate_components_names::Union{Nothing, Vector{String}}: Default: nothing\nstate_name::Union{Nothing, String}: Default: nothing\nstate::Union{Nothing, Function}: Default: nothing\nvariable_dimension::Union{Nothing, Integer}: Default: nothing\nvariable_components_names::Union{Nothing, Vector{String}}: Default: nothing\nvariable_name::Union{Nothing, String}: Default: nothing\nvariable::Union{Nothing, Real, AbstractVector{<:Real}}: Default: nothing\ncostate::Union{Nothing, Function}: Default: nothing\nobjective::Union{Nothing, Real}: Default: nothing\niterations::Union{Nothing, Integer}: Default: nothing\nstopping::Union{Nothing, Symbol}: Default: nothing\nmessage::Union{Nothing, String}: Default: nothing\nsuccess::Union{Nothing, Bool}: Default: nothing\ninfos::Dict{Symbol, Any}: Default: Dict{Symbol, Any}()\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.ParsingError","page":"CTBase API","title":"CTBase.ParsingError","text":"struct ParsingError <: CTException\n\nException thrown for syntax error during abstract parsing.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.PrintCallback","page":"CTBase API","title":"CTBase.PrintCallback","text":"mutable struct PrintCallback <: CTCallback\n\nCallback for printing.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.PrintCallback-Tuple","page":"CTBase API","title":"CTBase.PrintCallback","text":"Call the callback.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.StateConstraint","page":"CTBase API","title":"CTBase.StateConstraint","text":"struct StateConstraint{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nSimilar to VectorField in the usage, but the dimension of the output of the function f is arbitrary.\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor StateConstraint returns a StateConstraint of a function. The function must take 1 to 3 arguments, x to (t, x, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], Autonomous, Fixed)\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], Autonomous, NonFixed)\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous, Fixed)\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed)\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], autonomous=true, variable=false)\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], autonomous=true, variable=true)\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false, variable=false)\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar.\n\nCall\n\nThe call returns the evaluation of the StateConstraint for given values.\n\nExamples\n\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], autonomous=true, variable=false)\njulia> S([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> S(t, [1, -1], v)\n[1, -2]\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], autonomous=true, variable=true)\njulia> S([1, -1], [1, 2, 3])\n[1, 1]\njulia> S(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false, variable=false)\njulia> S(1, [1, -1])\n[2, -2]\njulia> S(1, [1, -1], v)\n[2, -2]\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> S(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.StateConstraint-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.StateConstraint","text":"StateConstraint(\n f::Function,\n dependencies::DataType...\n) -> StateConstraint{Autonomous, Fixed}\n\n\nReturn the StateConstraint of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], Autonomous, Fixed)\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], Autonomous, NonFixed)\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous, Fixed)\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.StateConstraint-Tuple{Function}","page":"CTBase API","title":"CTBase.StateConstraint","text":"StateConstraint(\n f::Function;\n autonomous,\n variable\n) -> StateConstraint{Autonomous, Fixed}\n\n\nReturn the StateConstraint of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], autonomous=true, variable=false)\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], autonomous=true, variable=true)\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false, variable=false)\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.StateConstraint-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.StateConstraint","text":"Return the value of the StateConstraint function.\n\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> StateConstraint(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> S = StateConstraint(x -> [x[1]^2, 2x[2]], autonomous=true, variable=false)\njulia> S([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> S(t, [1, -1], v)\n[1, -2]\njulia> S = StateConstraint((x, v) -> [x[1]^2, 2x[2]+v[3]], autonomous=true, variable=true)\njulia> S([1, -1], [1, 2, 3])\n[1, 1]\njulia> S(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> S = StateConstraint((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false, variable=false)\njulia> S(1, [1, -1])\n[2, -2]\njulia> S(1, [1, -1], v)\n[2, -2]\njulia> S = StateConstraint((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> S(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.States","page":"CTBase API","title":"CTBase.States","text":"Type alias for a vector of states.\n\njulia> const States = AbstractVector{<:State}\n\nSee also: State, Costates, Controls.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.StopCallback","page":"CTBase API","title":"CTBase.StopCallback","text":"Stopping callback.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.StopCallback-Tuple","page":"CTBase API","title":"CTBase.StopCallback","text":"Call the callback.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Time","page":"CTBase API","title":"CTBase.Time","text":"Type alias for a time.\n\njulia> const Time = ctNumber\n\nSee also: ctNumber, Times, TimesDisc.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.TimeDependence","page":"CTBase API","title":"CTBase.TimeDependence","text":"abstract type TimeDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.Times","page":"CTBase API","title":"CTBase.Times","text":"Type alias for a vector of times.\n\njulia> const Times = AbstractVector{<:Time}\n\nSee also: Time, TimesDisc.\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.UnauthorizedCall","page":"CTBase API","title":"CTBase.UnauthorizedCall","text":"struct UnauthorizedCall <: CTException\n\nException thrown when a call to a function is not authorized.\n\nFields\n\nvar::String\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.VariableConstraint","page":"CTBase API","title":"CTBase.VariableConstraint","text":"struct VariableConstraint\n\nFields\n\nf::Function\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor VariableConstraint returns a VariableConstraint of a function. The function must take 1 argument, v.\n\nExamples\n\njulia> V = VariableConstraint(v -> [v[1]^2, 2v[2]])\n\nwarning: Warning\nWhen the variable is of dimension 1, consider v as a scalar.\n\nCall\n\nThe call returns the evaluation of the VariableConstraint for given values.\n\nExamples\n\njulia> V = VariableConstraint(v -> [v[1]^2, 2v[2]])\njulia> V([1, -1])\n[1, -2]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.VariableConstraint-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.VariableConstraint","text":"Return the value of the VariableConstraint function.\n\njulia> V = VariableConstraint(v -> [v[1]^2, 2v[2]])\njulia> V([1, -1])\n[1, -2]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.VariableDependence","page":"CTBase API","title":"CTBase.VariableDependence","text":"abstract type VariableDependence\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.VectorField","page":"CTBase API","title":"CTBase.VectorField","text":"struct VectorField{time_dependence, variable_dependence} <: CTBase.AbstractVectorField{time_dependence, variable_dependence}\n\nFields\n\nf::Function\n\nThe default values for time_dependence and variable_dependence are Autonomous and Fixed respectively.\n\nConstructor\n\nThe constructor VectorField returns a VectorField of a function. The function must take 1 to 3 arguments, x to (t, x, v), if the function is variable or non autonomous, it must be specified. Dependencies are specified either with :\n\nbooleans, autonomous and variable, respectively true and false by default \nDataType, Autonomous/NonAutonomous and NonFixed/Fixed, respectively Autonomous and Fixed by default.\n\nExamples\n\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true)\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false)\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], NonFixed)\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous)\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed)\n\nwarning: Warning\nWhen the state is of dimension 1, consider x as a scalar.\n\nCall\n\nThe call returns the evaluation of the VectorField for given values.\n\nExamples\n\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> V(t, [1, -1])\nMethodError\njulia> V([1, -1], v)\nMethodError\njulia> V(t, [1, -1], v)\n[1, -2]\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true)\njulia> V([1, -1], [1, 2, 3])\n[1, 1]\njulia> V(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false)\njulia> V(1, [1, -1])\n[2, -2]\njulia> V(1, [1, -1], v)\n[2, -2]\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> V(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.VectorField-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.VectorField","text":"VectorField(\n f::Function,\n dependencies::DataType...\n) -> VectorField{Autonomous, Fixed}\n\n\nReturn a VectorField of a function. Dependencies are specified with DataType, Autonomous, NonAutonomous and Fixed, NonFixed.\n\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], NonFixed)\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], NonAutonomous)\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], NonAutonomous, NonFixed)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.VectorField-Tuple{Function}","page":"CTBase API","title":"CTBase.VectorField","text":"VectorField(\n f::Function;\n autonomous,\n variable\n) -> VectorField{Autonomous, Fixed}\n\n\nReturn a VectorField of a function. Dependencies are specified with a boolean, variable, false by default, autonomous, true by default.\n\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true)\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false)\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.VectorField-Tuple{Union{Real, AbstractVector{<:Real}}}","page":"CTBase API","title":"CTBase.VectorField","text":"Return the value of the VectorField.\n\nExamples\n\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> VectorField(x -> [x[1]^2, 2x[2]], Int64)\nIncorrectArgument\njulia> V = VectorField(x -> [x[1]^2, 2x[2]]) # autonomous=true, variable=false\njulia> V([1, -1])\n[1, -2]\njulia> t = 1\njulia> v = Real[]\njulia> V(t, [1, -1])\nMethodError\njulia> V([1, -1], v)\nMethodError\njulia> V(t, [1, -1], v)\n[1, -2]\njulia> V = VectorField((x, v) -> [x[1]^2, 2x[2]+v[3]], variable=true)\njulia> V([1, -1], [1, 2, 3])\n[1, 1]\njulia> V(t, [1, -1], [1, 2, 3])\n[1, 1]\njulia> V = VectorField((t, x) -> [t+x[1]^2, 2x[2]], autonomous=false)\njulia> V(1, [1, -1])\n[2, -2]\njulia> V(1, [1, -1], v)\n[2, -2]\njulia> V = VectorField((t, x, v) -> [t+x[1]^2, 2x[2]+v[3]], autonomous=false, variable=true)\njulia> V(1, [1, -1], [1, 2, 3])\n[2, 1]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctNumber","page":"CTBase API","title":"CTBase.ctNumber","text":"Type alias for a real number.\n\njulia> const ctNumber = Real\n\n\n\n\n\n","category":"type"},{"location":"api-ctbase.html#CTBase.:⋅-Tuple{Function, Function}","page":"CTBase API","title":"CTBase.:⋅","text":"⋅(X::Function, f::Function) -> Function\n\n\nLie derivative of a scalar function along a function. In this case both functions will be considered autonomous and non-variable.\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> f = x -> x[1]^2 + x[2]^2\njulia> (φ⋅f)([1, 2])\n0\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> (φ⋅f)(1, [1, 2], [2, 1])\nMethodError\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.:⋅-Tuple{VectorField{Autonomous, <:VariableDependence}, Function}","page":"CTBase API","title":"CTBase.:⋅","text":"⋅(\n X::VectorField{Autonomous, <:VariableDependence},\n f::Function\n) -> CTBase.var\"#97#99\"{VectorField{Autonomous, var\"#s98\"}, <:Function} where var\"#s98\"<:VariableDependence\n\n\nLie derivative of a scalar function along a vector field : L_X(f) = X⋅f, in autonomous case\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> X = VectorField(φ)\njulia> f = x -> x[1]^2 + x[2]^2\njulia> (X⋅f)([1, 2])\n0\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.:⋅-Tuple{VectorField{NonAutonomous, <:VariableDependence}, Function}","page":"CTBase API","title":"CTBase.:⋅","text":"⋅(\n X::VectorField{NonAutonomous, <:VariableDependence},\n f::Function\n) -> CTBase.var\"#101#103\"{VectorField{NonAutonomous, var\"#s98\"}, <:Function} where var\"#s98\"<:VariableDependence\n\n\nLie derivative of a scalar function along a vector field : L_X(f) = X⋅f, in nonautonomous case\n\nExample\n\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> X = VectorField(φ, NonAutonomous, NonFixed)\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> (X⋅f)(1, [1, 2], [2, 1])\n10\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Tuple{Function, Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Lie","text":"Lie(\n X::Function,\n f::Function,\n dependences::DataType...\n) -> Function\n\n\nLie derivative of a scalar function along a vector field or a function. Dependencies are specified with DataType : Autonomous, NonAutonomous and Fixed, NonFixed.\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> f = x -> x[1]^2 + x[2]^2\njulia> Lie(φ,f)([1, 2])\n0\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> Lie(φ, f, NonAutonomous, NonFixed)(1, [1, 2], [2, 1])\n10\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Tuple{Function, Function}","page":"CTBase API","title":"CTBase.Lie","text":"Lie(\n X::Function,\n f::Function;\n autonomous,\n variable\n) -> Function\n\n\nLie derivative of a scalar function along a function. Dependencies are specified with boolean : autonomous and variable.\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> f = x -> x[1]^2 + x[2]^2\njulia> Lie(φ,f)([1, 2])\n0\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> Lie(φ, f, autonomous=false, variable=true)(1, [1, 2], [2, 1])\n10\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Tuple{VectorField, Function}","page":"CTBase API","title":"CTBase.Lie","text":"Lie(X::VectorField, f::Function) -> Function\n\n\nLie derivative of a scalar function along a vector field.\n\nExample\n\njulia> φ = x -> [x[2], -x[1]]\njulia> X = VectorField(φ)\njulia> f = x -> x[1]^2 + x[2]^2\njulia> Lie(X,f)([1, 2])\n0\njulia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]\njulia> X = VectorField(φ, NonAutonomous, NonFixed)\njulia> f = (t, x, v) -> t + x[1]^2 + x[2]^2\njulia> Lie(X, f)(1, [1, 2], [2, 1])\n10\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Union{Tuple{V}, Tuple{VectorField{Autonomous, V}, VectorField{Autonomous, V}}} where V<:VariableDependence","page":"CTBase API","title":"CTBase.Lie","text":"Lie(\n X::VectorField{Autonomous, V<:VariableDependence},\n Y::VectorField{Autonomous, V<:VariableDependence}\n) -> VectorField\n\n\nLie bracket of two vector fields: [X, Y] = Lie(X, Y), autonomous case\n\nExample\n\njulia> f = x -> [x[2], 2x[1]]\njulia> g = x -> [3x[2], -x[1]]\njulia> X = VectorField(f)\njulia> Y = VectorField(g)\njulia> Lie(X, Y)([1, 2])\n[7, -14]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lie-Union{Tuple{V}, Tuple{VectorField{NonAutonomous, V}, VectorField{NonAutonomous, V}}} where V<:VariableDependence","page":"CTBase API","title":"CTBase.Lie","text":"Lie(\n X::VectorField{NonAutonomous, V<:VariableDependence},\n Y::VectorField{NonAutonomous, V<:VariableDependence}\n) -> VectorField{NonAutonomous}\n\n\nLie bracket of two vector fields: [X, Y] = Lie(X, Y), nonautonomous case\n\nExample\n\njulia> f = (t, x, v) -> [t + x[2] + v, -2x[1] - v]\njulia> g = (t, x, v) -> [t + 3x[2] + v, -x[1] - v]\njulia> X = VectorField(f, NonAutonomous, NonFixed)\njulia> Y = VectorField(g, NonAutonomous, NonFixed)\njulia> Lie(X, Y)(1, [1, 2], 1)\n[-7,12]\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lift-Tuple{Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Lift","text":"Lift(\n X::Function,\n dependences::DataType...\n) -> HamiltonianLift\n\n\nReturn the HamiltonianLift of a VectorField or a function. Dependencies are specified with DataType : Autonomous, NonAutonomous and Fixed, NonFixed.\n\nExample\n\njulia> H = Lift(x -> 2x)\njulia> H(1, 1)\n2\njulia> H = Lift((t, x, v) -> 2x + t - v, NonAutonomous, NonFixed)\njulia> H(1, 1, 1, 1)\n2\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lift-Tuple{Function}","page":"CTBase API","title":"CTBase.Lift","text":"Lift(X::Function; autonomous, variable) -> HamiltonianLift\n\n\nReturn the HamiltonianLift of a function. Dependencies are specified with boolean : autonomous and variable.\n\nExample\n\njulia> H = Lift(x -> 2x)\njulia> H(1, 1)\n2\njulia> H = Lift((t, x, v) -> 2x + t - v, autonomous=false, variable=true)\njulia> H(1, 1, 1, 1)\n2\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Lift-Tuple{VectorField}","page":"CTBase API","title":"CTBase.Lift","text":"Lift(X::VectorField) -> HamiltonianLift\n\n\nReturn the HamiltonianLift of a VectorField.\n\nExample\n\njulia> HL = Lift(VectorField(x -> [x[1]^2,x[2]^2], autonomous=true, variable=false))\njulia> HL([1, 0], [0, 1])\n0\njulia> HL = Lift(VectorField((t, x, v) -> [t+x[1]^2,x[2]^2+v], autonomous=false, variable=true))\njulia> HL(1, [1, 0], [0, 1], 1)\n1\njulia> H = Lift(x -> 2x)\njulia> H(1, 1)\n2\njulia> H = Lift((t, x, v) -> 2x + t - v, autonomous=false, variable=true)\njulia> H(1, 1, 1, 1)\n2\njulia> H = Lift((t, x, v) -> 2x + t - v, NonAutonomous, NonFixed)\njulia> H(1, 1, 1, 1)\n2\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Model-Tuple{Vararg{DataType}}","page":"CTBase API","title":"CTBase.Model","text":"Model(\n dependencies::DataType...\n) -> OptimalControlModel{Autonomous, Fixed}\n\n\nReturn a new OptimalControlModel instance, that is a model of an optimal control problem.\n\nThe model is defined by the following argument:\n\ndependencies: either Autonomous or NonAutonomous. Default is Autonomous. And either NonFixed or Fixed. Default is Fixed.\n\nExamples\n\njulia> ocp = Model()\njulia> ocp = Model(NonAutonomous)\njulia> ocp = Model(Autonomous, NonFixed)\n\nnote: Note\nIf the time dependence of the model is defined as nonautonomous, then, the dynamics function, the lagrange cost and the path constraints must be defined as functions of time and state, and possibly control. If the model is defined as autonomous, then, the dynamics function, the lagrange cost and the path constraints must be defined as functions of state, and possibly control.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Model-Tuple{}","page":"CTBase API","title":"CTBase.Model","text":"Model(\n;\n autonomous,\n variable\n) -> OptimalControlModel{Autonomous, Fixed}\n\n\nReturn a new OptimalControlModel instance, that is a model of an optimal control problem.\n\nThe model is defined by the following optional keyword argument:\n\nautonomous: either true or false. Default is true.\nvariable: either true or false. Default is false.\n\nExamples\n\njulia> ocp = Model()\njulia> ocp = Model(autonomous=false)\njulia> ocp = Model(autonomous=false, variable=true)\n\nnote: Note\nIf the time dependence of the model is defined as nonautonomous, then, the dynamics function, the lagrange cost and the path constraints must be defined as functions of time and state, and possibly control. If the model is defined as autonomous, then, the dynamics function, the lagrange cost and the path constraints must be defined as functions of state, and possibly control.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Tuple{Function, Function, Vararg{DataType}}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::Function,\n g::Function,\n dependences::DataType...\n) -> Hamiltonian\n\n\nPoisson bracket of two functions : {f, g} = Poisson(f, g) Dependencies are specified with DataType : Autonomous, NonAutonomous and Fixed, NonFixed.\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> Poisson(f, g)([1, 2], [2, 1])\n-20 \njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> Poisson(f, g, NonAutonomous, NonFixed)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Tuple{Function, Function}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::Function,\n g::Function;\n autonomous,\n variable\n) -> Hamiltonian\n\n\nPoisson bracket of two functions : {f, g} = Poisson(f, g) Dependencies are specified with boolean : autonomous and variable.\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> Poisson(f, g)([1, 2], [2, 1])\n-20 \njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> Poisson(f, g, autonomous=false, variable=true)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{AbstractHamiltonian{Autonomous, V}, AbstractHamiltonian{Autonomous, V}}} where V<:VariableDependence","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::AbstractHamiltonian{Autonomous, V<:VariableDependence},\n g::AbstractHamiltonian{Autonomous, V<:VariableDependence}\n) -> HamiltonianLift\n\n\nPoisson bracket of two Hamiltonian functions (subtype of AbstractHamiltonian) : {f, g} = Poisson(f, g), autonomous case\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> F = Hamiltonian(f)\njulia> G = Hamiltonian(g)\njulia> Poisson(f, g)([1, 2], [2, 1])\n-20 \njulia> Poisson(f, G)([1, 2], [2, 1])\n-20\njulia> Poisson(F, g)([1, 2], [2, 1])\n-20\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{AbstractHamiltonian{NonAutonomous, V}, AbstractHamiltonian{NonAutonomous, V}}} where V<:VariableDependence","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::AbstractHamiltonian{NonAutonomous, V<:VariableDependence},\n g::AbstractHamiltonian{NonAutonomous, V<:VariableDependence}\n) -> HamiltonianLift\n\n\nPoisson bracket of two Hamiltonian functions (subtype of AbstractHamiltonian) : {f, g} = Poisson(f, g), non autonomous case\n\nExample\n\njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> F = Hamiltonian(f, autonomous=false, variable=true)\njulia> G = Hamiltonian(g, autonomous=false, variable=true)\njulia> Poisson(F, G)(2, [1, 2], [2, 1], [4, 4])\n-76\njulia> Poisson(f, g, NonAutonomous, NonFixed)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{T}, Tuple{AbstractHamiltonian{T, V}, Function}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::AbstractHamiltonian{T<:TimeDependence, V<:VariableDependence},\n g::Function\n) -> Hamiltonian\n\n\nPoisson bracket of an Hamiltonian function (subtype of AbstractHamiltonian) and a function : {f, g} = Poisson(f, g), autonomous case\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> F = Hamiltonian(f)\njulia> Poisson(F, g)([1, 2], [2, 1])\n-20\njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> F = Hamiltonian(f, autonomous=false, variable=true)\njulia> Poisson(F, g)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{T}, Tuple{Function, AbstractHamiltonian{T, V}}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::Function,\n g::AbstractHamiltonian{T<:TimeDependence, V<:VariableDependence}\n) -> Hamiltonian\n\n\nPoisson bracket of a function and an Hamiltonian function (subtype of AbstractHamiltonian) : {f, g} = Poisson(f, g)\n\nExample\n\njulia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2\njulia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]\njulia> G = Hamiltonian(g) \njulia> Poisson(f, G)([1, 2], [2, 1])\n-20\njulia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]\njulia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]\njulia> G = Hamiltonian(g, autonomous=false, variable=true)\njulia> Poisson(f, G)(2, [1, 2], [2, 1], [4, 4])\n-76\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.Poisson-Union{Tuple{V}, Tuple{T}, Tuple{HamiltonianLift{T, V}, HamiltonianLift{T, V}}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.Poisson","text":"Poisson(\n f::HamiltonianLift{T<:TimeDependence, V<:VariableDependence},\n g::HamiltonianLift{T<:TimeDependence, V<:VariableDependence}\n) -> HamiltonianLift\n\n\nPoisson bracket of two HamiltonianLift functions : {f, g} = Poisson(f, g)\n\nExample\n\njulia> f = x -> [x[1]^2+x[2]^2, 2x[1]^2]\njulia> g = x -> [3x[2]^2, x[2]-x[1]^2]\njulia> F = Lift(f)\njulia> G = Lift(g)\njulia> Poisson(F, G)([1, 2], [2, 1])\n-64\njulia> f = (t, x, v) -> [t*v[1]*x[2]^2, 2x[1]^2 + + v[2]]\njulia> g = (t, x, v) -> [3x[2]^2 + -x[1]^2, t - v[2]]\njulia> F = Lift(f, NonAutonomous, NonFixed)\njulia> G = Lift(g, NonAutonomous, NonFixed)\njulia> Poisson(F, G)(2, [1, 2], [2, 1], [4, 4])\n100\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.__OCPModel-Tuple","page":"CTBase API","title":"CTBase.__OCPModel","text":"__OCPModel(args...; kwargs...) -> OptimalControlModel\n\n\nRedirection to Model to avoid confusion with other functions Model from other packages if imported. This function is used by @def.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.add-Tuple{Tuple{Vararg{Tuple{Vararg{Symbol}}}}, Tuple{Vararg{Symbol}}}","page":"CTBase API","title":"CTBase.add","text":"add(\n x::Tuple{Vararg{Tuple{Vararg{Symbol}}}},\n y::Tuple{Vararg{Symbol}}\n) -> Tuple{Tuple{Vararg{Symbol}}}\n\n\nConcatenate the description y to the tuple of descriptions x if x does not contain y and return the new tuple of descriptions. Throw an error if the description y is already contained in x.\n\nExample\n\njulia> descriptions = ()\njulia> descriptions = add(descriptions, (:a,))\n(:a,)\njulia> descriptions = add(descriptions, (:b,))\n(:a,)\n(:b,)\njulia> descriptions = add(descriptions, (:b,))\nERROR: IncorrectArgument: the description (:b,) is already in ((:a,), (:b,))\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.add-Tuple{Tuple{}, Tuple{Vararg{Symbol}}}","page":"CTBase API","title":"CTBase.add","text":"add(\n x::Tuple{},\n y::Tuple{Vararg{Symbol}}\n) -> Tuple{Tuple{Vararg{Symbol}}}\n\n\nReturn a tuple containing only the description y.\n\nExample\n\njulia> descriptions = ()\njulia> descriptions = add(descriptions, (:a,))\n(:a,)\njulia> print(descriptions)\n((:a,),)\njulia> descriptions[1]\n(:a,)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.constraint!-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Symbol}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.constraint!","text":"constraint!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol;\n rg,\n f,\n lb,\n ub,\n label\n)\n\n\nAdd a constraint to an optimal control problem, denoted ocp.\n\nnote: Note\nThe state, control and variable dimensions must be set before. Use state!, control! and variable!.\nThe initial and final times must be set before. Use time!.\nWhen an element is of dimension 1, consider it as a scalar.\n\nYou can add an :initial, :final, :control, :state or :variable box constraint (whole range). \n\nRange constraint on the state, control or variable\n\nYou can add an :initial, :final, :control, :state or :variable box constraint on a range of it, that is only on some components. If not range is specified, then the constraint is on the whole range. We denote by x, u and v respectively the state, control and variable. We denote by n, m and q respectively the dimension of the state, control and variable. The range of the constraint must be contained in 1:n if the constraint is on the state, or 1:m if the constraint is on the control, or 1:q if the constraint is on the variable.\n\nExamples\n\njulia> constraint!(ocp, :initial; rg=1:2:5, lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :initial; rg=2:3, lb=[ 0, 0 ], ub=[ 1, 2 ])\njulia> constraint!(ocp, :final; rg=1, lb=0, ub=2)\njulia> constraint!(ocp, :control; rg=1, lb=0, ub=2)\njulia> constraint!(ocp, :state; rg=2:3, lb=[ 0, 0 ], ub=[ 1, 2 ])\njulia> constraint!(ocp, :variable; rg=1:2, lb=[ 0, 0 ], ub=[ 1, 2 ])\njulia> constraint!(ocp, :initial; lb=[ 0, 0, 0 ]) # [ 0, 0, 0 ] ≤ x(t0), dim(x) = 3\njulia> constraint!(ocp, :initial; lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ]) # [ 0, 0, 0 ] ≤ x(t0) ≤ [ 1, 2, 1 ], dim(x) = 3\njulia> constraint!(ocp, :final; lb=-1, ub=1) # -1 ≤ x(tf) ≤ 1, dim(x) = 1\njulia> constraint!(ocp, :control; lb=0, ub=2) # 0 ≤ u(t) ≤ 2, t ∈ [t0, tf], dim(u) = 1\njulia> constraint!(ocp, :state; lb=[ 0, 0 ], ub=[ 1, 2 ]) # [ 0, 0 ] ≤ x(t) ≤ [ 1, 2 ], t ∈ [t0, tf], dim(x) = 2\njulia> constraint!(ocp, :variable; lb=[ 0, 0 ], ub=[ 1, 2 ]) # [ 0, 0 ] ≤ v ≤ [ 1, 2 ], dim(v) = 2\n\nFunctional constraint\n\nYou can add a :boundary, :control, :state, :mixed or :variable box functional constraint.\n\nExamples\n\n# variable independent ocp\njulia> constraint!(ocp, :boundary; f = (x0, xf) -> x0[3]+xf[2], lb=0, ub=1)\n\n# variable dependent ocp\njulia> constraint!(ocp, :boundary; f = (x0, xf, v) -> x0[3]+xf[2]*v[1], lb=0, ub=1)\n\n# time independent and variable independent ocp\njulia> constraint!(ocp, :control; f = u -> 2u, lb=0, ub=1)\njulia> constraint!(ocp, :state; f = x -> x-1, lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :mixed; f = (x, u) -> x[1]-u, lb=0, ub=1)\n\n# time dependent and variable independent ocp\njulia> constraint!(ocp, :control; f = (t, u) -> 2u, lb=0, ub=1)\njulia> constraint!(ocp, :state; f = (t, x) -> t * x, lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :mixed; f = (t, x, u) -> x[1]-u, lb=0, ub=1)\n\n# time independent and variable dependent ocp\njulia> constraint!(ocp, :control; f = (u, v) -> 2u * v[1], lb=0, ub=1)\njulia> constraint!(ocp, :state; f = (x, v) -> x * v[1], lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :mixed; f = (x, u, v) -> x[1]-v[2]*u, lb=0, ub=1)\n\n# time dependent and variable dependent ocp\njulia> constraint!(ocp, :control; f = (t, u, v) -> 2u+v[2], lb=0, ub=1)\njulia> constraint!(ocp, :state; f = (t, x, v) -> x-t*v[1], lb=[ 0, 0, 0 ], ub=[ 1, 2, 1 ])\njulia> constraint!(ocp, :mixed; f = (t, x, u, v) -> x[1]*v[2]-u, lb=0, ub=1)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.constraint-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Symbol}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.constraint","text":"constraint(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n label::Symbol\n) -> Any\n\n\nRetrieve a labeled constraint. The result is a function associated with the constraint computation (not taking into account provided value / bounds).\n\nExample\n\njulia> constraint!(ocp, :initial, 0, :c0)\njulia> c = constraint(ocp, :c0)\njulia> c(1)\n1\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.constraint_type-NTuple{7, Any}","page":"CTBase API","title":"CTBase.constraint_type","text":"constraint_type(\n e,\n t,\n t0,\n tf,\n x,\n u,\n v\n) -> Union{Symbol, Tuple{Symbol, Any}}\n\n\nReturn the type constraint among :initial, :final, :boundary, :control_range, :control_fun, :state_range, :state_fun, :mixed, :variable_range, :variable_fun (:other otherwise), together with the appropriate value (range, updated expression...) Expressions like u(t0) where u is the control and t0 the initial time return :other.\n\nExample\n\njulia> t = :t; t0 = 0; tf = :tf; x = :x; u = :u; v = :v\n\njulia> constraint_type(:( ẏ(t) ), t, t0, tf, x, u, v)\n:other\n\njulia> constraint_type(:( ẋ(s) ), t, t0, tf, x, u, v)\n:other\n\njulia> constraint_type(:( x(0)' ), t, t0, tf, x, u, v)\n:boundary\n\njulia> constraint_type(:( x(t)' ), t, t0, tf, x, u, v)\n:state_fun\n\njulia> constraint_type(:( x(0) ), t, t0, tf, x, u, v)\n(:initial, nothing)\n\njulia> constraint_type(:( x[1:2:5](0) ), t, t0, tf, x, u, v)\n(:initial, 1:2:5)\n\njulia> constraint_type(:( x[1:2](0) ), t, t0, tf, x, u, v)\n(:initial, 1:2)\n\njulia> constraint_type(:( x[1](0) ), t, t0, tf, x, u, v)\n(:initial, 1)\n\njulia> constraint_type(:( 2x[1](0)^2 ), t, t0, tf, x, u, v)\n:boundary\n\njulia> constraint_type(:( x(tf) ), t, t0, tf, x, u, v)\n(:final, nothing)\nj\njulia> constraint_type(:( x[1:2:5](tf) ), t, t0, tf, x, u, v)\n(:final, 1:2:5)\n\njulia> constraint_type(:( x[1:2](tf) ), t, t0, tf, x, u, v)\n(:final, 1:2)\n\njulia> constraint_type(:( x[1](tf) ), t, t0, tf, x, u, v)\n(:final, 1)\n\njulia> constraint_type(:( 2x[1](tf)^2 ), t, t0, tf, x, u, v)\n:boundary\n\njulia> constraint_type(:( x[1](tf) - x[2](0) ), t, t0, tf, x, u, v)\n:boundary\n\njulia> constraint_type(:( u[1:2:5](t) ), t, t0, tf, x, u, v)\n(:control_range, 1:2:5)\n\njulia> constraint_type(:( u[1:2](t) ), t, t0, tf, x, u, v)\n(:control_range, 1:2)\n\njulia> constraint_type(:( u[1](t) ), t, t0, tf, x, u, v)\n(:control_range, 1)\n\njulia> constraint_type(:( u(t) ), t, t0, tf, x, u, v)\n(:control_range, nothing)\n\njulia> constraint_type(:( 2u[1](t)^2 ), t, t0, tf, x, u, v)\n:control_fun\n\njulia> constraint_type(:( x[1:2:5](t) ), t, t0, tf, x, u, v)\n(:state_range, 1:2:5)\n\njulia> constraint_type(:( x[1:2](t) ), t, t0, tf, x, u, v)\n(:state_range, 1:2)\n\njulia> constraint_type(:( x[1](t) ), t, t0, tf, x, u, v)\n(:state_range, 1)\n\njulia> constraint_type(:( x(t) ), t, t0, tf, x, u, v)\n(:state_range, nothing)\n\njulia> constraint_type(:( 2x[1](t)^2 ), t, t0, tf, x, u, v)\n:state_fun\n\njulia> constraint_type(:( 2u[1](t)^2 * x(t) ), t, t0, tf, x, u, v)\n:mixed\n\njulia> constraint_type(:( 2u[1](0)^2 * x(t) ), t, t0, tf, x, u, v)\n:other\n\njulia> constraint_type(:( 2u[1](0)^2 * x(t) ), t, t0, tf, x, u, v)\n:other\n\njulia> constraint_type(:( 2u[1](t)^2 * x(t) + v ), t, t0, tf, x, u, v)\n:mixed\n\njulia> constraint_type(:( v[1:2:10] ), t, t0, tf, x, u, v)\n(:variable_range, 1:2:9)\n\njulia> constraint_type(:( v[1:10] ), t, t0, tf, x, u, v)\n(:variable_range, 1:10)\n\njulia> constraint_type(:( v[2] ), t, t0, tf, x, u, v)\n(:variable_range, 2)\n\njulia> constraint_type(:( v ), t, t0, tf, x, u, v)\n(:variable_range, nothing)\n\njulia> constraint_type(:( v^2 + 1 ), t, t0, tf, x, u, v)\n:variable_fun\njulia> constraint_type(:( v[2]^2 + 1 ), t, t0, tf, x, u, v)\n:variable_fun\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.constraints_labels-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.constraints_labels","text":"constraints_labels(\n ocp::OptimalControlModel\n) -> Base.KeySet{Symbol, Dict{Symbol, Tuple}}\n\n\nReturn the labels of the constraints as a Base.keys.\n\nExample\n\njulia> constraints_labels(ocp)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.control!","page":"CTBase API","title":"CTBase.control!","text":"control!(ocp::OptimalControlModel, m::Integer)\ncontrol!(ocp::OptimalControlModel, m::Integer, name::String)\ncontrol!(\n ocp::OptimalControlModel,\n m::Integer,\n name::String,\n components_names::Vector{String}\n)\n\n\nDefine the control dimension and possibly the names of each coordinate.\n\nnote: Note\nYou must use control! only once to set the control dimension.\n\nExamples\n\njulia> control!(ocp, 1)\njulia> ocp.control_dimension\n1\njulia> ocp.control_components_names\n[\"u\"]\n\njulia> control!(ocp, 1, \"v\")\njulia> ocp.control_dimension\n1\njulia> ocp.control_components_names\n[\"v\"]\n\njulia> control!(ocp, 2)\njulia> ocp.control_dimension\n2\njulia> ocp.control_components_names\n[\"u₁\", \"u₂\"]\n\njulia> control!(ocp, 2, :v)\njulia> ocp.control_dimension\n2\njulia> ocp.control_components_names\n[\"v₁\", \"v₂\"]\n\njulia> control!(ocp, 2, \"v\")\njulia> ocp.control_dimension\n2\njulia> ocp.control_components_names\n[\"v₁\", \"v₂\"]\n\n\n\n\n\n","category":"function"},{"location":"api-ctbase.html#CTBase.ct_repl-Tuple{}","page":"CTBase API","title":"CTBase.ct_repl","text":"ct_repl(; debug, verbose)\n\n\nCreate a ct REPL.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ct_repl_update_model-Tuple{Expr}","page":"CTBase API","title":"CTBase.ct_repl_update_model","text":"ct_repl_update_model(e::Expr)\n\n\nUpdate the model adding the expression e. It must be public since in the ct repl, this function is quoted each time an expression is parsed and is valid. \n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctgradient-Tuple{Function, Any}","page":"CTBase API","title":"CTBase.ctgradient","text":"ctgradient(f::Function, x; backend) -> Any\n\n\nReturn the gradient of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctgradient-Tuple{Function, Real}","page":"CTBase API","title":"CTBase.ctgradient","text":"ctgradient(f::Function, x::Real; backend) -> Any\n\n\nReturn the gradient of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctgradient-Tuple{VectorField, Any}","page":"CTBase API","title":"CTBase.ctgradient","text":"ctgradient(X::VectorField, x) -> Any\n\n\nReturn the gradient of X at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctindices-Tuple{Integer}","page":"CTBase API","title":"CTBase.ctindices","text":"ctindices(i::Integer) -> String\n\n\nReturn i > 0 as a subscript.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctinterpolate-Tuple{Any, Any}","page":"CTBase API","title":"CTBase.ctinterpolate","text":"ctinterpolate(x, f) -> Any\n\n\nReturn the interpolation of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctjacobian-Tuple{Function, Any}","page":"CTBase API","title":"CTBase.ctjacobian","text":"ctjacobian(f::Function, x; backend) -> Any\n\n\nReturn the Jacobian of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctjacobian-Tuple{Function, Real}","page":"CTBase API","title":"CTBase.ctjacobian","text":"ctjacobian(f::Function, x::Real; backend) -> Any\n\n\nReturn the Jacobian of f at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctjacobian-Tuple{VectorField, Any}","page":"CTBase API","title":"CTBase.ctjacobian","text":"ctjacobian(X::VectorField, x) -> Any\n\n\nReturn the Jacobian of X at x.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.ctupperscripts-Tuple{Integer}","page":"CTBase API","title":"CTBase.ctupperscripts","text":"ctupperscripts(i::Integer) -> String\n\n\nReturn i > 0 as an upperscript.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_boundary_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_boundary_constraints","text":"dim_boundary_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of the boundary constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_control_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_control_constraints","text":"dim_control_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of nonlinear control constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_control_range-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_control_range","text":"dim_control_range(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of range constraints on control (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_mixed_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_mixed_constraints","text":"dim_mixed_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of nonlinear mixed constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_path_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_path_constraints","text":"dim_path_constraints(ocp::OptimalControlModel) -> Any\n\n\nReturn the dimension of nonlinear path (state + control + mixed) constraints (nothing if one of them is not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_state_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_state_constraints","text":"dim_state_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of nonlinear state constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_state_range-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_state_range","text":"dim_state_range(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of range constraints on state (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_variable_constraints-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_variable_constraints","text":"dim_variable_constraints(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of nonlinear variable constraints (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dim_variable_range-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.dim_variable_range","text":"dim_variable_range(\n ocp::OptimalControlModel\n) -> Union{Nothing, Integer}\n\n\nReturn the dimension of range constraints on variable (nothing if not knonw). Information is updated after nlp_constraints! is called.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.dynamics!-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Function}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.dynamics!","text":"dynamics!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n f::Function\n)\n\n\nSet the dynamics.\n\nnote: Note\nYou can use dynamics! only once to define the dynamics.The state, control and variable dimensions must be set before. Use state!, control! and variable!.\nThe times must be set before. Use time!.\nWhen an element is of dimension 1, consider it as a scalar.\n\nExample\n\njulia> dynamics!(ocp, f)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.getFullDescription-Tuple{Tuple{Vararg{Symbol}}, Tuple{Vararg{Tuple{Vararg{Symbol}}}}}","page":"CTBase API","title":"CTBase.getFullDescription","text":"getFullDescription(\n desc::Tuple{Vararg{Symbol}},\n desc_list::Tuple{Vararg{Tuple{Vararg{Symbol}}}}\n) -> Tuple{Vararg{Symbol}}\n\n\nReturn a complete description from an incomplete description desc and a list of complete descriptions desc_list. If several complete descriptions are possible, then the first one is returned.\n\nExample\n\njulia> desc_list = ((:a, :b), (:b, :c), (:a, :c))\n(:a, :b)\n(:b, :c)\n(:a, :c)\njulia> getFullDescription((:a,), desc_list)\n(:a, :b)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.get_priority_print_callbacks-Tuple{Tuple{Vararg{CTCallback}}}","page":"CTBase API","title":"CTBase.get_priority_print_callbacks","text":"get_priority_print_callbacks(\n cbs::Tuple{Vararg{CTCallback}}\n) -> Tuple{Vararg{CTCallback}}\n\n\nGet the highest priority print callbacks.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.get_priority_stop_callbacks-Tuple{Tuple{Vararg{CTCallback}}}","page":"CTBase API","title":"CTBase.get_priority_stop_callbacks","text":"get_priority_stop_callbacks(\n cbs::Tuple{Vararg{CTCallback}}\n) -> Tuple{Vararg{CTCallback}}\n\n\nGet the highest priority stop callbacks.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.has_free_final_time-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.has_free_final_time","text":"has_free_final_time(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined with free final time.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.has_free_initial_time-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.has_free_initial_time","text":"has_free_initial_time(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined with free initial time.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.has_lagrange_cost-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.has_lagrange_cost","text":"has_lagrange_cost(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined with lagrange cost.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.has_mayer_cost-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.has_mayer_cost","text":"has_mayer_cost(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined with mayer cost.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_autonomous-Tuple{OptimalControlModel{Autonomous}}","page":"CTBase API","title":"CTBase.is_autonomous","text":"is_autonomous(ocp::OptimalControlModel{Autonomous}) -> Bool\n\n\nReturn true if the model is autonomous.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_fixed-Tuple{OptimalControlModel{<:TimeDependence, Fixed}}","page":"CTBase API","title":"CTBase.is_fixed","text":"is_fixed(\n ocp::OptimalControlModel{<:TimeDependence, Fixed}\n) -> Bool\n\n\nReturn true if the model is fixed (= has no variable).\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_max-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_max","text":"is_max(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the criterion type of ocp is :max.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_min-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_min","text":"is_min(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the criterion type of ocp is :min.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_time_dependent-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_time_dependent","text":"is_time_dependent(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined as time dependent.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_time_independent-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_time_independent","text":"is_time_independent(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined as time independent.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_variable_dependent-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_variable_dependent","text":"is_variable_dependent(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined as variable dependent.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.is_variable_independent-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.is_variable_independent","text":"is_variable_independent(ocp::OptimalControlModel) -> Bool\n\n\nReturn true if the model has been defined as variable independent.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.nlp_constraints!-Tuple{OptimalControlModel}","page":"CTBase API","title":"CTBase.nlp_constraints!","text":"nlp_constraints!(\n ocp::OptimalControlModel\n) -> Tuple{Tuple{Any, CTBase.var\"#ξ#91\", Vector{Real}}, Tuple{Any, CTBase.var\"#η#92\", Vector{Real}}, Tuple{Any, CTBase.var\"#ψ#93\", Vector{Real}}, Tuple{Any, CTBase.var\"#ϕ#94\", Vector{Real}}, Tuple{Any, CTBase.var\"#θ#95\", Vector{Real}}, Tuple{Vector{Real}, Vector{Int64}, Vector{Real}}, Tuple{Vector{Real}, Vector{Int64}, Vector{Real}}, Tuple{Vector{Real}, Vector{Int64}, Vector{Real}}}\n\n\nReturn a 6-tuple of tuples:\n\n(ξl, ξ, ξu) are control constraints\n(ηl, η, ηu) are state constraints\n(ψl, ψ, ψu) are mixed constraints\n(ϕl, ϕ, ϕu) are boundary constraints\n(θl, θ, θu) are variable constraints\n(ul, uind, uu) are control linear constraints of a subset of indices\n(xl, xind, xu) are state linear constraints of a subset of indices\n(vl, vind, vu) are variable linear constraints of a subset of indices\n\nand update information about constraints dimensions of ocp.\n\nnote: Note\nThe dimensions of the state and control must be set before calling nlp_constraints!.\nFor a Fixed problem, dimensions associated with constraints on the variable are set to zero.\n\nExample\n\njulia> (ξl, ξ, ξu), (ηl, η, ηu), (ψl, ψ, ψu), (ϕl, ϕ, ϕu), (θl, θ, θu),\n (ul, uind, uu), (xl, xind, xu), (vl, vind, vu) = nlp_constraints!(ocp)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.objective!-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Symbol, Function, Function}, Tuple{OptimalControlModel{T, V}, Symbol, Function, Function, Symbol}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.objective!","text":"objective!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol,\n g::Function,\n f⁰::Function\n)\nobjective!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol,\n g::Function,\n f⁰::Function,\n criterion::Symbol\n)\n\n\nSet the criterion to the function g and f⁰. Type can be :bolza. Criterion is :min or :max.\n\nnote: Note\nYou can use objective! only once to define the objective.The state, control and variable dimensions must be set before. Use state!, control! and variable!.\nThe times must be set before. Use time!.\nWhen an element is of dimension 1, consider it as a scalar.\n\nExample\n\njulia> objective!(ocp, :bolza, (x0, xf) -> x0[1] + xf[2], (x, u) -> x[1]^2 + u^2) # the control is of dimension 1\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.objective!-Union{Tuple{V}, Tuple{T}, Tuple{OptimalControlModel{T, V}, Symbol, Function}, Tuple{OptimalControlModel{T, V}, Symbol, Function, Symbol}} where {T<:TimeDependence, V<:VariableDependence}","page":"CTBase API","title":"CTBase.objective!","text":"objective!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol,\n f::Function\n)\nobjective!(\n ocp::OptimalControlModel{T<:TimeDependence, V<:VariableDependence},\n type::Symbol,\n f::Function,\n criterion::Symbol\n)\n\n\nSet the criterion to the function f. Type can be :mayer or :lagrange. Criterion is :min or :max.\n\nnote: Note\nYou can use objective! only once to define the objective.The state, control and variable dimensions must be set before. Use state!, control! and variable!.\nThe times must be set before. Use time!.\nWhen an element is of dimension 1, consider it as a scalar.\n\nExamples\n\njulia> objective!(ocp, :mayer, (x0, xf) -> x0[1] + xf[2])\njulia> objective!(ocp, :lagrange, (x, u) -> x[1]^2 + u^2) # the control is of dimension 1\n\nwarning: Warning\nIf you set twice the objective, only the last one will be taken into account.\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.remove_constraint!-Tuple{OptimalControlModel, Symbol}","page":"CTBase API","title":"CTBase.remove_constraint!","text":"remove_constraint!(ocp::OptimalControlModel, label::Symbol)\n\n\nRemove a labeled constraint.\n\nExample\n\njulia> remove_constraint!(ocp, :con)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.replace_call-Tuple{Any, Symbol, Any, Any}","page":"CTBase API","title":"CTBase.replace_call","text":"replace_call(e, x::Symbol, t, y) -> Any\n\n\nReplace calls in e of the form (...x...)(t) by (...y...).\n\nExample\n\n\njulia> t = :t; t0 = 0; tf = :tf; x = :x; u = :u;\n\njulia> e = :( x[1](0) * 2x(tf) - x[2](tf) * 2x(0) )\n:((x[1])(0) * (2 * x(tf)) - (x[2])(tf) * (2 * x(0)))\n\njulia> x0 = Symbol(x, 0); e = replace_call(e, x, t0, x0)\n:(x0[1] * (2 * x(tf)) - (x[2])(tf) * (2x0))\n\njulia> xf = Symbol(x, \"f\"); replace_call(ans, x, tf, xf)\n:(x0[1] * (2xf) - xf[2] * (2x0))\n\njulia> e = :( A*x(t) + B*u(t) ); replace_call(replace_call(e, x, t, x), u, t, u)\n:(A * x + B * u)\n\njulia> e = :( F0(x(t)) + u(t)*F1(x(t)) ); replace_call(replace_call(e, x, t, x), u, t, u)\n:(F0(x) + u * F1(x))\n\njulia> e = :( 0.5u(t)^2 ); replace_call(e, u, t, u)\n:(0.5 * u ^ 2)\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.replace_call-Tuple{Any, Vector{Symbol}, Any, Any}","page":"CTBase API","title":"CTBase.replace_call","text":"replace_call(e, x::Vector{Symbol}, t, y) -> Any\n\n\nReplace calls in e of the form (...x1...x2...)(t) by (...y1...y2...) for all symbols x1, x2... in the vector x.\n\nExample\n\n\njulia> t = :t; t0 = 0; tf = :tf; x = :x; u = :u;\n\njulia> e = :( (x^2 + u[1])(t) ); replace_call(e, [ x, u ], t , [ :xx, :uu ])\n:(xx ^ 2 + uu[1])\n\njulia> e = :( ((x^2)(t) + u[1])(t) ); replace_call(e, [ x, u ], t , [ :xx, :uu ])\n:(xx ^ 2 + uu[1])\n\njulia> e = :( ((x^2)(t0) + u[1])(t) ); replace_call(e, [ x, u ], t , [ :xx, :uu ])\n:((xx ^ 2)(t0) + uu[1])\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.state!","page":"CTBase API","title":"CTBase.state!","text":"state!(ocp::OptimalControlModel, n::Integer)\nstate!(ocp::OptimalControlModel, n::Integer, name::String)\nstate!(\n ocp::OptimalControlModel,\n n::Integer,\n name::String,\n components_names::Vector{String}\n)\n\n\nDefine the state dimension and possibly the names of each component.\n\nnote: Note\nYou must use state! only once to set the state dimension.\n\nExamples\n\njulia> state!(ocp, 1)\njulia> ocp.state_dimension\n1\njulia> ocp.state_components_names\n[\"x\"]\n\njulia> state!(ocp, 1, \"y\")\njulia> ocp.state_dimension\n1\njulia> ocp.state_components_names\n[\"y\"]\n\njulia> state!(ocp, 2)\njulia> ocp.state_dimension\n2\njulia> ocp.state_components_names\n[\"x₁\", \"x₂\"]\n\njulia> state!(ocp, 2, :y)\njulia> ocp.state_dimension\n2\njulia> ocp.state_components_names\n[\"y₁\", \"y₂\"]\n\njulia> state!(ocp, 2, \"y\")\njulia> ocp.state_dimension\n2\njulia> ocp.state_components_names\n[\"y₁\", \"y₂\"]\n\n\n\n\n\n","category":"function"},{"location":"api-ctbase.html#CTBase.time!-Union{Tuple{OptimalControlModel{<:TimeDependence, VT}}, Tuple{VT}} where VT","page":"CTBase API","title":"CTBase.time!","text":"time!(\n ocp::OptimalControlModel{<:TimeDependence, VT};\n t0,\n tf,\n ind0,\n indf,\n name\n)\n\n\nSet the initial and final times. We denote by t0 the initial time and tf the final time. The optimal control problem is denoted ocp. When a time is free, then one must provide the corresponding index of the ocp variable.\n\nnote: Note\nYou must use time! only once to set either the initial or the final time, or both.\n\nExamples\n\njulia> time!(ocp, t0=0, tf=1 ) # Fixed t0 and fixed tf\njulia> time!(ocp, t0=0, indf=2) # Fixed t0 and free tf\njulia> time!(ocp, ind0=2, tf=1 ) # Free t0 and fixed tf\njulia> time!(ocp, ind0=2, indf=3) # Free t0 and free tf\n\nWhen you plot a solution of an optimal control problem, the name of the time variable appears. By default, the name is \"t\". Consider you want to set the name of the time variable to \"s\".\n\njulia> time!(ocp, t0=0, tf=1, name=\"s\") # name is a String\n# or\njulia> time!(ocp, t0=0, tf=1, name=:s ) # name is a Symbol \n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.variable!","page":"CTBase API","title":"CTBase.variable!","text":"variable!(ocp::OptimalControlModel, q::Integer)\nvariable!(\n ocp::OptimalControlModel,\n q::Integer,\n name::String\n)\nvariable!(\n ocp::OptimalControlModel,\n q::Integer,\n name::String,\n components_names::Vector{String}\n)\n\n\nDefine the variable dimension and possibly the names of each component.\n\nnote: Note\nYou can use variable! once to set the variable dimension when the model is NonFixed.\n\nExamples\n\njulia> variable!(ocp, 1, \"v\")\njulia> variable!(ocp, 2, \"v\", [ \"v₁\", \"v₂\" ])\n\n\n\n\n\n","category":"function"},{"location":"api-ctbase.html#CTBase.∂ₜ-Tuple{Any}","page":"CTBase API","title":"CTBase.∂ₜ","text":"∂ₜ(f) -> CTBase.var\"#106#108\"\n\n\nPartial derivative wrt time of a function.\n\nExample\n\njulia> ∂ₜ((t,x) -> t*x)(0,8)\n8\n\n\n\n\n\n","category":"method"},{"location":"api-ctbase.html#CTBase.@Lie-Tuple{Expr, Any, Any}","page":"CTBase API","title":"CTBase.@Lie","text":"Macros for Poisson brackets\n\nExample\n\njulia> H0 = (x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2)\njulia> H1 = (x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2)\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7]) autonomous=true variable=false\n#\njulia> H0 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2)\njulia> H1 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7]) autonomous=false variable=false\n#\njulia> H0 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v)\njulia> H1 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v)\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7], 2) autonomous=true variable=true\n#\njulia> H0 = (t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v)\njulia> H1 = (t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7], 2) autonomous=false variable=true\n\n\n\n\n\n","category":"macro"},{"location":"api-ctbase.html#CTBase.@Lie-Tuple{Expr, Any}","page":"CTBase API","title":"CTBase.@Lie","text":"Macros for Lie and Poisson brackets\n\nExample\n\njulia> H0 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2)\njulia> H1 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7]) autonomous=false\n#\njulia> H0 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v)\njulia> H1 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v)\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7], 2) variable=true\n#\n\n\n\n\n\n","category":"macro"},{"location":"api-ctbase.html#CTBase.@Lie-Tuple{Expr}","page":"CTBase API","title":"CTBase.@Lie","text":"Macros for Lie and Poisson brackets\n\nExample\n\njulia> F0 = VectorField(x -> [x[1], x[2], (1-x[3])])\njulia> F1 = VectorField(x -> [0, -x[3], x[2]])\njulia> @Lie [F0, F1]([1, 2, 3])\n[0, 5, 4]\n#\njulia> F0 = VectorField((t, x) -> [t+x[1], x[2], (1-x[3])], autonomous=false)\njulia> F1 = VectorField((t, x) -> [t, -x[3], x[2]], autonomous=false)\njulia> @Lie [F0, F1](1, [1, 2, 3])\n#\njulia> F0 = VectorField((x, v) -> [x[1]+v, x[2], (1-x[3])], variable=true)\njulia> F1 = VectorField((x, v) -> [0, -x[3]-v, x[2]], variable=true)\njulia> @Lie [F0, F1]([1, 2, 3], 2)\n#\njulia> F0 = VectorField((t, x, v) -> [t+x[1]+v, x[2], (1-x[3])], autonomous=false, variable=true)\njulia> F1 = VectorField((t, x, v) -> [t, -x[3]-v, x[2]], autonomous=false, variable=true)\njulia> @Lie [F0, F1](1, [1, 2, 3], 2)\n#\njulia> H0 = Hamiltonian((x, p) -> 0.5*(2x[1]^2+x[2]^2+p[1]^2))\njulia> H1 = Hamiltonian((x, p) -> 0.5*(3x[1]^2+x[2]^2+p[2]^2))\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7])\n3.0\n#\njulia> H0 = Hamiltonian((t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2), autonomous=false)\njulia> H1 = Hamiltonian((t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2), autonomous=false)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7])\n#\njulia> H0 = Hamiltonian((x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v), variable=true)\njulia> H1 = Hamiltonian((x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v), variable=true)\njulia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7], 2)\n#\njulia> H0 = Hamiltonian((t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v), autonomous=false, variable=true)\njulia> H1 = Hamiltonian((t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v), autonomous=false, variable=true)\njulia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7], 2)\n#\n\n\n\n\n\n","category":"macro"},{"location":"api-ctbase.html#CTBase.@def","page":"CTBase API","title":"CTBase.@def","text":"Define an optimal control problem. One pass parsing of the definition.\n\nExample\n\n@def ocp begin\n tf ∈ R, variable\n t ∈ [ 0, tf ], time\n x ∈ R², state\n u ∈ R, control\n tf ≥ 0\n -1 ≤ u(t) ≤ 1\n q = x₁\n v = x₂\n q(0) == 1\n v(0) == 2\n q(tf) == 0\n v(tf) == 0\n 0 ≤ q(t) ≤ 5, (1)\n -2 ≤ v(t) ≤ 3, (2)\n ẋ(t) == [ v(t), u(t) ]\n tf → min\nend\n\n\n\n\n\n","category":"macro"},{"location":"api-ctbase.html","page":"CTBase API","title":"CTBase API","text":"","category":"page"},{"location":"tutorial.html#First-example:-Goddard-problem","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"(Image: R.H. Goddard)","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"We start with the well-known so-called Goddard problem, which models the ascent of a rocket through the atmosphere (see for instance [1],[2]). We restrict here ourselves to vertical (monodimensional) trajectories, and the state variables are the altitude, speed and mass of the rocket during the flight, for a total dimension of 3. The rocket is subject to gravity, thrust and drag forces. The final time is free, and the objective is here to reach a maximal altitude with a given fuel consumption, i.e a fixed final mass. We also impose a speed limit. All units are renormalized.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"[1]: R.H. Goddard. A Method of Reaching Extreme Altitudes, volume 71(2) of Smithsonian Miscellaneous Collections. Smithsonian institution, City of Washington, 1919.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"[2]: H. Seywald and E.M. Cliff. Goddard problem in presence of a dynamic pressure limit. Journal of Guidance, Control, and Dynamics, 16(4):776–781, 1993.","category":"page"},{"location":"tutorial.html#Problem-definition","page":"First example: Goddard problem","title":"Problem definition","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"First import the necessary modules","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"using CTDirect\nusing CTBase\nusing NLPModelsIpopt\nusing Plots","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Then define the OCP for the Goddard problem. Note that the free final time is modeled as an optimization variable, and has both a lower bound to prevent the optimization getting stuck at tf=0. In this particular case an upper bound is not needed for the final time since the final mass is prescribed.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Cd = 310\nβ = 500\nTmax = 3.5\nb = 2\nvmax = 0.1\nfunction F0(x)\n r, v, m = x\n D = Cd * v^2 * exp(-β*(r - 1))\n return [ v, -D/m - 1/r^2, 0 ]\nend\nfunction F1(x)\n r, v, m = x\n return [ 0, Tmax/m, -b*Tmax ]\nend\n@def ocp begin\n tf ∈ R, variable\n t ∈ [ 0, tf ], time\n x ∈ R^3, state\n u ∈ R, control\n 0.1 ≤ tf ≤ Inf\n r = x[1]\n v = x[2]\n m = x[3]\n x(0) == [1, 0, 1]\n m(tf) == 0.6\n 1 ≤ r(t) ≤ 1.1\n 0 ≤ v(t) ≤ vmax\n 0 ≤ u(t) ≤ 1\n ẋ(t) == F0(x(t)) + u(t)*F1(x(t))\n r(tf) → max\nend\nnothing # hide","category":"page"},{"location":"tutorial.html#Basic-solve","page":"First example: Goddard problem","title":"Basic solve","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"We can solve the problem directly using the default options.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"sol1 = solve(ocp, print_level=5)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Then plot the solution with the state and control variables, as well as the costate recovered from the lagrange multipliers of the discretized problem. ","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"plot(sol1)","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"The most common option for solve is the number of time steps for the discretized problem (default 100), that can be set with the argument grid_size. A larger grid size will increase the computational cost, while a smaller value may lead to a very coarse solution.","category":"page"},{"location":"tutorial.html#Initial-guess-options","page":"First example: Goddard problem","title":"Initial guess options","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"The function solve uses a default constant initialisation of 0.1 for all variables. More advanced options include constant and/or functional initialisation for each individual state or control component, as well as reusing an existing solution, also known as warm start[3].","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"[3]: Currently only the primal variables are reused for the warm start, not the lagrange multipliers. It should be noted that previous experiments with the Bocop software seemed to indicate that initializing also the multipliers gave little benefit.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Let us start with the simplest case, constant initialisation.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"x_const = [1.05, 0.2, 0.8]\nu_const = 0.5\nv_const = 0.15\ninit1 = (state=x_const, control=u_const, variable=v_const)\nsol2 = solve(ocp, print_level=0, init=init1)\nprintln(\"Objective \", sol2.objective, \" after \", sol2.iterations, \" iterations\")","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Now we illustrate the functional initialisation, with some random functions. Note that we only consider the state and control variables, since the optimization variables are scalar and therefore a functional initialisation is not relevant. In the example notice that the initialization does not provide an argument for the optimization variables, therefore the default initial guess will be used. ","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"x_func = t->[1+t^2, sqrt(t), 1-t]\nu_func = t->(cos(t)+1)*0.5\ninit2 = (state=x_func, control=u_func)\nsol3 = solve(ocp, print_level=0, init=init2)\nprintln(\"Objective \", sol3.objective, \" after \", sol3.iterations, \" iterations\")","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"More generally, the default, constant and functional initialisations can be mixed, as shown in the example below that uses a functional initial guess for the state, a constant initial guess for the control, and the default initial guess for the optimization variables. ","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"init3 = (state=x_func, control=u_const)\nsol4 = solve(ocp, print_level=0, init=init3)\nprintln(\"Objective \", sol4.objective, \" after \", sol4.iterations, \" iterations\")","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Finally, we can also use a so-called warmstart strategy and use an existing solution as initial guess (note that the OCP solution returned by the solve call is functional, thus it is not necessary to use the same time grid). Notice that the objective and constraint violation values start much closer to the solution than with the previous initialisations.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"sol4 = solve(ocp, grid_size=200, print_level=5, init=sol1)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"plot(sol4)","category":"page"},{"location":"tutorial.html#The-discretized-problem","page":"First example: Goddard problem","title":"The discretized problem","text":"","category":"section"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Instead of calling solve directly on the OCP problem, you can first obtain the discretized problem (DOCP) by calling directTranscription, then call solve on the DOCP. The resulting solution of the discretized problem can be used to generate the corresponding OCP solution with OCPSolutionFromDOCP.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"docp = directTranscription(ocp, grid_size=100)\ndsol = solve(docp, print_level=5)\nsol5 = OCPSolutionFromDOCP(docp, dsol)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"The initial guess can be passed to solve same as before.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"dsol = solve(docp, print_level=0, init=sol1)\nsol6 = OCPSolutionFromDOCP(docp, dsol)\nprintln(\"Objective \", sol6.objective, \" after \", sol6.iterations, \" iterations\")","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Another possibility is to set the initial guess associated to the DOCP, using the function setInitialGuess.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"setInitialGuess(docp, sol1)\ndsol = solve(docp, print_level=5)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"Finally, the direct transcription also accept an initial guess.","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"docp = directTranscription(ocp, grid_size=100, init=sol1)\ndsol = solve(docp, print_level=5)\nnothing # hide","category":"page"},{"location":"tutorial.html","page":"First example: Goddard problem","title":"First example: Goddard problem","text":"","category":"page"}] } diff --git a/dev/tutorial-511a0a73.svg b/dev/tutorial-092c1114.svg similarity index 85% rename from dev/tutorial-511a0a73.svg rename to dev/tutorial-092c1114.svg index fe283dd..343a17b 100644 --- a/dev/tutorial-511a0a73.svg +++ b/dev/tutorial-092c1114.svg @@ -1,234 +1,234 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/tutorial-c8b96e1b.svg b/dev/tutorial-334f7511.svg similarity index 84% rename from dev/tutorial-c8b96e1b.svg rename to dev/tutorial-334f7511.svg index f634f79..c11a0df 100644 --- a/dev/tutorial-c8b96e1b.svg +++ b/dev/tutorial-334f7511.svg @@ -1,234 +1,234 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/tutorial.html b/dev/tutorial.html index aeef74e..7655cd2 100644 --- a/dev/tutorial.html +++ b/dev/tutorial.html @@ -104,9 +104,9 @@ Number of equality constraint Jacobian evaluations = 34 Number of inequality constraint Jacobian evaluations = 0 Number of Lagrangian Hessian evaluations = 32 -Total seconds in IPOPT = 2.020 +Total seconds in IPOPT = 2.003 -EXIT: Optimal Solution Found.

Then plot the solution with the state and control variables, as well as the costate recovered from the lagrange multipliers of the discretized problem. 

plot(sol1)
Example block output

The most common option for solve is the number of time steps for the discretized problem (default 100), that can be set with the argument grid_size. A larger grid size will increase the computational cost, while a smaller value may lead to a very coarse solution.

Initial guess options

The function solve uses a default constant initialisation of 0.1 for all variables. More advanced options include constant and/or functional initialisation for each individual state or control component, as well as reusing an existing solution, also known as warm start[3].

Let us start with the simplest case, constant initialisation.

x_const = [1.05, 0.2, 0.8]
+EXIT: Optimal Solution Found.

Then plot the solution with the state and control variables, as well as the costate recovered from the lagrange multipliers of the discretized problem. 

plot(sol1)
Example block output

The most common option for solve is the number of time steps for the discretized problem (default 100), that can be set with the argument grid_size. A larger grid size will increase the computational cost, while a smaller value may lead to a very coarse solution.

Initial guess options

The function solve uses a default constant initialisation of 0.1 for all variables. More advanced options include constant and/or functional initialisation for each individual state or control component, as well as reusing an existing solution, also known as warm start[3].

Let us start with the simplest case, constant initialisation.

x_const = [1.05, 0.2, 0.8]
 u_const = 0.5
 v_const = 0.15
 init1 = (state=x_const, control=u_const, variable=v_const)
@@ -170,9 +170,9 @@
 Number of equality constraint Jacobian evaluations   = 16
 Number of inequality constraint Jacobian evaluations = 0
 Number of Lagrangian Hessian evaluations             = 15
-Total seconds in IPOPT                               = 3.995
+Total seconds in IPOPT                               = 4.031
 
-EXIT: Optimal Solution Found.
plot(sol4)
Example block output

The discretized problem

Instead of calling solve directly on the OCP problem, you can first obtain the discretized problem (DOCP) by calling directTranscription, then call solve on the DOCP. The resulting solution of the discretized problem can be used to generate the corresponding OCP solution with OCPSolutionFromDOCP.

docp = directTranscription(ocp, grid_size=100)
+EXIT: Optimal Solution Found.
plot(sol4)
Example block output

The discretized problem

Instead of calling solve directly on the OCP problem, you can first obtain the discretized problem (DOCP) by calling directTranscription, then call solve on the DOCP. The resulting solution of the discretized problem can be used to generate the corresponding OCP solution with OCPSolutionFromDOCP.

docp = directTranscription(ocp, grid_size=100)
 dsol = solve(docp, print_level=5)
 sol5 = OCPSolutionFromDOCP(docp, dsol)
This is Ipopt version 3.14.14, running with linear solver MUMPS 5.6.2.
 
@@ -246,7 +246,7 @@
 Number of equality constraint Jacobian evaluations   = 34
 Number of inequality constraint Jacobian evaluations = 0
 Number of Lagrangian Hessian evaluations             = 32
-Total seconds in IPOPT                               = 1.994
+Total seconds in IPOPT                               = 2.209
 
 EXIT: Optimal Solution Found.

The initial guess can be passed to solve same as before.

dsol = solve(docp, print_level=0, init=sol1)
 sol6 = OCPSolutionFromDOCP(docp, dsol)
@@ -300,7 +300,7 @@
 Number of equality constraint Jacobian evaluations   = 12
 Number of inequality constraint Jacobian evaluations = 0
 Number of Lagrangian Hessian evaluations             = 11
-Total seconds in IPOPT                               = 0.712
+Total seconds in IPOPT                               = 0.787
 
 EXIT: Optimal Solution Found.

Finally, the direct transcription also accept an initial guess.

docp = directTranscription(ocp, grid_size=100, init=sol1)
 dsol = solve(docp, print_level=5)
This is Ipopt version 3.14.14, running with linear solver MUMPS 5.6.2.
@@ -352,11 +352,11 @@
 Number of equality constraint Jacobian evaluations   = 12
 Number of inequality constraint Jacobian evaluations = 0
 Number of Lagrangian Hessian evaluations             = 11
-Total seconds in IPOPT                               = 0.751
+Total seconds in IPOPT                               = 0.880
 
 EXIT: Optimal Solution Found.
  • 1R.H. Goddard. A Method of Reaching Extreme Altitudes, volume 71(2) of Smithsonian Miscellaneous Collections. Smithsonian institution, City of Washington, 1919.
  • 2H. Seywald and E.M. Cliff. Goddard problem in presence of a dynamic pressure limit. Journal of Guidance, Control, and Dynamics, 16(4):776–781, 1993.
  • 3Currently only the primal variables are reused for the warm start, not the lagrange multipliers. It should be noted that previous experiments with the Bocop software seemed to indicate that initializing also the multipliers gave little benefit.
+
  • 1R.H. Goddard. A Method of Reaching Extreme Altitudes, volume 71(2) of Smithsonian Miscellaneous Collections. Smithsonian institution, City of Washington, 1919.
  • 2H. Seywald and E.M. Cliff. Goddard problem in presence of a dynamic pressure limit. Journal of Guidance, Control, and Dynamics, 16(4):776–781, 1993.
  • 3Currently only the primal variables are reused for the warm start, not the lagrange multipliers. It should be noted that previous experiments with the Bocop software seemed to indicate that initializing also the multipliers gave little benefit.