Merge branch 'main' into 0.10.0

Author: PierreMartinon

docs/make.jl

@@ -20,6 +20,7 @@ makedocs(
         prettyurls = false,
         size_threshold_ignore = [
             "api-ctbase/types.md",
+            "tutorial-plot.md",
         ],
         assets = [
             asset("https://control-toolbox.org/assets/css/documentation.css"),

docs/src/tutorial-iss.md

@@ -1,4 +1,4 @@
-# Indirect simple shooting
+# [Indirect simple shooting](@id iss)
 
 In this tutorial we present the indirect simple shooting method on a simple example.
 
@@ -160,7 +160,16 @@ end # hide
 nothing # hide
 ```
 
-We get:
+The solution can be plot calling first the flow.
+
+```@example main
+sol = φ( (t0, tf), x0, p0_sol )
+plot(sol, size=(800, 600))
+```
+
+In the indirect shooting method, the research of the optimal control is replaced by the computation
+of its associated extremal. This computation is equivalent to finding the initial covector solution
+to the shooting function.
 
 ```@example main
 exp(p0; saveat=[]) = φ((t0, tf), x0, p0, saveat=saveat).ode_sol # hide

docs/src/tutorial-plot.md

@@ -66,6 +66,33 @@ plot(sol;
      control_style = (color=:red, linewidth=2))
 ```
 
+## From Flow
+
+The previous resolution of the optimal control problem was done with the `solve` function.
+If you use an indirect shooting method and solve shooting equations, you may want to plot the 
+associated solution. To do so, you need to use the `Flow` function to 
+reconstruct the solution. See the [Indirect Simple Shooting](@ref iss) tutorial for an example.
+In our example, you must provide the maximising control $(x, p) \mapsto p_2$ together with the 
+optimal control problem.
+
+```@example main
+t0 = 0
+tf = 1
+x0 = [ -1, 0 ]
+p0 = sol.costate(t0)
+f  = Flow(ocp, (x, p) -> p[2])
+sol_flow = f( (t0, tf), x0, p0 )
+plot(sol_flow)
+```
+
+You can notice that the time grid has very few points. To have a better visualisation (the accuracy 
+won't change), you can give a finer grid.
+
+```@example main
+sol_flow = f( (t0, tf), x0, p0; saveat=range(t0, tf, 100) )
+plot(sol_flow)
+```
+
 ## Split versus group layout
 
 If you prefer to get a more compact figure, you can use the `layout` optional keyword argument with `:group` value. It will group the state, costate and control trajectories in one subplot for each.