refactor ok

Author: ocots

docs/Project.toml

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+[deps]
+DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
+Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
+ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
+MINPACK = "4854310b-de5a-5eb6-a2a5-c1dee2bd17f9"
+OptimalControl = "5f98b655-cc9a-415a-b60e-744165666948"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"

docs/make.jl

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+using Documenter
+
+makedocs(
+    warnonly = :cross_references,
+    sitename = "Geometric preconditioner",
+    format = Documenter.HTML(
+        prettyurls = false, 
+        #size_threshold_ignore = [""],
+        assets=[
+            asset("https://control-toolbox.org/assets/css/documentation.css"),
+            asset("https://control-toolbox.org/assets/js/documentation.js"),
+        ],
+    ),
+    pages = [
+        "Introduction" => "index.md",
+        "2D example"   => "2D-example.md",
+    ]
+)
+
+deploydocs(
+    repo = "github.com/control-toolbox/preconditioning.git",
+    devbranch = "main"
+)

docs/src/2D-example.md

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+### Optimal control problem
+
+We consider the following optimal control problem 
+
+```math
+    \left\{ \begin{array}{ll}
+    \displaystyle \min_{x,u} \int_{t_0}^{t_f} x(t) ~\mathrm dt \\[1em]
+    \text{s.c.}~\dot x(t) = u(t), & t\in [t_0, t_f]~\mathrm{a.e.}, \\[0.5em]
+    \phantom{\mathrm{s.c.}~} u(t) \in [-1,1], & t\in [t_0, t_f], \\[0.5em]
+    \phantom{\mathrm{s.c.}~} x(t_0) = x_0, \quad x(t_f) = x_f,
+    \end{array} \right.
+```
+
+with $x_0$, $t_0$, $x_f$ and $t_f$ fixed. This problem is simple, and can be analytically solve without the use of numerical method. However, the goal is to solve this problem by indirect shooting.  
+
+### Indirect method
+
+We introduce the pseudo-Hamiltonian 
+
+```math
+    h(x,p,p^0,u) = p^0 x + p u.
+```
+
+For the sake of simplicity, we consider in this notebook only the normal case, and we fix $p^0 = -1$. According to the Pontryagin maximum principle, the maximizing control is given by $u(x,p) \to \mathrm{sign}(p)$. This function is non-differentiable, and may lead to numerical issues.  
+
+Let us import the necessary package and define the studied optimal control problem with some fixed initial and final time and state values.
+
+```@example main
+using OptimalControl
+using Plots
+using ForwardDiff
+using DifferentialEquations
+using MINPACK
+
+t0 = 0
+x0 = 0
+tf = 5
+xf = 0                      # initial and final time and state
+
+@def ocp begin              # problem definition
+
+    t ∈ [ t0, tf ], time
+    x ∈ R, state
+    u ∈ R, control
+
+    x(t0) == x0
+    x(tf) == xf
+
+    ẋ(t) == u(t)      
+
+    ∫( x(t) ) → min
+
+end
+nothing # hide
+```

docs/src/index.md

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+# Geometric preconditioner for indirect shooting
+
+blabla