Update lecture.md

course/lecture/lecture.md

@@ -22,6 +22,8 @@ The goal of this presentation is that at the end, you will be able to implement
 
 **_Contents_**
 
+[[_TOC_]]
+
 * I) Statement of the optimal control problem and necessary conditions of optimality
     * a) Definition of the optimal control problem - [Video](https://youtu.be/QUbeyLNZR8A)
     * b) Application of the Pontryagin Maximum Principle - [Video](https://youtu.be/xedLNn08Kn4)
@@ -62,3 +64,55 @@ with $`U \subset \mathrm{R}^m`$ an arbitrary control set and with $`c`$ a smooth
 Jacobian $`c'(x)`$ (or $`J_c(x)`$) is of full rank for any $`x`$ satisfying the constraint $`c(x)=0`$.
 The solution $`u`$ belongs to the *set of control laws* $`L^\infty([0, t_f], \mathrm{R}^m)`$.
 
+[Additional comments (video)](http://www.youtube.com/watch?v=QUbeyLNZR8A)
+
+### b) Application of the Pontryagin Maximum Principle
+
+Let us denote by
+
+```math
+    H(x,p,u) := p \, f(x,u) + p^0\, L(x,u),
+```
+
+the *pseudo-Hamiltonian* (that is the non-maximized Hamiltonian) associated to the optimal control problem.
+
+>>>
+
+**_Pontryagin Maximum Principle_**
+    
+According to the PMP, if $`u`$ is solution of the problem (with $`x`$ the *associated trajectory*),
+then there exists a *covector* $`p`$ (which is 
+[absolutely continuous](https://en.wikipedia.org/wiki/Absolute_continuity)),
+a scalar $`p^0 \in \{-1, 0\}`$, a *Lagrange multiplier* $`\lambda`$, such that: 
+
+1. $`(p, p^0) \ne (0,0)`$,
+
+2. $`\displaystyle \dot{x}(t) = \nabla_p H(x(t),p(t),u(t))`$, 
+$`\displaystyle \dot{p}(t) = -\nabla_x H(x(t),p(t),u(t))`$, a.e on $`[0, t_f]`$,
+
+3. $`\displaystyle  H(x(t),p(t),u(t)) = \max_{w \in U} H(x(t), p(t), w)`$ a.e on $`[0, t_f]`$
+ (maximization condition),
+
+4. $`\displaystyle p(t_f) = J_c^T(x(t_f)) \lambda = \sum_{i=1}^k \lambda_i \nabla c_i(x(t_f))`$
+ (transversality condition).
+
+>>>
+
+>>>
+    
+**_Assumptions_**
+    
+We assume the following:
+    
+* $`U = \mathrm{R}^m`$,
+* $`\forall (x,p) \in \mathrm{R}^n \times \mathrm{R}^n`$, $`u \mapsto H(x,p,u)`$ has a unique maximum denoted $`\varphi(x,p)`$ (or $`u[x,p]`$ to recall the fact that it is the control law in feedback form),
+* $`\varphi`$ is smooth, that is at least $`C^1`$.
+    
+>>>
+
+Under these assumptions, the maximization condition (3) is equivalent to the first order necessary condition of optimality and we have:
+
+```math
+    \forall (x,p), \quad \nabla_u H(x,p, \varphi(x,p)) = 0.
+```
+