From a5f9b5f399bd3e536bbd8d51dc7e8862bb8caeed Mon Sep 17 00:00:00 2001
From: Olivier Cots <61-ocots@users.noreply.022e47118ec0>
Date: Fri, 26 Mar 2021 10:35:26 +0000
Subject: [PATCH] Update lecture2.md
---
course/lecture/lecture2.md | 14 +++++++++++++-
1 file changed, 13 insertions(+), 1 deletion(-)
diff --git a/course/lecture/lecture2.md b/course/lecture/lecture2.md
index 14b75ba..d337c4e 100644
--- a/course/lecture/lecture2.md
+++ b/course/lecture/lecture2.md
@@ -5,6 +5,9 @@
------
+**_Contents_**
+
+[[_TOC_]]
## III) Indirect simple shooting
@@ -66,7 +69,7 @@ The *indirect simple shooting method* consists in solving this equation.
>>>
-In order to solve the shooting equations, we need to compute the control law $u[\cdot]$, the Hamiltonian system $`\vec{H}`$ (or $`\vec{h}`$), we need an [integrator method](https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations) to compute the *exponential map* $\exp(t \vec{H})$ defined by
+In order to solve the shooting equations, we need to compute the control law $`u[\cdot]`$, the Hamiltonian system $`\vec{H}`$ (or $`\vec{h}`$), we need an [integrator method](https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations) to compute the *exponential map* $`\exp(t \vec{H})`$ defined by
```math
\exp(t \vec{H})(x_0, p_0) := z(t, x_0, p_0),
```
@@ -77,3 +80,12 @@ and we need a [Newton-like](https://en.wikipedia.org/wiki/Newton%27s_method) sol
**_Remark:_**
It is important to understand that if $`(p_0^*, \lambda^*)`$ is solution of $`S=0`$, then the control $`u(\cdot) := u[z(\cdot, x_0, p_0^*)]`$ is a candidate as a solution of the optimal control problem. It is only a candidate and not a solution of the OCP since the PMP gives necessary conditions of optimality. We would have to go further to check whether the control is locally or globally optimal.
+<div align="center">
+ <img src="attachment:21f9cc6f-5061-4fbe-ab9b-b905c2389dc9.png" width="400">
+</div>
+<div align="center">
+<i>
+Figure: Illustration of the shooting method in the state-time space. The blue trajectory reaches the target in red. The shooting method consists in finding the right impulse to reach the target.
+</i>
+</div>
+
--
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