From 1d2310847e0e36bff2b37b2fbbdf87cac8b61769 Mon Sep 17 00:00:00 2001
From: Olivier Cots <61-ocots@users.noreply.022e47118ec0>
Date: Fri, 26 Mar 2021 10:46:18 +0000
Subject: [PATCH] Update lecture2.md

---
 course/lecture/lecture2.md | 24 +++++++++++++++++++++++-
 1 file changed, 23 insertions(+), 1 deletion(-)

diff --git a/course/lecture/lecture2.md b/course/lecture/lecture2.md
index d337c4e..71c30f9 100644
--- a/course/lecture/lecture2.md
+++ b/course/lecture/lecture2.md
@@ -81,7 +81,7 @@ and we need a [Newton-like](https://en.wikipedia.org/wiki/Newton%27s_method) sol
 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">
+    <img src="../images/simple_shooting.png" width="400">
 </div>
 <div align="center">
 <i>
@@ -89,3 +89,25 @@ Figure: Illustration of the shooting method in the state-time space. The blue tr
 </i>
 </div>
 
+<div align="center">
+    <img src="../images/simple_shooting_2.png" width="420">
+</div>
+<div align="center">
+<i>
+Figure: Illustration of the shooting method in the cotangent space. The blue extremal reaches the target in red.
+</i>
+</div>
+
+[Additional comments (video)](http://www.youtube.com/watch?v=YVG2Z_TEkBQ)
+
+### b) The iteration of the Newton solver and the Jacobian of the shooting function
+
+[Additional comments (video)](http://www.youtube.com/watch?v=hVbi9kShR90)
+
+### c) A word on the Lagrange multiplier
+
+[Additional comments (video)](http://www.youtube.com/watch?v=3VEi-UHAS6w)
+
+## IV) Numerical resolution of the shooting equations with the nutopy package
+
+This part is there: [notebook lecture](lecture.ipynb). Please follow the procedure given [here](../../README.md) to install all packages and use `jupyter-lab` to execute the code and do the exercices.
-- 
GitLab