Mathematics

The course of 20 lectures will provide an introduction to geometric control and sub-Riemannian geometry. The first part of the course will be devoted to controllability, the second part will discuss stabilization and Optimal Control, while the last part will focus sub-Riemannian geometry. No prior knowledge of control theory is required.

 

Course program:

 

PART 1.

1. Some basic questions in the control formalism, some examples of

control systems.

2. Controllability of linear systems. Lie brackets and their relation

with controlled motions.

3. Krener's theorem, Rashevskii-Chow's theorem, and the orbit theorem.

4. Compatible vector fields, the strong bracket generating condition,

recurrence & controllability.

 

PART 2.

5. Stabilization of linear systems and the pole-placement theorem.

Stabilization of nonlinear systems by control Lyapunov functions.

6. Existence of minimizer in optimal control problems: Filippov's theorem.

7. First-order necessary conditions for optimality: Pontryagin's

maximum principle.

8. Minimum time problems with bounded controls.

 

PART 3.

8. Sub-Riemannian manifolds: definitions and examples.

9. Continuity of the distance.

10. Length minimizers and equivalence with energy minimizers.

11. Minimality of short pieces of normal extremals.

12. Examples: Heisenberg, Grushin, Martinet.

13. The sub-Riemannian Laplacian.

14. Heat: the Gaveau kernel

14. Schroedinger: Quantum confinement

 

 

References:

[1] Andrei A. Agrachev, Yuri L. Sachkov. Control theory from the geometric viewpoint. Springer-Verlag, 2004.

[2] Andrei A. Agrachev, Davide Barilari, Ugo  Boscain. A comprehensive introduction to sub-Riemannian geometry. Cambridge university press 2020.

[3] Ugo Boscain, Dario Prandi, Mario Sigalotti, Introduction to Geometric Control Theory. Preprint. https://dprn.github.io/docs/poly.pdf

[4] Jean-Michel Coron. Control and nonlinearity. American Mathematical Society, 2007.

[5] Eduardo Sontag. Mathematical control theory. Springer-Verlag, 1998.