Mathematics

The course of 10 lectures will provide an introduction to geometric control theory. The first part of the course will be devoted to controllability, the second part will discuss stabilization, while the last part will focus on optimal control. No prior knowledge of control theory is required.

Course program:
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.

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.

References:
[1] Andrei A. Agrachev, Yuri L. Sachkov. Control theory from the geometric viewpoint. Springer-Verlag, 2004.
[2] Jean-Michel Coron. Control and nonlinearity. American Mathematical Society, 2007.

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

 

• A-134