Mathematics

In this course (1 cycle), I will discuss the basic tools of geometric control theory. The program is the following:

• Definition of control systems on  smooth manifolds. Feedback. Families of vector fields. Differential inclusions. The classical problems in geometric control: controllability, stabilizability, optimal control.
• The controllability problem: the Kalman condition for Linear systems, the Krener Theorem, the Chow Theorem. Compatible vector fields: Systems with recurrent drift, systems satisfying the strong Hormander condition.
• The problem of finding a smooth stabilizing feedback. The Brockett condition.
• Optimal control: the Filippov theorem; the Pontryagin Maximum Principle; minimum time for linear systems. Minimum time for 2D non-linear systems. Systems with quadratic cost.
• Extensions: Connection to problem of sub-Riemannian geometry. What happens if we put in place of the control a Brownian motion.