Geometric control theory

Mario Sigalotti and Ugo Boscain

Goals of the UE :

Geometric theory of control studies, from a qualitative and geometric point of view, the properties of dynamical systems for which some parameters can be controlled as functions of time. The objective of the course is to introduce the fundamental notions of this theory and to show how they can be applied to systems stemming from applications or other mathematical theories.

Topics :

• accessibility of nonlinear systems and controllability of symmetric systems (Krener and Chow theorems)
• controllability of nonlinear systems (recurrent flows, strong Hörmander condition, convexification, invariant systems on Lie groups with applications to the controllability of quantum systems)
• optimal control (existence by Filippov's theorem, Pontryagin maximum principle, abnormal and singular trajectories)
• time optimization (introduction to optimal two-dimensional syntheses, applications to quantum systems)
• symmetric systems with quadratic cost: introduction to sub-Riemannian geometry