8/17/2019 Controle Geometrique

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Contrôle géométrique Ugo Boscain, Frederic Jean, Yacine Chitour  

Résumé:

Le but de ce cours est de fournir une introduction à la théorie du contrôle avec le point de vue

géométrique, c'est-à-dire en utilisant les outils de la géométrie différentielle. Le cours est divisé en

deux parties. Dans la première, on étudiera la géométrie de familles de systèmes dynamiques,

exhibant en particulier la structure de l'ensemble des trajectoires. Dans la seconde partie, nous

traiterons de la théorie du contrôle optimal, en mettant l'accent sur le Principe du Maximum de

Pontryagin et sa géométrie sous-jacente, en connexion avec les propriétés des opérateurs

hypoelliptique.

Title: Geometric Control

Abstract:

The purpose of this course is to provide an introduction to control theory with the geometric point

of view, i.e. by using the tools of differential geometry. The course is divided into two parts. The

first will study the geometry of families of dynamical systems, showing in particular the structure of

the set of trajectories. In the

second part, we discuss the optimal control theory, focusing on the Pontryagin Maximum Principle

and its underlying geometry, in connection with the properties of hypoelliptic operators.

Outline:

I. Controllability

-) definition of control systems as families of dynamical systems

-) structural properties

-) Accessibility, Controllability and Stabilization

-) Lie bracket, the Lie algebraic condition and the Brockett obstruction

-) Krener theorem, Chow theorem, the orbit theorem

-) Poisson stability and recurrence

-) Controllability for systems under the strong Hörmander condition and unbounded controls

-) the Frobenius Theorem

-) non-holonomic mechanics

-) examples: Linear systems, control of satellites, rolling bodies, quantum systems

II. Optimal Control

-) The classical approach of the calculus of variations (Euler-Lagrange Equations)

-) Legendre transformation and the Hamiltonian formalism

-) existence of solutions: Filippov Theorem

-) the general Statement of the Pontryagin Maximum Principle

-) Singular trajectories and abnormal extremals

-) Proof of the PMP in the case of systems affine in the control with quadratic cost

-) Minimum time with bounded controls

-) Sub-Riemannian geometry and the hypoelliptic Laplacian