L'équation de Hamilton-Jacobi en contrôle optimal

Abstract

The main object of this thesis is the application of new methods from nonsmooth analysis and which use the Hamilton-Jacobi equation for the study of certain problems in control theory. There are three parts in our work: * In the first part we develop a new duality result in control theory. This result generalizes, in a number of ways, the Vinter's duality (1993) and gives a new characterization of the minimal time function. * The second part is devoted to the study of the Hamilton-Jacobi equation of minimal time, but in a domain which contains the origin. We prove the existence of (minimal) solutions of this equation and we show that these solutions are closely linked to global geodesics trajectories. * In the third part, we study the existence of minimal loop trajectories for a control system. We give a necessary and sufficient conditions for the existence of this type of trajectories at a given point.