The ergodic problem for the Hamilton-Jacobi-Bellman equation and regularizing effects of certain classes of Hamilton-Jacobi equations.

Summary The work presented in this thesis can be grouped into two themes. 1. The ergodic problem for the Hamilton-Jacobi-Bellman (H-J-B) equations. 2. Regularizing effects for a class of Hamilton-Jacobi equations. The ergodic problem concerns the long time average behavior of deterministic or stochastic controlled systems. We study here deterministic systems (including those defined in infinite dimension) and the corresponding H-J-B equations, using the theory of viscosity solutions. We first establish in chapter 2 the ergodicity of infinite dimensional systems under classical finite dimensional assumptions. Then, we focus on necessary or sufficient conditions to ensure the ergodicity of finite dimensional systems. In chapter 3, we prove the existence of an ergodic attractor on which the system is controllable. And in chapter 4, we give a kind of reciprocal, the estimate of the controllability on the ergodic attractor. Controllability also plays an essential role in the regularizing effects of the first order Hamilton-Jacobi equations. We show in chapter 5, three types of regularizing effects: lipschitzian regularization, semi-concave regularization and regularization in c#1#,#1#l#o#c.