École Nationale Supérieure de
Techniques Avancées

Résumé

In this master thesis we analyse the asymptotic behavior of optimal control problems and its impact on Hamilton-Jacobi-Bellman equation. We focus on Linear-Quadratic problems in a finite time-horizon and with a non-zero target. We first construct a Riccati operator that characterizes the value function for such problems and operates jointly on the initial state and the target. We then prove a convergence result for such operator and deduce the asymptotic behavior of the value function. We will show that the latter is indeed a unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation and that its asymptotic behavior satisfies an ergodic problem. Finally, we will study regularity of such solution in the more general class of Lagrange problems, and make the link with previous results in literature. Key words: optimal control problems, long time behavior, Hamilton-Jacobi-Bellman equation