Limit Value for Optimal Control with general means

Résumé

We consider optimal control problems where the running cost of the trajectory is evaluated by a probability measure on R. As a particular case, we take the Cesàro average of the running cost over a fixed horizon. The limit of the value with Cesàro average when the horizon tends to infinity is widely studied in the literature. We address the more general question of the existence of a limit for values defined by general evaluations satisfying certain long-term condition. For this aim, we introduce an asymptotic regularity condition for a sequence of probability measures on R. Our main result is that, for any sequence of probability measures on R satisfying this condition, the associated value functions converge uniformly if and only if this family is totally bounded for the uniform norm. As a byproduct, we obtain the existence of a limit value (for general evaluations) for control systems defined on a compact invariant domain and satisfying suitable nonexpansive property.