Limit value for optimal control with general means

TL;DR

This paper investigates the existence of a limit value in optimal control problems with general mean-based costs, providing necessary and sufficient conditions for uniform convergence of the value functions as the averaging parameter varies.

Contribution

It introduces a comprehensive framework for analyzing limit values in control problems with general means, extending beyond Cesàro averages, and establishes conditions based on total variation of probability measures.

Findings

Established necessary and sufficient conditions for uniform convergence.

Proved existence of a limit value for systems with compact invariant sets.

Extended classical results to more general mean-based cost functions.

Abstract

We consider optimal control problem with an integral cost which is a mean of a given function. As a particular case, the cost concerned is the Ces\`aro average. The limit of the value with Ces\`aro mean when the horizon tends to infinity is widely studied in the literature. We address the more general question of the existence of a limit when the averaging parameter converges, for values defined with means of general types. We consider a given function and a family of costs defined as the mean of the function with respect to a family of probability measures -- the evaluations -- on R_+. We provide conditions on the evaluations in order to obtain the uniform convergence of the associated value function (when the parameter of the family converges). Our main result gives a necessary and sufficient condition in term of the total variation of the family of probability measures on R_+. As a byproduct, we obtain the existence of a limit value (for general means) for control systems having a compact invariant set and satisfying suitable nonexpansive property.