Uniform value for some nonexpansive optimal control problems with general evaluations

TL;DR

This paper investigates the long-term behavior of continuous-time optimal control problems with general evaluations, establishing the existence of a uniform value under regularity conditions on the evaluation measures.

Contribution

It proves the existence of a general uniform value for nonexpansive control systems with regular evaluations, extending previous results on limit values.

Findings

Existence of a general uniform value under regularity conditions.

Uniform convergence of value functions for regular evaluations.

Applicability to control systems on compact domains with nonexpansive conditions.

Abstract

We study the long-run properties of optimal control problems in continuous time, where the running cost of a control problem is evaluated by a probability measure over R_+. Li, Quincampoix and Renault [DCDS-A, 2016] introduced an asymptotic regularity condition for a sequence of probability measures to study the limit properties of the value functions with respect to the evaluation. In the particular case of t-horizon Ces\`aro mean or rho-discounted Abel mean, this condition implies that the horizon t tends to infinity or the discount factor rho tends to zero. For the control system defined on a compact domain and satisfying some nonexpansive condition, Li, Quincampoix and Renault [DCDS-A, 2016] proved the existence of general limit value, i.e. the value function uniform converges as the evaluation becomes more and more regular. Within the same context, we prove the existence of general uniform value, i.e. for any epsilon>0, there is an optimal control that guarantees the general limit value up to epsilon for all control problems where the cost is evaluated by sufficiently regular probability measures.