Continuity of Optimal Control Costs and its application to Weak KAM Theory

TL;DR

This paper establishes the continuity of specific optimal control cost functions for affine systems, providing conditions for this property, and applies these results to weak KAM theory and Aubry-Mather problems.

Contribution

It proves continuity conditions for control costs and applies them to weak KAM theory, linking optimal control with dynamical systems.

Findings

Proved continuity of control cost functions under sharp conditions

Established a version of the weak KAM theorem for affine control systems

Analyzed Aubry-Mather problems in this context

Abstract

We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the Aubry-Mather problems corresponding to these systems.