Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations

TL;DR

This paper investigates the long-term behavior of weakly coupled first-order Hamilton-Jacobi systems in a periodic setting, extending scalar case results through PDE techniques and control interpretations.

Contribution

It introduces new comparison, existence, and regularity results for coupled systems and extends scalar convergence results to systems using PDE methods.

Findings

Established large time convergence for weakly coupled systems

Developed new comparison and regularity results for systems

Provided an optimal control interpretation of solutions

Abstract

We show a large time behavior result for class of weakly coupled systems of first-order Hamilton-Jacobi equations in the periodic setting. We use a PDE approach to extend the convergence result proved by Namah and Roquejoffre (1999) in the scalar case. Our proof is based on new comparison, existence and regularity results for systems. An interpretation of the solution of the system in terms of an optimal control problem with switching is given.