Some results on the large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations

TL;DR

This paper investigates the large time behavior of weakly coupled first-order Hamilton-Jacobi systems in a periodic setting, extending known convergence results to more general cases including nonconvex Hamiltonians.

Contribution

It extends the convergence results for large time behavior of Hamilton-Jacobi systems to broader classes, including nonconvex Hamiltonians, using a PDE approach.

Findings

Established convergence results for systems with nonconvex Hamiltonians.

Extended scalar case results to coupled systems.

Provided clearer understanding of long-term dynamics of Hamilton-Jacobi systems.

Abstract

Systems of Hamilton-Jacobi equations arise naturally when we study the optimal control problems with pathwise deterministic trajectories with random switching. In this work, we are interested in the large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations in the periodic setting. The large time behavior for systems of Hamilton-Jacobi equations have been obtained by Camilli-Loreti-Ley and the author (2012) and Mitake-Tran (2012) under quite strict conditions. In this work, we use a PDE approach to extend the convergence result proved by Barles-Souganidis (2000) in the scalar case. This general result permits us to treat lot of general cases, for instance, systems with nonconvex Hamiltonians and systems with strictly convex Hamiltonians. We also obtain some other convergence results under different assumptions, these results give a clearer view on the large time behavior for systems of Hamilton-Jacobi equations.