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

Abstract

Systems of Hamilton–Jacobi equations arise naturally when we study 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. First results have been obtained by Camilli et al. (NoDEA Nonlinear Diff Eq Appl, 2012) and Mitake and Tran (Asymptot Anal, 2012) under quite strict conditions. Here, we use a PDE approach to extend the convergence result proved by Barles and Souganidis (SIAM J Math Anal 31(4):925–939 (electronic), 2000) in the scalar case. This result permits us to treat general cases, for instance, systems of nonconvex Hamiltonians and systems of 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.