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

Abstract

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-Loreti-Ley and the author (2012) and Mitake-Tran (2012) under quite strict conditions. Here, we use a PDE approach to extend the convergence result proved by BarlesSouganidis (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.

A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians

“…Their proof is based on the dynamical approach, inspired by the papers [7,12], together with a new representation formula for the solutions. See [19] for a PDE approach inspired by [2]. We also refer to [16] for a related work on homogenization of weakly coupled systems of Hamilton-Jacobi equations with fast switching rates.…”

Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations

“…In [6], motivated by earlier publications [2,7,11], the authors extended the notion of viscosity solution to these problems and proved that their value functions are viscosity solutions of a weakly coupled system of Hamilton-Jacobi equations. Recently, several authors have investigated random switching problems, their weakly coupled Hamilton-Jacobi equations [19], the corresponding extensions of the weak KAM and Aubry-Mather theories [10,17], the long-time behavior of solutions [3,4,18,20,22], and homogenization questions [21]. In these references, as in the present paper, the state of the systems has different modes.…”

Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems

“…In this framework, the gradient bounds are not a difficult step but the proof of the convergence is more delicate since one does not have any strong maximum principle. Such kind of results were extended to systems of Hamilton-Jacobi equations in [9,27,29,26]. For second order nonlinear equations, the asymptotics results of [7] were recently generalized in [22] to some superlinear degenerate equations which are totally degenerate on some subset Σ of T N and uniformly parabolic outside Σ using the gradient bound of Theorem 2.6 and some strong maximum principle type ideas.…”