A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton--Jacobi equations

TL;DR

This paper studies the long-term behavior of solutions to weakly coupled Hamilton--Jacobi systems on the torus, using a dynamical approach to establish convergence to asymptotic solutions under general conditions.

Contribution

It introduces a dynamical method to analyze the large-time behavior of viscosity solutions for general weakly coupled Hamilton--Jacobi systems, extending previous special-case results.

Findings

Proves convergence of solutions to asymptotic states as time approaches infinity.

Establishes the result under broad, general assumptions.

Uses dynamical properties of value functions to analyze PDE solutions.

Abstract

We investigate the large-time behavior of the value functions of the optimal control problems on the $n$-dimensional torus which appear in the dynamic programming for the system whose states are governed by random changes. From the point of view of the study on partial differential equations, it is equivalent to consider viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations. The large-time behavior of viscosity solutions of this problem has been recently studied by the authors and Camilli, Ley, Loreti, and Nguyen for some special cases, independently, but the general cases remain widely open. We establish a convergence result to asymptotic solutions as time goes to infinity under rather general assumptions by using dynamical properties of value functions.