The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling

Hitoshi Ishii

doi:10.3934/mine.2021032

Abstract

We establish a convergence theorem for the vanishing discount problem for a weakly coupled system of Hamilton-Jacobi equations. The crucial step is the introduction of Mather measures and their relatives for the system, which we call respectively viscosity Mather and Green-Poisson measures. This is done by the convex duality and the duality between the space of continuous functions on a compact set and the space of Borel measures on it. This is part 1 of our study of the vanishing discount problem for systems, which focuses on the linear coupling, while part 2 will be concerned with nonlinear coupling.

Highlights

We consider the weakly coupled m-system of Hamilton-Jacobi equations (Pλ) λvλ + Bvλ + H[vλ] = 0 in Tn, where m ∈ N, λ is a nonnegative constant, called the discount factor in terms of optimal control
Tn denotes the n-dimensional flat torus, H = (Hi)i∈I is a family of Hamiltonians given by (H)
The unknown in (Pλ) is an Rm-valued function vλ =i∈I on Tn, B : C(Tn)m → C(Tn)m is a linear map represented by a matrix B =i,j∈I ∈ C(Tn)m×m, that is, (Bu)i(x) = (B(x)u(x))i := bij(x)uj(x) for (x, i) ∈ Tn × I

Summary

Introduction

This paper is part 1 of our study of the vanishing discount problem for weakly coupled systems of Hamilton-Jacobi equations and deals only with the linear coupling and with compact control sets Ξ. These restrictions make the presentation of our results clear and transparent. (See (6) below.) According to [22, Theorems 3.3, Lemma 4.8], there is a function vλ = (viλ)i∈I : Tn → Rm such that the upper and lower semicontinuous envelopes (vλ)∗ and v∗λ are a subsolution and a supersolution of (Pλ), respectively. We study the ergodic problem (P0) in the cases when B is irreducible, and B is a constant matrix, respectively, in Sections 5 and 6, and combine the results with the analysis on the vanishing discount problem of Section 4

Monotone matrices

Viscosity Green-Poisson measures

A convergence result for the vanishing discount problem

The ergodic problem for irreducible matrix B

The ergodic problem for constant matrix B