Discounted Hamilton-Jacobi Equations on Networks and Asymptotic Analysis

TL;DR

This paper investigates discounted Hamilton-Jacobi equations on networks, establishing comparison principles, solution representations, and analyzing the asymptotic behavior of solutions and special vertex sets as the discount factor approaches zero.

Contribution

It extends the analysis of Hamilton-Jacobi equations to networks without geometric restrictions, introducing a novel approach linking differential problems to discrete functional equations.

Findings

Comparison principle established for solutions.

Representation formulas for solutions derived.

Asymptotic behavior of solutions and Aubry sets analyzed as discount factor vanishes.

Abstract

We study discounted Hamilton Jacobi equations on networks, without putting any restriction on their geometry. Assuming the Hamiltonians continuous and coercive, we establish a comparison principle and provide representation formulae for solutions. We follow the approach introduced in 11, namely we associate to the differential problem on the network, a discrete functional equation on an abstract underlying graph. We perform some qualitative analysis and single out a distinguished subset of vertices, called lambda Aubry set, which shares some properties of the Aubry set for Eikonal equations on compact manifolds. We finally study the asymptotic behavior of solutions and lambda Aubry sets as the discount factor lambda becomes infinitesimal.