The Selection problem for discounted Hamilton-Jacobi equations: some non-convex cases

TL;DR

This paper investigates the selection problem for the vanishing discount approximation in non-convex Hamilton-Jacobi equations, extending understanding beyond the well-studied convex case through new methods and specific case analyses.

Contribution

It introduces new techniques to analyze the selection problem for non-convex Hamiltonians, including generalized equations and transformations, and addresses cases not covered by existing theory.

Findings

Established convergence for certain non-convex Hamiltonians with flat parts.

Applied exponential transformation to handle quasi-convex and some non-convex cases.

Extended the selection problem analysis beyond convex Hamiltonians.

Abstract

Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton-Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton-Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton-Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not directly apply. In this case, we establish the convergence by a direct approach.