Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample

TL;DR

This paper presents a counterexample demonstrating that solutions of a discounted Hamilton-Jacobi equation do not necessarily converge as the discount factor approaches zero, challenging previous assumptions about their asymptotic behavior.

Contribution

It provides the first known counterexample showing non-convergence of solutions in a 1D continuous coercive Hamiltonian setting.

Findings

Solutions do not always converge as discount vanishes

Counterexample applies to 1D continuous coercive Hamiltonians

Challenges existing beliefs about asymptotic behavior

Abstract

This paper provides a counterexample about the asymptotic behavior of the solutions of a discounted Hamilton-Jacobi equation, as the discount factor vanishes. The Hamiltonian of the equation is a 1-dimensional continuous and coercive Hamiltonian.