The vanishing discount problem for monotone systems of Hamilton-Jacobi equations: a counterexample to the full convergence

TL;DR

This paper constructs an explicit example of nonlinear monotone Hamilton-Jacobi systems with convex Hamiltonians where solutions do not fully converge as the discount factor approaches zero, challenging previous assumptions.

Contribution

It provides the first known counterexample of non-converging solutions for monotone systems with convex Hamiltonians in the vanishing discount problem.

Findings

Full convergence fails in the constructed example

Counterexample applies to monotone systems with convex Hamiltonians

Challenges previous beliefs about convergence in such systems

Abstract

In recent years there has been intense interest in the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto has recently given an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give here an explicit example of nonlinear monotone systems of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the full convergence of the solutions fails as the discount factor goes to zero.