A PDE formulation of Lyapunov stability for contact-type Hamilton-Jacobi equations

TL;DR

This paper develops a PDE-based framework to analyze the Lyapunov stability of solutions to contact-type Hamilton-Jacobi equations, replacing measure-based conditions with criteria involving the Hamiltonian's critical value and viscosity subsolutions.

Contribution

It introduces verifiable PDE criteria for stability and instability of Hamilton-Jacobi equations with continuous, convex, coercive Hamiltonians, extending previous measure-based approaches.

Findings

Established PDE criteria for stability and instability.

Replaced Mather measure conditions with Hamiltonian critical value criteria.

Connected stability analysis with asymptotic behaviors of viscosity solutions.

Abstract

We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume $C^3$ Tonelli Hamiltonians and characterize stability in terms of Mather measures. In this paper, we consider continuous, convex and coercive Hamiltonians and establish verifiable PDE-type criteria for both stability and instability. In particular, the dynamical conditions involving Mather measures are replaced by conditions expressed in terms of the critical value of the Hamiltonian and viscosity subsolutions. This provides a PDE-based framework for stability analysis and reveals connections with various asymptotic behaviors of viscosity solutions.