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

Abstract: We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume $C<sup>3$ 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.

• The paper establishes explicit, verifiable PDE criteria for Lyapunov and asymptotic stability of contact-type Hamilton-Jacobi equations using viscosity solution theory.
• It replaces traditional dynamical methods based on Mather measures with a continuous, convex, and coercive Hamiltonian framework, ensuring rigorous stability and instability conditions.
• The framework guarantees uniqueness of stationary solutions and quantifies convergence rates in homogenization, paving the way for practical stability verifications in low-regularity settings.

PDE-Based Lyapunov Stability for Contact-Type Hamilton–Jacobi Equations

Overview

This paper introduces a partial differential equation (PDE) framework for analyzing Lyapunov stability of stationary solutions to contact-type Hamilton–Jacobi equations on compact manifolds. Unlike earlier works that rely on Tonelli Hamiltonians and the dynamical characterization via Mather measures, the authors employ continuous, convex, and coercive Hamiltonians, providing stability and instability conditions directly amenable to PDE techniques. The main achievements are the derivation of explicit, verifiable PDE-type criteria for Lyapunov (and asymptotic) stability and instability, the establishment of uniqueness results, and the quantification of convergence rates in homogenization contexts.

Problem Formulation and Setting

The core equation studied is a contact-type Hamilton–Jacobi PDE: $$u_t(x,t) + H(x, Du(x, t), u(x, t)) = 0 \quad \text{on} \quad M \times (0, +\infty),$$ with $M$ a compact smooth manifold and $H$ a continuous, convex, and coercive Hamiltonian of the form $H(x, p, u) = G(x, p) + W(x, u)$. Stationary solutions ($u_-$) satisfy the critical equation

$$H(x, Du(x), u(x)) = 0 \quad \text{for} \quad x \in M.$$

The analysis uses viscosity solution theory, replacing the dynamical Mather measure framework with PDE-based quantities: critical values of Hamiltonians and viscosity subsolutions.

PDE Criteria for Lyapunov Stability and Instability

The main results are a series of equivalence theorems and stability characterizations, founded on critical value perturbations and differential inequalities:

• Lyapunov Stability: Stability is established under conditions such as $$c(H(x, p, u_-(x)) - \zeta \partial_u W(x, u_-(x))) < 0$$, for some $\zeta > 0$, with $c(\cdot)$ denoting the critical value. The authors prove equivalence between this condition, the existence of an explicit subsolution violating the perturbed inequality everywhere, and a positive left derivative of the critical value with respect to perturbation. Asymptotic stability is shown via exponential decay of perturbations, with rate constants determined by the PDE coefficients.
• Lyapunov Instability: Instability is characterized with similar PDE criteria, but with the perturbation inverted ($+\zeta$ term), and right derivative of $M$0 being negative. These provide sharp thresholds where stability is lost.
• Uniqueness and Global Stability: Under strict convexity and superlinearity of $M$1, and corresponding PDE inequalities for all subsolutions, uniqueness of stationary solutions is proven, and global asymptotic stability holds for any initial data.

These criteria are rigorous, verifiable, and do not invoke smoothness, Mather measures, or dynamical flows—making them suitable for lower-regularity settings or applications with non-monotonic Hamiltonian dependence. Notably, the analysis covers systems where classical monotonicity fails, as in models for dislocation dynamics.

Connections with Aubry–Mather Theory and Applications

A key theoretical advance is the integration of PDE-based critical value analysis with concepts from Aubry–Mather theory. The authors demonstrate direct links between their Lyapunov-type inequalities and the behavior of viscosity solutions in asymptotic regimes (large time behavior, vanishing discount limits, homogenization):

• Vanishing Discount Limit: The PDE conditions here are closely related to convergence results in discounted Hamilton–Jacobi equations, previously characterized by Mather measure integrals. The paper offers new PDE-type criteria, facilitating analysis of uniqueness and existence in these limits.
• Homogenization: Quantitative convergence rates for nonlinear multiscale homogenization problems are established. Explicit bounds of order $M$2 are derived for solutions to periodically homogenized equations, generalizing earlier results that handled only convergence without rates. The approach combines nonlinear semigroup methods and PDE techniques.
• Aubry Set and Uniqueness: The uniqueness result is extended to equations with coefficients dominating the projected Aubry set. The authors develop a PDE proof, independent of previous comparison-principle-based arguments, further underscoring the theoretical utility of their framework.

Numerical Results and Explicit Examples

Sharp numerical estimates are presented where possible. For instance, exponential rates are specified for asymptotic stability, and explicit examples are given for non-monotone equations to illustrate the checkability of the PDE-type conditions. In homogenization, convergence rates for the effective equation are quantified as $M$3.

Implications and Future Directions

Practically, the work provides a robust and accessible set of PDE-based tools for the stability analysis of contact-type Hamilton–Jacobi equations, applicable in settings where dynamical and measure-theoretic methods are obstructed by low regularity or non-monotonicity. The theoretical implications extend to a deeper understanding of the connections between viscosity solutions, critical values, and asymptotic dynamical behavior—especially in the context of Aubry–Mather theory.

Future research may expand these PDE-based techniques to more general classes of Hamiltonians (beyond additive or convex structures), to weakly coupled systems, and to higher-order or stochastic Hamilton–Jacobi equations. Further refinement of convergence rates in homogenization problems, potential extension to noncompact manifolds, and characterization of stability phenomena in systems with state-dependent switching are suggested directions. The development of computational algorithms leveraging the PDE criteria for practical stability verification is also anticipated.

Conclusion

The paper provides a rigorous PDE formulation for Lyapunov stability of contact-type Hamilton–Jacobi equations, replacing dynamical system-based assumptions with explicit, verifiable PDE conditions. It achieves equivalence theorems for stability and instability, proves uniqueness and global stability, and quantifies convergence rates for nonlinear homogenization. It forges new connections with Aubry–Mather theory and establishes a foundation for further theoretical and practical advancements in PDE-based stability analysis for broad classes of Hamilton–Jacobi-type systems (2604.24329).