Lyapunov stability of the equilibrium of the non-local continuity equation

TL;DR

This paper develops Lyapunov methods to analyze the stability of equilibrium measures in non-local continuity equations, providing new criteria and applying them to gradient flows and coupled pendulums.

Contribution

It introduces novel Lyapunov-based stability criteria for non-local continuity equations and simplifies stability analysis for linear dynamics using quadratic forms.

Findings

Derived sufficient stability conditions for equilibrium distributions.

Reduced linear stability analysis to quadratic form study.

Applied results to gradient flow and coupled pendulum systems.

Abstract

The paper is concerned with the development of Lyapunov methods for the analysis of equilibrium stability in a dynamical system on the space of probability measures driven by a non-local continuity equation. We derive sufficient conditions of stability of an equilibrium distribution relying on an analysis of a non-smooth Lyapunov function. For the linear dynamics we reduce the stability analysis to a study of a quadratic form on a tangent space to the space of probability measures. These results are illustrated by the studies of the stability of the equilibrium measure for gradient flow in the space of probability measures and Gibbs measure for a system of coupled mathematical pendulums.