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

2024/10/11 by Yurii Aveboukh, Aveboukh, Yurii, A. N. Volkov +1
Engineering · Mathematics · Medicine · #34D20 #35B35 #35F20 #35Q70 #35R06 #82C22 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2410.08913

openalex publication_date 2024/10/11 · openalex created_date 2024/10/16 · openalex updated_date 2026/07/28

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.

Related

• Stabilization of solutions of the controlled non-local continuity equation
• On forward invariance in Lyapunov stability theorem for local stability
• Intrinsic Construction of Lyapunov Functions on Riemannian Manifold
• Lyapunov functions obtained from first order approximations
• Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes