This article is cited in 3 papers

Lyapunov stability of an equilibrium of the nonlocal continuity equation

Yu. V. Averboukh, A. M. Volkov

N. N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

Abstract: The paper is devoted to developing Lyapunov's methods for analyzing the stability of an equilibrium of a dynamical system in the space of probability measures that is defined by a nonlocal continuity equation. Sufficient stability conditions are obtained based on the basis of an analysis of the behaviour of a nonsmooth Lyapunov function in a neighbourhood of the equilibrium and the investigation of a certain quadratic form defined on the tangent space of the space of probability measures. The general results are illustrated by the study of the stability of an equilibrium for a gradient flow in the space of probability measures and the Gibbs measure for a system of connected simple pendulums.
Bibliography: 28 titles.

Keywords: nonlocal continuity equation, Lyapunov's second method, nonsmooth Lyapunov function, stability, derivatives in the space of measures.

MSC: Primary 34D20; Secondary 35B35, 35F20, 35Q70, 35R06, 82C22

Received: 15.02.2024 and 09.10.2024

DOI: 10.4213/sm10084

 English version: Sbornik: Mathematics, 2025, 216:2, 140–167

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