Geodesic
Lyapunov stability of an equilibrium of the nonlocal continuity equation
Sbornik. Mathematics, Tome 216 (2025) no. 2, pp. 140-167

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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. 
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Yu. V. Averboukh; A. M. Volkov. Lyapunov stability of an equilibrium of the nonlocal continuity equation. Sbornik. Mathematics, Tome 216 (2025) no. 2, pp. 140-167. http://geodesic.mathdoc.fr/item/SM_2025_216_2_a0/