Lyapunov Stability for nonautonomous systems on Manifolds
Li Deng, Xin Li
TL;DR
This work extends Lyapunov stability theory to nonautonomous systems on complete Riemannian manifolds, establishing Lyapunov-type theorems and providing explicit domain-of-attraction estimates tied to the equilibrium’s injectivity radius and the manifold’s curvature. The core method transforms the dynamics to the tangent space at the equilibrium via the exponential map, enabling the application of Euclidean Lyapunov results and then transferring conclusions back to the manifold. The paper also demonstrates that the domain of attraction and stability properties depend on the chosen Riemannian metric and offers refined, metric-aware bounds that improve upon classical Euclidean results; it further provides sufficient conditions for exponential stability. Practical implications include more precise safety and robustness guarantees for mechanical and geometric control systems described on manifolds, where curvature and injectivity radius naturally constrain stability regions.
Abstract
This paper studies the uniformly asymptotic stability of nonautonomous systems on Riemannian manifolds. We establish corresponding Lyapunov-type theorems (Theorems 2.1 and 2.2), extending classical Euclidean results (e.g., [9, Theorems 4.9 and 4.10]) to curved spaces. Our main contributions are: (i) an estimate for the domain of attraction linked to the equilibrium point's injectivity radius, where, under suitable conditions, this radius can be bounded using the sectional curvature (Proposition 2.1); (ii) a demonstration that this estimate depends on the choice of the Riemannian metric (Examples 2.1 and 2.2 and Remark 2.4); and (iii) a refined estimate compared to the Euclidean case, as detailed in item (6) of Remark 2.1 and in Remark 2.3.