Uniform asymptotic stability and regions of attraction for systems on Riemannian manifolds

Abstract

This paper studies uniform asymptotic stability of equilibrium points of ordinary differential equations on complete Riemannian manifolds and estimates of the corresponding regions of attraction. We first refine the classical Lyapunov stability theorem in Euclidean space and obtain a sharper estimate of the region of attraction. We then extend this result to systems on complete Riemannian manifolds and establish sufficient conditions for uniform asymptotic stability together with an explicit estimate of the corresponding region of attraction. The resulting estimate is constrained by both the injectivity radius at the equilibrium point and the Lyapunov comparison functions. Compared with a previous result, the obtained theorem removes the monotonicity assumption on the squared distance function along system trajectories. An example on the two-dimensional hyperbolic space illustrates the result and shows that this monotonicity condition may fail.