A Local Characterization of Lyapunov Functions and Robust Stability of Perturbed Systems on Riemannian Manifolds

TL;DR

This paper extends Lyapunov stability theory to nonlinear dynamical systems on Riemannian manifolds, providing local characterizations and conditions for robustness under perturbations using differential geometry tools.

Contribution

It introduces Converse Lyapunov Theorems for systems on Riemannian manifolds and characterizes Lyapunov functions in normal neighborhoods, advancing stability analysis in geometric settings.

Findings

Lyapunov functions can be constructed on Riemannian manifolds using differential geometry.

Stability of perturbed systems is established via properties of Lyapunov functions.

Normal neighborhoods and injectivity radius are key in local stability analysis.

Abstract

This paper proposes several Converse Lyapunov Theorems for nonlinear dynamical systems defined on smooth connected Riemannian manifolds and characterizes properties of corresponding Lyapunov functions in a normal neighborhood of an equilibrium. We extend the methods of constructing of Lyapunov functions for ordinary differential equations on $\mathds{R}^{n}$ to dynamical systems defined on Riemannian manifolds by employing the differential geometry. By employing the derived properties of Lyapunov functions, we obtained the stability of perturbed dynamical systems on Riemannian manifolds. The results are obtained by employing the notions of normal neighborhoods, the injectivity radius on Riemannian manifolds and existence of bump functions on manifolds.