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

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.