Lyapunov and converse Lyapunov theorems for semistability

This paper develops Lyapunov and converse Lyapunov theorems for semistable nonlinear dynamical systems. Semistability is the property whereby the solutions of a dynamical system converge to (not necessarily isolated) Lyapunov stable equilibrium points determined by the system initial conditions. Specifically, we provide necessary and sufficient conditions for semistability and show that semistability implies the existence of a smooth Lyapunov function that decreases along the dynamical system trajectories such that the Lyapunov function satisfies inequalities involving the distance to the set of equilibria.