Weak Converse Lyapunov Theorems and Control-Lyapunov Functions

Abstract

Given a weakly uniformly globally asymptotically stable closed (not necessarily compact) set ${\cal A}$ for a differential inclusion that is defined on $\mathbb{R}^n$, is locally Lipschitz on $\mathbb{R}^n \backslash {\cal A}$, and satisfies other basic conditions, we construct a weak Lyapunov function that is locally Lipschitz on $\mathbb{R}^n$. Using this result, we show that uniform global asymptotic controllability to a closed (not necessarily compact) set for a locally Lipschitz nonlinear control system implies the existence of a locally Lipschitz control-Lyapunov function, and from this control-Lyapunov function we construct a feedback that is robust to measurement noise.