Weak converse Lyapunov theorems and control-Lyapunov functions

Given a weakly uniformly globally asymptotically stable closed (not necessarily compact) set A for a differential inclusion that is defined on R-n, is locally Lipschitz on R-n\A, and satisfies other basic conditions, we construct a weak Lyapunov function that is locally Lipschitz on 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.