Existence of Lipschitz and Semiconcave Control-Lyapunov Functions

Given a locally Lipschitz control system which is globally asymptotically controllable at the origin, we construct a control-Lyapunov function for the system which is Lipschitz on bounded sets, and we deduce the existence of another one which is semiconcave (and so locally Lipschitz) outside the origin. The proof relies on value functions and nonsmooth calculus.