Feedback stabilization and Lyapunov functions

Given a locally defined, nondifferentiable but Lipschitz Lyapunov function, we employ it in order to construct a ( discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A converse result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish that the feedback in question possesses a robustness property relative to measurement error, despite the fact that it may not be continuous.