Remarks on Control Lyapunov Functions for Discontinuous Stabilizing Feedback

We present a formula for a stabilizing feedback law under the assumption that a piecewise smooth controlLyapunov function exists. The resulting feedback is continuous at the origin and smooth everywhere except on a hypersurface of codimension 1. We provide an explicit and "universal" formula. Finally, we mention a general result connecting asymptotic controllability and the existence of control-Lyapunov functions in the sense of nonsmooth optimization. 1. Introduction In this work we want to focus on nonsmooth Lyapunov functions that can be obtained by "pasting together" smooth ones (see [6] for other approaches). For these functions the verification of the Lyapunov property can be carried out using gradients, as in the smooth case. A result by Artstein [1] guarantees the existence of a globally stabilizing feedback law provided the system has a smooth control Lyapunov function. Sontag [5] gave a constructive proof providing a formula for the feedback law in terms of Lie derivatives ..