Smooth Approximation of Patchy Lyapunov Functions for Switched Systems

Starting with a locally Lipschitz (patchy) Lyapunov function for a given switched system, we provide the construction of a continuously differentiable (smooth) Lyapunov function, obtained via a convolution-based approach. This smooth function approximates the patchy function when working with Clarke’s generalized gradient. The convergence rate inherited by the smooth approximations, as a by-product of our construction, is useful in establishing the robustness with respect to additive inputs. With the help of an example, we address the limitations of our approach for other notions of directional derivatives, which generally provide less conservative conditions for stability of switched systems than the conditions based on Clarke's generalized gradient.