Smooth Lyapunov functions for discontinuous stable systems (Q1570039)

This paper is devoted to a deep study of the existence of Lyapunov functions for stable, time-varying systems and their regularity. Since the author is interested in applications to systems with discontinuous right-hand side, the main results are formulated in terms of differential inclusions. First of all, it is proved that in the time-varying case, even if the origin is globally asymptotically stable a \(C^1\) Lyapunov function need not to exist. Then, the existence of a \(C^\infty\) Lyapunov function is proved for robustly stable differential inclusions which satisfy certain assumptions. These assumptions are actually fulfilled by large classes of differential inclusions associated to discontinuous systems. The existence of smooth Lyapunov functions is also related to robustness with respect to measurement errors.