Smooth Lyapunov Functions for Homogeneous Differential Inclusions

Abstract

This paper provides a construction method of a smooth homogeneous Lyapunov function associated with a discontinuous homogeneous system, which is locally asymptotically stable. First, we analyze two similar converse Lyapunov theorems for differential inclusions and unify them into a simple theorem. Next, we propose a new definition of homogeneous differential inclusion. Then, we construct a smooth homogeneous Lyapunov function associated with the homogeneous differential inclusion. Lastly, we show that the order of homogeneity of a homogeneous system indicates the speed of convergence.