A Converse Liapunov Theorem for Uniformly Locally Exponentially Stable Systems Admitting Carathéodory Solutions

Abstract

This paper provides a converse Liapunov theorem for uniformly locally exponentially stable, locally Lipschitz, non-linear, time-varying, possibly non-smooth systems that admit Carathéodory solutions. The main result proves that a critical point of such a system is uniformly locally exponentially stable if and only if the system admits a local (possibly non-smooth, time-varying) Liapunov function.