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, timevarying) Liapunov function.

ℎ-Stability of set differential equations

“…Moreover, some theorems prove, at least conceptually with respect to several Lyapunov stability theorems, that the conditions given are indeed necessary. These theorems are generally known as converse theorems (see [10,16,22,35]). Converse theorems are generally the most difficult part of the theory and the first overall outcomes for nonlinear systems were achieved by Massera [28] and Kurzweil [19].…”

converse theorem for practical $h$-stability of time-varying nonlinear systems

“…It gives sucient conditions for stability, asymptotic stability, and so on. There are theorems which establish ([5, 17,22]), at least conceptually, that for many of Lyapunov stability theorems the given conditions are indeed necessary. Such theorems are usually called converse theorems.…”