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

Abstract

This paper treats the concept of practical uniform $h$-stability for such perturbed dynamical systems as an extension of practical uniform exponential stability. We present a converse Lyapunov theorem and we give sufficient conditions that guarantee practical uniform $h$-stability for a time-varying perturbed system using the Gronwall-Bellman inequality and Lyapunov's theory. Some examples are introduced to illustrate the applicability of the main results.