On stability analysis of nonlinear time-delay systems on time scales

Abstract

This paper is concerned with the stability problems of a more general class of nonlinear time-delay systems, called time-scale time-delay systems, which can include not only the traditional continuous and discrete ones, but also some other cases, such as systems on hybrid time domains. As far as we know, the only existing approach for analyzing stability of such systems is the Razumikhin-type stability theorem. To enrich the stability results for such systems, this paper proposes the Lyapunov–Krasovskii theorem. Firstly, a conservative stability criterion is provided by requiring the time-scale time derivative of relating Lyapunov functional to be negative. Then, by introducing the time-scale type uniformly stable function and uniformly asymptotically stable function, a more relaxed stability theorem is proposed, in which the time-scale time derivative of Lyapunov functional is allowed to be non-negative. A numerical example about non-continuous and non-discrete time-delay systems is given to illustrate the effectiveness and general applicability of the theoretic results.