Input-to-state stability theorem and strict Lyapunov functional constructions for time-delay systems on time scales

Abstract

ABSTRACT In this paper, input-to-state stability of nonlinear time-delay systems on time scales is investigated. Due to the advantages of the strict Lyapunov functionals in uncertainty quantification and robustness analysis, one always prefers to construct the strict Lyapunov functionals to analyse stability of time-delay systems. However, it may be not an easy task to do this for some time-delay systems. This paper proposes an input-to-state stability theorem based on a time-scale uniformly asymptotically stable function. The advantage of this theorem is that it is dependent on the non-strict Lyapunov functional, whose time-scale derivative can be non-negative on some time intervals. Then, some approaches are established to construct the strict Lyapunov functionals based on the non-strict ones. It is shown that input-to-state stability theorems can be also formulated in terms of these strict Lyapunov functionals. Finally, to illustrate the effectiveness of the main results, an example is given.