Input-to-state exponents and related ISS for delayed discrete-time systems with application to impulsive effects

Abstract

This paper aims to investigate the input-to-state exponents (IS-e) and the related input-to-state stability (ISS) for delayed discrete-time systems (DDSs). By using the method of variation of parameters and introducing notions of uniform and weak uniform M-matrix, the estimates for 3 kinds of IS-e are derived for time-varying DDSs. The exponential ISS conditions with parts suitable for infinite delays are thus established, by which the difference from the time-invariant case is shown. The exponential stability of a time-varying DDS with zero external input cannot guarantee its ISS. Moreover, based on the IS-e estimates for DDSs, the exponential ISS under events criteria for DDSs with impulsive effects are obtained. The results are then applied in 1 example to test synchronization in the sense of ISS for a delayed discrete-time network, where the impulsive control is designed to stabilize such an asynchronous network to the synchronization.