Input-to-state stability for time-varying delayed systems in Halanay-type inequality forms

This paper studies the input-to-state stability (ISS) for time-varying delayed systems (TVDS) in Halanay-type inequality forms. The time-delay in TVDS is allowed to be time-varying and unbounded. By introducing the notion of a uniform M-matrix, exponential ISS theorems are established respectively for continuous-time, discrete-time, and zero-order TVDS. The convergence rates of exponential ISS and ISS gains and their relation are subsequently estimated. These ISS theorems are less conservative and generalize the results of stability and ISS for Halanay-type inequalities in the literature. Moreover, necessary conditions of ISS are given for TVDS in Halanay-type equality forms. By specializing the ISS results to linear time-invariant delayed systems, the necessary and sufficient conditions of ISS are derived respectively. Three examples are given throughout the paper to illustrate the theoretical results. (C) 2022 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Automated summary

This paper investigates the input-to-state stability (ISS) of time-varying delayed systems (TVDS) in Halanay-type inequality forms. By introducing the concept of a uniform M-matrix, exponential ISS theorems are established for continuous-time, discrete-time, and zero-order TVDS. The convergence rates of exponential ISS, ISS gains, and their relationship are estimated. These ISS theorems are less conservative and extend the existing results on stability and ISS for Halanay-type inequalities. Furthermore, necessary conditions for ISS of TVDS in Halanay-type equality forms are given, and the necessary and sufficient conditions for ISS are derived for linear time-invariant delayed systems. Three examples are provided to illustrate the theoretical findings.