Elsevier

Systems & Control Letters

Volume 107, September 2017, Pages 22-27
Systems & Control Letters

Razumikhin-type theorems for time-delay systems with Persistent impulses

https://doi.org/10.1016/j.sysconle.2017.06.007Get rights and content

Abstract

The main results of the paper are generalizations of the Razumikhin classical theorems for stability analysis of impulsive time-delay systems. By employing Lyapunov–Razumikhin method and impulsive control theory, several Razumikhin-type theorems for uniform stability and global exponential stability are obtained. The significance and novelty of the results lie in that the criteria for stability of impulsive time-delay systems admit the existence of persistent impulses and unbounded time-varying delays. Two examples are provided to show the effectiveness of the proposed results.

Introduction

The stability problem of impulsive time-delay systems has been investigated intensively during the past decades due to their potential applications in many fields such as electronics, medicine, economics, biology, and engineering, see [1], [2], [3], [4], [5] and the references therein. The earlier works for stability problems of impulsive delay systems were done by Anokhin [6] and Gopalsamy and Zhang [7]. Now there are many different methods such as LMI tools, Laplace transform, inequality technique and Lyapunov functions (or Lyapunov functionals) combined with the Razumikhin technique to deal with the stability problem of such systems, see  [8], [9], [10], [11], [12], [13], for recent works.

On the other hand, from the viewpoint of impulsive effects, the investigation of stability problem for impulsive time-delay systems can be generally classified into two groups: impulsive stabilization and impulsive perturbation. In the case where a given system without impulses is unstable or stable and can be turned into uniformly stable, uniformly asymptotically stable, and even exponentially stable under proper impulsive control, it is regarded as impulsive stabilization problem. Now it has been shown that impulsive stabilization problem can be widely applied to many fields such as orbital transfer of satellite, dosage supply in pharmacokinetics, ecosystems management, and synchronization in chaotic secure communication systems [11], [12], [13], [14], [15]. Alternatively, in the case where a given system without impulses is stable and can remain stable under certain impulsive interference, it is regarded as impulsive perturbation problem. To date, many interesting results dealing with impulsive perturbations of time-delay systems have been reported, see [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. However, one may observe that most of the existing results on impulsive perturbations, such as those in [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], are usually based on the assumption that there just exist slight impulsive perturbations at infinity. In other words, in the sense of Lyapunov it always holds that V(tn)V(tn) or V(tn)(1+βn)V(tn) with βn< (it implies that βn0 as n). However, in reality many systems are usually affected by impulsive perturbations which may be caused by external or internal factors and in many cases it is of great intermittence and persistence. For example, in electronic networks, persistent impulsive perturbations on neurons may be unavoidable due to the intermittent switching, frequency change or other sudden noise [26]. In the process of partial discharge on line Monitoring of high-voltage transformer, periodic impulsive perturbations which can be regarded as a kind of persistent impulsive perturbations will occur unavoidably due to the frequent switching of Silicon Controlled Rectifier trigger and the influence of unknown factors in ground networks [27]. Thus the assumption on impulses in Refs. [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] restricts the applications of those stability results to real problems and cannot reflect a more realistic dynamics. Hence, it is significant and important to consider persistent impulsive perturbations to the stability problem of impulsive time-delay systems.

Motivated by the above discussion, in this paper, we shall investigate the stability problem of time-delay systems with persistent impulses. Since in many practical problems time delays are time-varying and sometimes may depend on the information of history deeply, we mainly focus on the discussion of systems with unbounded time-varying delays. To overcome the difficulties created by the special features possessed by persistent impulses and unbounded time-varying delays, as we will see, a more complicated analysis is required. This paper is organized as follows. In Section 2, we introduce some preliminary knowledge. In Section 3, several Razumikhin theorems are presented. In Section 4, two numerical examples are provided and a conclusion is finally given in Section 5.

Section snippets

Preliminaries

Notations

Let R denote the set of real numbers, R+ the set of positive real numbers, Z+ the set of positive integers and Rn the n-dimensional real space equipped with the Euclidean norm ||. The impulse times tk satisfy 0t0<t1<<tk as k. For any interval JR, set SRk(1kn),C(J,S)={φ:JSis continuous} and PC(J,S)={φ:JSis continuous everywhere except at finite number of pointst,at whichφ(t+),φ(t)exist andφ(t+)=φ(t)}. K={aC(R+,R+)|a(0)=0anda(s)>0fors>0andais

Razumikhin theorems

Here we present our main stability results.

Theorem 1

System (1)- (2)is uniformly stable if there exist functions w1,w2K,cC(R+,R+),pPC(R+,R+), V(t,x)ν0, and constants μ1,TμZ+,βk0,kZ+, such that

    (i)

    w1(|x|)V(t,x)w2(|x|),(t,x)[σα,)×Rn;

    (ii)

    D+V(t,ψ(0))p(t)c(V(t,ψ(0))) for every ψPC([α,0],Rn)  if V(t+s,ψ(s))μV(t,ψ(0)), s[α,0],tσ,ttk;

Examples

In this section, we give two examples to illustrate the results obtained in the previous sections.

Example 1

Consider a linear time-delay system ẋ(t)=a(t)x(t)+b(t)x(th(t))+u,t>0,x(s)=ϕ(s),s[α,0]with the control input u=δ(ttn)Knx(t),nZ+,where δ is Dirac Delta function, a,hC([0,),R+), bC([0,),R), and Kn>0 are some constants.

It is easy to check that system (12) has an equivalent form (see [33]): ẋ(t)=a(t)x(t)+b(t)tanh(x(th(t))),ttn,x(tn)=(1+Kn)x(tn),nZ+,x(s)=ϕ(s),s[α,0].

Conclusion

In this paper, we derived several stability theorems for time-delay systems with persistent impulses and unbounded time-varying delays by employing Lyapunov–Razumikhin method and impulsive control theory. Two numerical examples were given to illustrate the effectiveness of the results obtained. Our main idea is to fetch the information of persistent impulses and then integrate it into the constraint on Razumikhin condition. However, our results apply only when the time-derivative of Lyapunov

References (35)

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This work was supported by National Natural Science Foundation of China (11301308, 61673247), and the Research Fund for Distinguished Young Scholars and Excellent Young Scholars of Shandong Province (ZR201702100145, ZR2016JL024). The paper has not been presented at any conference.

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