Finite-time Stability of Continuous-time Systems with Time-varying Delays∗

LI Yingke,TENG Zhidong,Nerbek Aizmah

(1.College of Mathematics and Physics,Xinjiang Agriculture University,Urumqi xinjiang 830052,China;2.College of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830046,China)

0 Introduction

In the past decades,the concept of Lyapunov asymptotic stability(LS)has been dominating in the field of theory of system stability[1–4].From a practical point of view,however,it is more valuable that the dynamic behavior of the system is realized in finite time rather than infinite time,this is so called finite-time stability(FTS)(or short-time stability)[5].FTS of systems and stabilization of them have been extensively studied recently[6–12].General speaking,given a bound on the initial condition,a system is said FTS if its state trajectories dose not exceed a certain threshold during a pre-proposed time interval[5].

Various results to check FTS as well as to solve the corresponding design problems have been provided for delay systems[5–7],switching and impulsive systems[8],linear system recently[9–12],and stochastic systems[12].Bhat and Bernstein obtained conditions for FTS of an autonomous systems in terms of scalar differential inequalities by using Lyapunov and converse Lyapunov[9].By applying different Lyapunov-Krasovskii functionals,some sufficient conditions for FTS of linear discrete-time with time-varying delay were obtained in the form of inequalities[11].Currently,Liu and Yu[1,2]analyzed continuous time positive systems with time-varying delays and discrete positive systems with bounded time-varying delays,the conditions on system LS are described by Hurwitz matrix or Schur matrix,respectively.The main object of this paper is to show sufficient condition of FTS for positive systems with time-dependent delays.

The paper is organized as follows:model description and some preliminary results are introduced in Section 2.The main results of this paper,namely,sufficient conditions for FTS of time-dependent delays systems are presented in Section 3.In Section 4,the effectiveness and feasibility of the proposed condition are shown by numerical examples.

NotationsFor the given positive integermandn,we denotem0:={0,1,···,m},={1,2,···,n}.R+=[0,∞),Rnrepresents then−dimensional space with the normstands for the set of realm×n−matrices.Foru=(ui),v=(vi)inRn,u≤vif and only ifui≤vi;u?vif and only ifLete=(1,1,···,1)T∈Rn.

1 Problem statement and Preliminaries

Let us consider the following time-varying delays system:

wherex(t)∈Rnis the state variable,are system matrices.Delay τl(t)is assumed to be bounded and continuous with respect tot,and satisfies

(t)=(ϕi(t))∈C([−τ,0],Rn)is the initial function,whereC([−τ,0],Rn)denotes the Banach space of all continuous functions mapping the interval[−τ,0]intoWe usex(t)to denote the

solutionx(t,ϕ)in the following if it does not make any confusion.

Definition 1For a given timeT>0 and positive numbersd1

We make the following preparation for introducing Metzler matrix.For any matrixM∈Rn×n,the spectrum ofMis denoted by σ(M)={λ:det(λIn−M)=0}and the spectral abscissa ofMis denoted byr(M)=max{Reλ:λ ∈σ(M)}.An×n−matrix is called Metzler matrix if all of f-diagonal elements ofMare non-negative.We quote here some properties of Metzler matrix[5].

Lemma 1[5]LetM∈Rn×nbe a Metzler matrix.The following statements are equivalent

(i)r(M)<0;

(ii)M is non-singular andM−1?0;

(iii)There exists 0?ξ∈Rnsuch thatMξ?0;

(iv)for any 0?b∈Rn,there existsx∈Rn+such thatMx+b=0;

(v)There exists 0?ξ∈Rnsuch thatMTξ?0;

(vi)For anyx∈Rn+{0},the row vectorxTMhas at least one negative entry.

We make assumptions on the system matrices in(1)as follows:

Let Mα=where α≥0,there exist Metzler matrixes such that

2 Main results

Theorem 1For givenT>0 andd2>d1>0,if(H1)and(H2)hold,assume there exist α ≥0,λ1>0,λ2>0,e=(1,1,···,1)T∈Rnand ξ∈Rnsatisfying

Then system(1)is FTS with respect to(d1,d2,T).

ProofFor α>0,let ξ∈Rnsatisfied the first inequality in(3),then we have

and hence

Firstly,for anyt≥0 andwe obtain the Dini upper-right derivative along system(1)

Secondly,define the functionsui(t)as follows

Obviously,for allandt≥−τ,we have

Thus,it follows from(4),(6)and(7)that

Therefore,from(8),we have

Next,we will prove that|xi(t)|≤ui(t),for anyt∈[0,T].Let ηi(t)=|xi(t)|−ui(t),t≥ −τ.For allt∈[−τ,0],we havewhich is that ηi(t)≤0 for allt∈[−τ,0]Assume that there exists ani0∈nandt0∈(0,T]such that ηi0(t0)=0,ηi0(t)>0,t∈(t0,t0+θ)for some θ>0 and ηi(t)≤0 for alltHenceD+ηi0(t0)>0.On the other hand,by(5)and(9),for allt∈[0,t0),we get

henceD+ηi0(t0)≤0,which is a contradiction.This means that ηi(t)≤0 for allt>We easily obtainTherefore,for allt∈[0,T],if kϕk∞≤d1,

According to Definition 1,system(1)is FTS with respect to(d1,d2,T).This completes the proof.

Remark 1From Lemma 1 and the first inequality in(3),we ascertain thatMαis a set of Metzler matrix.

Remark 2It is easily confirmed that,if the first inequality in(3)is satisfied with α=0,then system(1)is exponentially stable in the sense of Lyapunov[3,5].Furthermore,in this case system(1)is FTS with respect to(d1,d2,T)for any 0

3 A numerical example

Let us consider the following system

where

and τ1(t)=|sint|,τ2(t)=|cost|.We easily obtain that system(10)satis fi ed(H1)and(H2)for allT>0.For simpli fication,let A0=A0,A1=A1,A2=A2.It is noted that system(10)does not satisfy the asymptotically stable condition proposed inLemma4[1]when set λ=(1,1)T.However,Mα=A0−αIsatis fi es the fi rst inequalities in(3)for any α>0 and its solution η=(η1,η2)T∈R2is defined byIf we setd1=0.7,d2=1 and then system(10)is FTS with respect to(d1,d2,T)for any time 0

Fig 1 Time sires of xi(t)in(10)with initial values xi(0)=0.2+0.1k,k=−5,−4,···,5,i=1,2

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