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Stability of impulsive delayed linear differential systemswith delayed impulses

Huamin Wanga,b, Shukai Duana,n, Chuandong Lia, Lidan Wanga,Tingwen Huangc

aCollege of Electronic and Information Engineering, Southwest University, Chongqing 400715, PR ChinabDepartment of Mathematics, Luoyang Normal University, Luoyang, Henan 471022, PR China

cDepartment of Electrical and Computer Engineering, Texas A&M University, Doha, PO Box 23874, Qatar

Received 17 April 2014; received in revised form 2 December 2014; accepted 17 December 2014

Abstract

In this paper, uniform stability problem of impulsive delayed differential systems with delayed impulsesis investigated. By means of the Lyapunov function method combined with Razumikhin technique andclassic analysis methods, the criteria of uniform stability, uniform asymptotic stability and exponentialstability for impulsive delay systems with delayed impulses are obtained. The numerical examples are givento illustrate the effectiveness of the proposed conclusions.& 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Over the past several decades, impulsive dynamical systems have attracted increasing interestbecause they provide a natural framework for mathematical modelling in the fields of science andengineering, such as biology, physics, control technology, industrial robotics, and communica-tion engineering (see, for example, [1–12]). Studying impulsive delayed systems is veryimportant for its inevitable in many practical problems. Stability is one of the most importantissues in the study of impulsive delay differential equations. In recent years, there have beenmany research works in the literature on impulsive delay differential equations (see, for example,

rg/10.1016/j.jfranklin.2014.12.0092015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

ding author.dress: duansk@swu.edu.cn (S. Duan).

this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]2

[13–43]) and many stability criteria for impulsive delay systems have been proposed. However,most research results on impulsive delay equations do not consider delayed impulses, which aremainly due to some theoretical difficulties in this kind of systems.Lyapunov–Razumikhin method is one of the most useful ways in dealing with the stability of

impulsive delayed differential systems, see [12–30] and the references cited therein. Recently, makinguse of Lyapunov functional and analysis technique, the stability issue has been studied in [12–28,32–34] for some impulsive control systems with time delay. Several criteria on uniformly stability,uniformly asymptotic stability and exponential stability were established, and those results show that asystem can be stabilized by impulses even if it contains an unstable system matrix. In [13,21], theauthors proved the criteria on uniform stability and exponential stability for linear impulsive delayedsystems based on Lyapunov function technique. Liu and Ballinger considered the uniform asymptoticstability of impulsive delay differential equations in [32]. Ho and Sun [14] investigated the uniformstability and the uniform asymptotic stability for the Takagi–Sugeno fuzzy delay systems. A unifiedsynchronization criterion for impulsive dynamical networks was investigated in [28]. In these papers,the impulses were usually assumed to take the form ΔxðtkÞ ¼ xðtþk Þ�xðt�k Þ ¼ Bkxðt�k Þ, whichindicates the state ‘jump’ at the impulse times tk, and there have been few available results on stabilityanalysis of impulsive time-delay systems with delayed impulses [17]. However, in the transmission ofthe impulse information, input delays are often encountered [43]. Based on this fact, it is necessary todeeply study the stability of impulsive delayed systems with delayed impulses. In recent years, bymeans of the fundamental matrix of differential systems and some restrictions on delays, Liu et al. [24]considered the stability problem of some particular classes of impulsive delayed systems with delayedimpulses. Uniform stability for nonlinear systems with delayed impulses was studied in [16]. In [25],Khadra et al. used exponential estimates to investigate the asymptotic stability for delay-freeautonomous impulsive control systems with delayed impulses. Making use of Lyapunov functionmethod combined with Razumikhin technique, Chen et al. [17–19,29,30] studied the exponentialstability for the time-delay nonlinear impulsive differential systems, nonlinear singularly perturbedsystems and Takagi–Sugeno fuzzy systems with delayed impulses. Although several results ofimpulsive delayed systems with delayed impulses have been obtained in [17–19,24–30], somerestrictions must be illustrated to get these results. Moreover, most of these results were discussed onexponential stability. Therefore, the existing research on stability of impulsive delayed systems withdelayed impulses may still be insufficient.In this paper, we will further discuss the stability for the impulsive linear and nonlinear

delayed systems with delayed impulses. By ways of Lyapunov functions, differential inequalitiesand Razumikhin technique, we will obtain the uniform stability criteria, uniform asymptoticstability criteria and exponential stability criteria of delayed systems with delayed impulsesrespectively. Three numerical examples will be given to illustrate the effectiveness of our results.From these examples, it can be seen that proper delayed impulse does not affect the stability, butit may change the convergence speed and process of impulsive system.The remainder of this paper is organized as follows. In Section 2, we introduce some basic

notation and definitions. We establish several stability criteria for linear and nonlinear impulsivedelayed systems with delayed impulses in Section 3. Finally, conclusions are given in Section 4.

2. Preliminaries

The following notations and definitions will be used in this paper. Let R be the set of realnumbers, Rþ the set of nonnegative real numbers and Rn the space of n-dimensional columnvectors x¼ colðx1; x2;…; xnÞ with the Euclidean norm. I is the identity matrix of order n.

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 3

For a, bAR with aob and for SDRn, we define the following classes of functions.PCð½a; b�; SÞ ¼ fϕ : ½a; b�-SjϕðtÞ ¼ ϕðtþÞ; 8 tA ½a; b�;ϕðt� Þ exists in S; 8 tA ½a; b� andϕðt� Þ ¼ ϕðtÞ for all but at most a finite number of points tA ½a; b�g. Let Jτ ¼ PCð½�τ; 0�;RnÞ,for ψAJτ, the norm of ψ is defined by Jψ J ¼ sup� τr sr 0 JψðsÞJ and xtAJτ is defined byxtðsÞ ¼ xðt þ sÞ for sA ½�τ; 0�, where τ40 is a constant. For any α40, let PCðαÞ ¼fϕAJτ : jϕjoαg.

For φAJτ, consider the following impulsive delay system with delayed impulse:

_xðtÞ ¼ AxðtÞ þ f ðt; xðtÞ; xðt�r1ðtÞÞÞ; taτk ; tZ t0;

ΔxðtÞ ¼ Ckxðt� Þ þ gðxðt� Þ; xðt� �r2ðtÞÞÞ; t¼ τk; kANþ;

xt0 ¼ φ;

8><>: ð2:1Þ

where xðtÞARn, A;CkARn�n, f AC½Rþ � Rn � Rn;Rn�, ΔxðtÞ9xðtÞ�xðt� Þ, gAC½Rn � Rn;Rn�,f ðt; 0; 0Þ ¼ 0, gð0; 0Þ ¼ 0, r1ðtÞ; r2ðtÞA ½Rþ;Rþ� are bounded and 0rr2ðtÞoτk�τk�1

o1ðt0 ¼ τ0Þ, τ¼maxfsuptARþr1ðtÞ; suptARþr2ðtÞg. From these conditions we know that thesolution of Eq. (2.1) exists and is unique. Then denote with xðtÞ ¼ xðt; t0;φÞ the solution of Eq.(2.1) such that xt0 ¼ φ.

Definition 2.1 (Yang [3]). The function V : ½t0;1Þ-Rþ belongs to class v0 if the following aretrue:

(i)

Plede

Vðt; xÞ is locally Lipschitzian in xARn and is continuous on each of the sets ½τk�1; τkÞ,Vðt; 0Þ � 0 for all tZ t0;

(ii)

For each k¼ 1; 2;…, there exist finite limits

limðt;yÞ-ðτ�k ;xÞ

Vðt; yÞ ¼ Vðτ�k ; xÞ;

limðt;yÞ-ðτþk ;xÞ

Vðt; yÞ ¼ Vðτþk ; xÞ;

with Vðτþk ; xÞ ¼ Vðτk; xÞ being satisfied.

Definition 2.2 (Yang [3]). Let VAv0 and _xðtÞ ¼ f ðt; xðtÞÞ, for tA ðτk�1; τkÞ, the upper right-handderivative of V is defined by

DþV t; x tð Þð Þ ¼ limh-0þ

sup1h

V t þ h; x tð Þ þ hf t; x tð Þð Þð Þ�V t; x tð Þð Þ� �:

Definition 2.3. The trivial solution of system (2.1) is

(i)

la

stable if for any ε40, there exists δ¼ δðt0; εÞ40 such that JφJoδ implies thatJxðt; t0;φÞJoε, and uniformly stable if δ is independent of t0;

(ii)

(uniformly) asymptotically stable if it is (uniformly) stable and there exists a γ40 such thatfor any ε40, there is a T ¼ Tðγ; εÞ such that JφJoγ, Jxðt; t0;φÞJoε holds for t4t0 þ T ;

(iii)

globally exponentially stable if for any xt0 ¼ φ, there exist constants λ40 and MZ1 suchthat Jxðt; t0;φÞJrM JφJe� λðt� t0Þ, tZ t0.

ase cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withyed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

3. Main result

Firstly, we consider the stability of the following linear impulsive delayed systems withdelayed impulses:

_xðtÞ ¼ AxðtÞ þ Bxðt�r1ðtÞÞ; taτk;

ΔxðtÞ ¼Ckxðt� Þ þ Dkxðt� �r2ðtÞÞ; t¼ τk;

xt0 ¼ φ;

8><>: ð3:1Þ

where B;DkARn�n.

Theorem 3.1. Let λ140 and λ240 be respectively the smallest and the largest eigenvalues ofsymmetric and positive matrix P, λ3 and λ4 be respectively the largest eigenvalues ofP�1ðA>Pþ PAþ PPÞ and P�1B>B, 0oλ5k and 0oλ6k be respectively the largest eigenvaluesof P�1½ðI þ CkÞ>PðI þ PÞðI þ CkÞ� and P�1½D>

k ðI þ PÞDk�. Then the origin solution of Eq.(3.1) is

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]4

(1)

(3)

Plede

uniformly stable if

λ3 þ λ4η

� �τkþ1�τkð Þo� ln η: ð3:2Þ

where η¼ supðλ5k þ λ6k Þo1ðkANÞ, 0oλ50 þ λ60 r1 and λ3 þ λ4=η40.

(2) uniformly asymptotically stable if

(i) the inequality (3.2) holds;(ii) there exists l41, such that xðt þ sÞ>Pxðt þ sÞolxðtÞ>PxðtÞ, �τrsr0, implies that

r¼ λ3 þ ljλ4jo0 and dk ¼ λ5k þ lλ6ko1.

aselayed

globally exponentially stable if there exist constants σZλ40(i) λ3 þ λ4=ðλ5k þ λ6k Þrσ�λ;(ii) lnðλ5k þ λ6k Þo�ðσ þ λÞðτkþ1�τkÞ.

Proof. Let xðtÞ ¼ xðt; t0;φÞ be any solution of the linear impulsive systems (3.1) through ðt0;φÞ,and construct a Lyapunov function Vðt; xðtÞÞ ¼ x> ðtÞPxðtÞAv0, then

λ1‖x‖2rVðt; xðtÞÞrλ2‖x‖2:

When taτk, kANþ, we have

DþVðt; xðtÞÞ ¼ x> ðtÞ½A>Pþ PA�xðtÞ þ 2x> ðt�r1ðtÞÞPBxðtÞrx> ðtÞ½A>Pþ PAþ PP�xðtÞ þ x> ðt�r1ðtÞÞB>Bxðt�r1ðtÞÞrλ3Vðt; xðtÞÞ þ λ4Vðt�r1ðtÞ; xðt�r1ðtÞÞÞ: ð3:3Þ

Vðτk; xðτkÞÞ ¼ x> ðτkÞPxðτkÞ¼ ½ðI þ CkÞxðτ�k Þ þ Dkxðτ�k �r2ðτ�k ÞÞ�>P½ðI þ CkÞxðτ�k ÞþDkxðτ�k �r2ðτ�k ÞÞ�

¼ x> ðτ�k Þ½ðI þ CkÞ>PðI þ CkÞ�xðτ�k Þþ2x> ðτ�k �r2ðτ�k ÞÞD>

k PðI þ CkÞxðτ�k Þþx> ðτ�k �r2ðτ�k ÞÞD>

k PDkxðτ�k �r2ðτ�k ÞÞrx> ðτ�k Þ½ðI þ CkÞ>PðI þ CkÞ�xðτ�k Þ þ x> ðτ�k Þ½ðI þ CkÞ>PPðI þ CkÞ�xðτ�k Þ

cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withimpulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 5

þx> ðτ�k �r2ðτ�k ÞÞD>k Dkxðτ�k �r2ðτ�k ÞÞ

þx> ðτ�k �r2ðτ�k ÞÞD>k PDkxðτ�k �r2ðτ�k ÞÞ

¼ x> ðτ�k ÞðI þ CkÞ>PðI þ PÞðI þ CkÞxðτ�k Þþx> ðτ�k �r2ðτ�k ÞÞD>

k ðI þ PÞDkxðτ�k �r2ðτ�k ÞÞrλ5k x

> ðτ�k ÞPxðτ�k Þ þ λ6k x> ðτ�k �r2ðτ�k ÞÞPxðτ�k �r2ðτ�k ÞÞ

¼ λ5kVðτ�k ; xðτ�k ÞÞ þ λ6kVðτ�k �r2ðτ�k Þ; xðτ�k �r2ðτ�k ÞÞÞ: ð3:4Þ(1) For any ε40, there is a δ¼ δðεÞ such that δo

ffiffiffiffiffiffiffiffiffiffiffiffiffiλ1η=λ2

pε. Obviously, for φAPCðδÞ and

tA ½t0�τ; t0�, there exists a αA ½�τ; 0� such that t¼ t0 þ α implies that

V t; x tð Þð Þ ¼ V t0 þ α; x t0 þ αð Þð Þrλ2‖φ‖2rλ2δ2o λ2

ηδ2:

We shall prove that

V t; x tð Þð Þr λ2ηδ2; τk�1r toτk; kANþ: ð3:5Þ

We first prove that

V t; x tð Þð Þr λ2ηδ2; tA τ0; τ1½ Þ: ð3:6Þ

If inequality (3.6) is not true, then there is a ξAðτ0; τ1Þ such that

V τ0; x τ0ð Þð Þrλ2δ2o λ2

ηδ2oV ξ; x ξð Þð Þ; ð3:7Þ

where we define

ξ4inf tA τ0; τ1ð Þ V t; x tð Þð Þ4 λ2ηδ2

�����:

Then by employing the continuance of Vðt; xðtÞÞ in ½τ0; τ1Þ, there exists a t1A ðτ0; ξ� such that

V t1; x t1ð Þð Þ ¼ λ2ηδ2;

V t; x tð Þð Þr λ2ηδ2; τ0�τr tr t1: ð3:8Þ

From the inequality (3.7) we know that there exists a s1A ½τ0; t1Þ such that

Vðs1; xðs1ÞÞ ¼ λ2δ2;

Vðt; xðtÞÞZλ2δ2; s1r tr t1: ð3:9Þ

Therefore, from inequalities (3.8) and (3.9), we have

V t þ s; x t þ sð Þð Þr λ2ηδ2r 1

ηV t; x tð Þð Þ; sA �τ; 0½ �;

for tA ½s1; t1�. As a result, Vðt�r1ðtÞ; xðt�r1ðtÞÞÞr ð1=ηÞVðt; xðtÞÞ, then for tA ½s1; t1�

DþV t; x tð Þð Þr λ3 þλ4η

� �V t; x tð Þð Þ: ð3:10Þ

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]6

Now, integrating inequality (3.10) in tA ½s1; t1�, we getZ t1

s1

DþVðt; xðtÞÞVðt; xðtÞÞ dt ¼

Z Vðt1;xðt1ÞÞ

Vðs1;xðs1ÞÞ

du

u

¼Z ðλ2=ηÞδ2

λ2δ2

du

u¼ � ln η

rZ t1

s1

λ3 þλ4η

� �dt

rZ τ1

τ0

λ3 þλ4η

� �dto� ln η:

It is obviously a contradiction. So inequality (3.6) is proved.Then, we assume that inequality (3.5) holds for any l¼ 1; 2;…; k, i.e.

V t; x tð Þð Þr λ2ηδ2; tA τl�1; τl½ Þ; l¼ 1; 2;…; k: ð3:11Þ

Now, we prove that inequality (3.5) holds for l¼ k þ 1, i.e.

V t; x tð Þð Þr λ2ηδ2; tA τk ; τkþ1½ Þ: ð3:12Þ

Firstly, from inequality (3.4), we obtain

V τk; x τkð Þð Þrλ5kV τ�k ; x τ�k � �þ λ6kV τ�k �r2; x τ�k �r2

� �rλ5k

λ2ηδ2 þ λ6k

λ2ηδ2

¼ λ5k þ λ6k � λ2

ηδ2

¼ λ2δ2

o λ2ηδ2: ð3:13Þ

If inequality (3.12) is not true, then there is a ξ1A ðτk ; τkþ1Þ such that

V τk; x τkð Þð Þrλ2δ2o λ2

ηδ2oV ξ1; x ξ1ð Þð Þ; ð3:14Þ

where we define

ξ14inf tA τk; τkþ1ð Þ V t; x tð Þð Þ4 λ2ηδ2

�����:

Then by employing the continuance of Vðt; xðtÞÞ in ½τk; τkþ1Þ, there exists a t2A ðτk ; ξ1� such that

V t2; x t2ð Þð Þ ¼ λ2ηδ2;

V t; x tð Þð Þr λ2ηδ2; τk�τr tr t2: ð3:15Þ

From the inequality (3.14) we know that there exists a s2A ½τk; t2Þ such that

Vðs2; xðs2ÞÞ ¼ λ2δ2;

Vðt; xðtÞÞZλ2δ2; s2r tr t2: ð3:16Þ

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 7

Therefore, from inequalities (3.15) and (3.16), we have

V t þ s; x t þ sð Þð Þr λ2ηδ2r 1

ηV t; x tð Þð Þ; sA �τ; 0½ �;

for tA ½s2; t2�. As a result, Vðt�r1ðtÞ; xðt�r1ðtÞÞÞr ð1=ηÞVðt; xðtÞÞ, then for tA ½s2; t2�

DþV t; x tð Þð Þr λ3 þλ4η

� �V t; x tð Þð Þ: ð3:17Þ

Now, integrating inequality (3.17) in tA ½s2; t2�, we getZ t2

s2

DþVðt; xðtÞÞVðt; xðtÞÞ dt¼

Z Vðt2;xðt2ÞÞ

Vðs2;xðs2ÞÞ

du

u

¼Z ðλ2=ηÞδ2

λ2δ2

du

u¼ � ln η

rZ t2

s2

λ3 þ λ4η

� �dt

rZ τkþ1

τk

λ3 þ λ4η

� �dto� ln η:

Contradiction is obvious and inequality (3.12) is true.As a result, we get the following inequality:

V t; x tð Þð Þr λ2ηδ2

for any tZ t0. It then follows that

λ1‖x‖2rV t; x tð Þð Þ ¼ x> tð ÞPx tð Þr λ2ηδ2; tZ t0;

which implies

JxJrffiffiffiffiffiffiffiλ2λ1η

sδoε; tZ t0:

So the origin solution of systems (3.1) is uniformly stable.(2) Clearly, by condition (i), the zero solution of systems (3.1) is uniformly stable. Therefore,

for given α40, we choose a δ40, such that φAPCð½�τ; 0�;RnÞ with JφJoδ implies that fortZ t0�τ, Jxðt; t0;φÞJoα and

V t; x tð Þð Þr λ2ηδ2:

Then, for any 0oϵoα, there exists a number ϱ¼ ϱðϵÞ40 such that for λ1ϵ2rVðt; xðtÞÞr ðλ2=ηÞδ2, ϱoðl�1ÞVðt; xðtÞÞ. Let N ¼NðϵÞ40 be the smallest integer such that ðλ2=ηÞδ2rλ1ϵ2 þ Nϱ. Set xðtÞ ¼ xðt; t0;φÞ, h¼minf�ðλ2=rηÞδ2; τg, and let T ¼ TðϵÞ ¼ ð2N�1Þh. We willprove that

Vðt; xðtÞÞrλ1ϵ2; tZ t0 þ T : ð3:18Þ

First, we will prove that

Vðt; xðtÞÞrλ1ϵ2 þ ðN�1Þϱ; tZ t0 þ h: ð3:19Þ

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]8

Now, we suppose that for all tA ½t0; t0 þ h�Vðt; xðtÞÞ4λ1ϵ

2 þ ðN�1Þϱ:Then, for any sA ½�τ; 0�,

V t þ s; x t þ sð Þð Þr λ2ηδ2rλ1ϵ

2 þ Nϱ

¼ λ1ϵ2 þ ðN�1Þϱþ ϱ

rVðt; xðtÞÞ þ ϱolVðt; xðtÞÞ:In view of condition (ii) and inequality (3.3), we have

DþVðt; xðtÞÞrλ3Vðt; xðtÞÞ þ λ4Vðt�r1; xðt�r1ÞÞr ðλ3 þ ljλ4jÞVðt; xðtÞÞ ¼ rVðt; xðtÞÞ: ð3:20Þ

From inequality (3.4), we obtain

Vðτk; xðτkÞÞrλ5kVðτ�k ; xðτ�k ÞÞ þ λ6kVðτ�k �r2; xðτ�k �r2ÞÞr ðλ5k þ lλ6k ÞVðτ�k ; xðτ�k ÞÞ¼ dkVðτ�k ; xðτ�k ÞÞ: ð3:21Þ

We choose a small number δ40 such that Vðt; xðtÞÞr ðλ2=ηÞδ2r1, integrate inequality (3.20)on tA ½t0; t0 þ h� and get

V t; x tð Þð ÞrV t0; x t0ð Þð Þ þ r t� t0ð Þ þX

t0oτjr t

V τj; x τj � ��V τ�j ; x τ�j

� � h i

r λ2ηδ2 þ r t� t0ð Þ;

which implies that

V t0 þ h; x t0 þ hð Þð Þr λ2ηδ2 þ rhr λ2

ηδ2 þ r � λ2

rηδ2

� �¼ 0

Obviously, contradiction is achieved, and so there is a ~tA ½t0; t0 þ h� such that

Vð~t ; xð~tÞÞrλ1ϵ2 þ ðN�1Þϱ:

Let m¼minfkAZþ : τk4~tg, we claim that

Vðt; xðtÞÞrλ1ϵ2 þ ðN�1Þϱ; tA ½~t ; τmÞ: ð3:22Þ

If it does not hold, then there is a u1A ð~t ; τmÞ such that

Vðu1; xðu1ÞÞ4λ1ϵ2 þ ðN�1ÞϱZVð~t ; xð~tÞÞ:

Thus, there exists a u2A ð~t ; u1Þ such that

DþVðu2; xðu2ÞÞ40;

and

Vðt; xðtÞÞrVðu2; xðu2ÞÞ; tA ½~t ; u2�;Vðu2; xðu2ÞÞZλ1ϵ

2 þ ðN�1Þϱ:

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 9

Since λ1ϵ2rVðu2; xðu2ÞÞr ðλ2=ηÞδ2, for sA ½�τ; 0�, we have

V u2 þ s; x u2 þ sð Þð Þr λ2ηδ2rλ1ϵ

2 þ Nϱ

¼ λ1ϵ2 þ ðN�1Þϱþ ϱ

rVðu2; xðu2ÞÞ þ ϱolVðr2; xðr2ÞÞ:In view of inequality (3.3) and condition (ii), we obtain DþVðu2; xðu2ÞÞr0, this is acontradiction and inequality (3.22) holds. From inequality (3.21) and inequality (3.22), weconclude that

Vðt; xðtÞÞrλ1ϵ2 þ ðN�1Þϱ; tA ½~t ; τm�:

Similarly, we may show that

Vðt; xðtÞÞrλ1ϵ2 þ ðN�1Þϱ; tA ½τm; τmþ1�:

By a simple induction, we can prove that

Vðt; xðtÞÞrλ1ϵ2 þ ðN�1Þϱ; tA ½τmþi; τmþiþ1�; i¼ 1; 2;…:

Therefore

Vðt; xðtÞÞrλ1ϵ2 þ ðN�1Þϱ; tZ ~tA ½t0; t0 þ h�;

and inequality (3.19) holds.Next, we prove that

Vðt; xðtÞÞrλ1ϵ2 þ ðN�2Þϱ; tZ t0 þ 3h: ð3:23Þ

Suppose that for all tA ½t0 þ 2h; t0 þ 3h�Vðt; xðtÞÞ4λ1ϵ

2 þ ðN�2Þϱ:Then, for any sA ½�τ; 0�,

V t þ s; x t þ sð Þð Þr λ2ηδ2rλ1ϵ

2 þ N�1ð Þϱ¼ λ1ϵ

2 þ ðN�2Þϱþ ϱrVðt; xðtÞÞ þ ϱolVðt; xðtÞÞ:

In view of condition (ii) and inequalities (3.3) and (3.4), we have

Vðτk ; xðτkÞÞrdkVðτ�k ; xðτ�k ÞÞ;DþVðt; xðtÞÞrrVðt; xðtÞÞ: ð3:24Þ

We choose a small number δ40 such that Vðt; xðtÞÞr ðλ2=ηÞδ2r1, integrate inequality (3.24)on tA ½t0 þ 2h; t0 þ 3h� and get

V t; x tð Þð ÞrV t0; x t0ð Þð Þ þ r t� t0�2hð Þ þX

t0þ2hoτj r t

V τj; x τj � ��V τ�j ; x τ�j

� � h i

r λ2ηδ2 þ r t� t0�2hð Þ;

which implies that

V t0 þ 3h; x t0 þ 3hð Þð Þr λ2ηδ2 þ rhr λ2

ηδ2 þ r � λ2

rηδ2

� �¼ 0:

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]10

Obviously, contradiction is achieved, and so there is a ~sA ½t0 þ 2h; t0 þ 3h� such that

Vð~s; xð~sÞÞrλ1ϵ2 þ ðN�2Þϱ:

Similarly, we can prove that

Vðt; xðtÞÞrλ1ϵ2 þ ðN�2Þϱ; tZ ~sA ½t0 þ 2h; t0 þ 3h�;

and inequality (3.23) holds. By simple induction, the following inequality is proved:

Vðt; xðtÞÞrλ1ϵ2 þ ðN� iÞϱ; tZ t0 þ ð2i�1Þh; i¼ 1; 2;…;N: ð3:25Þ

Choosing i¼N,

Vðt; xðtÞÞrλ1ϵ2; tZ t0 þ ð2N�1Þh¼ t0 þ T :

Thus we have

JxJoε; tZ t0 þ T :

Then the origin solution of systems (3.1) is uniformly asymptotically stable.(3) Letting γk ¼ 1=ðλ5k þ λ6k Þ, the following inequality can be obtained from the condition (ii):

ln γk�ðσ þ λÞðτkþ1�τkÞ40:

Then we can choose 1rM such that

1oeðσþλÞðτ1 � τ0ÞrMrγ0e�ðσþλÞðτ1� τ0ÞeðσþλÞðτ1� τ0Þ:

Therefore, we can get

‖φ‖2o‖φ‖2eσðτ1� τ0ÞrM‖φ‖2e� λðτ1� τ0Þ:

We then shall prove that

Vðt; xðtÞÞrλ2M‖φ‖2e� λðt� τ0Þ; τk�1r toτk; kANþ: ð3:26ÞWe first prove that

Vðt; xðtÞÞrλ2M‖φ‖2e� λðt� τ0Þ; tA ½τ0; τ1Þ: ð3:27ÞObviously, we only need to prove that

Vðt; xðtÞÞrλ2M‖φ‖2e� λðτ1� τ0Þ; tA ½τ0; τ1Þ: ð3:28ÞIf inequality (3.28) is not true, then there is a ζAðτ0; τ1Þ such that

Vðτ0 þ s; xðτ0 þ sÞÞrλ2‖φ‖2oλ2‖φ‖2eσðτ1 � τ0Þrλ2M‖φ‖2e� λðτ1 � τ0ÞoVðζ; xðζÞÞ ð3:29Þfor any sA ½�τ; 0�, where we define

ζ4inf ftA ðτ0; τ1ÞjVðt; xðtÞÞ4λ2M‖φ‖2e� λðτ1 � τ0Þg:Then by employing the continuance of Vðt; xðtÞÞ in ½τ0; τ1Þ, there exists a tnA ðτ0; ζ� such that

Vðtn; xðtnÞÞ ¼ λ2M‖φ‖2e� λðτ1� τ0Þ;

Vðt; xðtÞÞrλ2M‖φ‖2e� λðτ1� τ0Þ; τ0�τr tr tn ð3:30Þand there exists a snA ½τ0; tnÞ such that

Vðsn; xðsnÞÞ ¼ λ2‖φ‖2;Vðt; xðtÞÞZλ2‖φ‖2; snr tr tn; ð3:31Þ

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 11

which implies that

Vðsn; xðsnÞÞrVðt; xðtÞÞrVðtn; xðtnÞÞ; tA ½sn; tn�:Therefore, for any tA ½sn; tn� and sA ½�τ; 0�, we have

V t þ s; x t þ sð Þð Þrλ2M‖φ‖2e� λðτ1 � τ0Þ

rλ2M‖φ‖2

r λ2λ50 þ λ60

‖φ‖2

¼ 1λ50 þ λ60

V sn; x sn � �

r 1λ50 þ λ60

V t; x tð Þð Þ:

As a result, Vðt�r1ðtÞ; xðt�r1ðtÞÞÞr1=ðλ50 þ λ60 ÞVðt; xðtÞÞ, then for tA ½sn; tn�

DþV t; x tð Þð Þr λ3 þλ4

λ50 þ λ60

� �V t; x tð Þð Þ

r ðσ�λÞVðt; xðtÞÞ: ð3:32ÞNow, integrating inequality (3.32) in tA ½sn; tn�, we getZ tn

sn

DþVðt; xðtÞÞVðt; xðtÞÞ dt¼

Z Vðtn ;xðtnÞÞ

Vðsn ;xðsnÞÞ

du

u

¼Z λ2M‖φ‖2e� λðτ1 � τ0 Þ

λ2‖φ‖2

du

u¼ ln Me� λðτ1� τ0Þ

rZ tn

snðσ�λÞ dt

¼ ðσ�λÞðtn�snÞoσðt1�τ0Þr ln Me� λðτ1� τ0Þ:

Contradiction is obtained and inequality (3.27) is proved.Then, we assume that inequality (3.26) holds for any l¼ 1; 2;…; k, i.e.

Vðt; xðtÞÞrλ2M‖φ‖2e� λðt� τ0Þ; tA ½τl�1; τlÞ: ð3:33ÞNow, we prove that inequality (3.26) holds for l¼ k þ 1, i.e.

Vðt; xðtÞÞrλ2M‖φ‖2e� λðt� τ0Þ; tA ½τk; τkþ1Þ: ð3:34ÞIf inequality (3.34) is not true, then there is a ζ1A ðτk ; τkþ1Þ such thatVðζ1; xðζ1ÞÞ4λ2M‖φ‖2e� λðζ1� τ0Þ. Then, from inequality (3.4), we obtain

Vðτk ; xðτkÞÞrλ5kVðτ�k ; xðτ�k ÞÞ þ λ6kVðτ�k �r2; xðτ�k �r2ÞÞrλ5kλ2M‖φ‖2e� λðτk � τ0Þ þ λ6kλ2M‖φ‖2e� λðτk � r2� τ0Þ

¼ ðλ5k þ λ6k eλr2Þλ2M‖φ‖2e� λðτk � τ0Þ

r ðλ5k þ λ6k Þeλr2λ2M‖φ‖2e� λðτk � τ0Þ

¼ ðλ5k þ λ6k Þeλr2eλðζ1 � τkÞλ2M‖φ‖2e� λðζ1� τ0Þ

r ðλ5k þ λ6k Þe2λðτkþ1� τkÞλ2M‖φ‖2e� λðζ1 � τ0Þ:oλ2M‖φ‖2e� λðζ1� τ0Þ: ð3:35Þ

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]12

As a result, we have

Vðτk; xðτkÞÞr ðλ5k þ λ6k Þe2λðτkþ1 � τkÞλ2M‖φ‖2e� λðζ1� τ0Þ

oλ2M‖φ‖2e� λðζ1 � τ0ÞoVðζ1; xðζ1ÞÞ: ð3:36ÞMaking use of the continuance of Vðt; xðtÞÞ in ½τk ; τkþ1Þ, there exists a tnnA ðτk ; ζ1� such that

Vðtnn; xðtnnÞÞ ¼ λ2M‖φ‖2e� λðζ1 � τ0Þ;

Vðt; xðtÞÞrλ2M‖φ‖2e� λðζ1 � τ0Þ; τk�τr tr tnn ð3:37Þand there exists a snnA ½τk; tnnÞ such that

Vðsnn; xðsnnÞÞ ¼ ðλ5k þ λ6k Þe2λðτkþ1 � τkÞλ2M‖φ‖2e� λðζ1 � τ0Þ;

Vðt; xðtÞÞZ ðλ5k þ λ6k Þe2λðτkþ1� τkÞλ2M‖φ‖2e� λðζ1� τ0Þ; snnr tr tnn: ð3:38Þwhich implies that

Vðsnn; xðsnnÞÞrVðt; xðtÞÞrVðtnn; xðtnnÞÞ; tA ½snn; tnn�:Therefore, for any tA ½snn; tnn� and sA ½�τ; 0�, we have

V t þ s; x t þ sð Þð Þrλ2M‖φ‖2e� λðζ1� τ0Þ

rλ2M‖φ‖2

r λ2λ5k þ λ6k

‖φ‖2

¼ 1λ5k þ λ6k

V snn; x snn � �

r 1λ5k þ λ6k

V t; x tð Þð Þ:

As a result, Vðt�r1ðtÞ; xðt�r1ðtÞÞÞr1=ðλ5k þ λ6k ÞVðt; xðtÞÞ, then for tA ½snn; tnn�

DþV t; x tð Þð Þr λ3 þ λ4λ5k þ λ6k

� �V t; x tð Þð Þ

r ðσ�λÞVðt; xðtÞÞ: ð3:39ÞNow, integrating inequality (3.39) in tA ½snn; tnn�, we getZ tnn

snn

DþVðt; xðtÞÞVðt; xðtÞÞ dt¼

Z Vðtnn;xðtnnÞÞ

Vðsnn;xðsnnÞÞ

du

u

¼Z λ2M‖φ‖2e� λðζ1 � τ0 Þ

ðλ5kþλ6k Þe2λðτkþ1 � τk Þλ2M‖φ‖2e� λðζ1 � τ0 Þ

du

u¼ � ln λ5k þ λ6k

��2λ τkþ1�τkð Þ

rZ tnn

snnðσ�λÞ dt

¼ ðσ�λÞðtnn�snnÞoðσ�λÞðτkþ1�τkÞo� lnðλ5k þ λ6k Þ�2λðτkþ1�τkÞ:

Contradiction is obvious and inequality (3.34) is true.As a result, we get the following inequality:

Vðt; xðtÞÞrλ2M‖φ‖2e� λðt� τ0Þ

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

�

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 13

for any tZ t0 ¼ τ0. It then follows that

λ1‖x‖2rVðt; xðtÞÞ ¼ x> ðtÞPxðtÞrλ2M‖φ‖2e� λðt� t0Þ; tZ t0;

which implies

JxJrffiffiffiffiffiλ2λ1

rM1=2 JφJe�ðλ=2Þðt� t0Þ; tZ t0:

So the origin solution of systems (3.1) is globally exponentially stable. □

Remark 3.1. The conditions of uniform stability, uniform asymptotic stability and exponentialstability of the impulsive linear delayed systems with delayed impulses are given in this theorem.It is worth pointing out that the two conditions of part (3) can derive the conditions of part (1) orpart (2) in this theorem. For example,

ln λ5k þ λ6k �

o� σ þ λð Þ τkþ1�τkð Þ⟹ λ3 þ

λ4λ5k þ λ6k

� �τkþ1�τkð Þo σ�λð Þ τkþ1�τkð Þo σ þ λð Þ τkþ1�τkð Þo� ln λ5k þ λ6k

⟹ λ3 þλ4η

� �τkþ1�τkð Þo� ln η;

which implies that the exponential stability can derive uniform stability. In some senses,exponential stability does not need to be satisfied or some restrictions are too strong to be easilyobtained. As a result, it is necessary to study others properties by weakening some restrictions.Parts (1) and (2) of Theorem 3.1 are obviously the results by weakening the conditions ofexponential stability in part (3).

Remark 3.2. In [17], the exponential stability of nonautonomous impulsive nonlinear delayedsystems with delayed impulses was discussed in Corollary 1, and the exponential stability ofautonomous linear systems with delayed impulses was studied by means of Lyapunov functionalmethods in the part (3) of Theorem 3.1. From the conditions of this theorem, we can concludethat these conditions are more easier to be satisfied in actual application than those in [17]. At thesame time, we obtain the conditions of uniform stability and uniform asymptotical stability of theautonomous linear systems with delayed impulses. In this theorem, if let Dk ¼ 0 or r2 ¼ 0, i.e.there are no impulse input delays, then the results of [13,14,21] can be easily obtained.

The results of Theorem 3.1 can further be extended to dealing with the stability problems forthe following impulsive nonlinear systems (2.1) with delayed impulses:

_xðtÞ ¼ AxðtÞ þ f ðt; xðtÞ; xðt�r1ðtÞÞÞ; taτk ; tZ t0;

ΔxðtÞ ¼ Ckxðt� Þ þ gðxðt� Þ; xðt� �r2ðtÞÞÞ; t¼ τk; kANþ;

xt0 ¼ φ;

8><>: ð3:40Þ

in which we assume that there exist constants K140, K240, Lk140 and Lk240 such that

‖f ðt; xðtÞ; xðt�r1ðtÞÞÞ‖2rK1‖xðtÞ‖2 þ K2‖xðt�r1ðtÞÞ‖2;‖gðxðt� Þ; xðt� �r2ðtÞÞÞ‖2rLk1‖xðt� Þ‖2 þ Lk2‖xðt� �r2ðtÞÞ‖2:

Theorem 3.2. Let λ140 and λ240 be respectively the smallest and the largest eigenvalues ofsymmetric and positive matrix P, λ3 and λ4 be respectively the largest eigenvalues of

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]14

P�1ðA>Pþ PAþ PPÞ þ K1P�1 and K2P�1, λ5k40 and λ6k40 be respectively the largesteigenvalues of P�1½ðI þ CkÞ>PðI þ PÞðI þ CkÞ� þ ðλ2 þ 1ÞLk1P�1 and ðλ2 þ 1ÞLk2P�1. Thenthe origin solution of Eq. (2.1) is

(1)

(3)

Plede

uniformly stable if

λ3 þλ4η

� �τkþ1�τkð Þo� ln η; ð3:41Þ

where η¼ supðλ5k þ λ6k Þo1ðkANÞ, 0oλ50 þ λ60 r1 and λ3 þ λ4=η40.

(2) uniformly asymptotically stable if

(i) the inequality (3.41) holds;(ii) there exists l41, such that xðt þ sÞ>Pxðt þ sÞolxðtÞ>PxðtÞ, �τrsr0, implies that

r¼ λ3 þ ljλ4jo0 and dk ¼ λ5 þ lλ6 o1.

aselayed

k k

globally exponentially stable if there exist constants σZλ40:(i) λ3 þ λ4=ðλ5k þ λ6k Þrσ�λ;(ii) lnðλ5k þ λ6k Þo�ðσ þ λÞðτkþ1�τkÞ.

Proof. Let xðtÞ ¼ xðt; t0;φÞ be any solution of the linear impulsive systems (2.1) through ðt0;φÞ,and construct a Lyapunov function Vðt; xðtÞÞ ¼ x> ðtÞPxðtÞAv0, then

λ1‖x‖2rVðt; xðtÞÞrλ2‖x‖2:

When taτk, kANþ, we have

DþVðt; xðtÞÞ ¼ x> ðtÞ½A>Pþ PA�xðtÞ þ 2f > ðt; xðtÞ; xðt�r1ðtÞÞÞPxðtÞrx> ðtÞ½A>Pþ PAþ PP�xðtÞ þ f > ðt; xðtÞ; xðt�r1ðtÞÞÞf ðt; xðtÞ; xðt�r1ðtÞÞÞrx> ðtÞ½A>Pþ PAþ PP�xðtÞ þ K1‖xðtÞ‖2 þ K2‖xðt�r1ðtÞÞ‖2rx> ðtÞ½A>Pþ PAþ PP�xðtÞ þ K1P

�1x> ðtÞPxðtÞþK2P

�1x> ðt�r1ðtÞÞPxðt�r1ðtÞÞrλ3Vðt; xðtÞÞ þ λ4Vðt�r1ðtÞ; xðt�r1ðtÞÞÞ: ð3:42Þ

Vðτk; xðτkÞÞ ¼ x> ðτkÞPxðτkÞ¼ ½ðI þ CkÞxðτ�k Þ þ gðxðτ�k Þ; xðτ�k �r2ðτ�k ÞÞÞ�>P½ðI þ CkÞxðτ�k Þþgðxðτ�k Þ; xðτ�k �r2ðτ�k ÞÞÞ� ¼ x> ðτ�k Þ½ðI þ CkÞ>PðI þ CkÞ�xðτ�k Þþ2g> ðxðτ�k Þ; xðτ�k �r2ðτ�k ÞÞÞPðI þ CkÞxðτ�k Þþg> ðxðτ�k Þ; xðτ�k �r2ðτ�k ÞÞÞPgðxðτ�k Þ; xðτ�k �r2ðτ�k ÞÞÞ

rx> ðτ�k Þ½ðI þ CkÞ>PðI þ CkÞ�xðτ�k Þ þ x> ðτ�k Þ½ðI þ CkÞ>PPðI þ CkÞ�xðτ�k Þþðλ2 þ 1Þg> ðxðτ�k Þ; xðτ�k �r2ðτ�k ÞÞÞgðxðτ�k Þ; xðτ�k �r2ðτ�k ÞÞÞ

rx> ðτ�k ÞðI þ CkÞ>PðI þ PÞðI þ CkÞxðτ�k Þ þ ðλ2 þ 1ÞLk1‖xðτ�k Þ‖2þðλ2 þ 1ÞLk2‖xðτ�k �r2ðτ�k ÞÞ‖2¼ x> ðτ�k ÞðI þ CkÞ>PðI þ PÞðI þ CkÞxðτ�k Þ þ ðλ2 þ 1ÞLk1P�1x> ðτ�k ÞPxðτ�k Þþðλ2 þ 1ÞLk2P�1x> ðτ�k �r2ðτ�k ÞÞPxðτ�k �r2ðτ�k ÞÞ

rλ5k x> ðτ�k ÞPxðτ�k Þ þ λ6k x

> ðτ�k �r2ðτ�k ÞÞPxðτ�k �r2ðτ�k ÞÞ¼ λ5kVðτ�k ; xðτ�k ÞÞ þ λ6kVðτ�k �r2ðτ�k Þ; xðτ�k �r2ðτ�k ÞÞÞ: ð3:43Þ

cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withimpulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 15

Then the rest of the proof is precisely the same as that for Eq. (3.1), and hence, it is omitted.This theorem is proved. □

Remark 3.3. This theorem deals with the stability problems of non-autonomous impulsivenonlinear systems with delayed impulses by means of Theorem 3.1. The conditions of uniformstability, uniform asymptotic stability and exponential stability for impulsive nonlinear systemswith delayed impulses are derived in Theorem 3.2. Similarly, if let gðxðt� Þ; xðt� �r2ÞÞ ¼ 0, theresult of Theorem 2 in [21] is obtained and Theorems 1 and 2 of [14] are generalized to the caseof nonlinear.

Remark 3.4. Exponential stability for nonautonomous impulsive nonlinear systems withdelayed impulses was discussed in the Theorems 1 and 2 of [17]. However, some restrictions ofthe results in [17] were difficult to be satisfied. In this paper, we investigated the exponentialstability of autonomous impulsive nonlinear systems with delayed impulses. Its conditions aresimple and convenient to construct a plastic plant. Furthermore, we also studied the uniformstability and uniform asymptotic stability of nonlinear systems with delayed impulses, which arenot discussed in other papers. On the other hand, exponential stability of delayed impulsiveTakasi–Sugeno fuzzy delay systems was discussed in [29]. In the future, we will investigateuniform stability and uniform asymptotic stability of this system.

Finally, the stability for the following general nonlinear delayed systems with delayedimpulses will be investigated:

_xðtÞ ¼ f ðt; xðtÞ; xðt�r1ðtÞÞÞ; taτk ; tZ t0;

ΔxðtÞ ¼ Ikðxðt� Þ; xðt� �r2ðtÞÞÞ; t¼ τk ; kANþ;

xt0 ¼ φ;

8><>: ð3:44Þ

Theorem 3.3. Assume that there exist functions VAv0 and several constants λ140, λ240,λ5k40;, λ6k40; kANþ, λ3 and λ4 such that

(I)

Pledel

λ1‖x‖2rVðt; xðtÞÞrλ2‖x‖2 for any tA ½t0�τ;1Þ;

(II) DþVðt; xðtÞÞrλ3Vðt; xðtÞÞ þ λ4Vðt�r1ðtÞ; xðt�r1ðtÞÞÞ for any tZ t0, taτk, kANþ.

Then the origin solution of systems (3.44) is(1) uniformly stable if

λ3 þλ4η

� �τkþ1�τkð Þo� ln η; ð3:45Þ

where η¼ supðλ5k þ λ6k Þo1ðkANÞ, 0oλ50 þ λ60 r1 and λ3 þ λ4=η40.(2) uniformly asymptotically stable if

(i) the inequality (3.45) holds;(ii) there exists l41, such that Vðt þ s; xðt þ sÞÞolVðt; xðtÞÞ, �τrsr0, implies that

r¼ λ3 þ ljλ4jo0 and dk ¼ λ5k þ lλ6ko1.(3) globally exponentially stable if there exist constants σZλ40

(i) λ3 þ λ4=ðλ5k þ λ6k Þrσ�λ;(ii) lnðλ5k þ λ6k Þo�ðσ þ λÞðτkþ1�τkÞ.

aseayed

citeimpu

this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withlses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]16

The proof of this Theorem is precisely the same as that for Eq. (3.1), and hence,it is omitted.

Remark 3.5. In [16], Theorems 1 and 2 gave the uniform stability criteria for nonlinear delayedsystem. When these results are applied to linear systems, it can be seen that the condition“Vðt þ s; xðt þ sÞÞrVðt; xðtÞÞ for sA ½�τ; 0� implies that DþVðt; xðtÞÞr0” was a very importantcondition, which was hard to be satisfied. We weakened this condition given byDþVðt; xðtÞÞrλ3Vðt; xðtÞÞ þ λ4Vðt�r1ðtÞ; xðt�r1ðtÞÞÞ in our paper. In order to get the uniformstability criteria, we limited the length of the impulse interval by inequalityðλ3 þ λ4=ηÞðτkþ1�τkÞo� ln η. At the same time, at impulsive points, the constraint conditionon Lyapunov function was different partly. Based on this comparison, they complemented eachother to some extent.Cheng [18] and Wang [19] mainly investigated the exponential stability for functional

differential systems with delayed impulses. The condition “DþVðt; xðtÞÞrcVðt; xðtÞÞ for alltZ t0; tA ½τk�1; τkÞ whenever Vðt þ s; xðt þ sÞÞrqVðt; xðtÞÞ for sA ½�τ; 0�” was needed [18,19]to obtain the exponential stability criteria for delayed linear systems with delayed impulses. Inthis paper, this condition was weakened and provided by DþVðt; xðtÞÞrλ3Vðt; xðtÞÞþλ4Vðt�r1ðtÞ; xðt�r1ðtÞÞÞ. And the length of the impulse interval was limited by inequalityλ3 þ λ4=ðλ5k þ λ6k Þrσ�λ and lnðλ5k þ λ6k Þo�ðσ þ λÞðτkþ1�τkÞ. The constraint condition onthe values of Lyapunov function at impulsive points is different partly. In this sense, theycomplemented each other at some degree. Meanwhile, the time-varying delayed systems werediscussed in this paper, which was different from the systems discussed in [18,19] at the impulse.

Example 3.1. Consider the following linear impulsive delay system with delay impulse:

_x tð Þ ¼2 5

312 4

!x tð Þ þ

3 513 1

!x t�0:1ð Þ; taτk ;

x tð Þ ¼35 0

0 35

!x t�ð Þ þ

15 0

0 15

!x t� �0:008ð Þ; t¼ τk ;

xðt0 þ sÞ ¼ φðsÞ;

8>>>>>>><>>>>>>>:

ð3:47Þ

where x¼ ðx1; x2Þ> AR2, τkþ1�τk ¼ 0:004, and

φ1ðtÞ ¼0; tA ½�0:1; 0�;1:5; t ¼ 0;

(

φ2ðtÞ ¼0; tA ½�0:1; 0�;�1:5; t¼ 0:

(

We choose P¼ 10

01

�, using the notations of Theorem 3.1, we obtain that λ3 ¼ 9:9486,

λ4 ¼ 35:0604, λ5k ¼ 0:72, λ6k ¼ 0:08, η¼ 0:8o1, thus

λ3 þλ4η

� �τkþ1�τkð Þo� ln η:

As a result, the system (3.1) is uniformly stable. If we choose λ¼ 0:3 and σ ¼ 54:0945, thenλ3 þ λ4=ðλ5k þ λ6k Þrσ�λ, lnðλ5k þ λ6k Þo�ðσ þ λÞðτkþ1�τkÞ. Therefore, the system (3.1) isglobally exponentially stable. It can be illustrated in Figs. 1–4.

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

0 0.2 0.4 0.6 0.8 1−300

−250

−200

−150

−100

−50

0

50

t

x

x1x2

Fig. 1. System without impulses.

0 0.05 0.1 0.15 0.2 0.25−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

x

x1x2

Fig. 2. Impulsive system with impulse.

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 17

Remark 3.6. From Figs. 1 and 2, we can see that the linear unstable system becomes stable ifwe choose proper impulses. Furthermore, if delay impulse is imposed on this system, it does notchange the stability. However, delay impulse will affect the speed of stability. Figs. 3 and 4 showthat delay impulse reduces the convergence speed of impulsive system comparing with Fig. 2.Here we respectively choose Dk ¼ 0:05

00

0:05

�and Dk ¼ 0:2

000:2

�in Figs. 3 and 4,we find that the

convergence speed of impulsive system with delay impulse is also in relation to the value of Dk.

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

0 0.05 0.1 0.15 0.2 0.25−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

x

x1x2

Fig. 3. Impulsive system with delayed impulses.

0 0.05 0.1 0.15 0.2 0.25−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

x

x1x2

Fig. 4. Impulsive system with delayed impulses.

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]18

Example 3.2. Consider the following linear impulsive delay system with delay impulse:

_x tð Þ ¼ �3 0

0 �2

� �x tð Þ þ

12

12

0 12

!x t�0:5ð Þ; taτk ;

x tð Þ ¼� 3

5 0

0 � 35

!x t�ð Þ þ

15 0

0 15

!x t� �0:08ð Þ; t¼ τk;

xðt0 þ sÞ ¼ φðsÞ;

8>>>>>>><>>>>>>>:

ð3:48Þ

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 19

where x¼ ðx1; x2Þ> AR2, τkþ1�τk ¼ 0:1, and

φ1ðtÞ ¼0; tA ½�0:5; 0�;1:5; t¼ 0;

(

φ2ðtÞ ¼0; tA ½�0:5; 0�;�1:5; t¼ 0:

(

We choose P¼ 10

01

�and l¼2, using the notations of Theorem 3.1, we obtain that λ3 ¼ �3,

λ4 ¼ 0:6545, λ5k ¼ 0:2400, λ6k ¼ 0:08, η¼ 0:32o1, thus r¼ λ3 þ ljλ4jo0 anddk ¼ λ5k þ lλ6ko1. As a result, the system (3.1) is uniformly asymptotically stable. It can beillustrated in Figs. 5–8.

Remark 3.7. In Figs. 7 and 8, we respectively choose Dk ¼ 0:050

00:05

�and Dk ¼ 0:2

000:2

�.

Example 3.3. Consider the following nonlinear impulsive delay system with delay impulse:

_x tð Þ ¼12 0

0 12

!x tð Þ þ

12 cos tð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijx1ðtÞx2ðt�0:2ð1� sin tÞÞj

p12 cos tð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijx2ðtÞx1ðt�0:2ð1� sin tÞÞj

p !

; taτk;

x tð Þ ¼15 0

0 15

!x t�ð Þ þ

15

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijx1ðt� Þx2ðt� �0:06ð1� cos tÞÞj

p15

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijx2ðt� Þx1ðt� �0:06ð1� cos tÞÞj

p0@

1A; t¼ τk ;

xðt0 þ sÞ ¼ φðsÞ;

8>>>>>>>>><>>>>>>>>>:

ð3:49Þ

where x¼ ðx1; x2Þ> AR2, τkþ1�τk ¼ 0:1, and

φ1ðtÞ ¼0; tA ½�0:2; 0�;�1:1; t ¼ 0;

(

0 5 10 15 20−5

−4

−3

−2

−1

0

1

2

t

x

x1x2

Fig. 5. System without impulses.

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

t

x

x1x2

Fig. 6. Impulsive system with impulse.

0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

t

x

x1x2

Fig. 7. Impulsive system with delayed impulses.

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]20

φ2ðtÞ ¼0; tA ½�0:2; 0�;1:1; t ¼ 0:

(

We choose P¼ 10

01

�, using the notations of Theorem 3.2, we obtain that λ3 ¼ 2:25,

λ4 ¼ 0:25, λ5k ¼ 0:16, λ6k ¼ 0:08, η¼ 0:24o1, thus

λ3 þλ4η

� �τkþ1�τkð Þo� ln η:

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

t

x

x1x2

Fig. 8. Impulsive system with delayed impulses.

0 0.5 1 1.5 2 2.5 3−5

−4

−3

−2

−1

0

1

2

3

4

5

t

x(t)

x1x2

Fig. 9. System without impulses.

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 21

As a result, the system (3.40) is uniformly stable. If we choose λ¼ 5 and σ ¼ 8:7814, thenλ3 þ λ4=ðλ5k þ λ6k Þrσ�λ, lnðλ5k þ λ6k Þo�ðσ þ λÞðτkþ1�τkÞ. Therefore, the system (3.40) isglobally exponentially stable. It can be illustrated in Figs. 9–12.

Remark 3.8. From Figs. 9 and 10, we can see that the nonlinear unstable system becomes stableif we choose proper impulses. Although, delay impulse does not change the stability, it willaffect the speed and process of stability (see the curve of x1 in Figs. 6 and 8 respectively).Figs. 11 and 12 show that a different delay impulse produces the convergence process of

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

0 0.02 0.04 0.06 0.08 0.1−1.5

−1

−0.5

0

0.5

1

1.5

t

x(t)

x1x2

Fig. 10. Impulsive system with impulse.

0 0.05 0.1 0.15 0.2−1.5

−1

−0.5

0

0.5

1

1.5

t

x(t)

x1x2

Fig. 11. Impulsive system with delayed impulses.

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]22

impulsive system, where we respectively choose

g x t�ð Þ; x t� �r2ð Þð Þ ¼12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijx1ðt� Þx2ðt� �0:06ð1� cos tÞÞj

p12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijx2ðt� Þx1ðt� �0:06ð1� cos tÞÞj

p !

and

g x t�ð Þ; x t� �r2ð Þð Þ ¼15

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijx1ðt� Þx2ðt� �0:06ð1� cos tÞÞj

p15

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijx2ðt� Þx1ðt� �0:06ð1� cos tÞÞj

p0@

1A:

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

0 0.05 0.1 0.15 0.2−1.5

−1

−0.5

0

0.5

1

1.5

t

x(t)

x1x2

Fig. 12. Impulsive system with delayed impulses.

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 23

Remark 3.9. From these three examples, it can be concluded that delayed impulses do notchange the stability but affect the speed of stability under some proper conditions.

4. Conclusions

We have investigated the stability of impulsive delayed differential systems with delayedimpulses. Making use of the Lyapunov function method combined with Razumikhin technique,we have obtained the conditions of uniform stability, uniform asymptotic stability andexponential stability of linear and nonlinear impulsive delayed systems with delayed impulses.The deduced new exponential stability criteria can remove some complex restrictions for delayedimpulses imposed by the existed results. Through the analyses of three numerical examples, itcan be seen that delayed impulse may change the convergence speed and process of impulsivesystem. The uniform stability and uniform asymptotic stability may be discussed in delayedimpulsive Takasi–Sugeno fuzzy systems with delayed impulses in the future by the same meansproposed in this paper.

Acknowledgments

The work was supported by Program for New Century Excellent Talents in University Grantnos. [2013]47, National Natural Science Foundation of China (Grant nos. 61372139, 61101233,60972155), Spring Sunshine Plan Research Project of Ministry of Education of China (Grant no.z2011148), Technology Foundation for Selected Overseas Chinese Scholars, Ministry ofPersonnel in China (Grant no. 2012-186), University Excellent Talents Supporting Foundationsin of Chongqing (Grant no. 2011-65), University Key Teacher Supporting Foundations ofChongqing (Grant no. 2011-65), Fundamental Research Funds for the Central Universities(Grant nos. XDJK2014A009, XDJK2013B011). This publication was made possible by NPRPGrant ♯ NPRP 4-1162-1-181 from the Qatar National Research Fund (a member of QatarFoundation). The statements made herein are solely the responsibility of the author[s].

Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009

H. Wang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]24

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Please cite this article as: H. Wang, et al., Stability of impulsive delayed linear differential systems withdelayed impulses, Journal of the Franklin Institute. (2015), http://dx.doi.org/10.1016/j.jfranklin.2014.12.009