# Exponential Synchronization for Impulsive Dynamical Networks

Authors: Lijun Pan; Jinde Cao
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

This paper is devoted to exponential synchronization for complex dynamical networks with delay and impulsive effects. The coupling configuration matrix is assumed to be irreducible. By using impulsive differential inequality and the Kronecker product techniques, some criteria are obtained to guarantee the exponential synchronization for dynamical networks. We also extend the delay fractioning approach to the dynamical networks by constructing a Lyapunov-Krasovskii functional and comparing to a linear discrete system. Meanwhile, numerical examples are given to demonstrate the theoretical results.


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## Body

## 1. Introduction

In the past two decades, complex dynamical networks have attracted lot of attention in different areas, such as physical science, engineering, mathematics, biology, and sociology [1–3]. The synchronization of all dynamical nodes is an important and interesting phenomena mostly because the synchronization can well explain many natural phenomena. Consequently, the synchronization has been actively investigated due to past physics and potential engineering applications. Recently, there has been an increasing interest in the investigation of synchronization of complex dynamical networks, then many synchronization results have been derived for complex dynamical networks [4–9].Impulsive effects widely exist in the networks. Such systems are described by impulsive differential systems which have been used efficiently in modelling many practical problems that arise in the fields of engineering, physics, and science as well. So the theory of impulsive differential equations is also attracting much attention in recent years [10–13]. Correspondingly, based on the theory of impulsive differential equations, a lot of synchronization results of dynamical networks with impulsive effects have been obtained [13–20].As is well known, two kinds of impulses in terms of synchronization in complex dynamical networks are considered. One is desynchronizing impulse, the other is synchronizing impulse. An impulsive sequence is said to be desynchronizing if the impulsive effect can suppress the synchronization of complex dynamical networks. An impulsive sequence is said to be synchronizing if a corresponding impulsive effect can enhance the synchronization of the complex dynamical networks. According to the previous literature, complex dynamical networks with delay and impulses can reach synchronization provided that delayed dynamical networks are synchronized. In this paper, by impulsive differential inequality [21], the Lyapunov functional method and the Kronecker product techniques, some sufficient conditions are derived for the globally exponential synchronization of dynamical networks. We also extend the delay fractioning method [22, 23] to dynamical networks by constructing Lyapunov-Krasovskii functional and comparing to a linear discrete system. Meanwhile, numerical simulations are given to show that our derived criteria can easily be used to make judgements on synchronization for the delayed dynamical networks with impulsive effects and show that impulsive effects play an important role in the delay dynamical networks. The rest of this paper is organized as follows. In Section 2, the network model is presented, together with some definitions and lemmas. In Section 3, some synchronization criteria are derived for general dynamical networks with delay and impulsive effects. In Section 4, two numerical examples are given to demonstrate that our results are relevant to not only linear coupling but also delay and impulsive effects. Finally, some conclusions are given in Section 5.Notations.
Throughout this paper, the superscriptT represents the transpose. In stands for the identity matrix of order n. For x=(x1,x2,…,xn)T∈Rn, the norm is defined as ∥x∥=(∑i=1nxi2)1/2. For matrix A, λmax(A) and λmin(A) denote the maximum and minimum eigenvalues of matrix A, respectively. For real symmetric matrices X and Y, the notation X≤Y (resp., X<Y) means that the matrix X-Y is negative semidefinite (resp., negative definite). For a sequence {tk,k=0,1,…} satisfying 0=t0<t1<⋯<tk<tk+1<⋯, let Δk≜tk+1-tk, Δsup=supk≥0{Δk}, Δinf=infk≥0{Δk}.


## 2. Model Description and Preliminaries

We consider a delayed complex dynamical network consisting ofN-coupled identical nodes. Each node is an n-dimensional dynamical system composed of linear term and nonlinear term. The ith node can be described as follows:
(2.1)x˙i=Cxi+B1f(xi(t))+B2g(xi(t-τ(t))),i=1,2,…,N,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T is the state vector of the ith node at time t, C, B1, B2∈Rn×n; 0<τ(t)≤τ, τ′(t)≤σ<1, τ>0, f(x),g(x)∈C(Rn,Rn), f(xi(t))=(f1(xi1(t)),f2(xi2(t)),…,fn(xin(t)))T, g(xi(t-τ(t)))=(g1(xi1(t-τ(t))),g2(xi2(t-τ(t))),…,gn(xin(t-τ(t))))T.The dynamical behavior of the dynamical network with delay can be described by the following linearly coupled systems:(2.2)x˙i=Cxi+B1f(xi(t))+B2g(xi(t-τ(t)))+c∑j=1,j≠iNaijΓ(xj(t)-xi(t)),i=1,2,…,N,
where Γ=diag{γ1,γ2,…,γn} is the inner coupling positive definite matrix between two connected nodes i and j, c is the coupling strength, and aij is defined as follows: if there is a connection from node j to node i(j≠i), then aij>0; otherwise, aij=0.In the process of signal transmission, due to the impulsive effects, the statesxi(t),i=1,2,…,N are suddenly changed in the form of impulses at discrete times tk. That is, xi(tk+)=dkxi(tk). Let aii=-∑j=1,j≠iNaij. Thus, the dynamical network with delay and impulsive effects can be obtained by the following form:
(2.3)x˙i=Cxi+B1f(xi(t))+B2g(xi(t-τ(t)))+c∑j=1NaijΓxj(t),t≥t0,t≠tk,xi(tk+)=dkxi(tk),k=1,2,…,xi(t)=φi(t),t∈[t0-τ,t0],i=1,2,…,N,
where xi(tk+)=limh→0+xi(tk+h),xi(tk)=limh→0-xi(tk+h),tk≥0 are impulsive moments satisfying tk<tk+1 and limk→+∞tk=+∞, dk,k=1,2,… are the impulsive gains at tk for ith unit, A=(aij)N×N is the Laplacian matrix of the corresponding network.By a solutionxi=xi(t) of system (2.3), we mean a real function on [t0-τ,∞) such that xi(t0)=φi(t) for t∈[t0-τ,t0], and xi(t) satisfies system (2.3) for t≥t0, and xi(t) is continuous everywhere except for some tk and left continuous at t=tk, and the right limit x(tk+)  k=1,2,… exists. Here, we always assume that system (2.3) has a unique solution.Remark 2.1.
If|dk|<1, the impulsive sequence is of synchronizing impulse, which may enhance the synchronization of the networks. But if |dk|>1, the impulsive sequence can suppress the synchronization, which is said to be desynchronizing impulse.Definition 2.2.
The dynamical networks (2.3) are said to be globally exponentially synchronized if there exist η>0 and M>0 such that for any initial values φi(t)  (i=1,2,…,N):
(2.4)‖xi(t)-xj(t)‖≤Me-η(t-t0)
hold all t>t0, and for any i,j=1,2,…,N.Definition 2.3.
ForA=(aij)m×n∈Rm×n, B=(bij)p×q∈Rp×q, the Kronecker product between two matrices is defined by
(2.5)A⊗B=(a11Ba12B⋯a1nBa21Ba22B⋯a2nB⋮⋮⋯⋮am1Bam2B⋯amnB)∈Rmp×nq.Assumption 2.4.
There exist constantsli,li′>0(i=1,2,…,N) such that |fi(x1)-fi(x2)|≤li|x1-x2| and |gi(x1)-gi(x2)|≤li′|x1-x2| hold for any x1,x2∈R.Assumption 2.5.
The coupling configuration matrixA is irreducible, and the real parts of the eigenvalues of A are all negative except an eigenvalue 0 with multiplicity 1.To derive our main results, we need the following lemmas.Lemma 2.6 (see [24]).
If an irreducible matrixA with nonnegative offdiagonal elements satisfies aii=-∑j=1,j≠iNaij,i=1,2,…,N, then the following propositions are obtained:(1)
ifλ is an eigenvalue of A and λ≠0, then Re(λ)<0;(2)
A has an eigenvalue 0 with multiplicity 1 and the right eigenvector (1,1,…,1)T;(3)
suppose thatξ=(ξ1,ξ2,…,ξN)T∈RN satisfying ∑i=1Nξi=1 is the normalized left eigenvector of A corresponding to eigenvalue 0. Then, ξi>0 hold for all i=1,2,…,N;(4)
furthermore, ifA is symmetric, then we have ξi=1/N for i=1,2,…,N.Lemma 2.7 (see [21]).
Letp, q, τ, dk,k=1,2,… be constants and q≥0, τ>0, dk≥0 and assume that u(t) is a piece continuous nonnegative function satisfying:
(2.6)D+u(t)≤pu(t)+qu¯(t)t≥t0,t≠tk,u(tk+)≤dk(u(tk)),k=1,2,…,u(t)=ϕ(t),t∈[t0-τ,t0].
If there exist α such that for k=1,2,…(2.7)ln⁡dktk-tk-1≤α,p+dq+α<0.
Then
(2.8)u(t)≤d(sup⁡t0-τ≤t≤t0|ϕ|)e-λ(t-t0),
where u¯(t)=supt-τ≤σ≤tx(σ), d=sup1≤k<+∞{eα(tk-tk-1),1/eα(tk-tk-1)}, λ is an unique positive solution of λ+p+dqeλτ+α=0.Remark 2.8.
The condition of Lemma2.7 does not need -p>q due to the effects α, which implies that the above inequality is less conservative than the results in [25].Lemma 2.9.
For any vectorsx,y∈Rn, scalar ϵ>0, and positive definite matrix Q∈Rn×n, the following inequality holds:
(2.9)2xTy≤ϵxTQx+ϵ-1yTQ-1y.Lemma 2.10.
LetA∈Rn×n be a positive definite matrix, then for x∈Rn,
(2.10)λmin⁡(A)xTx≤xTAx≤λmax⁡(A)xTx.


## 3. Synchronization Analysis

In this section, the globally exponential synchronization will be analyzed for delayed dynamical networks with impulsive effects. We assume that the network topology is strongly connected, then the corresponding Laplacian coupling matrixA is irreducible.Letx(t)=(x1T(t),x2T(t),…,xNT(t))T, F(x(t))=(fT(x1(t)),fT(x2(t)),…,fT(xN(t)))T, G(x(t))=(gT(x1(t)),gT(x2(t)),…,gT(xN(t)))T and φ(t)=(φ1(t),φ2(t),…,φN(t))T. Then, the delayed dynamical network (2.3) can be rewritten in the following Kronecker product form:
(3.1)x˙=(IN⊗C)x(t)+(IN⊗B1)F(x(t))+(IN⊗B2)G(x(t-τ(t)))+c(A⊗Γ)x(t),t≥t0,t≠tk,x(tk+)=dkx(tk),k=1,2,…,x(t)=φ(t),t∈[t0-τ,t0].Suppose thatξ=(ξ1,ξ2,…,ξN)T is the left eigenvector of the configuration coupling matrix A with respect to eigenvalue 0 satisfying ∑i=1Nξi=1. Since the coupling configuration matrix A is irreducible, by Lemma 2.6, we can see that ξi>0 for i=1,2,…,N. Let Ξ=diag{ξ1,ξ2,…,ξN}>0, L=diag{l1,l2,…,ln}, L′=diag{l1′,l2′,…,ln′}, W=Ξ-ξξT and A¯=ΞA+ATΞ.Theorem 3.1.
Suppose that Assumptions2.4 and 2.5 hold. Also suppose that there exist a diagonal positive-definite matrix P and scalars η>0, ε>0, γ>0, μ2≥0, μ1, δ such that(H1)
Θ1=PC+CTP+εPB1B1TP+γPB2B2TP+ε-1L2-cηPΓ-μ1P≤0;(H2)
Θ2=γ-1L′2-μ2P≤0;(H3)
forall  k=1,2,…,  2ln|dk|/(tk-tk-1)≤δ;(H4)
μ1+dμ2+δ<0;(H5)
ηλmax(W)+λ2(A¯)≤0.Then the complex dynamical networks (3.1) are exponentially synchronized, where d=sup1≤k<+∞{eδ(tk-tk-1),1/eδ(tk-tk-1)}, λ2(A¯) is defined to be the second largest eigenvalue of A¯.Proof.
We define a Lyapunov functionV(t)=xT(t)(W⊗P)x(t). Since W=Ξ-ξξT, we have wij=-ξiξj for i≠j and wii=ξi-ξi2. In view of ∑j=1Nξj=1, it follows that ∑j=1Nwij=ξi-∑j=1Nξiξj=0. Therefore, we can conclude that V(t)=∑i=1N∑j=1,j≠iN-(1/2)wij(xi(t)-xj(t))TP(xi(t)-xj(t)). Calculating the Dini derivative of V(t) along the trajectories of the systems (3.1), we have for t≠tk,k=1,2,…:
(3.2)D+V(t)=2xT(t)(W⊗P)×(IN⊗C)x(t)+2xT(W⊗P)×(IN⊗B1)F(x(t))+2xT(W⊗P)×(IN⊗B2)G(x(t-τ(t)))+2cxT(t)(W⊗P)×(A⊗Γ)x(t).
By adding -cxT(t)(W⊗ηPΓ)x(t)+cxT(t)(W⊗ηPΓ)x(t) to (3.2) and noting that WA=(Ξ-ξξT)A=ΞA-ξ(ξTA)=ΞA, we can obtain that
(3.3)D+V(t)≤-∑i=1N∑j=1,j≠iNwij[(xi(t)-xj(t))T(PC-12cηPΓ)(xi(t)-xj(t))∑i=1N∑j=1,j≠iNwij+(xi(t)-xj(t))TPB1(f(xi(t))-f(xj(t)))∑i=1N∑j=1,j≠iNwij+(xi-xj)TPB2(g(xi(t-τ(t)))-g(xj(t-τ(t))))12]+cxT(t)×[(ΞA+ATΞ)⊗PΓ+W⊗ηPΓ]x(t).
Since the matrix A¯=ΞA+ATΞ has the following property:
(3.4)A¯=(A¯ij)N×N,A¯ii=2ξiAii<0,i=1,2,…,N,A¯ij=ξiAij+ξjAji=A¯ji,i≠j,∑j=1NA¯ij=ξi∑j=1NA¯ij+∑j=1NξjA¯ji=0.
By Perron-Frobenius theorem (see [24]), we can arrange the eigenvalues of matrix A¯ as follows: 0=λ1(A¯)>λ2(A¯)≥⋯≥λN(A¯). Applying matrix decomposition theory (see [24]), there exists unitary matrix U, such that A¯=UΛUT, where Λ=diag{0,λ2(A¯),…,λN(A¯)} and U={u1,u2,…,uN} with u1=(1/N,1/N,…,1/N)T and UTU=IN.
Lety(t)=(UT⊗In)x(t), where y(t)=(y1T(t),y2T(t),…,yNT(t))T, yi(t)∈Rn,i=1,2,…,N. Then we have x(t)=(U⊗In)y(t). Thus, we have
(3.5)xT(t)[(ΞA+ATΞ)⊗PΓ]x(t)=yT(t)(UT⊗In)(A¯⊗PΓ)(U⊗In)y(t)=∑i=2Nλi(A¯)yiT(t)PΓyi(t)≤λ2(A¯)∑i=2NyiT(t)PΓyi(t).
In view of matrix W is a zero row sum irreducible symmetric matrix with negative off-diagonal elements, we see that λmax(W)>0 and Wu1=(0,0,…,0))T. Hence by Lemma 2.10, we have
(3.6)xT(t)(W⊗ηPΓ)x(t)=ηyT(t)(UTWU⊗PΓ)y(t)≤ηλmax⁡(W)∑i=2NyiT(t)PΓyi(t).
It follows from condition λ2(A¯)+ηλmax(W)≤0 that
(3.7)cxT(t)[(ΞA+ATΞ)⊗PΓ+W⊗ηPΓ]x(t)≤c(λ2(A¯)+ηλmax⁡(W))∑i=2NyiT(t)Pyi(t)≤0.
By Assumption 2.4 and Lemma 2.10, there exists ε>0 such that
(3.8)2(xi(t)-xj(t))TPB1(f(xi(t))-f(xj(t)))≤ε(xi(t)-xj(t))TPB1B1TP(xi(t)-xj(t))+ε-1(f(xi(t))-f(xj(t)))T(f(xi(t))-f(xj(t)))≤ε(xi(t)-xj(t))TPB1B1TP(xi(t)-xj(t))+ε-1(xi(t)-xj(t))TL2(xi(t)-xj(t))=(xi(t)-xj(t))T(εPB1B1TP+ε-1L2)(xi(t)-xj(t)).
Similarly, we have the following estimation:
(3.9)2(xi(t)-xj(t))TB2(g(xi(t-τ(t)))-g(xj(t-τ(t))))≤(xi(t)-xj(t))TγPB2B2TP(xi(t)-xj(t))+(xi(t-τ(t))-xj(t-τ(t)))T(γ-1L′2)(xi(t-τ(t))-xj(t-τ(t))),
where γ>0. Substituting these into (3.3), we have for t≠tk(3.10)V˙(t)≤-∑i=1N∑j=1,j≠iN12wij×[(xi(t)-xj(t))T(PC+CTP+εPB1B1TP+γPB2B2TP+ε-1L2-cηPΓ)(xi(t)-xj(t))]-∑i=1N∑j=1,j≠iN12wij×[(xi(t-τ(t))-xj(t-τ(t)))T(γ-1L′2)(xi(t-τ(t))-xj(t-τ(t)))]=-∑i=1N∑j=1,j≠iN12wij×[(xi(t)-xj(t))T(PC+CTP+εPB1B1TP+γPB2B2TP+ε-1L2-cηPΓ-μ1P)×(xi(t)-xj(t))(xi(t)-xj(t))T]-μ1∑i=1N∑j=1,j≠iN12wij(xi(t)-xj(t))TP(xi(t)-xj(t))-∑i=1N∑j=1,j≠iN12wij×[(xi(t-τ(t))-xj(t-τ(t)))T(γ-1L′2-μ2P)(xi(t-τ(t))-xj(t-τ(t)))]-μ2∑i=1N∑j=1,j≠iN12wij[(xi(t-τ(t))-xj(t-τ(t)))TP(xi(t-τ(t))-xj(t-τ(t)))]≤μ1V(t)+μ2V(t-τ(t)).
For t=tk, we have
(3.11)V(tk+)=-12∑i=1N∑j=1,j≠iNwij(xi(tk+)-xj(tk+))TP(xi(tk+)-xj(tk+))=-dk22∑i=1N∑j=1,j≠iNwij(xi(tk)-xj(tk))TP(xi(tk)-xj(tk))=dk2V(tk).
By Lemma 2.7, there exist M>0 such that
(3.12)V(t)≤M(sup⁡-τ≤s≤0V(t0+s))e-η(t-t0),
which implies that
(3.13)12ξiξjλmin⁡(P)‖xi(t)-xj(t)‖2≤12∑i=1,j=1Nξiξj(xi(t)-xj(t))TP(xi(t)-xj(t))=V(t)=O(e-η(t-t0)).
Consequently, the complex dynamical network (3.1) can reach globally exponential synchronization.Remark 3.2.
When the impulsive effects are desynchronizing, that is,|dk|>1, the condition (H4) in Theorem 3.1 yields -μ1>μ2, which means that the delayed complex networks without impulsive effects of (2.2) is exponentially synchronized. But when the impulsive effects are synchronizing, that is, |dk|<1, we do not need the condition -μ1>μ2 due to the effect of impulses.Theorem 3.3.
Suppose that Assumptions2.4 and 2.5 hold and Δsup<∞. Also suppose that there exist a diagonal positive definite matrix P and scalars η¯>0,  ε¯>0,  γ¯>0, μ¯1>0, μ¯2≥0 such that(H1)
Θ1=PC+CTP+ε¯PB1B1TP+γ¯PB2B2TP+ε¯-1L2-cη¯PΓ-μ¯1P≤0;(H2)
Θ2=γ¯-1L′2-μ¯2P≤0;(H3)
for allk=1,2,…, |dk|<1;(H4)
there exists a integerm≥1 such that tk-m≤tk-τ≤tk+1-m for all k≥m, and the discrete system:
(3.14)θ(k+1)=Jk(m)θ(k)is globally exponentially stable with decayλ>0, where
(3.15)Jk(m)≜(010⋯00001⋯00⋮⋮⋮⋯⋮⋮000⋯01βk+1-mβk+2-mβk+3-m⋯βk-1αk-1),ζ=μ1+μ2/(1-σ), αk=dk2eζΔk-1+βk-1, βk-j=(β¯/(1-σ))Δk-jeζΔk-j,j=1,2,…,m-1;(H5)
there exists a constant𝒯0 such that the average dwell time 𝒯a satisfies
(3.16)N[t0,t]≥-T0+t-t0Ta,t≥t0,where𝒩[t0,t] is the number of impulsive times of the impulsive sequence on the interval [t0,t];(H6)
η¯λmax(W)+λ2(A¯)≤0.
Then the complex dynamical networks (3.1) are exponentially synchronized with decay rate λ/2𝒯a.Proof.
Consider a Lyapunov-Krasovskii functional:(3.17)V(t)=V1(t)+V2(t),
with
(3.18)V1(t)=xT(t)(W⊗P)x(t),V2(t)=μ¯21-σ∫t-τ(t)txT(s)(W⊗P)x(s)ds.
Similar to the proof of Theorem 3.1, for t∈(tk,tk+1], we get
(3.19)D+V1(t)≤μ¯1xT(t)(W⊗P)x(t)+μ¯2xT(t-τ(t))(W⊗P)x(t-τ(t)).
For t∈(tk,tk+1], we have
(3.20)D+V2(t)≤μ¯21-σxT(t)(W⊗P)x(t)-μ¯2xT(t-τ(t))(W⊗P)x(t-τ(t)).
Then
(3.21)D+V(t)=D+V1(t,x(t))+D+V2(t)≤(μ¯1+μ¯21-σ)V1(t)≤ζV(t).
Thus
(3.22)V(t)≤V(tk+)eζ(t-tk),t∈(tk,tk+1].
By (3.11), for t=tk, we have
(3.23)V1(tk+)≤dk2V1(tk).
It follows from condition (iii) that there exists some t^k-j+1∈(tk-j,tk-j+1] such that
(3.24)V2(tk+)≤μ¯21-σ∫tk-τ(tk)tkV1(s)ds≤μ¯21-σ∫tk-mtkV1(s)ds=μ¯21-σ∑j=1m∫tk-j+tk-j+1V1(s)ds=μ¯21-σ∑j=1mΔk-jV1(t^k-j+1).
Then from (3.22), we have
(3.25)V2(tk+)≤μ¯21-σ∑j=1mΔk-jV(t^k-j+1)≤μ¯21-σ∑j=1mΔk-jeζΔk-jV(tk-j+).
Together with (3.22), (3.23) and the above inequality, we have
(3.26)V(tk+)≤(dk2+μ¯21-σΔk-1)eζΔk-1V(tk-1+)+μ¯21-σ∑j=2mΔk-jeζΔk-jV(tk-j+)≜αk-1V(tk-1+)+∑j=1m-1βk-j-1V(tk-j-1+).
Set Z(k)=(z1(k),z2(k),…,zm(k))T and z1(k)=V(tk+1+), z2(k)=V(tk+2+),…,zm(k)=V(tk+m+). Then
(3.27)(z1(k+1-m)z2(k+1-m)⋮zm(k+1-m))≤Jk(m)(z1(k-m)z2(k-m)⋮zm(k-m)),
that is,
(3.28)Z(k-m+1)≤Jk(m)Z(k-m).
We consider the discrete system:
(3.29)θ(k+1)=Jk(m)θ(k),θ(m-1)=Z(-1).
Then, by the comparison principle, we see that for k≥m-1(3.30)Z(k-m)≤θ(k).
Note that the system (3.29) is globally exponential stable with decay λ>0, then there exists constant M>0 such that
(3.31)‖Z(k-m)‖≤Me-λ(k-m+1)‖Z(-1)‖,k≥m-1,
where ∥Z(-1)∥=[∑j=0m-1V2(tj)]1/2,∥Z(k-m)∥=[∑j=1mV2(tj+k-m)]1/2. From (3.17) and (3.22), we have
(3.32)V2(tj+)=μ¯21-σ∫tj-τ(tj)tjxT(s)(W⊗P)x(s)ds≤μ¯21-σ∫t0-τtjxT(s)(W⊗P)x(s)ds=μ¯21-σ∫t0-τt0xT(s)(W⊗P)x(s)ds+μ¯21-σ∑s=0j-1∫ts+ts+1xT(s)(W⊗P)x(s)ds≤μ¯2τ1-σsup⁡-τ≤s≤0V(t0+s)+μ¯21-σ∑s=0j-1ΔsV(ts+)eζΔs,j=0,1,…,m-1.
Furthermore, it follows that
(3.33)V(tj+)=V1(tj+)+V2(tj+)≤μ¯2τ1-σsup⁡-τ≤s≤0V(t0+s)+dj2eζΔj-1V(tj-1+)+μ¯21-σ∑s=0j-1ΔsV(ts+)eζΔs-1=μ¯2τ1-σsup⁡-τ≤s≤0V(t0+s)+αjV(tj-1+)+∑s=0j-2βsV(ts+),j=1,2,…,m-1,V(t0)≤(1+μ¯2τ1-σ)sup⁡-τ≤s≤0V(t0+s).
By induction, there exists a constant ϑ>0, which is dependent on τ, σ, μ¯1, μ¯2,  Δj,  j=0,1,…,m-1 such that
(3.34)V(tj+)≤ϑ‖ξ‖2,
which yields that
(3.35)‖Z(-1)‖=[∑j=0m-1V2(tj+)]1/2≤mϑsup⁡-τ≤s≤0V(t0+s).
From (3.31) and the above inequality, we see that for all k=0,1,…,
(3.36)V(tk+)≤‖Z(k-m)‖≤Mmϑe-λ(k-m+1)sup⁡-τ≤s≤0V(t0+s).
Therefore, by (3.17), (3.22), and (3.36), we conclude that for t∈(tk,tk+1],k=0,1,…,
(3.37)V(t)≤eζ(t-tk)V(tk+)≤Υe-λksup⁡-τ≤s≤0V(t0+s),
where Υ=Mmϑeλ(m-1)+ζΔk. For all t∈(tk,tk+1],k=0,1,…, we obtain that 𝒩[t0,t]=k. Then
(3.38)V(t)≤ΥeλT0e-(λ/Ta)(t-t0)sup⁡-τ≤s≤0V(t0+s),
which means that
(3.39)12ξiξjλmin⁡(P)‖xi(t)-xj(t)‖2≤12∑i=1,j=1N‍ξiξj(xi(t)-xj(t))TP(xi(t)-xj(t))=V(t)=O(e-(λ/Ta)(t-t0)).
This completes the proof of the theorem.Remark 3.4.
Theorem3.3 presents a new delay-dependent exponential synchronization criterion for complex dynamical networks by using the Lyapunov-Krasovskii functional. Note that, for dk<1, σ=0, the proposed result demonstrates its superiority to Theorem 3.1, which will be well illustrated via an example in the next section.Corollary 3.5.
Suppose that Assumptions2.4 and 2.5 hold and Δsup<∞ and τ<tk-tk-1 for all k=1,2,…. If there exist positive definite matrix P and scalars η¯>0, ε¯>0, γ¯>0,  μ¯1>0, μ¯2≥0 such that (H1)–(H3) and (H5)-(H6) of Theorem 3.3 hold, and condition (H4) of Theorem 3.3 is replaced by the following condition:(H4′)
there exists a constantλ>0 such that
(3.40)ln⁡(dk2+μ¯2τ1-σ)+ζΔk-1≤-λ,whereζ=μ¯1+μ¯2/(1-σ), then the complex dynamical networks (3.1) are exponentially synchronized with decay rate λ/2𝒯a.Proof.
Choose a Lyapunov-Krasovskii functional candidateV(x(t)) as
(3.41)V(x(t))=V1(x(t))+V2(x(t)),
with
(3.42)V1(x(t))=xT(t)(W⊗P)x(t),V2(x(t))=μ¯21-σ∫t-τ(t)txT(s)(W⊗P)x(s)ds.
By the proof of Theorem 3.3, for t∈(tk,tk+1], we have
(3.43)D+V(t)≤ζV(t),V(t)≤V(tk+)eζ(t-tk),t∈(tk,tk+1].
Note that τ<tk-tk-1, then there exists some t^k∈[tk-τ,tk] such that
(3.44)V2(tk+)≤μ¯21-σ∫tk-τtkV1(s)ds=μ¯2τ1-σV1(t^k).
Thus
(3.45)V(tk+)≤dk2V1(tk)+μ¯2τ1-σV1(t^k)≤dk2eζΔk-1V1(tk-1+)+μ¯2τ1-σeζΔk-1V1(tk-1+)=eln⁡(dk2+μ¯2τ/(1-σ))+ζΔk-1V(tk-1+).
Then from condition (H4′), we obtain
(3.46)V(tk+)≤e-λV(tk-1+)≤⋯≤e-λkV(t0),
for all k=1,2,…. The remainder proof of the theorem is similar to Theorem 3.3.Remark 3.6.
By Corollary3.5, under the case that τ<tk-tk-1 for all k=1,2,…, we see that the estimations of maximal time-delay τ′ and maximal dwell time Δsup as
(3.47)τ′<sup⁡k≥1{(1-σ)e-ζΔk-1-λ-dk2ζ},Δsup⁡<sup⁡k≥1{-λ-ln⁡⁡(dk2+μ¯2τ/(1-σ))ζ}.Remark 3.7.
By Corollary3.5, if we take the impulsive gains dk as
(3.48)0<dk<e-ζΔk-1-λ-μ¯2τ1-σ,k=1,2,…,
then network (3.1) achieves exponential synchronization.Corollary 3.8.
Suppose that Assumptions2.4 and 2.5 hold and Δsup<∞. If there exist positive definite matrix P and scalars η¯>0, ε¯>0, γ¯>0, μ¯1>0, μ¯2≥0 such that (H1)–(H3) and (H5)-(H6) of Theorem 3.3 hold and condition (H4) of Theorem 3.3 is replaced by one of the following two conditions:(H4′′)
there exists a constantm>1 such that tk-m<tk-τ≤tk+1-m  for all  k≥m, and the matrix J(m) satisfies the spectral radius condition for some λ>0(3.49)ρ(J(m))<e-λ,where(3.50)J(m)≜(010⋯00001⋯00⋮⋮⋮⋯⋮⋮000001ϵ1ϵ1ϵ1⋯ϵ1ϵ1+ϵ2),ϵ1=(μ¯2/(1-σ))ΔsupeζΔsup, ϵ2=deζΔsup, d=supk≥1{dk2}, ζ=μ¯1+μ¯2/(1-σ);(iii′′′)
there exists a positive integerm≥1 such that tk-m<tk-τ≤tk+1-m for all k≥m, and there exists a constant 0<ϱ<1 such that all roots λj(j=1,2,…,m) of the characteristic polynomial:
(3.51)Ψk(λ)≜λm-μk-1λm-1-νk-1λm-2-⋯-νk+2-mλ-νk+1-msatisfy that|λj|≤ϱ<1,then the complex dynamical networks (3.1) are exponentially synchronized.Remark 3.9.
From Theorems3.1 and 3.3, when the delayed network dynamics are desynchronizing and the impulsive effects are synchronizing, in order to ensure synchronization, it should be naturally assumed that the frequency of impulses should not be too low. Usually, we always use condition tk-tk-1≤T1  (T1>0) to ensure that the frequency of impulses should not be too low. Conversely, when the delayed network dynamics are synchronizing but the impulsive effects are desynchronizing, the impulses should not occur too frequently in order to guarantee synchronization. To ensure that the impulses do not occur too frequently, we always assume that tk-tk-1≥T2  (T2>0).


## 4. Examples and Simulations

In this section, some examples and numerical simulations are provided to illustrate our results.Example 4.1.
Consider the following delayed neural networks [26]:
(4.1)x′(t)=Cx(t)+B1f(x(t))+B2g(x(t-1)),
where x=(x1,x2)T, f(x)=(f(x1),f(x2))T,g(x)=(g(x1),g(x2))T, f(x)=g(x)=tanh(x), C=(-100-1), B1=(2-0.1-5.01.5), and B2=(-1.5-0.1-0.2-1). These neural networks (4.1) are chaotic, and chaotic attractor is shown in Figure 1.
We consider the following linear coupled delayed networks:(4.2)xi′(t)=Cxi(t)+B1f(xi(t))+B2g(xi(t-1))+c∑j=14aijΓxj(t),i=1,2,3,4,
where xi(t)=(xi1(t),xi2(t))T,  c=1.4, Γ=(4.18004.9) and A=(-20.410.60.4-302.610-2.41.40.62.61.4-4.6).Figure 1
Chaotic and chaotic attractor.Figure2 shows the synchronization of networks of (4.3).Figure 2
The state variablesxi1(t) and xi2(t) without impulsive effects.At last, we consider the following linear coupled delayed networks with impulsive effects:(4.3)xi′(t)=Cxi(t)+B1f(xi(t))+B2g(xi(t-1))+c∑j=14aijΓxj(t),t≥0,t≠k,k=1,2,…,xi(tk+)=dkxi(tk),t=k,i=1,2,3,4,
where dk=1.2,λmax(W)=0.25,andλ2(A¯)=-1.0439. Letting L=L′=(1/2)I,η=4.1756,ε=γ=μ2=1 and solving the LMIs in (i), (ii) in Theorem 3.1, we get that μ1=-6.4461 and P=diag{0.8432,0.8774}. By Theorem 3.1, we see that the complex dynamical networks (4.3) are exponentially synchronized. Figure 3 shows the synchronization of networks with delay and impulsive effects.Figure 3
The state variablesxi1(t) and xi2(t) with impulsive effects.Example 4.2.
Consider the following neural networks with delay and impulse:(4.4)xi′(t)=Cxi(t)+B1f(xi(t))+B2g(xi(t-τ(t)))+c∑j=15aijΓxj(t),t≥0,t≠k,k=1,2,…,xi(tk+)=dkxi(tk),t=k,i=1,2,…,5,
where xi(t)=(xi1(t),xi2(t),xi3(t))T,  τ(t)=0.01,dk=0.25,c=0.8, C=diag{-0.3,-0.6,-1},
(4.5)B1=(0.4-1.20.40.3-1-0.41.40.5-0.8),B2=(0.60.70.2-0.3-0.3-0.41.10.50.4),A=(-0.60.300.20.10.3-0.500.10.100-0.30.10.20.20.10.1-0.400.10.10.20-0.4),Γ=(1.10001.20002.1),f(xi(t))=(f1(xi1(t)),f2(xi2(t)),f3(xi3(t)))T,g(xi(t))=(g1(xi1(t)),g2(xi2(t)),g3(xi3(t)))T, fj(x)=x,gj(x)=(1/10)(|x+1|-|x-1|),j=1,2,3. λmax(W)=0.56,λ2(A¯)=-0.4693. Letting L=I,L=(1/10)I,η=0.838,ε=γ=μ2=1 and solving the LMIs in (i), (ii) in Theorem 3.1, we get that μ1=1.5626 and P=diag{1.3782,0.9991,1.4307}. We can verify that the synchronization criteria proposed by Theorem 3.1 are not satisfied. However, we conclude that the complex dynamical networks (4.4) are exponentially synchronized by Corollary 3.5. Figure 4 depicts the synchronization state variables xi1(t), xi2(t), and xi3(t) with impulsive effects. Figure 5 depicts the synchronization state variables xi1(t), xi2(t), and xi3(t) without impulsive effects.Figure 4
The state variablesxi1(t), xi2(t), and xi3(t) with impulses.Figure 5
The state variablesxi1(t), xi2(t), and xi3(t) without impulsive effects.Remark 4.3.
In Example4.2, if we take tk-tk-1=0.1 and dk=0.2,τ(t)=0.9sint, it is easy to see that the synchronization criteria proposed by Corollary 3.8 are not satisfied. However, we conclude that the networks (4.4) are exponentially synchronized by Theorem 3.1.


## 5. Conclusions

In this paper, by establishing some lemmas of new impulsive differential inequality and by using the Lyapunov functional method and the Kronecker product techniques, exponential synchronization for impulsive dynamical networks with irreducible coupling matrix is derived. Some criteria are obtained not only relevant to delay but also to impulsive effects. In particular, the results can be extended to the case of one reducible coupling matrixA, which implies that the network topology may be a weakly connected graph containing a rooted spanning tree.



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*Source: 232794-2012-08-09.xml*