# Synchronization of Complex Networks with Random Coupling Strengths and Mixed Probabilistic Time-Varying Coupling Delays Using Sampled Data

Authors: Jian-An Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The sampled-data synchronization problem for complex networks with random coupling strengths, probabilistic time-varying coupling delay, and distributed delay (mixed delays) is investigated. The sampling period is assumed to be time varying and bounded. By using the properties of random variables and input delay approach, new synchronization error dynamics are constructed. Based on the delay decomposition method and reciprocally convex approach, a delay-dependent mean square synchronization condition is established in terms of linear matrix inequalities (LMIs). According to the proposed condition, an explicit expression for a set of desired sampled-data controllers can be achieved by solving LMIs. Numerical examples are given to demonstrate the effectiveness of the theoretical results.


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## Body

## 1. Introduction

It is well known that many large-scale systems, such as the Internet, World Wide Web, metabolic pathways, food webs, electric power grids, and social networks, can be modeled by complex networks [1, 2]. A real-world network often consists of a large number of interconnected nodes, in which each node represents an element with certain dynamical system and edge represents the interactions among them. Synchronization, as a typical collective behavior, is an important topic in complex networks. In the past few decades, much effort has been devoted to the study of synchronization in complex networks [3–12]. The main reason for this is that network synchronization can not only explain many natural phenomena but also has many potential applications, such as secure communication, synchronous information exchange over the Internet, and the synchronous transfer of digital signals in communication networks.With the rapid development of modern microprocessor, sampled-data control system has been investigated extensively by various researchers [13–16]. In the sampled-data control system, the control signals are allowed to change only at discrete sampling instants, which can drastically reduce the amount of information transmitted and increase the efficiency of bandwidth usage. The input delay approach [13], which is based on modeling the sampled-data system as a continuous-time system with a time-varying sawtooth delay in the control input, is an important approach in the study of sampled control system. Furthermore, by constructing a time-dependent Lyapunov functional, a refined input delay approach [14] was proposed. Based on the input delay and refined input delay approach, the sampled-data synchronization problems in Lur’e system or delayed neural networks were studied [17–19]. Recently, in the framework of the input delay approach, the authors in [20] investigated the sampled-data synchronization problem for a class of general complex networks with time-varying coupling delays. By considering the neglected terms in the derivative of Lyapunov functional in [20], some improved sampled-data synchronization criterion was derived in [21], and the desired sampled-data feedback controllers were designed in terms of the solution to certain linear matrix inequalities (LMIs). In [22], by using the refined input delay approach and convex combination technique, the sampled-data exponential synchronization for complex dynamical networks with time-varying coupling delay and uncertain sampling has been considered.Time delay is ubiquitous in many physical systems. In order to give a more precise description of dynamical network, time delay should be given more attention inevitably. Therefore, the synchronization for a complex network with coupling delays has been investigated by many researchers. In general, coupling delay of complex networks is deterministic. However, as illustrated in [23], coupling delay often exists in a random form; that is to say, some values of the delay are very large, but the probability of the delay taking such large values is very small. This may lead to a more conservative result if only the information of variation range of the delay is considered. It is worth mentioning that time delays can be generally categorized as discrete ones and distributed ones. Moreover, it has been observed that they usually have a spatial nature due to the presence of a number of parallel pathways of a variety of axon sizes and lengths in a network. Therefore, it is interesting and meaningful to investigate the synchronization in a dynamical network with probabilistic time-varying coupling delay as well as distributed delay (mixed delays).On the other hand, because of the effects of environment and artificial factor, the coupling strength of complex network may randomly vary around some constants [23, 24]. Thus, random phenomena in coupling strength should be taken into account when dealing with the synchronization of complex networks. If only the upper or lower bound of the random coupling strength is considered, some conservative result will be derived. Furthermore, the normal distribution characteristic of random variables can be easily obtained by statistical methods. Therefore, it is interesting to investigate the synchronization of complex networks with random coupling strength described by normal distribution. To the best of our knowledge, the synchronization problem of complex networks with mixed coupling delays and random coupling strengths based on sampled-data control has not been studied in the literature. This is the motivation of our paper.Motivated by the aforementioned discussion, in this paper, the problem of sampled-data synchronization is investigated for complex network with random coupling strengths and mixed probabilistic time-varying coupling delays. The sampling period is assumed to be time varying but bounded by a known positive constant. On the basis of the properties of random variables and the input delay approach, new error dynamics is obtained. By using the delay decomposition method and reciprocally convex approach, a mean square synchronization condition is derived in terms of LMIs. Based on the proposed condition, the corresponding desired sampled-data controllers are designed, which can be solved effectively by using MATLAB LMI Toolbox. Finally, numerical examples are given to illustrate the effectiveness of the obtained results.Notations. Throughout this paper, Rn and Rm×n denote the n-dimensional Euclidean space and the set of all m×n real matrices, respectively. P>0 or P<0 means that P is symmetric and positive or negative definite. The superscript “T” represents the transpose, and “I” and “0” denote the identity and zero matrices with appropriate dimensions. diag⁡{l1,l2,…,ln} stands for a block diagonal matrix. ∥·∥ denotes the Euclidean norm of a vector. The symmetric terms in a symmetric matrix are denoted by *. λmin⁡(A) is the minimum eigenvalue of symmetric matrix A. The notation A⊗B represents the Kronecker product of matrices A and B.


## 2. Preliminaries and Model Description

Consider a hybrid coupled complex network consisting ofN identical coupled nodes as follows:
(1)x˙i(t)=f(xi(t))+c1(t)∑j=1NGijDxj(t)+c2(t)∑j=1NGijAxj(t-τ(t))+c3(t)∑j=1NGijB∫t-r(t)txj(s)ds+ui(t),i=1,2,…,N,
where xi=(xi1,xi2,…,xin)∈Rn and ui(t)∈Rn are, respectively, the state variable and the control input of the node i. f:Rn→Rn is a continuous vector-valued function. τ(t)∈[0,h2] and r(t)∈[0,r] denote the time-varying coupling delay and distributed coupling delay, respectively. ck(t), k=1,2,3 are mutually independent random variables, which denote the coupling strengths of no-delayed coupling, time-delayed coupling, and distributed coupling, respectively. D,A,B∈Rn×n are the inner-coupling matrices. The coupling configuration matrix G=(Gij)N×N is defined as follows: if there is a connection between node i and node j (i≠j), then Gij=Gji=1; otherwise, Gij=Gji=0, and the diagonal elements of matrix G are defined by Gii=-∑j=1,j≠iNGij=-∑j=1,j≠iNGji, i=1,2,…,N.Throughout this paper, the following assumptions are made.Assumption 1.
The time-varying coupling delayτ(t) satisfies 0≤h1≤τ(t)≤h2. Furthermore, the probability distribution of τ(t) taking values in [0,h1) and [h1,h2] is known a priori.Under Assumption1, the probability distribution of τ(t) is assumed to be Prob{τ(t)∈[0,h1)}=β0, and Prob{τ(t)∈[h1,h2]}=1-β0, where 0≤β0≤1 is a constant. Therefore, a stochastic Bernoulli variable can be defined as
(2)β(t)={1,τ(t)∈[0,h1),0,τ(t)∈[h1,h2].
It can be derived from (2) that
(3)Prob{β(t)=1}=E{β(t)}=β0,Prob{β(t)=0}=1-E{β(t)}=1-β0.
Furthermore, from the definition of β(t), it is easy to check that
(4)E{β0-β(t)}=0,E{(β0-β(t))2}=β0(1-β0).
Now, we introduce two time-varying delays τ1(t) and τ2(t) such that
(5)τ(t)={τ1(t),τ(t)∈[0,h1),τ2(t),τ(t)∈[h1,h2].Assumption 2.
There exist constantsμ1 and μ2 such that τ˙1(t)≤μ1<∞ and τ˙2(t)≤μ2<∞.By using the new functionsτ1(t), τ2(t), and β(t), system (1) can be rewritten as
(6)x˙i=f(xi(t))+c1(t)∑j=1NGijDxj(t)+β(t)c2(t)∑j=1NGijAxj(t-τ1(t))+(1-β(t))c2(t)∑j=1NGijAxj(t-τ2(t))+c3(t)∑j=1NGijB∫t-r(t)txj(s)ds+ui(t),i=1,2,…,N.For the random coupling strengthsck(t), k=1,2,3, similar to [23, 24], we assume that almost all the values of them are taken on some nonnegative intervals; that is, ck(t)∈(σk,ρk), where σk and ρk are nonnegative constants. It should be pointed out that almost all the values of ck(t) satisfying ck(t)∈(σk,ρk) just imply that Prob{ck(t)∈(σk,ρk)}=1, Prob{ck(t)<σk}=0, and Prob{ck(t)>ρk}=0. It does not mean that the minimum and maximum allowable coupling strength bounds of ck(t) are σk and ρk, respectively. Thus, the actual lower bound and upper bound of ck(t) may be very small and be very large, respectively. Clearly, it is very different from the constant coupling strength.Assumption 3.
The mathematical exception and variance ofck(t) are E{ck(t)}=ck0 and E{(ck(t)-ck0)2}=δk2, respectively, where ck0 and δk are nonnegative constants.Based on the property of variablesck(t), system (6) can be further expressed by
(7)x˙i=f(xi(t))+c10∑j=1NGijDxj(t)+(c1(t)-c10)∑j=1NGijDxj(t)+β(t)c20∑j=1NGijAxj(t-τ1(t))+β(t)(c2(t)-c20)∑j=1NGijAxj(t-τ1(t))+(1-β(t))c20∑j=1NGijAxj(t-τ2(t))+(1-β(t))(c2(t)-c20)∑j=1NGijAxj(t-τ2(t))+c30∑j=1NGijB∫t-r(t)txj(s)ds+(c3(t)-c30)∑j=1NGijB∫t-r(t)txj(s)ds+ui(t),i=1,2,…,N.Remark 4.
The network model considered here is more general. It is worth noting that whenβ(t)=1, c1(t)=c10, c2(t)=c20, and the distributed coupling is missing, system (7) includes the models considered in [20, 21] as a special case.Assumption 5.
The nonlinear continuous vector-valued functionf is assumed to satisfy the following sector-bounded condition:
(8)[f(x)-f(y)-U(x-y)]T[f(x)-f(y)-V(x-y)]≤0
for all x,y∈Rn, where U and V are constant matrices of appropriate dimensions.Letei(t)=xi(t)-s(t) be the synchronization error, where s(t)∈Rn is the state trajectory of the unforced isolate node s˙(t)=f(x(t)). Then, the error dynamics can be obtained as follows:
(9)e˙i=g(ei(t))+c10∑j=1NGijDej(t)+(c1(t)-c10)∑j=1NGijDej(t)+β(t)c20∑j=1NGijAej(t-τ1(t))+β(t)(c2(t)-c20)∑j=1NGijAej(t-τ1(t))+(1-β(t))c20∑j=1NGijAej(t-τ2(t))+(1-β(t))(c2(t)-c20)∑j=1NGijAej(t-τ2(t))+c30∑j=1NGijB∫t-r(t)tej(s)ds+(c3(t)-c30)∑j=1NGijB∫t-r(t)tej(s)ds+ui(t),i=1,2,…,N,
where g(ei(t))=[f1(ei1(t))f2(ei2(t))⋯fn(ein(t))]T, fj(eij(t))=fj(xij(t))-fj(sj(t)).The control signal is assumed to be generalized by using a zero-order hold (ZOH) function with a sequence of hold times0=t0<t1<⋯<tk<⋯. Therefore, the state feedback controller takes the following form:
(10)ui=Kiei(tk),tk≤t<tk+1,i=1,2,…,N,
where Ki is the feedback gain matrix to be designed, ei(tk) is the discrete measurement of ei(t) at sampling instant tk, and limk→+∞tk=+∞. In this paper, the sampling is not required to be periodic, and the only assumption is that the distance between any two consecutive sampling instants is less than a given bound. It is assumed that tk+1-tk=hk≤p for any integer k≥0, where p>0 represents the largest sampling interval.By using the input delay approach and the Kronecker product, error dynamics (9) can be rewritten in the following compact form:
(11)e˙(t)=g-(e(t))+c10(G⊗D)e(t)+(c1(t)-c10)(G⊗D)e(t)+β(t)c20(G⊗A)e(t-τ1(t))+β(t)(c2(t)-c20)(G⊗A)e(t-τ1(t))+(1-β(t))c20(G⊗A)e(t-τ2(t))+(1-β(t))(c2(t)-c20)(G⊗A)e(t-τ2(t))+c30(G⊗B)∫t-r(t)te(s)ds+(c3(t)-c30)(G⊗B)∫t-r(t)te(s)ds+Ke(t-d(t)),
where e(t)=col{e1(t),e2(t),…,eN(t)}, g-(e(t))=[gT(e1(t))gT(e2(t))⋯gT(en(t))]T, K=diag⁡{K1,K2,…,KN}, d(t)=t-tk.The purpose of this paper is focusing on designing a set of sampled-data controllers (10) with sampling period as big as possible to ensure synchronizing the delayed complex network (6) in mean square sense. Obviously, the synchronization problem has been converted into the mean square asymptotical stability problem of error system (11). Therefore, there are two main issues to be considered in this paper: one is to find some sufficient conditions for the error system (11) to be globally stable in mean square for given Ki, and the other is to derive the design method of sampled-data controllers. To obtain the main results, the following definition and lemmas will be needed.Definition 6.
The coupled complex network (6) is said to be globally synchronized in mean square sense if lim⁡t→∞E{∥ei(t)∥2}=0,i=1,2,…,N holds for any initial values.Lemma 7 (see [25] (Jensen’s inequality)).
For any positive symmetric constant matrixZ=ZT>0, scalar h1<h2 and vector function ω:[h1,h2]→Rn such that the integrations concerned are well defined; then
(12)-(h2-h1)∫t-h2t-h1ωT(s)Zω(s)ds≤-∫t-h2t-h1ωT(s)dsZ∫t-h2t-h1ω(s)ds.Lemma 8 (see [26] (reciprocally convex approach)).
Letf1,f2,…,fN:Rm↦R have positive values in an open subset D of Rm. Then, the reciprocally convex combination of fi over D satisfies
(13)min⁡{αi∣αi>0,∑iαi=1}∑ifi(t)=∑ifi(t)+max⁡gi,j(t)∑i≠jgi,j(t)
subject to
(14){gi,j:Rm⟼R,gj,i(t)≜gi,j(t),[fi(t)gi,j(t)gj,i(t)fj(t)]≥0}.


## 3. Main Results

In this section, we will first give a sufficient condition, which can guarantee the error system (11) to be globally stable in mean square sense. Then, based on the given synchronization condition, we propose an explicit expression of the sampled-data controllers. Before presenting the main results, for the sake of presentation simplicity, we denote
(15)U-=(IN⊗U)T(IN⊗V)2+(IN⊗V)T(IN⊗U)2V-=-(IN⊗U)T+(IN⊗V)T2.Theorem 9.
Suppose that Assumptions1, 2, 3, and 5 hold. For given controller gain matrices Ki, the error system (11) is globally asymptotically stable in mean square sense if there exist matrices P>0, [Q1Q2*Q3]>0, Qi>0  (i=4,…,7), Zi>0  (i=1,…,5), Si  (i=1,…,4) of appropriate dimensions and a scalar ε>0 such that the following LMIs hold:
(16)[ZkSk*Zk]≥0,k=1,…,4,(17)[ΦiΓ*Ψ]<0,i=1,2,
where(18)Φ1=Σ-Δ1TZ1Δ1-Δ2TZ1Δ2-Δ1TS1Δ2-Δ2TS1tΔ1-Δ5TS3Δ6-Δ6TS3TΔ5-Δ7TS4Δ8-Δ8TS4TΔ7-Δ9TZ2Δ9,Φ2=Σ-Δ3TZ2Δ3-Δ4TZ2Δ4-Δ3TS2Δ4-Δ4TS2TΔ3-Δ5TS3Δ6-Δ6TS3TΔ5-Δ7TS4Δ8-Δ8TS3TΔ7-Δ10TZ1Δ10,Σ=[Σ11PKQ20Σ15Σ1600Σ19Σ110*000000000**Σ33-Q2000000***-Q3000000****Σ550Z3000*****Σ66Z4Z400******Σ77000*******-Q4-Z400********-εI0*********-Z5],Σ11=c10P(G⊗D)+c10(G⊗D)TP+Q1+Q4+Q6+Q7+r2Z5-εU--Z3,Σ15=β0c20P(G⊗A)+Z3,Σ16=(1-β0)c20P(G⊗A),Σ33=-Q1+Q3,Σ19=P-εV-,Σ110=c30P(G⊗B),Σ55=-(1-μ1)Q5-2Z3,Σ66=-(1-μ2)Q6-2Z4,Σ77=-Q3+Q4-Z3-Z4,Γ=[β0Ω1T1-β0Ω2Tδ1Ω3Tβ0δ2Ω4T1-β0δ2Ω5Tδ3Ω6T],Ψ=diag⁡{-Z,-Z,-Z,-Z,-Z,-Z},Z=14p2(Z1+Z2)+h12Z3+(h2-h1)2Z4,Δ1=[I-I00000000],Δ2=[0I-I0000000],Δ3=[0-II0000000],Δ4=[0I0-I000000],Δ5=[I000-I00000],Δ6=[0000I0-I000],Δ7=[00000-II000],Δ8=[00000I0-I00],Δ9=[00II000000],Δ10=[I0I0000000],Ω1=[c10Z(G⊗D)ZK00c20Z(G⊗A)000Zc30Z(G⊗B)],Ω2=[c10Z(G⊗D)ZK000c20Z(G⊗A)00Zc30Z(G⊗B)],Ω3=[Z(G⊗D)000000000],Ω4=[0000Z(G⊗A)00000],Ω5=[00000Z(G⊗A)0000],Ω6=[000000000Z(G⊗B)].Proof.
Consider the following Lyapunov-Krasovskii (LK) functional:(19)V(t)=V1(t)+V2(t),
where
(20)V1(t)=eT(t)Pe(t)+∫t-(1/2)ptζT(s)[Q1Q2*Q3]ζ(s)ds+∫t-h1teT(s)Q4e(s)ds+∫t-h2t-h1eT(s)Q5e(s)ds+∫t-τ1(t)teT(s)Q6e(s)ds+∫t-τ2(t)teT(s)Q7e(s)ds,V2(t)=12p∫-(1/2)p0∫t+θte˙T(s)Z1e˙(s)dsdθ+12p∫-p-(1/2)p∫t+θte˙T(s)Z2e˙(s)dsdθ+h1∫-h10∫t+θte˙T(s)Z3e˙(s)dsdθ+(h2-h1)∫-h2-h1∫t+θte˙T(s)Z4e˙(s)dsdθ+r∫-r0∫t+θteT(s)Z5e(s)dsdθ,ζ(s)=col{e(s),e(s-12p)}.
We use LV(t) to denote the infinitesimal operator of V(t), which is defined as
(21)LV(t)=lim⁡Δ→0+Δ-1[E{V(t+Δ)∣e(t)}-V(t)].
It follows from (19) and (21) that
(22)LV(t)=LV1(t)+LV2(t).
Now, taking the derivative of (22) along the solution of system (11) yields
(23)LV1(t)≤2eT(t)P[∫t-r(t)te(s)dsg-(e(t))+c10(G⊗D)e(t)+β0c20(G⊗A)e(t-τ1(t))+(1-β0)c20(G⊗A)e(t-τ2(t))+c30(G⊗B)∫t-r(t)te(s)ds+Ke(t-d(t))]+ζT(t)[Q1Q2*Q3]ζ(t)-ζT(t-12p)[Q1Q2*Q3]ζ(t-12p)+eT(t)Q4e(t)-eT(t-h1)Q4e(t-h1)+eT(t-h1)Q5e(t-h1)-eT(t-h2)Q5e(t-h2)+eT(t)Q6e(t)-(1-μ1)eT(t-τ1(t))Q6e(t-τ1(t))+eT(t)Q7e(t)-(1-μ2)eT(t-τ2(t))Q7e(t-τ2(t)),(24)LV2(t)=e˙T(t)Ze˙(t)+r2eT(t)Z4e(t)-12p∫t-(1/2)pte˙T(s)Z1e˙(s)ds-12p∫t-pt-(1/2)pe˙T(s)Z2e˙(s)ds-h1∫t-h1te˙T(s)Z3e˙(s)ds-(h2-h1)∫t-h2t-h1e˙T(s)Z4e˙(s)ds-r∫t-rteT(s)Z5e(s)ds.
According to Lemma 7, we have
(25)-r∫t-rteT(s)Z5e(s)ds≤-r(t)∫t-r(t)teT(s)Z5e(s)ds≤-∫t-r(t)teT(s)dsZ5∫t-r(t)te(s)ds.
Denote η1(t)=∫t-d(t)te˙(s)ds, η2(t)=∫t-(1/2)pt-d(t)e˙(s)ds, η3(t)=∫t-d(t)t-(1/2)pe˙(s)ds, η4(t)=∫t-pt-d(t)e˙(s)ds, v1(t)=∫t-τ1(t)te˙(s)ds, v2(t)=∫t-h1t-τ1(t)e˙(s)ds, v3(t)=∫t-τ2(t)t-h1e˙(s)ds, and v4(t)=∫t-h2t-τ2(t)e˙(s)ds. According to Lemma 8, if (16) holds, we have
(26)-h1∫t-h1te˙T(s)U1e˙(s)ds≤-[v1(t)v2(t)]T[Z3S3*Z3][v1(t)v2(t)],-(h2-h1)∫t-h2t-h1e˙T(s)U2e˙(s)ds≤-[v3(t)v4(t)]T[Z4S4*Z4][v3(t)v4(t)].
Depending on whether the delay d(t) belongs to the interval 0≤d(t)≤(1/2)p or (1/2)p≤d(t)≤p, we will prove the result in two cases.
Case
1 (0≤d(t)≤(1/2)p). Using Lemmas 7 and 8, one can obtain
(27)-12p∫t-(1/2)pte˙T(s)Z1e˙(s)ds≤-[η1(t)η2(t)]T[Z1S1*Z1][η1(t)η2(t)],-12p∫t-pt-(1/2)pe˙T(s)Z2e˙(s)ds≤-[e(t-12p)e(t-12p)]T[Z2Z2*Z2][e(t-12p)e(t-12p)].
Denote ξ(t)=col{e(t),e(t-d(t)),e(t-(1/2)p),e(t-p),e(t-τ1(t)),e(t-τ2(t)),e(t-h1),e(t-h2),g-(e(t)),∫t-r(t)te(s)ds}, Ω=β(t)Ω10+(1-β(t))Ω20+(c1(t)-c10)Ω30+β(t)(c2(t)-c20)Ω40+(1-β(t))(c2(t)-c20)Ω50+(c3(t)-c30)Ω60, Ω10=[c10(G⊗D)K00c20(G⊗A)000Ic30(G⊗B)], Ω20=[c10(G⊗D)K00c20(G⊗A)000Ic30(G⊗B)], Ω30=[(G⊗D)000000000], Ω40=[0000(G⊗A)00000], Ω50=[00000(G⊗A)0000], Ω60=[000000000(G⊗B)]. Because β(t) and ck(t) are mutually independent variables, it can be obtained that
(28)E{e˙T(t)Ze˙(t)}=E{ξT(t)ΩTZΩξ(t)}=ξT(t)Ω-ξ(t),
where Ω-=β0Ω10TZΩ10+(1-β0)Ω20TZΩ20+δ12Ω30TZΩ30+β0δ22Ω40TZΩ40+(1-β0)δ22Ω50TZΩ50+δ32Ω60TZΩ60.
In addition, based on Assumption5, for any scalar ε>0, we have
(29)y(t)=ε[e(t)g-(e(t))]T[U-V-*I][e(t)g-(e(t))]≤0.
Combining (23)–(29) and taking mathematical exceptions on both sides of (22) give
(30)E{LV(t)}≤ξT(t)[Φ1+Ω-]ξ(t).
Case
2 ((1/2)p≤d(t)≤p). Using Lemmas 7 and 8, one can obtain
(31)-12p∫t-(1/2)pt-(1/2)pe˙T(s)Z2e˙(s)ds≤-[η3(t)η4(t)]T[Z2S2*Z2][η3(t)η4(t)],-12p∫t-pte˙T(s)Z1e˙(s)ds≤-[e(t)e(t-12p)]T[Z1-Z1*Z1][e(t)e(t-12p)].
Similar to the above process, we also can obtain
(32)E{LV(t)}≤ξT(t)[Φ2+Ω-]ξ(t).
By the Schur complement, it is easy to derive that (17) is equivalent to Φi+Ω-<0. Let λ=min⁡{λmin⁡(-(Φi+Ω-))}. It follows from (30) and (32) that
(33)E{LV(t)}≤-λE{∥ξ(t)∥2}≤-λE{∥e(t)∥2}.
Then, by the generalized Itô formula, we have
(34)E{V(t)}-E{V(0)}=E{∫0tLV(s)ds}≤-λ∫0tE{∥e(s)∥2}ds.
Therefore, by virtue of (34), the discussion in [27], and Definition 6, system (11) is globally asymptotically stable in mean-square sense. This completes the proof.Remark 10.
It is well known that delay decomposition method is helpful in the reduction of conservatism for the stability of delayed systems. In this paper, for the sake of obtaining some less conservative sufficient conditions and saving time consumed, a new LKF of form (19) is constructed, where the interval [-p,0] is divided into two divisions: [-p,-(1/2)p] and [-(1/2)p,0]. It is obvious that the LKF (19) can make good use of the information of additional error state e(t-(1/2)p) sufficiently. Therefore, such an LKF is expected to be effective in the reduction of conservatism.Remark 11.
In the previous paper [21], by setting α=d(t)/p and β=τ(t)/h, based on Jensen’s inequality technique, the integral terms -p∫t-pte˙T(s)Z1e˙(s)ds and -h∫t-hte˙T(s)Z2e˙(s)ds were upper-bounded by
(35)-p∫t-pte˙T(s)Z1e˙(s)ds≤-(2-α)[e(t)-e(t-d(t))]T×Z1[e(t)-e(t-d(t))]-(1+α)[e(t-d(t))-e(t-p)]T×Z1[e(t-d(t))-e(t-p)],-h∫t-hte˙T(s)Z2e˙(s)ds≤-(2-β)[e(t)-e(t-τ(t))]T×Z2[e(t)-e(t-τ(t))]-(1+β)[e(t-τ(t))-e(t-h)]T×Z2[e(t-τ(t))-e(t-h)],
which result in a convex combination on α and β, respectively. As discussed in [26], reciprocally convex approach is an effective approach in handling the double integral terms of the LK functional for delayed systems, which can achieve performance behavior identical to the approaches based on the integral inequality lemma but with much less decision variables, comparable to those based on the Jensen inequality lemma. By utilizing the result of Lemma 8, Theorem 9 directly handles convex combination of quadratic terms of some integral terms, which leads to a less conservative result than [21].In the following, we will explore how to design the desired sampled-data controllers to guarantee the complex network (6) synchronizing in mean square sense. Based on Theorem 9, we can easily derive the following theorem.Theorem 12.
Suppose that Assumptions1, 2, 3, and 5 hold.The complex network (6) is globally asymptotically synchronized in mean square sense by the sampled-data controllers of form (10) if there exist matrices P=diag⁡{P1,P2,…,PN}>0, [Q1Q2*Q3]>0, Qi>0  (i=4,…,7), Zi>0  (i=1,…,4), Si(i=1,2,3), X=diag⁡{X1,X2,…,XN} of appropriate dimensions and a scalar ε>0 such that (16) and the following LMIs hold:
(36)[Φ^iΓ^*Ψ^]<0,i=1,2,
where(37)Φ^1=Σ^-Δ1TZ1Δ1-Δ2TZ1Δ2-Δ1TS1Δ2-Δ2TS1Δ1-Δ5TS3Δ6-Δ6TS3Δ5-Δ7TS4Δ8-Δ8TS4Δ7-Δ9TZ2Δ9,Φ^2=Σ^-Δ3TZ2Δ3-Δ4TZ2Δ4-Δ3TS2Δ4-Δ4TS2Δ3-Δ5TS3Δ6-Δ6TS3Δ5-Δ7TS4Δ8-Δ8TS4Δ7-Δ10TZ1Δ10,Σ=[Σ11X00Σ15Σ1600Σ19Σ110*000000000**Σ220000000***-Q2000000****Σ550Z3000*****Σ66Z4Z400******Σ77000*******-Q4-Z400********-εI0*********-Z5],Γ^=[β0Υ1T1-β0Υ2Tδ1Υ3Tβ0δ2Υ4T1-β0δ2Υ5Tδ3Υ6T],Ψ^=diag⁡{-2P+Z,-2P+Z,-2P+Z,-2P+Z,-2P+Z,-2P+Z},Υ1=[c10P(G⊗D)X00c20P(G⊗A)000Pc30P(G⊗B)],Υ2=[c10P(G⊗D)X000c20P(G⊗A)00Pc30P(G⊗B)],Υ3=[P(G⊗D)000000000],Υ4=[0000P(G⊗A)00000],Υ5=[00000P(G⊗A)0000],Υ6=[000000000P(G⊗B)]
and the other terms follow the same definitions as those in Theorem9. Moreover, the desired controllers gain matrices are given by
(38)Ki=Pi-1Xi,i=1,2,…,N.Proof.
Define matricesJ=diag⁡{I,I,I,I,I,I,I,I,I,I,PZ-1,PZ-1,PZ-1,PZ-1,PZ-1,PZ-1} and X=PK. Note that -PZ-1P≤-2P+Z is true for Z>0. Then, performing a congruence transformation of J to (17), and considering the relation X=PK, we can obtain that if LMIs (36) is satisfied, then LMIs (17) holds. This completes the proof.Remark 13.
According to Theorem12, though mixed probabilistic time-varying delays and random coupling strengths coexist in the considered complex network, the desired sampled-data controllers have been designed in terms of the solution to LMIs that can be solved effectively by using available software. In the next section, the effectiveness of the proposed method will be verified by some numerical examples.


## 4. Numerical Examples

In this section, two numerical examples are given to illustrate the effectiveness of the results proposed above.Example 1.
Consider complex network (6) with three nodes. We assume that the coupling strengths are some constants, time-varying coupling delay is deterministic, and the distributed coupling term vanishes. That is to say, β(t)=1, c1(t)=c10, c2(t)=c20, and B=0. The coupling configuration matrix is assumed to be
(39)G=[-1010-1111-2].
The nonlinear function f is taken as
(40)f(xi(t))=[-0.5xi1+tanh(0.2xi2)+0.2xi20.95xi2-tanh(0.75xi2)].
It can be verified that f satisfies (8) with
(41)U=[-0.50.200.95],V=[-0.30.200.2].
The time-varying coupling delay is chosen as τ(t)=0.2+0.05sin(10t). A straightforward calculation gives h=0.25 and μ=0.5. The inner-coupling matrices are given as D=0 and A=[1001]. Thus, we only need to consider the effect of c20. For different coupling strength c20, Table 1 lists the maximum sampling interval p obtained by Theorem 12 and [20, 21]. From Table 1, we can see that our result is less conservative than the existing ones.Table 1
Maximum sampling intervalp for different c20.
Methods
0.5
0.75
[20]
0.5409
0.1653
[21]
0.5573
0.2277
Theorem12
0.5891
0.2809Example 2.
The isolated node of the dynamical networks and the coupling configuration matrixG are the same as Example 1. Let c1(t), c2(t), and c3(t) are two mutually independent random variables satisfying normal distribution with c10=5, c20=2, c20=1, δ1=0.25, δ2=0.2, and δ3=0.1. According to the property of normal distribution, almost all the values of ck(t), k=1,2,3, satisfy ci(t)∈(ci0-3δi,ci0+3δi); that is, c1(t)∈(4.25,5.75), c2(t)∈(1.4,2.6), and c3(t)∈(0.7,1.3). Figures 1, 2, and 3 depict the random coupling strengths c1(t), c2(t), and c3(t), respectively. Take β0=0.9, τ1(t)=0.2+0.1sin(t), and τ2(t)=0.55+0.24sin(t). It followed that h1=0.3, h2=0.79, μ1=0.1, and μ2=0.24. Figure 4 shows the random coupling delay. The distributed coupling delay is chosen as r(t)=0.5sin2(t), so we have r=0.5.
The inner-coupling matrices are given as(42)D=A=[0.1000.1],B=[0.4000.4].
Based on Theorem 12, the maximum value of sampling period is p=0.4623. Moreover, the gain matrices of the desired controllers can be obtained as follows:
(43)K1=[-0.0647-0.1537-0.0048-1.1621],K2=[-0.0647-0.1537-0.0048-1.1621],K3=[0.1637-0.1603-0.0067-0.8554].
In the numerical simulation, the initial values are given by x1(0)=[-24]T, x2(0)=[4-2]T, x3(0)=[-23]T, and s(0)=[32]T. The state trajectories of the synchronization error and the control inputs ui(t) are given in Figures 5 and 6, respectively. Clearly, the synchronization errors are globally asymptotically stable in mean square under the proposed sampled-data scheme.Figure 1
Random coupling strengthc1(t).Figure 2
Random coupling strengthc2(t).Figure 3
Random coupling strengthc3(t).Figure 4
Random coupling delayτ(t).Figure 5
Synchronization error states.Figure 6
Sampled-data control inputs.


## 5. Conclusions

In this paper, the sampled-data synchronization problem has been studied for complex networks with random coupling strengths and mixed probabilistic time-varying coupling delays. Based on random variables and the input delay approach, synchronization error dynamics are obtained. By using the delay decomposition method and reciprocally convex approach, a mean square synchronization criterion is derived, and the corresponding sampled-data controllers are designed in terms of the solution to LMIs. Numerical examples have shown the validity of the presented results. The LK functional considered here neglects the characteristic of sampled-data system, so it inevitably leads to some conservatism. Designing a less conservative sampled-data controller via taking the characteristic of sampled-data system into account constitutes a future research direction.



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