# Adaptive Pinning Synchronization of Complex Networks with Stochastic Perturbations

Authors: Xinsong Yang; Jinde Cao
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

The adaptive pinning
synchronization is investigated for complex networks with
nondelayed and delayed couplings and vector-form stochastic
perturbations. Two kinds of adaptive pinning controllers are
designed. Based on an Lyapunov-Krasovskii functional and the
stochastic stability analysis theory, several sufficient
conditions are developed to guarantee the synchronization of the
proposed complex networks even if partial states of the nodes are
coupled. Furthermore, three examples with their numerical
simulations are employed to show the effectiveness of the
theoretical results.


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## Body

## 1. Introduction

Recently, synchronization of all dynamical nodes in a network is one of the hot topics in the investigation of complex networks. It is well known that there are many useful synchronization phenomena in real life, such as the synchronous transfer of digital or analog signals in communication networks. Adaptive feedback control has witnessed its effectiveness in synchronizing a complex network [1–4]. By using the adaptive feedback control scheme, Chen and Zhou [1] studied synchronization of complex nondelayed networks, Cao et al. [2] investigated the complete synchronization in an array of linearly stochastically coupled identical networks with delays. In [3], Zhou et al. considered complex dynamical networks with uncertain couplings. In [4], Lu et al. studied the synchronization in arrays of delay-coupled neural networks. However, it is assumed that all the nodes need to be controlled in [1–4]. As we know, the real-world complex networks normally have a large number of nodes; it is usually impractical and impossible to control a complex network by adding the controllers to all nodes. To overcome this difficulty, pinning control, in which controllers are only applied to a small fraction of nodes, has been introduced in recent years [5–9]. By using adaptive pinning control method, Zhou et al. [10] studied local and global synchronizations of complex networks without delays; authors of [11, 12] considered the global synchronizations of the complex networks with nondelayed and delayed couplings. Note that, in most of existing results of complex networks' synchronization, all of the states are coupled for connected nodes. However, adaptive pinning synchronization results in which only partial states of the nodes are coupled are few. Hence, in this paper, we consider two different adaptive pinning controllers, which synchronize complex networks with partial or complete couplings of the nodes' states.In the process of studying synchronization of complex networks, two main factors should be considered: time delays and stochastic perturbations. Time delays commonly exist in the real world and even vary according to time. In the subsystems, time delay can give rise to chaos, such as delayed neural network and delayed Chua's circuit system in Section4 of this paper. Moreover, time-delayed couplings between subsystems cannot be ignored, such as in long-distance communication and traffic congestions, and so forth. In [8], authors investigated synchronization of complex networks with delayed subsystems, while in [4, 12, 13], authors investigated synchronization of complex networks with coupling delays. On the other hand, in real world, due to random uncertainties, such as stochastic forces on the physical systems and noisy measurements caused by environmental uncertainties, a stochastic behavior should be produced instead of a deterministic one. In fact, signals transmitted between subsystems of complex networks are unavoidably subject to stochastic perturbations from environment, which may cause information contained in these signals to be lost [2]. Therefore, transmitted signals may not be fully detected and received by other subsystems. This can have a great influence on the behavior of complex networks. There are some works in the field of synchronization of complex networks [2, 14–16]. Noise perturbations in [2, 14, 15] are all one-dimensional, which means that the signal transmitted by subsystems is influenced by the same noise. In [16], Yang and Cao considered stochastic synchronization of coupled neural networks with intermittent control, in which noise perturbations are vector forms. Vector-form perturbation means that different subsystem is influenced by different noise, and hence is practical in the real world.Based on the above analysis, in this paper, we study the synchronization of complex networks with time-varying delays and vector-form stochastic elements. Two different adaptive pinning controllers according to the different properties of inner couplings are considered. To obtain our main results, we first formulate a new complex network with nondelayed and delayed couplings and vector-form Wiener processes. Then, by virtue of an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, we develop several theoretical results guaranteeing the synchronization of the new complex networks. Our adaptive pinning controllers are simple. The coupling matrices can be symmetric or asymmetric. Numerical simulations testify the effectiveness of our theoretical results.Notations
In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions.IN denotes the N×N identity matrix. The Euclidean norm in ℝn is denoted as ∥·∥; accordingly, for vector x∈ℝn, ∥x∥=xTx, where T denotes transposition. G=(gij)N×N denotes a matrix of N-dimension, ∥G∥=λmax(GTG), where λmax(A) (respectively, λmin(A)) means the largest (respectively, smallest) eigenvalue of matrix A. Gs=(1/2)(G+GT). G>0 or G<0 denotes that the matrix G is symmetric and positive or negative definite matrix, respectively. Gl denotes the matrix of the first l row-column pairs of G. GN-l denotes the minor matrix of matrix G by removing all the first l row-column pairs of G. E[·] is the mathematical expectation.The rest of this paper is organized as follows. In Section2, new model of delayed complex networks with nondelayed and delayed couplings and vector-form stochastic Weiner processes is presented. Some necessary assumptions, definitions, and lemmas are also given in this section. Our main results and their rigorous proofs are described in Section 3. In Section 4, two directed networks and one Barabási-Albert (BA) network [17] with their numerical simulations are employed to show the effectiveness of our results. Finally, in Section 5, conclusions are given.


## 2. Preliminaries

Consider complex a network consisting ofN identical nodes with nondelayed and delayed linear couplings and vector-form stochastic perturbations, which is described as(2.1)dxi(t)=[f(t,xi(t),xi(t-τ(t)))+c∑j=1NgijΓxj(t)+d∑j=1NuijDxj(t-τ(t))]dt+hi(x1(t),…,xN(t))dωi(t),i=1,2,…,N,
where xi(t)=[xi1(t),xi2(t),…,xin(t)]T∈ℝn represents the state vector of the ith node of the network at time t, τ(t)≥0 is time-varying delay, and f(t,xi(t),xi(t-τ(t)))=[f1(t,xi(t),xi(t-τ(t))),f2(t,xi(t),xi(t-τ(t))),…,fn(t,xi(t),xi(t-τ(t)))]T is a continuous vector-form function. Constants c>0, d>0 are nondelayed coupling strength and delayed coupling strength, respectively. Matrices Γ=diag(γ1,γ2,…,γn) and D=diag(d1,d2,…,dn) are inner couplings of the networks at time t and t-τ(t), respectively, satisfying γj≥0, not all γj=0, dj≥0, j=1,2,…,n. Matrices G=(gij)N×N, and U=(uij)N×N are outer couplings of the networks at time t and t-τ(t), respectively, satisfying gij≥0 for i≠j, gii=-∑j=1,j≠iNgij, and uij≥0 for i≠j, uii=-∑j=1,j≠iNuij, i=1,2,…,N. Matrix hi(x1(t),x2(t),…,xN(t))∈ℝn×n describes the unknown coupling of the complex networks satisfying hi(x(t),x(t),…,x(t))=0 (zero matrix of n dimension). ωi(t)=(ωi1(t),ωi2(t),…,ωin(t))T is a bounded vector-form Weiner process. In this paper, we always assume that ωi(t) and ωj(t) are independent processes of one another for i≠j, and matrix G=(gij)N×N is irreducible in the sense that there is no isolated node.The initial condition of system (2.1) is given in the following form: (2.2)xi(s)=φi(s),-τ≤s≤0,i=1,2,…,N,
where τ=max{τ(t)}, φi∈Lℱ02([-τ,0],ℝn), Lℱ02([-τ,0],ℝn) is the family of all ℱ0-measurable ℜ([-τ,0],ℝn)-valued random variables satisfying sup-τ≤s≤0E[∥φi(s)∥]<∞, and ℜ([-τ,0],ℝn) denotes the family of all continuous ℝn-valued functions φi(s) on [-τ,0] with the norm ∥φi∥=sup-τ≤s≤0φiT(s)φi(s). We always assume that system (2.1) has a unique solution with respect to initial condition.Our objective of synchronization is to control network (2.1) to the trajectory s(t)∈ℝn of the uncoupled system:(2.3)ds(t)=f(t,s(t),s(t-τ(t)))dt,
where s(t)=(s1(t),s2(t),…,sn(t))T can be any desired state: equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit. To achieve this goal, some adaptive pinning controllers will be added to part of its nodes. Without loss of generality, rearrange the order of the nodes in the network, and let the first l nodes be controlled. Thus, the pinning controlled network can be described by(2.4)dxi(t)=[f(t,xi(t),xi(t-τ(t)))+c∑j=1NgijΓxj(t)+d∑j=1NuijDxj(t-τ(t))]dt+hi(t,x1(t),…,xN(t))dωi(t)+Ridt,i=1,2,…,l,dxi(t)=[f(t,xi(t),xi(t-τ(t)))+c∑j=1NgijΓxj(t)+d∑j=1NuijDxj(t-τ(t))]dt+hi(t,x1(t),…,xN(t))dωi(t),i=l+1,2,…,N,
where Ri, i=1,2,…,l, are control inputs.For convenience of writing, in the sequel, we denotehi(t,x1(t),x2(t),…,xN(t)) as hi(t,x(t)). Accordingly, denote hi(t,s(t),s(t),…,s(t)) as hi(t,s(t)). Let ei(t)=xi(t)-s(t), e(t)=(e1T(t),e2T(t),…,eNT(t))T, i=1,2,…,N.Subtracting (2.3) from (2.4) yields the following error dynamical system:(2.5)dei(t)=[F(t,ei(t),ei(t-τ(t)))+c∑j=1NgijΓej(t)+d∑j=1NuijDej(t-τ(t))]dt+h̃i(t,e(t))dωi(t)+Ridt,i=1,2,…,l,dei(t)=[F(t,ei(t),ei(t-τ(t)))+c∑j=1NgijΓej(t)+d∑j=1NuijDej(t-τ(t))]dt+h̃i(t,e(t))dωi(t),i=l+1,2,…,N,
where F(t,ei(t),ei(t-τ(t)))=f(t,xi(t),xi(t-τ(t)))-f(t,s(t),s(t-τ(t))), and h̃i(t,e(t))=hi(t,x(t))-hi(t,s(t)).Then the objective here is to find some appropriate adaptive pinning controllersRi, i=1,2,…,l, such that the trivial solution of error system (2.5) is globally asymptotically stable in the mean square, that is, (2.6)limt→∞E[∥ei(t)∥2]=0,i=1,2,…,N.Definition 2.1.
Function class QUAD1(P,Δ,η,θ): let P=diag(p1,p2,…,pn) be a positive-definite diagonal matrix, Δ=diag(δ1,δ2,…,δn) a diagonal matrix, and constants η>0, θ≥0. QUAD1(P,Δ,η,θ) denotes a class of continuous functions f(t,x,z)∈[0,+∞)×ℝn×ℝn→ℝn satisfying
(2.7)(x-y)TP[f(t,x,z)-f(t,y,w)-Δ(x-y)]≤-η(x-y)T(x-y)+θ(z-w)T(z-w)
for all t≥0, for all x,y,z,w∈ℝn.Remark 2.2.
The function class QUAD1(P,Δ,η,θ) includes almost all the well-known chaotic systems with delays or without delays such as Lorenz system, Rössler system, Chen system, delayed Chua's circuit as well as logistic delay differential system, delayed Hopfield neural networks and delayed CNNs, and so on. In fact, all those systems mentioned above satisfy the following condition [8, 18]: there exist positive kij>0 such that for
(2.8)|fi(t,x(t),x(t-τ(t)))-fi(t,y(t),y(t-τ(t)))|≤∑j=1nkij(|xj(t)-yj(t)|+|xj(t-τ(t))-yj(t-τ(t))|),i=1,2,…,n
for any x(t)=(x1(t),x2(t),…,xn(t))T∈ℝn, y(t)=(y1(t),y2(t),…,yn(t))T∈ℝn.From condition (2.8), we get (2.9)(x-y)TP(f(t,x(t),x(t-τ(t)))-f(t,y(t),y(t-τ(t))))≤∑i=1n∑j=1npikij|xi(t)-yi(t)|(|xj(t)-yj(t)|+|xj(t-τ(t))-yj(t-τ(t))|)=∑i=1n∑j=1npi[kij1/2|xi(t)-yi(t)|kij1/2|xj(t)-yj(t)|+kij1/2|xi(t)-yi(t)|kij1/2|xj(t-τ(t))-yj(t-τ(t))|]≤∑i=1n∑j=1npi[kij(xi(t)-yi(t))2+12kij(xj(t)-yj(t))2+12kij(xj(t-τ(t))-yj(t-τ(t)))2]=∑i=1n∑j=1npi[kij(xi(t)-yi(t))2+12kji(xi(t)-yi(t))2+12kji(xi(t-τ(t))-yi(t-τ(t)))2]=∑i=1n∑j=1npi[(kij+12kji)(xi(t)-yi(t))2+12kji(xi(t-τ(t))-yi(t-τ(t)))2]=∑i=1npi[(-δi+∑j=1n(kij+12kji))(xi(t)-yi(t))2+12∑j=1nkji(xi(t-τ(t))-yi(t-τ(t)))2]+(x(t)-y(t))TPΔ(x(t)-y(t))≤-η(x(t)-y(t))T(x(t)-y(t))+(x(t)-y(t))TPΔ(x(t)-y(t))+θ(x(t-τ(t))-y(t-τ(t)))T(x(t-τ(t))-y(t-τ(t))),
where η=min1≤i≤n{pi(δi-∑j=1n(kij+(1/2)kji))}, θ=max1≤i≤n{(1/2)pi∑j=1nkji}, and 2ab≤a2+b2 is used in the above inequality. Hence, condition (2.7) is satisfied. Note that function class QUAD1(P,Δ,η,θ) is more general than the usual function class QUAD(P,Δ,η) (see [13, 19]) for chaotic systems and includes it as a special class when θ=0.Before giving our main results, we present the following lemmas, which are needed in the next section.Lemma 2.3 (see [12]).
IfA=(aij)N×N is an irreducible matrix satisfying aij=aji≥0, for i≠j, and aii=-∑i=1,i≠jNaij, for 1=1,2,…,N, then all eigenvalues of the matrix A¯ = A-diag(k1,…,kl,0,…,0) are negative, where k1,…,kl are positive constants.Lemma 2.4 (see [19]).
Suppose thatG=(gij)N×N is an irreducible matrix satisfying gij≥0, for i≠j, and gii=-∑i=1,i≠jNgij, for 1=1,2,…,N. Let ξ=(ξ1,ξ2,…,ξN)T be the left eigenvector corresponding to the eigenvalue 0, that is, ξTG=0. Then, ξi>0, i=1,2,…,N, and its multiplicity is 1.Lemma 2.5 (see [20]; (Schur Complement)).
The linear matrix inequality (LMI)(2.10)S=[S11S12S12TS22]<0
is equivalent to anyone of the following two conditions:(L1)
S11<0, S22-S12TS11-1S12<0,(L2)
S22<0, S11-S12S22-1S12T<0,where S11=S11T and S22=S22T.Now we list assumptions as follows.(H1)
There exist nonnegative constantsμij, i,j=1,2,…,N, such that
(2.11)trace((h̃i(t,e(t)))Th̃i(t,e(t)))≤∑j=1Nμij∥ej(t)∥2,
where h̃i(t,e(t))=hi(t,x(t))-hi(t,s(t)), i=1,2,…,N.(H2)
τ(t) is a continuous differentiable function on ℝ+ with τ̇(t)≤h<1.


## 3. Main Results

In this section, two different adaptive pinning feedback controllers corresponding to two kinds of properties of the inner couplingΓ are designed, and several sufficient conditions for synchronization of the complex networks (2.1) are derived.For the the inner coupling matrixΓ, there exist the following two cases:(a)
γj≥0, j=1,2,…,n, and there exists at least one j0∈{j=1,2,…,n} such that γj0>0.(b)
γj>0, j=1,2,…,n.Obviously, case (b) is the special case of case (a). Case (b) means that all the states of connected nodes are coupled, while in case (a) this is not necessary. However, most of existing papers only consider case (b); for example, see references [7, 12, 13, 19]. In this paper, we will consider both of the two cases. We first study the general case (a).Theorem 3.1.
Suppose that the assumptions(H1)-(H2) hold and f∈ QUAD1(P,Δ,η,θ). For the case (a), if 2cγ¯ĜN-ls+WN-l<0, where Ĝ=(ĝij)N×N, ĝij=gij,i≠j, ĝii=(γ̲/γ¯)gii, γ¯=max{γj,j=1,2,…,n}, γ̲=min{γj,j=1,2,…,n}, W=2(-η+δ¯)IN+2Φ+((5-h)/2(1-h))dd¯∥V∥IN, δ¯=max{pjδj,j=1,2,…,n}, Φ=diag(ϕ1,ϕ2,…,ϕN), ϕj=(1/2)∑i=1Nμij, d¯=max{dj,j=1,2,…,n}, V=(|uij|)N×N, then the trivial solution of (2.5) is globally asymptotically stable in the mean square with the adaptive pinning controllers
(3.1)Ri=-cɛiei(t),i=1,2,…,l,ɛ̇i=vieiT(t)Pei(t),i=1,2,…,l.Proof.
According to(H1) and (H2), the origin is an equilibrium point of the error system (2.5). We define the Lyapunov-Krasovskii functional as
(3.2)V(t)=V1(t)+V2(t),
where
(3.3)V1(t)=12∑i=1NeiT(t)Pei(t)+12∑i=1lc(ɛi-ki)2vi,V2(t)=∫t-τ(t)tζT(s)Qζ(s)ds,ζ(t)=(∥e1(t)∥,∥e2(t)∥,…,∥eN(t)∥)T, Q is a positive definite matrix, ki, i=1,2,…,l, are positive constants, and Q and ki, i=1,2,…,l, are to be determined.
DifferentiatingV1(t) along the solution of (2.5) and using the properties of vector-form Weiner process [16], we get
(3.4)dV1(t)=∑i=1N[eiT(t)Pdei(t)+12(dei(t))TPdei(t)]+c∑i=1l(ɛi-ki)eiT(t)Pei(t)dt=∑i=1NeiT(t)P[F(t,ei(t),ei(t-τ(t)))+c∑j=1NgijΓej(t)+d∑j=1NuijDej(t-τ(t))]dt-c∑i=1lkieiT(t)Pei(t)dt+∑i=1NeiT(t)h̃i(t,e(t))dωi(t)+12∑i=1Ntrace[(h̃i(t,e(t)))Th̃i(t,e(t))]dt={∑i=1NeiT(t)P(F(t,ei(t),ei(t-τ(t)))-Δei(t))+∑i=1NeiT(t)PΔei(t)-c∑i=1lkieiT(t)Pei(t)+∑i=1NeiT(t)P(c∑j=1NgijΓej(t)+d∑j=1NuijDej(t-τ(t)))+12∑i=1Ntrace[(h̃i(t,e(t))Th̃i(t,e(t))]}dt+∑i=1NeiT(t)h̃i(t,e(t))dωi(t)≤{-η∑i=1N∥ei(t)∥2+θ∑i=1N∥ei(t-τ(t))∥2+δ¯∑i=1N∥ei(t)∥2-cp¯∑i=1lki∥ei(t)∥2+c∑i=1N∑j=1,j≠iNgijγ¯∥ei(t)∥∥ej(t)∥+c∑i=1Ngiiγ̲∥ei(t)∥2+12∑i=1N∑j=1Nμij∥ej(t)∥2+d∑i=1N∑j=1,j≠iN|uij|d¯∥ei(t)∥∥ej(t-τ(t))∥}dt+∑i=1NeiT(t)h̃i(t,e(t))dωi(t),
where p¯=max{pj,j=1,2,…,n}.
Similarly, fromV2(t) we have
(3.5)dV2(t)=[ζT(t)Qζ(t)-(1-τ̇(t))ζT(t-τ(t))Qζ(t-τ(t))]≤[ζT(t)Qζ(t)-(1-h)ζT(t-τ(t))Qζ(t-τ(t))]dt.
Hence,
(3.6)dV(t)≤[(δ¯-η)ζT(t)ζ(t)+θζT(t-τ(t))ζ(t-τ(t))-cp¯ζT(t)Kζ(t)+cγ¯ζT(t)Ĝsζ(t)+dd¯ζT(t)Vζ(t-τ(t))+ζT(t)(R+Q)ζ(t)-(1-h)ηT(t-τ(t))Qη(t-τ(t))]dt+∑i=1NeiT(t)h̃i(t,e(t))dωi(t)=12ϖT(t)Πϖ(t)d(t)+∑i=1NeiT(t)h̃i(t,e(t))dωi(t),
where K=diag(k1,k2,…,kl,0,…,0), ϖT(t)=(ζT(t),ζT(t-τ(t)))T, and
(3.7)Π=[2((-η+δ¯)IN-cp¯K+cγ¯Ĝs+Φ+Q)dd¯VTdd¯V2θIN-2(1-h)Q].
In view of Lemma2.5, Π<0 is equivalent to
(3.8)Υ=2((-η+δ¯)IN-cp¯K+cγ¯Ĝs+Φ+Q)+12d2d¯2VT((1-h)Q-θIN)-1V<0,-(1-h)Q+θIN<0.
LetQ=(1/(1-h))(dd¯∥V∥+θ)IN, we obtain
(3.9)Υ≤2(-η+δ¯)IN-2cp¯K+2cγ¯Ĝs+2Φ+5-h2(1-h)dd¯∥V∥IN=[-2cp¯Kl+2cγ¯Ĝls+Wl2cγ¯(Ĝ*s)T2cγ¯Ĝ*s2cγ¯ĜN-ls+WN-l],
where Ĝ*s is matrix with appropriate dimension.
Since2cγ¯ĜN-ls+WN-l<0, obviously, there exist positive constants k1,k2,…,kl such that
(3.10)-2cp¯Kl+2cγ¯Ĝls+Wl-2c2γ¯2(Ĝ*s)T(2cγ¯ĜN-ls+WN-l)-1Ĝ*s<0.
Again, from Lemma 2.5 we obtain Υ<0. Hence, (3.8) holds, which means that Π<0. Taking the mathematical expectation of both sides of (3.6), we have
(3.11)dEV(t)dt≤E[12ϖT(t)Πϖ(t)]≤-12λmin(-Π)E[∥ζ(t)∥2+∥ζ(t-τ(t))∥2]≤-12λmin(-Π)E[∥ζ(t)∥2]=-12λmin(-Π)E[∑i=1N∥ei(t)∥2],
where λmin(-Π) is positive. In view of the LaSalle invariance principle of stochastic differential equation, which was developed in [21], we have limt→∞E[∑i=1N∥ei(t)∥2]=0, which in turn illustrates that limt→∞E[∥ei(t)∥2]=0, and at the same time, limt→∞ɛi(t)=ɛi (constants). This completes the proof.Next, we consider the case (b). Obviously, the adaptive pinning controllers (3.1) can also synchronize (2.1) for this case. Here we use another kind of controllers, which is related to the inner matrix Γ, to synchronize the complex networks (2.1).Assume thatξ=(ξ1,ξ2,…,ξN)T is defined in Lemma 2.4; we have ξi>0, i=1,2,…,N. Define Ξ=diag(ξ1,ξ2,…,ξN). It is easy to verify that ΞG+GTΞ is a symmetric and zero row sum matrix. Moreover, if G is irreducible, then ΞG is also irreducible; hence, ΞG+GTΞ is irreducible. Therefore, by Lemma 2.3, (ΞG¯)s=(1/2)(ΞG¯+G¯TΞ)=(1/2)(ΞG+GTΞ)-ΞK is negative definite, where G¯=G-K, K=diag(k1,…,kl,0,…,0), and ki, i=1,…,l, are positive constants.Theorem 3.2.
Suppose that the assumptions(H1)-(H2) hold and f∈ QUAD1(P,Δ,η,θ). If γj>0 and 2cpjγj(ΞG)N-ls+(Wj)N-l<0, j=1,2,…,n, where Wj=-2ηΞ+2pjδjΞ+Φ̃+((2-h)/(1-h))dpjdj∥ΞU∥IN+(2θ/(1-h))Ξ, Φ̃=diag(ϕ̃1,ϕ̃2,…,ϕ̃N), and ϕ̃j=∑i=1Nξiμij, then the trivial solution of (2.5) is globally asymptotically stable in the mean square with the adaptive pinning controllers
(3.12)Ri=-cɛiΓei(t),i=1,2,…,l,ɛ̇i=vieiT(t)PΓei(t),i=1,2,…,l.Proof.
Letẽj(t)=(e1j,e2j,…,eNj)T, j=1,2,…,n. We define the Lyapunov-Krasovskii functional as
(3.13)V(t)=12∑i=1NξieiT(t)Pei(t)+12∑j=1n∫t-τ(t)tẽjT(s)Qjẽj(s)ds+12∑i=1lcξi(ɛi-ki)2vi,
where Qj, j=1,2,…,n, are positive definite matrices, ki, i=1,2,…,l, are positive constants, and Qj and ki,i=1,2,…,l, are to be determined.
Differentiating both sides of (3.13) along the solution of (2.5) yields
(3.14)dV(t)=∑i=1Nξi[eiT(t)Pdei(t)+12(dei(t))TPdei(t)]+{12∑j=1n[ẽjT(t)Qjẽj(t)-(1-τ̇(t))ẽjT(t-τ(t))Qjẽj(t-τ(t))]+c∑i=1lξi(ɛi-ki)eiT(t)PΓei(t)dt}=∑i=1NξieiT(t)P[F(t,ei(t),ei(t-τ(t)))+c∑j=1NgijΓej(t)+d∑j=1NuijDej(t-τ(t))]dt-c∑i=1lξikieiT(t)PΓei(t)dt+∑i=1NξieiT(t)h̃i(t,e(t))dωi(t)+12∑i=1Nξitrace[(h̃i(t,e(t)))Th̃i(t,e(t))]dt+12∑j=1n[ẽjT(t)Qjẽj(t)-(1-τ̇(t))ẽjT(t-τ(t))Qjẽj(t-τ(t))]dt={∑i=1NξieiT(t)P(F(t,ei(t),ei(t-τ(t)))-Δei(t))+∑i=1NξieiT(t)PΔei(t)-c∑i=1lξikieiT(t)PΓei(t)+∑i=1NξieiT(t)P(c∑j=1NgijΓej(t)+d∑j=1NuijDej(t-τ(t)))+12∑i=1Nξitrace[(h̃i(t,e(t))Th̃i(t,e(t))]+12∑j=1n[ẽjT(t)Qjẽj(t)-(1-τ̇(t))ẽjT(t-τ(t))Qjẽj(t-τ(t))]}dt+∑i=1NξieiT(t)h̃i(t,e(t))dωi(t)≤{-η∑j=1nẽjT(t)Ξẽj(t)+θ∑j=1nẽjT(t-τ(t))Ξẽj(t-τ(t))+∑j=1npjδjẽjT(t)Ξẽj(t)+c∑j=1npjγjẽjT(t)(ΞG¯)sẽj(t)+d∑j=1npjdjẽjT(t)ΞUẽj(t-τ(t))+12∑j=1nẽjT(t)Φ̃ẽj(t)+12∑j=1n[ẽjT(t)Qjẽj(t)-(1-h)ẽjT(t-τ(t))Qjẽj(t-τ(t))]}dt+∑i=1NξieiT(t)h̃i(t,e(t))dωi(t)=12∑j=1nζjTΠjζjdt+∑i=1NξieiT(t)h̃i(t,e(t))dωi(t),
where ζj=(ẽjT(t),ẽjT(t-τ(t)))T and
(3.15)Πj=[-2ηΞ+2pjδjΞ+2cpjγj(ΞG¯)s+Φ+QjdpjdjΞUdpjdj(ΞU)T-(1-h)Qj+2θΞ],j=1,2,…,n.
In view of Lemma 2.5, Πj<0 is equivalent to
(3.16)Υj=-2ηΞ+2pjδjΞ+2cpjγj(ΞG¯)s+Φ+Qj+(dpjdj)2(ΞU)[(1-h)Qj-2θΞ]-1(ΞU)T<0-(1-h)Qj+2θΞ<0.
Let Qj=(1/(1-h))[dpjdj∥ΞU∥IN+2θΞ]; we have
(3.17)Υj≤-2ηΞ+2pjδjΞ+2cpjγj(ΞG¯)s+Φ+2-h1-hdpjdj∥ΞU∥IN+2θ1-hΞ=[2cpjγj(ΞG¯)ls-2cpjγj(ΞK)l+(Wj)l2cpjγj((ΞG¯)*s)T2cpjγj(ΞG¯)*s2cpjγj(ΞG¯)N-ls+(Wj)N-l],
where (ΞG¯)*s is the matrix with appropriate dimension.
Since2cpjγj(ΞG)N-ls+(Wj)N-l<0, (ΞG¯)N-ls=(ΞG)N-ls, there exist positive constants k1,k2,…,kl such that
(3.18)2cpjγj(ΞG¯)ls-2cpjγj(ΞK)l+(Wj)l-4c2pj2γj2((ΞG¯)*s)T[2cγj(ΞG¯)N-ls+(Wj)N-l]-1(ΞG¯)*s<0.
Again, from Lemma 2.5 we obtain Υj<0. Hence, (3.16) holds, which implies Πj<0. The rest steps are the same as those in the proof of Theorem 3.1. This completes the proof.Remark 3.3.
If there is no noise perturbation in (2.1), then Φ=0 in Theorem 3.1 and Φ̃=0 in Theorem 3.2, respectively. Hence, if (2.1) is synchronized with the first l nodes to be controlled by adding controllers (3.1) or (3.12), then (2.1) with no noise perturbation can sure be synchronized by using adapting pinning controllers (3.1) or (3.12) with no more than l nodes to be controlled.Remark 3.4.
Theorems3.1 and 3.2 are applicable to directed networks and undirected networks. Although they do not point out which nodes should be controlled first; however, according to [22, 23], for the graph corresponding to G, nodes whose out-degrees are larger than their in-degrees [24] should be controlled first for directed networks, while the controlled nodes can be randomly selected and it is better to successively pin the most-highly connected nodes for undirected networks.


## 4. Numerical Examples

In this section, we provide three examples to illustrate the effectiveness of the results obtained above.Example 4.1.
Consider the following chaotic delayed neural networks:(4.1)ds(t)=[-Cs(t)+Af(s(t))+Bf(s(t-τ))]dt,
where f(s)=tanh(s), τ=1,
(4.2)C=[1001],A=[2-0.1-54.5],B=[-1.5-0.1-0.2-4].
In the case that the initial condition is chosen as s1(t)=0.4, s2(t)=0.6, for all t∈[-1,0], the chaotic attractor can be seen in Figure 1.
TakingP=diag(1,1) and Δ=diag(5,11.5) and we have η=0.15, θ=3.25. Hence the condition (2.7) is satisfied.
In order to verify our new results, consider the following coupled networks:(4.3)dxi(t)=[-Cxi(t)+Af(xi(t))+Bf(xi(t-τ))+c∑j=13gijΓxj(t)+d∑j=13uijDxj(t-τ)]dt+hi(x1(t),x2(t),x3(t))dωi(t),i=1,2,3,
where Γ and D are 2-dimensional identity matrices, c=4, d=2, and G and U are asymmetric and zero-row sum matrices as the following:
(4.4)G=[-1010-1146-10],U=[-2110-1120-2].
Through simple computation, we get that the left eigenvector ofG corresponding to eigenvalue 0 is (0.5494,0.8242,0.1374)T and ∥ΞU∥=1.3901.
Lethi(x1(t),x2(t),x3(t)) = 0.2diag(xi,1(t)-xi+1,1(t),xi,2(t)-xi+1,2(t)), i=1,2,3, where x4(t)=x1(t). Obviously,
(4.5)trace((h̃i(e(t)))Th̃i(e(t)))=0.04[(xi1(t)-xi+1,1(t))2+(xi2-xi+1,2(t))2]=0.04[(ei1(t)-ei+1,1(t))2+(ei2(t)-ei+1,2(t))2]≤0.08(∥ei(t)∥2+∥ei+1(t)∥2),i=1,2,3,
where e4(t)=e1(t). Hence, the assumption conditions (H1) and (H2) are satisfied. Selecting the first two nodes to be controlled, we have 2cp1γ1(ΞG)3-2s+(W1)3-2<2cp2γ2(ΞG)3-2s+(W2)3-2=-1.2986<0. According to Theorem 3.2, the complex networks (4.3) can be controlled to the state of (4.1) under the adaptive pinning controllers (3.12) with the first two nodes to be controlled.
The initial conditions of the numerical simulations are as follows:xi(s)=(-4+12i),-10+12i)T, 1≤i≤3, for all s∈[-1,0], ɛ1(0)=ɛ2(0)=1, v1=v2=0.04. Figure 2 shows the time evolutions of synchronization errors eij(t)=xij(t)-sj(t), i=1,2,3, j=1,2, with the first two nodes being controlled, which verify Theorem 3.2 perfectly. The trajectories of the gains are shown in Figure 3.Figure 1
Chaotic trajectory of system (4.1).Time evolutions of synchronization errors with the first two nodes being controlled.
(a)(b)Figure 3
Trajectories ofɛ1(t) and ɛ2(t) for (4.3) with adaptive pinning control.Example 4.2.
Consider the following coupled neural networks:(4.6)dxi(t)=[-Cxi(t)+Af(xi(t))+Bf(xi(t-τ))+c∑j=110gijΓxj(t)+d∑j=110uijDxj(t-τ)]dt+hi(x1(t),x2(t),…,x10(t))dωi(t),i=1,2,3,…,10,
where c=1.7, d=0.3, hi(x1(t),x2(t),…,x10(t)) = 0.1diag(xi,1(t)-xi+1,1(t),xi,2(t)-xi+1,2(t)), i=1,2,…,10, x11(t)=x1(t), and
(4.7)G=U=[-11000000001-10000000010-10000000010-10000001100-20000011000-20000110000-20001100000-20011000000-20110000000-2],Γ=D=[1000].
Take the first two nodes to be pinned. Through simple computation, we obtain that all the conditions of Theorem 3.1 are satisfied. Hence, system (4.6) is synchronized with adaptive pinning controllers (3.1).
The initial conditions of the numerical simulations are as follows:xi(s)=(-4+12i),-10+12i)T, 1≤i≤10, for all s∈[-1,0], ɛ1(0)=ɛ2(0)=1, v1=v2=1. Figure 4 shows the time evolutions of synchronization errors with adaptive pinning control. The trajectories of the adaptive pinning control gains are shown in Figure 5.Trajectories ofɛ1(t) and ɛ2(t) for (4.6) with adaptive pinning control.
(a)(b)Figure 5
Trajectories ofɛ1(t) and ɛ2(t) for (4.6) with adaptive pinning control.Example 4.3.
Consider the delayed Chua's circuit:(4.8)ds(t)=[C¯s(t)+g1(s(t))+g2(s(t-τ))]dt,
where s(t)=(s1(t),s2(t),s3(t))T, g1(s(t))=(-(1/2)δ(a-b)(|s1(t)+1|-|s1(t)-1|),0,0)T, g2(s(t-τ))=(0,0,-ξϵsin(νs1(t-τ)))T,
(4.9)C¯=[-δ(1+b)δ01-110-ξ-ω],
and δ=10,ξ=19.53,ω=0.1636,a=-1.4325,b=-0.8731,ν=0.5,ϵ=0.2,τ=0.02.
In the case that the initial condition is chosen asst(t)=2; s2(t)=0.2; s3(t)=0.3, for all t∈[-0.02,0], the chaotic attractor can be seen in Figure 6.
Takek11=(1/2)δ(b-1)+δ(1+b)=5.416, k12=δ=10, k13=0; k21=k22=k23=1, k31=ξϵν=1.953, k32=ξ=19.53; k33=ω=0.1636, P = diag(1,1,1), Δ = diag(19,18,23), we have η=0.035, θ=15.265. Hence the condition (2.7) is satisfied.
Now we construct a complex network, which obeys the scale-free distribution of the Barabási-Albert model [17]. The parameters in the process of constructing are the following: initial graph is complete with m0=5 nodes, m=5 edges are added to the network when a new node is introduced, and the final number of nodes is N=100. See Figure 7 for the BA scare-free network.
Consider the following complex networks:(4.10)dxi(t)=[C¯xi(t)+g1(xi(t))+g2(xi(t-τ))+6∑j=1100gijxj(t)+0.35∑j=1100gijxj(t-τ)]dt+0.4hi(x1(t),x2(t),…,x100(t))dωi(t),i=1,2,…,100,
where G=U=(gij)100×100 is the graph laplacian of the BA scale-free network, hi(x1(t),x2(t),…,x100(t))=diag(xi,1(t)-xi+1,1(t),xi,2(t)-xi+1,2(t),xi,3(t)-xi+1,3(t)) with x101(t)=x1(t), i=1,2,…,100.
Without loss of generality, we take the first 10 nodes to be controlled. The initial conditions of the numerical simulations are as follows:xi(s)=(-8+i,-5+i,-10+i)T, s∈[-0.02,0], 1≤i≤100, ɛj(0)=1, vj=0.04. For the synchronization errors eij(t)=xij(t)-sj(t), 1≤i≤100, j=1,2,3 and trajectories of the control gains; see Figures 8 and 9.Figure 6
Chaotic trajectory of system (4.8).Figure 7
BA scale-free network initial graph is complete withm0=5 nodes; m=5 edges are added in the network when a new node is introduced and the final number of nodes is N=100.Synchronization errorsei1(t) (a), ei2(t) (b), and ei3(t) (c) (1≤i≤100) of (4.10) with the first 10 nodes being controlled.
(a)(b)(c)Figure 9
Trajectories of control gainsɛi(t), 1≤i≤10 for (4.10).Remark 4.4.
In Example4.2, only the first state of all nodes is coupled; however, the complex network (4.6) is synchronized by adaptive pinning of the first two nodes, which verifies the effectiveness of our theoretical results. In [7, 12, 13, 19], however, all states of the nodes should be coupled. Therefore, our results extend some of existing results. On the other hand, Examples 4.1 and 4.2 are directed networks, while Example 4.3 is undirected network with 100 nodes. Examples with numerical simulations show that our theoretical results are applicable to directed and undirected networks, even networks of large size.


## 5. Conclusions

In this paper, we have studied the adaptive pinning synchronization of the complex networks with delays and vector-form stochastic perturbations, in which dynamical behaviors are more realistic and complicated. Since the new condition QUAD1, our proof is very simple. Via two simple adaptive pinning feedback control schemes, several sufficient conditions guaranteeing the synchronization of the proposed complex networks are obtained. Our results are valid even if only partial states of nodes are coupled. Numerical simulations verified the effectiveness of our results. Our results improve and extend some existing results. Models and results in this paper provide possible new applications for network designers.



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