# Synchronization and Pinning Control in Complex Networks with Interval Time-Varying Delay

Authors: Hai-Feng Jiang; Tao Li
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The problems on synchronization and pinning control for complex dynamical
networks with interval time-varying delay are investigated and two less conservative criteria are established based on reciprocal convex technique. Pinning control strategies are designed to make the complex
networks synchronized. Moreover, the problem of designing controllers can be converted into solving a
series of NMIs (nonlinear matrix inequalities) and LMIs (linear matrix inequalities), which reduces the
computation complexity when comparing with those present results. Finally, numerical simulations can
verify the effectiveness of the derived methods.


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## Body

## 1. Introduction

During the past decades, complex dynamical networks have increasingly become a focal research topic and received much attention in various fields such as physics, chemistry, and computer science [1–4]. Such systems in the real world usually consist of a large number of highly interconnected dynamical units. Transportation networks, coupled engineering systems, artificial neural networks in human brains, and the Internet are only the typically realistic examples. In order to capture their evolving properties, several famous models have been proposed, such as scale-free networks, random networks, and small world networks.Recently, the research on synchronization and dynamical behaviors of complex networks has become an important direction, and it was mainly based on the techniques including analyzing topological structure and studying the origin of the intrinsic self-organized dynamics; see [5–7] and references therein. Authors in [5–7] have introduced one uniform dynamic network model and studied the phenomena of synchrony of all network states, with some derived conditions that can guarantee the asymptotical synchronization or projective cluster synchronization. Meanwhile, since there inevitably exists communication delay which is the main source of poor dynamical behaviors in various networks, some efforts have been applied to the synchronization of complex systems [8–21]. More recently, some researchers have considered the global or robust or cluster synchronization for various coupled delayed neural networks based on some effective techniques including LMI one in [8–12]. As for complex networks with linear or nonlinear couplings, some delay-dependent criteria are presented to guarantee the synchronization in [13–16], and moreover in [17, 18], some synchronization results are achieved for complex networks based on linear feedback control or adaptive feedback control. Additionally, to avoid the conservatism, some authors employed delay-partitioning idea to tackle the constant delay or time-variant one in [19, 20]. Yet, it has come to our attention that though some delay-dependent methods have appeared, the techniques above still require great improvements owing to ignoring some important terms when estimating the derivative of Lyapunov functional, especially for the case of interval time-varying delay [21].Presently, the control problem on scale-free dynamical networks was investigated by applying local linear feedback injections to a small fraction of network nodes [22, 23]. Meanwhile, some researchers have considered pinning control of delayed complex networks in [24–31]. In [24], cluster synchronization for stochastic delayed neural networks was studied based on pinning control and LMI approach. By adopting pinning periodically intermittent control, the work [25] studied the exponential synchronization and yet gave some uneasy-to-check results. With existence of various couplings, the synchronization has been analyzed for delayed complex networks using pinning adaptive control in [26]. In [27], the authors have presented linear feedback and adaptive feedback pinning control to synchronize the delayed complex networks. Yet, we notice that as for constructions of Lyapunov functional and analyzed techniques described in [24–29], there still exists much room waiting for the further improvement, which can be fully considered and tackled in this presented work.In this paper, the problems on synchronization and pinning control for delayed dynamical networks are studied based on pinning control strategies. Then the decentralized feedback controllers are designed to make the addressed networks synchronized. Moreover, since the most improved techniques including reciprocal convex technique have been applied, the design of controllers can be converted into solving optimal solution of an array of NMIs and LMIs. These conditions can be extended to delayed complex networks with different topology and different sizes.Notations 1.
Rn denotes the n-dimensional Euclidean space, and Rn×m is the set of all n×m real matrices. AT stands for the transpose of matrix A; I represents the identity matrix of an appropriate dimension; A⊗B indicates the Kronecker product of the m×n matrix A and p×q matrix B; [XYYTZ]=[XY*Z] with * denoting the symmetric term in a symmetric matrix.


## 2. Problem Formulations

Suppose that the nodes are coupled with statesxi(·), i∈{1,…,N}, then the complex dynamical networks can be described by(2.1)ẋi(t)=f(xi(t))+g(xi(t-τ(t)))+∑j=1Nlij1Exj(t)+∑j=1Nlij2Fxj(t-τ(t)),
where xi(t)=[xi1(t),xi2(t),…,xin(t)]T are the state vectors, and f(·),g(·):Rn→Rn are the continuously differentiable functions; here, E=[eij]n×n and F=[fij]n×n are the inner coupling matrices between the connected nodes i and j at time t and t-τ(t), respectively.For the dynamical system (2.1), the following assumptions are utilized throughout this paper:(A1)τ(t) denotes the interval time-varying delay satisfying(2.2)0≤τ0≤τ(t)≤τm,τ̇(t)≤μ<+∞,
and we set τ̅m=τm-τ0,(A2) here,Lk=[lijk]N×N(k=1,2) is the configuration matrix that is irreducible and satisfies(2.3)liik=-∑j=1,j≠iNlijk=-∑j=1,j≠iNljik,k=1,2;i=1,2,…,N,
with lijk=ljik>0 if there is a connection between node i and the one j, and otherwise, lijk=0. Furthermore, suppose that the eigenvalue λi(νi) of the matrix L1(L2) can be ordered as follows:(2.4)0=λ1>λ2≥⋯≥λN(0=ν1>ν2≥⋯≥νN).Prior to addressing the synchronization results, the following definition and lemmas are introduced.Definition 2.1 (see [28]).
The complex networks described by (2.1) are said to achieve the asymptotical synchronization
(2.5)x1(t)=x2(t)=⋯=xN(t)=s(t),
where s(t) is the solution of the local dynamics of an isolate node satisfying
(2.6)ṡ(t)=f(s(t))+g(s(t-τ(t))).Lemma 2.2 (see [32]).
For any constant matrixW∈Rn×n, W=WT>0, scalar functional 0≤r(t)≤rM, and a vector function ė:[-rM,0]→Rn such that the following integration is well defined, then
(2.7)-rM∫-r(t)0ėT(t+s)Wė(t+s)ds≤[e(t)-e(t-r(t))]TW[e(t)-e(t-r(t))].Lemma 2.3 (see [33]).
Let the functionsf1(t),f2(t),…,fN(t):Rm→R have the positive values in an open subset D of Rm, then for αi>0 satisfying ∑iαi=1 and gi,j(t):Rm→R satisfying [fi(t)gi,j(t)gj,i(t)fj(t)]≥0, one has that
(2.8)∑i1αifi(t)≥∑ifi(t)+∑i≠jgi,j(t).As for the complex networks (2.1) and symmetric matrices L1, L2, if the following N-1 systems of n-dimensional linear delayed differential systems(2.9)ω̇i(t)=(J(t)+λiE)ωi(t)+(Γ(t)+νiF)ωi(t-τ(t)),i=2,…,N
are asymptotically stable with respect to their zero solutions, the complex networks (2.1) can achieve the synchronization at s(t), where J(t)Ḃf′(s(t)) is the Jacobian matrix of f(x(t)) at s(t), and Γ(t)Ḃg′(x(t-τ(t))) is the Jacobian matrix of g(x(t-τ(t))) at s(t-τ(t)).


## 3. Synchronization Criterion for Complex Networks

In this section, we will derive one delay-derivative-dependent synchronization criterion for the complex networks in (2.1), which can be much more effective than those present ones based on some most developed techniques in [28–33].Theorem 3.1.
Considering the time delay in (2.2), the complex networks (2.1) can achieve the asymptotical synchronization if there exist the appropriately dimensional matrices Pi>0, Qi>0, Ri>0, Ti>0, Ui>0, Vi, Si, Hi, Mi, and Ni  for i=2,…,N, satisfying the matrix inequalities in (3.1),
(3.1)[ViSiSiTVi]≥0,i=2,…,N,[Ωi11UiΩi130Ωi15*Ωi22Vi-SiSi0**Ωi33Vi-SiΩi35***-Ri-Vi0****Ωi55]<0,
with
(3.2)Ωi11=-Ui+Qi+HiT(J(t)+λiE)+(J(t)+λiE)THi,Ωi13=HiT(Γ(t)+νiF)+(J(t)+λiE)TNi,Ωi15=Pi-HiT+(J(t)+λiE)TMi,Ωi22=-Qi+Ri+Ti-Vi-Ui,Ωi33=-(1-μ)Ti-NiT(Γ(t)+νiF)+(Γ(t)+νiF)TNi-2Vi+Si+SiT,Ωi35=(Γ(t)+νiF)TMi-NiT,Ωi55=-MiT-Mi+τ02Ui+τ̅m2Vi.Proof.
As for theith node, we choose the following Lyapunov-Krasovskii functional:
(3.3)Vi(ωit)=ωiT(t)Piωi(t)+∫t-τ0tωiT(s)Qiωi(s)ds+∫t-τmt-τ0ωiT(s)Riωi(s)ds+∫t-τ(t)t-τ0ωiT(s)Tiωi(s)ds+∫-τ00∫t+θtτ0ω̇iT(s)Uiω̇i(s)dsdθ+∫-τm-τ0∫t+θtτ̅mω̇iT(s)Viω̇i(s)dsdθ,
with symmetric matrices Pi, Qi, Ri, Ti, Ui, and Vi waiting to be determined. Then for any n×n matrices Hi, Mi, and Ni, the time derivative of Vi(ωi(t)) along the system (2.9) can be derived as
(3.4)V̇i(ωit)=2ωiT(t)Piω̇i(t)+ωiT(t)Qiωi(t)-ωiT(t-τ0)(Qi-Ri-Ti)ωi(t-τ0)-ωiT(t-τm)Riωi(t-τm)-(1-τ̇(t))ωiT(t-τ(t))Tiωi(t-τ(t))+τ02ω̇iT(t)Uiω̇i(t)-∫t-τ0tτ0ω̇iT(s)Uiω̇i(s)ds+τ̅m2ω̇iT(t)Viω̇i(t)-∫t-τ(t)t-τ0τ̅mω̇iT(s)Viω̇i(s)ds-∫t-τmt-τ(t)τ̅mω̇iT(s)Viω̇i(s)ds+[ωiT(t)HiT+ω̇iT(t)MiT+ωiT(t-τ(t))NiT]×[-ω̇i(t)+(J(t)+λiE)ωi(t)+(Γ(t)+νiF)ωi(t-τ(t))].
Through employing Lemmas 2.2 and 2.3, we can yield that
(3.5)-∫t-τ0tτ0ω̇iT(s)Uiω̇i(s)ds≤[ωi(t)-ωi(t-τ0)]TUi[ωi(t)-ωi(t-τ0)],-∫t-τ(t)t-τ0τ̅mω̇iT(s)Viω̇i(s)ds-∫t-τmt-τ(t)τ̅mω̇iT(s)Viω̇i(s)ds≤-τ̅mτ(t)-τ0[ωi(t-τ0)-ωi(t-τ(t))]TVi[ωi(t-τ0)-ωi(t-τ(t))]-τ̅mτm-τ(t)[ωi(t-τ(t))-ωi(t-τm)]TVi[ωi(t-τ(t))-ωi(t-τm)]≤-[ωi(t-τ0)-ωi(t-τ(t))ωi(t-τ(t))-ωi(t-τm)]T[ViSiSiTVi][ωi(t-τ0)-ωi(t-τ(t))ωi(t-τ(t))-ωi(t-τm)].
Then together with the terms (3.4)-(3.5), it can be deduced that
(3.6)V̇i(ωit)≤[ωi(t)ωi(t-τ0)ωi(t-τ(t))ωi(t-τm)ω̇i(t)]T[Ωi11UiΩi130Ωi15*Ωi22Vi-SiSi0**Ωi33Vi-SiΩi35***-Ri-Vi0****Ωi55][ωi(t)ωi(t-τ0)ωi(t-τ(t))ωi(t-τm)ω̇i(t)].
Then there must exist one scalar ε>0 such that V̇i(ωit)≤-ε∥Pωi(t)P∥<0 for any ωi(t)≠0. Based on the Lyapunov-Krasovskii stability theory, all nodes in complex networks (2.1) are asymptotically synchronized, and it completes the proof.Remark 3.2.
Presently, the convex combination technique has been widely employed to tackle time-varying delay owing to that it could reduce the conservatism more effectively than the previous methods; see [21, 32]. In [33], the authors put forward the reciprocal convex approach, which can reduce the conservatism more effectively than convex combination ones. Yet, it has come to our attention that no researchers have utilized reciprocal convex idea to tackle the synchronization for delayed complex networks.


## 4. Pinning Control of Delayed Complex Networks

Suppose that we want to stabilize the system (2.1) onto a homogeneous stationary state. In order to achieve the goal, we will apply the pinning control strategy on a small fraction δ(0<δ≪1) of the nodes in network (2.1). Suppose that the nodes i1,i2,…,il are selected to be under pinning control, and il+1,il+2,…,iN are the unselected ones, where l=[δN] stands for the smaller but nearest to the real number δN, then l is no less than 0 for pinning control, and the controlled networks can be described by(4.1)ẋi(t)=f(xi(t))+g(xi(t-τ(t)))+∑j=1Nlij1Exj(t)+∑j=1Nlij2Fxj(t-τ(t))+ui(t),i=1,2,…,l,ẋi(t)=f(xi(t))+g(xi(t-τ(t)))+∑j=1Nlij1Exj(t)+∑j=1Nlij2Fxj(t-τ(t)),i=l+1,l+2,…,N.
For a special case that τ(t) is available, we adopt the following local linear feedback control law:(4.2)ui(t)=-kiE[xi(t)-s(t)]-liF[xi(t-τ(t))-s(t-τ(t))],
with the feedback gains ki>0,li>0.The aim of this section is to stabilize the network (4.1) to be of homogenous stationary state defined by(4.3)x1(t),x2(t),…,xN(t)⟶s(t),ṡ(t)=f(s(t))+g(s(t-τ(t))),ast⟶+∞.
Denoting ki=li=0(i=l+1,l+2,…,N), then we can derive that(4.4)ẋi(t)=f(xi(t))+g(xi(t-τ(t)))+∑j=1Nlij1Exj(t)+∑j=1Nlij2Fxj(t-τ(t))-kiE[xi(t)-s(t)]-liF[xi(t-τ(t))-s(t-τ(t))],i=1,2,…,N.
If we let ei(t)=xi(t)-s(t) and e(t)=[e1(t),e2(t),…,eN(t)]T, then we can linearize the network (4.4) as(4.5)ėi(t)=J(t)ei(t)+Γ(t)ei(t-τ(t))+∑j=1Nlij1Eej(t)+∑j=1Nlij2Fej(t-τ(t))-kiEei(t)-liFei(t-τ(t)),
where J(t) and Γ(t) are the Jacobian matrices of the functions f,g at s(t) and s(t-τ(t)), respectively.In what follows, we defineK=diag(k1,k2,…,kN), L=diag(l1,l2,…,lN), and(4.6)L1=[lij1]N×N,L2=[lij2]N×N,e(t)=[e1T(t),e2T(t),…,eNT(t)]T.
Then based on Kronecker product, we can deduce(4.7)ė(t)=(IN⊗J(t))e(t)+(IN⊗Γ(t))e(t-τ(t))+((L1-K)⊗E)e(t)+((L2-L)⊗F)e(t-τ(t)).Assumption 4.1.
The matricesL1-K and L2-L satisfy [(L1-K)(L2-L)]T=(L1-K)(L2-L).Then there does exist an orthogonal matrixU such that U(L1-K)UT=Λ̅ and U(L2-L)UT=Π̅, in which Λ̅=diag(λ̅1,λ̅2,…,λ̅N) and Π̅=diag(ν̅1,ν̅2,…,ν̅N) satisfy 0>λ̅1≥λ̅2≥⋯≥λ̅N and 0>ν̅1≥ν̅2≥⋯≥ν̅N.Denotingη(t)=(UT⊗In)e(t)=[η1T(t),η2T(t),…,ηNT(t)]T and employing the property of Kronecker product (A1⊗B1)(A2⊗B2)=(A1A2)⊗(B1B2), one can derive(4.8)η̇(t)=(IN⊗J(t))η(t)+(IN⊗Γ(t))η(t-τ(t))+(Λ̅⊗E)η(t)+(Π̅⊗F)η(t-τ(t)),
that is,(4.9)η̇i(t)=[J(t)+λ̅iE]ηi(t)+[Γ(t)+ν̅iF]ηi(t-τ(t)),i=1,2,…,N.Theorem 4.2.
Considering the time delay in (2.2), the controlled networks (4.9) can achieve local asymptotical stability at s(t) if there exist the appropriately dimensional matrices Pi>0, Qi>0,Ri>0,Ti>0,Ui>0, Vi>0,Si,Hi, Mi, and Ni for i=1,2,…,N, satisfying the following inequalities in (4.10):
(4.10)[ViSiSiTVi]≥0,i=1,2,…,N,[Ξi11UiΞi130Ξi15*Ξi22Vi-SiSi0**Ξi33Vi-SiΞi35***-Ri-Vi0****Ξi55]<0,
with
(4.11)Ξi11=-Ui+Qi+HiT(J(t)+λ̅iE)+(J(t)+λ̅iE)THi,Ξi13=HiT(Γ(t)+ν̅iF)+(J(t)+λ̅iE)TNi,Ξi15=Pi-HiT+(J(t)+λ̅iE)TMi,Ξi22=-Qi+Ri+Ti-Vi-Ui,Ξi33=-(1-μ)Ti-NiT(Γ(t)+ν̅iF)+(Γ(t)+ν̅iF)TNi-2Vi+Si+SiT,Ξi35=(Γ(t)+ν̅iF)TMi-NiT,Ξi55=-MiT-Mi+τ02Ui+τ̅m2Vi,
where λ̅i and ν̅i are the ith eigenvalues of the matrices L1-K and L2-L, respectively.Proof.
As for theith node, we choose the following Lyapunov-Krasovskii functional:
(4.12)Vi(ηit)=ηiT(t)Piηi(t)+∫t-τ0tηiT(s)Qiηi(s)ds+∫t-τmt-τ0ηiT(s)Riηi(s)ds+∫t-τ(t)t-τ0ηiT(s)Tiηi(s)ds+∫-τ00∫t+θtτ0η̇iT(s)Uiη̇i(s)dsdθ+∫-τm-τ0∫t+θtτ̅mη̇iT(s)Viη̇i(s)dsdθ,
with appropriately dimensional matrices Pi>0,Qi>0, Ri>0, Ti>0,Ui>0,Vi>0 waiting to be determined.
Together with similar proof of Theorem3.1, one can yield
(4.13)V̇i(ηit)≤[ηi(t)ηi(t-τ0)ηi(t-τ(t))ηi(t-τm)η̇i(t)]T[Ξi11UiΞi130Ξi15*Ξi22Vi-SiSi0**Ξi33Vi-SiΞi35***-Ri-Vi0****Ξi55][ηi(t)ηi(t-τ0)ηi(t-τ(t))ηi(t-τm)η̇i(t)].
Then there does exist one scalar ε>0 such that V̇i(ηi(t))≤-ε∥Pηi(t)P∥<0 for any ηi(t)≠0. Based on the theory of Lyapunov stability, the controlled network (4.9) is locally stable for any time-delay satisfying (2.2). The proof is completed.Remark 4.3.
Based on Theorem4.2, for the controlled dynamical network with fixed network structure and coupling strength as in (4.1), it is easy to determine the number of pinned nodes for network synchronization and select nodes to pin since there are no restrictions for the positions of the pinned nodes. Moreover, the reciprocal convex technique can reduce the conservatism more effectively than those earlier methods.


## 5. Numerical Examples

Now, two numerical examples with simulations will be presented to illustrate efficiency of our results.Example 5.1.
Consider one 3-node complex network, in which each node is a simple 3-dimensional nonlinear system [28],
(5.1)[ẋi1(t)ẋi2(t)ẋi3(t)]=[-xi1(t)+xi22(t)-2xi2(t)-3xi3(t)+xi2(t)xi3(t)],i=1,2,3,
which is asymptotically stable at s(t)=0, and its Jacobian matrix is J=diag(-1,-2,-3). Assuming that the inner coupling matrices and the coupling configuration ones can be represented as
(5.2)E=F=[100010001],L1=L2=[-10.50.50.5-10.50.50.5-1],
then the eigenvalues of matrices L1, L2 can be computed as 0,-1.5,-1.5, respectively.In the case ofτ0=μ=0, using [28, Theorem 1] in  gives τmax=0.615, while applying theorem in this paper can verify that the maximum delay upper bound guaranteeing the synchronized state to be asymptotically stable is τmax=0.867. This means that our results can be less conservative than some existing ones.Whenτ(t)=sin2(2t), it follows from Theorem 3.1 that the feasible solution to the condition in (3.1) can be existent. Defining the total error of the array of the system (5.1) as follows:(5.3)error(t)=∑i=13[x1i(t)-x2i(t)]2+[x2i(t)-x3i(t)]2,
then the synchronization error can be seen in Figure 1, and it shows that system (5.1) can achieve the asymptotical synchronization. During the process of simulation, the initial conditions of nodes are selected as x1=[-0.5,-0.3,0.3]T, x2=[0.7,-0.5,-0.6]T, and x3=[1,0.5,0.3]T.Figure 1
The total synchronous error of system (5.1).Example 5.2.
Together with local linear feedback pinning control in (4.2) to the first node and setting feedback gains k1=l1=1, we consider the following nonlinear system,
(5.4)[ẋi1(t)ẋi2(t)]=[-xi1(t)+xi22(t)-3xi1(t)+xi1(t)xi2(t)]+[2xi1(t-τ(t))-xi2(t-τ(t))xi1(t-τ(t))+3xi2(t-τ(t))],i=1,2,3,
with τ(t)=cos2(2t). Its Jacobians are J=diag(-1,-3) and Γ=[2-113], respectively. The inner coupling matrices and the coupling configuration ones can be represented as
(5.5)E=F=[1001],L1=L2=[-10.50.50.5-10.50.50.5-1].
Then we can stabilize the network onto the original equilibrium point s̅=[0,0,0]T based on Theorem 4.2, and the feasible solution to (4.10) can be existent with the eigenvalues of matrices L1-K and L2-L being {-2.9808,-1.5000,-0.2192}. We also define the total error of the array of system (5.4) as follows:
(5.6)error(t)=∑i=12[x1i(t)-x2i(t)]2+[x2i(t)-x3i(t)]2,
then the synchronization error can be seen in Figure 2, and it shows that system (5.4) can achieve the asymptotical synchronization. During the process of simulation, the initial conditions of nodes are selected as x1=[-0.5,-0.3]T, x2=[0.7,-0.5]T, and x3=[0.4,0.3]T.Figure 2
The total synchronous error of system (5.4).


## 6. Conclusions

In this paper, the asymptotical synchronization and pinning control for delayed dynamical networks have been studied based on Lyapunov-Krasovskii functional theory, and the decentralized feedback controller has been designed to make the networks synchronized. Moreover, since the most improved techniques including reciprocal convex one have been applied, the established results can be more applicable than those present ones, and the design of controllers can be achieved by solving an array of NMIs and LMIs. Finally, two illustrative examples can show the efficiency of the proposed methods.



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