# New Pinning Synchronization of Complex Networks with Time-Varying Coupling Strength and Nondelayed and Delayed Coupling

Authors: Guoliang Wang; Zhongbao Yue; Feng Wang
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

The pinning synchronization problem for a class of complex networks
is studied by a stochastic viewpoint, in which both time-varying coupling strength and nondelayed and delayed coupling are included. Different from the traditionally similar methods, its interval is separated into two subintervals and described by a Bernoulli variable. Both bounds and switching probability of such
subintervals are contained. Particularly, the nondelayed and delayed couplings occur alternately in which another independent Bernoulli variable is introduced. Then, a new kind of pinning controller without time-varying coupling strength signal is developed, in which only its bounds and probabilities are contained. When such probabilities are unavailable, two different kinds of adaption laws are established to make the complex network globally synchronous. Finally, the validity of the presented methods is proved through a numerical example.


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## Body

## 1. Introduction

In the recent years, because of many systems in all fields of sciences and society, complex networks (CNs) have drawn a lot of attention. The property of CNs is that a large set of interconnected nodes are contained, and each node is an essential unit. Many natural and manmade systems can be found in our surroundings, such as ecosystems, internet, World Wide Web, social networks, and power grids, which are very important in our daily life. Due to the broad applications, the research of complex behaviors in complex networks has become a hot topic across many fields such as in [1–7].In complex networks, such as ER random, small-world, and scale-free CNs [8–10], one of the most important collective behaviors is named as synchronization and has been studied extensively. Because of a large number of nodes existing in CNs, it is unrealistic to control CN by adding controllers to all nodes. Then, it is natural to exploit a practical control strategy reducing the number of controllers usually referred to as pining control [11, 12]. Up to now, a lot of results on pinning synchronization of various CNs have emerged, for example, [13–21]. When there are some parameters unknown, the problem of adaptive synchronization was reported in [22–24]. If there are constant or time-varying delays in coupling, some delay-dependent criteria for synchronization stability were established in [25–27]. As for CN with nondelayed and delayed coupling, the pinning synchronization problem was considered in [28]. It is seen that the underlying system in this reference has the nondelayed and delayed couplings occurring simultaneously. If such two couplings happen separately with some probability, how to consider the pinning synchronization problem will be necessary and interesting. On the other hand, by investigating the aforementioned works, it is known that the coupling strength (CS) in most of them is constant. When CS is time-varying in a limited range, the global synchronization was realized by adjusting time-varying coupling strength (TCS) [29]. Based on the proposed methods, it is concluded that the adjusted CS will vary in a range, whose lower and upper bounds are finite. Though the given adjusting TCS is time-varying, its bounds used in realizing global synchronization may be different from the ones of the original TCS. Moreover, even if such bounds can be equal in theory, they may not be realized in practice. That is because the ability of instruments or actuators is limited in many practical applications. Thus, the method without considering the upper bound of CS may be conservative. In addition, the synchronization in [29] achieved by adjusting TCS needs every node state. It will make the realization not easy when CS has a great deal of nodes. Via exploiting the existing pinning algorithms, a state feedback controller will be constructed as one with TCS signal. In this case, the implementation of the designed pinning controller requires such a signal exactly available online. Unfortunately, from the point of application, this assumption is very ideal and hard to be satisfied. To the best of authors’ knowledge, the pinning control problem of complex networks with TCS and nondelayed and delayed coupling has not yet been investigated, which motivates the current research.In this paper, the pinning synchronization of complex networks with TCS and nondelayed and delayed coupling is studied by a stochastic viewpoint. Compared with the existing results of pinning control methods, the main contributions of this paper are as follows:(1) The pinning synchronization is realized by using a stochastic method. The original range of TCS is separated into two different subintervals, whose switching between such subintervals is described by a Bernoulli variable; (2) based on the proposed method, a kind of local state feedback controller is developed such that the resulting system is global synchronization in probability. It is seen that, without depending on TCS signal directly, the desired pinning controller is only related to the bounds and probabilities of such separated subintervals; (3) instead of the nondelayed and delayed couplings existing simultaneously, such couplings act on the underlying system asynchronously, which is modeled by another independent Bernoulli variable; (4) by the proposed results in this paper, it is concluded that such bounds and probabilities play important roles in the pinning synchronization, whose relationships are given and demonstrated in detail; (5) when such probabilities are unknown or inaccessible, two different adaptation laws for global synchronization are proposed, respectively.Notations.
R
n denotes the n dimensional Euclidean space, Rm×n is the set of all m×n real matrices. C([-τ,0],Rn) is the Banach space of continuous functions mapping the interval [τ,0] into Rn with the norm |ϕ|=sup-τ≤θ≤0|ϕ(θ)|, where |x| denotes the usual Euclidean norm of a vector. E{·} is the expectation operator with respect to some probability measure. λmax(M) denote largest eigenvalues of a matrix M. L is the infinitesimal generator of V(x,t) from Rn×R+ to Rn.


## 2. Problem Formulation

Consider a class of complex networks consisting ofN identical linearly and diffusively coupled nodes; it is described as(1)x˙it=fxit+βtct∑j=1Naijxjt+1-βtct∑j=1Nbijxjt-τ,i∈S,where xi(t)=(xi1(t),xi2(t),xi3(t),…,xin(t))T∈Rn, f:Rn×R+→Rn is a differentiable function and is continuous, τ>0 is the coupling delay, c(t)≥0 is the time-varying coupling strength and varies in Ω≜[c1,c2] with 0≤c1≤c2<∞, and matrices A≜(aij)N×N and B≜(bij)N×N are the coupling configuration matrices and irreducible, whose elements aij and bij are described as follows: aij=aji≥0, for i≠j, aij=-∑j=1,i≠jNaij, and bij=bji≥0, for i≠j, and bij=-∑j=1,i≠jNbij, i,j∈S≜{1,2,…,N}. The initial conditions are given by xi0(t)=ϕi(t)∈C([-τ,0],Rn), ∀t∈[-τ,0]. β(t) is a Bernoulli variable and is described as follows:(2)βt=1,if nondelayed coupling acts,0,if delayed coupling actswhose probability is(3)Pβt=1=Eβt=β∗,Pβt=0=1-β∗.In this paper, TCSc(t) varies in [c1,c2]. That is, for any given constant c^∈[c1,c2], c(t) will take value in [c1,c^] or (c^,c2] randomly. Without loss of generality, c^ can be chosen as c^=(1/2)(c1+c2). So for any c(t), two sets are described as(4)Ω1≜ct:ct∈c1,c^,Ω2≜ct:ct∈c^,c2.A Bernoulli variable α(t) is described as(5)αt=0,ct∈Ω1,1,ct∈Ω2with(6)Pαt=1=Eαt=α∗,Pαt=0=1-α∗,where 1-α∗ and α∗ denote the probabilities of c(t) taking values in Ω1 and Ω2, respectively. Moreover, it can be readily verified that(7)Eαt-α∗=0.Then, c(t) is expressed as(8)ct=1-αtc1t+αtc2t,where c1(t) and c2(t) are described as (9)c1t≜ct,ifαt=0,c^,ifαt=1,c2t≜c^,ifαt=0,ct,ifαt=1.Then, system (1) is equivalent to(10)x˙it=fxit+βtθt∑j=1Naijxjt+1-βtθt∑j=1Nbijxjt-τ,i∈S,where (11)θt=θ1t+θ2t,θ1t=αt-α∗c2t-c1t,θ2t=1-α∗c1t+α∗c2t.Let s(t) be a solution to the isolated node of CN; we have(12)s˙t=fst,where s(t) may be an equilibrium point, a nontrivial periodic orbit, or even a chaotic attractor. For complex network (10) with stochastic variable, some assumptions, definitions, and lemmas are needed to derive our main results.Assumption 1 (see [15]).
SupposingP=diag⁡(p1,p2,…,pn) is a positive definite diagonal matrix and Δ=diag⁡(δ1,δ2,…,δn) is a diagonal matrix, then, for some η>0, all x,y∈Rn, and t>0, f(x,t) can make the following inequality establish: (13)x-yTPfx-fy-Δx+Δy≤-ηx-yTx-y.Definition 2.
Complex network (5) is said to achieve globally asymptotical synchronization almost surely, for any initial conditions xi0 and s0, if the following condition (14)Plimt→∞⁡xit;xi0-st,s0=0=1,i∈Sholds.Definition 3.
The solutionx(t) to complex network (5) is said to be bounded almost surely if (15)supt≥0⁡xt;x0<∞a.s.holds for all x0∈RnN with x(t)=(x1T(t),x2T(t),…,xNT(t))T.Lemma 4 (see [15]).
SupposeA∈RN×N is in accordance with the following conditions: (1)
aij≥0, for i≠j, and aij=-∑j=1,i≠jNaij, ∀i,j∈S;(2)
A is irreducible; then one has the following:
(i)
Real parts of the eigenvalues ofA are all negative, in addition to an eigenvalue 0 with the multiplicity 1.(ii)
A has the right eigenvector (1,1,…,1)T corresponding to the eigenvalue 0.(iii)
Letξ=(ξ1,ξ2,…,ξN)T be the left eigenvector of A corresponding to the eigenvalue 0; then one can let ξi>0, i∈S.Lemma 5.
IfA=(aij)N×N is irreducible with aij=aji≥0 for i≠j and ∑j=1N‍aij=0 for i∈S, then all the eigenvalues of the matrix (16)A~≜a~ijN×N=A-Kare negative, where K≜diag{k1,k2,…,kN} with ki>0, i=1,2,…,l, and ki=0, i∈Sl≜{1,2,…,l}.Proof.
Based on the result [12], it can be obtained directly. Thus it is omitted here.
IfA is an irreducible coupling matrix, let ξ=(ξ1,ξ2,…,ξN)T be the left eigenvector of A corresponding to the eigenvalue 0 and with ξi>0, one has that matrix Ξ=diag(ξ1,ξ2,…,ξN) is positive definite. If A is irreducible, we have(17)ΞA+ATΞ1⋮1=ΞA1⋮1+ATΞ1⋮1=ΞA1⋮1+ATξ1⋮ξN=0which is irreducible. Letting (ΞA~)s=(1/2)(ΞA~+A~TΞ), it is known from Lemma 5 that (ΞA~)s is negative definite.Lemma 6.
For any given appropriate matricesΩ, Φ1, and Φ2 and scalars λ_1≤λ¯1 and λ_2≤λ¯2, then(18)Ω+λ1tΦ1+λ2tΦ2<0,λ1t∈λ_1,λ¯1,λ2t∈λ_2,λ¯2if and only if(19)Ω+λ_1Φ1+λ_2Φ2<0,(20)Ω+λ_1Φ1+λ¯2Φ2<0,(21)Ω+λ¯1Φ1+λ_2Φ2<0,(22)Ω+λ¯1Φ1+λ¯2Φ2<0.Proof.
The necessity is obvious, since (18) holds. Next, we prove the sufficiency. From λ1(t)∈[λ_1,λ¯1] and  λ2(t)∈[λ_2,λ¯2], it is concluded that(23)λ1t=t1λ_1+1-t1λ¯1,∀t1∈0,1,λ2t=t2λ_2+1-t2λ¯2,∀t2∈0,1.Then, from (19) ×t1 plus (21) ×(1-t1), one gets(24)Ω+λ_1tΦ1+λ_2Φ2≤0.Similarly,(25)Ω+λ_1tΦ1+λ¯2Φ2≤0.By applying similar method to (24) and (25), we obtain(26)Ω+λ1tΦ1+λ2tΦ2≤0,where “=” is satisfied if and only if t1=0 or t1=1 and t2=0 or t2=1 hold simultaneously. However, without loss of generality, if t1=0 and t2=0, it is seen that(27)Ω+λ¯1Φ1+λ¯2Φ2≤0which is contradicted with (22). This completes the proof.Here, the objective of pinning synchronization is to achieveP{limt→∞|xi(t;xi0)-s(t,s0)|=0}=1, i∈S, by using pinning control strategy on a small fraction δ  (0<δ≪1) of the total nodes in (5). Suppose that nodes i1,i2,…,il are selected, where l=⌊δN⌋ is the smaller, but a nearest integer to the real number δN. Without loss of generality, assume that the first l nodes are selected to be pinned. A kind of local pinning controller without c(t) but containing its bounds and probability distributions is designed as(28)x˙it=fxit+βtθt∑j=1Naijxjt+1-βtθt∑j=1Nbijxjt-τ+uit,i∈Sl,x˙it=fxit+βtθt∑j=1Naijxjt+1-βtθt∑j=1Nbijxjt-τ,i∈S¯,where (29)uit=-β∗c~kixit-st,ki>0,c~=1-α∗c^+α∗c2,S¯=S-Sl.Letting ei(t)=xi(t)-s(t), we have the error system as(30)e˙it=fxit-fst+1-βtθt∑j=1Nbijxjt-τ+βtθt∑j=1Naijxjt-keit,i∈S,where ki=0, i∈S¯.Remark 7.
It is worth mentioning that, compared with some existing results such as [5, 11, 18, 28, 29], complex network (28) in this paper is more general and has its properties. Firstly, the underlying system is more general in terms of having time-varying coupling strength, where nondelayed and delayed couplings take place synchronously. Secondly, a new kind of controller depending on its lower and upper bounds in addition to probabilities will be designed to realize the pinning synchronization, which is without TCS signal. However, in order to achieve this aim, new problems will be confronted, which will be mentioned below and should be dealt with.


## 3. Main Results

Theorem 8.
Suppose Assumption1 holds and A is a irreducible coupling matrix. If k1,k2,…,kl are positive constants and(31)2δjΞ+2β∗c~ΞA~s+IN+1-β∗2c~2λmax2ΞB<0,j∈T2δjΞ+2β∗c¯ΞA~s+IN+1-β∗2c¯2λmax2ΞB<0,j∈T,where λmax(ΞB) is the spectral radius of  ΞB, c¯=(1-α∗)c1+α∗c^, and T≜{1,…,n}, complex network (28) achieves globally asymptotical synchronization almost surely.Proof.
The Lyapunov function is defined as(32)Vt=12∑i=1NξieiTtPeit+12∑i=1npj∫t-τte~jTse~jsds,where ξi>0 is the element of the left eigenvector of A corresponding the eigenvalue 0, and e~j(t)=(e1j(t),e2j(t),…,eNj(t))T∈RN, j∈T. Then we have(33)LVt=∑i=1NξieiTtPfxit-fst-Δeit+∑i=1NξieiTPΔeit+∑i=1Nξiθ2teiTt·Pβ∗∑j=1Naijejt+1-β∗∑j=1Nbijejt-τ-c~β∗∑q=1lξqkqeqTtPeqt+12·∑j=1npje~jTte~jTt-e~jTt-τe~jTt-τ≤∑i=1NξieiTtPfxit-fst-Δeit+∑i=1NξieiTPΔeit+∑i=1Nξiθ2teiTt·Pβ∗∑j=1Naijejt+1-β∗∑j=1Nbijejt-τ-θ2tβ∗∑q=1lξqkqeqTtPeqt+12·∑j=1npje~jTte~jTt-e~jTt-τe~jTt-τ≤-η∑i=1NξieiTteit+∑j=1npjδje~jTtΞe~jt+β∗θ2t∑j=1npje~jTtΞA~e~jTt+1-β∗θ2t·∑j=1npje~jTtΞBe~jTt-τ+12·∑j=1npje~jTte~jTt-e~jTt-τe~jTt-τ≤-η∑i=1NξieiTteit+∑j=1npjδje~jTtΞe~jt+β∗θ2t∑j=1npje~jTtΞA~se~jTt+1-β∗θ2t·∑j=1npje~jTtΞBe~jTt-τ+12·∑j=1npje~jTte~jTt-e~jTt-τe~jTt-τ=-η∑i=1NξieiTteit+12∑j=1npje~jTt,e~jTt-τ·Zjte~jt,e~jt-τ<0,where (34)Zjt=2δjΞ+2β∗θ2tΞA~s+IN1-β∗θ2tΞB1-β∗θ2tΞBT-IN,j∈T.By Lemma 6, it is concluded that Zj(t)<0 if(35)2δjΞ+2β∗c~ΞA~s+IN+1-β∗2c~2ΞBΞBT<0,j∈T,2δjΞ+2β∗c¯ΞA~s+IN+1-β∗2c¯2ΞBΞBT<0,j∈Thold, which could be obtained by (31). Then, (33) holds. Thus, there is always a small scalar μ>0 such that(36)EVet,t-EVe0,0≤-μE∫0teTsesdswhich implies(37)E∫0teTsesds≤1μEVe0,0<∞.By Dynkins formula, one gets(38)limt→∞⁡Eet2=0which implies(39)Plimt→∞⁡et=0=1.This completes the proof.Remark 9.
From Theorem8, it is seen that both the lower bounds of the separated subintervals and the probabilities α∗ and β∗ are involved. By using the existing methods dealing with TCS c(t), it is known that only its lower bound is considered. Thus, for any given K, it is concluded that the lower bound c1 obtained by Theorem 8 will be smaller than one obtained by the traditional methods only using its lower bound. In this sense, it is said that our result is less conservative. In other words, the range of c(t) determined by (28) is usually larger than one obtained by considering c1 only. Particularly, it is worth mentioning that such Bernoulli variables can also be replaced with Markov processes such as [30], since Bernoulli process is a special case of Markov process.It is worth mentioning that the pinning controller in (28) needs α∗ and β∗ to be known. When either of α∗ and β∗ is unavailable, how to synchronize the complex network via using a pinning controller without c(t) signal and to estimate such probabilities should be considered. In the following, adaptive pinning control laws will be developed to deal with them separately.Theorem 10.
Suppose Assumption1 holds and the coupling matrix A is irreducible. If k1,…,kl are positive constants and(40)2δjΞ+2β∗c1ΞA~s+IN+1-β∗2c12λmax2ΞBIN+c2IN<0,2δjΞ+2β∗c^ΞA~s+IN+1-β∗2c^2λmax2ΞBIN+c2IN<0,β∗ΞA~s+141-β∗2λmax2ΞBIN<0hold for ∀j∈T, complex network (28) achieves globally asymptotical synchronization almost surely under the adaptive pinning controller(41)uit=-β∗c^kieit+vit,i∈Sl,where vi(t)=-α^(t)ei(t), and the updating law(42)α^˙t=0,ifα^t=1ρ∑i=1lξieiTtPeit,otherwisewith ∀ρ>0 and α^0∈[0,1].Proof.
Choose a Lyapunov function(43)Vt=12∑i=1NξieiTtPeit+12∑i=1npj∫t-τte~jTse~jsds+12ρα~tα~t,where α~(t)=α^(t)-α∗. Then, for ∀ζ(t)=[eT(t),α~T(t)]T≠0, we have(44)LVt=∑i=1NξieiTtPfxit-fst-Δeit+∑i=1NξieiTtPΔeit+β∗c1t∑i=1NξieiTtP·∑j=1Naijejt+β∗θ~t∑i=1NξieiTtP∑j=1Naijejt+1-β∗c1t∑i=1NξieiTtP∑j=1Nbijejt-τ+1ρα^t-α∗α^˙t+1-β∗θ~t∑i=1NξieiTtP·∑j=1Nbijejt-τ-β∗c^∑q=1lξqkqeqTtPeqt+12·∑j=1npje~jTte~jt-e~jTt-τe~jt-τ+∑i=1NξieiTtPvit≤-η∑i=1NξieiTtPeit+∑j=1npjδje~jTtΞe~jt+β∗c1t·∑j=1npje~jTtΞA~e~jt+1-β∗c1t·∑j=1npje~jTtΞBe~jt-τ+β∗θ~t·∑j=1npje~jTtΞA~e~jt+1-β∗θ~t·∑j=1npje~jTtΞBe~jt-τ+∑i=1NξieiTtPvit+12·∑j=1npje~jTte~jt-e~jTt-τe~jt-τ+1ρα^t-α∗α^˙t≤-η∑i=1NξieiTtPeit+β∗+∑j=1npjδje~jTtΞe~jt+β∗c1t·∑j=1npje~jTtΞA~se~jt+1-β∗c1t·∑j=1npje~jTtΞBe~jt-τθ~t·∑j=1npje~jTtΞA~se~jt+1-β∗θ~t·∑j=1npje~jTtΞBe~jt-τ+∑i=1NξieiTtPvit+12·∑j=1npje~jTte~jt-e~jTt-τe~jt-τ+1ρα^t-α∗α^˙t=-η∑i=1NξieiTtPeit+∑j=1npje~jTtδjΞ+β∗c1tΞA~s+12INe~jt+1-β∗c1t∑j=1npje~jTtΞBe~jt-τ-12·∑j=1npje~iTt-τe~it-τ+β∗θ~t·∑j=1npje~jTtΞA~se~jt+1-β∗θ~t·∑j=1npje~jTtΞBe~jt-τ+∑i=1NξieiTtPvit+1ρα^t-α∗α^˙t-θ~t·∑i=1npje~jTt-τe~jt-τ+θ~t·∑i=1npje~jTt-τe~jt-τ≤-η∑i=1NξieiTtPeit+∑j=1npje~jTtδjΞ+β∗c1tΞA~s+12INe~jt+1-β∗c1t∑j=1npje~jTtΞBe~jt-τ-12·∑j=1npje~jTt-τe~jt-τ+β∗θ~t·∑j=1npje~jTtΞA~se~jt+1-β∗θ~t·∑j=1npje~jTtΞBe~jt-τ+∑i=1NξieiTtPvit+1ρα^t-α∗α^˙t-θ~t·∑i=1npje~jTt-τe~jt-τ+c2∑i=1npje~jTt-τe~jt-τ=-η+α∗·∑i=1NξieiTtPeit+12·∑j=1npje~jTt,e~jTt-τZjte~jt,e~jt-τ+θ~t·∑j=1npje~jTt,e~jTt-τMje~jt,e~jt-τ≤0,where (45)Zjt=2δjΞ+2β∗c1tΞA~s+IN+c2IN1-β∗c1tΞB1-β∗c1ΞBT-IN,j∈TMj=β∗ΞA~s121-β∗ΞB121-β∗ΞBT-IN,j∈Tθ~t=α∗c2t-c1t.By Lemma 6, it is known that Zj(t)<0 if(46)2δjΞ+2β∗c1ΞA~s+IN+1-β∗2c12ΞBΞBT+c2IN<0,2δjΞ+2β∗c^ΞA~s+IN+1-β∗2c^2ΞBΞBT+c2IN<0hold for j∈S. Similarly, Mj<0 is equivalent to(47)β∗ΞA~s+141-β∗2ΞBΞBT<0.From conditions (31)-(32), it is obtained that (46)-(47) hold. Then, one has (44) which implies(48)EVζt,t≤EVζ0,0.For any t≥0, k>0, when sup0≤s≤t|ζ(s)|≥k, and from (48), one obtains(49)Psup0≤s≤t⁡ζs≥kinfζ≥k,t≥0⁡Vζt,t≤EVζt,t≤EVζ0,0which implies(50)Psup0≤s≤t⁡ζs≥k≤EVζ0,0infζ≥k,t≥0⁡Vζt,t.From (43), it is concluded that inf|ζ|≥k,t≥0V(ζ(t),t)→∞, as k→∞. Based on this and taking into account (50), one concludes that ζ(t) is bounded almost surely. On the other hand, because of (44), there is μ>0 such that(51)EVζt,t-EVζ0,0≤-μE∫0teTsesds.Then, it is directly obtained that(52)E∫0teTsesds≤1μEVζ0,0<∞.Similar to the proof of (37), it is concluded that complex network (28) plus adaptive pinning controller (41) is asymptotical synchronization almost surely and the unknown probability α∗ is estimated. This completes the proof.Remark 11.
Compared with some existing references, the adaptive pinning controller (33) has its properties which are listed as follows. Firstly, the adaptive controller is used to estimate a scalar in a range and having an upper finite bound, while the other results such as [20, 22, 28] are used to estimate control gains and without this restriction. Secondly, when c(t) varies in a range, a new class of pinning controller in this paper is designed without signal c(t), which is different from that in [29]. Moreover, instead of c(t) without an upper bound, it is more practical that it is in a finite range. Thirdly, some adaption controllers such as in [17, 22, 24] should be added to all the nodes of complex network. It is impossible to realize these controllers for complex network with large nodes. In addition, the adaption controller in [17] can cause chattering as the error state can reach the discontinuity region before the complex network achieves the global synchronization. However, this is not true for adaption controller (33) which is also on a small fraction of nodes in complex networks.Theorem 12.
Suppose Assumption1 holds and the coupling matrix A is irreducible. If k1,…,kl are positive constants and satisfy(53)2δjΞ+1-2c~IN+c~2λmax2ΞBIN<0,j∈T2δjΞ+1-2c~IN+c¯2λmax2ΞBIN<0,j∈TΞA~s+14ΞBΞBT<0the complex network achieves globally asymptotical synchronization almost surely under the adaptive pinning controller(54)uit=-c~kieit+vit,i∈Sl,vit=0,if∑i=1leit2=0,-P-1eit∑i=1lξieit2c~Φet+β^tΦet-τ,otherwisewith Φ(e(t))=∑i=1NeiT(t)Pei(t) and the updating laws(55)β^˙t=0,ifβ^t=1ρc~∑i=1NeiTt-τPeit-τ,otherwisewith ∀ρ>0 and β^0∈[0,1].Proof.
Define the Lyapunov function as(56)Vt=12∑i=1NξieiTtPeit+12∑i=1npj∫t-τte~jTse~jsds+12ρβ~tβ~t,where ξi is the element of the left eigenvector of A corresponding to the eigenvalue 0 and β~(t)=β^(t)-β∗. Then, for ∀ζ(t)=[eT(t),α~T(t)]T≠0, we have(57)LVt=∑i=1NξieiTtPfxit-fst-Δeit+∑i=1NξieiTtPΔeit+β∗θ2t∑i=1NξieiTtP·∑j=1Naijejt+1-β∗θ2t∑i=1NξieiTtP·∑j=1Nbijejt-τ-c~∑q=1lξqkqeqTtPeqt+12·∑j=1npje~jTte~jt-e~jTt-τe~jt-τ+1ρβ^t-β∗β^˙t+∑i=1NξieiTtPvit≤∑i=1NξieiTtPfxit-fst-Δeit+∑i=1NξieiTtPΔeit+β∗θ2t∑i=1NξieiTtP·∑j=1Naijejt+1-β∗θ2t∑i=1NξieiTtP·∑j=1Nbijejt-τ-β∗θ2t∑q=1lξqkqeqTtPeqt+12·∑j=1npje~jTte~jt-e~jTt-τe~jt-τ+1ρβ^t-β∗β^˙t+∑i=1NξieiTtPvit≤-η∑i=1NξieiTtPeit+∑j=1npjδje~jTtΞe~jt+β∗θ2t∑j=1npje~jTtΞA~e~jt+1-β∗θ2t·∑j=1npje~jTtΞBe~jt-τ+12·∑j=1npje~jTte~jt-e~jTt-τe~jt-τ+1ρβ^t-β∗β^˙t+∑i=1NξieiTtPvit=-η∑i=1NξieiTtPeit+β∗θ2t·∑j=1npje~jTtΞA~e~jt-∑j=1npje~jTtΞBe~jt-τ+∑j=1npje~jTtδjΞ+12INe~jt+θ2t·∑j=1npje~jTtΞBe~jt-τ-12·∑j=1npje~jTt-τe~jt-τ+1ρβ^t-β∗β^˙t+∑i=1NξieiTtPvit=-η∑i=1NξieiTtPeit+12·∑j=1npje~jTt,e~jTt-τZ¯jte~jt,e~jt-τ+β∗θ2t·∑j=1npje~jTtΞA~e~jt-∑j=1npje~jTtΞBe~jt-τ+1ρβ^t-β∗β^˙t+∑i=1NξieiTtPvit=-η∑i=1NξieiTtPeit+12·∑j=1npje~jTt,e~jTt-τZ¯jte~jt,e~jt-τ+β∗θ2t∑j=1npje~jTtΞA~e~jt-β∗θ2t·∑j=1npje~jTtΞBe~jt-τ+β∗θ2t·∑j=1npje~jTt-τe~jt-τ-β∗θ2t·∑j=1npje~jTt-τe~jt-τ+1ρβ^t-β∗β^˙t+∑i=1NξieiTtPvit≤-η∑i=1NξieiTtPeit+12·∑j=1npje~jTt,e~jTt-τZ¯jte~jt,e~jt-τ+β∗θ2t∑j=1npje~jTtΞA~se~jt-β∗θ2t·∑j=1npje~jTtΞBe~jt-τ+β∗θ2t·∑j=1npje~jTt-τe~jt-τ-β∗θ2t·∑j=1npje~jTt-τe~jt-τ+1ρβ^t-β∗β^˙t+∑i=1NξieiTtPvit=-η∑i=1NξieiTtPeit+12·∑j=1npje~jTt,e~jTt-τZ¯jte~jt,e~jt-τ+β∗θ2t·∑j=1npje~jTt,e~jTt-τM¯je~jt,e~jt-τ≤0,where(58)Z¯jt=2δjΞ+1-2c~INθ2tΞBθ2tΞBT-IN,j∈T,M¯j=ΞA~s-12ΞB-12ΞBT-IN,j∈T.Similarly, one has that Z¯j(t)<0 is guaranteed by(59)2δjΞ+1-2c~IN+c~2ΞBΞBT<0,j∈T,2δjΞ+1-2c~IN+c¯2ΞBΞBT<0,j∈T,where M¯j<0 is obtained by(60)ΞA~s+14ΞBΞBT<0.Based on (53), we have (59)-(60). The next is similar to the proof of Theorem 10. This completes the proof.


## 4. Numerical Examples

In this section, a simulation example is used to illustrate the usefulness of our main results.Example 1.
Consider that a dynamical node of complex network is a Lorenz system. It is described as(61)x˙1t=ax2t-x1t,x˙2t=cx1t-x1txxt-x2t,x˙3t=x1tx2t-bx3t.When the parameters are given as a=10, b=8/3, and c=28, Lorenz system has a chaotic attractor shown in Figure 1. We take P=diag⁡{24,0.002,12} and Δ=diag⁡{3,3,3} such that Lorenz system (61) satisfies Assumption 1. Suppose an undirected E-R random network consisting of N=10 nodes with p=0.8, where the coupling matrix is (62)A=-61001011111-60111001100-61111110011-71011111111-70110101100-61111101111-71011011111-60011110100-50110111100-6,B=-51010010111-50011001100-41100110101-50110010110-40010101010-51011100101-30000010100-30111100100-40110111010-6.In this example, we randomly choose l=δN nodes with a fraction δ=0.2. For this E-R network with β∗=0.4 and given control gains k1=k2=⋯=kl=0.1, when the upper bound c2=3 and c^=(1/2)c1+c2, by Theorem 8, we could get the curve of c1 along with α∗ varying in 0,1. Based on Figure 2, it is known that the larger α∗ results in the smaller c1. This conclusion is consistent with constant coupling strength case that the larger coupling strength results in the easier synchronization of complex network. In order to establish the correlation between c^ and α∗, we select c1=20 and c2=100. Under the similar conditions and by Theorem 8, one can have the correlation between c^ and α∗, which is illustrated in Figure 3. It is also concluded that the smaller α∗ leads to the larger c^. Moreover, when α∗ is smaller, α∗≤0.2, the value of c^ changes actually and is very near to the upper bound c2. Particularly, it is worth mentioning that when α∗ is very small, α∗≪0.1, there will be no solvable solution to c^ in terms of c^ being larger than its upper bound c2=100. In this sense, it is seen that such probabilities play important roles in pinning synchronization of complex networks. Next, we will demonstrate the correlation between probabilities α∗ and β∗. Letting ct∈c1,c2 with c1=15 and c2=55, a simulation of probabilities α∗ and β∗ will be obtained by Theorem 8, which is shown in Figure 4. Based on Figure 4, one knows that the larger β∗ results in the larger α∗. In other words, the larger β∗ means that the probability of delayed coupling taking place is larger, which needs the desired pinning controller to be added with a higher probability. When such probabilities are unavailable, we will exploit the adaptive control to realize the purpose of pinning synchronization. Without loss of generality, here it is assumed that β∗=0.3 is known, while α∗=0.2 is inaccessible. If α∗ is unknown, let the initial condition x0=[-10.50.1]T; by Theorem 10, we have the state responses of synchronization error and estimation α∗ shown in Figures 5 and 6. Particularly, Figure 5 is the simulation of state error |ei(t)|, i=1,2,…,10, instead of ei(t). The reason is we want to demonstrate the state response of ei(t) in detail. Unfortunately, there are 10 nodes, where every node has 3 variables. Then, there will be 30 variables to be illustrated in one figure. This will make the state response of ei(t), i=1,2,…,10, difficult to be distinguished. However, if we plot |ei(t)| instead of ei(t), there will be 10 lines in one figure. Then, we will identify them easily. But, the effects of |ei(t)| and ei(t) implying that e(t) is stable are same. All the simulations show the utility of the proposed methods in this paper.Figure 1
Chaotic orbit of Lorenz’s circuit.Figure 2
The curve ofc1 along with α∗.Figure 3
The curve ofc^ along with α∗.Figure 4
The curve ofβ∗ along with α∗.Figure 5
Synchronization error|ei(t)| by adaption controller (41).Figure 6
Response curve of the estimation ofα∗.


## 5. Conclusion

In this paper, based on the stochastic viewpoint, we have investigated the pinning synchronization problem of complex networks with time-varying coupling strength and nondelayed and delayed coupling. Here, the lower and upper bounds in addition to the probability distribution of each subinterval are firstly considered in designing a pinning controller. Moreover, the nondelayed and delayed coupling here are asynchronous, whose switchings are described by a Bernoulli variable. Then, a new kind of local state feedback controller is developed without using time-varying coupling strength signal, which is dependent on such bounds and probabilities. When such probabilities are unavailable, two different kinds of adaption laws are proposed to estimate the unknown parameters, respectively, and also guarantee the global synchronization of complex networks. Finally, a numerical example is given to show the utility of the presented theories. Future research could be considered to the pinning synchronization of complex networks with nondelayed and time-varying delayed couplings, stochastic pinning controller added to some nodes in term of Brownian noise, and such Bernoulli variablesα(t) and β(t) replaced by Markov process.



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*Source: 989201-2015-08-24.xml*