# Semiglobal Finite-Time Synchronization of Complex Dynamical Networks via Periodically Intermittent Control

Authors: Yihan Fan; Hongmei Liu; Jun Mei
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

This paper studies the finite-time synchronization problem for a class of complex
dynamical networks by means of periodically intermittent control. Based
on some analysis techniques and finite-time stability theory, some novel and
effective finite-time synchronization criteria are given in terms of a set of
linear matrix inequalities. Particularly, the previous synchronization problem
by using periodically intermittent control has been extended in this paper.
Finally, numerical simulations are presented to verify the theoretical results.


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## Body

## 1. Introduction

A complex dynamical network consists of a number of nodes, which are dynamic systems, and links between the nodes. Complex networks exist in various fields of science, engineering, and society and have been deeply investigated in recent years [1]. As the major collective behavior, synchronization is one of the key issues that has been extensively addressed. Several books and reviews [2–6] have appeared which deal with this topic.Up till now, the synchronization for nonlinear systems especially dynamical networks [7, 8] has been one of the extensive research subjects and many important and fundamental results have been reported on the synchronization and control of nonlinear systems. Meanwhile, lots of control approaches have been developed to synchronize nonlinear systems such as adaptive control [9, 10], feedback control [11, 12], observer control [13, 14], impulsive control [15–19], and intermittent control [20–26]. Among these control approaches and other control methods, the discontinuous control methods, such as impulsive control and intermittent control, have received much interest because they are practical and easily implemented in engineering fields such as transportation and communication. Though the two control methods are discontinuous control, the intermittent control is different from the impulsive control since impulsive control is activated only at some isolated instants, while intermittent control has a nonzero control width. Under some circumstances, using intermittent control is more effective and robust [27]. Hence, some synchronization criteria for nonlinear systems with or without time delays via intermittent control have been presented in recent years; see [21, 22, 28, 29].Nevertheless, to our best knowledge, the previous results only focus on asymptotical or exponential synchronization of networks through intermittent control; there are few results concerned with finite-time synchronization via intermittent control. In view of this, the purpose of this paper is to study the synchronization of a class of systems by designing reasonable intermittent control. In addition, some previous work views the finite-time synchronization via intermittent control in [30], which will be extended in this paper. Besides, many superiority in finite-time stability has no emphasis in this paper (see [30–32]).The main contribution of this paper lies in the following aspects. Firstly, a new central lemma is proved by using analysis method. Additionally, an intermittent controller is designed to synchronize the addressed complex networks and some new and useful finite-time synchronization criteria are obtained. Besides, the derivative of the Lyapunov functionV(t) is smaller than -αVη(t)+βV(t), which enriches the previous results in [30], when controllers are added into the network. Finally, numerical examples are given to show the effectiveness of the theoretical results.The paper is organized as follows. In Section2, the problem statement and synchronization scheme to be studied are formulated, and some useful lemmas and preliminaries are presented. In Section 3, some finite-time synchronization criteria for the complex dynamical networks are rigorously derived. In Section 4, the effectiveness of the developed methods is shown by numerical examples. Conclusions are finally drawn in Section 5.


## 2. Preliminaries

Consider a complex dynamical network consisting ofN linearly and diffusively coupled identical nodes, with each node being an n-dimensional dynamical system. The state equation of the entire network is designed as follows:(1)x˙it=fxit+c∑j=1NaijΓxjt,i=1,2,…,N,where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn is the state vector of the ith dynamical node, f:Rn→Rn is a smooth nonlinear vector-value function, and the constant c>0 is a coupling strength. Γ∈Rn×n is the inner-coupling matrix of the network. Matrix A=(aij)∈RN×N represents the coupling configuration of the network, in which aij is defined as follows: if there is a connection from the nodes j to i, then aij≠0; otherwise, aij=0, and the diagonal elements of matrices A are defined as (2)aii=-∑j=1,j≠iNaij.To achieve the aim of this paper, the following assumptions and some lemmas are necessary.Assumption 1.
Assume that there exists a positive definite diagonal matrixP=diag⁡(p1,…,pn) and a diagonal matrix Θ=diag⁡(θ1,…,θn), such that f(·) satisfies the following inequality: (3)y-xTPfy-fx-Θy-x≤-ξy-xTy-x,for some ξ>0, all x,y∈Rn, and t>0.Lemma 2 (see [33]).
Assume that a continuous, positive-definite functionV(t) satisfies the following differential inequality:(4)V˙t≤-αVηt,∀t≥t0,Vt0≥0,where α>0 and 0<η<1 are all constants. Then, for any given t0,V(t) satisfies the following inequality:(5)V1-ηt≤V1-ηt0-α1-ηt-t0,t0≤t≤t1,Vt≡0,∀t≥t1,with t1 given by(6)t1=t0+V1-ηt0α1-η.Lemma 3.
Assume that a continuous, positive-definite functionV(t) satisfies the following differential inequality:(7)V˙t≤-αVηt+pVt,∀t≥t0,V1-ηt0≤αp,where α>0, 0<η<1, and p>0 are three constants. Then, for any given t0,V(t) satisfies the following inequality: (8)V1-ηtexp⁡-1-ηpt≤V1-ηt0exp⁡-1-ηpt0+αpexp⁡-1-ηpt-exp⁡-1-ηpt0,t0≤t≤t1,Vt≡0,∀t≥t1,with t1 given by (9)t1=ln⁡1-p/αV1-η0pη-1for t0=0.Proof.
Consider the following differential equation:(10)X˙t=-αXηt+pXt,Xt0=Vt0.By multiplying exp⁡{-pt}, we have (11)dexp⁡-ptXtdt=-αexp⁡-ptXtηexp⁡-1-ηpt.Although this differential equation does not satisfy the global Lipschitz condition, the unique solution to this equation can be found as (12)X1-ηtexp⁡-1-ηpt=X1-ηt0exp⁡-1-ηpt0+αpexp⁡-1-ηpt-exp⁡-1-ηpt0,Xt≡0,∀t≥t1.It is direct to prove that x(t) is differential for t>t0. From the comparison lemma, one obtains(13)V1-ηtexp⁡-1-ηpt≤V1-ηt0exp⁡-1-ηpt0+αpexp⁡-1-ηpt-exp⁡-1-ηpt0,t0≤t≤t1,Vt≡0,∀t≥t1with t1 given in (9) with t0=0.Remark 4.
Lemma3 is similar to Lemma 2, but our result can enrich the famous differential inequality [33] to general differential inequality, and give a direction to proof the following Lemma 5.Lemma 5.
Suppose that functionV(t) is continuous and nonnegative when t∈[0,∞) and satisfies the following conditions: (14)V˙t≤-αVηt+pVt,lT≤t<lT+θT,V˙t≤0,lT+θT≤t<l+1T,where α>0, T>0, p>0, 0<η, θ<1, l∈ι={1,2,…,ς}, and ς is a nature number; then the following inequality holds: (15)V1-ηtexp⁡-1-ηpt≤V1-η0+αpexp⁡-1-ηpθt-1,0≤t≤T~.Proof.
DenoteM0=V1-η(0)-α/p and W(t)=V1-ηtexp⁡{-(1-η)pt}, where t≥0. Let Q(t)=W(t)-M0-α/pexp⁡{-(1-η)pt}. It is easy to see that (16)Qt<0,fort=0.
In the following, we will prove that(17)Qt<0,∀t∈0,θT.Otherwise, there exists a t0∈[0,θT) such that (18)Qt0=0,Q˙t0>0,(19)Qt<0,0≤t<t0.From (16), (18), and (19), we obtain (20)Q˙t0=1-ηV-ηt0V˙t0exp⁡-1-ηpt0+V1-ηt0-1-ηpexp⁡-1-ηpt0-αpexp⁡-1-ηpt0-1-ηp≤1-η·V-ηt0-αVηt0+pVt0·exp⁡-1-ηpt0-p1-ηV1-ηt0·exp⁡-1-ηpt0+α1-η·exp⁡-1-ηpt0=-α1-η·exp⁡-1-ηpt0+α1-η·exp⁡-1-ηpt0+p1-ηV1-ηt0·exp⁡-1-ηpt0-p1-ηV1-ηt0·exp⁡-1-ηpt0=0,which leads to a contradiction with (18). Hence inequality (17) holds.
Now, we prove that fort∈[θT,T)(21)Ht=Wt-M0-αpexp⁡-1-ηptexp⁡1-ηpt-θT<0.Otherwise, there exists a t1∈[θT,T) such that (22)Ht1=0,H˙t1>0,(23)Ht<0,θT≤t<t1.By (22) and (23), we have (24)H˙t1=W˙t1-αpexp⁡-1-ηpt1-1-ηp·exp⁡1-ηpt1-θT-αpexp⁡-1-ηpt1·exp⁡1-ηpt1-θT1-ηp≤α1-η·exp⁡-1-ηpt1exp⁡1-ηpt1-θT-α1-ηexp⁡-1-ηpt1·exp⁡1-ηpt1-θT=0,which contradicts (22). Hence (21) holds.
Consequently, on the one hand, fort∈[θT,T), (25)Wt<M0+αpexp⁡-1-ηptexp⁡1-ηpt-θT≤M0+αpexp⁡-1-ηptexp⁡1-ηp1-θT.On the other hand, it follows from (16) and (17) that for t∈[0,θT)(26)Wt<M0+αpexp⁡-1-ηpt≤M0+αpexp⁡-1-ηptexp⁡1-ηp1-θT.So (27)Wt<M0+αpexp⁡-1-ηptexp⁡1-ηp1-θT,∀t∈0,T.Similarly, we can prove the following results for t∈[T,(1+θ)T), (28)Wt<M0+αpexp⁡-1-ηptexp⁡1-ηp1-θT,and for t∈[(1+θ)T,2T)(29)Wt<M0+αpexp⁡-1-ηpt·exp⁡1-ηp1-θT·exp⁡1-ηpt-θT-T=M0+αp·exp⁡-1-ηptexp⁡1-ηpt-2θT.
Now, using mathematical induction method, suppose that the following statements are true; for any integersm, we can obtain W(t).
FormT≤t<(m+θ)T,(30)Wt<M0+αpexp⁡-1-ηpt·exp⁡1-ηp1-θmT,and for (m+θ)T≤t<(m+1)T, (31)Wt<M0+αpexp⁡-1-ηpt·exp⁡1-ηpt-m+1θT.Since, for any t≥0, there exists a positive integer k, such that kT≤t<(k+1)T, we can conclude the following estimation of W(t) by (30) and (31).
ForkT≤t<(k+θ)T, (32)Wt<M0+αpexp⁡-1-ηptexp⁡1-ηp1-θkT≤M0+αpexp⁡-1-ηptexp⁡1-ηp1-θt,and for (k+θ)T≤t<(k+1)T, (33)Wt<M0+αpexp⁡-1-ηpt·exp⁡1-ηpt-k+1θT≤M0+αp·exp⁡-1-ηptexp⁡1-ηp1-θt.From the previous definition of W(t), we have (34)V1-ηtexp⁡-1-ηpt≤V1-η0-αp+αpexp⁡-1-ηptexp⁡1-ηp1-θt=V1-η0-αp+αpexp⁡-1-ηθpt=V1-η0+αpexp⁡-1-ηpθt-1,t≥0.The proof of Lemma 3 is completed.Remark 6.
Lemmas3 and 5 played an important role in the finite-time synchronization analysis of dynamical networks via intermittent control in this paper, because it shows the utilization of finite-time intermittent control.Lemma 7 (see [30]).
Letx1,x2,…,xn∈Rm be any vectors and 0<q<2 is a real number satisfying (35)x1q+x2q+⋯+xnq≥x12+x22+⋯+xn2q/2.


## 3. Criteria for Finite-Time Synchronization

In this section, we study finite-time synchronization of system (1) with system (36) under the following intermittent controller (37).In order to drive system (1) to achieve finite-time synchronization by means of periodically intermittent control, the corresponding response system is designed as follows: (36)y˙it=fyit+c∑j=1NaijΓyjt+uit,i=1,2,…,N,where yi(t)=(yi1(t),yi2,…,yin(t))T∈Rn, i=1,2,…,N, denote the response state vector of the node i of system (36). u(t)=(u1(t),u2(t),…,uN(t))T is an intermittent controller defined as follows:(37)uit=-ηieit-k¯λmaxPλminPsign⁡ei,lT≤t<lT+δ,uit=0,ei=0orlT+δ≤t<l+1T,where ηi>0 is a positive constant control gain and k¯>0 is a tunable constant. Denote λmax(P)(λmin(P)) as the maximum (minimum) eigenvalue of the matrix P. T>0 is the control period, δ>0 is called the control width (control duration), and θ=δ/T is the ratio of the control width δ to the control period T called control rate. ι={1,2,…,ς} is a finite natural number set and l∈ι.Letei(t)=yi(t)-xi(t)(1≤i≤N) be synchronization errors between the states of drive system (1) and response system (37); then the following error system can be obtained:(38)e˙it=fyit-fxit+c∑j=1NaijΓejt+uit,lT≤t<lT+θT,i=1,2,…,N,e˙it=fyit-fxit+c∑j=1NaijΓejt,lT+θT≤t<l+1T,i=1,2,…,N.The main results are stated as follows.Theorem 8.
Let Assumption1 hold. Suppose that positive constants η1,η2,…,ηN, ξ, β, and a positive defined diagonal matrix P>0 satisfy (39)θjIN-Ξ+cγjA-βIN≤0,j=1,2,…,n,(40)θjIN-ξλmaxPIN+cγjA≤0,j=1,2,…,n,where Γ=diag⁡(γ1,γ2,…,γn), Ξ=diag⁡(η1,η2,…,ηn), Θ=diag⁡(θ1,θ2,…,θn), and IN is the N×N identity matrix. Then under the periodically intermittent controllers (37), the error system (38) is synchronized in a finite time: (41)T1=-2ln⁡1-β/2k¯V1/20θβ,where V(0)=1/2∑i=1N‍eiT(0)Pei(0) and ei(0) is the initial condition of ei(t).Proof.
Consider the following Lyapunov function:(42)Vt=12∑i=1NeiTtPeit.Then the time derivative of (42) along the trajectories of the first subsystem of system (38) is calculated and estimated as follows.
WhenlT≤t<(l+θ)T, for l∈ι,(43)V˙t=∑i=1NeiTtPe˙it=∑i=1NeiTtPfyit-fxit+c∑j=1NaijΓejt+uit=∑i=1NeiTtPfyit-fxit-Θeit+eiTtPΘeit+eiTtPuit+ceiTtPΓ∑j=1Naijejt≤-ξ∑i=1NeiTteit+∑i=1NeiTtΘP-ηiPeit+c∑i=1NeiTΓ∑j=1NaijPejt-k¯∑i=1NλmaxPλminPeiTtPsign⁡ei-β∑i=1NeiTt·eit+β∑i=1NeiTteit≤∑j=1npje~jTtθjIN-Ξ+cγjA-βINe~jt+β∑i=1NeiTteit-k¯∑i=1NλmaxPλminPeiTtPsign⁡ei=∑j=1npje~jTt·θjIN-Ξ+cγjA-βINe~jt+β∑i=1NeiTteit-k¯∑i=1NλmaxPλminPeiTtPsign⁡ei.Defining sign⁡(ei)=(sign⁡ei1,sign⁡ei2,…,sign⁡(ein))T, we have (44)V˙t≤β∑i=1NeiTtPeit+∑j=1npje~jTtZje~jTt-k¯∑i=1NλmaxPei,where e~j(t)=[e~j1,e~j2,…,e~jN] is a column vector of ej(t) and Zj is defined as (45)Zj=θjIN-Ξ+cγjA-βIN,j=1,2,…,n.Using Lemma 7, we have (46)V˙t≤β∑i=1NeiTtPeit+∑j=1npje~jTtZje~jTt-2k¯12∑i=1NeiTtPeit1/2.It follows from inequality (39) that(47)Zj≤0,which shows that V˙(t)≤-2k¯V1/2(t)+βV(t).
When(l+θ)T≤t<(l+1)T, for l∈ι, we have (48)V˙t=∑i=1NeiTtPe˙it=∑i=1NeiTtPfyit-fxit-Θeit+eiTtPΘeit+c∑j=1NaijeiTtPΓejt≤-ξ∑i=1NeiTteit+∑j=1npje~jTtθjINe~jt+c∑j=1npje~jTtγjAe~jt≤-∑j=1npje~jTt·ξλmaxPINe~jt+∑j=1npje~jTtθjINe~jt+c∑j=1npje~jTtγjAe~jt=∑j=1npje~jTtSje~jTt,where e~j(t)=[e~j1,e~j2,…,e~jN] is a column vector of ej(t) and Sj is defined as (49)Sj=θjIN-ξλmaxPIN+cγjA,j=1,2,…,n.It follows from inequality (40) that (50)Sj≤0,which shows that V˙(t)≤0.
Namely, definingα=2k¯, η=1/2, and p=β, we get(51)D+Vt≤-αVηt+pVt,lT≤t≤lT+θT,D+Vt≤0,lT+θT≤t<l+1T.According to Lemma 5, we have (52)V1-ηtexp⁡-1-ηpt≤V1-η0+αpexp⁡-1-ηpθt-1,0≤t≤T1.By Lemma 3, we have(53)t≤ln⁡1-p/αV1-η0pθη-1=-2ln⁡1-β/2k¯V1/20θβ,0≤t≤T1.The proof of Theorem 8 is completed.Remark 9.
Obviously, whenθ=1, the intermittent control (37) is degenerated to a continuous control input which has been extensively proposed in previous work (see [34, 35]) and focuses on [13]. However, this trivial case is not to be discussed in this paper.
If the Lyapunov functionV˙(t)≤-αV(t)(p=0) when controllers are added into the network, then it is easy to see that Theorem 8 can be restated as the following corollary.Corollary 10.
Let Assumption1 hold. Suppose that positive constants η1,η2,…,ηN, ξ, and a positive define diagonal matrix P>0 satisfy (54)θjIN-Ξ+cγjA≤0,j=1,2,…,n,(55)θjIN-ξλmaxPIN+cγjA≤0,j=1,2,…,n,where Γ=diag⁡(γ1,γ2,…,γn), Ξ=diag⁡(η1,η2,…,ηn), Θ=diag⁡(θ1,θ2,…,θn), and IN is the N×N identity matrix. Then under the periodically intermittent controllers (37), the error system (38) is synchronized in a finite time: (56)T2=V1-η02k¯θ,where V(0)=1/2∑i=1N‍eiT(0)Pei(0) and ei(0) is the initial condition of ei(t).Remark 11.
Corollary10 in this paper is the main result of Theorem 2 in [30] and the main results in [36].Remark 12.
According to (41) and (56) and the convergence time T1,T2, we can conclude that the convergence time satisfies T2≤T1. We can analyse that the term βV(t) should impede the convergence time. But compared with the control gain matrix Ξ in (54), it is easy to seek an appropriate control gain matrix Ξ in (39) for which synchronization happens.Remark 13.
It is clear to see that inequality (39) is more easily satisfied compared with inequality (54) under the same controllers and conditions via LMI Toolbox, which reveals a very interesting phenomenon; that is, the control gain under condition (39) is more easily designed than condition (54), though it can impede the convergence time.Remark 14.
We can find that if condition (55) is satisfied, condition (54) easily holds when the positive definite control gain matrix Ξ is anything. Then we have the following corollary.Corollary 15.
Let Assumption1 hold. Suppose that positive constants η1,η2,…,ηN, ξ, and a positive defined diagonal matrix P>0 satisfy(57)θjIN-ξλmaxPIN+cγjA≤0,j=1,2,…,n,where Γ=diag⁡(γ1,γ2,…,γn), Θ=diag⁡(θ1,θ2,…,θn), and IN is the N×N identity matrix. Then under the periodically intermittent controllers (37), the error system (38) is synchronized in a finite time: (58)T3=V1-η02k¯θ,where V(0)=1/2∑i=1N‍eiT(0)Pei(0) and ei(0) is the initial condition of ei(t).


## 4. Numerical Examples

In this section, we give some numerical examples to show the validity and effectiveness of the derived results for finite-time synchronization via periodically intermittent control.In this case, to demonstrate the results above, we consider general complex dynamical networks, in which each subsystem is a Lorenz system. The dynamics of Lorenz system is described as follows:(59)s˙=fs=s˙1s˙2s˙3=-aa0c-1000-bs1s2s3+0-s1s3s1s2≜Cs+ψs,where the parameters are selected as a=10, c=30, and b=8/3; then the Lorenz system has a chaotic attractor (see Figure 1). Moreover, it is known that |xi1|≤18.6360=r1, |xi2|≤25.3679=r2, and |xi3|≤45.9792=r3. Now we will show that there exists a positive definite diagonal matrix P  that satisfies Assumption 1. Let x=[x1,x2,x3]T, y=[y1,y2,y3]T, δ=x-y=[δ1,δ2,δ3]T=[x1-y1,x2-y2,x3-y3]T, and |ϵ|=[|ϵ1|,|ϵ2|,|ϵ3|]T; then (60)ψx-ψy=0-x1x3+y1y3x1x2-y1y2=0-x1ϵ3-y3ϵ1x1ϵ2+y2ϵ1.Hence, (61)x-yTψx-ψy=-y3ϵ1ϵ2+y2ϵ1ϵ3≤12ϵT0r3r2r300r200ϵ≜12ϵTRϵ.Then (62)x-yTPfx-fy-Θx-y≤λmaxC+1/2λmaxR-λminΘλminPx-yT·x-y,where λmax(C),λmax(R),λmin(Θ) are the maximum eigenvalues of C,R,Θ, respectively. Let ξ=-λmax(C)+1/2λmax(R)-λmin(Θ)/λmin(P)=43.48; therefore Assumption 1 is satisfied with Θ=diag⁡(50,50,50) and P=diag⁡(0.5,0.4,0.2).Figure 1
The chaotic trajectories of the Lorenz system.Consider the complex dynamical network (1) consisting of 50 identical Lorenz oscillators nodes, which is described by(63)x˙it=fxit+c∑j=150aijΓxjt,i=1,2,…,50,where Γ=diag⁡(1,1,1) and A=(aij)50×50 is a symmetrically diffusive coupling matrix with aij=0 or 1(j≠i) and the coupling strength c=1.The corresponding controlled response system of (36) is of the form (64)y˙it=fyit+c∑j=150aijΓyjt+uit,i=1,2,…,50,where aij and fi(·) are the same as (63) and(65)uit=-ηieit-k¯λmaxPλminPsign⁡ei,lT≤t<lT+δ,uit=0,ei=0orlT+δ≤t<l+1T,where k¯=5.The initial states of the numerical simulations in the master and slave systems are as follows:xi(0)=(2+0.2i,0.2+0.3i,0.3+0.1i)T and yi(0)=(-8+0.7i,-5+0.8i,-10+0.6i)T. The initial conditions of the error system are ei(0)=(-10+0.5i,-5.2+0.5i,-10.3+0.5i)T, where 1≤i≤50. In addition, the values of the parameters for the controllers (37) are selected as T=1 and δ=0.8. Choosing Θ=diag⁡(50,50,50), P=diag⁡(0.5,0.4,0.2), and λmax(A)=-1.6414, one can easily obtain that inequality (40) holds. Besides, we can obtain the parameters of the intermittent controllers as Ξ=diag⁡[57.9071,…,57.9071]50×50 and β=1.304 by using LMI toolbox in Matlab to solve (39) and (40). Therefore from Theorem 8 it can be obtained that the response system (36) will synchronize the drive system (1) under the periodically intermittent controllers (37) within the time T=1.234 (the average time). The synchronous error ei(t) is illustrated in Figures 2–4.Figure 2
Synchronization errorsei1, 1≤i≤50, with control parameters k¯=5, T=0.5, and θ=0.8.Figure 3
Synchronization errorsei2, 1≤i≤50, with control parameters k¯=5, T=0.5, and θ=0.8.Figure 4
Synchronization errorsei3, 1≤i≤50, with control parameters k¯=5, T=0.5, and θ=0.8.Takek¯=5, T=0.5, and θ=0.4. By calculating (41), we can obtain the convergence time t=2.242. The time responses of the error variables eij, 1≤i≤50, j=1,2,3, are illustrated in Figures 5–7, with k¯=5, T=0.5, and θ=0.4. Let k¯=5, T=0.5, and θ=0.1. By calculating (41), we can obtain the convergence time t=3.278. Figures 8–10 describe the time responses or the error variables eij, 1≤i≤50, j=1,2,3, respectively, with different parameters of k¯,T,θ.Figure 5
Synchronization errorsei1, 1≤i≤50, with control parameters k¯=5, T=0.5, and θ=0.4.Figure 6
Synchronization errorsei2, 1≤i≤50, with control parameters k¯=5, T=0.5, and θ=0.4.Figure 7
Synchronization errorsei3, 1≤i≤50, with control parameters k¯=5, T=0.5, and θ=0.4.Figure 8
Synchronization errorsei1, 1≤i≤50, with control parameters k¯=5, T=0.5, and θ=0.1.Figure 9
Synchronization errorsei2, 1≤i≤50, with control parameters k¯=5, T=0.5, and θ=0.1.Figure 10
Synchronization errorsei3, 1≤i≤50, with control parameters k¯=5, T=0.5, and θ=0.1.Remark 16.
From the above analysis and figures, we can conclude that the control rate influences the convergence time; in addition, the tunablek¯ can also influence the convergence time, which is discussed in [34]; we should omit it here. And then, it can be seen that the continuous controller can synchronize the network at t=0.984. Therefore, the convergence time is shorter than the periodically intermittent control.
To further verify the effectiveness of the proposed control design, we take the control rateθ=1, namely, general continuous full control. Figures 11–13 demonstrate time response of the error variables eij, 1≤i≤50, j=1,2,3, respectively, with k¯=5, T=0.5, and θ=1. When the control variables ui=0, 1≤i≤50, namely no controlled is added to the system. The time response of the error variables eij, 1≤i≤50, j=1,2,3, is displayed in Figures 14–16, respectively, with no control input ui.Figure 11
Synchronization errorsei1, 1≤i≤50, with full control parameters k¯=5, T=0.5, and θ=1.Figure 12
Synchronization errorsei2, 1≤i≤50, with full control parameters k¯=5, T=0.5, and θ=1.Figure 13
Synchronization errorsei3, 1≤i≤50, with full control parameters k¯=5, T=0.5, and θ=1.Figure 14
Synchronization errorsei1, 1≤i≤50, with no control input.Figure 15
Synchronization errorsei2, 1≤i≤50, with no control input.Figure 16
Synchronization errorsei3, 1≤i≤50, with no control input.


## 5. Conclusion

This paper has dealt with the finite-time synchronization problem for a class of complex dynamical networks by means of periodically intermittent control. Some novel and useful synchronization criteria, given in terms of a set of linear matrix inequalities, have been obtained by some analysis techniques and finite-time stability theory. Our results reduce the previous works on the controllers that the derivative of the Lyapunov functionV(t) is smaller than -αVη(t). Several simulations are presented to verify the effectiveness of the proposed synchronization criteria finally. The future work in this endeavor will focus on the global problem.



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*Source: 381078-2015-05-20.xml*