# Pinning Synchronization of Complex Dynamical Networks with Variable-Delayed Coupling by Periodically Intermittent Control

Authors: Ranran Cheng; Xiaoyong Tian; Mingshu Peng; Jinchen Yu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper studied the adaptive pinning synchronization in complex networks with variable-delay coupling via periodically intermittent control. Theoretical analysis is included by means of Lyapunov functions and linear matrix inequalities (LMI) to make all nodes reach complete synchronization. Moreover, the synchronization criteria do not impose any restriction on the size of time delay. Numerical examples including the regular, Watts–Strogatz and scale-free BA random topological architecture are provided to illustrate the importance of our theoretical analysis.


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## Body

## 1. Introduction

Complex dynamical networks have been intensively studied [1, 2] over the last few years due to their potential applications in various fields of the real world, such as in the biological systems, scientific citation web, social networks, and electrical power grids and thus became an important part of our daily life.Since Pecora and Carroll [3] found chaos synchronization in 1990, it has become a focal research topic recently. Due to the synchronization not only can well explain many natural phenomena observed, such as synchronized firefly flashing and swarming of fishes, but also has many practical potential applications such as image processing, the operation of unmanned aerial vehicle, and secure communication. However, the complex network cannot synchronize by itself, so many control methods have been developed, such as linear and nonlinear feedback control, time-delay feedback control, sliding mode control [4], adaptive control [5], pinning control [6], impulsive control [7, 8], and intermittent control [9–12]. Especially, discontinuous feedback controls, including impulsive control [13] and intermittent control, have been attracting much attention since they are practically and easily implemented in some engineering domains. However, the intermittent control is different from the impulsive control because impulsive control is activated only at some isolated moments, while the intermittent control has a nonzero control width.Intermittent control was first introduced to control chaos systems by Zochowski [9] in 2000. It is more efficient when the system output is measured intermittently rather than continuously. There are some novel synchronization criteria in complex networks with delay or nondelay coupling by intermittent control. Li et al. [14] studied the inner synchronization of delayed nonlinear chaotic systems by intermittent control. Further, Mei et al. [10] used diver and response systems to produce the finite-time synchronization for the complex systems without delay. In [12], the authors investigated the synchronization problem of stochastic perturbed complex systems with time-varying delays. On the contrary, complex networks have a large number of nodes in the real world, and it is usually impractical to control a complex network by adding the controllers to all nodes. To reduce the number of controlled nodes, pinning control, in which controllers are only applied to partial nodes, is introduced [15–20]. In [21], the authors discussed the synchronization for coupled dynamical networks with mixed delays and uncertain parameters using pinning control and intermittent control. By means of intermittent control, the authors [22] studied finite-time synchronization for a class of reaction-diffusion neural networks by small domains on their spatial boundaries, associated with an interaction graph. Li et al. [23] and Xu et al. [24]discussed the synchronization problem of general complex networks with fractional-order dynamical networks by periodically intermittent pinning control.Moreover, in [11, 15], the authors investigated the synchronization of complex networks with delays by pinning periodically intermittent control; especially, they assumed that the control width needs to be larger than the time delay or the time-varying delays should be differentiable, and their derivatives are less than 1. Motivated by the above discussions, we removed these constraints in our results. In this paper, we give the complex systems with both time-varying delays and nondelay couplings, and using pinning control and periodically intermittent control methods, some novel criteria for pinning synchronization are derived. Numerical examples including the regular, Watts–Strogatz and scale-free BA random topological architecture are provided to illustrate the importance of our theoretical analysis.The rest of the paper is organized as follows: in Section2, we propose a general complex dynamical network model; some preliminaries and lemmas are given. In Section 3, some pinning adaptive synchronization criteria for the general complex dynamical networks with delay coupling are given. Numerical examples are given in Section 4. Finally, we draw our conclusion in Section5.


## 2. Preliminaries and Mathematical Models

Consider a generally controlled complex delayed dynamical system consisting ofN nodes, with each node being of n dimensions, which is described by(1)x˙it=Axit+fxit+c0∑j=1NbijΓxjt+c1∑j=1Nb¯ijΓxjt−τt,where 1≤i≤N, xit=xi1t,…,xintT∈ℝn is the state variable of node i, A∈ℝn×n is a given constant matrix, and f:ℝn⟶ℝn is a continuously differentiable function describing the nonlinear dynamics of the single node. Here, c0 and c1 are two parameters of the nondelay and delay coupling strength, respectively, Γ=γij∈ℝn×n is an inner-coupling matrix, and τt is the coupled delay and bounded by a known constant, i.e., 0<τt<τ. B=bijN×N and B¯=b¯ijN×N represent the adjacency configuration of the network with the nondelay and delay couplings, respectively; if there is a link from node i to node j, then bij>0 (or b¯ij>0) j≠i; otherwise, bij=0 (or b¯ij=0). Moreover, bii=−∑j≠ibij and b¯ii=−∑j≠ib¯ij.As we know, the complex network cannot synchronize by itself; then, we add the adaptive controllervit as follows:(2)x˙it=Axit+fxit+c0∑j=1NbijΓxjt+c1∑j=1Nb¯ijΓxjt−τt+vit.Hereafter, letst=st;t0,x0∈Rm be a solution of the node system x˙t=fxt. Then, st is a synchronous solution of controlled complex delayed dynamical system (2). Note that st may be an equilibrium point, a periodic orbit, or a chaotic attractor.Define error vectors as(3)eit=xit−st,1≤i≤N.According to system (2), the error system is described by(4)ei.t=Aei+fxit−fyit+c0∑j=1NbijΓejt+c1∑j=1Nb¯ijΓejt−τt+vit,where 1≤i≤N.Lemma 1 (Schur complement, see [25]).
The linear matrix inequality (LMI) is as follows:(5)AxBxBTxCx>0,where Ax=ATx,Cx=CTx is equivalent to one of the following conditions:(1)
Ax>0,Cx−BTxA−1xBx>0(2)
Cx>0,Ax−BxC−1xBTx>0Lemma 2 (see [26]).
LetX and Y be arbitrary n-dimensional real vectors, K be a positive definite matrix, and P∈ℝm×m. Then, the following matrix inequality holds:(6)2XTPY≤XTPK−1X+YTKY.


## 3. Main Results

In the following, assume thatΓ≠0 and Γ2=γ>0. Denote ρii as the minimum eigenvalue of the matrix Γ+ΓT/2. Let B^s=B^+B^T/2, and its eigenvalues are expressed as λ1≥λ2≥…≥λN, where B^ is a modifying matrix of B via replacing the diagonal elements bii by ρmin/γbii. Note that, generally, B^ does not possess the property of zero row sums.To realize the network synchronization, the controllersvi should guide the error vectors eit to approach zero as t goes to infinity as(7)limt⟶∞eit2=0,1≤i≤N.Choose the adaptive controllers as follows:(8)vit=−piteit,1≤i≤l,t∈mT,m+hT,0,l+1≤i≤N,t∈mT,m+hT,0,1≤i≤N,t∈m+hT,m+1T,where pi.t=expa1tqieiTteit, qi, and a1 are positive constants. T>0 denotes the control period, 0<h<1 is the rate of control duration called control rate, and m=0,1,2,….Theorem 1.
Suppose thatfxi−fyi2≤βei2. If there exists a positive constant a1,a2, and k such that(9)α+β+a12+c1γN−1b¯1+kb¯22kIN+c0γB^s−D<0,α+β−a2−a12+c1γN−1b¯1+kb¯22k+c0γλ1<0,ξ−a21−h>0,where A2=α, D=diagp,…p︸l,0,…0︸N−l, p>0, q=c1N−1kb¯1+b¯2γ, b¯1=maxb¯ij,j≠i, a2>a1>q, b¯2=maxb¯ii, and ξ>0 is the smallest real root of the equation −a1+ξ+qexpξτ=0, then the synchronized manifold st,st,…,st︷NT of controlled complex delayed dynamical system (2) is globally asymptotically stable under periodical intermittent controllers (8).Proof.
Construct the candidate Lyapunov function as follows:(10)Vt=12∑i=1NeiTteit+12∑i=1lexp−a1tpit−p2qi,and when mT≤t≤mT+h, calculating the time derivative of Vt along the trajectories of (4), one has(11)V˙t=∑i=1NeiTteit.+∑i=1lexp−a1tpit−ppit.qi−a12∑i=1lexp−a1tpit−p2qi=∑i=1NeiTtAei+fxi−fs+c0∑i=1N∑j=1NbijeiTtΓejt+c1∑i=1N∑j=1Nb¯ijeiTΓejt−τt−∑i=1lpiteiTteit+∑i=1lexp−a1tpit−ppi.tqi−a12∑i=1lexp−a1tpit−p2qi≤α+β∑i=1NeiTteit+c0∑i=1N∑j=1NbijeiTΓejt+c1∑i=1N∑j=1Nb¯ijeiTΓejt−τt−∑i=1lpeiTteit−a12∑i=1lexp−a1tpit−p2qi.
From Lemma2, one can find(12)2∑i=1N∑j=1Nb¯ijeiTΓejt−τt≤∑i=1N∑j=1,j≠iNb¯ijeiTtΓK−1eit+ejt−τtTKejt−τt+2∑i=1Nb¯iiγeit2eit−τ2≤∑i=1N∑j=1,j≠iNb¯1γkeiTteit+kb¯1ejTt−τtejt−τt+∑i=1Nb¯iiγeiTteiTt+eiTt−τteiTt−τt=N−1b¯1γk+b¯2γ∑i=1NeiTteit+N−1kb¯1+b¯2γ∑i=1NeiTt−τteit−τt,(13)c0∑i=1N∑j=1NbijeiTΓejt≤c0∑i=1N∑j=1,j≠iNγbijeit2ejt2+c0∑i=1NbiiρmineiTteit,where K2=k,Γ2=γ,b¯1=maxb¯ij,j≠i,b¯2=maxb¯ii, and ρii is the minimum eigenvalue of the matrix Γ+ΓT/2. It follows from (12) and (13) that(14)V˙t≤eTtα+β+c1γN−1b¯1+kb¯22kIN+c0γB^s−Det+c1N−1kb¯1+b¯2γ2eTt−τtet−τt−a12∑i=1lexp−a1tpit−p2qi≤eTtα+β+a12+c1γN−1b¯1+kb¯22kIN+c0γB^s−Det+q2eTt−τtet−τt−a12∑i=1NeTtet+∑i=1lexp−a1tpit−p2qi≤−a1Vt+qVt−τt,where et=e1t2,e2t2,…eNt2T, D=diagp,…p︸l,0,…0︸N−l, and q=c1N−1kb¯1+b¯2γ.
WhenmT+h≤t≤m+1T,(15)V˙t=∑i=1NeiTtei.t−a12∑i=1lexp−a1tpit−p2qi=∑i=1NeiTtAei+fxi−fs+c0∑i=1N∑j=1NbijeiTtΓejt+c1∑i=1N∑j=1Nb¯ijeiTΓejt−τt−a12∑i=1lexp−a1tpit−p2qi≤α+β∑i=1NeiTteit+c0∑i=1N∑j=1NbijeiTΓejt+c1∑i=1N∑j=1Nb¯ijeiTΓejt−τt−a12∑i=1lexp−a1tpit−p2qi≤eTtα+β+c1γN−1b¯1+kb¯22kIN+c0γB^set+q2eTt−τet−τt−a12∑i=1lexp−a1tpit−p2qi≤eTtα+β−a2−a12+c1γN−1b¯1+kb¯22kIN+c0γB^set+q2eTt−τtet−τt+a2−a12∑i=1NeTtet+∑i=1lexp−a1tpit−p2qi≤a2−a1Vt+qVt−τt.
Namely, we have(16)V˙t≤−a1Vt+qVt−τt,t∈mT,m+hT,V˙t≤a2−a1Vt+qVt−τt,t∈m+hT,m+1T.
In the following, we will prove that(17)Vt≤sup−τ≤θ≤0Vθexp−ξ−a21−ht,t≥0.
Denotegξ=−a1+ξ+qexpξτ; as a1>q>0, we have g0<0, g∞>0, and g′ξ>0. Using the continuity and the monotonicity of the function, gξ=0 has a unique positive solution ξ>0. Let M=sup−τ≤θ≤0Vθ and Wt=expξtVt.
Q
1
t
=
W
t
−
ϵ
M, ϵ>1 is a constant. Obviously,(18)Q1t<0,forallt∈−τ,0.
Next, we will prove that(19)Q1t<0,forallt∈0,hT.
Otherwise, there existst1∈0,hT such that(20)Q1t1=0,Q˙1t1≥0,andQ1t<0,−τ≤t<t1.
Using (16) and (20), one obtains(21)Q˙1t1=ξexpξt1Vt1+expξtV˙t1≤ξ−a1Wt+qexpξτt1Wt1−τt1<−a1+ξ+qexpξτεM=0.
This leads to a contradiction with (20); hence, (19) holds.
Now, we prove that, fort∈hT,T,(22)Q2t=Wt−εMexpa2t−hT<0.
Otherwise, there existst2∈hT,T such that(23)Q2t2=0,Q˙2t2≥0,andQ2t<0,hT≤t<t2.
Then,(24)Q˙2t2=ξWt2+expξt2V˙t2−a2εMexpa1t−hT≤ξ+a2−a1Wt+qexpξτt2Wt2−τt2−a2εMexpa2t−hT.
For0<τt<τ, if hT≤t2−τt2<t2, then from (23), one has(25)Wt2−τt2<εMexpa1t2−hT,and if −τ≤t2−τt2<hT, from (19), one obtains(26)Wt2−τt2<εM<εMexpa2t2−hT.
So,(27)Q˙2t2<−a1+ξ+qexpξτεMexpa2t2−hT=0.
This leads to a contradiction with (23); hence, (22) holds.
According (19) and (22), one obtains(28)Wt<εMexpa21−hT,forallt∈−τ,T.
Similarly, we can prove that(29)Wt<εMexpa21−hT,t∈T,1+hT,Wt<εMexpa2t−2hT,t∈1+hT,2T.
By induction, we can derive the following estimation ofWt for any integer m:(30)Wt<εMexpa2m1−hT,t∈mT,m+hT,Wt<εMexpa2t−m+1hT,t∈m+hT,m+1T.
Since for anyt≥0, there exists a nonnegative integer r such that rT≤t≤r+1T; then, we have(31)Wt<εMexpa2r1−hT≤εMexpa21−ht,t∈rT,r+hT,Wt<εMexpa2t−r+1hT≤εMexpa21−ht,t∈r+hT,r+1T.
Letε⟶1, and from the definition of Wt, one has(32)Vt≤Mexp−ξ−a21−ht.
AsVt=1/2∑i=1NeiTteit+1/2∑i=1lexp−a1tpit−p2/qi, one has(33)et2≤2M1/2exp−ξ−a21−h2t.
In view ofξ−a21−h>0, we can draw the conclusion.
The proof is thus completed.
Let(34)η=α+β+a12+c1γN−1b¯1+kb¯22k,M=ηIN+c0γB^s−D=Q−D˜GGTM˜,where D˜=diagp,…p︸l, M˜ is obtained by removing the 1,2,…,l row-column pairs of matrix M, and Q and G are matrices with appropriate dimensions.
According to Lemma1, one can easily see that M<0 is equivalent to M˜<0 because of Q−D˜−GM˜−1GT<0 by choosing p>λmaxQ−GM˜−1GT, i.e., M<0 is equivalent to M˜<0 for sufficiently large p. Note that M˜=ηIN−l+cγB^s¯, where B^s¯i,j=B^l+i,l+js,i,j=1,2,…,N−l. Then, one can immediately get the following corollary:Corollary 1.
Suppose thatfxi−fyi2≤βei2, τ≤h and τ≤T−h. If there exists a positive constant a1,a2, and k such that(35)λ¯l=λmaxB^s¯<−ηc0γ,α+β−a2−a12+c1γN−1b¯1+kb¯22k+c0γλ1<0,ξ−a21−h>0,where A2=α, q=c1N−1kb¯1+b¯2γ, b¯1=maxb¯ij,j≠i, a2>a1>q, b¯2=maxb¯ii, and Γ2=γ, ξ>0 is the smallest real root of the equation −a1+ξ+qexpξτ=0, then the synchronized manifold st,st,…,st︷NT of controlled complex delayed dynamical system (2) is globally asymptotically stable under periodical intermittent controllers (8).Remark 1.
The number of control nodesl can be chosen properly to adjust the synchronization efficiency of networks. However, it is noted that l by the theoretical prediction is only a sufficient condition but not a necessary one. In simulations, we will show that a small value of l can also lead to synchronization.


## 4. Numerical Simulation

In this section, a numerical example is used to verify the effectiveness of the proposed network synchronization criteria.Consider the Lorenz oscillator model described by the following equation:(36)x˙it=Axi+fxi,where xit=xi1t,xi2t,xi3tT∈R3, fxi=0,−xi1txi3t,xi1txi2tT, and(37)A=−1010030−1000−3/8.It has been known that the Lorenz oscillators exhibit chaotic behavior, and Figure 1 shows it clearly.Figure 1
The chaotic attractor of the Lorenz system.Then, we consider controlled complex delayed dynamical system (2) consisting of 200 identical Lorenz systems, which are described by(38)x˙it=Axi+fxi+c0∑j=1200bijΓxjt+c1∑j=1200b¯ijΓxjt−τ+vit,1≤i≤200,where Γ=diag1,1.2,1 and B=bij200×200 and B¯=b¯ij200×200 are symmetrically diffusive coupling matrices with bij=0 (or b¯ij=0 ) or 1j≠i. Here, the coupling coefficient c0=30.As we know, Lorenz system is bounded. Here, we supposexi1≤29, xi2≤29, −1≤xi3≤57, s1≤29, s2≤29, −1≤s3≤57, and 1≤i≤200.(39)fxi−fs2=−xi1xi3+s1s32+xi1xi2−s1s22=−xi3ei1−s1ei32+xi2ei1+s1ei22≤75.97ei2.Suppose that the network structure of equation (38) obeys the regular (p = 0, 2m = 8)/small-world (p = 0.3, 2m = 8) [1]/scale-free (SF) [2] distribution, respectively. The number of the nodes N=200. Obviously, γ=Γ2=1.2, α=A2=31.91, β=75.97, b¯1=1, and ρmin=1. Choosing the coupling coefficient c0=30, c1=0.01, k=γ, T=0.2, h=0.9, and τt=1+0.1sint<τ=1.1. By using the MATLAB LMI Toolbox and the corollary, one can obtain Table 1.Table 1
Parameters in system (38) with different topological architecture.
Parameter
Regularp=0
Small-worldp=0.1
Scale-free
b
¯
2
8
14
47
λ
1
0.8561
0.9388
0.6623
q
2.27601
2.3113
2.7439Please note that the average number of neighbours is the same between the small-world network with the connection probabilityp=0.1 and the regular network (p = 0), the number of neighbours becomes rand in the scale-free/small-world topological structure, and maxb¯ii can be estimated by the statistic method.Asa1−8<ξ<a1−9, then if we choose a1=45 and a2=325, it is easy to verify that the criteria in Theorem 1 are satisfied.For the regular network (p = 0), one has η=131.580; from corollary, one can find λl∗<−3.6531. As λ25=−3.5984 and λ26=−4.0041, we can choose l∗=26. By numerical simulation, the synchronization error quickly tends to zero for l=15<l∗. This further indicates that l>l∗ is only a sufficient condition, and a small value of l can also lead to synchronization.The initial conditions of the numerical simulations are as follows:xi0=4+0.5i,5+0.5i,6+0.5iT and s0=4,5,6T. Synchronous errors are shown in Figure 2. And we choose different control periods, and the synchronous errors are shown in Figure 3. It is seen from the figures that small-world or scale-free networks can reach complete synchronization by pin-controlling fewer nodes than regular systems.Figure 2
Synchronizability in system (38) with different topological architecture (N = 200) by pinning control: (a) regular networks with 8 neighbours, l = 15, (b) WS small-world networks with the connection probability p=0.1, l = 5, and (c) scale-free networks, l = 2.
(a)
(b)
(c)Figure 3
Synchronizability in system (38) with different periods: (a) T=0.2; (b) T=0.5.
(a)
(b)


## 5. Conclusions

In this paper, we investigate adaptive pinning synchronization in complex delayed dynamical networks with time-varying delays by intermittent control. Based on the Lyapunov stability theory and chaos control method, several adaptive synchronization criteria are obtained. Our results show that the control width does not need to be larger than the time delays, and there is no restriction on the size of time delays. Moreover, we also find that small-world or scale-free networks can reach complete synchronization by pin-controlling fewer nodes than regular systems.



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