# Pinning Synchronization for Complex Networks with Interval Coupling Delay by Variable Subintervals Method and Finsler’s Lemma

Authors: Dawei Gong; Frank L. Lewis; Liping Wang; Dong Dai; Shuang Zhang
Journal: Complexity
 (2017)

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## Abstract

The pinning synchronous problem for complex networks with interval delays is studied in this paper. First, by using an inequality which is introduced from Newton-Leibniz formula, a new synchronization criterion is derived. Second, combining Finsler’s Lemma with homogenous matrix, convergent linear matrix inequality (LMI) relaxations for synchronization analysis are proposed with matrix-valued coefficients. Third, a new variable subintervals method is applied to expand the obtained results. Different from previous results, the interval delays are divided into some subdelays, which can introduce more free weighting matrices. Fourth, the results are shown as LMI, which can be easily analyzed or tested. Finally, the stability of the networks is proved via Lyapunov’s stability theorem, and the simulation of the trajectory claims the practicality of the proposed pinning control.


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## Body

## 1. Introduction

Complex networks have large size and nontrivial complex topological features have been intensively studied by many researchers in recent years. Such networks have connections which are neither purely regular nor purely random. These networks are used to understand and predict the behavior of many structures, for example, Internet, medicine, society, and biology.It has been found that lots of phenomena in real world can be studied by complex networks (such as [1–5] and references therein). Amongst all the topics which are studied by complex networks, synchronization phenomena play an important role due to their real world potential applications. There are many interesting synchronization phenomena in nature world. Lots of efforts have been put into the development of the synchronization problems in complex networks [6–11].It should be noticed that time-varying delays occur commonly in connection topology of networks which are more realistic and cover more situations in practice. Therefore, various kinds of delay methods have been proposed, and synchronization problems for networks with delay have been extensively studied [12–14]. However, the methods to deal with the delay in these papers always need large amount of calculation. So how to remove the redundant computation and improve networks’ performance is still a challenging objective.Normally, complex networks cannot synchronize by themselves, and some controllers are designed to force the system to be synchronized. However, it is hard to design or realize controllers for all nodes of large network structure. Therefore, pinning controllers have been widely used to synchronize complex networks. In [15], an adaptive predictive pinning control is proposed to suppress the cascade in coupled map lattices (CMLs). In [16], by using piecewise Lyapunov theory, some less conservative criteria are deduced for exponential synchronization of the complex networks. In [17], a new adaptive intermittent scheme is used to deduce some novel criteria by utilizing a piecewise auxiliary and other relative references [18–21]. However, in the above papers, many useful situations such as some novel delay processing methods and Finsler’s Lemma which can introduce more matrix-valued coefficients to synchronization criteria are not utilized. As far as I know, such pinning synchronization methodology for complex networks has not been proposed yet.Motivated by the former discussions, we elaborate pinning synchronization results for complex networks via subintervals delay method and Finsler’s Lemma. By constructing a novel Lyapunov-Krasovskii function (LKF) and using some mathematical skills proposed in this paper, complex networks can achieve synchronization.Notations
R
n
↔
n-dimensional Euclidean space
R
m
×
n
↔
m
×
n real matrices
I
n
↔
n-dimensional identity matrix
A
⊗
B
↔ Kronecker product of matrices A and B
diag
⁡
(
⋯
)
↔ block-diagonal matrix
X
Y
∗
Z
↔
X
Y
Y
T
Z


## 2. Preliminaries

Consider the system which consists ofN nodes, and each node has an n-dimensional subsystem; then the pinning control system can be written as (1)x˙it=fxit+c∑j=1NgijAxjt-τt+ui,i=1,2,…,l,(2)x˙it=fxit+c∑j=1NgijAxjt-τt,i=l+1,l+2,…,N,xi(t)=xi1t,xi2t,…,xintT∈Rn. f(xi(t))∈Rn is the activation function. The constants c(c>0), A=(aij)n×n, and G=(gij)N×N are, respectively, representing coupling strength, inner-coupling matrix, and outer-coupling connections.τ
(
t
) satisfies (3)0≤τt≤τ,τ˙t≤μ<1.u
i
,
i
=
1,2
,
…
,
N, are the pinning controllers, which are designed as (4)uit=-cηiAxit-τt-st-τt∈Rn.Letηi>0, for i=1,2,…,l, and ηi=0, for i=l+1,l+2,…,N, ηi are the control gains. Then we can get (5)x˙it=fxit-fst+c∑j=1NgijAxjt-τt-st-τt-cηiAxit-τi-st-τi,i=1,2,…,N.In this following, we will introduce some elementary situations.Assumption 1.
The outer-coupling matricG satisfies (6)gij=gji≥0,i≠jgii=-∑j=1,j≠iNgij,i,j=1,2,…,N.Definition 2.
System (2) is synchronized if (7)x1t=⋯=xNt=st,t⟶∞,where s˙(t)=f(s(t)) and s(t) is a solution of an isolate node.Lemma 3 (see [22]).
The eigenvalues ofG in system (2) is defined by (8)λN≤λN-1≤⋯≤λ3≤λ2≤λ1=0.On the other hand, if N-1 ofn-dimensional differential equations of their 0 solution are asymptotically stable (9)w˙kt=cλkAwkt-τt+Jtwkt.The Jacobian matrix of f(x(t)) at s(t) is J(t); then the synchronized states (2) are the same as the asymptotically stable results of system (9).Lemma 4 (Jensen’s inequality).
For positive definite symmetric matricesϖ∈Rn×n,ϖ:[0,ρ]→Rn, scalar ρ>0, we have(10)ρ∫0ρϖTsΥϖsds≥∫0ρϖsdsTΥ∫0ρϖsds.Lemma 5 (Finsler’s Lemma).
Letς∈Rn,ϕT=ϕ∈Rn×n, and B∈Rm×n, rank⁡(B)<n. Then(1)
ςTϕς<0,Bς=0,ς≠0(2)
(B⊥)TϕB⊥<0,(3)
∃L∈Rn×m,Φ+LB+BTLT<0, where B⊥ is a right orthogonal complement of B.Lemma 6 (see [23]).
Letx(t)∈Rn,M1,M2∈Rn×n,χT=χ>0∈Rn×n,Z∈R2n×2n, a constant τ≥0, and then the integral inequality is defined as follows:(11)-∫t-τtx˙Tsχx˙sds≤ξTtΥξt+τξTtZξt,where (12)Υ≔M1T+M1-M1T+M2∗-M2T-M2,ξt≔xtxt-τ,Z=M1TM2Tχ-1M1M2.


## 3. Main Results

Theorem 7.
For positive definite symmetric matricesPk, Qk, and Sk, real matrices Mik,(i=1,2), and the following LMIs hold for all 2≤k≤N: (13)Ξ1k=H1kτΓ1TSkτ1-μΓ2T∗-τSk0∗∗-τ1-μSk<0,where (14)Γ1=Jt,cλkA,Γ2=M1k,M2k,H1k=Ω11Ω12∗Ω22.Ω11=PkJt+JtTPk+Qk+1-μM1kT+M1kΩ12=PkcλkA-1-μM1kT-M2kΩ22=-1-μQk-1-μM2kT+M2k.Then system (2) is synchronized.Proof.
The Lyapunov function is confined in the following:(15)Vt=wkTtPkwkt+∫t-τttwkTsQkwksds+∫t-τtt∫θtw˙kTsSkw˙ksdsdθ.Then V˙(t) can be expressed as (16)V˙t=2wkTtPkJtwkt+cλkAwkt-τt+wkTtQkwkt-1-τ˙twkTt-τtQkwkt-τt+τtw˙kTtSkw˙kt-1-τ˙t∫t-τttw˙kTsSkw˙ksds≤2wkTtPkJtwkt+cλkAwkt-τt+wkTtQkwkt-1-μwkTt-τtQkwkt-τt+τw˙kTtSkw˙kt-1-μ∫t-τttw˙kTsSkw˙ksds.
From Lemma6, for any constant matrices M1k,M2k(17)-∫t-τttw˙kTsSkw˙ksds≤ηkTtM1kT+M1k-M1kT+M2k∗-M2kT-M2kηkt+τηkTtM1kTM2kTSk-1M1kM2kηkt,where (18)ηkTt=wkTt,wkTt-τt.
Then(19)V˙t≤ηkTtH1k+τΓ1TSkΓ1+τ1-μΓ2TSk-1Γ2ηkt.
By Schur complement,H1k+τΓ1TSkΓ1+τ(1-μ)Γ2TSk-1Γ2 is equivalent to expression (13). Then the proof is completed.In the following criteria, we will introduce Finsler’s Lemma. Combining with the Finsler’s Lemma, convergent LMI relaxations for synchronization analysis are proposed.Theorem 8.
From Lemma5, dynamical system (2) is asymptotically synchronized if there exist Pk=PkT>0,Qk=QkT>0,Sk=SkT>0,Yk=YkT>0, and any real matrices Mik,(i=1,2), and the following LMIs hold for all 2≤k≤N: (20)B⊥TΞ2kB⊥<0,where (21)B=Jt,cλkA,-IN,Ξ2k=H2kτΓ1TYkTτ1-μΓ2T∗-Yk-YkT+τSk0∗∗-τ1-μSk<0,Γ1=Jt,cλkA,Γ2=M1k,M2k,H2k=Ω11Ω12∗Ω22.Ω11=PkJt+JtTPk+Qk+1-μM1kT+M1kΩ12=PkcλkA-1-μM1kT-M2kΩ22=-1-μQk-1-μM2kT+M2k.Proof.
Choose the same LKF in Theorem7. From system (9), the following equation holds for any matrices Yk(2≤k≤N), (22)0=2w˙kTtYk-w˙kt+Jtwkt+cλkAwkt-τt.
Combing (16), (17), and (22), we can obtain (23)V˙kt≤ζkTtΞ2kζkt,where (24)ζkTt=wkTt,wkTt-τt,w˙kTt.
NoteBζk(t)=0; it follows from Lemma 5 that (B⊥)TΞ2kB⊥<0 is equivalent to ζkT(t)Ξ2kζk(t)<0.Remark 9.
Convergent LMI relaxations are introduced by Finsler’s Lemma with homogenous matrix. Then more matrix-valued coefficients can be introduced to reduce conservatism. Moreover, our methods can also be applied to most of the existing synchronization results, such as [6–21].Remark 10.
Different from [24, 25] which divide the constant delay part into many more same size delay, the interval [0,τ(t)] can be chosen arbitrarily into smaller variable subintervals [0,ρτ(t)], [ρτ(t),τ(t)], where ρ∈(0,1).Theorem 11.
From Lemma5, for given scalar 0<ρ<1, system (2) is synchronized if there exist Pk=PkT>0, Qk=QkT>0, Rk=RkT>0, Wk=WkT>0, Sik=SkT>0,(i=1,2), and real matrices Yk, Mik, and Tik,(i=1,2), and the following LMIs hold for all 2≤k≤N: (25)B⊥TΞ3kB⊥<0,where (26)B=Jt,cλkA,0,0,-IN,Ξ3k=H3k10τΓ1TYkTτ1-ρμΓ2T0∗H3k200τ1-ρ1-ρμΓ3T∗∗H3k300∗∗∗-τ1-ρμS1k0∗∗∗∗-τ1-ρ1-ρμS2k<0,H3k1=Ω111Ω121∗Ω221.Ω111=PkJt+JtTPk+Qk+Rk+Wk+1-ρμM1kT+M1kΩ121=PkcλkA-1-ρμM1kT-M2kΩ221=-1-μQk-1-μM2kT+M2kH3k2=Ω112Ω122∗Ω222.Ω112=1-ρμT1kT+T1k-1-ρμRkΩ122=-1-ρμT1kT-T2kΩ222=-1-ρμT2kT+T2k-Wk.Γ1=Jt,cλkA,Γ2=M1k,M2k,Γ3=T1k,T2kH3k3=-Yk-YkT+ρτS1k+τ-ρτS2k.Proof.
The LKF is confined in the following inequality:(27)Vkt=V1kt+V2kt+V3kt+V4kt+V5kt,where (28)V1kt=wkTtPkwktV2kt=∫t-τttwkTsQkwksdsV3kt=∫t-ρτttwkTsRkwksds+∫t-τtwkTsWkwksdsV4kt=∫t-ρτtt∫θtw˙kTsS1kw˙ksdsdθV5kt=∫t-τt-ρτt∫θtw˙kTsS2kw˙ksdsdθ.Then V˙(t) can be expressed as (29)V˙1kt=2wkTtPkJtwkt+cλkAwkt-τtV˙2kt=wkTtQkwkt-1-τ˙twkTt-τtQkwkt-τtV˙3kt=wkTtRkwkt-1-ρτ˙twkTt-ρτtRkwkt-ρτt+wkTtWkwkt-wkTt-τWkwkt-τV˙4kt=ρτtw˙kTtS1kw˙kt-1-ρτ˙t∫t-τttw˙kTsS1kw˙ksdsV˙5kt=τ-ρτtw˙kTtS2kw˙kt-1-ρτ˙t∫t-τtt-ρτtw˙kTsS2kw˙ksds.
From Lemma6, for any constant matrices M1k,M2k(30)-∫t-τttw˙kTsS1kw˙ksds≤η1kTtM1kT+M1k-M1kT+M2k∗-M2kT-M2kη1kt+τη1kTtM1kTM2kTS1k-1M1kM2kη1kt,where (31)η1kTt=wkTt,wkTt-τt.
For any appropriate dimension matricesT1k, T2k, we have (32)-∫t-τt-ρτtw˙kTsS2kw˙ksds≤η2kTtT1kT+T1k-T1kT+T2k∗-T2kT-T2kη2kt+τ1-ρη2kTtT1kTT2kTS2k-1T1kT2kη2kt,where (33)η2kTt=wkTt-ρτt,wkTt-τ.
From system (9), the following equation holds for any matrices Yk(2≤k≤N): (34)0=2w˙kTtYk-w˙kt+Jtwkt+cλkAwkt-τt.
Substituting the proposed equalities,(35)V˙kt≤ΘkTtΞ3kΘkt,where (36)ΘkTt=wkTt,wkTt-τt,wkTt-ρτt,wkTt-τ,w˙kTt.
From Lemma5, ΘkT(t)Ξ3kΘk(t)<0 can be acquired. This completes the proof.Remark 12.
Obviously, when we choose different values ofρ, the synchronization criteria can also be changed. Through the choice of appropriate parameters ρ, different stability results can be obtained.Remark 13.
By using Lemma6 and introducing Finsler’s Lemma, some delay-dependent conditions are acquired in complex networks. The criteria in this paper can be easily used in many existing references and obtain better results, such as [26–30].


## 4. Numerical Example

Consider the Lorenz system in this example(37)x˙1=ax2-x1x˙2=c^x1-x1x3-x2x˙3=x1x2-bx3,where a>0,b>0,c^>0. When a=10,b=8/3,c^=28, the net is chaotic, and its behavior is shown in Figure 1.Figure 1
The chaotic trajectory of Lorenz systems.We assumec=1, and the derivatives of time delays are μ=0,τ=0.2,Λ=6∗diag⁡(1,1,1,1,1), and its Jacobian is (38)Jt=-aa0c^-1000-b,G=-31111-31111-31111-3,A=2.402.404.802.402.4.Figures2 and 3 show the states of network (2) without and with control, respectively. Figure 4 shows the synchronization errors between every node, ej(t)=x1j(t)-xij(t), for i=2,3,4,j=1,2,3. From Figures 3 and 4, it is easy to see that the system cannot be synchronized by itself, and it can achieve synchronization by pinning control.Figure 2
States of network (2) without control: xi(t),i=1,2,3,4.Figure 3
States of network (2) with control: xi(t),i=1,2,3,4.Figure 4
Synchronization errors in an example:ej(t)=x1j(t)-xij(t) for i=2,3,4,j=1,2,3.


## 5. Conclusion

During the past decades, there has been a rapid development of the techniques about complex networks. Numerous studies have shown that complex network is good at dealing with the problem of function approximation and uncertainties. In the paper, a novel analytical method is provided to ensure the synchronization rigorously for complex system. By using Newton-Leibniz formula and introducing Finsler’s Lemma, the obtained synchronization criteria are divided in terms of LMI inequalities, and such method has not been obtained until now. On the other hand, in the studies of network-based motion control in actual models [31–34], a novel integral barrier function is first employed for control design of the constrained distributed parameter system modeled as PDEs [35–38], fault tolerance control in complex systems [39], have been hot topics in recent time. Therefore, how to extend our results into these control fields is still a challenging problem.



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*Source: 2137103-2017-06-08.xml*