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

1School of Mechatronics Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China 2UTA Research Institute, The University of Texas at Arlington, Arlington, TX 76118, USA 3Northeastern University, Shenyang 110036, China 4Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China 5School of Electric Power, South China University of Technology, Guangzhou 510641, China 6School of Astronautics and Aeronautic, University of Electronic Science and Technology of China, Chengdu 611731, China


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][2][3][4][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][7][8][9][10][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][13][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 2 Complexity 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][19][20][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.

Preliminaries
Consider the system which consists of  nodes, and each node has an n-dimensional subsystem; then the pinning control system can be written as ,  = 1, 2, . . ., , are the pinning controllers, which are designed as Let   > 0, for  = 1, 2, . . ., , and   = 0, for  =  + 1,  + 2, . . ., ,   are the control gains.Then we can get In this following, we will introduce some elementary situations.
Lemma 3 (see [22]).The eigenvalues of  in system ( 2) is defined by On the other hand, if  − 1 of n-dimensional differential equations of their 0 solution are asymptotically stable The Jacobian matrix of (()) at () is (); then the synchronized states (2) are the same as the asymptotically stable results of system (9).
Proof.The Lyapunov function is confined in the following: Then V() can be expressed as From Lemma 6, for any constant matrices  1 ,  2 ]   () where Then By Schur complement, (13).Then the proof is completed.
Proof.The LKF is confined in the following inequality: where Then V() can be expressed as From Lemma 6, for any constant matrices  1 ,  2 where For any appropriate dimension matrices  1 ,  2 , we have where From system (9), the following equation holds for any matrices   (2 ≤  ≤ ): Substituting the proposed equalities, From Lemma 5, Θ   ()Ξ 3 Θ  () < 0 can be acquired.This completes the proof.
Remark 12. Obviously, when we choose different values of , the synchronization criteria can also be changed.Through the choice of appropriate parameters , different stability results can be obtained.Remark 13.By using Lemma 6 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][27][28][29][30].

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][32][33][34], a novel integral barrier function is first employed for control design of the constrained distributed parameter system modeled as PDEs [35][36][37][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.