Pinning Cluster Synchronization in Linear Hybrid Coupled Delayed Dynamical Networks

Bao, Anping; Wang, Ting; Fei, Shumin; Tian, Xiaomin

doi:https://doi.org/10.1155/2016/6021497

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Appendix Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2016 | Article ID 6021497 | https://doi.org/10.1155/2016/6021497

Pinning Cluster Synchronization in Linear Hybrid Coupled Delayed Dynamical Networks

Anping Bao,¹Ting Wang,²Shumin Fei,¹and Xiaomin Tian¹

Academic Editor: Xiangyu Meng

Received30 Dec 2015

Accepted17 Apr 2016

Published09 Jun 2016

Abstract

The problem on cluster synchronization will be investigated for a class of delayed dynamical networks based on pinning control strategy. Through utilizing the combined convex technique and Kronecker product, two sufficient conditions can be derived to ensure the desired synchronization when the designed feedback controller is employed to each cluster. Moreover, the inner coupling matrices are unnecessarily restricted to be diagonal and the controller design can be converted into solving a series of linear matrix inequalities (LMIs), which greatly improve the present methods. Finally, two numerical examples are provided to demonstrate the effectiveness and reduced conservatism.

1. Introduction

In past decade, the synchronization of various chaotic systems has received considerable attention since the pioneering works have appeared [1, 2]. Presently, it is widely known that many benefits of having synchronization can be existent. In particular, the synchronization in language emergence and development results can come up with the common vocabulary and agents’ synchronization in organization management can improve their work efficiency. Thus recently, the synchronization has been widely studied owing to its great potential applications. Furthermore, since chaos synchronization in arrays of coupled dynamical networks was initially studied [3], various coupled networks have received the attention because they can exhibit some interesting phenomena [4, 5], and many elegant results have been reported [6–32]. In particular, in [6, 7, 33], time-delay is unavoidable and delayed neural networks (DNNs) are verified to exhibit some complex and unpredictable behaviors, such as periodic oscillations, bifurcation, and chaotic attractors; then, the impulsive and adaptive synchronization has also been studied [8–11], and some uneasy-to-test results have been presented. Most recently, through using Kronecker product, the global synchronization has been studied and elegant criteria have been obtained in terms of LMIs [12–20, 23]. Yet it is worth noting that, in the above works, some most developed techniques were not utilized and the addressed networks seemed to be of simple forms. Thus, researchers have used some effective tools to give less conservative results ensuring the synchronization for more general coupled DNNs [23].

In 1992, as the truth that the effective coupling among neurons varies temporally in a rather short time scale has been found [34], some researchers have mentioned that the degree of synchronization among pairs of neurons changed both temporally and by the choice of pairs. Therefore, the cluster synchronization has been imposed to various dynamical networks [21, 22, 24–31, 35]. However, due to the existence of embedding of invariant synchronization manifolds, it may occur that the system can reach different clustering patterns from the different initial conditions [24–29]. Thus, together with pinning control, some suitable methods have appeared but have been independent of initial states [30]. In [30], the pinning control strategy has been used to realize the cluster synchronization for stochastic coupled DNNs, in which the upper bound of delay variation was less than . Later, some effective techniques were used to overcome the shortcoming during tackling the delayed dynamical networks [31, 35]. Yet though these results above were elegant, there still exist some points waiting for the improvements. Firstly, most works above have not contained lower bound of delay variation and, in fact, its information can play an important role in reducing the conservatism. Secondly, in [30, 31], the inner coupling matrices had to be diagonal, which unavoidably limits the application areas. Thirdly, as for delay , since the triple integral LKF terms such as were firstly put forward [35], it has been used and improved owing to the fact that it could help reduce the conservatism greatly [36]. Yet the authors noticed that some important terms have been ignored when estimating its derivative [35, 36], which also induces the conservatism. Therefore, the tighter estimation should be given. Overall, as for the pinning cluster synchronization of coupled networks, the mentioned points above have not been considered, which remains important and motivates this work.

Inspired by the above discussions, this paper aims to study the problem on cluster synchronization for a class of coupled time-delay networks with linear hybrid coupling by means of pinning control. Through choosing two augmented Lyapunov-Krasovskii functionals (LKFs) and using the combined convex technique, some novel sufficient conditions are presented via Kronecker product and LMIs, whose feasibility can be easily checked by resorting to Matlab LMI Toolbox. In particular, we will give the tighter upper bounds on time derivative of LKF terms. The efficiency and less conservatism can be verified on the basis of two numerical examples.

Notations. is the set of all real matrices; represents the identity matrix and denotes the zero matrix; represents Kronecker product of matrices and .

2. Problem Formulations and Preliminaries

Firstly, suppose the nodes are coupled with states , ; we consider the dynamical networks with each node being an -dimensional DNN with linear hybrid coupling aswhere are the state vectors; here , , , and are the activation functions; also here we assume , , denote the inner coupling matrices, is the control input, and is the input vector.

Remark 1. In system (1), the hybrid coupling is utilized in model (1) and it should be emphasized that the inner coupling matrices , , and are not necessarily restricted to be of diagonal form, which can represent more general cases than the ones in [30, 31].

Suppose that networks (1) will be controlled onto some desired inhomogeneous state as , ; that is, is the desired cluster synchronization pattern under the pinning control, where means that for with and . The function is defined as

For the dynamical networks described by (1), the following assumptions are utilized.

() Here denotes the interval time-varying delay satisfyingMoreover, we give the denotations as , , and .

() For and the configuration matriceswith and , , assume that every matrix for satisfies , , and the sums of all rows in every are zeros.

() There exist constants such that the bounded functions satisfy Here we set , , and

In this paper, we consider one special case that the accurate information on time-delay is available. Without loss of generality, to achieve the goal of cluster synchronization in this work, we will apply the pinning control strategy on the nodes set and adopt the following pinning controller as Let ; one can check that , , and . Then, combining (1) and (2) with (7) yields where andThen, we can easily check that the functions satisfy assumption () and we set

In what follows, some useful basic definition and denotations will be introduced.

Definition 2 (see [30]). The dynamical network (1) with nodes is said to achieve the cluster synchronization, if the nodes are split into clusters as such that the nodes synchronize with each other in the same cluster; namely, for the states and of the arbitrary nodes and in the same cluster , holds, in which stands for the Euclidean norm.

Denotation 1. Denote the constant matrix as in which the matrix can be expressed as follows:with the identity matrix denoting the th one in the matrix vector .

Denotation 2. Denote

3. Pinning Cluster Synchronization

Prior to addressing the main results, the following lemmas will be useful in the proof.

Lemma 3 (see [35]). For any constant matrix , , two scalars , such that the following integrations are well defined; then

Lemma 4 (see [37]). For any vectors and , constant matrices and , and real scalars and satisfying that and , the inequality holds:

Lemma 5 (see [23]). Suppose that , , and are the constant matrices of appropriate dimensions, ; then holds, if and only if ,

Lemma 6 (see [37]). For the symmetric appropriately dimensional matrices , and matrix , the two following statements are equivalent: (i) ; (ii) there exists a matrix of appropriate dimension such that

Now, together with the pinning control strategy, two less conservative criteria will be presented for the cluster synchronization based on Kronecker product and LMI approach.

Theorem 7. Suppose that assumptions are true; then, the controlled dynamical networks (1) can achieve the desired cluster synchronization, if there exist two matrices making , constant matrices , , , , , and guaranteeing and diagonal matrices , , , and such that the LMIs in (17)-(18) hold: where is expressed in Denotation 1 and and with

In what follows, based on Theorem 7, we can consider the pinning cluster synchronization for the dynamical networks composed of time-delay Lur’e systems [32]:where , , and are constant matrices; here denotes the nonlinear function satisfying () and . Then, by using the pinning controller (7), we can derive the following theorem.

Theorem 8. Suppose that assumptions are true; then, the controlled dynamical networks (21) can achieve the desired cluster synchronization, if there exist two matrices making , constant matrices , , , , , and guaranteeing , and diagonal matrices , , and such that the LMIs in (22) hold: where is expressed in Denotation 1, is expressed in Theorem 7, and with

4. Numerical Examples

Two numerical examples will be provided to illustrate the derived results with some typical cases.

Example 1. Consider one 2-dimensional delayed dynamical network (1) described by with the following parameters: Then, through adopting the pinning controller in (7), with and , we assume that the desired cluster synchronization states of DNNs (25) are and , which can satisfywith different initial conditions. In order to reduce the number of controllers and realize the cluster synchronization, we can use the controlled networks sets and, respectively, choose the configuration matrices as with In what follows, two cases will be given to illustrate the efficiency and reduced conservatism of our results.

Case 1. Given , choose three inner coupling matrices of diagonal form as Then, through, respectively, setting , and unavailable , we can compute the corresponding maximum allowable upper bounds (MAUBs) in Table 1 based on Theorem 7 and Remark A.2 by resorting to Matlab LMI Toolbox.
In Table 1, the term “—” means that the corresponding value is unavailable. Based on the MAUBs in Table 1, one can verify that our results can be less conservative than some existent ones. In particular, as the inner coupling matrices , , and are not diagonal, our theorems still can be applicable while [30, 31] fail.

Case 2. Choosing and the inner coupling matrices , , and as Case 1, one can derive , , , and . Owing to the fact that and , , and are not diagonal, the methods in [30, 31] fail to verify the synchronization. Yet, by resorting to Matlab LMI Toolbox, Theorem 7 can guarantee the pinning cluster synchronization and the feasible solution to LMIs in (17)-(18) can be obtained as follows:

Example 2. In this example, we consider the well-known Chua’s circuit to illustrate our synchronization results, which can be expressed as with and the parameters , , , , and . Then, the circuit model can be represented as the Lur’e system: with and belonging to the sector . Now we consider the cluster synchronization of the dynamical networks with each node being a -dimensional system (33) with linear coupling as In order to reduce the number of controllers and realize the cluster synchronization, we adopt the pinning controller (7) as and the desired cluster synchronization states of (A.15) satisfy with the initial conditions and . Now through setting , , and the configuration matrices as with together with Theorem 8 and LMI in Matlab Toolbox, we can easily verify that network (34) can achieve the desired cluster synchronization, which can be further supported by the synchronization error states in Figure 1.

5. Conclusions

This paper has investigated the problem on pinning cluster synchronization for delayed dynamical networks with linearly hybrid coupling. Two novel conditions have been derived by employing the Lyapunov-Krasovskii stability theory. It is worth pointing out that some most recently developed techniques such as combined convex technique and triple integral LKF terms have been employed, which can help extend the application areas. The synchronization criteria are presented in the forms of LMIs, which can be checked easily by referring to Matlab LMI Toolbox. Finally, two numerical examples can illustrate the less conservatism of our theorems based on some comparing results.

Appendix

Proof of Theorem 7. Based on assumptions ()–(), we can choose the Lyapunov-Krasovskii functional aswherewith constant matrices , , , and and diagonal matrices and waiting to be determined.
Then, the time derivative of along system (8) can be directly computed as Through employing Lemmas 3 and 4 and , we can estimate two terms in (A.4) as with denoting Furthermore, together with Lemmas 3 and 4 and , we can compute For any constant matrices and , it follows from (8) and Denotation 2 that Together with () and any diagonal matrices and , one can easily deriveThen, through employing Denotations 1 and 2 and the right terms in (A.3), (A.4), (A.5), (A.7), (A.8), and (A.9), the term can be estimated as where , the denotations , , , , and are expressed in (17)-(18), and Now as for the terms in (A.10), if the two following inequalities can be true simultaneously, then we can obtain for any .
Together with Lemma 5, the LMIs in (18) can make hold. Meanwhile, with the denotations and in (17), the LMI results in (17) mean the following inequalities to be true. Then, based on Lemma 5, the terms in (A.13) can guarantee Furthermore, one can easily check Then, based on Lemma 6, the matrix inequality in (A.15) can make the following term true: Therefore, the LMIs in (17)-(18) can make for any . Then, based on Lyapunov stability theory and Definition 2, the error system (8) is asymptotically stable; that is, the controlled networks (1) can achieve the desired cluster synchronization.

Remark A.1. During the proof procedure in Theorem 7, one can easily check that, firstly, two triple integral Lyapunov terms in could play an essential role in reducing the conservatism, and some tighter upper bounds on the derivatives were presented; secondly, we combined reciprocal convex technique with normal convex one to tackle time-delay issue; thirdly, through employing the Kronecker product in (A.10), (A.13), and (A.15), the forms of the inner coupling matrices , , and can be described more generally than the ones in [30, 31].

Remark A.2. As for time-delay issue, most present works on delayed dynamical networks have only considered the case . Thus, through replacing by in (A.1) and using the similar proving procedure, we also can derive some relevant results.

Proof of Theorem 8. Based on assumptions ()–(), we can choose the Lyapunov-Krasovskii functional as wherewith setting diagonal matrices , , and identical to the ones in Theorem 7.

Remark A.3. Theorems 7 and 8 provide two less conservative criteria that ensure the cluster synchronization of network models (1) and (28) via pinning control, which can be easily checked by resorting to LMI in the Matlab Toolbox and do not require the inner coupling matrices to be of diagonal form. Moreover, as for , the derived theorems can be reduced to guarantee the pinning global synchronization.

Remark A.4. Presently, the reciprocal convex technique in [37] has been widely put forward to tackle time-delay systems, owing to the fact that it could reduce the conservatism more efficiently than some previous ones [23, 31]. Yet it comes to our attention that the reciprocal technique has not been utilized to tackle the pinning cluster synchronization on delayed complex networks and it has been fully considered in this work.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (nos. 61374116 and 61473079), Jiangsu Natural Science Foundation (no. BK20150888), and Natural Science Foundation for Jiangsu’s Universities (no. 15KJB12004).

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Copyright © 2016 Anping Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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