Abstract and Applied Analysis

Volume 2013 (2013), Article ID 728606, 16 pages

http://dx.doi.org/10.1155/2013/728606

Research Article

Synchronization of Switched Complex Bipartite Neural Networks with Infinite Distributed Delays and Derivative Coupling

Qiuxiang Bian,¹ Jinde Cao,2,3 Jie Wu,¹ Hongxing Yao,⁴ Tingfang Zhang,¹ and Xiaoxu Ling⁵

¹Department of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang 212003, China

²Department of Mathematics, Southeast University, Nanjing 210096, China

³Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

⁴Department of Mathematics, Jiangsu University, Zhenjiang 212003, China

⁵Faculty of Economics and Management, Nanjing University of Aeronautics & Astronautics, Nanjing 212096, China

Received 7 March 2013; Accepted 8 April 2013

Academic Editor: Hongli Dong

Copyright © 2013 Qiuxiang Bian 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.

Abstract.

A new model of switched complex bipartite neural network (SCBNN) with infinite distributed delays and derivative coupling is established. Using linear matrix inequality (LMI) approach, some synchronization criteria are proposed to ensure the synchronization between two SCBNNs by constructing effective controllers. Some numerical simulations are provided to illustrate the effectiveness of the theoretical results obtained in this paper.

1. Introduction

In recent years, neural networks have been intensively studied due to their potential applications in many different areas such as signal and image processing, content-addressable memory, optimization, and parallel computation [1–3]. Bidirectional associative memory (BAM) neural networks were first proposed by Kosko in [4, 5]. This class of networks has good applications in pattern recognition, solving optimization problems, and automatic control engineering. A large number of results on the dynamical behavior of BAM neural networks have been reported [6–9].

Switched systems, as an important kind of hybrid systems, have drawn considerable attention of researchers because of their theoretical significance and practical applications [10–12]. Switched systems are composed of a family of continuous-time or discrete-time subsystems and a rule that specifies the switching among them [13, 14]. Recently, the switched neural networks, whose individual subsystems are a set of neural networks, have found applications in the field of high speed signal processing, artificial intelligence, and biology, so there are many theoretical results about the switched neural networks [15–17].

Complex networks, which are a set of interconnected nodes with specific dynamics, have sparked the interest of many researchers from various fields of science and engineering such as the World Wide Web, electrical power grids, global economic markets, sensor networks; for example, see [18–20] and references therein. Bipartite networks are an important kind of complex networks, whose nodes can be divided into two disjoint nonempty sets such that every edge only connects a pair of nodes, which belong to different sets. Many real-world networks are naturally bipartite networks, such as the papers-scientists networks [21] and producer-consumer networks [22]. Recently, authors [23] have introduced a bipartite-graph complex dynamical network model that is only linearly coupled and has no delays. It is well known that time delays exist commonly in real-world systems. Therefore, many models of coupled networks with coupling delays are proposed, for example, constant single time delay [24], time-varying delays [25], and mix-time delays [26]. On the other hand, the coupled network often occurs in other forms, for example, nonlinearly coupled networks [27] and linearly derivative coupled networks [28]. In [29], a general model of bipartite dynamical network (BDN) with distributed delays and nonlinear derivative coupling was introduced. Synchronization of complex networks has been intensively investigated since they can be applied in power system control, secure communication, automatic control, chemical reaction, and so on [30–32]. The study of synchronization of coupled neural networks is an important step for both understanding brain science and designing coupled neural networks for practical use. Yu et al. [33] consider the synchronization of switched linearly coupled neural networks with constant delays, but the controllers are complex and changed with the switched rule. Synchronization of two coupled BDNs was investigated by adaptive method [29], but the controllers are complicated and the model does not include infinite distributed delays coupling and switching. Extending BAM neural networks to complex networks, we get complex bipartite dynamical networks (CBDNs). The dynamics of individual node in CBDNs is switched system and the switched coupling is considered; switched complex bipartite neural network (SCBNN) can be obtained. To the best of our knowledge, up to now, there is not any work that discusses the synchronization problem in SCBNN.

Motivated by the previous discussion, we first proposed a model of SCBNN, and then investigated the synchronization between two SCBNNs with infinite distributed delays and derivative coupling. Using adaptive controllers and linear matrix inequality (LMI) approach, some synchronization criteria are proposed to ensure the synchronization between two coupled SCBNNs. In our paper, the proposed controllers are simpler and do not change with the switched rule, which can be realize more easily.

The paper is organized as follows. In Section 2, a model of SCBNN with infinite distributed delays and derivative coupling is presented, and some hypotheses and lemmas are given too. In Section 3, several synchronization criteria on the SCBNNs are deduced. In Section 4, numerical examples are given to demonstrate the effectiveness of the proposed controller design methods in Section 3. Finally, conclusions are given in Section 5.

Notations. Throughout this paper, and denote the maximum eigenvalue and minimum eigenvalue of a real symmetric matrix, respectively. The notation denotes the symmetric block.

2. Model Description, Assumptions, and Lemmas

Consider a complex bipartite dynamical network (CBDN) consisting of two disjoint nonempty node sets and . Suppose that and , , are integer. The coupled network is described as follows:

where , denotes the state variables of nodes and , respectively. and are diagonal matrices with . are weight matrices, are delayed weight matrices, , , , , , , corresponds to the boundedness activation functions of neurons. , are the delay kernel functions. , and are time delays. and express infinite distributed delays. and are the constant external input vectors. The matrix is the delayed weight coupling matrix denoting coupling strength between nodes. If there is a connection from node to , then ; otherwise, and the matrix satisfies the sum of every row being zero. The definitions of the other coupling matrixes , , ,, and are similar to that of matrix ; hence, they are omitted here.

In this paper, we consider a class of switched complex bipartite neural network with infinite distributed delays and derivative coupling, which is described as follows:

where switching signal is piecewise constant functions, which is a value in the finite set . This means that the matrices are allowed to take values at particular time, in a finite set . We define the function as follows:

It follows that under any switching rules . Model (2) can be written as

The response network of the drive network (4) is

where and are the control inputs.

Let , , , and . The error dynamical system of (4) and (5) is given by

where

In this paper, the following assumptions and lemmas are needed.

There exist diagonal matrices and , such that

  and , , and . There exist diagonal matrices and , such that

  and , , , and  ., , , , , and are differential functions with , , , , , and . are real-value nonnegative continuous functions defined in satisfying

Lemma 1 (see [34]). Given any real matrices , , and of appropriate dimensions and a scalar such that , then the following inequality holds:

Lemma 2 (see [35]). Given a positive definite matrixand a symmetric matrix , then

Lemma 3 (Schur complement). Given constant symmetric matrices , , and , where and , then if and only if

For convenience, let

3. Main Results

Theorem 4. Under assumptions , the two coupled SCBNNs (4) and (5) can be synchronized, if there exist positive constants, , positive matrices , , , , , and diagonal positive matrices , , such that

and the adaptive feedback controllers are designed as

where , , , , and .

Proof. For the error dynamical system (6), we design the following Lyapunov-Krasovskii function:

where

Calculating the derivative of (22) along the trajectories of (6), we have

By Lemma 1, we can get from

By assumptions and , it is obvious that

Observe that

Using inequality

we have

Using Lemma 2 and condition (17), we get

Substituting (20) into (24) and combining (24)–(31), it can be derived by condition (18) that

where   .

Meanwhile, by a similar process, the following inequality can be true:

where .

By condition (17), we have

By (15)-(16) and Lemma 3 (Schur complement), it can be obtained that ,. Set , where

then , and

Therefore, is nonincreasing in . One has bounded since , so exists and

From (22)-(23) and conditions , and we have , so is bounded. According to error system (6), is bounded for due to the boundedness of activation functions. From the above we can see that and . By using Barbǎlat lemma (see [36]), one has , so the two SBNNs (4) and (5) can obtain synchronization under the controllers (20). This completes the proof.

We take CBDN (1) as drive network. The response network of the drive network (1) is

where are the control inputs.

From Theorem 4, we can get the following corollary.

Corollary 5. Under assumptions , the two coupled CBDNs (1) and (38) can be synchronized, if there exist positive constants , positive matrices , , , , , and diagonal positive matrices , , , such that

and the adaptive feedback controllers are designed as

where ,  .

Remark 6. From Corollary 5, we can easily get that the controllers in this paper are simpler than those of Theorem 1 in [29].

Remark 7. If the coupling matrix of the SCBNN is not a diffusive matrix satisfying the sum of every row being zero, we can still obtain the same result from the proof of Theorem 4.

Theorem 8 presents another sufficient condition to ascertain that the two networks (4) and (5) can be synchronized, using the following simple adaptive feedback controllers:

where , , and are positive constants.

Let

then the error dynamical system of (6) becomes

Theorem 8. Under assumptions and using the adaptive feedback controllers (41), the two coupled SCBNNs (4) and (5) can be synchronized, if there exist positive matrices ,,, and diagonal positive matrices , , , , , , , such that for , the following matrix inequalities hold:

with

Proof. For the error dynamical system (43), we define the following Lyapunov-Krasovskii function:

By (26), we have

Using (30), we get

From and (46)–(48), we have

In the same way, we have

From and ,

With the aid of (43) and (51), we have

where

Let, where , , then and

The following proof is similar to that of Theorem 4 and is omitted here.

4. Simulations

In this section, numerical examples are provided to demonstrate the validity of the synchronization criteria obtained in the previous sections. Consider the following network as drive network:

where , , and . , , , and . Choose time delays , , and , , . We define a switching rule . The other parameters are as follows:

The response network of drive network (55) is

where .

Let , , , , and the feasible solution of the matrix inequalities (15)–(19) by employing MATLAB LMI Toolbox be as follows:

The initial values are chosen as , , , , and . Clearly, the two coupled networks (55) and (57) satisfy the conditions of Theorem 4. Figure 1 presents the synchronization errors of the state variables between the two networks. The simulation result shows that the synchronization is achieved under the proposed controllers (20). Thus, the proposed synchronization control scheme in Theorem 4 is valid.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Synchronization errors of BDN (55) and (57) with adaptive feedback controllers (20).

Let , , then the feasible solution of the matrix inequalities (44) in Theorem 8 by employing MATLAB LMI Toolbox is as follows:

Using the controllers (41), the simulation result is given in Figure 2, which shows that the proposed synchronization control scheme in Theorem 8 is effective.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Synchronization errors with adaptive feedback controllers (41).

5. Conclusions

In this paper, we have proposed a general SCBNN with distributed delays and derivative coupling and investigated the synchronization problem in the two coupled SCBNNs. Using linear matrix inequality (LMI) approach and Barbǎlat lemma, we have deviated some useful synchronization criteria to ensure the synchronization of these two SCBNNs by constructing effective controllers. Compared with relative previous jobs, the controllers proposed by us are more simple and feasible. Some simulation results have been presented to demonstrate our theoretical results. In our future work, we will consider using pinning control to realize the synchronization of SCBNNs and identify the network topology of the unknown SCBNNs.

Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grants nos. 61272530 and 11072059, the Natural Science Foundation of Jiangsu Province of China under Grants no. BK2012741, Specialized Research Fund for the Doctoral Program of Higher Education under Grants no. 20110092110017, and the National Social Science Fund of China under Grant no. 11BGL039.

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