Abstract

The problem of adaptive asymptotical synchronization is discussed for the stochastic complex dynamical networks with time-delay and Markovian switching. By applying the stochastic analysis approach and the M-matrix method for stochastic complex networks, several sufficient conditions to ensure adaptive asymptotical synchronization for stochastic complex networks are derived. Through the adaptive feedback control techniques, some suitable parameters update laws are obtained. Simulation result is provided to substantiate the effectiveness and characteristics of the proposed approach.

1. Introduction

As is known to all, complex networks widely exist in nature, such as brain structures, protein interactions, social networks, electrical power, and World Wide Web. Recently, the dynamical behaviors of complex networks have attracted ever-increasing research interest from a variety of communities such as mathematicians, computer scientists, and control engineers. As a result, a number of dynamic analysis issues have been extensively investigated for complex networks, such as synchronization, consensus, and flocking phenomenon, in which synchronization is one of the most important and has attracted special attention of researchers in different fields [1–13]. In [4], by using Lyapunov method and some properties of Kronecker product, a sufficient condition is proposed to ensure that the dynamics of the considered network globally exponentially synchronizes with the desired solution in the mean square sense. In particular, the proposed criteria for network synchronization are in terms of linear matrix inequalities (LMIs). In [13], a modified Lyapunov-Krasovskii functional is constructed by employing the more general decomposition approach; the novel delay-dependent synchronization conditions are derived in terms of LMIs, which can be easily solved by various convex optimization algorithms.

Meanwhile, the stability and synchronization of complex networks can be applied to secure communication systems [14], information science [15], and brain science [16], and so on. The synchronization of complex networks is to achieve the accordance of the states of the drive complex network and the response complex network in a moment. That is to say, the state of the error system can achieve zero eventually when the time approaches infinity. In particular, the adaptive synchronization for a complex network is such synchronization that the parameters of the drive complex network need to be estimated and the synchronization control law needs to be updated in real time when the complex network evolves. Furthermore, the stochastic complex dynamic network contains inherent time delay, which may cause instability or oscillation.

It should be pointed out that, up to now, the problem of adaptive asymptotical synchronization for stochastic complex networks with time-delay and Markovian switching has received very little research attention.

Summarizing the above discussions, the focus of this paper is on the adaptive asymptotical synchronization problem for stochastic delayed complex networks with Markovian switching. The main purpose of this paper is to establish stability criteria for testing whether the stochastic complex network is adaptive asymptotical synchronization. By using the stochastic analysis approach and the -matrix method, several sufficient conditions to ensure adaptive synchronization for stochastic complex networks are derived. Via the adaptive feedback technique, some suitable parameters update laws are obtained. Moreover, a simulation example is provided to show the effectiveness of the proposed controller design scheme. The main novelty of our contribution is threefold: (1) adaptive asymptotical synchronization control is addressed for stochastic complex networks with time-delay and Markovian switching; (2) using the adaptive feedback control techniques, adaptive feedback controller is designed; (3) the -matrix method of adaptive synchronization controller is given by employing a new nonnegative function.

The organization of this paper is as follows. In Section 2, the mathematical model of the stochastic complex networks with time-delay and Markovian switching is presented and some preliminaries are given. The main results of adaptive asymptotical synchronization are proved in Section 3. In Section 4, a simple example is given to demonstrate the effectiveness of the proposed results. Finally, the conclusions are presented in Section 5.

2. Problem Formulation and Preliminaries

The coupled complex networks can be called drive complex network and described as follows: where , is the state vector of the th node, is a nonlinear vector-valued function, is an inner-coupling matrix, and are the connection weight and the delayed connection weight matrices, and and are the weight or coupling strength. If there exists a link from node to   , then and . Otherwise, and . is the time-varying delay satisfying and , where and are constants.

Given a probability space ,  and  ,   is a homogeneous finite-state Markovian process with right continuous trajectories and taking values in finite set with the initial model . Let generator , , be the transition rate matrix with transition probability where and is the transition rate from to if , while

Let , ; the drive complex network (1) with Markovian switching can be rewritten as where, for the purpose of simplicity, we denote , , , and , respectively.

For the drive complex network (4), a response complex network is constructed in the following form: where is the state vector of the response complex network (5). is a control input vector with the form of is an -dimensional Brown moment defined on a complete probability space ,  and   with a natural filtration (i.e., is a -algebra) and is independent of the Markovian process , and is the noise intensity matrix and can be regarded as a result from the occurrence of eternal random fluctuation and other probabilistic causes.

Let and  . As a matter of convenience, we mark and . From the complex networks (4) and (5), the error system is arranged as

The main purpose of the rest of this paper is to set up a criterion of adaptive asymptotical synchronization for the system (4)–(7) via employing adaptive control and -matrix methods. Next, we firstly introduce assumptions, definitions, and lemmas which will be used in the proofs of main results.

Assumption 1. The activation function satisfies the Lipschitz condition. That is to say, there exists a constant such that

Assumption 2. The noise intensity matrix satisfies the linear growth condition. That is to say, there exist positives and , such that

Definition 3 (see [17]). The trivial solution of the error system (7) is said to be almost surely asymptotically stable if for any .

The response system (5) and the drive system (4) are said to be asymptotically synchronized if the error system (7) is asymptotically stable.

Definition 4 (see [18]). Consider an -dimensional stochastic delayed differential equation (SDDE, for short) with Markovian switching: on with the initial data given by
For , define an operator from to by where

Lemma 5 (see [18]). Let and ; then , for any .

Lemma 6 (see [18]). If with , then the following statements are equivalent. (i)is a nonsingular -matrix.(ii)Every real eigenvalue of is positive.(iii) is positive stable. That is, exists and (i.e., and at least one element of is positive).

Lemma 7 (see [17]). Assume that there are functions , , and such that Then the solution of (11) is almost surely asymptotically stable.

3. Main Results

In this section, some criteria of adaptive asymptotical synchronization will be obtained for the system (4), (5), and (7).

Theorem 8. Assume that is a nonsingular -matrix, where
Let and . That is to say, all elements of are positive. According to Lemma 6, . In addition, assume also that where .
Under Assumptions 1 and 2, the response complex network (5) can be adaptively synchronized with the drive complex network (4), if the feedback gain with the update law is chosen as

Proof. Choose a nonnegative function candidate as
The computation of along with the solution of error system (7) and using (20) is
Now, according to Assumptions 1 and 2 together with Lemma 5, one obtains
Substituting (23) into (22), one gets where by .
Let , , and . Then inequality (24) holds such that inequality (15) holds. Consider and when and , and inequality (19) implies . So (16) holds. Moreover, (17) holds when and . By Lemma 7, the error system (7) is adaptive almost surely asymptotically stable, and hence the noise-perturbed response complex network (5) can be adaptively almost surely asymptotically synchronized with the drive complex network (4). This completes the proof.

Remark 9. For complex networks (1), the method in the paper can be used in some systems, such as multiagent systems [19–21] and wireless sensor networks [22], which are the next research topic for us.

Now, we are in a position to consider two cases of the complex networks (4)-(5), which have the following corollaries.

The Markovian switching is removed from the complex networks. That is to say, the drive complex network, the response complex network, and the error system can be represented, respectively, as follows: For this case, one can get the following result analogous to Theorem 8.

Corollary 10. Assume that is a nonsingular -matrix, where ,  , and
Under Assumptions 1 and 2, the noise-perturbed response complex network can be adaptively asymptotically synchronized with the drive complex network, if the feedback gain of the controller (6) with the update law is chosen as

Proof. Choose the following nonnegative function:
The remaining proof is similar to that of Theorem 8 and hence omitted.

The noise-perturbation is removed from the response complex network (5); then the response complex network and the error system can be represented, respectively, as follows: which can lead to the following results.

Corollary 11. Assume that is a nonsingular -matrix, where and where .
Under Assumptions 1 and 2, the noiseless-perturbed response complex network can be adaptively asymptotically synchronized with the drive complex network, if the feedback gain of the controller (6) with the update law is chosen as (20).

Proof. The proof is similar to that of Theorem 8 and hence omitted.

4. Illustrative Example

In this section, an illustrative example will be given to demonstrate the effectiveness of the proposed methods.

Example 1. The Lorenz system is described by where ,  , and  .
According to Theorem 8, the complex networks (drive complex network and response complex network) with four nodes are described as follows:
In the simulation, let
These parameters fully satisfy Assumptions 1 and 2 and condition (19). Therefore, it will prove the main result to be correct if the error system can be adaptively asymptotically synchronized satisfying Theorem 8.
To illustrate the effectiveness of the developed theory, we employ the nonnegative function to solve the solutions for stochastic complex networks and to simulate the dynamics of error system and the adaptive feedback gain. The simulation figures are shown in Figures 1, 2, 3, and 4. Among them, Figures 1–3 plot the error states of complex networks ,  , and . Figure 4 depicts the adaptive feedback gain. From all these figures, one can find that the stochastic complex networks are adaptively asymptotically synchronized.