Adaptive almost surely asymptotically synchronization for stochastic delayed neural networks with Markovian switching
© Ding et al.; licensee Springer 2013
Received: 19 February 2013
Accepted: 15 June 2013
Published: 13 July 2013
In this paper, the problem of the adaptive almost surely asymptotically synchronization for stochastic delayed neural networks with Markovian switching is considered. By utilizing a new nonnegative function and the M-matrix approach, we derive a sufficient condition to ensure adaptive almost surely asymptotically synchronization for stochastic delayed neural networks. Some appropriate parameters analysis and update laws are found via the adaptive feedback control techniques. We also present an illustrative numerical example to demonstrate the effectiveness of the M-matrix-based synchronization condition derived in this paper.
As we know, the stochastic delayed neural networks (SDNNs) with Markovian switching have played an important role in the fields of science and engineering for their many practical applications, including image processing, pattern recognition, associative memory, and optimization problems [1, 2]. In the past several decades, the characteristics of the SDNNs with Markovian switching, such as the various stability [3, 4], have received a lot of attention from scholars in various fields of nonlinear science. Wang et al. in  considered exponential stability for delayed recurrent neural networks with Markovian jumping parameters. Zhang et al. investigated stochastic stability for Markovian jumping genetic regulatory networks with mixed time delays . Huang et al. investigated robust stability for stochastic delayed additive neural networks with Markovian switching . The researchers presented a number of sufficient conditions to achieve the global asymptotic stability and exponential stability for the SDNNs with Markovian switching [8–11]. As is well known, time delays, as a source of instability and oscillations, always appear in various aspects of neural networks. Recently, the time delays of neural networks have received a lot of attention [12–15]. The linear matrix inequality (LMI, for short) approach is one of the most extensively used in recent publications [16, 17].
In recent years, it has been found that the synchronization of the coupled neural networks has potential applications in many fields such as biology and engineering [18–21]. In the coupled nonlinear dynamical systems, many neural networks may experience abrupt changes in their structure and parameters caused by some phenomena such as component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbances. The synchronization may help to protect interconnected neurons from the influence of random perturbations which affect all neurons in the system. Therefore, from the neurophysiological as well as theoretical point of view, it is important to investigate the impact of synchronization on the SDNNs. Moreover, in the adaptive synchronization for the neural networks, the control law needs to be adapted or updated in realtime. So, the adaptive synchronization for neural networks has been used in real neural networks control such as parameter estimation adaptive control, model reference adaptive control, etc. Some stochastic synchronization results have been investigated. For example, in , an adaptive feedback controller is designed to achieve complete synchronization for unidirectionally coupled delayed neural networks with stochastic perturbation. In , via adaptive feedback control techniques with suitable parameters update laws, several sufficient conditions are derived to ensure lag synchronization for unknown delayed neural networks with or without noise perturbation. In , a class of chaotic neural networks is discussed, and based on the Lyapunov stability method and the Halanay inequality lemma, a delay independent sufficient exponential synchronization condition is derived. The simple adaptive feedback scheme has been used for the synchronization for neural networks with or without time-varying delay in . A general model of an array of N linearly coupled delayed neural networks with Markovian jumping hybrid coupling is introduced in  and some sufficient criteria have been derived to ensure the synchronization in an array of jump neural networks with mixed delays and hybrid coupling in mean square.
It should be pointed out that, to the best of our knowledge, the adaptive almost surely asymptotically synchronization for the SDNNs with Markovian switching is seldom mentioned although it is of practical importance. Motivated by the above statements, in this paper, we aim to analyze the adaptive almost surely asymptotically synchronization for the SDNNs with Markovian switching. M-matrix-based criteria for determining whether adaptive almost surely asymptotically synchronization for the SDNNs with Markovian switching are developed. An adaptive feedback controller is proposed for the SDNNs with Markovian switching. A numerical simulation is given to show the validity of the developed results.
The rest of this paper is organized as follows: in Section 2, the problem is formulated and some preliminaries are given; in Section 3, a sufficient condition to ensure the adaptive almost surely asymptotically synchronization for the SDNNs with Markovian switching is derived; in Section 4, an example of numerical simulation is given to illustrate the validity of the results; Section 5 gives the conclusion of the paper.
2 Problem formulation and preliminaries
Throughout this paper, ℰ stands for the mathematical expectation operator, is used to denote a vector norm defined by , ‘T’ represents the transpose of a matrix or a vector, is an n-dimensional identical matrix.
We denote .
where is the time, is the state vector associated with n neurons, denote the activation functions of the neurons, is the transmission delay satisfying that and , where , are constants. As a matter of convenience, for , we denote and , , , , respectively. In model (1), furthermore, , (i.e., is a diagonal matrix) has positive and unknown entries , and are the connection weight and the delayed connection weight matrices, respectively. is the constant external input vector.
is an n-dimensional Brown moment defined on a complete probability space with a natural filtration (i.e., is a σ-algebra) and is independent of the Markovian process , and is the noise intensity matrix and can be regarded as a result of the occurrence of eternal random fluctuation and other probabilistic causes.
for any , where is the family of all -measurable -value random variables satisfying that , and denotes the family of all continuous -valued functions on with the norm .
To obtain the main result, we need the following assumptions.
for all .
Remark 1 Under Assumption 1∼Assumption 3, the error system (4) admits an equilibrium point (or trivial solution) , .
The following stability concept and synchronization concept are needed in this paper.
for any .
The response system (2) and the drive system (1) are said to be almost surely asymptotically synchronized if the error system (4) is almost surely asymptotically stable.
The main purpose of the rest of this paper is to establish a criterion of the adaptive almost surely asymptotically synchronization of system (1) and response system (2) by using the adaptive feedback control and M-matrix techniques.
To this end, we introduce some concepts and lemmas which will be frequently used in the proofs of our main results.
Definition 2 
A square matrix is called a nonsingular M-matrix if M can be expressed in the form with some (i.e., each element of G is nonnegative) and , where is the spectral radius of G.
Lemma 1 
M is a nonsingular M-matrix.
Every real eigenvalue of M is positive.
M is positive stable. That is, exists and (i.e., and at least one element of is positive).
for any .
For the SDDE with Markovian switching, we have the Dynkin formula as follows.
For the SDDE with Markovian switching again, the following hypothesis is imposed on the coefficients f and g.
Now we cite a useful result given by Yuan and Mao .
Lemma 4 
Then the solution of Eq. (5) is almost surely asymptotically stable.
3 Main results
In this section, we give a criterion of the adaptive almost surely asymptotically synchronization for the drive system (1) and the response system (2).
where () are arbitrary constants.
Proof Under Assumptions 1∼3, it can be seen that the error system (4) satisfies Assumption 4.
Then it is obvious that condition (8) holds.
where by .
Let , . Then inequalities (6) and (7) hold by using (9), where in (6). By Lemma 4, the error system (4) is adaptive almost surely asymptotically stable, and hence the noise-perturbed response system (2) can be adaptive almost surely asymptotically synchronized with the drive delayed neural network (1). This completes the proof. □
Remark 2 In Theorem 1, condition (9) of the adaptive almost surely asymptotically synchronization for the SDNN with Markovian switching obtained by using M-matrix and the Lyapunov functional method is generator-dependent and very different to other methods such as the linear matrix inequality method. And it is easy to check the condition if the drive system and the response system are given and the positive constant m is well chosen. To the best of the authors’ knowledge, this method is the first development in the research area of synchronization for neural networks.
Now, we are in a position to consider two special cases of the drive system (1) and the response system (2).
For this case, one can get the following result that is analogous to Theorem 1.
where () are arbitrary constants.
The rest of the proof is similar to that of Theorem 1, and hence omitted. □
In this case, one can get the following results.
where are arbitrary constants.
The rest of the proof is similar to that of Theorem 1, and hence omitted. □
4 Numerical example
In the section, an illustrative example is given to support our main results.
In this paper, we have proposed the concept of adaptive almost surely asymptotically synchronization for the stochastic delayed neural networks with Markovian switching. Making use of the M-matrix and Lyapunov functional method, we have obtained a sufficient condition, under which the response stochastic delayed neural network with Markovian switching can be adaptive almost surely asymptotically synchronized with the drive delayed neural networks with Markovian switching. The method to obtain the sufficient condition of the adaptive synchronization for neural networks is different to that of the linear matrix inequality technique. The condition obtained in this paper is dependent on the generator of the Markovian jumping models and can be easily checked. Extensive simulation results are provided to demonstrate the effectiveness of our theoretical results and analytical tools.
We would like to thank the referees and the editor for their valuable comments and suggestions, which have led to a better presentation of this paper. This work is supported by the National Natural Science Foundation of China (61075060), the Innovation Program of Shanghai Municipal Education Commission (12zz064) and the Fundamental Research Funds for the Central Universities.
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