By using a Lyapunov-Krasovskii functional method and the stochastic analysis technique, we investigate the problem of synchronization for discrete-time stochastic neural networks (DSNNs) with random delays. A control law is designed, and sufficient conditions are established that guarantee the synchronization of two identical DSNNs with random delays. Compared with the previous works, the time delay is assumed to be existent in a random fashion. The stochastic disturbances are described in terms of a Brownian motion and the time-varying delay is characterized by introducing a Bernoulli stochastic variable. Two examples are given to illustrate the effectiveness of the proposed results. The main contribution of this paper is that the obtained results are dependent on not only the bound but also the distribution probability of the time delay. Moreover, our results provide a larger allowance variation range of the delay, and are less conservative than the traditional delay-independent ones.
Synchronization is one of the most important dynamic behavior of complex networks, which means if two or more systems have something in common, they will adjust each other to give rise to a common dynamical behavior. It has been found applications in many fields such as synchronous information exchange in the Internet WWW, crickets chirping in synchrony, rhythmic applause, and synchronous transfer of digital or analog signals in the communication networks.
Since the pioneering works of Pecora and Carroll [
Itô-type stochastic systems are well known for their important impact on practical applications such as chemistry, biology, ecology, control, and information systems. In real complex networks, the signal transmission could be a noisy process brought by random fluctuations from the release of probabilistic causes such as neurotransmitters. Stochastic neural networks, as a special case of complex networks, have gained much more researchers' interests; see for example, [
Discrete-time neural networks play a more and more important role in engineering application. As pointed out in [
Time delays occur frequently in practical situations, it can cause undesirable dynamic network behaviors such as oscillation and instability. Therefore, dynamical behavior [
Inspired by the above discussion, the aim of this paper is to study the synchronization problem for a class of DSNNs with random delay. The effect of both variation range and distribution probability of the time delay are taken into account in the proposed approach. The stochastic disturbances are described in terms of a Brownian motion, and the time-varying delay is characterized by introducing a Bernoulli stochastic variable. By employing a Lyapunov-Krasovskii functional, sufficient delay-distribution-dependent conditions are established in terms of linear matrix inequalities (LMIs) that guarantee the exponentially mean square synchronization of two identical DSNNs with random delays, which can be checked readily by Matlab toolbox.
This paper is organized as follows. In Section
Throughout this paper,
Consider the following
In this paper, we consider the model (
Throughout this paper, the following assumptions are made.
For
The time delay
It is noted that the introduction of binary stochastic variable was first introduced in [
In order to describe the probability distribution of the time delay, we define two sets
then system (
It is further assumed that the variables
Under the Assumption
Letting
The initial condition associated with (
where
Let
The drive system (
For brevity of the following analysis, we denote
In order to realize the synchronization between the drive system (
For the noised-perturbed response system (
Suppose that Assumptions
And then the feedback gains can be designed as
See the Appendix
We would like to point out that there is still enough room to improve the result. Because of (1) in real-time systems, time delays always exist in a stochastic fashion, so we also can consider the time delays satisfy other distributions. (2) And we can also extend this method to the dynamics of discrete-time stochastic complex networks. (3) The results can be improved by combining with delay-fractioning method to reduce conservatism.
Our results are less conservative than some other existed results because they are dependent on not only the bound but also the distribution probability of the time delays, and we obtain a larger allowance variation range of the delay, while the delay-fractioning or delay-partitioning approach [
When
Now, (
Suppose that Assumptions
And then the feedback gains can be designed as
The model proposed in this paper takes some well-studied models as special cases such as the model given in [
If we neglect the effect of the stochastic term
Suppose that Assumptions
And then the feedback gains can be designed as
Two numerical examples are presented to demonstrate the effectiveness of our results.
Consider the DSNNs (
Therefore, it can be seen from Theorem
State trajectories of
Error state trajectories of
Consider the DSNNs (
Therefore, it can be seen from Theorem
State trajectories of
Error state trajectories of
In recent years, synchronization in networks has become a hot research subject. However, the corresponding results are none for DSNNs with random delay. This paper has addressed the problem of exponential synchronization for the drive-response systems, where the drive system describes a class of discrete-time stochastic neural networks (DSNNs) with random delays and the response system is disturbed by some stochastic motions. Sufficient conditions have been established in terms of LMIs, which can be checked readily by Matlab toolbox. Two examples have been given to illustrate the effectiveness of the proposed results.
It is worth noting that the following two interesting and important issues should be addressed in our future work. Firstly, within the same LMI framework, it is not difficult to extend our main results to the synchronization problem for DSNNs with randomly mixed time-varying delays. Another future work may be how to extend the obtained results to DSNNs with Markovian jump and so on.
To obtain our main results, we need the following lemmas.
Let
For a given matrix,
(i) (ii)
For notation convenience, in the sequel, we denote
Now, in order to ensure that (
where
where
Calculating the difference of
Combining (
From (
Since
The authors appreciate the editor's work and the reviewer's helpful comments and suggestions. This work was jointly supported by the National Natural Science Foundation of China under Grant no. 11072059 and the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2009271.