Synchronization of Discrete-Time Stochastic Neural Networks with Random Delay

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.


Introduction
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 1, 2 , the control and synchronization problems have become an active topic that attracts a lot of researchers' interest, including general complex dynamical networks 3-8 , and the array of coupled neural networks with or without delays 6, 9-11 .Several different approaches have been proposed for the synchronization of chaotic systems, such as linear and nonlinear feedback control 12, 13 , adaptive control 14 , impulsive control 15 , and intermittent linear state feedback 16 .
It ô-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, 4, 5, 13, 15, 17, 18 and the references therein.
Discrete-time neural networks play a more and more important role in engineering application.As pointed out in 19-21 , the discretization cannot preserve the dynamics of continuous-time counterpart even for a small sampling period.Recently, the synchronization problem for discrete-time networks has received more attention 4, 5, 15, 22 .Time delays occur frequently in practical situations, it can cause undesirable dynamic network behaviors such as oscillation and instability.Therefore, dynamical behavior 23, 24 , especially synchronization problem for discrete-time neural networks with constant and time-varying delays has gained interesting research attention; see, for example, 4, 5, 15, 22 .It is worth mentioning that as a particular kind of time delays, random delays have also received much researchers' attention 25-31 .This is mainly because in many real systems, some values of the delay are very large, but the variation range of time delay taking such large values are very small.In this case, if only the variation range of time delay is employed to derive the criteria, the results may be somewhat more conservative.
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 timevarying 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 2, the model formulation and some preliminaries are given.The main results are stated in Section 3. Two illustrative examples are given to demonstrate the effectiveness of the proposed results in Section 4. Finally, the conclusions are made in Section 5.

Notation
Throughout this paper, R n and R n×m , respectively, denote the n-dimensional Euclidean and the set of all n × m matrices.I is the identity matrix of appropriate dimensions.The superscript "T " denotes matrix transposition.The notation X > 0 resp., X ≥ 0 , where X is a real symmetric matrix, means X is positive definite resp., positive semidefinite .
, where λ max A resp., λ min A means the largest resp., smallest eigenvalue of A. Z ≥0 denotes the set including zero and positive integers.The asterisk * in a matrix is used to denote term that is induced by symmetry.E{•} denotes the expectation.Moreover, let Ω, F, {F t } t≥0 , P be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions.Denote by L 2 F 0 −τ, 0 , R n the family of all F 0 -measurable C −τ, 0 : R n -valued random variables φ {φ s , −τ ≤ s ≤ 0} with the norm

Problem Formulation
Consider the following n-neuron DSNNs with time delay In this paper, we consider the model 2.1 as the drive system, and the noise-perturbed response system is given as where u k u 1 k , u 2 k , . . ., u n k T ∈ R n is the state feedback controller given to achieve the global exponental synchronization between the drive-response system; σ : R × R n → R n is a continuous function, ω k is a scalar Wiener process on a probability space Ω, F, P with Throughout this paper, the following assumptions are made.
Assumption 2.1.For i ∈ {1, 2, . . ., n}, the neuron activation functions f i • and g i • are continuous and bounded and satisfy the following conditions:

2.4
As first discussed in 32 , for brevity of the following representation, we denote
Remark 2.4.It is noted that the introduction of binary stochastic variable was first introduced in 25 and then successfully used in 26-31 .Under the Assumption 2.3, we know that δ 0 is dependent on the values of τ m , τ 0 , τ M .In addition, Prob{τ k ∈ τ 0 : In order to describe the probability distribution of the time delay, we define two sets where τ 0 is an integer satisfying τ m ≤ τ 0 < τ M .Define two mapping functions

2.8
It follows from 2.7 that Defining a stochastic variable as 2.9 then system 2.1 and 2.2 can be equivalently rewritten as It is further assumed that the variables δ k and ω k are mutually independent.
Letting e k x k − x k be the synchronization error, we can derive error system as follows: where

2.13
The initial condition associated with 2.12 is given as where . Let e k, φ be the state trajectory of system 2.12 under the initial condition.It is obvious that e k, 0 0 is a trivial solution of DSNNs 2.12 .
Definition 2.6.The drive system 2.10 and the response system 2.11 are said to be exponentially synchronized if, for a suitably designed feedback controller, the trivial solution of the error system 2.12 is globally exponentially stable in the mean square.That is, there exist constants ϑ > 0, and ν ∈ 0, 1 such that for sufficiently integer T > 0, the inequality holds for all k > T.

Main Results
In order to realize the synchronization between the drive system 2.10 and the noiseperturbed response system 2.11 with different initial conditions, we will design a suitable feedback controller and develop theoretical results of the synchronization scheme.In fact, if the trivial solution of the controlled error system 2.12 is exponentially stable in the mean square, 2.10 and 2.11 can achieve globally synchronization.
For the noised-perturbed response system 2.11 , the feedback controller u k is designed as where K and K 1 are the feedback gains to be determined. where And then the feedback gains can be designed as Proof.See the Appendix B.
Remark 3.2.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.
Remark 3.3.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 22 can reduce conservatism lie in the methods increase the number of fraction of the time delay.The delay-fractioning approach is a very effective approach to reduce conservatism.And we will combine with the free-weighting methods or delay partitioning approaches to reduce the conservativeness of the results in the future.
Remark 3.4.When δ k ≡ 1 ∀k ∈ Z ≥0 , which means τ 1 k ≡ τ k , 2.10 reduces to 2.1 .By setting δ 0 1, Q 2 0 in Theorem 3.1 and deleting the third and sixth rows and the corresponding third and sixth columns of 3.2 , we can obtain the following results.Now, 2.11 reduces to where And then the feedback gains can be designed as 3.9 Remark 3.6.The model proposed in this paper takes some well-studied models as special cases such as the model given in 4 .
If we neglect the effect of the stochastic term ω k in 2.11 , then 2.11 reduces to where And then the feedback gains can be designed as 3.13

Two Numerical Examples
Two numerical examples are presented to demonstrate the effectiveness of our results.

4.3
Therefore, it can be seen from Theorem 3.1, the response system 2.11 is globally exponentially synchronized with the drive system 2.10 .The result is further verified by the simulation results given by Figures 1 and 2. Figures 1 a , 1 b , and 1 c represent the trajectories of x 1 k , x 1 k , x 2 k , x 2 k , and x 3 k , x 3 k , respectively, and the red line represents the drive system state, the blue line stands for the response system state, and the initial conditions are taken as x k 1.0, 2.5, 0.5 T , x k −0.5, −0.5, −0.5 T .Figure 2 shows that the error state goes to zero after a short period time.Furthermore, if we increase δ 0 , the maximum allowance value of τ M will increase subsequently.Specially, if δ 0 0.99, we get the maximum allowance value of τ M 13.Setting τ m 1, τ 0 2, τ M 3, and δ 0 0.85 in Theorem 3.

4.6
Therefore, it can be seen from Theorem 3.1, the response system 2.11 is globally exponentially synchronized with the drive system 2.10 .The result is further verified by the simulation results given by Figures 3 and 4. Figures 3 a and 3 b represent the trajectories of x 1 k , x 1 k and x 2 k , x 2 k , respectively, and the red line represents the drive system state, the blue line stands for the response system state, the initial conditions are taken as x k 1.0, 2.5 T , x k −0.5, −0.5 T .Figure 4 shows that the error state goes to zero after a short period time.

Conclusions
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.
is equivalent to any one of the following conditions:

B. Proof of Theorem 3.1
For notation convenience, in the sequel, we denote Now, in order to ensure that 2.11 is globally exponentially synchronized with 2.10 , we just need to show that the error system 2.12 or 2.16 is globally exponentially stable in the mean square.To this end, we construct the following Lyapunov-Krasovskii functional V k by where where Calculating the difference of V k along the solution of DSNN 2.16 and taking the mathematical expectation, we have where E{ξ k Ω 0 ξ k }, B.9 where where It can be conclude from B.11 that there exists Λ 1 diag λ 11 , λ 12 , . . ., λ 1n > 0, where α i denote a column vector having "1" element on its ith row and zeros elsewhere.In a similar way, from B.12 and B.13 , we get where Λ 2 diag λ 21 , λ 22 , . . ., λ 2n > 0, Λ 3 diag λ 31 , λ 32 , . . ., λ 3n > 0.

B.23
Then, from B.22 and B. 23 we have E|e Since N is an any positive integer, it can be concluded from B.28 and Definition 2.6 that the response system 2.11 can be globally exponentially synchronized with the drive model 2.10 , and this completes the proof of the theorem.

Example 4 . 1 .
Consider the DSNNs 2.10 with the following parameters:

Figure 1 :
Figure 1: State trajectories of x k , x k .

2 Figure 4 :
Figure 4: Error state trajectories of e k .