Synchronization of Discrete-Time Stochastic Neural Networks with Random Delay

Abstract

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.

1. 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].

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, [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 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 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.

1.1. 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). |⋅| is the Euclidean norm in ℝ^𝑛. If 𝐴 is a matrix, ‖𝐴‖ denotes its operator norm. That is, ‖𝐴‖=sup{|𝐴𝑥|∶|𝑥|=1}=𝜆_max(𝐴^𝑇𝐴), 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 ℒ²_ℱ0([−𝜏,0],ℝ^𝑛) the family of all ℱ₀-measurable 𝐶([−𝜏,0]∶ℝ^𝑛)-valued random variables 𝜙={𝜙(𝑠),−𝜏≤𝑠≤0} with the norm ‖𝜙‖=sup_{−𝜏≤𝑠≤0}|𝜙(𝑠)|²<∞.

2. Problem Formulation

Consider the following 𝑛-neuron DSNNs with time delay𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝐹(𝑥(𝑘))+𝐷𝐺(𝑥(𝑘−𝜏(𝑘))), (2.1) where 𝑥(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘))^𝑇∈ℝ^𝑛 is the state vector associated with the 𝑛 neurons, 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛) with |𝑎_𝑖|<1; 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, 𝐷=(𝑑_𝑖𝑗)_𝑛×𝑛 denote the connection weights matrix and the delayed connection weights matrix, respectively, 𝐹(𝑥(𝑘))=[𝑓₁(𝑥₁(𝑘)),𝑓₂(𝑥₂(𝑘)),…,𝑓_𝑛(𝑥_𝑛(𝑘))]^𝑇 and 𝐺(𝑥(𝑘))=[𝑔₁(𝑥₁(𝑘)),𝑔₂(𝑥₂(𝑘)),…,𝑔_𝑛(𝑥_𝑛(𝑘))]^𝑇 denote the neuron activation functions; 𝜏(𝑘) denotes the time-varying delay.

In this paper, we consider the model (2.1) as the drive system, and the noise-perturbed response system is given as𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝐹(𝑥(𝑘))+𝐷𝐺(𝑥(𝑘−𝜏(𝑘)))+𝑢(𝑘)+𝜎(𝑘,𝑥(𝑘)−𝑥(𝑘))𝜔(𝑘), (2.2) where 𝑢(𝑘)=(𝑢₁(𝑘),𝑢₂(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 𝔼{𝜔(𝑘)}=0,𝔼𝜔²(𝑘)=1,𝔼𝜔(𝑖)𝜔𝑗=0𝑖≠𝑗. (2.3)

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: 𝑙_𝑖≤𝑓_𝑖𝑠₁−𝑓_𝑖𝑠₂𝑠₁−𝑠₂≤𝐿_𝑖,𝜔_𝑖≤𝑔_𝑖𝑠₁−𝑔_𝑖𝑠₂𝑠₁−𝑠₂≤𝑊_𝑖,∀𝑠₁,𝑠₂∈ℝ𝑠₁≠𝑠₂,𝑖=1,2,…,𝑛. (2.4) As first discussed in [32], for brevity of the following representation, we denote 𝒞₁=diag𝑙₁𝐿₁,…,𝑙_𝑛𝐿_𝑛,𝒞₂−𝑙=diag¹+𝐿₁2𝑙,…,−^𝑛+𝐿_𝑛2,ℋ₁=diag𝜔₁𝑊₁,…,𝜔_𝑛𝑊_𝑛,ℋ₂−𝜔=diag¹+𝑊₁2𝜔,…,−^𝑛+𝑊_𝑛2. (2.5)

Assumption 2.2. 𝜎(𝑘,𝑥(𝑘)) is a continuous function satisfying 𝜎(0,0)=0 and 𝜎(𝑘,𝑥(𝑘))^𝑇𝜎(𝑘,𝑥(𝑘))≤𝜌𝑥(𝑘)^𝑇𝑥(𝑘), (2.6) where 𝜌>0 is a known constant scalar.

Assumption 2.3. The time delay 𝜏(𝑘) is bounded, 0≤𝜏_𝑚<𝜏(𝑘)≤𝜏_𝑀, and its probability distribution can be observed; that is, suppose that 𝜏(𝑘) takes values in [𝜏_𝑚∶𝜏₀] or (𝜏₀∶𝜏_𝑀] and Prob{𝜏(𝑘)∈[𝜏_𝑚∶𝜏₀]}=𝛿₀, where 𝜏_𝑚≤𝜏₀<𝜏_𝑀, and 0≤𝛿₀≤1.

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 𝛿₀ is dependent on the values of 𝜏_𝑚, 𝜏₀, 𝜏_𝑀. In addition, Prob{𝜏(𝑘)∈(𝜏₀∶𝜏_𝑀]}=1−𝛿₀≜𝛿₀.

In order to describe the probability distribution of the time delay, we define two sets ℬ₁=𝑘∣𝜏(𝑘)∈𝜏_𝑚,𝜏₀,ℬ₂=𝑘∣𝜏(𝑘)∈𝜏₀,𝜏_𝑀, (2.7) where 𝜏₀ is an integer satisfying 𝜏_𝑚≤𝜏₀<𝜏_𝑀. Define two mapping functions 𝜏₁(𝑘)=𝜏(𝑘),𝑘∈ℬ₁,𝜏_𝑚𝜏,else,²(𝑘)=𝜏(𝑘),𝑘∈ℬ₂,𝜏₀,else. (2.8) It follows from (2.7) that ℬ₁∪ℬ₂=ℤ_≥0 and ℬ₁∩ℬ₂=∅. From (2.8), it can be seen that 𝑘∈ℬ₁ implies the event 𝜏(𝑘)∈[𝜏_𝑚∶𝜏₀] occurs and 𝑘∈ℬ₂ implies the event 𝜏(𝑘)∈(𝜏₀∶𝜏_𝑀] occurs. Defining a stochastic variable as 𝛿(𝑘)=1,𝑘∈ℬ₁,0,𝑘∈ℬ₂, (2.9)

then system (2.1) and (2.2) can be equivalently rewritten as 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝐹(𝑥(𝑘))+𝛿(𝑘)𝐷𝐺𝑥𝑘−𝜏₁(𝑘)+(1−𝛿(𝑘))𝐷𝐺𝑥𝑘−𝜏₂(𝑘)𝑥()𝑥(𝑘)𝐹(𝑥(𝑘(𝑘)𝐷𝐺𝑥,𝑘+1=𝐴+𝐵))+𝛿𝑘−𝜏₁(𝑘)+(1−𝛿(𝑘))𝐷𝐺𝑥𝑘−𝜏₂(𝑘)(𝑘)(𝑥(𝑘)(𝑘𝜔(𝑘).+𝑢+𝜎𝑘,−𝑥)) (2.10)(2.11)

It is further assumed that the variables 𝛿(𝑘) and 𝜔(𝑘) are mutually independent.

Remark 2.5. Under the Assumption 2.3 and the definition of 𝛿(𝑘), it can be seen that 𝛿(𝑘) is a Bernoulli distributed white sequence with Prob{𝛿(𝑘)=1}=𝔼{𝛿(𝑘)}=𝛿₀ and Prob{𝛿(𝑘)=0}=𝛿₀. Furthermore, we can show that 𝔼{𝛿(𝑘)−𝛿₀}=0 and 𝔼{(𝛿(𝑘)−𝛿₀)²}=𝛿₀𝛿₀.

Letting 𝑒(𝑘)=𝑥(𝑘)−𝑥(𝑘) be the synchronization error, we can derive error system as follows:𝑒(𝑘+1)=𝐴𝑒(𝑘)+𝐵𝐹(𝑒(𝑘))+𝛿(𝑘)𝐷𝐺𝑒𝑘−𝜏₁(𝑘)+(1−𝛿(𝑘))𝐷𝐺𝑒𝑘−𝜏₂(𝑘)+𝜎(𝑘,𝑒(𝑘))𝜔(𝑘)+𝑢(𝑘), (2.12) where 𝐹(𝑒(𝑘))=𝑓₁𝑒₁(𝑘),𝑓₂𝑒₂(𝑘),…,𝑓_𝑛𝑒_𝑛(𝑘)^𝑇=𝐹(𝑥(𝑘−𝐹(𝑥(𝑘,𝐺(𝑒(𝑘=𝑔))))))₁𝑒₁(𝑘),𝑔₂𝑒₂(𝑘),…,𝑔_𝑛𝑒_𝑛(𝑘)^𝑇=𝐺(𝑥(𝑘))−𝐺(𝑥(𝑘)). (2.13)

The initial condition associated with (2.12) is given as 𝑒(𝑠)=𝜙(𝑠),𝑠∈ℕ−𝜏_𝑀,0, (2.14)

where 𝜙(𝑠)∈ℒ²_ℱ0([−𝜏_𝑀,0],ℝ^𝑛), and ℕ[−𝜏_𝑀,0]={−𝜏_𝑀,−𝜏_𝑀+1,…,0}.

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 𝔼|𝑒(𝑘)|²≤𝜗𝜈^𝑘sup_{𝑠∈ℕ−𝜏𝑀,0}𝔼|𝑒(𝑠)|² (2.15) holds for all 𝑘>𝑇.

For brevity of the following analysis, we denote 𝑒(𝑘), 𝑒(𝑘−𝜏₁(𝑘)), 𝑒(𝑘−𝜏₂(𝑘)), 𝛿(𝑘), 1−𝛿(𝑘), 𝜔(𝑘) by 𝑒_𝑘, 𝑒_𝜏,1, 𝑒_𝜏,2, 𝛿_𝑘, 𝛿_𝑘, 𝜔_𝑘, respectively. Then, (2.12) can be rearranged as𝑒_𝑘+1=𝐴𝑒_𝑘+𝐵𝐹𝑒_𝑘+𝛿_𝑘𝐷𝐺𝑒_𝜏,1+𝛿_𝑘𝐷𝐺𝑒_𝜏,2+𝜎𝑘,𝑒_𝑘𝜔_𝑘+𝑢(𝑘). (2.16)

3. Main Results

In order to realize the synchronization between the drive system (2.10) and the noise-perturbed 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 as𝑢(𝑘)=𝐾𝑒(𝑘)+𝛿₀𝐾₁𝑒𝑘−𝜏₁(𝑘)+1−𝛿₀𝐾₁𝑒𝑘−𝜏₂(𝑘), (3.1) where 𝐾 and 𝐾₁ are the feedback gains to be determined.

Theorem 3.1. Suppose that Assumptions 2.1–2.3 hold, and let the state-feedback controller (3.1) be given. The noised-perturbed response system (2.11) is globally exponentially synchronized with the drive model (2.10) if there exist three positive definite matrices 𝑃, 𝑄₁, 𝑄₂, three diagonal matrices Λ_𝑖=diag(𝜆_𝑖1,𝜆_𝑖2,…,𝜆_𝑖𝑛)>0(𝑖=1,2,3), two real matrices 𝑅, 𝑅₁ and a scalar 𝜆^∗>0 such that the following two LMIs hold: ΞΩ=¹00−Λ₁𝒞₂00𝐴^𝑇𝑃+𝑅^𝑇∗Ξ₂00−Λ₂ℋ₂0𝛿₀𝑅^𝑇₁∗∗Ξ₃00−Λ₃ℋ₂𝛿₀𝑅^𝑇₁∗∗∗−Λ₁00𝐵^𝑇𝑃∗∗∗∗−Λ₂0𝛿₀𝐷^𝑇𝑃∗∗∗∗∗−Λ₃𝛿₀𝐷^𝑇𝑃∗∗∗∗∗∗−𝑃<0,𝑃<𝜆^∗𝐼, (3.2)(3.3) whereΞ₁=−𝑃+𝜆^∗𝜌𝐼+(𝜏₀−𝜏_𝑚+1)𝑄₁+(𝜏_𝑀−𝜏₀+2)𝑄₂−Λ₁𝒞₁, Ξ₂=−𝑄₁−Λ₂ℋ₁, Ξ₃=−𝑄₂−Λ₃ℋ₁.
And then the feedback gains can be designed as 𝐾=𝑃⁻¹𝑅,𝐾₁=𝑃⁻¹𝑅₁. (3.4)

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 𝜏₁(𝑘)≡𝜏(𝑘), (2.10) reduces to (2.1). By setting 𝛿₀=1, 𝑄₂=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 to𝑥()𝑥(𝑘)𝐹(𝑥(𝑘𝐺(𝑥((𝑘(𝑘)(𝑥(𝑘)(𝑘𝜔(𝑘).𝑘+1=𝐴+𝐵))+𝐷𝑘−𝜏)))+𝑢+𝜎𝑘,−𝑥)) (3.5) Equation (2.16) reduces to𝑒_𝑘+1=𝐴𝑒_𝑘+𝐵𝐹𝑒_𝑘+𝐷𝐺𝑒_𝜏+𝜎𝑘,𝑒_𝑘𝜔_𝑘+𝑢(𝑘). (3.6)

Corollary 3.5. Suppose that Assumptions 2.1-2.2 hold, and let the state-feedback controller (3.1) be given. The noised-perturbed response system (3.5) is globally exponentially synchronized with the drive model (2.1) if there exist two positive definite matrices 𝑃,𝑄₁, two diagonal matrices Λ_𝑖=diag(𝜆_𝑖1,𝜆_𝑖2,…,𝜆_𝑖𝑛)>0(𝑖=1,2), a real matrix 𝑅, and a scalar 𝜆^∗>0, such that the following two LMIs hold: Ω₂=Γ₁0−Λ𝒞₂0𝐴^𝑇𝑃+𝑅^𝑇∗Γ₂0−Λ₂ℋ₂𝑅^𝑇₁∗∗−Λ₁0𝐵^𝑇𝑃∗∗∗−Λ₂𝐷^𝑇𝑃∗∗∗∗−𝑃<0,𝑃<𝜆^∗𝐼, (3.7)(3.8) where Γ₁=−𝑃+𝜆^∗𝜌𝐼+(𝜏₀−𝜏_𝑚+1)𝑄₁−Λ₁𝒞₁, Γ₂=−𝑄₁−Λ₂ℋ₁.
And then the feedback gains can be designed as 𝐾=𝑃⁻¹𝑅. (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 to𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝐹(𝑥(𝑘))+𝛿(𝑘)𝐷𝐺𝑥𝑘−𝜏₁(𝑘)+(1−𝛿(𝑘))𝐷𝐺𝑥𝑘−𝜏₂(𝑘)+𝑢(𝑘). (3.10) Equation (2.16) reduces to𝑒_𝑘+1=𝐴𝑒_𝑘+𝐵𝐹𝑒_𝑘+𝛿_𝑘𝐷𝐺𝑒_𝜏,1+𝛿_𝑘𝐷𝐺𝑒_𝜏,2+𝑢(𝑘). (3.11) In this case, we have the following results.

Corollary 3.7. Suppose that Assumptions 2.1 and 2.3 hold, and let the state-feedback controller (3.1) be given. The response system (3.10) is globally exponentially synchronized with the drive model (2.10) if there exist three positive definite matrices 𝑃,𝑄₁,𝑄₂, three diagonal matrices Λ_𝑖=diag(𝜆_𝑖1,𝜆_𝑖2,…,𝜆_𝑖𝑛)>0(𝑖=1,2,3), two real matrices 𝑅,𝑅₁, such that the following LMI holds: Ω₃=Π₁00−Λ₁𝒞₂00𝐴^𝑇𝑃+𝑅^𝑇∗Π₂00−Λ₂ℋ₂0𝛿₀𝑅^𝑇₁∗∗Π₃00−Λ₃ℋ₂𝛿₀𝑅^𝑇₁∗∗∗−Λ₁00𝐵^𝑇𝑃∗∗∗∗−Λ₂0𝛿₀𝐷^𝑇𝑃∗∗∗∗∗−Λ₃𝛿₀𝐷^𝑇𝑃∗∗∗∗∗∗−𝑃<0, (3.12) where Π₁=−𝑃+(𝜏₀−𝜏_𝑚+1)𝑄₁+(𝜏_𝑀−𝜏₀+2)𝑄₂−Λ₁𝒞₁, Π₂=−𝑄₁−Λ₂ℋ₁, Π₃=−𝑄₂−Λ₃ℋ₁.
And then the feedback gains can be designed as 𝐾=𝑃⁻¹𝑅,𝐾₁=𝑃⁻¹𝑅₁. (3.13)

4. Two Numerical Examples

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

Example 4.1. Consider the DSNNs (2.10) with the following parameters: ,𝑓𝐴=0.40000.50000.4,𝐵=0.3−0.10.20−0.30.2−0.1−0.1−0.2,𝐷=0.20.10.1−0.20.30.10.1−0.20.3₁(𝑠)=tanh(0.6𝑠)−0.2sin(𝑠),𝑓₂(𝑠)=tanh(−0.4𝑠),𝑓₃(𝑠)(),𝑔=tanh−0.2𝑠₁(𝑠)=tanh(−0.4𝑠)+0.2sin(𝑠),𝑔₂(𝑠)=tanh(0.2𝑠),𝑔₃(𝑠)=tanh(0.4𝑠), (4.1) It is easy to verify that 𝒞₁=diag(−0.08,0,0),𝒞₂(),ℋ=diag0.1,0.2,0.1₁=diag(−0.06,0,0),ℋ₂=diag(0.06,−0.1,−0.2). (4.2) Setting 𝜏_𝑚=1, 𝜏₀=2, 𝜏_𝑀=5, and 𝛿₀=0.89, 𝜌=0.1 in Theorem 3.1, we can verify by Matlab toolbox that LMIs (3.2) and (3.3) are solved and the solutions are given as follows: 𝑃=15.7114−0.86100.1575−0.861011.95851.05100.15751.051014.8306,𝑄₁=,𝑄1.9720−0.15970.0202−0.15972.06450.20550.02020.20552.2099₂=0.6299−0.01910.0012−0.01910.34600.02510.00120.02510.5228,Λ₁=,Λ369.9501000369.9501000369.9501₂=369.9501000369.9501000369.9501,Λ₃=,369.9501000369.9501000369.9501𝑅=−5.74160.31310.0554−0.0464−6.5348−0.35670.0554−0.3567−6.4303,𝑅₁=0.1675−0.2223−0.1017−0.2223−0.4297−0.2061−0.1017−0.2061−0.7492,𝜆^∗=16.0757,𝐾=𝑃⁻¹𝑅=−0.3656−0.01030.0084−0.0008−0.54850.00890.00770.0419−0.4343,𝐾₁=𝑃⁻¹𝑅₁=.0.0098−0.0160−0.0067−0.0174−0.0361−0.0134−0.0057−0.0112−0.495 (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 𝑥₁(𝑘), 𝑥₁(𝑘), 𝑥₂(𝑘), 𝑥₂(𝑘), and 𝑥₃(𝑘), 𝑥₃(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 𝛿₀, the maximum allowance value of 𝜏_𝑀 will increase subsequently. Specially, if 𝛿₀=0.99, we get the maximum allowance value of 𝜏_𝑀=13.

(a)

(b)

(c)

Figure 1 

State trajectories of 𝑥(𝑘), 𝑥(𝑘).

Figure 2 

Error state trajectories of 𝑒(𝑘).

Example 4.2. Consider the DSNNs (2.10) with the following parameters: 𝑓𝐴=−0.1000.4,𝐵=0.1−0.20−0.1,𝐷=00.20.2−0.1,𝜌=0.02,₁(𝑠)=sin(0.2𝑠)−0.6cos(𝑠),𝑓₂(𝑠)(),𝑔=tanh−0.4𝑠₁(𝑠)=tanh(0.83𝑠)+0.6cos(𝑠),𝑔₂(𝑠)=tanh(0.2𝑠). (4.4) It is easy to verify that 𝒞₁=diag(−0.64,0),𝒞₂=diag(0,0.2),ℋ₁=diag(−0.6,0),ℋ₂=diag(−0.2,−0.1). (4.5) Setting 𝜏_𝑚=1, 𝜏₀=2, 𝜏_𝑀=3, and 𝛿₀=0.85 in Theorem 3.1, we can verify by Matlab toolbox that LMIs (3.2) and (3.3) are solved and the solutions are given below: 𝑃=48.7972−2.3255−2.325542.7033,𝑄₁=4.2255−0.4620−0.46208.3487,𝑄₂=,Λ4.0513−0.1026−0.10261.8888₁=880.773500880.7735,Λ₂=880.773500880.7735,Λ₃=,880.773500880.7735𝑅=4.8936−0.7368−0.7368−17.8357,𝑅₁=0.0860−1.2452−1.24520.3222,𝜆^∗=49.8783,𝐾=𝑃⁻¹𝑅=0.0997−0.0351−0.0118−0.4196,𝐾₁=𝑃⁻¹𝑅₁=.0.0004−0.0252−0.02910.0062 (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 𝑥₁(𝑘), 𝑥₁(𝑘) and 𝑥₂(𝑘), 𝑥₂(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.

(a)

(b)

Figure 3 

State trajectories of 𝑥(𝑘), 𝑥(𝑘).

Figure 4 

Error state trajectories of 𝑒(𝑘).

5. 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.

Appendices

A. Preliminary Lemmas

To obtain our main results, we need the following lemmas.

Lemma A.1 (see [33]). Let 𝑋∈ℝ^𝑛, 𝑌∈ℝ^𝑛, 𝜖>0, then we have 𝑋^𝑇𝑌+𝑌^𝑇𝑋≤𝜖𝑋^𝑇𝑋+𝜖⁻¹𝑌^𝑇𝑌. (A.1)

Lemma A.2 (Schur Complement [34]). For a given matrix, 𝑆𝑆=¹¹𝑆₁₂𝑆^𝑇₁₂𝑆₂₂<0 (A.2) is equivalent to any one of the following conditions: (i) 𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0, (ii) 𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0.

B. Proof of Theorem 3.1

For notation convenience, in the sequel, we denoteℰ_𝑘=𝑒^𝑇_𝑘,𝑒^𝑇_𝑘−1,…,𝑒^𝑇_{(𝑘)𝑘−𝜏},𝜉_𝑘=𝑒^𝑇_𝑘,𝑒^𝑇_𝜏,1,𝑒^𝑇_𝜏,2,𝐹^𝑇𝑒_𝑘,𝐺^𝑇𝑒_𝜏,1,𝐺^𝑇𝑒_𝜏,2^𝑇,Ψ=(𝐴+𝐾),𝛿₀𝐾₁,𝛿₀𝐾₁,𝐵,𝛿₀𝐷,𝛿₀𝐷. (B.1)

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 𝑉_𝑘=𝑉_𝑘,1+𝑉_𝑘,2+𝑉_𝑘,3, (B.2)

where 𝑉_𝑘,1=𝑒^𝑇_𝑘𝑃𝑒_𝑘,𝑉_𝑘,2=^𝑘−1_{𝑖=𝑘−𝜏𝑘1}𝑒^𝑇_𝑖𝑄₁𝑒_𝑖+^𝜏0−1_{𝑖=𝜏𝑚}^𝑘−1_{𝑗=𝑘−𝑖}𝑒^𝑇_𝑗𝑄₁𝑒_𝑗,𝑉_𝑘,3=^𝑘−1_{𝑖=𝑘−𝜏𝑘2}𝑒^𝑇_𝑖𝑄₂𝑒_𝑖+^𝜏𝑀−1_𝑖=𝜏0+1^𝑘−1_{𝑗=𝑘−𝑖}𝑒^𝑇_𝑗𝑄₂𝑒_𝑗, (B.3)

where 𝑃>0, 𝑄₁>0, 𝑄₂>0.

Calculating the difference of 𝑉_𝑘 along the solution of DSNN (2.16) and taking the mathematical expectation, we have 𝔼Δ𝑉_𝑘=𝔼Δ𝑉_𝑘,1+𝔼Δ𝑉_𝑘,2+𝔼Δ𝑉_𝑘,3, (B.4) where 𝔼Δ𝑉_𝑘,1=𝔼𝔼𝑉_𝑘+1,1∣ℰ_𝑘−𝑉_𝑘,1=𝔼𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃(𝐴+𝐾)−𝑃𝑒_𝑘+2𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐵𝐹𝑒_𝑘+2𝛿₀𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐷𝐺𝑒_𝜏,1+2𝛿₀𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐷𝐺𝑒_𝜏,2+2𝛿₀𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐾₁𝑒_𝜏,1+2𝛿₀𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐾₁𝑒_𝜏,2+𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐵𝐹𝑒_𝑘+2𝛿₀𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐷𝐺𝑒_𝜏,1+2𝛿₀𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐷𝐺𝑒_𝜏,2+𝛿₀𝐺^𝑇𝑒_𝜏,1𝐷^𝑇𝑃𝐷𝐺𝑒_𝜏,1+𝛿₀𝐺^𝑇𝑒_𝜏,2𝐷^𝑇𝑃𝐷𝐺𝑒_𝜏,2+𝜎^𝑇𝑘,𝑒_𝑘𝑃𝜎𝑘,𝑒_𝑘+2𝛿₀𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐾₁𝑒_𝜏,1+2𝛿₀𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐾₁𝑒_𝜏,2+2𝛿²₀𝐺^𝑇𝑒_𝜏,1𝐷^𝑇𝑃𝐾₁𝑒_𝜏,1+𝛿₀𝛿₀𝐺^𝑇𝑒_𝜏,1𝐷^𝑇𝑃𝐾₁𝑒_𝜏,2+𝛿₀𝛿₀𝐺^𝑇𝑒_𝜏,2𝐷^𝑇𝑃𝐾₁𝑒_𝜏,1+2𝛿²₀𝐺^𝑇𝑒_𝜏,2𝐷^𝑇𝑃𝐾₁𝑒_𝜏,2+𝛿²₀𝑒^𝑇_𝜏,1𝐾^𝑇₁𝑃𝐾₁𝑒_𝜏,1+𝛿²₀𝑒^𝑇_𝜏,2𝐾^𝑇₁𝑃𝐾₁𝑒_𝜏,2,𝔼Δ𝑉_𝑘,2=𝔼𝔼𝑉_𝑘+1,2∣ℰ_𝑘−𝑉_𝑘,2=𝔼𝜏₀−𝜏_𝑚+1𝑒^𝑇_𝑘𝑄₁𝑒_𝑘−𝑒^𝑇_𝜏,1𝑄₁𝑒_𝜏,1+^{𝑘−𝜏𝑚}_{𝑖=𝑘−𝜏1𝑘+1+1}+^𝑘−1_{𝑖=𝑘−𝜏𝑚+1}−^𝑘−1_{𝑖=𝑘−𝜏𝑘1+1}−^{𝑘−𝜏𝑚}_{𝑖=𝑘−𝜏0+1}𝑒^𝑇_𝑖𝑄₁𝑒_𝑖≤𝔼𝜏₀−𝜏_𝑚+1𝑒^𝑇_𝑘𝑄₁𝑒_𝑘−𝑒^𝑇_𝜏,1𝑄₁𝑒_𝜏,1,𝔼Δ𝑉_𝑘,3=𝔼𝔼𝑉_𝑘+1,3∣ℰ_𝑘−𝑉_𝑘,3≤𝔼𝜏_𝑀−𝜏₀+2𝑒^𝑇_𝑘𝑄₂𝑒_𝑘−𝑒^𝑇_𝜏,2𝑄₂𝑒_𝜏,2. (B.5)(B.6)(B.7) It can be easily seen from Assumption 2.2 and (3.3) that𝔼𝜎^𝑇𝑘,𝑒_𝑘𝑃𝜎𝑘,𝑒_𝑘≤𝔼𝜆_max(𝑃)𝜎^𝑇𝑘,𝑒_𝑘𝜎𝑘,𝑒_𝑘≤𝔼𝜆^∗𝜌𝑒^𝑇_𝑘𝑒_𝑘. (B.8)

Combining (B.4)–(B.8), we obtain𝔼Δ𝑉_𝑘≤𝔼𝑒^𝑇_𝐾(𝐴+𝐾)^𝑇𝑃(𝐴+𝐾)−𝑃+𝜏₀−𝜏_𝑚+1+𝜏_𝑀−𝜏₀+2+𝜆^∗𝜌𝐼𝑒_𝑘−𝑒^𝑇_𝜏,1𝑄₁𝑒_𝜏,1−𝑒^𝑇_𝜏,2𝑄₂𝑒_𝜏,2+2𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐵𝐹𝑒_𝑘+2𝛿₀𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐷𝐺𝑒_𝜏,1+2𝛿₀𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐷𝐺𝑒_𝜏,2+2𝛿₀𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐾₁𝑒_𝜏,1+2𝛿₀𝑒^𝑇_𝑘(𝐴+𝐾)^𝑇𝑃𝐾₁𝑒_𝜏,2+𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐵𝐹𝑒_𝑘+2𝛿₀𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐷𝐺𝑒_𝜏,1+2𝛿₀𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐷𝐺𝑒_𝜏,2+2𝛿₀𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐾₁𝑒_𝜏,1+2𝛿₀𝐹^𝑇𝑒_𝑘𝐵^𝑇𝑃𝐾₁𝑒_𝜏,2+𝛿₀𝐺^𝑇𝑒_𝜏,1𝐷^𝑇𝑃𝐷𝐺𝑒_𝜏,1+𝛿₀𝐺^𝑇𝑒_𝜏,2𝐷^𝑇𝑃𝐷𝐺𝑒_𝜏,2+2𝛿²₀𝐺^𝑇𝑒_𝜏,1𝐷^𝑇𝑃𝐾₁𝑒_𝜏,1+𝛿₀𝛿₀𝐺^𝑇𝑒_𝜏,1𝐷^𝑇𝑃𝐾₁𝑒_𝜏,2+𝛿₀𝛿₀𝐺^𝑇𝑒_𝜏,2𝐷^𝑇𝑃𝐾₁𝑒_𝜏,1+2𝛿²₀𝐺^𝑇𝑒_𝜏,2𝐷^𝑇𝑃𝐾₁𝑒_𝜏,2+𝛿²₀𝑒^𝑇_𝜏,1𝐾^𝑇₁𝑃𝐾₁𝑒_𝜏,1+𝛿²₀𝑒^𝑇_𝜏,2𝐾^𝑇₁𝑃𝐾₁𝑒_𝜏,2=𝔼𝜉_𝑘Ω₀𝜉_𝑘, (B.9) whereΩ₀=Θ₁𝛿₀(𝐴+𝐾)^𝑇𝑃𝐾₁𝛿₀(𝐴+𝐾)^𝑇𝑃𝐾₁(𝐴+𝐾)^𝑇𝑃𝐵𝛿₀(𝐴+𝐾)^𝑇𝑃𝐷𝛿₀(𝐴+𝐾)^𝑇𝑃𝐷∗Θ₂𝛿₀𝛿₀𝐾^𝑇₁𝑃𝐾₁𝛿₀𝐾^𝑇₁𝑃𝐵𝛿²₀𝐾^𝑇₁𝑃𝐷𝛿₀𝛿₀𝐾^𝑇₁𝑃𝐷∗∗Θ₃𝛿₀𝐾^𝑇₁𝑃𝐵𝛿₀𝛿₀𝐾^𝑇₁𝑃𝐷𝛿²₀𝐾^𝑇₁𝑃𝐷∗∗∗𝐵^𝑇𝑃𝐵𝛿₀𝐵^𝑇𝑃𝐷𝛿₀𝐵^𝑇𝑃𝐷∗∗∗∗𝛿₀𝐷^𝑇𝑃𝐷0∗∗∗∗∗𝛿₀𝐷^𝑇𝑃𝐷<0, (B.10) where Θ₁=(𝐴+𝐾)^𝑇𝑃(𝐴+𝐾)−𝑃+𝜆^∗𝜌𝐼+(𝜏₀−𝜏_𝑚+1)𝑄₁+(𝜏_𝑀−𝜏₀+2)𝑄₂, Θ₂=−𝑄₁+𝛿²₀𝐾^𝑇₁𝑃𝐾₁, Θ₃=−𝑄₂+𝛿²₀𝐾^𝑇₁𝑃𝐾₁.

From (2.4), we can conclude that𝑓_𝑖𝑒_𝑖(𝑘)−𝑙_𝑖𝑒_𝑖(𝑘)×𝑓_𝑖𝑒_𝑖(𝑘)−𝐿_𝑖𝑒_𝑖(𝑘𝑔)≤0,_𝑖𝑒_𝑖𝑘−𝜏₁(𝑘)−𝜔_𝑖𝑒_𝑖𝑘−𝜏₁(𝑘)×𝑔_𝑖𝑒_𝑖𝑘−𝜏₁(𝑘)−𝑊_𝑖𝑒_𝑖𝑘−𝜏₁(𝑘)𝑔≤0,_𝑖𝑒_𝑖𝑘−𝜏₂(𝑘)−𝜔_𝑖𝑒_𝑖𝑘−𝜏₂(𝑘)×𝑔_𝑖𝑒_𝑖𝑘−𝜏₂(𝑘)−𝑊_𝑖𝑒_𝑖𝑘−𝜏₂(𝑘)≤0. (B.11)(B.12)(B.13) It can be conclude from (B.11) that there exists Λ₁=diag(𝜆₁₁,𝜆₁₂,…,𝜆_1𝑛)>0,^𝑛_𝑖=1𝜆_1𝑖𝑒_𝑘𝐹𝑒_𝑘^𝑇𝑙_𝑖𝐿_𝑖𝛼_𝑖𝛼^𝑇_𝑖−𝑙₁+𝐿_𝑖2𝛼_𝑖𝛼^𝑇_𝑖−𝑙₁+𝐿_𝑖2𝛼_𝑖𝛼^𝑇_𝑖𝛼_𝑖𝛼^𝑇_𝑖𝑒_𝑘𝐹𝑒_𝑘=𝑒_𝑘𝐹𝑒_𝑘^𝑇Λ₁𝒞₁Λ₁𝒞₂Λ₁𝒞₂Λ₁𝑒_𝑘𝐹𝑒_𝑘≤0, (B.14) 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 get𝑒_𝜏,1𝐺𝑒_𝜏,1^𝑇Λ₂ℋ₁Λ₂ℋ₂Λ₂ℋ₂Λ₂𝑒_𝜏,1𝐺𝑒_𝜏,1𝑒≤0,_𝜏,2𝐺𝑒_𝜏,2^𝑇Λ₃ℋ₁Λ₃ℋ₂Λ₃ℋ₂Λ₃𝑒_𝜏,2𝐺𝑒_𝜏,2≤0, (B.15)(B.16) where Λ₂=diag(𝜆₂₁,𝜆₂₂,…,𝜆_2𝑛)>0, Λ₃=diag(𝜆₃₁,𝜆₃₂,…,𝜆_3𝑛)>0.

Since Ω₀<0 implies Ω₁+Ψ^𝑇𝑃Ψ<0, combing (B.10) and (B.14)–(B.16), we can deduce𝔼Δ𝑉_𝑘≤𝔼𝜉^𝑇_𝑘Ω₀𝜉_𝑘−𝑒_𝑘𝐹𝑒_𝑘^𝑇Λ₁𝒞₁Λ₁𝒞₂Λ₁𝒞₂Λ₁𝑒_𝑘𝐹𝑒_𝑘−𝑒_𝜏,1𝐺𝑒_𝜏,1^𝑇Λ₂ℋ₁Λ₂ℋ₂Λ₂ℋ₂Λ₂𝑒_𝜏,1𝐺𝑒_𝜏,1−𝑒_𝜏,2𝐺𝑒_𝜏,2^𝑇Λ₃ℋ₁Λ₃ℋ₂Λ₃ℋ₂Λ₃𝑒_𝜏,2𝐺𝑒_𝜏,2≤𝔼𝜉^𝑇_𝑘Ω₁+Ψ^𝑇𝑃Ψ𝜉_𝑘≜𝔼𝜉^𝑇_𝑘Ψ𝜉_𝑘, (B.17) whereΩ₁=Ξ₁00−Λ₁𝒞₂00∗Ξ₂00−Λ₂ℋ₂0∗∗Ξ₃00−Λ₃ℋ₂∗∗∗−Λ₁00∗∗∗∗−Λ₂0∗∗∗∗∗−Λ₃<0. (B.18) By applying Lemma A.2 (in the Appendix A) to (3.2), we get Ψ<0. Let 𝑐₀=𝜆_max(Ψ), then 𝑐₀<0, and from (B.17), we get𝔼Δ𝑉_𝑘=𝔼𝔼𝑉_𝑘+1∣ℰ_𝑘−𝑉_𝑘≤𝑐₀𝔼𝑒_𝑘². (B.19) From the definition of 𝑉_𝑘, it can be easily derived that𝔼𝑉_𝑘≤𝜂₁𝔼𝑒_𝑘²+𝜂₂^𝑘−1_{𝑖=𝑘−𝜏𝑀}𝔼𝑒_𝑖², (B.20) where 𝜂₁=𝜆_max(𝑃),  𝜂₂=(𝜏₀−𝜏_𝑚+1)𝜆_max(𝑄₁)+(𝜏_𝑀−𝜏₀+2)𝜆_max(𝑄₂). For any scalar 𝜇>1, (B.20) together with (B.19), implies that𝔼𝜇^𝑘+1𝑉_𝑘+1−𝜇^𝑘𝑉_𝑘=𝜇^𝑘+1𝔼Δ𝑉_𝑘+𝜇^𝑘𝜇−1𝔼𝑉_𝑘≤ℎ₁𝜇𝜇^𝑘𝔼𝑒_𝑘²+ℎ₂𝜇^𝑘−1_{𝑖=𝑘−𝜏𝑀}𝜇^𝑘𝔼𝑒_𝑖², (B.21) where ℎ₁(𝜇)=𝜇𝑐₀+(𝜇−1)𝜂₁, ℎ₂(𝜇)=(𝜇−1)𝜂₂. Furthermore, for any integer 𝑁≥𝜏_𝑀+1, summing up both sides of (B.21) from 0 to 𝑁−1 with respect to 𝑘, we get𝔼𝜇^𝑁𝑉_𝑁−𝔼𝑉₀≤ℎ₁𝜇^𝑁−1_𝑘=0𝜇^𝑘𝔼𝑒_𝑘²+ℎ₂𝜇^𝑁−1_𝑘=0^𝑘−1_{𝑖=𝑘−𝜏𝑀}𝜇^𝑘𝔼𝑒_𝑖². (B.22) Note that for 𝜏_𝑚≥1,^𝑁−1_𝑘=0^𝑘−1_{𝑖=𝑘−𝜏𝑀}𝜇^𝑘𝔼𝑒_𝑖²≤⁻¹_{𝑖=−𝜏𝑀}^{𝑖+𝜏𝑀}_𝑘=0+^{𝑁−1−𝜏𝑀}_𝑖=0^{𝑖+𝜏𝑀}_𝑘=𝑖+1+^𝑁−1_{𝑖=𝑁−𝜏𝑀}^𝑁−1_𝑘=𝑖+1𝜇^𝑘𝔼𝑒_𝑖²≤𝜏_𝑀⁻¹_{𝑖=−𝜏𝑀}𝜇^{𝑖+𝜏𝑀}𝔼𝑒_𝑖²+𝜏_𝑀^{𝑁−1−𝜏𝑀}_𝑖=0𝜇^{𝑖+𝜏𝑀}𝔼𝑒_𝑖²+𝜏_𝑀^𝑁−1_{𝑖=𝑁−1−𝜏𝑀}𝜇^{𝑖+𝜏𝑀}𝔼𝑒_𝑖²≤𝜏_𝑀𝜇^𝜏𝑀max_{−𝜏𝑀≤𝑖≤0}𝔼𝑒_𝑖²+𝜏_𝑀𝜇^{𝜏𝑀𝑁−1}_𝑖=0𝜇^𝑖𝔼𝑒_𝑖². (B.23) Then, from (B.22) and (B.23) we have𝔼𝜇^𝑁𝑉_𝑁≤𝔼𝑉₀+ℎ₁𝜇+𝜏_𝑀𝜇^𝜏𝑀ℎ₂𝜇^𝑁−1_𝑘=0𝜇^𝑘𝔼𝑒_𝑘²+𝜏_𝑀𝜇^𝜏𝑀ℎ₂𝜇max_{−𝜏𝑀≤𝑖≤0}𝔼𝑒_𝑖². (B.24) Let 𝜂₀=𝜆_min(𝑃), and 𝜂=max{𝜂₁,𝜂₂}. Obviously, we have𝔼𝑉_𝑁≥𝜂₀𝔼𝑒_𝑁². (B.25) It also follows easily from (B.20) that𝔼𝑉₀≤𝜂max_{−𝜏𝑀≤𝑖≤0}𝔼𝑒_𝑖². (B.26) In addition, we can verify that there exists a scalar 𝜇₀>1 such thatℎ₁𝜇₀+𝜏_𝑀𝜇^𝜏𝑀₀ℎ₂𝜇₀=0. (B.27) Substituting (B.25)–(B.27) into (B.24), we obtain𝔼𝑒_𝑁²≤1𝜇₀𝜂+𝜏_𝑀𝜇^𝜏𝑀₀ℎ₂𝜇₀1𝜇₀^𝑁max_{−𝜏𝑀≤𝑖≤0}𝔼𝑒_𝑖². (B.28) 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.

Acknowledgments

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.