Global Synchronization of Neutral-Type Stochastic Delayed Complex Networks

and Applied Analysis 3 It is still very difficult to calculate expectations of these stochastic cross terms up to now. The results in 28–31 resorted to bounding techniques, which obviously can bring the conservatism. Some papers such as 32–34 considered that expectations of these stochastic cross terms are all equal to zero. However, these results are not given by strict mathematical proofs, and we can find examples to illustrate that expectations of some stochastic cross terms are not equal to zero in Remark 3.3. Therefore, in order to obtain the delay-dependent synchronization criterion with less conservatism for neutral-type stochastic delayed complex networks, there is a strong need to investigate the expectations of stochastic cross terms containing the Itô integral firstly. Motivated by the discussion mentioned above, this paper investigates the delaydependent synchronization problem for neutral-type stochastic delayed complex networks. The main contributions of this paper are summarized as follows. 1 Expectations of stochastic cross terms containing the Itô integral are investigated by stochastic analysis techniques in Lemma 3.1 and Corollary 3.2. We prove that the expectation of x t − h K ∫ t t−h μ s, xs dw s is equal to zero and expectations of other stochastic cross terms are not. 2 Based on this conclusion, this paper establishes a delay-dependent synchronization criterion that guarantees the globally asymptotic synchronization of neural-type stochastic delayed complex networks. In the derivation process, the mathematical development avoids bounding stochastic cross terms. Thus, thismethod leads to a criterionwith less conservatism. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach. Notation. Throughout the paper, unless otherwise specified, we will employ the following notation. Let Ω,F, {Ft}t≥0,P be a complete probability space with a natural filtration {Ft}t≥0, and let E · be the expectation operator with respect to the probability measure. If A is a vector or matrix, its transpose is denoted by A . If P is a square matrix, then P > 0 P < 0 means that it is a symmetric positive negative definite matrix of appropriate dimensions while P ≥ 0 P ≤ 0 is a symmetric positive negative semidefinite matrix. I stands for the identity matrix of appropriate dimensions. Denote by λmin · the minimum eigenvalue of a given matrix. Let | · | denote the Euclidean norm of a vector and its induced norm of a matrix. Unless explicitly specified, matrices are assumed to have real entries and compatible dimensions. L2 Ω denotes the space of all random variables X with E|X|2 < ∞, it is a Banach space with norm ‖X‖2 E|X|2 . Let h > 0 and C −h, 0 ;Rn denote the family of all continuous Rn-valued functions φ on −h, 0 with the norm ‖φ‖ sup{|φ θ | : −h ≤ θ ≤ 0}. Let LF0 −h, 0 ;Rn be the family of all F0-measurable C −h, 0 ;Rn -valued random variables φ such that E ‖φ‖2 < ∞, and let L2 a, b ;Rn be the family of all Rn-valued Ftadapted processes {f t }a≤t≤b such that ∫b a |f t |2dt < ∞ a.s. Let M2 a, b ;Rn be the family of processes {f t }a≤t≤b in L2 a, b ;Rn such that E ∫b a |f t |2dt < ∞, and M2 a, b is the 1-dimensional case of M2 a, b ;Rn . 2. Problem Formulation and Preliminaries In this paper, we consider the following neutral-type stochastic delayed complex networks consisting of N identical nodes: 4 Abstract and Applied Analysis d xi t −Dxi t − h ⎡ ⎣Axi t Bf xi t Cf xi t − h N ∑


Introduction
In the real world, many systems can be described as complex networks such as Internet networks, biological networks, epidemic spreading networks, collaborative networks, social networks, neural networks, and so forth 1-4 .Thus, during the past years, the study of complex networks has become a very active area, see, for example, 5, 6 and the references therein.In particular, for complex networks, the major collective behavior is the synchronization phenomena, because many problems in practice have close relationships with synchronization 7 .Recently, growing research results, that focused on synchronization problems for complex networks, have been reported in 8-12 and the references therein.method 25, 26 and the free-weighting matrix method 27 , to give the delay-dependent condition for stochastic delay systems including stochastic delayed complex or neural networks, the following stochastic cross terms containing the It ô integral will appear: x t T  μ s, x s dw s .

1.1
It is still very difficult to calculate expectations of these stochastic cross terms up to now.
The results in 28-31 resorted to bounding techniques, which obviously can bring the conservatism.Some papers such as 32-34 considered that expectations of these stochastic cross terms are all equal to zero.However, these results are not given by strict mathematical proofs, and we can find examples to illustrate that expectations of some stochastic cross terms are not equal to zero in Remark 3.3.Therefore, in order to obtain the delay-dependent synchronization criterion with less conservatism for neutral-type stochastic delayed complex networks, there is a strong need to investigate the expectations of stochastic cross terms containing the It ô integral firstly.Motivated by the discussion mentioned above, this paper investigates the delaydependent synchronization problem for neutral-type stochastic delayed complex networks.The main contributions of this paper are summarized as follows. 1 Expectations of stochastic cross terms containing the It ô integral are investigated by stochastic analysis techniques in Lemma 3.1 and Corollary 3.2.We prove that the expectation of x t − h T K t t−h μ s, x s dw s is equal to zero and expectations of other stochastic cross terms are not. 2 Based on this conclusion, this paper establishes a delay-dependent synchronization criterion that guarantees the globally asymptotic synchronization of neural-type stochastic delayed complex networks.In the derivation process, the mathematical development avoids bounding stochastic cross terms.Thus, this method leads to a criterion with less conservatism.Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach.
Notation.Throughout the paper, unless otherwise specified, we will employ the following notation.Let Ω, F, {F t } t≥0 , P be a complete probability space with a natural filtration {F t } t≥0 , and let E • be the expectation operator with respect to the probability measure.If A is a vector or matrix, its transpose is denoted by A T .If P is a square matrix, then P > 0 P < 0 means that it is a symmetric positive negative definite matrix of appropriate dimensions while P ≥ 0 P ≤ 0 is a symmetric positive negative semidefinite matrix.I stands for the identity matrix of appropriate dimensions.Denote by λ min • the minimum eigenvalue of a given matrix.Let | • | denote the Euclidean norm of a vector and its induced norm of a matrix.Unless explicitly specified, matrices are assumed to have real entries and compatible dimensions.L 2 Ω denotes the space of all random variables X with E|X| 2 < ∞, it is a Banach space with norm X 2 E|X| 2 1/2 .Let h > 0 and C −h, 0 ; R n denote the family of all continuous R n -valued functions ϕ on −h, 0 with the norm ϕ

Problem Formulation and Preliminaries
In this paper, we consider the following neutral-type stochastic delayed complex networks consisting of N identical nodes: where x i t x i1 t , x i2 t , . . ., x in t T ∈ R n represents the state vector of the ith node; the scalar h > 0 is the time delay; A is a known connection matrix; B and C denote, respectively, the connection weight matrix and the delayed connection weight matrix; Γ, Υ ∈ R n×n are matrices describing the inner coupling between the subsystems at time t and t − h, respectively; G g ij N×N and H h ij N×N are called the outer-coupling configuration matrices representing the coupling strength and the topological structure of the complex networks; D is a known real matrix, and the spectrum radius of the matrix D, ρ D , satisfies ρ D < 1. σ i •, •, • : R × R n × R n → R n which is the noise intensity function vector; w t is a scalar standard Brownian motion defined on a complete probability space Ω, F, {F t } t≥0 , P with a natural filtration {F t } t≥0 .f x i t f 1 x i1 t , . . ., f n x in t T , is an unknown but sector-bounded nonlinear function.
The initial conditions associated with system 2.1 are given by where

2.3
With the Kronecker product "⊗" for matrices, system 2.1 can be rearranged as

2.4
Before stating our main results, we need the following definitions, assumptions, and propositions.
Definition 2.1.The neutral-type stochastic delayed complex network 2.1 is globally asymptotically synchronized in the mean square if, for all Assumption 2.4.The outer-coupling configuration matrices of the complex networks 2.1 satisfy

2.8
Assumption 2.5.The noise intensity function vector σ i : R × R n × R n → R n satisfies the Lipschitz condition, that is, there exist constant matrices W 1 and W 2 of appropriate dimensions such that Assumption 2.6.For all x, y ∈ R n , the nonlinear function f • is assumed to satisfy the following condition: where U and V are real constant matrices with U-V being symmetric and positive definite.
Proposition 2.7 see 14 .The Kronecker product has the following properties:

Main Results
Then, we give the following lemma and corollary which will play a key role in the proof of our main results.Most important of all, since is a bounded and F a -measurable random variable, it is easy to verify { ν t } a≤t≤b ∈ M 2 a, b .Then, we will prove 3.2 by the following two steps.
Step 1.If {ν t } a≤t≤b is a step stochastic process, then we let, without loss of generality,

3.4
Step • satisfy the local Lipschitz condition and the linear growth condition.If x t is the solution of 3.12 and K is any compatible dimensional matrix, then

3.13
Especially when D 0 in 3.12 , that is, dx t κ t, x t dt μ t, x t dw t .

3.14
Equation 3.14 is a common stochastic functional equation.For this case, 3.13 is also tenable.
Proof.Since κ •, • and μ •, • satisfy the local Lipschitz condition and the linear growth condition, we can know that, for all T > 0, 3.12 has a unique continuous solution on −h, T denoted by {x t } −h≤t≤T that is adapted to {F t } −h≤t≤T and {x t } −h≤t≤T ∈ M 2 −h, T 37 .Therefore, it can be derived that for t ≥ h, x t − h is a bounded random variable and x t − h is F t−h -measurable.Then, by Lemma 3.1, it is easy to obtain 3.13 .If D 0 in 3.12 that is a common stochastic functional equation, then we can easily prove that 3.13 is also tenable for this case.

3.15
However, for any compatible dimensional matrix J or L, the following results are not correct: We should point out that in recent years, some papers such as 32-34 considered that the expectations of these stochastic terms are all equal to zero.However, this is not the case.
From the above examples and Corollary 3.2, we can see that x t − h T K t t−h μ s, x s dw s is the only one whose expectation is equal to zero.
Then, we are in the position to present our main result for the synchronization criterion of the neutral-type delayed complex networks with stochastic disturbances.Theorem 3.6.Under the Assumptions 2.4-2.6, the dynamical system 2.1 is globally asymptotically synchronized in the mean square if there exist matrices P > 0, Q 1 > 0, Q 2 > 0, R > 0, Z > 0, S and scalars > 0, λ > 0 such that the following LMIs hold for all 1 ≤ i < j ≤ N: where

3.27
Consider the following Lyapunov functional for the system 3.26 :

3.28
where Then, by the It ô's formula, the stochastic differential dV x t , t can be obtained where

3.31
By 3.27 , we have

3.32
From Corollary 3.2, it follows that

3.33
Abstract and Applied Analysis 13 By 3.25 , it is easy to know that for any matrix S, we have

3.34
From 3.31 -3.34 and by the Propositions 2.7 and 2.8, it is easy to get

3.35
According to Assumptions 2.5 and 3.22 , it is clear that By Assumption 2.6, we can obtain

3.37
Abstract and Applied Analysis 15 Combining 3.35 -3.37 , we have where Since Ξ < 0, it is guaranteed that all the subsystems in 2.1 are globally asymptotically synchronized in the mean square.The proof is completed.
Remark 3.7.We note here that if D 0 in 2.1 , then system 2.1 describes a kind of stochastic delayed complex networks considered in 32 .Our result can be applied to this case, and we have pointed out that 32 made a mistake when dealing with expectations of stochastic cross terms in Remark 3.3.If we let A be a diagonal and negative matrix and let D 0 in 2.1 , the system 2.1 will be an array of coupled neural networks consisting of N nodes, in which each node is an n-dimensional stochastic delayed Hopfield neural network.As to stochastic Hopfield neural networks with time delays, 30, 38 have investigated the stability problems, respectively.Furthermore, if we don't consider stochastic disturbances and time delays in stochastic delayed Hopfield neural networks, then this kind of neural networks is the famous Hopfield neural network.
Remark 3.8.If we do not consider the stochastic disturbances in 2.1 , then the system will be a kind of determinate neutral-type delayed complex networks, that have been considered in the 18-20 .If we let A be a diagonal and negative matrix in this kind of determinate neutraltype delayed complex networks, each node will be an n-dimensional neutral-type delayed neural network.For neutral-type neural networks with time delays, 39, 40 have discussed the stability problems and presented the new and effective stability conditions, respectively.
Remark 3.9.For neutral stochastic delay systems, a very active topic is to obtain the delaydependent condition.For example, 28, 29 considered delay-dependent stability problems for neutral stochastic delay systems.However, these two papers used bounding techniques including the Jensen inequality to deal with stochastic cross terms contain the It ô integral.
Obviously, bounding techniques will increase the conservatism.In the derivation process of Theorem 3.6, we don't use any bounding technique to deal with stochastic cross terms.Therefore, this method can show less conservatism and can also be extended to solve delaydependent stability problems for neutral stochastic delay systems.
Remark 3.10.In Theorem 3.6, we give a delay-dependent synchronization criterion by the linear matrix inequalities LMIs , because LMIs can be easily solved by using the Matlab LMI toolbox and no tuning of parameters is required.Moreover, we can easily get the maximum possible upper bound on the delay by the LMI toolbox.The maximum possible upper bound on the delay is the main criterion for judging the conservatism of a delaydependent condition.

Numerical Example
In this section, we present a simulation example to illustrate the effectiveness of our approach.
Example 4.1.Consider the following complex network consisting of three identical nodes: for all i 1, 2, 3, where x i t x i1 t , x i2 t T ∈ R 2 is the state vector of the ith subsystem, and According to Theorem 3.6, the allowable maximum delay h, that can guarantee the globally asymptotic mean-square synchronization of the neutral-type stochastic delayed complex networks, is 0.33.When we randomly choose the the initial states in 0, 1 × 0, 1 , the synchronization errors are plotted in Figures 1 and 2, which can confirm that the neutraltype stochastic delayed complex system is globally synchronized in the mean square.

Conclusions
This paper has investigated the problem of delay-dependent synchronization criterion for neutral-type stochastic delayed complex networks.Most important of all, this paper is concerned with expectations of stochastic cross terms containing the It ô integral.By stochastic analysis techniques, we prove that among these stochastic cross terms, x t − h T K t t−h μ s, x s dw s is the only one whose expectation is equal to zero.Then, this paper has utilized this conclusion to give a delay-dependent synchronization criterion for neutraltype stochastic delayed complex networks.In the derivation process, the mathematical development avoids bounding stochastic cross terms.Thus, the method in our paper can lead to a criterion with less conservatism, and a numerical example is provided to demonstrate the effectiveness of the proposed approach.

x 12 (Figure 2 :
Figure 2: State error of x 12 t − x i2 t , i 2, 3. J T L t t−h

5
Definition 2.2 see 35 .If a stochastic process {ν t } a≤t≤b belongs to M 2 a, b , then its It o integral from a to b is defined by n t } a≤t≤b n 1, 2, . . .are the step stochastic processes and belong to M 2 a, b Definition 2.3 see 36 .Let {F t } t∈T be an increasing family of σ-algebras of subset of Ω.A stochastic process {X t } t∈T is said to be adapted to {F t } t∈T if for each t, the random variable X t is F t -measurable.
Proof.Firstly, in order to prove the above results, we will prove that if {ν t } a≤t≤b ∈ M 2 a, b and is a bounded and F a -measurable random variable, then Lemma 3.1.If a stochastic process {ν t } a≤t≤b ∈ M 2 a, b and is a bounded and F a -measurable random variable, then 2. If {ν t } a≤t≤b ∈ M 2 a, b is not a step stochastic process, then by Definition 2.2, we can find a sequence of step stochastic processes in M 2 a, b : {ν 1 t } a≤t≤b , {ν 2 t } a≤t≤b , . .., {ν n t } a≤t≤b , . . .such that Due to { ν t } a≤t≤b ∈ M 2 a, b , then by Proposition 2.9, we can know that where {ν t } a≤t≤b and {ν n t} a≤t≤b satisfy lim n → ∞ E b a |ν t − ν n t | 2 dt 0. 3.6Because is bounded, by Definition 2.2 and 3.5 -3.6 , it is easy to prove that From Step 1, it follows that for each step stochastic process {ν n t } a≤t≤b , we have Thus, the matrices U, V, W 1 , W 2 , in the Assumptions 2.5 and 2.6 are