# Robust Synchronization Criterion for Coupled Stochastic Discrete-Time Neural Networks with Interval Time-Varying Delays, Leakage Delay, and Parameter Uncertainties

Authors: M. J. Park; O. M. Kwon; Ju H. Park; S. M. Lee; E. J. Cha
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The purpose of this paper is to investigate a delay-dependent robust synchronization analysis for coupled stochastic discrete-time neural networks with interval time-varying delays in networks coupling, a time delay in leakage term, and parameter uncertainties. Based on the Lyapunov method, a new delay-dependent criterion for the synchronization of the networks is derived in terms of linear matrix inequalities (LMIs) by constructing a suitable Lyapunov-Krasovskii’s functional and utilizing Finsler’s lemma without free-weighting matrices. Two numerical examples are given to illustrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

In recent years, the problem of synchronization of coupled neural networks which is one of hot research fields of complex networks has been a challenging issue due to its potential applications such as physics, information sciences, biological systems, and so on. Here, complex networks, which are a set of interconnected nodes with specific dynamics, have been studied from various fields of science and engineering such as the World Wide Web, social networks, electrical power grids, global economic markets, and so on. Many mathematical models were proposed to describe various complex networks [1, 2]. Also, in the real applications of systems, there exists naturally time delay due to the finite information processing speed and the finite switching speed of amplifiers. It is well known that time delay often causes undesirable dynamic behaviors such as performance degradation and instability of the systems. So, some sufficient conditions for synchronization of coupled neural networks with time delay have been proposed in [3–5]. Moreover, the synchronization of delayed systems was applied in practical systems such as secure communication [6]. Furthermore, these days, most systems use digital computers (usually microprocessor or microcontrollers) with the necessary input/output hardware to implement the systems. The fundamental character of the digital computer is that it takes compute answers at discrete steps. Therefore, discrete-time modeling with time delay plays an important role in many fields of science and engineering applications. In this regard, various approaches to synchronization stability criterion for discrete-time complex networks with time delay have been investigated in the literature [7–9].On the other hand, in implementation of many practical systems such as aircraft, chemical and biological systems, and electric circuits, there exist occasionally stochastic perturbations. It is not less important than the time delay as a considerable factor affecting dynamics in the fields of science and engineering applications. Therefore, the study on the problems for various forms of stochastic systems with time-delay has been addressed. For more details, see the literature [10–13] and references therein. Furthermore, on the problem of synchronization of coupled stochastic neural networks with time delay, various researches have been conducted [14–17]. Li and Yue [14] studied the synchronization stability problem for a class of complex networks with Markovian jumping parameters and mixed time delays. The model considered in [14] has stochastic coupling terms and stochastic disturbances to reflect more realistic dynamical behaviors of the complex networks that are affected by noisy environment. In [15], by utilizing novel Lyapunov-Krasovskii's functional with both lower and upper delay bounds, the synchronization criteria for coupled stochastic discrete-time neural networks with mixed delays were presented. Tang and Fang [16] derived several sufficient conditions for the synchronization of delayed stochastically coupled fuzzy cellular neural networks with mixed delays and uncertain hybrid coupling based on adaptive control technique and some stochastic analysis methods. In [17], by using Kronecker product as an effective tool, robust synchronization problem of coupled stochastic discrete-time neural networks with time-varying delay was investigated. Moreover, Song [18–20] addressed synchronization problem for the array of asymmetric, chaotic, and coupled connected neural networks with time-varying delay or nonlinear coupling. Also, in [21], robust exponential stability analysis of uncertain delayed neural networks with stochastic perturbation and impulse effects was investigated.Very recently, a time delay in leakage term of the systems is being put to use in the problem of stability for neural networks as a considerable factor affecting dynamics for the worse in the systems [22, 23].  Li et al. [22] studied the existence and uniqueness of the equilibrium point of recurrent neural networks with time delays in the leakage term. By use of the topological degree theory, delay-dependent stability conditions of neural networks of neutral type with time delays in the leakage term were proposed in [23]. Unfortunately, to the best of authors’ knowledge, delay-dependent synchronization analysis of coupled stochastic discrete-time neural networks with time-varying delay in network coupling and leakage delay has not been investigated yet. Thus, by attempting the synchronization analysis for the model of coupled stochastic discrete-time neural networks with time delay in the leakage term, the model for coupled neural networks and its applications are closed to the practical networks. Here, delay-dependent analysis has been paid more attention than delay-independent one because the sufficient conditions for delay-dependent analysis make use of the information on the size of time delay [24]. That is, the former is generally less conservative than the latter.Motivated by the above discussions, the problem of a new delay-dependent robust synchronization criterion for coupled stochastic discrete-time neural networks with interval time-varying delays in network coupling, the time delay in leakage term, and parameter uncertainties is considered for the first time. The coupled stochastic discrete-time neural networks are represented as a simple mathematical model by the use of Kronecker product technique. Then, by construction of a suitable Lyapunov-Krasovskii's functional and utilization of Finsler’s lemma without free-weighting matrices, a new synchronization criterion is derived in terms of LMIs. The LMIs can be formulated as convex optimization algorithms which are amenable to computer solution [25]. In order to utilize Finsler’s lemma as a tool of getting less conservative synchronization criteria on the number of decision variables, it should be noted that a new zero equality from the constructed mathematical model is devised. The concept of scaling transformation matrix will be utilized in deriving zero equality of the method. In [26], the effectiveness of Finsler’s lemma was illustrated by the improved passivity criteria of uncertain neural networks with time-varying delays. Finally, two numerical examples are included to show the effectiveness of the proposed method.Notation.
ℝ
n is the n-dimensional Euclidean space, and ℝm×n denotes the set of all m×n real matrices. For symmetric matrices X and Y, X>Y (resp., X≥Y) means that the matrix X-Y is positive definite (resp., nonnegative). X⊥ denotes a basis for the null-space of X. In and 0n and 0m×n denote n×n identity matrix and n×n and m×n zero matrices, respectively. ∥·∥ refers to the Euclidean vector norm or the induced matrix norm. λmax(·) means the maximum eigenvalue of a given square matrix. diag{⋯} denotes the block diagonal matrix. ⋆ represents the elements below the main diagonal of a symmetric matrix. Let (Ω,ℱ,{Ft}t≥0,𝒫) be complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all 𝒫-pull sets). 𝔼{·} stands for the mathematical expectation operator with respect to the given probability measure 𝒫.


## 2. Problem Statements

Consider the following discrete-time delayed neural networks:(1)y(k+1)=(A+ΔA)y(k-τ)+(W1+ΔW1)g(y(k))+(W2+ΔW2)g(y(k-h(k)))+b,
where n denotes the number of neurons in a neural network, y(·)=[y1(·),…,yn(·)]T∈ℝn is the neuron state vector, g(·)=[g1(·),…,gn(·)]T∈ℝn denotes the neuron activation function vector, b=[b1,…,bn]T∈ℝn means a constant external input vector, A=diag{a1,…,an}∈ℝn×n(0<aq<1,q=1,…,n) is the state feedback matrix, Wq∈ℝn×n  (q=1,2) are the connection weight matrices, and ΔA and ΔWq(q=1,2) are the parameter uncertainties of the form
(2)[ΔA,ΔW1,ΔW2]=DF(k)[Ea,E1,E2],
where F(k) is a real uncertain matrix function with Lebesgue measurable elements satisfying
(3)FT(k)F(k)≤I.
The delays h(k) and τ are interval time-varying delays and leakage delay, respectively, satisfying
(4)0<hm≤h(k)≤hM,0<τ,
where hm and hM are positive integers.The neuron activation functions,gp(yp(·))(p=1,…,n), are assumed to be nondecreasing, bounded, and globally Lipschitz; that is,
(5)lp-≤gp(ξp)-gp(ξq)ξp-ξq≤lp+,∀ξp,ξq∈ℝ,ξp≠ξq,
where lp- and lp+ are constant values.For simplicity, in stability analysis of the network (1), the equilibrium point y*=[y1*,…,yn*]T is shifted to the origin by the utilization of the transformation y~(·)=y~(·)-y*, which leads the network (1) to the following form:
(6)y~(k+1)=(A+ΔA)y~(k-τ)+(W1+ΔW1)g~(y~(k))+(W2+ΔW2)g~(y~(k-h(k))),
where y~(·)=[y~1(·),…,y~n(·)]T∈ℝn is the state vector of the transformed network, and g~(y~(·))=[g~1(y~1(·)),…,g~n(y~n(·))]T is the transformed neuron activation function vector with g~q(y~q(·))=gq(y~q(·)+yq*)-gq(yq*)  (q=1,…,n) satisfies, from (5), lp-≤g~p(ξp)/ξp≤lp+,∀ξp≠0, which is equivalent to
(7)[g~p(y~p(k))-lp-y~p(k)][g~p(y~p(k))-lp+y~p(k)]≤0.In this paper, a model of coupled stochastic discrete-time neural networks with interval time-varying delays in network coupling, leakage delay, and parameter uncertainties is considered as(8)y~i(k+1)=(A+ΔA)y~i(k-τ)+(W1+ΔW1)g~(y~i(k))+(W2+ΔW2)g~(y~i(k-h(k)))+∑j=1NgijΓy~j(k-h(k))(1+ω1(k))+σi(k,y~i(k),y~i(k-h(k)))ω2(k),i=1,2,…,N,
where N is the number of couple nodes, y~i(k)=[y~i1(k),…,y~in(k)]T∈ℝn is the state vector of the ith node, Γ∈ℝn×n is the constant inner-coupling matrix of nodes, which describe the individual coupling between the subnetworks, G=[gij]N×N is the outer-coupling matrix representing the coupling strength and the topological structure of the network satisfies the diffusive coupling connections
(9)gij=gji≥0(i≠j),gii=-∑j=1,i≠jNgij(i,j=1,2,…,N),
and ωq(k)(q=1,2) are m-dimensional Wiener processes (Brownian Motion) on (Ω,ℱ,{Ft}t≥0,𝒫) which satisfy
(10)𝔼{ωq(k)}=0,𝔼{ωq2(k)}=1,𝔼{ωq(i)ωq(j)}=0(i≠j).
Here, ω1(k) and ω2(k), which are mutually independent, are the coupling strength disturbance and the system noise, respectively. And the nonlinear uncertainties σi(·,·,·)∈ℝn×m(i=1,…,N) are the noise intensity functions satisfying the Lipschitz condition and the following assumption:
(11)σiT(k,y~i(k),y~i(k-h(k)))σi(k,y~i(k),y~i(k-h(k)))≤∥H1y~i(k)∥2+∥H2y~i(k-h(k))∥2,
where Hq(q=1,2) are constant matrices with appropriate dimensions.Remark 1.
According to the graph theory [27], the outer-coupling matrix G is called the negative Laplacian matrix of undirected graph. A physical meaning of the matrix G is the bilateral connection between node i and j. If the matrix G cannot satisfy symmetric, the unidirectional connection between nodes i and j is expressed. At this time, the matrix G is called the negative Laplacian matrix of directed graph. Therefore, new numerical model and strong sufficient condition guaranteed to the stability for networks are needed. Moreover, in order to analyze the consensus problem for multiagent systems, the Laplacian matrix of directed graph was used [28].
For the convenience of stability analysis for the network (8), the following Kronecker product and its properties are used.Lemma 2 (see [29]).
Let⊗ denote the notation of Kronecker product. Then, the following properties of Kronecker product are easily established:(i)
(αA)⊗B=A⊗(αB),(ii)
(A+B)⊗C=A⊗C+B⊗C,(iii)
(A⊗B)(C⊗D)=(AC)⊗(BD),(iv)
(A⊗B)T=AT⊗BT.Let us define(12)x(k)=[y~1(k),…,y~N(k)]T,f(x(k))=[g~(y~1(k)),…,g~(y~N(k))]T,σ(t)=[σ1(·,·,·),…,σN(·,·,·)]T.
Then, with Kronecker product in Lemma 2, the network (8) can be represented as
(13)x(k+1)=(IN⊗A(k))x(k-τ)+(IN⊗W1(k))f(x(k))+(IN⊗W2(k))f(x(k-h(k)))+(G⊗Γ)x(k-h(k))(1+ω1(k))+σ(t)ω2(t),
where A(k)=A+DF(k)Ea, W1(k)=W1+DF(k)E1, and W2(k)=W2+DF(k)E2.In addition, for stability analysis, (13) can be rewritten as follows:
(14)x(k+1)=η(k)+ϱ(k)ω(k),
where
(15)η(k)=(IN⊗A)x(k-τ)+(IN⊗W1)f(x(k))+(IN⊗W2)f(x(k-h(k)))+(G⊗Γ)x(k-h(k))+(IN⊗D)p(k),p(k)=(IN⊗F(k))q(k),q(k)=(IN⊗Ea)x(k-τ)+(IN⊗E1)f(x(k))+(IN⊗E2)f(x(t-h(k))),ϱ(k)=[(G⊗Γ)x(k-h(k)),σ(k)],ωT(k)=[ω1T(k),ω2T(k)].The aim of this paper is to investigate the delay-dependent synchronization stability analysis of the network (14) with interval time-varying delays in network coupling, leakage delay, and parameter uncertainties. In order to do this, the following definition and lemmas are needed.Definition 3 (see [7]).
The network (8) is said to be asymptotically synchronized if the following condition holds:
(16)limt→∞∥xi(k)-xj(k)∥=0,i,j=1,2,…,N.Lemma 4 (see [3]).
LetU=[uij]N×N, P∈ℝn×n, xT=[x1,x2,…,xn]T, and yT=[y1,y2,…,yn]T. If U=UT and each row sum of  U is zero, then
(17)xT(U⊗P)y=-∑1≤i<j≤Nuij(xi-xj)TP(yi-yj).Lemma 5 (see [30]).
For any constant matrix0<M=MT∈ℝn×n, integers hm and hM satisfying 1≤hm≤hM, and vector function x(k)∈ℝn, the following inequality holds:
(18)-(hM-hm+1)∑k=hmhMxT(k)Mx(k)≤-(∑k=hmhMx(k))TM(∑k=hmhMx(k)).Lemma 6 (see [31] (Finsler's lemma)).
Letζ∈ℝn, Φ=ΦT∈ℝn×n, and Υ∈ℝm×n such that rank(Υ)<n. The following statements are equivalent: (i)
ζTΦζ<0,
∀Υζ=0,
ζ≠0,(ii)
Υ⊥TΦΥ⊥<0.


## 3. Main Results

In this section, a new synchronization criterion for the network (14) will be proposed. For the sake of simplicity on matrix representation, ei(i=1,…,9)∈ℝ9n×n are defined as block entry matrices (e.g., e2=[0n,In,0n,0n,0n,0n,0n,0n,0n]T). The notations of several matrices are defined as follows:
(19)ζT(k)=[(x(k-h(k)))xT(k),xT(k-τ),xT(k-hm),xT(k-h(k)),xT(k-hM),(η(k)-x(k))T,fT(x(k)),fT(x(k-h(k))),pT(k)],zij(k)=xi(k)-xj(k),f(zij(k))=f(xi(k))-f(xj(k)),ηij(k)=ηi(k)-ηj(k),pij(k)=pi(k)-pj(k),ζijT(k)=[(zij(k-h(k)))zijT(k),zijT(k-τ),zijT(k-hm),zijT(k-h(k)),zijT(k-hM),(ηij(k)-zij(k))T,fT(zij(k)),fT(zij(k-h(k))),pijT(k)],Υij=[-In,A,0n,-(NgijΓ),0n,-In,W1,W2,D],Σ=P+hm2R1+(hM-hm)2R2+τ2S2,Ξ1=e1Pe6T+e6Pe1T+e6Pe6T,Ξ2=e1Q1e1T-e3(Q1-Q2)e3T-e5Q2e5T,Ξ3=e6(hm2R1+(hM-hm)2R2)e6T-(e1-e3)R1(e1-e3)T-(e3-e4)R2(e3-e4)T-(e4-e5)R2(e4-e5)T-(e3-e4)TT(e4-e5)T-(e4-e5)T(e3-e4)T,Ξ4=e1S1e1T-e2S1e2T+e6(τ2S2)e6T-(e1-e2)S2(e1-e2)T,Ξ5=e4(N∑l=1NgilgljΓTΣΓ)e4T+e1(ρH1TH1)e1T+e4(ρH2TH2)e4T,Ξ6=-e1(2LmD1Lp)e1T+e1(Lm+Lp)D1e7T+(e1(Lm+Lp)D1e7T)T-e7(2D1)e7T-e4(2LmD2Lp)e4T+e4(Lm+Lp)D2e8T+(e4(Lm+Lp)D2e8T)T-e8(2D2)e8T,Ξ7=-e9(ϵIn)e9T,Ψ=[0n,Ea,0n,0n,0n,0n,E1,E2,0n].Then, the main result of this paper is presented as follows.Theorem 7.
For given positive integershm, hM and τ, diagonal matrices Lm=diag{l1-,…,ln-} and Lp=diag{l1+,…,ln+}, the network (14) is asymptotically synchronized for hm≤h(k)≤hM, if there exist positive scalars ρ, ϵ, positive definite matrices P, Q1, Q2, R1, R2, S1, S2, positive diagonal matrices D1, D2, and any matrix T satisfying the following LMIs for 1≤i<j≤N:
(20)Σ-ρIn≤0,(21)[R2T⋆R2]≥0,(22)[[(j-i)Υij]⊥09n×n0n×8nIn]T[∑l=17‍ΞlϵΨT⋆-ϵIn]×[[(j-i)Υij]⊥09n×n0n×8nIn]<0,
where Σ, Υij, Ξl  (l=1,…,7), and Ψ are defined in (19).Proof.
Define a matrixU as
(23)U=[uij]N×N=[N-1-1⋯-1-1N-1-1⋮⋮-1⋱-1-1⋯-1N-1]
and the forward difference of x(k) and V(k) as
(24)Δx(k)=x(k+1)-x(k)=η(k)-x(k)+ϱ(k)ω(t),ΔV(k)=V(k+1)-V(k).
Let us consider the following Lyapunov-Krasovskii's functional candidate as
(25)V(k)=V1(k)+V2(k)+V3(k)+V4(k),
where
(26)V1(k)=xT(k)(U⊗P)x(k),V2(k)=∑s=k-hmk-1xT(s)(U⊗Q1)x(s)+∑s=k-hMk-hm-1xT(s)(U⊗Q2)x(s),V3(k)=hm∑s=-hm-1∑u=k+sk-1ΔxT(u)(U⊗R1)Δx(u)+(hM-hm)∑s=-hM-hm-1∑u=k+sk-1ΔxT(u)(U⊗R2)Δx(u),V4(k)=∑s=k-τk-1xT(s)(U⊗S1)x(s)+τ∑s=-τ-1∑u=k+sk-1ΔxT(u)(U⊗S2)Δx(u).
The mathematical expectation of ΔV(k) is calculated as follows:
(27)𝔼{ΔV1(k)}=𝔼{xT(k+1)(U⊗P)x(k+1)-xT(k)(U⊗P)x(k)}=𝔼{(Δx(k)+x(k))T(U⊗P)(Δx(k)+x(k))-xT(k)(U⊗P)x(k)(Δx(k)+x(k))T}=𝔼{ΔxT(k)(U⊗P)Δx(k)+2ΔxT(k)(U⊗P)x(k)}=𝔼{(η(k)-x(k))T(U⊗P)(η(k)-x(k))+(ϱ(k)ω(k))T(U⊗P)(ϱ(k)ω(k))+2(η(k)-x(k))T(U⊗P)x(k)}=𝔼{(η(k)-x(k))T(U⊗P)(η(k)-x(k))+xT(t-h(k))(G⊗Γ)T(U⊗P)(G⊗Γ)x(t-h(k))︸Θ1+σT(k)(U⊗P)σ(k)︸Ω1+2(η(k)-x(k))T(U⊗P)x(k)},𝔼{ΔV2(k)}=𝔼{(U⊗(Q1-Q2))xT(k)(U⊗Q1)x(k)-xT(k-hm)(U⊗(Q1-Q2))x(k-hm)-xT(k-hM)(U⊗Q2)x(k-hM)(U⊗(Q1-Q2))},𝔼{ΔV3(k)}=𝔼{∑s=k-hMk-hm-1ΔxT(s)(U⊗R2)Δx(s)ΔxT(k)(U⊗(hm2R1+(hM-hm)2R2))Δx(k)-hm∑s=k-hmk-1ΔxT(s)(U⊗R1)Δx(s)-(hM-hm)∑s=k-hMk-hm-1ΔxT(s)(U⊗R2)Δx(s)}=𝔼{∑s=k-hMk-hm-1ΔxT(s)(U⊗R2)Δx(s)(η(k)-x(k))T(U⊗(hm2R1+(hM-hm)2R2))×(η(k)-x(k))+(xT(t-h(k))(G⊗Γ)T×(U⊗(hm2R1+(hM-hm)2R2))×(G⊗Γ)x(t-h(k)))︸Θ2+σT(k)(U⊗(hm2R1+(hM-hm)2R2))σ(k)︸Ω2-hm∑s=k-hmk-1ΔxT(s)(U⊗R1)Δx(s)-(hM-hm)∑s=k-hMk-hm-1ΔxT(s)(U⊗R2)Δx(s)},𝔼{ΔV4(k)}=𝔼{∑s=k-τk-1ΔxT(s)(U⊗S2)Δx(s)xT(k)(U⊗S1)x(k)-xT(k-τ)(U⊗S1)x(k-τ)+ΔxT(k)(U⊗τ2S2)Δx(k)-τ∑s=k-τk-1ΔxT(s)(U⊗S2)Δx(s)}=𝔼{∑s=k-τk-1ΔxT(s)(U⊗S2)Δx(s)xT(k)(U⊗S1)x(k)-xT(k-τ)(U⊗S1)x(k-τ)+(η(k)-x(k))T(U⊗τ2S2)(η(k)-x(k))+xT(t-h(k))(G⊗Γ)T(U⊗τ2S2)(G⊗Γ)x(t-h(k))︸Θ3+σT(k)(U⊗τ2S2)σ(k)︸Ω3-τ∑s=k-τk-1ΔxT(s)(U⊗S2)Δx(s)}.
By Lemmas 4 and 5, the sum terms of 𝔼{ΔV3(k)} are bounded as follows:
(28)-hm∑s=k-hmk-1ΔxT(s)(U⊗R1)Δx(s)≤-(∑s=k-hmk-1Δx(s))T(U⊗R1)(∑s=k-hmk-1Δx(s))=-∑1≤i<j≤NζijT(k)(e1T-e3T)TR1(e1T-e3T)ζij(k),-(hM-hm)∑s=k-hMk-hm-1ΔxT(s)(U⊗R2)Δx(s)(29)≤-[∑s=k-hMk-h(k)-1Δx(s)∑s=k-h(k)k-hm-1Δx(s)]T[1αk(U⊗R2)0Nn0Nn11-αk(U⊗R2)]×[∑s=k-hMk-h(k)-1Δx(s)∑s=k-h(k)k-hm-1Δx(s)]=-∑1≤i<j≤NζijT(k)[e4T-e5Te3T-e4T]T×[1αkR20n0n11-αkR2][e4T-e5Te3T-e4T]ζij(k),
where αk=(hM-h(k))(hM-hm)-1, which satisfies 0<αk<1.
Also, by Theorem7 in [32], the following inequality for any matrix T holds
(30)[1-αkαkIn0n0n-αk1-αkIn][R2T⋆R2]×[1-αkαkIn0n0n-αk1-αkIn]≥0,
which implies
(31)[1αkR20n0n11-αkR2]≥[R2T⋆R2],
then, an upper bound of the sum term (29) of 𝔼{ΔV3(k)} can be rebounded as
(32)-(hM-hm)∑s=k-hMk-hm-1ΔxT(s)(U⊗R2)Δx(s)≤-∑1≤i<j≤NζijT(k)[e4T-e5Te3T-e4T]T[R2T⋆R2]×[e4T-e5Te3T-e4T]ζij(k).
Similarly, the sum term of 𝔼{ΔV4(k)} is bounded as
(33)-τ∑s=k-τk-1ΔxT(s)(U⊗S2)Δx(s)≤-∑1≤i<j≤NζijT(k)(e1-e2)S2(e1-e2)Tζij(k).
Also, by properties of Kronecker product in Lemma 2 and UG=GU=NG, the terms Θq(q=1,2,3) in (27) are calculated as follows:
(34)∑l=13Θl=xT(t-h(k))(G⊗Γ)T(U⊗Σ)(G⊗Γ)x(t-h(k))=xT(t-h(k))(NGTG⊗ΓTΣΓ)x(t-h(k)),
where Σ is defined in (19), and, if Σ≤ρIn, then, from (11), the upper bound of terms Ωq(q=1,2,3) in (27) is calculated as follows:
(35)∑l=13Ωl=σT(k)(U⊗Σ)σ(k)≤ρ{(t-h(k))xT(k)(U⊗H1TH1)x(k)+xT(t-h(k))(U⊗H2TH2)x(t-h(k))}.
Then, by utilizing Lemma 4, an upper bound of 𝔼{ΔV(k)=∑l=14‍ΔVl(k)} can be written as follows:
(36)𝔼{ΔV(k)}≤𝔼{∑1≤i<j≤NζijT(k)(∑l=15Ξl)ζij(k)}.
From (7), for any positive diagonal matrices Dq(q=1,2), the following inequalities hold. 							(37)0≤∑1≤i<j≤NζijT(k)Ξ6ζij(k).
Since the relational expression between p(k) and q(k), pT(k)p(k)≤qT(k)q(k), holds from the second equality of the system (14), there exists a positive scalar ϵ satisfying the following inequality:
(38)0≤∑1≤i<j≤NζijT(k)(ϵΨTΨ+Ξ7)ζij(k).
From (36)–(38), by S-procedure [25], the 𝔼{ΔV(k)} has a new upper bound as follows:
(39)𝔼{ΔV(k)}≤𝔼{∑1≤i<j≤NζijT(k)(∑l=17Ξl+ϵΨTΨ)ζij(k)}.
Also, the network (14) with the augmented matrix ζij(k) can be rewritten as follows:
(40)𝔼{∑1≤i<j≤N(j-i)Υijζij(k)}=0n×1.
Here, in order to illustrate the process of obtaining (40), let us define the following:
(41)Λ=[Λ1,Λ2,…,ΛN]=[N,N-1,…,1]⊗In∈ℝn×Nn.
By (14), (23), and properties of Kronecker product in Lemma 2, we have the following zero equality:
(42)0n×1=𝔼{(x(k-h(k)))Λ(U⊗A)x(k-τ)+Λ(NG⊗Γ)x(k-h(k))-Λ(U⊗In)(η(k)-x(k))+Λ(U⊗W1)f(x(k))+Λ(U⊗W2)f(x(k-h(k)))+Λ(U⊗D)p(k)}.
By Lemma 4, the first term of (42) can be obtained as follows:
(43)Λ(U⊗A)x(k-τ)=[NIn,…,In]︸n×Nn(U⊗A)︸Nn×Nn[x1(k-τ),…,xN(k-τ)]T︸Nn×1=-∑1≤i<j≤Nuij(Λi-Λj)A(xi(k-τ)-xj(k-τ))=∑1≤i<j≤N(Λi-Λj)Azij(k-τ)=∑1≤i<j≤N((N+1-i)In-(N+1-j)In)Azij(k-τ)=∑1≤i<j≤N(j-i)Azij(k-τ).
Similarly, the other terms of (42) are calculated as follows:
(44)Λ(NG⊗Γ)x(k-h(k))=-∑1≤i<j≤NNgij(Λi-Λj)Γ×(xi(t-h(k))-xj(t-h(k)))=-∑1≤i<j≤N(j-i)(NgijΓ)zij(k-h(k)),-Λ(U⊗In)(η(k)-x(k))=∑1≤i<j≤Nuij(Λi-Λj)In×((ηi(k)-xi(k))-(ηj(k)-xj(k)))=-∑1≤i<j≤N(j-i)(ηij(k)-zij(k)),Λ(U⊗W1)f(x(k))=-∑1≤i<j≤Nuij(Λi-Λj)W1(f(xi(k))-f(xj(k)))=∑1≤i<j≤N(j-i)W1f(zij(k)),Λ(U⊗W2)f(x(k-h(k)))=-∑1≤i<j≤Nuij(Λi-Λj)W2×(f(xi(t-h(k)))-f(xj(t-h(k))))=∑1≤i<j≤N(j-i)W2f(zij(k-h(k))),Λ(U⊗D)p(k)=-∑1≤i<j≤Nuij(Λi-Λj)D(pi(k)-pj(k))=∑1≤i<j≤N(j-i)Dpij(k).
Then, (42) can be rewritten as follows:
(45)0n×1=𝔼{∑1≤i<j≤N(j-i)×[-In,A,0n,-(NgijΓ),0n,-In,W1,W2,D]︸Υij×ζij(k)∑1≤i<j≤N(j-i)}.
Therefore, if the zero equality (40) holds, then a synchronization condition for the network (14) is
(46)𝔼{∑1≤i<j≤NζijT(k)(∑l=17Ξl+ϵΨTΨ)ζij(k)}<0
subject to
(47)𝔼{∑1≤i<j≤N(j-i)Υijζij(k)}=0n×1.
Here, if inequality (47) holds, then there exists a positive scalar ε such that ∑l=17‍Ξl+ϵΨTΨ<-εI9n. From (39) and (47), we have 𝔼{ΔV(k)}≤𝔼{-ε∑1≤i<j≤N‍∥xi(k)-xj(k)∥2}. Thus, by Lyapunov theorem and Definition 3, it can be guaranteed that the subnetworks in the coupled discrete-time neural networks (14) are asymptotically synchronized. Also, condition (47) is equivalent to the following inequality:
(48)∑1≤i<j≤NζijT(k)(∑l=17‍Ξl+ϵΨTΨ)ζij(k)<0
subject to
(49)∑1≤i<j≤N(j-i)Υijζij(k)=0n×1.
Finally, by the use of Lemma 6, condition (49) is equivalent to the following inequality:
(50)∑1≤i<j≤N[(j-i)Υij]⊥T(∑l=17Ξl+ϵΨTΨ)[(j-i)Υij]⊥<0,
and applying Schur complement [25] leads to
(51)∑1≤i<j≤N[[(j-i)Υij]⊥T(∑l=17Ξl)[(j-i)Υij]⊥⋆ϵΨ[(j-i)Υij]⊥-ϵIn]<0,
which can be rewritten by
(52)∑1≤i<j≤N[[(j-i)Υij]⊥09n×n0n×8nIn]T[∑l=17ΞlϵΨT⋆-ϵIn]×[[(j-i)Υij]⊥09n×n0n×8nIn]<0.
From inequality (52), if the LMIs (22)  are satisfied, then stability condition (47) holds. This completes our proof.Remark 8.
In order to induce a new zero equality (40), the matrix Λ in (41) was defined. It is inspired by the concept of scaling transformation matrix. To reduce the decision variable, Finsler’s lemma (ii) Υ⊥TΦΥ⊥<0 without free-weighting matrices was used. At this time, a zero equality is required. If the matrix Λ is not considered, then the following description (see only (43) as an example)
(53){}(U⊗A)x(k-τ)={}(U⊗A)[x1(k-τ),…,xN(k-τ)]T=∑1≤i<j≤N{·}A(xi(k-τ)-xj(k-τ))
as shown in (53) does not hold. Thus, the derivation of zero equality in (40) is impossible. Here, to use Lemma 4, a suitable vector or matrix in the empty parentheses {} is needed. Therefore, by defining the matrix Λ, the induction of the zero equality (40) is possible.Remark 9.
In this paper, the problem of new delay-dependent synchronization for coupled stochastic discrete-time neural networks with leakage delay and parameter uncertainties is considered. By using Finsler's lemma without free-weighting matrices, the proposed robust synchronization criterion for the network is established in terms of LMIs. Here, as mentioned in the Introduction, the leakage delay is the time delay in leakage or forgetting term of the systems and a considerable factor affecting dynamics for the worse in the network. The effect of the leakage delay which cannot be negligible is shown in Figure2. Also, the stochastic discrete-time systems with parameter uncertainties do not formulate like as the network (14) in any other literature. To do this, the vector (η(k)-x(k)) is added in the augmented vector ζ(k). It is just like as x˙(t) in continuous-time systems. This form for the systems may give more less conservative results for stability analysis. As a case of stochastic continuous-time systems with parameter uncertainties, Kwon [13] derived the delay-dependent stability criteria for uncertain stochastic dynamic systems with time-varying delays via the Lyapunov-Krasovskii's functional approach with two delay fraction numbers.


## 4. Numerical Examples

In this section, we provide two numerical examples to illustrate the effectiveness of the proposed synchronization criterion in this paper.Example 10.
Consider the following coupled neural networks by complex model in Figure1:
(54)y~i(k+1)=(A+ΔA)y~i(k-τ)+(W1+ΔW1)g~(y~i(k))+(W2+ΔW2)g~(y~i(k-h(k)))+∑j=15gijΓy~j(k-h(k))(1+ω1(k))+σi(k,y~i(k),y~i(k-h(k)))ω2(k),
with g~(x)=0.5tanh(x), where
(55)A=[0.2000.3],W1=[0.001000.005],W2=[-0.10.01-0.2-0.1],Γ=0.01I2,G=[-210011-311001-210011-311001-2],Lm=02,Lp=0.5I2,D=0.1I2,Ea=[0.300-0.1],E1=[-0.400.3-0.7],E2=E1,H1=0.2I2,H2=H1.
For the network above, the maximum allowable delay bounds with differenthm and fixed τ=3 by Theorem 7 are listed in Table 1. In order to confirm the obtained results with the conditions of the time delays as listed in Table 2, the simulation results for the trajectories of state responses, xi(k)  (i=2,3,4,5), and synchronization errors, zi1(k)=xi(k)-x1(k), of the network (54) are shown in Figures 2, 3, 4, and 5. These figures show that the network with the errors converge to zero for given initial values of the state by x1T(0)=[1,-3], x2T(0)=[-1,2], x3T(0)=[4,-5], x4T(0)=[3,-1], and x5T(0)=[4,2]. Specially, the simulation results in Figure 2 show state response trajectories for the values of leakage delay, τ, by 3, 15, and 30 with fixed values hm=5 and hM=7. It is easy to illustrate that the larger value of leakage delay gives the worse dynamic behaviors of the network (54).Table 1
Maximum allowable delay bounds,hM, with different hm and fixed τ=3 (Example 10).
h
m
1
5
10
50
100
h
M
3
7
12
52
102Table 2
The conditions of simulation in Example10.
Number
τ
h
m
h
M
h
(
k
)
3
C1-1
15
5
7
sin
(
k
π
/
2
)
+
6
30
C1-2
3
50
52
sin
(
k
π
/
2
)
+
51Figure 1
The structure of complex networks withN=5 (Example 10).State responses with C1-1 (Example10): (a) τ=3, (b) τ=15, and (c) τ=30.
(a)
(b)
(c)Synchronization errors trajectories with C1-1 (τ=3) (Example 10).
(a)
(b)
(c)
(d)Figure 4
State responses with C1-2 (Example10).Synchronization errors trajectories with C1-2 (Example10).
(a)
(b)
(c)
(d)Example 11.
Consider the following coupled neural networks by BA scale-free model [33] in Figure 6:
(56)y~i(k+1)=(A+ΔA)y~i(k-τ)+(W1+ΔW1)g~(y~i(k))+(W2+ΔW2)g~(y~i(k-h(k)))+∑j=150gijΓy~j(k-h(k))(1+ω1(k))+σi(k,y~i(k),y~i(k-h(k)))ω2(k),
with g~(x)=0.1tanh(x), where
(57)A=[0.01000.02],W1=[0.2-0.10.3-0.2],W2=[0.30.1-0.30.2],Γ=0.001I2,Lm=02,Lp=0.1I2,D=0.1I2,Ea=[0.7-0.200.4],E1=[0.2-0.500.3],E2=E1,H1=0.2I2,H2=H1.
The results of maximum allowable delay bounds with differenthm and fixed τ=3 by Theorem 7 are listed in Table 3. For lack of space, the outer-coupling matrix G is omitted. It is easy that the matrix G  was expressed from Figure 6. Figures 7 and 8 show the state response trajectories, xi(t)  (i=1,…,50), of the network (56) with the condition of the time delays as listed in Table 4 for random initial values of the state. These figures show that the network (56) with the state responses converge to zero. This means the synchronization stability of the network (56).Table 3
Maximum allowable delay bounds,hM, with different hm and fixed τ=5 (Example 11).
h
m
1
5
10
25
30
h
M
5
9
14
29
34Table 4
The conditions of simulation in Example11.
Number
h
m
h
M
h
(
k
)
C2-1
5
9
Random integer variable with5≤h(k)≤9
C2-2
30
34
Random integer variable with30≤h(k)≤34Figure 6
The structure of BA scale-free networks withN=50 (Example 11).State responses and time-delayh(k) with C2-1 (Example 11).
(a)
(b)State responses and time-delayh(k) with C2-2 (Example 11).
(a)
(b)


## 5. Conclusions

In this paper, the delay-dependent robust synchronization criterion for the coupled stochastic discrete-time neural networks with interval time-varying delays in network coupling, leakage delay, and parameter uncertainties has been proposed. To do this, the suitable Lyapunov-Krasovskii’s functional was used to investigate the feasible region of stability criterion. By utilization of Finsler’s lemma with a new zero equality, a sufficient condition for guaranteeing asymptotic synchronization for the concerned networks has been derived in terms of LMIs. Two numerical examples have been given to show the effectiveness and usefulness of the presented criterion.



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*Source: 814692-2013-02-28.xml*