# Stability of Stochastic Discrete-Time Neural Networks with Discrete Delays and the Leakage Delay

Authors: Liyuan Hou; Hong Zhu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper investigates the stability of stochastic discrete-time neural networks (NNs) with discrete
time-varying delays and leakage delay. As the partition of time-varying and leakage delay is brought in the discrete-time system, we construct a novel Lyapunov-Krasovskii function based on stability theory. Furthermore sufficient conditions are derived to guarantee the global asymptotic stability of the equilibrium point. Numerical example is given to demonstrate the effectiveness of the proposed method and the applicability of the proposed method.


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## Body

## 1. Introduction

Neural networks (NNs) have received much interest owing to their wide application in the areas of signal processing, pattern recognition, and static image processing in [1, 2]. One of the most important and challenging questions in theoretical analysis of neural networks (NNs) is dynamical behaviors of the neural networks, for example, stability, instability, periodic oscillatory, and chaos. Among them analysis of stability has received much attention and various stability conditions have been obtained in [3–10]. The main problem of stability analysis is how to construct appropriate Lyapunov functions which are widely used in various fields [11–18]. Then the construction of Lyapunov functions is determined by the structure of neural networks. The study of neural networks generally considers the following factors.It is well known that time delay is inherent in various systems, including artificial neural networks, owing to the finite speed of signal transmission [3]. Delays in a system may cause oscillation and divergence and may degrade the performance. Hence, stability analysis of systems with time delay is widespread. Scholars classify the analysis of stability into delay-independent and delay-dependent ones. It has been proved in [4, 5] that the delay-dependent criterion is less conservative than the delay-independent one, especially for smaller values of delay. Delay-dependent stability condition for continuous-time NNs with time-varying delays has been reported in literatures [4–10]. The approaches to handle the time-varying delay in most literatures are based on introducing free weighting matrices [19], model transformation method [20], and linear matrix inequality (LMI) approach and employing the delay partitioning approach in [21, 22].In the digital life, most of the signals including the continuous-time NNs need to be processed, experimentalized, or computed by the computer, such that we must discretize the continuous-time signals before delivering them to the computer. Therefore, the stability of discrete-time neural networks is necessary, and more and more literature about it was published [22–28].As pointed out in [29], discrete-time delays were introduced into bidirectional associative memory (BAM) neural networks which were known as leakage (or forgetting) terms. From literatures [30–36] it can be found that the leakage terms had a tendency to destabilize the neural networks. In the literatures [30, 31], the system with leakage delays had been studied. However, different from [29], the leakage delays changed to be time varying. Not only that, different kinds of neural networks with time delays in the leakage terms were studied, especially the effect of leakage term on the dynamical behavior of various kinds of neural networks [32–42]. In [33], passivity analysis of neural networks with time-varying delays and leakage delay was considered. And they used the free-weighting matrix method and stochastic analysis technique. Furthermore, Li and others in [34, 35] researched stability of various differential systems including neural networks in leakage terms, by using contraction mapping theorem, Brouwers fixed point theorem, Lyapunov-Krasovskii functional method, and free weighting matrix technique. Unfortunately, neural networks with the leakage terms in most literatures are continuous systems, and there is not any thesis about discrete neural networks with the leakage delay. On the other hand, it has now been well recognized that stochastic disturbances are mostly inevitable owing to noise in electronic implementations. It has also been revealed that certain stochastic inputs could make a neural network unstable.In this paper, the stability problem is considered for discrete-time neural networks with time-varying delays in the leakage terms. Firstly, the mathematical models are established. Secondly, a less conservative and new stability criterion is derived by using a novel Lyapunov-Krasovskii functional which depends on the circumstance of the delay partition. Thirdly, a numerical example is provided to show the effectiveness of the main result.Notation. Throughout this paper, Rn denotes the n-dimensional Euclidean space, and Rn×m denotes the set of all real matrices. The superscript T denotes matrix transposition and the notation X≥Y (X>Y, resp.), where X and Y are symmetric matrices, means that X-Y is positive semidefinite (positive definite, resp.). In symmetric block matrices, the symbol * is used as an ellipsis for terms induced by symmetry. · stands for the Euclidean vector norm in Rn. E{x} and E{x∣y} denote the expectation of x and the expectation of x conditional on y. (Ω,F,P) is a probability space, where Ω is the sample space, F is the σ-algebra of subsets of the sample space, and P is the probability measure on F.


## 2. Preliminaries

Discretizing the continuous-time NNs with leakage terms [39], we will consider the following discrete-time systems with time-varying delays in leakage terms:(1)xk+1=xk-Cxk-σ+AFxk+BGxk-τk,where x(k)=[x1(k),x2(k),…,xn(k)]T is the state vector at time k; C=diag[c1,c2,…,cn] with C>0 is the state feedback coefficient matrix; the n×n matrices A=[aij]n×n and B=[bij]n×n are the connection weight matrix and the discretely delayed connection weight matrix, respectively; F(x(k)), G(x(k)) are the neuron activation functions, which satisfy F(x(k))=[F1(x1(k)),F2(x2(k)),…,Fn(xn(k))]T, G(x(k))=[G1(x1(k)),G2(x2(k)),…,Gn(xn(k))]T; τ(k) denotes the discrete time-varying delay; σ means the leakage delays.Assumption 1.
For anyx,y∈R, (x≠y), i∈{1,2,…,n}, the activation functions satisfy(2)f^i-≤Fix-Fiyx-y≤f^i+;g^i-≤Gix-Giyx-y≤g^i+,where f^i-, f^i+, g^i-, and g^i+ are constants.Remark 2.
The condition on the activation functions in Assumption1 was originally employed in [6, 7] and has been subsequently used in recent papers with the problem of stability of neural networks; see [21–28].The system (1) has equilibrium points under Assumption 1. Let x* be the equilibrium point of system (1), and shift it to the origin by letting y(k)=x(k)-x*, and then system (1) with stochastic disturbances can be rewritten as(3)yk+1=yk-Cyk-σ+Afyk+Bgyk-τk+δk,ykωk,where y(k)=[y1(k),y2(k),…,yn(k)]T is the state vector of the transformed system, y(k-σ)=x(k-σ)-x*, and the transformed neuron activation functions are f(y(k))=F(x(k))-F(x*), g(y(k))=G(x(k))-G(x*), and ω(k) is a scalar Wiener process on a probability space (Ω,F,P) with E{ω(k)}=0, E{ω2(k)}=1, Eωiωj=0  (i≠j). Then f^i-≤(fi(x)-fi(y))/(x-y)≤f^i+; g^i-≤(gi(x)-gi(y))/(x-y)≤g^i+ can be verified from Remark 2.Assumption 3.
The noise intensity function vectorδ(·,·):N×Nn→Nn satisfies the Lipschitz condition; that is, the nonlinear function δ(k,x) satisfies the following inequality:(4)δk,xTδk,x≤ξxTx,where ξ is a known scalar constant.Assumption 4.
The time-varying delayτ(k) is bounded, 0<τm≤τ(k)≤τM, and its probability distribution can be observed. Assume that integer τ(k) is satisfied in τi≤τ(k)≤τi+1, di=τi-τi-1≠0, i=1,2,…,l, which means we divide [τm,τM] into l parts, and Prob{τ(k)∈[τi-1,τi)}=ρi=1-ρ~i, where 0≤ρi≤1, ∑i‍ρi=1, i=1,…,l, and τ0=τm, τl=τM.
To describe the probability distribution of time-varying delay, we define the following setAi=(τi-1,τi], i=1,2,…,l. Define mapping functions(5)τik=τk,τk∈Aiτi-1,else,ρik=1,τk∈Ai0,else.Remark 5.
ConsiderProb{ρi(k)=1}=E{ρi(k)}=ρi, Prob{ρi(k)=0}=ρ~i, E{ρi(k)*ρj(k)}=ρi,i=j0,else.Remark 6.
In the literature [21], the delay-partitioning projection technique is used. In [21], they partition τ(t) into several components, that is, τ(t)=∑i=1m‍τi(t), where m is a positive integer, in order to derive some less restrictive stability criteria. Different from the literature [21], we use the knowledge of probability to describe the partition of time delay. This way has the advantage of reducing the amount of calculation.
Then the system (3) can be rewritten as(6)yk+1=y(k)-Cy(k-σ)+Afyk+∑i=1l‍ρikBgyk-τik+δk,ykωk.


## 3. New Stability Criteria

In this section, we will establish new stability criteria for system (1). The following lemma is needed in order to derive our main results.Lemma 7 (Zhu and Yang [23] discrete Jenson inequality).
For any constant matrixM∈Rn×n, M=MT>0, integers γ2≥γ1, vector function ω:[γ1,γ1+1,…,γ2]→Rn such that the sums in the following are well defined, and then(7)-γ2-γ1+1∑i=γ1γ2‍ωTiMωi≤-∑i=γ1γ2‍ωiTM∑i=γ1γ2‍ωi.The next theorem can be obtained if the stochastic termω(k) is removed in the system (6).Theorem 8.
Under Assumptions1 and 4, the system (6) with ω(k)=0 is globally asymptotically stable, if there exist positive matrices P, R1, R2, R3, Q, Si (i=1,…,l), W=diag[w1,w2,…,wl], U=diag[u1,u2,…,ul], and free matrix T1,T2,…,Tl+1, V1, V2, V3 with any appropriate dimensions, such that the following LMI holds:(8)Γ1Γ1,20Γ1,4Γ1,5Γ1,60Γ1,8Γ1,900*Γ200000000Γ2,11**Γ300Γ3,600000***Γ40Γ4,6000Γ4,100****Γ5000000*****Γ6000Γ6,100******Γ70000*******Γ8Γ8,9Γ8,100********Γ900*********-Γ100**********Γ11≤0,Γ1=CTPC+σ2R3-R1-UF1-WG1-V1T-V1,Γ1,2=R1,Γ1,4=-CTPA+UF2,Γ1,5=WG2,Γ1,6=-ρ1CTPB,…,-ρlCTPB,Γ1,8=V1-V2T,Γ1,9=V1-V3T,Γ2=-2R1R1R1-2R1⋱⋱⋱⋱⋱-2R1R1R1-R1-0WG1⋱WG10+∑i=1l+1‍Φi+ΦiT,Γ2,6=diag0,WG2,…,WG2,0,Γ2,11=diagT1,T2,…,Tl+1,Γ3=-R3,Γ4=ATPA-U,Γ4,6=ρ1ATPB,…,ρlATPB,Γ4,10=ATΓ10,Γ5=∑i=1l‍τi-τi-1+1Si-W,Γ6=Ψ1-diagW+S1,…,W+Sl,Ψ1=diagρ1BTPB,…,ρlBTPB,Γ6,10=ρ1Γ10TB,ρ2Γ10TB,…,ρlΓ10TBT,Γ7=-P,Γ8=V2+V2T,Γ8,9=V2+V3T,Γ9=V3+V3T-1σQ,Γ10=τM-τmhR1+τM-τmR2+σQ,Γ11=-1hdiagR2,R2,…,R2,Φi=TiI~i,F1=diagf^1-f^1+,f^2-f^2+,…,f^n-f^n+,F2=diagf^1-+f^1+2,f^2-+f^2+2,…,f^n-+f^n+2,G1=diagg^1-g^1+,g^2-g^2+,…,g^n-g^n+,G2=diagg^1-+g^1+2,g^2-+g^2+2,…,g^n-+g^n+2,h=max⁡i=2,3,…,lτi-τi-2,where I~i∈Rn×(l+2)n, (i=1,2,…,l,l+1) denotes a matrix with the ith element In×n while the i+1th element is -In×n, and other elements are zero matrixes. For example, Φ2=[0I-I0⋯0].Proof.
For convenience, we denoteΔ(y(k))=y(k+1)-y(k); that is, Δ{y(k)-C∑i=k-σk-1‍y(i)}=y(k+1)-y(k)-Cy(k)+Cy(k-σ), and then systems (6) with ω(k)=0 can be rewritten as follows:(9)ζk+1=Δyk-C∑i=k-σk-1‍yi=-Cy(k)+Afyk+∑i=1l‍ρikBgyk-τik.
We construct a new Lyapunov-Krasovskii functionalV(yk,k) as(10)Vyk,k=∑i=16‍Viyk,k,where(11)V1yk,k=ζTkPζk,V2yk,k=h∑i=-τM-1‍∑j=k+ik-1‍ηTjR1ηj,V3yk,k=∑i=-τM-τm-1‍∑j=k+ik-1‍ηTjR2ηj,V4yk,k=σ∑i=-σ-1‍∑j=k+ik-1‍yTjR3yj,V5yk,k=∑i=1l‍∑j=k-τikk-1‍gTyjSigyj+∑i=1l‍∑j=τi-1τi-1‍∑m=k-jk-1‍gTymSigym,V6yk,k=∑i=-σ-1‍∑j=k+ik-1‍ηTjQηj,ηk=yk+1-yk.
Taking the difference of the functional along the solution of (9), we obtain(12)EΔVyk,k=∑i=16‍EΔViyk,k,(13)EΔV1yk,k=EV1yk+1,k+1-V1yk,k=EζTk+1Pζk+1-ζTkPζk=EyTkCTPCyk-2yTkCTPAfyk∑i=1l‍ρikBgyk-τikTz-2∑i=1l‍ρikyTkCTPBgyk-τikz+fTykATPAfykz+2∑i=1l‍ρikfTykATPBgyk-τikz+∑i=1l‍ρikBgyk-τikTz·P∑j=1l‍ρjkBgyk-τjkz-∑i=1l‍ρikBgyk-τikTζTkPζk.According to Remark 5, it is easy to get(14)EΔV1yk,k=yTkCTPCyk-2yTkCTPAfyk-2∑i=1l‍ρiyTkCTPBgyk-τik+fTykATPAfyk+2∑i=1l‍ρifTykATPBgyk-τik+α2TkΨ1α2k-ζTkPζk,where α2k=[gT(y(k-τ1(k)))⋯gT(y(k-τl(k)))]T.
Consider(15)EΔV2yk,k=EV2yk+1,k+1-V2yk,k=Eh∑i=-τM-1‍∑j=k+i+1k‍ηTjR1ηj-h∑i=-τM-1‍∑j=k+ik-1‍ηTjR1ηj=EhτMηTkR1ηk-h∑i=-τM-1‍ηTk+iR1ηk+i,EΔV3yk,k=EV3yk+1,k+1-V3yk,k=E∑i=-τM-τm-1‍∑j=k+i+1k‍ηTjR2ηj-∑i=-τM-τm-1‍∑j=k+ik-1‍ηTjR2ηj=EτM-τmηTkR2ηk∑i=-τM-τm-1‍ηTk+iR2ηk+i-∑i=-τM-τm-1‍ηTk+iR2ηk+i,EΔV4yk,k=EEV4yk+1,k+1∣yk-V4yk,k=Eσ∑i=-σ-1‍∑j=k+i+1k‍yTjR3yj-σ∑i=-σ-1‍∑j=k+ik-1‍yTjR3yj=σ2yTkR3yk-σ∑i=k-σk-1‍yTiR3yi,ΔV5yk,k=V5yk+1,k+1-V5yk,k=∑i=1l‍∑j=k+1-τik+1k‍gTyjSigyj+∑i=1l‍∑j=τi-1τi-1‍∑m=k-j+1k‍gTymSigym-∑i=1l‍∑j=k-τikk-1‍gTyjSigyj+∑i=1l‍∑j=τi-1τi-1‍∑m=k-jk-1‍gTymSigym=∑i=1l‍gTykSigykgTyk-τikSigyk-τik-gTyk-τikSigyk-τik+∑j=k-τik+1+1k-1‍gTyjSigyj-∑j=k-τik+1k-1‍gTyjSigyj+∑j=τi-1τi-1‍gTykSigyk-gTyk-jSigyk-j≤∑i=1l‍τi-τi-1+1gTykSigyk-gTyk-τikSigyk-τik,ΔV6yk,k=V6(yk+1,k+1)-V6yk,k=∑i=-σ-1‍∑j=k+1+ik‍ηTjQηj-∑i=-σ-1‍∑j=k+ik-1‍ηTjQηj=∑i=-σ-1‍ηTkQηk+∑j=k+1+ik-1‍ηTjQηj-∑j=k+1+ik-1‍ηTjQηj-ηTk+iQηk+i=σηTkQηk-∑i=-σ-1‍ηTk+iQηk+i.
Using the algebraic expression-h∑i=-τM-1‍ηT(k+i)R1η(k+i) in ΔV2(yk,k) and by using Lemma 7, we can get(16)-h∑i=-τM-1‍ηTk+iR1ηk+i=-h∑i=k-τmk-1‍ηTiR1ηi-h∑i=k-τ1kk-τm-1‍ηTiR1ηi-h∑i=k-τ2kk-τ1k-1‍ηTiR1ηi-⋯-h∑i=k-τMk-τlk-1‍ηTiR1ηi≤-∑i=k-τmk-1‍ηiTR1∑i=k-τmk-1‍ηi-∑i=k-τ1kk-τm-1‍ηiTR1∑i=k-τ1kk-τm-1‍ηi-∑i=k-τ2kk-τ1k-1‍ηiTR1∑i=k-τ2kk-τ1k-1‍-⋯-∑i=k-τMk-τlk-1‍ηiTR1∑i=k-τMk-τlk-1‍ηi=-yk-yk-τmTR1yk-yk-τm-yk-τm-yk-τ1kT·R1yk-τm-yk-τ1k-yk-τ1k-yk-τ2kT·R1yk-τ1k-yk-τ2k-⋯-yk-τlk-yk-τMT·R1yk-τlk-yk-τM.
Letα1(k)=(yT(k-τm), yTk-τ1k, yTk-τ2k,…,yT(k-τl(k)), yT(k-τM))T; then(17)-h∑i=k-τMk-1‍ηTiR1ηi≤yT(k)α1T(k)·-R1R10⋯0*-2R1R1⋯0⋮⋮⋱⋮⋮***-2R1R1****-R1ykα1k≜yTkα1TkΨ2ykα1k.
As is well known, for any vectorsx, y and any symmetric matrix Z>0, the following inequality holds:(18)-2xTy≤xTZ-1x+yTZy.Denoting x=Tl+1Tα1(k), y=∑j=k-τMk-τl(k)-1‍η(j), Z=R2/(τl-τl(k)), where Tl+1 is free matrix, then the last inequality can be expressed as follows:(19)-2α1TkTl+1∑i=k-τMk-τlk-1‍ηi≤α1TkTl+1Z-1Tl+1α1Tk+τl-τlk·∑i=k-τMk-τlk-1‍ηiTR2∑i=k-τMk-τlk-1‍ηi≤α1TkTl+1Z-1Tl+1Tα1Tk+∑i=k-τMk-τlk-1‍ηiTR2ηiwhich is equal to(20)-∑i=k-τMk-τlk-1‍ηiTR2ηi≤τl-τlkα1TkTl+1R2-1Tl+1Tα1Tk+2α1TkTl+1∑j=k-τMk-τlk-1‍ηj.Similarly, we can get(21)-∑j=k-τi+1kk-τik-1‍ηjTR2ηj≤τi+1k-τikα1TkTi+1-1R2Ti+1Tα1Tk+2α1TkTi+1∑j=k-τi+1kk-τik-1‍ηj.Combining those inequalities, we obtain(22)-∑i=-τM-τm-1‍ηTk+iR2ηk+i=-∑i=k-τ1kk-τm-1‍ηTiR2ηi-∑i=k-τ2kk-τ1k-1‍ηTiR2ηi-⋯-∑i=k-τMk-τlk-1‍ηTiR2ηi≤τl-τlkα1TkTl+1R2-1Tl+1Tα1Tk+2α1TkTl+1∑j=k-τMk-τlk-1‍ηj+∑i=1l-1‍τi+1k-τikα1TkTi+1R2-1Ti+1Tα1Tk∑j=k-τi+1kk-τik-1‍+2α1TkTi+1∑j=k-τi+1kk-τik-1‍ηj+τ1k-τmα1TkT1R2-1T1Tα1Tk+2α1TkT1∑j=k-τ1kk-τm-1‍ηj.
According to the equation in Assumption1, we have(23)fyk-f^i-ykfyk-f^i+yk≤0,gyk-τik-g^i-yk-τik·gyk-τik-g^i+yk-τik≤0.It can be deduced that there exist U=diag[u1,u2,…,un]>0 and W=diag[w1,w2,…,wn]>0, such that(24)∑i=1n‍uiykfykTf^i-f^i+eieiT-f^i-+f^i+2eieiT-f^i-+f^i+2eieiTeieiT·ykfyk=ykfykTUF1-UF2-UF2Uykfyk≤0,∑i=1n‍wiykgykTg^i-g^i+eieiT-g^i-+g^i+2eieiT-g^i-+g^i+2eieiTeieiT·ykgyk=ykgykTWG1-WG2-WG2Wykgyk≤0,∑i=1n‍wiyk-τjkgyk-τjkTg^i-g^i+eieiT-g^i-+g^i+2eieiT-g^i-+g^i+2eieiTeieiT·yk-τjkgyk-τjk=yk-τjkgyk-τjkTWG1-WG2-WG2W·yk-τjkgyk-τjk≤0,where ei denotes the unit column vector having one element on its rth row and zeros elsewhere. Derived from ∑i=k-σk-1‍η(i)=y(k)-y(k-σ), we can get the following equation for any matrices V=(V1T,V2T,V3T)T with appropriate dimensions:(25)μk=2yTkyTk-σ∑i=k-σk-1‍ηTi·V-yk+yk-σ+∑i=k-σk-1‍ηi=0.Combining (12)–(25), we obtain(26)EΔVyk,k≤EyTkCTPCyk-2yTkCTPAfyk∑i=-σ-1‍ηk+i+μkz-2∑i=1l‍ρiyTkCTPBgyk-τikz+fTykATPAfykz+2∑i=1l‍ρifTykATPBgyk-τikz+α2TkΨ1α2k-ζTkPζk-α1TkΨ2α1kz+∑i=1l‍τi-τi-1+1gTykSigyk-gTyk-τikSigyk-τikz+∑i=1l‍τi-τi-1+1yTkQiyk-yTk-τikQiyk-τikz+τl-τlkα1TkTl+1R2-1Tl+1Tα1kz+∑i=1l-1‍τi+1k-τikα1TkTi+1R2-1Ti+1Tα1kz+τ1k-τmα1TkT1R2-1T1Tα1kz+∑i=1l-1‍2α1TkTi+1yk-τik-yk-τi+1kz+2α1TkT1yk-τm-yk-τ1kz+2α1TkTl+1yk-τlk-yk-τMz+ηTkτM-τmhR1+τM-τmR2+σQηkz-1σ∑i=-σ-1‍ηTk+iQ∑i=-σ-1‍ηk+i+μk≤θTkΞ+ΦTΓ10Φ+τM-τlkT~l+1R1T~l+1T∑i=1l-1‍τi+1k-τik+∑i=1l-1‍τi+1k-τikT~i+1R1T~i+1T∑i=1l-1‍τi+1k-τik+τ1k-τmT~1R1T~1Tθk,(27)θk=yTk,α1Tk,∑i=k-σk-1‍yi,fTyk,gTyk,α2Tk,ζTk,yTk-σ,∑i=k-σk-1‍ηTiT,Ξ=Γ100Γ1,4Γ1,5Γ1,60Γ1,8Γ1,9*Γ2000Γ2,6000**Γ3000000***Γ40Γ4,6000****Γ50000*****Γ6000******Γ700*******Γ8Γ8,9********Γ9,Φ=0,0,0,A,0,ρ1B,ρ2B,…,ρlB,0,-C,0,T~i=0I0⋯0TTi.Then if(28)Ξ+ΦTΓ10Φ+τl-τlkT~l+1R1T~l+1T+∑i=1l-1‍τi+1k-τikT~i+1R1T~i+1T+τ1k-τmT~1R1T~1T≤0we can conclude that ∑k=1+∞‍E{∥yk∥2} is convergent and limk→+∞E{∥yk∥2}=0, which implies that the system (6) is globally asymptotically stable. Using Schur complement and the boundary condition τi(k)∈[τi-1,τi], we can get the inequality in this theorem.
This completes the proof of the theorem.Remark 9.
The stability analysis problem of a general class of discrete-time neural networks with leakage delays is dealt with in Theorem8. The stability condition can be expressed by a set of LMIs, which can be checked by resorting to MATLAB LMI Toolbox.Remark 10.
A similar study to this paper has been investigated in [21, 22, 33]. We note that our new stability criterion for stochastic discrete system benefits from the idea of delay partitioning. Based on the general assumption of time delay, we represent τ(k) as l parts, such that the condition is not only delay dependent but also dependent on the partitioning number.Remark 11.
From the proof of Theorem8, it can be easily found that we enlarged (τ1(k)-τm), τi+1(k)-τi(k), (τl-τl(k)) into h. We can get the next theorem in a different way to handle (τ1(k)-τm), τi+1(k)-τi(k), (τl-τl(k)).Corollary 12.
Under Assumptions1 and 4, the system (6) with ω(k)=0 is globally asymptotically stable, if there exist positive matrices P, R1, R2, R3, Q, Si (i=1,…,l), W=diag[w1,w2,…,wl], U=diag[u1,u2,…ul], and free matrix Ti, (i=1,2,…,2l), V1, V2, V3 with any appropriate dimensions, such that the following LMI holds:(29)Πi=Γ1Γ1,20Γ1,4Γ1,5Γ1,60Γ1,8Γ1,9000*Γ20000000000**Γ300Γ3,6000000***Γ40Γ4,60000Γ4,110****Γ50000000*****Γ60000Γ6,110******Γ700000*******Γ8Γ8,90Γ8,110********Γ9000*********Γ100Γ10,12**********-Γ110***********Γ12≤0,(30)Ξi=Γ1Γ1,20Γ1,4Γ1,5Γ1,60Γ1,8Γ1,9000*Γ20000000000**Γ300Γ3,6000000***Γ40Γ4,60000Γ4,110****Γ50000000*****Γ60000Γ6,110******Γ700000*******Γ8Γ8,90Γ8,110********Γ9000*********Ω100Ω10,12**********-Γ110***********Γ12≤0
(31)
Γ
10
=
∑
j
=
1
l
‍
T
2
j
-
1
I
^
j
+
∑
j
=
1
l
‍
I
^
j
T
T
2
j
-
1
T
,
Γ
10,12
=
diag
T
1
,
T
3
,
…
,
T
2
l
-
1
,
Ω
10
=
∑
j
≠
i
l
‍
T
2
j
-
1
I
^
j
+
∑
j
≠
i
l
‍
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^
j
T
T
2
j
-
1
T
+
T
2
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^
i
+
I
^
i
T
T
2
i
T
,
Ω
10,12
=
diag
T
1
,
T
3
,
…
,
T
2
i
-
1
,
T
2
i
,
T
2
i
+
1
,
…
,
T
2
l
-
1
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12
=
-
diag
R
2
d
1
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R
2
d
2
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…
,
R
2
d
l
,and the definition of I^i is the same as the definition in Theorem 8, and Γ11, Γ4,11, Γ6,11, Γ8,11 in inequality (29) are Γ10, Γ4,10, Γ6,10, Γ8,10 in Theorem 8, respectively.Proof.
-
∑
i
=
-
τ
M
-
τ
m
-
1
‍
η
T
(
k
+
i
)
R
2
η
(
k
+
i
) in ΔV3(k) is handled in the following way:(32)-∑i=-τM-τm-1‍ηTk+iR2ηk+i=-∑i=1l‍∑j=k-τikk-τi-1-1‍+∑j=k-τik-τik-1‍ηTiR2ηi≤∑i=1l‍τi-τikα3TkT2i-1R2-1T2i-1Tα3k+2α3TkT2i-1yk-τik-yk-τi+τik-τi-1α3TkT2iR2-1T2iTα3k+2α3TkT2iyk-τi-1-yk-τik,α3Tk=yk-τm,yk-τ1,…,yk-τM.
Whenτ(k)∈Ai, τi(k)=τ(k), τj(k)=τj-1, (j≠i), the last inequality is equivalent to(33)-∑i=-τM-τm-1‍ηTk+iR2ηk+i≤τi-τikα3TkT2i-1R2-1T2i-1Tα3k+2α3TkT2i-1yk-τik-yk-τi+τik-τi-1α3TkT2iR2-1T2iTα3k+2α3TkT2iyk-τi-1-yk-τik+∑j≠il‍τj-τj-1α3TkT2j-1R2-1T2j-1Tα3k+2α3TkT2j-1yk-τj-1-yk-τj.Then(34)EΔVyk,k≤θTkΞ+ΦTΓ11Φθk+τi-τikα3TkT2i-1R2-1T2i-1Tα3k+2α3TkT2i-1yk-τik-yk-τi+τik-τi-1α3TkT2iR2-1T2iTα3k+2α3TkT2iyk-τi-1-yk-τik+∑j≠il‍τj-τj-1α3TkT2j-1R2-1T2j-1Tα3k+2α3TkT2j-1yk-τj-1-yk-τj.
By solving the convex LMI condition at boundary conditionsτi(k)∈[τi-1,τi], we can get the LMIs in Corollary 12. This completes the proof of Corollary 12.Remark 13.
The difference between Theorem8 and Corollary 12 is obvious. The LMIs in Corollary 12 are derived by property of τi(k), which is more simple than the LMIs in Theorem 8.The next theorem can be obtained if the stochastic termω(k) is not removed in (6).Corollary 14.
Under Assumptions1 and 4, the system (6) with ω(k)=0 is globally asymptotically stable, if there exist positive matrices P=PT, R1, R2, Qi (i=1,…,l), Si (i=1,…,l), W=diag(w1,w2,…,wl), U=diag[u1,u2,…ul], and free matrix T1,T2,…,Tl with any appropriate dimensions, such that the following LMI holds:(35)P+τMhR1+τM-τmR2<λI,(36)Πi=Γ1Γ1,20Γ1,4Γ1,5Γ1,60Γ1,8Γ1,9000*Γ20000000000**Γ300Γ3,6000000***Γ40Γ4,60000Γ4,110****Γ50000000*****Γ60000Γ6,110******Γ700000*******Γ8Γ8,90Γ8,110********Γ9000*********Γ100Γ10,12**********-Γ110***********Γ12≤0,Ξi=Γ1Γ1,20Γ1,4Γ1,5Γ1,60Γ1,8Γ1,9000*Γ20000000000**Γ300Γ3,6000000***Γ40Γ4,60000Γ4,110****Γ50000000*****Γ60000Γ6,110******Γ700000*******Γ8Γ8,90Γ8,110********Γ9000*********Ω100Ω10,12**********-Γ110***********Γ12≤0,(37)Γ10=∑j=1l‍T2j-1I^j+∑j=1l‍I^jTT2j-1T,Γ10,12=diagT1,T3,…,T2l-1,Ω10=∑j≠il‍T2j-1I^j+∑j≠il‍I^jTT2j-1T+T2iI^i+I^iTT2iT,Ω10,12=diagT1,T3,…,T2i-1,T2i,T2i+1,…,T2l-1,Γ12=-diagR2d1,R2d2,…,R2dl,and the definition of I^i is the same as it in Theorem 8, and Γ11, Γ4,11, Γ6,11, Γ8,11 are Γ10, Γ4,10, Γ6,10, Γ8,10, respectively, in Theorem 8.Proof.
Asω(k)≠0, Δ{V1(yk,k)}, and Δ{V3(yk,k)} are different from Theorem 8,(38)EΔV1yk,k=EEV1yk+1,k+1∣yk-V1yk,k=EζTk+1Pζk+1-ζTkPζk=yTkCTPCyk-2yTkCTPAfyk-2∑i=1l‍ρiyTkCTPBgyk-τik+fTykATPAfyk+2∑i=1l‍ρifTykATPBgyk-τik+α2TkΨ1α2k-ζTkPζk+δTk,ykPδk,yk.Derived from Assumption 3, we can get(39)δTk,ykP+τMhR1+τM-τmR2δk,yk≤λmax⁡P+τMhR1+τM-τmR2δTk,ykδk,yk≤λξyTkyk.Based on Theorem 8, the desired condition can be obtained. This is the end of proof.


## 4. Examples

In this section, a numerical example is given to illustrate the effectiveness and benefits of the developed methods.Example 1.
We consider the delayed stochastic DNNs (6) with the following parameters, which have been considered in [33]:(40)C=1.5001.3,A=0.50.20.40.3,B=0.4-0.10.10.2.And the activation functions satisfy Assumption 1 with(41)F1=0000,F2=0.5000.5;G1=0000,G2=0.5000.5.For the parameters listed above, we can obtain the following feasible solution. Due to the limitation of the length of this paper, we only provide a part of the feasible solution here:(42)Q1=0.75570.02830.02830.0014,Q2=0.58320.02170.02170.0011,S1=22.52291.51641.51640.1059,S2=0.59030.00550.00550.0010.


## 5. Conclusions

The stability analysis of stochastic discrete-time NNs with leakage delay has been investigated in this paper via the Lyapunov functional method. By employing delay partition and introducing a new Lyapunov functional, a delay-dependent stability condition is proposed. What is more, more general LMIs conditions on the stability of the stochastic discrete-time NNs are established by choosing a new novel Lyapunov-Krasovskii functional. Finally, the feasibility and effectiveness of the developed methods have been shown by the numerical simulation example. The foregoing results have the potential to be useful for the study of leakage delay. And the results can be extended to complex networks with mixed time-varying delays.



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*Source: 306806-2015-09-16.xml*