# New Results on Passivity for Discrete-Time Stochastic Neural Networks with Time-Varying Delays

Authors: Wei Kang; Jun Cheng; Xiangyang Cheng
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The problem of passivity analysis for discrete-time stochastic neural networks
with time-varying delays is investigated in this paper. New delay-dependent passivity
conditions are obtained in terms of linear matrix inequalities. Less conservative conditions are
obtained by using integral inequalities to aid in the achievement of criteria ensuring the positiveness of the Lyapunov-Krasovskii functional.
At last, numerical examples
are given to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

Neural networks have been greatly applied in many areas in the past few decades, such as static processing, pattern recognition, and combinatorial optimization [1–3]. In practice, time-delays are frequently encountered in neural networks. As the finite signal propagation time and the finite speed of information processing, the existence of the delays may cause oscillation, instability, and divergence in neural networks. Moreover, stochastic perturbations and parameter uncertainties are two main resources which could reduce the performances of delayed neural networks. Due to the importance in both theory and practice, the problem of stability for stochastic delayed neural networks with parameter uncertainties is one of hot issues. Therefore, there have been lots of important and interesting results in this field [3, 6–17].It should be noticed that most neural networks are focused on continuous-time case [3, 7–12]. However, discrete-time systems play crucial roles in today’s information society. Particulary, when implementing the delayed continuous-time neural networks for computer simulation, it needs to formulate discrete-time system. Thus, it is necessary to research the dynamics of discrete-time neural networks. In recent years, a lot of important results have been published in the literatures [13–17]. Kwon et al. [14] discussed the stability criteria for the discrete-time system with time varying delays. Wang et al. [16] researched the exponential stability of discrete-time neural networks with distributed delays by means of Lyapunov-Krasovskill functional theory and linear matrix inequalities technology. In [17], the authors are concerned with the robust state estimation for discrete neural networks with successive packet dropouts, linear fractional uncertainties, and mixed time-delays.On the other hand, passivity is a significant concept that represents input-output feature of dynamic systems, which can offer a powerful tool for analyzing mechanical systems, nonlinear systems, and electrical circuits [18]. The passivity theory was firstly presented in the circuit analysis [19]. During the past several decades, the passivity theory has found successful applications in various areas such as complexity, signal processing, stability, chaos control, and fuzzy control. Thus, the problem of passivity for time-delay neural networks has received much attention and lots of effective approaches have been proposed in this research area [20–27]. The authors [21, 22] discussed the problem of passivity for neural networks with time-delays. Recently, Lee et al. [23] further studied the problem of dissipative analysis for neural networks with times-delays by using reciprocally convex approach and linear matrix inequality technology. Very recently, in [24], the problem of passivity criterion of discrete-time stochastic bidirectional associative memory neural networks with time-varying delays has been developed. In [25], some delay-dependent sufficient passivity conditions have been obtained for stochastic discrete-time neural networks with time-varying delays in terms of linear matrix inequalities technology and free-weighting matrices approach. A less conservative passivity criterion for discrete-time stochastic neural networks with time-varying delays was derived in [26]. However, there is still a room for decreasing the conservatism.Motivated by the above discussion, the problem of passivity for discrete-time stochastic neural networks with time-varying delays is studied. The major contribution of this paper lies in that, first of all, different from the traditional way, a new inequality is introduced to deal with terms∑i=k-τmk-1‍ηT(i)T1η(i) and ∑i=k-τMk-τm-1‍ηT(i)T1η(i). This method can effectively reduce the conservatism. Secondly, we do not need all the symmetric matrices in the Lyapunov functional to be positive definite and take advantage of the relationships of τ(k)-τm and τM-τ(k). New passivity conditions are presented in terms of matrix inequalities. Finally, numerical examples are given to indicate the effectiveness of the proposed method.Notations. Throughout this paper, the superscripts “-1” and “T” stand for the inverse and transpose of a matrix, respectively; P>0  (P⩾0,P<0,P⩽0) means that the matrix P is symmetric positive definite (positive semidefinite, negative definite, and negative semidefinite); E{·} stands for the mathematical expectation operator with respect to the given probability measure; · refers to the Euclidean vector norm; (Ω,F,P) denotes a complete probability space with a filtration containing all p-null sets and is right conditions; N[a,b] denotes the discrete interval given N[a,b]={a,a+1,…,b-1,b}; Rn denotes n-dimensional Euclidean space; Rm×n is the set of m×n real matrices; ∗ denotes the symmetric block in symmetric matrix; λmax(Q) and λmin(Q) denote, respectively, the maximal and minimal eigenvalue of matrix Q.


## 2. Problem Statement and Preliminaries

Consider the following DSNN with time-varying delays:(1)xk+1=Axk+B1gxk+B2gxk-τk+uk+δxk,xk-τkωk,yk=gxk,xj=ψj,j=-τM,-τM+1,…,0,where x(k)=[x1(k),x2(k),…,xn(k)]T∈Rn is the neuron state vector of the system; y(k) is the output of the neural networks; u(k)=[u1(k),u2(k),…,un(k)]T is the input vector; ψ(j) is the initial condition; A=diag(a1,a2,…,an) is the state feedback coefficient matrix; B1 and B2 are the connection weight matrices and the delayed connection weight matrices, respectively; g(x(k))=[g1(x1(k)),g2(x2(k)),…,gn(xn(k))]T∈Rn represents the neuron activation functions; τ(k) denotes the known time-varying delay and satisfies 0<τm≤τ(k)≤τM; δ is diffusion coefficient vector and ω(k) is a scalar Brownian motion defined on the probability space (Ω,F,P) with (2)Eωk=0,Eωkωk=1,Eωiωj=0,i≠j.Assumption 1.
The neuron activation functiong(·) satisfies (3)ls-⩽gsa-gsba-b⩽ls+,gs0=0,s=1,2,…,n,for all a,b∈R, a≠b, where ls- and ls+ are known real constants.Remark 2.
In Assumption1, ls- and ls+ can be positive, negative, or zero. Moreover, when ls-=0, then ls+>0.Assumption 3.
δ
(
x
(
k
)
,
x
(
k
-
τ
(
k
)
)
) is the continuous function satisfying (4)δTxk,xk-τkδxk,xk-τk≤ρ1xTkxk+ρ2xTk-τkxk-τk,where ρ1, ρ2 are known constant scalars.The following lemmas and definition will be used in proof of main results.Lemma 4.
For integersτ(k) and vector function x(k+·):N[-τM,-τm]↦Rn, η(k)=x(k+1)-x(k), for any positive semidefinite matrix (5)X=X11X12X13X12TX22X23X13TX23TX33≥0,the following inequality holds: (6)-∑i=k-τkk-1ηTiX33ηiW≤∑i=k-τkk-1xTkxTk-τkηTiWWWiiW·X11X12X13X12TX22X23X13TX23T0xkxk-τkηi.Proof.
In fact, we have(7)0≤∑i=k-τkk-1xTkxTk-τkηTiWWWW·X11X12X13X12TX22X23X13TX23TX33xkxk-τkηi=∑i=k-τkk-1ηTiX33ηiX11X12X13X12TX22X23X13TX23T0WWWWW+xTkxTk-τkηTiWWWWW·X11X12X13X12TX22X23X13TX23T0xkxk-τkηi.Thus, one can easily obtain (8)-∑i=k-τkk-1ηTiX33ηiWW≤∑i=k-τkk-1xTkxTk-τkηTiWWWWWW·X11X12X13X12TX22X23X13TX23T0xkxk-τkηi.The proof is completed.Remark 5.
The new inequality was proposed in [5, 6] for continuous-time systems; it is worth noting that we firstly extend this method to study discrete-time neural networks in this paper.Lemma 6 (see [4, 13]).
LetM∈Rn×n be a positive-definite matrix, Xi∈Rn; then (9)-m-n∑i=k-mk-n-1XiTMXi⩽-∑i=k-mk-n-1XiTM∑i=k-mk-n-1Xi.Lemma 7 (see [24]).
LetA=AT,D,E be real matrices with appropriate dimensions, with matrix F(k) satisfying FT(k)F(k)≤I; there exists a scalar ε>0 such that (10)A+DFtE+ETFTtDT≤A+εDDT+ε-1ETE.Lemma 8 (see [3]).
For any constant matricesΩ, Υ1, Υ2 with appropriate dimensions, and a function τ(k) satisfying τm≤τ(k)≤τM, then (11)Ω+τk-τmΥ1+τM-τkΥ2<0,if and only if (12)Ω+τM-τmΥ1<0,Ω+τM-τmΥ2<0.Definition 9 (see [25]).
The system (1) is said to be passive if there exists a scalar γ>0 satisfying (13)2∑i=0k0EyTiui≥-γ∑i=0k0EuTiui,for all k0∈N and for all solution of (1) with ψ(0)=0.


## 3. Main Results

In this section, the passivity of discrete-time stochastic neural networks with time-varying delays will be investigated by use of the new integral inequality and Lyapunov method. In the paper, some of symmetric matrices in Lyapunov-Krasovskii functional are not necessarily required to be positive definite.Denote(14)L1=diagl1-l1+,…,ln-ln+,L2=diagl1-+l1+2,…,ln-+ln+2,τMm=τM-τm,ηk=xk+1-xk.Main results are given in the following theorems.Theorem 10.
Under Assumptions1 and 3, the discrete-time stochastic neural network (1) is passive, if there exist matrices P, R1, R2, Q=Q11Q12∗Q22>0, T1>0, T2>0, X=X11X12X13∗X22X23∗∗X33≥0, Y=Y11Y12Y13∗Y22Y23∗∗Y33≥0, Z=Z11Z12Z13∗Z22Z23∗∗Z33≥0, the positive diagonal matrices Sk=diag{s1k,s2k,…,snk}  (k=1,2), and scalars λ>0, γ>0, such that the following matrix inequalities hold: (15)T1+τmτMP-T1∗T1+τmR1>0,T2+P-T2∗T2+τMR2>0,(16)T1-X33≥0,T2-Y33≥0,T2-Z33≥0,Θ≤λI,(17)Ω+Υ1<0,Ω+Υ2<0,where(18)Θ=P+τmT1+τMmT2,Ω=Ω11⋯Ω17∗⋱⋮∗∗Ω77,Υ1=d11⋯d17∗⋱⋮∗∗d77,Υ2=e11⋯e17∗⋱⋮∗∗e77,Ω11=τMm+1Q11+R1+τmX11+X13+X13TΩ11=+τmA-IT1A-I+τMmA-IT2A-IΩ11=+ρ1λI+APA-P-S1L1,Ω12=τmX12-X13+X23T,Ω15=τMm+1Q12+L2S1+τmA-IT1B1Ω11=+τMmA-IT2B1+ATPB1,Ω16=τmA-IT1B1+τMmA-IT2B1+APB1,Ω17=τmA-IT1+τMmA-IT2+AP,Ω22=-R1+R2+τmX22-X23-X23T+Y13+Y13T,Ω23=-Y13+Y23T,Ω33=-Q11-S2L1+ρ2λI-Y23-Y23T+Z13+Z13T,Ω34=-Z13+Z23T,Ω36=-Q12+L2S2,Ω44=-R2-Z23-Z23T,Ω55=B1TPB1+τmB1TT1B1+τMmB1TT2B1Ω11=+τMm+1Q22-S1,Ω56=B1TPB2+τmB1TT1B2+τMmB1TT2B2,Ω57=B1TP+τmB1TT1+τMmB1TT2-I,Ω66=B2TPB2+τmB2TT1B2+τMmB2TT2B2Ω11=-Q22-S2,Ω67=B2TP+τmB2TT1+τMmB2TT2,Ω77=P+τmT1+τMmT2-γI,d22=Y11,d23=Y12,d33=Y22,e33=Z11,e34=Y12,d44=Z22,elsewhere,Ωij=0,dij=0,eij=0,i,j=1,2,…,8.Proof.
Define a new augmented of Lyapunov-Krasovskii functional as follows:(19)Vk=V1k+V2k+V3k+V4k,where (20)V1k=xTkPxk,V2k=∑i=k-τmk-1xTiR1xi+∑i=k-τMk-τm-1xTiR2xi,V3k=∑i=k-τkk-1xigxiTQ11Q12∗Q22xigxiV3k=+∑j=-τM-τm-1∑i=k+jk-1xigxiTQ11Q12∗Q22xigxi,V4k=∑j=-τm-1∑i=k+jk-1ηTiT1ηiV3k=+∑j=-τM-τm-1∑i=k+jk-1ηTiT2ηi.Firstly, we show that the Lyapunov-Krasovskii functional V(k) is positive definite. By using Lemma 6, one can obtain (21)∑j=-τm-1∑i=k+jk-1ηTiT1ηi≥∑j=-τm-1-1j∑i=k+jk-1ηiTT1∑i=k+jk-1ηi=∑j=-τm-1-1jxkxk+jTT1-T1∗T1xkxk+j≥1τm∑i=k-τmk-1xkxiTT1-T1∗T1xkxi,∑j=-τM-τm-1∑i=k+jk-1ηTiT2ηi≥∑j=-τM-τm-1-1j∑i=k+jk-1ηiTZ2∑i=k+jk-1ηi=∑j=-τM-τm-1-1jxkxk+jTT1-T1∗T1xkxk+j≥1τM∑i=k-τMk-τm-1xkxiTT2-T2∗T2xkxi,(22)xTkPxk=∑i=k-τmk-11τMxTkPxk+∑i=k-τMk-τm-11τMxTkPxk.Then, it follows from (19)–(21) that (23)Vk≥∑i=k-τmk-11τMxTkPxk+∑i=k-τMk-τm-11τMxTkPxkW+∑i=k-τmk-1xTiR1xi+∑i=k-τMk-τm-1xTiR2xiW+1τm∑i=k-τmk-1xkxiTT1-T1∗T1xkxiW+1τM∑i=k-τMk-τm-1xkxiTT2-T2∗T2xkxi≥1τm∑i=k-τmk-1xkxiTT1+τmτMP-T1-T1T1+τmR1xkxiW+1τM∑i=k-τMk-τm-1xkxiTT2+P-T2-T2T2+τmR2xkxi.From condition (15), there exists a scalar δ1>0, for any x(k)≠0, such that (24)Vk≥δ1xk2>0.Now, taking the forward difference of V(k) along the trajectories of system (1), it yields that (25)EΔV1kW=Eηk+xkTPηk+xk-xTkPxkW=EηTkPηk+2xTkPηk,EΔV2kW=ExTkR1xk-xTk-τmR1-R2xk-τmWWWW-xTk-τMR2xk-τM,EΔV3kW≤EτMm+1xkgxkTQ11Q12∗Q22xkgxkWWWW-xk-τkgxk-τkTQ11Q12∗Q22WWWWxkgxkT·xk-τkgxk-τkW=EτMm+1xTkQ11xk+2xTkQ12gxkWWWW+gTxkQ22gxkWWWW-xTk-τkQ11xk-τkWWWW-2xTk-τkQ12gxk-τkWWWW-gTxk-τkQ22gxk-τk,EΔV4kW=EηTkτmT1+τMmT2ηk-∑i=k-τmk-1ηTiT1ηiWWWW-∑i=k-τkk-τm-1ηTiT2ηi-∑i=k-τMk-τk-1ηTiT2ηi,E-∑i=k-τmk-1ηTiT1ηi-∑i=k-τkk-τm-1ηTiT2ηiWW-∑i=k-τMk-τk-1ηTiT2ηiW=E-∑i=k-τmk-1ηTiT1-X33ηiWWWW-∑i=k-τkk-τm-1ηTiT2-Y33ηiWWWW-∑i=k-τMk-τk-1ηTiT2-Z33ηi-∑i=k-τmk-1ηTiX33ηiWWWW-∑i=k-τkk-τm-1ηTiY33ηi-∑i=k-τMk-τk-1ηTiZ33ηi.Form the new inequality of Lemma 4, one can get (26)-∑i=k-τmk-1ηTiX33ηi≤∑i=k-τmk-1xTkxTk-τmηTiWWWW·X11X12X13X12TX22X23X13TX23T0xkxk-τmηi=xTkτmX11+X13T+X13xkW+xTkτmX12-X13+X23Txk-τmW+xTk-τmτmX12T-X13T+X23xkW+xTk-τmτmX22-X23T-X23xk-τm,-∑i=k-τkk-τm-1ηTiY33ηi≤∑i=k-τkk-τm-1xTk-τmxTk-τkηTiWWWW·X11X12X13X12TX22X23X13TX23T0xk-τmxk-τkηi=xTk-τmτk-τmY11+Y13T+Y13xk-τmW+xTk-τmτk-τmY12-Y13+Y23Txk-τkW+xTk-τkτk-τmY12T-Y13T+Y23xk-τmW+xTk-τkτk-τmY22-Y23T-Y23xk-τk,-∑i=k-τMk-τk-1ηTiZ33ηi≤∑i=k-τMk-τk-1xTk-τkxTk-τMηTiWWWW·X11X12X13X12TX22X23X13TX23T0xk-τkxk-τMηi=xTk-τkτM-τkZ11+Z13T+Z13xk-τkW+xTk-τkτM-τkZ12-Z13+Z23Txk-τMW+xTk-τMτM-τkZ12T-Z13T+Z23xk-τkW+xTk-τMτM-τkZ22-Z23T-Z23xk-τM.It is easy to get (27)EηTkΘηkW=EA-Ixk+B1gxk+B2gxk-τkxk-τk3WWWWxk-τk3+uk+δxk,xk-τkωkTWWWWxk-τk3·ΘA-Ixk+B1gxkxk-τk3WWWWWWxk-τk3+B2gxk-τkWWWWWWxk-τk3+uk+δxk,xk-τkωk.From Assumption 3 and inequality (16), we have (28)δxk,xk-τkΘδxk,xk-τk≤λρ1xTkxk+ρ2xTk-τkxk-τk.From Assumption 1, it follows that (29)gixik-li+xikgixik-li-xik≤0,gixik-τk-li+xik-τkgixik-τk-li-xik-τk≤0.Thus, for the diagonal matrices Sk=diag{s1k,s2k,…,snk}  (k=1,2), one can receive the following inequalities: (30)-xkgxkTS1L1-S1L2∗S1xkgxk≥0,-xk-τkgxk-τkTS2L1-S2L2∗S2xk-τkgxk-τk≥0.Combining (25)–(30), it yields (31)EΔVk-2yTkuk-γuTkukW≤EαTkΩ+τk-τmΥ1+τM-τkΥ2αk∑i=k-τMk-τk-1ηTiT2-Z33ηiWiiWiW-∑i=k-τmk-1ηTiT1-X33ηiWWiiiW-∑i=k-τkk-τm-1ηTiT2-Y33ηiWWiiiW-∑i=k-τMk-τk-1ηTiT2-Z33ηi,where (32)αTk=xTk,xTk-τm,xTk-τk,xTk-τM,WWgTxk,gTxk-τk,uTk.From (15)–(17), observing that T1-X33≥0, T2-Y33≥0, and T2-Z33≥0, one can conclude that (33)EΔVk-2yTkuk-γuTkuk≤0.Then, (34)2∑i=0k0EyTiui≥∑i=0k0EΔVi-γ∑i=0k0EuTiui.By the definition of V(k) and inequality (23), one can find that (35)∑i=0k0EΔVi=EVk0+1-V0≥0.So, one can have (36)2∑i=0k0EyTiui≥-γ∑i=0k0EuTiui,for all k0∈N. This completes the proof.Remark 11.
It should be pointed out that the new inequality is introduced to deal with∑i=k-τmk-1‍ηT(i)T1η(i) and ∑i=k-τMk-τm-1‍ηT(i)T1η(i), which is immensely different from traditional ways. This method can effectively reduce the conservatism of the results.Remark 12.
In this paper, not all the matrices in the Lyapunov functional need to be positive definite. In fact, the conditions in (15) assure the positive definiteness of the Lyapunov functional; this is greatly different from traditional ways for passivity researches of discrete-time neural network, because the traditional methods always need Lyapunov matrices to be positive definite.Remark 13.
It can be seen that the term∑i=k-τMk-τm-1‍ηT(i)T1η(i) is divided into two parts; the aim is to make full use of the relationship of τM-τ(k) and τ(k)-τm, and then, taking advantage of the integral inequality and Lemma 8, new passivity conditions are obtained in terms of LMIs.Corollary 14.
Under Assumptions1 and 3, the discrete-time stochastic neural network (1) is passive, if there exist scalars λ>0, γ>0, matrices P>0, R1>0, R2>0, Q=Q11Q12∗Q22>0, T1>0, T2>0, X=X11X12X13∗X22X23∗∗X33≥0, Y=Y11Y12Y13∗Y22Y23∗∗Y33≥0, Z=Z11Z12Z13∗Z22Z23∗∗Z33≥0, and the positive diagonal matrices Sk=diag{s1k,s2k,…,snk}  (k=1,2), such that linear matrix inequalities (16) and (17) hold.
Now, we will consider the stochastic discrete-time neural networks with time-varying delay and parameter uncertainties as follows:(37)xk+1=A+ΔAkxk+B1+ΔB1kgxk+B2+ΔB2kgxk-τk+uk+δxk,xk-τkωk,where ΔA(k), ΔB1(k), and ΔB2(k) denote the parameter uncertainties that are assumed to be of the form (38)ΔAkΔB1kΔB2k=MFkE1E2E3,where M, Ei(i=1,2,3) are known constant matrices, and F(k) is the unknown matrix valued function subject to (39)FkTFk≤I.Theorem 15.
Under Assumptions1 and 3, the discrete-time stochastic uncertain neural network (37) is robustly passive, if there exist scalars λ>0, γ>0, ε>0, matrices P, R1, R2, Q=Q11Q12∗Q22>0, T1>0, T2>0, X=X11X12X13∗X22X23∗∗X33≥0, Y=Y11Y12Y13∗Y22Y23∗∗Y33≥0, Z=Z11Z12Z13∗Z22Z23∗∗Z33≥0, and the positive diagonal matrices Sk=diag{s1k,s2k,…,snk}  (k=1,2), such that the following matrix inequalities hold: (40)T1+τmτMP-T1∗T1+τmR1>0,T2+P-T2∗T2+τMR2>0,T1-X33≥0,T2-Y33≥0,T2-Z33≥0,Θ≤λI,Σ+ΥiΞ1τmΞ2τMmΞ30εΞ4∗-P00PM0∗∗-T10τmT1M0∗∗∗-T2τMmT2M0∗∗∗∗-εI0∗∗∗∗∗-εI<0,i=1,2,where(41)Ξ1=PA000PB1PB2PT,Ξ2=T1A-I000T1B1T1B2T1T,Ξ3=T2A-I000T2B1T2B2T2T,Ξ4=E1000E2E30T,Θ=P+τmT1+τMmT2,Σ=Σ11Σ1200τMm+1Q12+L2S100∗Σ22-Y13+Y23T0000∗∗Σ33-Z13+Z23T0-Q12+L2S20∗∗∗-Z23-Z23T-R2000∗∗∗∗τMm+1Q22-S10-I∗∗∗∗∗-Q22-S20∗∗∗∗∗∗Σ77,Σ11=τMm+1Q11+R1+τmX11+X13+X13T+ρ1λI-P-S1L1,Σ12=τmX12-X13+X23T,Σ22=-R1+R2+τmX22-X23-X23T+Y13+Y13T,Σ33=-Q11-S2L1+ρ2λI-Y23-Y23T+Z13+Z13T,Σ77=P+τmT1+τMmT2-γI.
Υ
1,Υ2 are the same as defined in Theorem 10.Proof.
By replacingA, B1, B2 in (17) with A+ΔA(k), B1+ΔB1(k), B2+ΔB2(k), respectively, then using Lemma 7, the desired results can be obtained immediately. The proof is completed.


## 4. Numerical Examples

In this section, some numerical examples are proposed to show the effectiveness of the results obtained in this paper.Example 1.
Consider the system (1) with the following parameters: (42)A=0.8000.9,B1=0.001000.005,B2=-0.10.01-0.2-0.1,ρ1=0.003,ρ2=0.002.The activation functions are taken as (43)f1x=f2x=tanh0.5x.It can be verified that (44)L1=0000,L2=0.25000.25.In this example, if τm=3 and τM=12, the optimal passivity performance obtained is γ=12.2439 by the method in [25] and γ=5.4007 by the method in [26], while by Theorem 10 in this paper, the optimal passivity performance γ=3.9546. The comparisons of γ are listed in Table 1, when τm=3, τM=8,9,10,11,12. Then, when we assume τM=13, the optimal passivity performance γ obtained by Theorem 10 for different τm can be found in Table 2. It can be seen that our results are less conservative than the ones in [25, 26].Table 1
Optimal passivity performanceγ for different τM.
Methods
8
9
10
11
12
Theorem  1 [25]
3.9708
4.8691
6.2501
8.3991
12.2439
Corollary  2 [26]
2.9660
3.3483
3.8428
4.4968
5.4007
Theorem10
2.8481
3.2154
3.5172
3.7438
3.9546Table 2
Optimal passivity performanceγ for different τm.
Methods
4
5
6
7
8
Theorem  1 [25]
16.1853
13.2125
11.0130
9.2978
7.9134
Corollary  2 [26]
6.1649
5.6743
5.2451
4.8664
4.5300
Theorem10
4.4126
4.3752
4.2563
4.1871
4.0802Example 2.
Consider the system (1) with the following parameters: (45)A=0.4000.5,B1=0.001000.005,B2=-0.10.01-0.2-0.1,ρ1=ρ2=0.003.The activation functions are taken as (46)f1x=f2x=tanhx.It can be verified that (47)L1=0000,L2=0.5000.5.For this example, when γ=3 and τm=1, by Theorem 10, we can get that the upper bound of the time-varying delay is τM=8. When γ=4 and τm=1, by Theorem 10, we can get τM=10; we can obtain upper bound of τM for different γ and τm, which are summarized in Table 3. It can be found from Table 3 that, for the same τm, a larger passivity performance γ corresponds to a larger τM; with the same τM, a smaller passivity performance γ corresponds to a larger τm.Table 3
Allowable upper bounds ofτM for different γ and τm.
τ
m
1
3
5
7
9
γ
=
3
8
10
12
14
16
γ
=
4
10
12
14
16
18
γ
=
6
12
13
15
17
19


## 5. Conclusions

In this paper, the problem of passivity analysis for discrete-time stochastic neural networks with time-varying delays has been investigated. The presented sufficient conditions are based on the Lyapunov-Krasovskii functional, a new inequality and linear matrix inequality approach. Numerical examples are given to demonstrate the usefulness and effectiveness of the proposed results. Finally, it should be worth noticing that the proposed method in this paper may be extensively applicable in many other areas, such as Markov jump neural networks, Markov jump neural networks with incomplete transition descriptions, and switched neural networks, which deserves further investigation.



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*Source: 341839-2015-09-17.xml*