# Delay-Dependent Exponential Stability  for Discrete-Time BAM Neural Networks  with Time-Varying Delays

Authors: Yonggang Chen; Weiping Bi; Yuanyuan Wu
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

This paper considers the delay-dependent exponential stability for discrete-time BAM neural networks
with time-varying delays. By constructing the new Lyapunov functional, the improved delay-dependent
exponential stability criterion is derived in terms of linear matrix inequality (LMI). Moreover, in order to
reduce the conservativeness, some slack matrices are introduced in this paper. Two numerical examples
are presented to show the effectiveness and less conservativeness of the proposed method.


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## Body

## 1. Introduction

Bidirectional associative memory (BAM) neural networks
were first introduced by Kosko [1, 2]. It generalizes the single-layer
autoassociative Hebbian correlator to a two-layer pattern-matched
heteroassociative circuits. This class of neural networks has been successfully
applied to pattern recognition and artificial intelligence. In those
applications, it is very important to guarantee the designed neural networks to
be stable. On the other hand, time delays are unavoidably encountered in the
implementation of neural networks, and its existence will lead to instability,
oscillation, and poor performances of neural networks. Therefore, the the
asymptotic or exponential stability analysis for BAM neural networks with time
delays has received great attention during the past years, see, for example,
[3–15]. The obtained results are classified into two categories:
delay-independent results [3–5, 7, 10] and delay-dependent results
[6, 9, 11, 12, 14–16]. Generally speaking, the delay-dependent results are less
conservative than the delay-independent ones, especially when the size of time
delay is small.It should be
pointed out that all of the above-mentioned references are concerned with
continuous-time BAM neural networks. However, when implementing the
continuous-time neural networks for computer
simulation, for experimental or computational purposes, it is essential to
formulate a discrete-time system that is an analogue of the continuous-time
recurrent neural networks [17]. Generally speaking, the stability analysis of
continuous-time neural networks is not applicable to the discrete version.
Therefore, the stability analysis for various discrete-time neural networks with
time delays is widely studied in recent years [17–28]. Using M-matrix method,
Liang and Cao [25] studied the exponential stability of continuous-time BAM
neural network with constant delays and its discrete analogue. Liang et al.
[26] also studied the dynamics of discrete-time BAM neural networks with
variable time delays based on unreasonably severe constraints on the delay
functions. By using the Lyapunov functional method and linear matrix inequality
technique, the exponential stability and robust exponential stability for
discrete-time BAM neural networks with variable delays were considered in
[27, 28], respectively. However, the results presented in [27, 28] are still
conservative, since the the Lyapunov functionals constructed in [27, 28] are so simple.
Therefore, there is much room to improve the results in [27, 28].In this paper,
we present a new exponential stability criterion for discrete-time BAM neural
networks with time-varying delays. Compared with the existing methods, the main
contributions of this paper are as follows. Firstly, more general Lyapunov
functional is employed to obtain the improved exponential stability criterion;
secondly, in order to reduce the conservativeness, some slack matrices, which
bring much flexibility in solving LMI, are introduced in this paper. The
proposed exponential stability criterion is expressed in LMI which can be
efficiently solved by LMI toolbox in Matlab. Finally, two numerical examples
are presented to show that our result is less conservative than some existing
ones [27, 28].The
organization of this paper is as follows. Section2 presents problem
formulation of discrete-time BAM neural networks with time delay-varying. In
Section 3, our main result of this paper is established and some remarks are given.
In Section 4, numerical examples are given to demonstrate the proposed method.
Finally, conclusion is drawn in Section 5.Notation
Throughout this paper, the superscript “T” stands for the transpose of a matrix. Rn and Rn×n denote the n-dimensional Euclidean space and set of all n×n real matrices, respectively. A real symmetric
matrix P>0(≥0) denotes P being a positive definite (positive
semidefinite) matrix. I is used to denote an identity matrix with
proper dimension. For integers a,b, and a<b,N[a,b]  denote the discrete interval N[a,b]={a,a+1,…,b−1,b}. λM(X) and λm(X) stand for the maximum and minimum eigenvalues
of the symmetric matrix X,
respectively. Matrices, if not explicitly stated, are assumed to have compatible
dimensions. The symmetric terms in a symmetric matrix are denoted by ∗.


## 2. Problem Formulation

Consider the
following discrete-time BAM neural network with time-varying
delays:(2.1)ui(k+1)=aiui(k)+∑j=1mwijf˜j(vj(k−τ(k)))+Ii,i=1,2,…,n,vj(k+1)=bjvj(k)+∑i=1nvjig˜i(ui(k−h(k)))+Jj,j=1,2,…,m,where ui(k) and vj(k) are the states of the ith neuron from the neural field FX and the jth neuron from the neural field FY at time k,
respectively; ai,bj∈(0,1) denote the stability of internal neuron
process on the X-layer and the Y-layer, respectively; wij,vji are real constants, and denote the synaptic
connection weights; f˜j(⋅)g˜i(⋅) denote the activation functions of the jth
neuron from the neural field FY and the jth neuron from the neural field FX,
respectively; Ii and Jj denote the external constant inputs from
outside the network acting on the ith neuron from the neural field FX and the jth neuron from the neural field FY,
respectively. τ(k) and h(k) represent time-varying interval delays
satisfying(2.2)τm≤τ(k)≤τM,hm≤h(k)≤hM,where τm,τM,hm,
and hM are positive integers.Throughout this
paper, we make the following assumption on the activation functionsf˜j(⋅),g˜i(⋅).(A1)
The
activation functionsf˜j(⋅),g˜i(⋅)(i=1,2,…,n,j=1,2,…,m)  are bounded on R.(A2)
For anyζ1,ζ2∈R, there exist positive scalars l1j,l2i such that(2.3)0≤|f˜j(ζ1)−f˜j(ζ2)|≤l1j|ζ1−ζ2|,j=1,2,…,m,0≤|g˜j(ζ1)−g˜j(ζ2)|≤l2i|ζ1−ζ2|,i=1,2,…,n.It is clear
that under Assumption (A1) and (A2), system (2.1) has at least one equilibrium.
In order to simplify our proof, we shift the equilibrium point u*=[u1*,u2*,…,un*]T,v*=[v1*,v2*,…,vm*]T of system (2.1) to the origin. Let xi(k)=ui(k)−ui*,yj(k)=vj(k)−vj*,fj(yj(k))=f˜j(vj(k))−f˜j(vj*),gi(xi(k))=g˜i(ui(k))−g˜i(ui*),
then system (2.1) can be transformed to(2.4)xi(k+1)=aixi(k)+∑j=1mwijfj(yj(t−τ(k))),i=1,2,…,nyj(k+1)=bjyj(k)+∑i=1nvjigi(xi(t−h(k))),j=1,2,…,m.Obviously, the the activation functions fj(⋅),gi(⋅) satisfy the following conditions.(A3)
For anyζ∈R, there exist positive scalars l1j,l2i such that(2.5)0≤|fj(ζ)|≤l1j|ζ|,j=1,2,…,m,0≤|gi(ζ)|≤l2i|ζ|,i=1,2,…,n.Also we assume that system (2.4) with initial value(2.6)xi(s)=ϕi(s),yj(s)=ψj(s),∀s∈N[−τ*,0],i=1,2,…,n,j=1,2,…,m,where τ*=max{τM,hM}.For
convenience, we rewrite system (2.4) in the form(2.7)x(k+1)=Ax(k)+Wf(y(k−τ(k))),y(k+1)=By(k)+Vg(x(k−h(k))),x(s)=ϕ(s),y(s)=ψ(s),∀s∈N[−τ*,0],where  x(k)=[x1(k),x2(k),…,xn(k)]T,y(k)=[y1(k),y2(k),…,ym(k)]T,f(y(k))=[f1(y1(k)),f2(y2(k)),…,fm(ym(k))]T,g(x(k))=[g1(x1(k)),g2(x2(k)),…,gn(xn(k))]T,A=diag{a1,a2,…,an},B=diag{b1,b2,…,bm},W=(wij)m×n,V=(vij)n×m,ϕ(s)=[ϕ1(s),ϕ2(s),…,ϕn(s)]T,ψ(s)=[ψ1(s),ψ2(s),…,ψm(s)]T.From above
analysis, we can see that the exponential stability problem of system (2.1) on
equilibrium x* is changed into the zero stability problem of
system (2.7). Therefore, in the following part, we will devote into the exponential
stability analysis problem of system (2.7).Before giving
the main result, we will firstly introduce the following definition and lemmas.Definition 2.1.
The trivial solution of BAM neural network (2.7) is
said to be globally exponentially stable, if there exist scalars r>1 and M>1 such that(2.8)∥x(k)∥2+∥y(k)∥2≤M(sups∈N[−hM,0]∥ϕ(s)∥2+sups∈N[−τM,0]∥ψ(s)∥2)r−k.Lemma 2.2 (see [29]).
For any real vectorsa,b,
and any matrix Q>0 with appropriate dimensions, it follows
that(2.9)2aTb≤aTQa+bTQ−1b.


## 3. Main Result

In this
section, we are in a position to present the global exponential stability
criterion of system (2.7).Theorem 3.1.
For given diagonal matricesL1=diag{l11,l12,…,l1m}andL2=diag{l21,l22,…,l2n}. Under Assumptions (A1) and
(A2), the system (2.7) is globally exponentially stable, if there exist matrices Pi>0,Qi>0,Ri>0,Si>0,Zi>0,(i=1,2),Lj,Mj,Nj,Tj,(j=1,2,…,8),
and diagonal matrices D1=diag{d11,d12,…,d1m}>0,D2=diag{d21,d22,…,d2n}>0,
such that the following LMI holds:(3.1)[ΩhMLhM−hmMτMNτM−τmT*−Z1000**−Z100***−Z20****−Z2]<0,whereω=[ω11ω12ω13ω14ω15ω16−m1+l7tl8t*ω22ω23ω24ω25ω26ω27t8t−n8t**ω33−t4tω35ω36−m3−t7t−t8t***ω44n4m4−l4−m40****ω55ω56−m5+n7tω58*****ω66ω67−l8t+m8t******ω77−m8t*******ω88],l=[l1tl2tl3tl4tl5tl6tl7tl8t]t,m=[m1tm2tm3tm4tm5tm6tm7tm8t]t,n=[n1tn2tn3tn4tn5tn6tn7tn8t]t,t=[t1tt2tt3tt4tt5tt6tt7tt8t]t,ω11=ratp1a+rhmhm(a−i)tz1(a−i)−p1+(hm−hm+1)q1+r1+l1+l1t,ω12=l2t+t1−n1,ω13=l3t−t1,ω14=ratp1w+rhmhm(a−i)tz1w+l4t,ω15=l5t+n1,ω16=−l1+l6t+m1,ω22=−r−τmq2+t2+t2t+d1−n2−n2t,ω23=−t2+t3t−n3t,ω24=t4t−n4t,ω25=t5t+n2−n5t,ω26=t6t+m2−l2−n6t,ω27=−m2+t7t−n7t,ω33=−r−τmr2−t3−t3t,ω35=n3−t5t,(3.2)ω36=m3−l3−t6t,ω44=rwtp1w+rhmhmwtz1w−l1−1d1l1−1,ω55=rbtp2b+rτmτm(b−i)tz2(b−i)−p2+(τm−τm+1)q2+r2+n5+n5t,ω56=m5−l5+n6t,ω58=rbtp2v+rτmτm(b−i)tz2v+n8t,ω66=−r−hmq1−l6−l6t+d2+m6+m6t,ω67=−l7t−m6+m7t,ω77=−r−hmr1−m7−m7t,ω88=rvtp2v+rτmτmvtz2v−l2−1d2l2−1.Proof.
Choose the following
Lyapunov-Krasovskii candidate function of system (2.7) as
follows:(3.3)V(k)=V1(k)+V2(k)+V3(k)+V4(k)+V5(k),where(3.4)v1(k)=rkxt(k)p1x(k)+rkyt(k)p2y(k),v2(k)=∑l=k−h(k)k−1rlxt(l)q1x(l)+∑l=k−τ(k)k−1rlyt(l)q2y(l),v3(k)=∑θ=−hm+1−hm∑l=k+θk−1rlxt(l)q1x(l)+∑θ=−τm+1−τm∑l=k+θk−1rlyt(l)q2y(l),v4(k)=∑l=k−hmk−1rlxt(l)r1x(l)+∑l=k−τmk−1rlyt(l)r2y(l),v5(k)=rhm∑θ=−hm−1∑l=k+θk−1rlη1t(l)z1η1(l)+rτm∑θ=−τm−1∑l=k+θk−1rlη2t(l)z2η2(l),and η1(l)=x(l+1)−x(l),η2(l)=y(l+1)−y(l).
Calculating the difference ofV(k) along the trajectories of system (2.7), we can
obtain (3.5)δv1(k)=rk+1[ax(k)+wf(y(k−τ(k)))]tp1[ax(k)+wf(y(k−τ(k)))]+rk+1[by(k)+vg(x(k−h(k)))]tp2[by(k)+vg(x(k−h(k)))]−rkxt(k)p1x(k)−rkyt(k)p2y(k)=rk{xt(k)(ratp1a−p1)x(k)+2rxt(k)atp1wf(y(k−τ(k)))+rft(y(k−τ(k)))wtp1wf(y(k−τ(k)))+yt(k)(rbtp2b−p2)y(k)+2ryt(k)btp2vg(x(k−h(k)))+rgt(x(k−h(k)))vtp2vg(x(k−h(k)))},(3.6)δv2(k)≤rkxt(k)q1x(k)−rk−h(k)xt(k−h(k))q1x(k−h(k))+rkyt(k)q2y(k)−rk−τ(k)yt(k−τ(k))q2y(k−τ(k))+∑l=k+1−h(k+1)k−hmrlxt(l)q1x(l)+∑l=k+1−τ(k+1)k−τmrlyt(l)q2y(l)≤rkxt(k)q1x(k)−rk−hmxt(k−h(k))q1x(k−h(k))+rkyt(k)q2y(k)−rk−τmyt(k−τ(k))q2y(k−τ(k))+∑l=k+1−hmk−hmrlxt(l)q1x(l)+∑l=k+1−τmk−τmrlyt(l)q2y(l),(3.7)δv3(k)=(hm−hm)rkxt(k)q1x(k)+(τm−τm)rkyt(k)q2y(k)−∑l=k+1−hmk−hmrlxt(l)q1x(l)−∑l=k+1−τmk−τmrlyt(l)q2y(l),(3.8)δv4(k)=rkxt(k)r1x(k)−rk−hmxt(k−hm)r1x(k−hm)+rkyt(k)r2y(k)−rk−τmyt(k−τm)r2y(k−τm),(3.9)δv5(k)=rkrhmhmη1t(k)z1η1(k)−rhm∑l=k−hmk−1rlη1t(l)z1η1(l)+rkrτmτmη2t(k)z2η2(k)−rτm∑l=k−τmk−1rlη2t(l)z2η2(l)≤rkrhmhm[(a−i)x(k)+wf(y(k−τ(k)))]tz1[(a−i)x(k)+wf(y(k−τ(k)))]+rkrτmτm[(b−i)y(k)+vg(x(k−h(k)))]tz2[(b−i)y(k)+vg(x(k−h(k)))]−rk∑l=k−h(k)k−1η1t(l)z1η1(l)−rk∑l=k−hmk−h(k)η1t(l)z1η1(l)−rk∑l=k−τ(k)k−1η2t(l)z2η2(l)−rk∑l=k−τmk−τ(k)η2t(l)z2η2(l).By Lemma 2.2, we
have (3.10)−∑l=k−h(k)k−1η1t(l)z1η1(l)≤2∑l=k−h(k)k−1η1t(l)ltξ(k)+∑l=k−h(k)k−1ξt(k)lz1−1ltξ(k)=2[xt(k)−xt(k−h(k))]ltξ(k)+h(k)ξt(k)lz1−1ltξ(k)≤ξt(k)(ψ1lt+lψ1)ξ(k)+hmξt(k)lz1−1ltξ(k),(3.11)−∑l=k−hmk−h(k)η1t(l)z1η1(l)≤2∑l=k−hmk−h(k)η1t(l)mtξ(k)+∑l=k−hmk−h(k)ξt(k)mz1−1mtξ(k)=2[xt(k−h(k))−xt(k−hm)]mtξ(k)+(hm−h(k))ξt(k)mz1−1mtξ(k)≤ξt(k)(ψ2mt+mψ2)ξ(k)+(hm−hm)ξt(k)mz1−1mtξ(k),(3.12)−∑l=k−τ(k)k−1η2t(l)z2η2(l)≤2∑l=k−τ(k)k−1η2t(l)ntξ(k)+∑l=k−τ(k)k−1ξt(k)nz2−1ntξ(k)=2[yt(k)−yt(k−τ(k))]ntξ(k)+τ(k)ξt(k)nz2−1ntξ(k)≤ξt(k)(ψ3nt+nψ3)ξ(k)+τmξt(k)nz2−1ntξ(k),(3.13)−∑l=k−τmk−τ(k)η2t(l)z2η2(l)≤2∑l=k−τmk−τ(k)η2t(l)ttξ(k)+∑l=k−τmk−τ(k)ξt(k)tz2−1ttξ(k)=2[yt(k−τ(k))−yt(k−τm)]ttξ(k)+(τm−τ(k))ξt(k)tz2−1ttξ(k)≤ξt(k)(ψ4tt+tψ4)ξ(k)+(τm−τm)ξt(k)tz2−1ttξ(k),where(3.14)l=[l1tl2tl3tl4tl5tl6tl7tl8t]t,m=[m1tm2tm3tm4tm5tm6tm7tm8t]t,n=[n1tn2tn3tn4tn5tn6tn7tn8t]t,t=[t1tt2tt3tt4tt5tt6tt7tt8t]t,ψ1=[i0000−i00]t,ψ2=[00000i−i0]t,ψ3=[0−i00i000]t,ψ4=[0i−i00000]t,and ξ(k)=[xT(k)yT(k−τ(k))yT(k−τM)fT(y(k−τ(k)))yT(k)xT(k−h(k))xT(k−hM)gT(x(k−h(k)))]. By inequalities (2.5), it is well known that
there exist positive diagonally matrices Di≥0(i=1,2) such that the following inequalities
hold:(3.15)rk[yt(t−τ(t))d1y(t−τ(t))−ft(y(t−τ(t)))l1−1d1l1−1f(y(t−τ(t)))]≥0,rk[xt(t−h(t))d2x(t−h(t))−gt(x(t−h(t)))l2−1d2l2−1g(x(t−h(t)))]≥0,where L1=diag{l11,l12,…,l1m},L2=diag{l21,l22,…,l2n}.
Substituting (3.10)–(3.13) into (3.9), we
have(3.16)δv(k)≤δv1(k)+δv2(k)+δv3(k)+δv4(k)+δv5(k)+rk[yt(t−τ(t))d1y(t−τ(t)))−ft(y(t−τ(t)))l1−1d1l1−1f(y(t−τ(t)))]+rk[xt(t−h(t))d2x(t−h(t)))−gt(x(t−h(t)))l2−1d2l2−1g(x(t−h(t)))]≤rkξt(k)[ω+hmlz1−1lt+(hm−hm)mz1−1mt+τmnz2−1nt+(τm−τm)tz2−1tt]ξ(k),where Ω,L,M,N, and T are defined in Theorem 3.1. Thus, if LMI (3.1)
holds, it can be concluded that ΔV(k)<0 according to Schur complement, which implies
that V(k)<V(0).
Note that(3.17)v1(0)=xt(0)p1x(0)+yt(0)p2y(0)≤λm(p1)sups∈n[−hm,0]∥ϕ(s)∥2+λm(p2)sups∈n[−τm,0]∥ψ(s)∥2,(3.18)v2(0)=∑l=−h(0)−1rlxt(l)q1x(l)+∑l=−τ(0)−1rlyt(l)q2y(l)≤λm(q1)1−r−hmr−1sups∈n[−hm,0]∥ϕ(s)∥2+λm(q2)1−r−τmr−1sups∈n[−τm,0]∥ψ(s)∥2,(3.19)v3(0)=∑θ=−hm+1−hm∑l=θ−1rlxt(l)q1x(l)+∑θ=−τm+1−τm∑l=θ−1rlyt(l)q2y(l)≤(hm−hm)∑l=hm−1rlxt(l)q1x(l)+(τm−τm)∑l=τm−1rlyt(l)q2y(l)≤(hm−hm)λm(q1)1−r−hmr−1sups∈n[−hm,0]∥ψ(s)∥2+(τm−τm)λm(q2)1−r−τmr−1sups∈n[−τm,0]∥ψ(s)∥2,(3.20)v4(0)=∑l=−hm−1rlxt(l)r1x(l)+∑l=−τm−1rlyt(l)r2y(l)≤λm(r1)1−r−hmr−1sups∈n[−hm,0]∥ϕ(s)∥2+λm(r2)1−r−τmr−1sups∈n[−τm,0]∥ψ(s)∥2,(3.21)v5(0)=rhm∑θ=−hm−1∑l=θ−1rlη1t(l)z1η1(l)+rτm∑θ=−τm−1∑l=θ1rlη2t(l)z2η2(l)≤rhmhmλm(z1)∑l=−hm−1rlη1t(l)η1(l)+rτmτmλm(z2)∑l=−τm−1rlη2t(l)η2(l)≤rhmhmλm(z1)1−r−hmr−1sups∈n[−hm,−1]∥η1(s)∥2+rτmτmλm(z2)1−r−τmr−1sups∈n[−τm,−1]∥η2(s)∥2,(3.22)sups∈n[−hm,−1]∥η1(s)∥2=sups∈n[−hm,−1]∥x(s+1)−x(s)∥2≤2sups∈n[−hm,−1](∥x(s+1)∥2+∥x(s)∥2)≤4sups∈n[−hm,0]∥ϕ(s)∥2,(3.23)sups∈n[−τm,−1]∥η2(s)∥2=sups∈n[−τm,−1]∥y(s+1)−y(s)∥2≤2sups∈n[−τm,−1](∥y(s+1)∥2+∥y(s)∥2)≤4sups∈n[−τm,0]∥ψ(s)∥2.Substituting (3.22), (3.23) into
(3.21), and combining (3.17)–(3.21), then we have(3.24)V(0)=∑i=15Vi(0)≤ρ1sups∈N[−hM,0]∥ϕ(s)∥2+ρ2sups∈N[−τM,0]∥ψ(s)∥2,where(3.25)ρ1=λm(p1)+[(hm−hm+1)λm(q1)+λm(r1)+4rhmhmλm(z1)]1−r−hmr−1,ρ2=λm(p2)+[(τm−τm+1)λm(q2)+λm(r2)+4rτmτmλm(z2)]1−r−τmr−1.On the other
hand,(3.26)V(k)≥rkxT(k)P1x(k)+rkyT(k)P2y(k)≥rk{λm(P1)∥x(k)∥2+λm(P2)∥y(k)∥2}.Therefore, we can
obtain(3.27)∥x(k)∥2+∥y(k)∥2≤αβ(sups∈N[−hM,0]∥ϕ(s)∥2+sups∈N[−τM,0]∥ψ(s)∥2)r−k,where α=max{ρ1,ρ2},β=min{λm(P1),λm(P2)}.
Obviously, α/β>1,
by Definition 2.1, the system (2.7) or (2.1) is globally exponentially stable.Remark 3.2.
Based on the linear matrix inequality (LMI) technique and Lyapunov stability
theory, the exponential stability for discrete-time delayed BAM neural network
(2.1) was investigated in [26–28]. However, it should be noted that
the results in [26] are based on the unreasonably severe constraints on the
delay functions 1<τ(k+1)<1+τ(k),1<h(k+1)<1+h(k),
and the results in [27, 28] are based on the simple Lyapunov functionals. In
this paper, in order to obtain the less conservative stability criterion, the
novel Lyapunov functional V(k) is employed, which contains V4(k),V5(k),
and more general than the traditional ones [26–28]. Moreover, some slack
matrices, which bring much flexibility in solving LMI, are introduced in this
paper.Remark 3.3.
By settingr=1 in Theorem 3.1, we can obtain the global
asymptotic stability criterion of discrete-time BAM neural network (2.7).Remark 3.4.
Theorem3.1 in this paper depends on both the delay upper bounds τM,hM and the delay intervals τM−τm and hM−hm.Remark 3.5.
The proposed method in this paper can be generalized to more complex neural
networks, such as delayed discrete-time BAM neural networks with parameter
uncertainties [30, 31] and stochastic perturbations [30–32], delayed interval
discrete-time BAM neural networks [28], and discrete-time analogues of BAM
neural networks with mixed delays [32, 33].


## 4. Numerical Examples

Example 4.1.
Consider the delayed discrete-time BAM neural network (2.7) with the following
parameters:(4.1)A=[0.8000.9],B=[0.5000.4],W=[0.1−0.01−0.2−0.1],V=[0.150−0.20.1].The activation functions satisfy
Assumptions (A1) and (A2) with(4.2)L1=[1001],L2=[1001].For this example, by Theorem3.1 in this paper with r=1,
we can obtain some maximum allowable delay bounds for guaranteeing the
asymptotic stability of this system. For a comparison with [27, Theorem 1] and
in [28, Corollary 1], we made Table 1. For this
example, it is obvious that our
result is less conservative than those in [27, 28].Table 1
Maximum allowable delay bounds for Example4.1.
τm=hm246810152025τM=hM [27, 28]68101214192429τM=hM Theorem 3.11112131516212530Example 4.2 (see [27]).
Consider the delayed
discrete-time BAM NN (2.7) with the following parameters:(4.3)A=[150015],B=[11000110],W=[018180],V=[−12000−120].The activation functions satisfy
Assumptions (A1) and (A2) with(4.4)L1=[1001],L2=[1001].
Now, we assumeτ(k)=h(k)=4−2sin((π/2)k),r=1.55.
It is obvious that 2≤τ(k)=h(k)≤6,
that is, hm=τm=2,hM=τm=6. In this case, it is found that the exponential
conditions proposed in [27, 28] is not satisfied. However, it can be concluded
that this system is globally exponentially stable by using Theorem 3.1 in this
paper. If we assume hm=τm=τM=2,r=2.7. Using
[27, Theorem 1] and [28, Corollary 1],
the achieved maximum allowable delays for guaranteeing the globally
exponentially stable of this system are hM=3,hM=3, respectively. However, we can obtain the delay
bound hM=4 by using Theorem 3.1 in this paper. For this
example, it is seen that our result improve some existing results [27, 28].


## 5. Conclusion

In this paper,
we consider the delay-dependent exponential stability for discrete-time BAM
neural networks with time-varying delays. Based on the Lyapunov stability
theory and linear matrix inequality technique, the novel delay-dependent
stability criterion is obtained. Two numerical examples show that the proposed
stability condition in this paper is less conservative than previously
established ones.



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