# Exponential Stability for Discrete-Time Stochastic BAM Neural Networks with Discrete and Distributed Delays

Authors: R. Raja; R. Sakthivel; S. Marshal Anthoni
Journal: ISRN Discrete Mathematics
 (2011)

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## Abstract

This paper deals with the stability analysis problem for a class of discrete-time stochastic
BAM neural networks with discrete and distributed time-varying delays. By constructing a suitable
Lyapunov-Krasovskii functional and employing M-matrix theory, we find some sufficient
conditions ensuring the global exponential stability of the equilibrium point for stochastic BAM
neural networks with time-varying delays. The conditions obtained here are expressed in terms
of LMIs whose feasibility can be easily checked by MATLAB LMI Control toolbox. A numerical
example is presented to show the effectiveness of the derived LMI-based stability conditions.


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## Body

## 1. Introduction

Recently, the study of Bidirectional associative memory neural networks has attracted the attention of many researchers due to its applications in many fields such as pattern recognition, automatic control, associative memory, signal processing, and optimization; see, for example, [1–9]. The (BAM) neural networks model, proposed by Kosko [10, 11], is a two layer nonlinear feedback network model and it was described that the neurons in one layer are fully interconnected to the neurons in the other layer, while there are no interconnections among neurons in the same layer.Furthermore, due to the finite switching speed of neuron amplifiers and the finite speed of signal propagation time delays are unavoidable in the implementation of neural networks [12–14]. According to the way it occurs, time delay can be classified as two types: discrete and distributed delays. Discrete time-delay is relatively easier to be identified in practice and hence the stability analysis for BAM with discrete delays has been an attractive subject of research in the past few years; see [15, 16]. On the other hand, due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, a neural network usually has a spatial nature. Therefore, it is necessary to introduce continuously distributed delays over a certain duration of time; see [17, 18].Moreover, in implementations of neural networks, stochastic disturbances are inevitable owing to thermal noise in electronic devices. Practically, the stochastic phenomenon usually appears in the electrical circuit design of neural networks. The stochastic effects can have the ability to destabilize a neural system. Therefore, it is significant and of importance to consider stochastic effects to the stability property of the neural networks with delays. It is noted that most of the BAM neural networks have been assumed to act in a continuous-time manner. However, when it comes to the implementation of discrete-time BAM networks, there are only few works appeared in the literature; see [6, 19–24] and the references cited therein. Therefore, there is a crucial need to study the dynamics of discrete-time BAM neural networks and it becomes more significant from practical point of view. In [19], Gao and Cui discussed the global robust exponential stability of discrete-time interval BAM neural networks with time-varying delays, and in [24], the authors investigated the global exponential stability for discrete-time BAM neural network with time variable delay. In the above said references the stability problem for BAM neural networks is considered only with discrete delay, and distributed delay has not been taken into account and remains challenging. So, our main aim in this work is to make the first attempt to shorten such a gap.Motivated by the above points, in this paper, we will study the exponential stability problem for a new class of discrete-time stochastic BAM neural networks with both discrete and distributed delays. The existence of the equilibrium point is proved under mild conditions on the activation functions. By constructing an appropriate Lyapunov-Krasovskii functional, a linear matrix inequality (LMI) approach is developed to establish sufficient conditions for the discrete-time BAM neural networks to be globally exponentially stable in the mean square. Here, we note that the LMIs can be easily solved by using Matlab LMI toolbox, and no tuning of parameters is involved. Finally, a numerical example is presented to show the usefulness of the derived LMI-based stability conditions.Notations. Throughout this paper, ℝn and ℝn×m denote, respectively, the n-dimensional Euclidean space and the set of all n×m real matrices. I denotes the identity matrix with appropriate dimensions and diag(·) denotes the diagonal matrix. For real symmetric matrices X and Y, the notation X≥Y (resp., X>Y) means that the matrix X-Y is positive semidefinite (resp., positive definite). ℕ={1,2,…,n} and ∥·∥ stands for the Euclidean norm in ℝn. λmax(X) (resp., λmin(X)) stands for the maximum (resp., minimum) eigenvalue of the matrix X. The symbol * within a matrix represents the symmetric term of the matrix.


## 2. Problem Description and Preliminaries

Consider the following discrete-time stochastic BAM neural networks with both discrete and distributed delays of the following form:(2.1)xi(k+1)=[aixi(k)+∑j=1ncjifj(yj(k))+∑j=1nwjigj(yj(k-τji(k)))+∑j=1nmji∑M=1+∞μMhj(yj(k-M))+Ii]+δji(xi(k),yj(k-τji(k)),k)w1(k),i∈N,yj(k+1)=[bjyj(k)+∑i=1ndijf̂i(xi(k))+∑i=1nvijĝi(xi(k-σij(k)))+∑i=1nnij∑N=1+∞ρNĥi(xi(k-N))+Jj]+χij(yj(k),xi(k-σij(k)),k)w2(k),j∈N,
or, in an equivalent form,(2.2)x(k+1)=[Ax(k)+Cf(y(k))+Wg(y(k-τ(k)))+M∑M=1+∞μMh(y(k-M))+I]+δ(x(k),y(k-τ(k)),k)w1(k),y(k+1)=[By(k)+Df̂(x(k))+Vĝ(x(k-σ(k)))+N∑N=1+∞ρNĥ(x(k-N))+J]+χ(y(k),x(k-σ(k)),k)w2(k),
for k=1,2,…, where x(k) and y(k) are the neural state vector; A=diag{a1,a2,…,an} and B=diag{b1,b2,…,bn} are the state feedback coefficient matrices; C=[cij]n×n,D=[dij]n×n,W=[wij]n×n,V=[vij]n×n,M=[mij]n×n, and N=[nij]n×n are, respectively, the connection weight matrices, the discretely delayed connection weight matrices, and distributed delayed connection weight matrices; τ(k) and σ(k) denote the discrete time-varying delays satisfying(2.3)τm≤τ(k)≤τM,σm≤σ(k)≤σM,
where τm,τM,σm, and σM are known positive integer; M,N denotes the distributed time-varying delays.Then
(2.4)f(y(k))=[f1(y1(k)),f2(y2(k)),…,fn(yn(k))]T,f̂(x(k))=[f̂1(x1(k)),f̂2(x2(k)),…,f̂n(xn(k))]T,g(y(k))=[g1(y1(k)),g2(y2(k)),…,gn(yn(k))]T,ĝ(x(k))=[ĝ1(x1(k)),ĝ2(x2(k)),…,ĝn(xn(k))]T,h(y(k))=[h1(y1(k)),h2(y2(k)),…,hn(yn(k))]T,ĥ(x(k))=[ĥ1(x1(k)),ĥ2(x2(k)),…,ĥn(xn(k))]T,
denote the neuron activation functions. The constant vectors J=[J1,J2,…,Jn]T and I=[I1,I2,…,In]T are the external inputs from outside the system; μℳ,(ℳ=1,2,…) and ρ𝒩,(𝒩=1,2,…) are scalar constants, where w1(k) and w2(k) are scalar Wiener process (Brownian motion) on the probability space (Ω,ℱ,𝔓) with(2.5)E[w1(k)]=0,E[w12(k)]=1,E[w1(i)w1(j)]=0(i≠j),E[w2(k)]=0,E[w22(k)]=1,E[w2(i)w2(j)]=0(i≠j),
with 𝔼(·) being the mathematical expectation operator; δ:ℝn×ℝn×ℝ→ℝn and  χ:ℝn×ℝn×ℝ→ℝn are the nonlinear vector function representing the disturbance intensities.In this paper, we make following assumptions for the neuron activation functions.Assumption 1.
Forj,i∈{1,2,…,n}, the neuron activation functions fj(·), f̂i(·), gj(·), ĝi(·), hj(·), and ĥi(·) in (2.2) are continuous as well as bounded on ℝ.Assumption 2.
Forj,i∈{1,2,…,n}, the neuron activation functions in (2.2) satisfy
(2.6)lj-≤fj(s1)-fj(s2)s1-s2≤lj+,∀s1,s2∈R,ui-≤f̂i(t1)-f̂i(t2)t1-t2≤ui+,∀t1,t2∈R,mj-≤gj(s1)-gj(s2)s1-s2≤mj+,∀s1,s2∈R,vi-≤ĝi(t1)-ĝi(t2)t1-t2≤vi+,∀t1,t2∈R,nj-≤hj(s1)-hj(s2)s1-s2≤nj+,∀s1,s2∈R,wi-≤ĥi(t1)-ĥi(t2)t1-t2≤wi+,∀t1,t2∈R,
where lj-,lj+,mj-,mj+,nj-,nj+,ui-,ui+,vi-,vi+,wi-, and wi+ are some constants.Remark 2.1.
Assumption2 was first introduced by Liu et al. [25]. The constants lj-,lj+,mj-,mj+,nj-,nj+,ui-,ui+,vi-,vi+,wi-, and wi+ in Assumption 2 are allowed to be positive, negative, or zero. So, the activation functions used in this paper may be nonmonotonic and more general than the usual sigmoid functions and Lipschitz functions. Such conditions are very rude in quantifying the lower and upper bounds of the activation functions; hence we use generalized activation functions, because it is very helpful for using LMI-based technique to reduce the possible conservatism.
In order to simplify our proof, we shift the equilibrium pointx*=(x1*,x2*,…,xn*)T and y*=(y1*,y2*,…,yn*)T of system (2.2) to the origin. Let u(k)=x(k)-x* and v(k)=y(k)-y*; then system (2.2) can be transformed to(2.7)u(k+1)=[Au(k)+Cf(v(k))+Wg(v(k-τ(k)))+M∑M=1+∞μMh(v(k-M))]+δ(u(k),v(k-τ(k)),k)w1(k),v(k+1)=[Bv(k)+Df̂(u(k))+Vĝ(u(k-σ(k)))+N∑N=1+∞ρNĥ(u(k-N))]+χ(v(k),u(k-σ(k)),k)w2(k),
where(2.8)u(k)=(u1(k),u2(k),…,un(k))T,v(k)=(v1(k),v2(k),…,vn(k))T,f(v(k))=[f1(v1(k)),f2(v2(k)),…,fn(vn(k))]T=f(y(k)+y*)-f(y*),g(v(k))=[g1(v1(k)),g2(v2(k)),…,gn(vn(k))]T=g(y(k)+y*)-g(y*),h(v(k))=[h1(v1(k)),h2(v2(k)),…,hn(vn(k))]T=h(y(k)+y*)-h(y*),f̂(u(k))=[f̂1(u1(k)),f̂2(u2(k)),…,f̂n(un(k))]T=f̂(x(k)+x*)-f̂(x*),ĝ(u(k))=[ĝ1(u1(k)),ĝ2(u2(k)),…,ĝn(un(k))]T=ĝ(x(k)+x*)-ĝ(x*),ĥ(u(k))=[ĥ1(u1(k)),ĥ2(u2(k)),…,ĥn(un(k))]T=ĥ(x(k)+x*)-ĥ(x*).Assumption 3.
Obviously, the activation functionsfj,f̂i,gj,ĝi,hj,andĥi(i,j∈ℕ) satisfy the following condition:
(2.9)lj-≤fj(s)s≤lj+,∀s∈R,ui-≤f̂i(t)t≤ui+,∀t∈R,mj-≤gj(s)s≤mj+,∀s∈R,vi-≤ĝi(t)t≤vi+,∀t∈R,nj-≤hj(s)s≤nj+,∀s∈R,wi-≤ĥi(t)t≤wi+,∀t∈R.Assumption 4.
The constantsμℳ,ρ𝒩≥0 satisfy the following convergent conditions:
(2.10)∑M=1+∞μM<+∞,∑M=1+∞MμM<+∞,∑N=1+∞ρN<+∞,∑N=1+∞NρN<+∞.Remark 2.2.
Assumption4 ensures that the terms M∑ℳ=1+∞μℳh(v(k-ℳ)) and N∑𝒩=1+∞ρ𝒩ĥ(u(k-𝒩)) are convergent, which is significant for the subsequent analysis.Assumption 5.
There exist constant matricesG and K such that
(2.11)δT(x,y,k)δ(x,y,k)≤|Gx|2,∀x,y∈Rn,χT(x,y,k)χ(x,y,k)≤|Ky|2,∀x,y∈Rn.The following definition and lemmas will be essential in employing the exponential stability conditions.Definition 2.3.
The delayed discrete-time stochastic BAM neural network (2.7) is said to be globally exponentially stable, if there exist two positive scalars ν>0 and 0<𝒢<1 such that
(2.12)‖u(k)‖+‖v(k)‖≤νGk(sup⁡-σM≤s≤0‖u(s)‖+sup⁡-τM≤s≤0‖v(s)‖).Lemma 2.4.
LetX and Y be any n-dimensional real vectors and let P be an n×n positive semidefinite matrix. Then, the following matrix inequality holds:
(2.13)2XTPY≤XTPX+YTPY.Lemma 2.5.
LetM∈ℝn×n be a positive semidefinite matrix, xi∈ℝn, and ai≥0,(i=1,2,…). If the series concerned are convergent, the following inequality holds:
(2.14)(∑i=1+∞aixi)TM(∑i=1+∞aixi)≤(∑i=1+∞ai)∑i=1+∞aixiTMxi.In the rest of the paper, we will focus on the stability analysis of SBAMNN (2.7). By choosing an appropriate Lyapunov-Krasovskii functional, we aim to develop an LMI approach for deriving sufficient conditions under which the SBAMNN (2.7) is globally exponentially stable.


## 3. Main Results

Now, we are in a position to state our main results in the following theorem.Theorem 3.1.
Under Assumptions1–5, the discrete-time stochastic BAM neural network (2.7) is globally exponentially stable in the mean square, if there exist constants λ0>0 and ϵ0>0 if there exist diagonal matrices Λ1=diag{λ1(1),λ2(1),…,λn(1)}>0, Λ2=diag{λ1(2),λ2(2),…,λn(2)}>0, Γ1=diag{γ1(1),γ2(1),…,γn(1)}>0, Γ2=diag{γ1(2),γ2(2),…,γn(2)}>0, Ω1=diag{ω1(1),ω2(1),…,ωn(1)}>0, and Ω2=diag{ω1(2),ω2(2),…,ωn(2)}>0 and positive definite matrices P1>0,P2>0,Q1>0,Q2>0,R1>0,andR2>0, such that the following LMIs hold:
(3.1)P1<λ0I,Ξ1=[Π110Λ2U2Γ2V20Ω2W20*-Q200000**Π330000***-Γ2000****Π5500*****-Ω20******Π77]<0,P2<ϵ0I,Ξ2=[Θ110Λ1L2Γ1M20Ω1N20*-Q100000**Θ330000***-Γ1000****Θ5500*****-Ω10******Θ77]<0,
where
(3.2)Π11=ATP1A-2P1-Λ2U1-Γ2V1-Ω2W1+(1+σM-σm)Q2+ρ¯R2+λ0GTG,Π33=BDP2DTBT-Λ2+2P2,Π55=BVP2VTBT+DVP2VTDT+P2,Π77=BNP2NTBT+DNP2NTDT+VNP2NTVT-ρ-1R2,Θ11=BTP2B-2P2-Λ1L1-Γ1M1-Ω1N1+(1+τM-τm)Q1+μ¯R1+ϵ0KTK,Θ33=ACP1CTAT-Λ1+2P1,Θ55=AWP1WTAT+CWP1WTCT+P1,Θ77=AMP1MTAT+CMP1MTCT+WMP1MTWT-μ-1R1,L1=diag⁡{l1+l1-,l2+l2-,…,ln+ln-},L2=diag⁡{l1++l1-2,l2++l2-2,…,ln++ln-2},M1=diag⁡{m1+m1-,m2+m2-,…,mn+mn-},M2=diag⁡{m1++m1-2,m2++m2-2,…,mn++mn-2},N1=diag⁡{n1+n1-,n2+n2-,…,nn+nn-},N2=diag⁡{n1++n1-2,n2++n2-2,…,nn++nn-2},U1=diag⁡{u1+u1-,u2+u2-,…,un+un-},U2=diag⁡{u1++u1-2,u2++u2-2,…,un++un-2},V1=diag⁡{v1+v1-,v2+v2-,…,vn+vn-},V2=diag⁡{v1++v1-2,v2++v2-2,…,vn++vn-2},W1=diag⁡{w1+w1-,w2+w2-,…,wn+wn-},W2=diag⁡{w1++w1-2,w2++w2-2,…,wn++wn-2},μ¯=∑k=1+∞μk,ρ¯=∑k=1+∞ρk.Proof.
Let us choose the Lyapunov-Krasovskii functional as(3.3)V1(k)=uT(k)P1u(k)+vT(k)P2v(k),V2(k)=∑i=k-τ(k)k-1vT(i)Q1v(i)+∑i=k-σ(k)k-1uT(i)Q2u(i),V3(k)=∑j=k-τM+1k-τm∑i=jk-1vT(i)Q1v(i)+∑j=k-σM+1k-σm∑i=jk-1uT(i)Q2u(i),V4(k)=∑i=1+∞μi∑j=k-ik-1vT(i)R1v(i)+∑i=1+∞ρi∑j=k-ik-1uT(i)R2u(i).
In order to analyze the global exponential stability of the stochastic BAM neural network, we calculate differencesΔV(k) of the Lyapunov function V(k), along with the trajectories of the BAM neural network (2.7); then we have(3.4)ΔV(k)=ΔV1(k)+ΔV2(k)+ΔV3(k)+ΔV4(k),
where
(3.5)ΔV1(k)=uT(k+1)P1u(k+1)-uT(k)P1u(k)+vT(k+1)P2v(k+1)-vT(k)P2v(k)=[Au(k)+Cf(v(k))+Wg(v(k-τ(k)))+M∑M=1+∞μMh(v(k-M))]T×P1[Au(k)+Cf(v(k))+Wg(v(k-τ(k)))+M∑M=1+∞μMh(v(k-M))]-uT(k)P1u(k)+[Bv(k)+Df̂(u(k))+Vĝ(u(k-σ(k)))+N∑N=1+∞ρNĥ(u(k-N))]T×P2[Bv(k)+Df̂(u(k))+Vĝ(u(k-σ(k)))+N∑N=1+∞ρNĥ(u(k-N))]-vT(k)P2v(k)+δT(u,v,k)P1δ(u,v,k)+χT(u,v,k)P2χ(u,v,k)≤uT(k)(AP1A-P1)u(k)+2uT(k)AP1Cf(v(k))+2uT(k)AP1Wg(v(k-τ(k)))+2uT(k)AP1M(∑M=1+∞μMh(v(k-M)))+fT(v(k))CP1Cf(v(k))+2fT(v(k))×CP1Wg(v(k-τ(k)))+2fT(v(k))CP1M(∑M=1+∞μMh(v(k-M)))+gT(v(k-τ(k)))WP1Wg(v(k-τ(k)))+2gT(v(k-τ(k)))WP1M×(∑M=1+∞μMh(v(k-M)))+(∑M=1+∞μMh(v(k-M)))TMP1M×(∑M=1+∞μMh(v(k-M)))+vT(k)(BP2B-P2)v(k)+2vT(k)BP2Df̂(u(k))+2vT(k)BP2Vf̂(u(k-σ(k)))+2vT(k)BP2N(∑N=1+∞ρNĥ(u(k-N)))+f̂T(u(k))DP2Df̂(u(k))+2f̂T(u(k))DP2Vĝ(u(k-σ(k)))+2f̂T(u(k))DP2N×(∑N=1+∞ρNĥ(u(k-N)))+ĝT(u(k-σ(k)))VP2Vĝ(u(k-σ(k)))+2ĝT(u(k-σ(k)))VP2N(∑N=1+∞ρNĥ(u(k-N)))+(∑N=1+∞ρNĥ(u(k-N)))N×P2N(∑N=1+∞ρNĥ(u(k-N)))+uT(k)λ0GTGu(k)+vT(k)ϵ0KTKv(k).
By using Lemma 2.4, we have
(3.6)2uT(k)AP1Cf(v(k))≤uT(k)P1u(k)+fT(v(k))ACP1CTATf(v(k)),2uT(k)AP1Wg(v(k-τ(k)))≤uT(k)P1u(k)+gT(v(k-τ(k)))AWP1WTATg(v(k-τ(k))),2uT(k)AP1M(∑M=1+∞μMh(v(k-M)))≤uT(k)P1u(k)+(∑M=1+∞μMh(v(k-M)))TAMP1MTAT(∑M=1+∞μMh(v(k-M))),2fT(v(k))CP1Wg(v(k-τ(k)))≤fT(v(k))P1f(v(k))+gT(v(k-τ(k)))CWP1WTCTg(v(k-τ(k))),2fT(v(k))CP1M(∑M=1+∞μMh(v(k-M)))≤fT(v(k))P1f(v(k))+(∑M=1+∞μMh(v(k-M)))T×CMP1MTCT(∑M=1+∞μMh(v(k-M))),2gT(v(k-τ(k)))WP1M(∑M=1+∞μMh(v(k-M)))≤gT(v(k-τ(k)))P1g(v(k-τ(k)))+(∑M=1+∞μMh(v(k-M)))T×WMP1MTWT(∑M=1+∞μMh(v(k-M))),2vT(k)BP2Df̂(u(k))≤vT(k)P2v(k)+f̂T(u(k))BDP2DTBTf̂(u(k)),2vT(k)BP2Vĝ(u(k-σ(k)))≤vT(k)P2v(k)+ĝT(u(k-σ(k)))BVP2VTBTĝ(u(k-σ(k))),2vT(k)BP2N(∑N=1+∞ρNĥ(u(k-N)))≤vT(k)P2v(k)+(∑N=1+∞ρNĥ(u(k-N)))TBNP2NTBT(∑N=1+∞ρNĥ(u(k-N))),2f̂T(u(k))DP2Vĝ(u(k-σ(k)))≤f̂T(u(k))P2f̂(u(k))+ĝT(u(k-σ(k)))DVP2VTDTĝ(u(k-σ(k))),2f̂T(u(k))DP2N(∑N=1+∞ρNĥ(u(k-N)))≤f̂T(u(k))P2f̂(v(k))+(∑N=1+∞ρNĥ(u(k-N)))T×DNP2NTDT(∑N=1+∞ρNĥ(u(k-N))),2ĝT(u(k-σ(k)))VP2N(∑N=1+∞ρNĥ(u(k-N)))≤ĝT(u(k-σ(k)))P2ĝ(u(k-σ(k)))+(∑N=1+∞ρNĥ(u(k-N)))TVNP2NTVT(∑N=1+∞ρNĥ(u(k-N))),ΔV2(k)=V2(k+1)-V2(k)=∑i=k+1-τ(k+1)kvT(i)Q1v(i)-∑i=k-τ(k)k-1vT(i)Q1v(i)+∑i=k+1-σ(k+1)kuT(i)Q2u(i)-∑i=k-σ(k)k-1uT(i)Q2u(i)=vT(k)Q1v(k)-vT(k-τ(k))Q1v(k-τ(k))+∑i=k+1-τ(k+1)k-1vT(i)Q1v(i)-∑i=k-τ(k)+1k-1vT(i)Q1v(i)+uT(k)Q2u(k)-uT(k-σ(k))Q2u(k-σ(k))+∑i=k+1-σ(k+1)k-1uT(i)Q2u(i)-∑i=k-σ(k)+1k-1uT(i)Q2u(i)=vT(k)Q1v(k)-vT(k-τ(k))Q1v(k-τ(k))+∑i=k-τ(k+1)+1k-τmvT(i)Q1v(i)+∑i=k-τm+1k-1vT(i)Q1v(i)-∑i=k-τ(k)+1k-1vT(i)Q1v(i)+uT(k)Q2u(k)-uT(k-σ(k))Q2u(k-σ(k))+∑i=k-σ(k+1)+1k-σmuT(i)Q2u(i)+∑i=k-σm+1k-1uT(i)Q2u(i)-∑i=k-σ(k)+1k-1uT(i)Q2u(i)=vT(k)Q1v(k)-vT(k-τ(k))Q1v(k-τ(k))+∑i=k-τM+1k-τmvT(i)Q1v(i)+uT(k)Q2u(k)-uT(k-σ(k))Q2u(k-σ(k))+∑i=k-σM+1k-σmuT(i)Q2u(i),(3.7)ΔV3(k)=V3(k+1)-V3(k)=∑j=k-τM+2k-τm+1∑i=jkvT(i)Q1v(i)-∑j=k-τM+1k-τm∑i=jkvT(i)Q1v(i)+∑j=k-σM+2k-σm+1∑i=jkuT(i)Q2u(i)-∑j=k-σM+1k-σm∑i=jkuT(i)Q2u(i)=∑j=k-τM+1k-τm∑i=jkvT(i)Q1v(i)-∑j=k-τM+1k-τm∑i=jk-1vT(i)Q1v(i)+∑j=k-σM+1k-σm∑i=jkuT(i)Q2u(i)-∑j=k-σM+1k-σm∑i=jk-1uT(i)Q2u(i)=∑j=k-τM+1k-τm[vT(k)Q1v(k)-vT(j)Q1v(j)]+∑j=k-σM+1k-σm[uT(k)Q2u(k)-uT(j)Q2u(j)]≤(τM-τm)vT(k)Q1v(k)-∑j=k-τM+1k-τmvT(i)Q1v(i)+(σM-σm)uT(k)Q2u(k)-∑j=k-σM+1k-σmuT(i)Q2u(i),(3.8)ΔV4(k)=V4(k+1)-V4(k)=∑i=1+∞μi∑j=k-i+1kvT(i)R1v(i)-∑i=1+∞μi∑j=k-ik-1vT(i)R1v(i)+∑i=1+∞ρi∑j=k-i+1kuT(i)R2u(i)-∑i=1+∞ρi∑j=k-ik-1uT(i)R2u(i)=∑i=1+∞μi[vT(k)R1v(k)-vT(k-i)R1v(k-i)]+∑i=1+∞ρi[uT(k)R2u(k)-uT(k-i)R2u(k-i)]≤μ¯vT(k)R1v(k)-1μ¯(∑M=1+∞μMh(v(k-M)))TR1(∑M=1+∞μMh(v(k-M)))+ρ¯uT(k)R2u(k)-1ρ¯(∑N=1+∞ρNĥ(u(k-N)))TR2(∑N=1+∞ρNĥ(u(k-N))).
It is clear from (2.9) that
(3.9)(fj(vj(k))-lj+vj(k))(fj(vj(k))-lj-vj(k))≤0,(3.10)(gj(vj(k))-mj+vj(k))(gj(vj(k))-mj-vj(k))≤0,(3.11)(hj(vj(k))-nj+vj(k))(hj(vj(k))-nj-vj(k))≤0,(3.12)(f̂i(ui(k))-ui+ui(k))(f̂i(ui(k))-ui-ui(k))≤0,(3.13)(ĝi(ui(k))-vi+ui(k))(ĝi(ui(k))-vi-ui(k))≤0,(3.14)(ĥi(ui(k))-wi+ui(k))(ĥi(ui(k))-wi-ui(k))≤0,
which is equivalent to
(3.15)[v(k)f(v(k))]T[lj+lj-ejejT-lj++lj-2ejejT-lj++lj-2ejejTejejT][v(k)f(v(k))]≤0,k=1,2,…,n,
where ek denotes the unit column vector having “1” element on its kth row and zeros elsewhere.
Consequently,(3.16)∑j=1nλj(1)[v(k)f(v(k))]T[lj+lj-ejejT-lj++lj-2ejejT-lj++lj-2ejejTejejT][v(k)f(v(k))]≤0,⟹[v(k)f(v(k))]T[Λ1L1-Λ1L2-Λ1L2Λ1][v(k)f(v(k))]≤0.
Similarly, from (3.10)–(3.14), we have
(3.17)[v(k)g(v(k))]T[Γ1M1-Γ1M2-Γ1M2Γ1][v(k)g(v(k))]≤0,(3.18)[v(k)h(v(k))]T[Ω1N1-Ω1N2-Ω1N2Ω1][v(k)h(v(k))]≤0,(3.19)[u(k)f̂(u(k))]T[Λ2U1-Λ2U2-Λ2U2Λ2][u(k)f̂(u(k))]≤0,(3.20)[u(k)ĝ(u(k))]T[Γ2V1-Γ2V2-Γ2V2Γ2][u(k)ĝ(u(k))]≤0,(3.21)[u(k)ĥ(u(k))]T[Ω2W1-Ω2W2-Ω2W2Ω2][u(k)ĥ(u(k))]≤0.
Then from (3.5)–(3.8) and (3.16)–(3.21), we obtain
(3.22)ΔV(k)≤uT(k)[AP1A-2P1-Λ2U1-Γ2V1-Ω2W1+(1+σM-σm)Q2+ρ¯R2+λ0GTG]u(k)+uT(k)Λ2U2f̂(u(k))+uT(k)Γ2V2ĝ(u(k))+uT(k)×Ω2W2ĥ(u(k))-uT(k-σ(k))Q2u(k-σ(k))+f̂T(u(k))×[BDP2DTBT-Λ2+2P2]f̂(u(k))-ĝT(u(k))Γ2ĝ(u(k))+ĝT(u(k-σ(k)))×[BVP2VTBT+DVP2VTDT+P2]ĝ(u(k-σ(k)))-ĥT(u(k))Ω2ĥ(u(k))+(∑N=1+∞ρNĥ(u(k-N)))T[BNP2NTBT+DNP2NTDT+VNP2NTVT-ρ-1R2]×(∑N=1+∞ρNĥ(u(k-N)))+vT(k)×[BP2B-2P2-Λ1L1-Γ1M1-Ω1N1+(1+τM-τm)Q1+μ¯R1+ϵ0KTK]+vT(k)Λ1L2f(v(k))+vT(k)Γ1M2g(v(k))+vT(k)Ω1N2h(v(k))-vT(k-τ(k))Q1v(k-τ(k))+fT(v(k))[ACP1CTAT-Λ1+2P1]f(v(k))-gT(v(k))Γ1g(v(k))+gT(v(k-τ(k)))[AWP1WTAT+CWP1WTCT+P1]×g(v(k-τ(k)))-hT(v(k))Ω1h(v(k))+(∑M=1+∞μMh(v(k-M)))T×[AMP1MTAT+CMP1MTCT+WMP1MTWT-μ-1R1]×(∑M=1+∞μMh(v(k-M)))-[v(k)f(v(k))]T[Λ1L1-Λ1L2-Λ1L2Λ1][v(k)f(v(k))]-[v(k)g(v(k))]T[Γ1M1-Γ1M2-Γ1M2Γ1][v(k)g(v(k))]-[v(k)h(v(k))]T[Ω1N1-Ω1N2-Ω1N2Ω1][v(k)h(v(k))]-[u(k)f̂(u(k))]T[Λ2U1-Λ2U2-Λ2U2Λ2][u(k)f̂(u(k))]-[u(k)ĝ(u(k))]T[Γ2V1-Γ2V2-Γ2V2Γ2][u(k)ĝ(u(k))]-[u(k)ĥ(u(k))]T[Ω2W1-Ω2W2-Ω2W2Ω2][u(k)ĥ(u(k))]=ξT(k)Ξ1ξ(k)+ηT(k)Ξ2η(k),
where ξT(k)=[uT(k)uT(k-σ(k))f̂T(u(k))ĝT(u(k))ĝT(u(k-σ(k)))ĥT(u(k))(∑𝒩=1+∞ρ𝒩ĥ(u(k-𝒩)))T]ηT(k)=[vT(k)vT(k-τ(k))fT(v(k))gT(v(k))gT(v(k-τ(k)))hT(v(k))(∑ℳ=1+∞μℳh(v(k-ℳ)))T].
Therefore, if the LMIs (3.1) hold, it can be concluded that ΔV(k)≤0. It follows that V(k)≤V(0). By (3.22), the SBAMNN is globally asymptotically stable in the mean square.
Now, we are in a position to establish the exponential stability of the SBAMNN (2.7).
Then, there exists a scalarβ>0 such that(3.23)ΔV(k)≤-β(‖u(k)‖2+‖v(k)‖2).
From (3.3), it can be verified that
(3.24)V(k)≤λmax⁡(P1)‖u(k)‖2+λmax⁡(P2)‖v(k)‖2+λmax⁡(Q1)∑i=k-τMk-1‖v(i)‖2+λmax⁡(Q2)∑i=k-σMk-1‖u(i)‖2+(τM-τm)λmax⁡(Q1)∑i=k-τMk-1‖v(i)‖2+(σM-σm)λmax⁡(Q2)∑i=k-σMk-1‖v(i)‖2=λmax⁡(P1)‖u(k)‖2+λmax⁡(P2)‖v(k)‖2+β1∑i=k-σMk-1‖u(i)‖2+β2∑i=k-τMk-1‖v(i)‖2,
where    β1=(1+σM-σm)λmax(Q1) and β2=(1+τM-τm)λmax(Q2).
Choose a scalarθ>1, satisfying(3.25)-βθ+(θ-1)(λmax⁡(P1)+λmax⁡(P2))+(θ-1)(β1τMθτM+β2σMθσM)=0.
Then by (3.23) and (3.24), we have
(3.26)θk+1V(k+1)-θkV(k)=θk+1V(k+1)-θk+1V(k)+θk+1V(k)-θkV(k)=θk+1ΔV(k)+θk(θ-1)V(k)≤β3θk(‖u(k)‖2+‖v(k)‖2)+β4θk(∑i=k-σMk-1‖u(i)‖2+∑i=k-τMk-1‖v(i)‖2),
where β3=-βθ+(θ-1)(λmax(P1)+λmax(P2)) and β4=(θ-1)(β1+β2).
Therefore, for any integerN≥τM+1 and N≥σM+1, summing up both sides of (3.26) from 0 to N-1 with respect to k, we have(3.27)θNV(N)-V(0)≤β3∑k=0N-1θk(‖u(k)‖2+‖v(k)‖2)+β4∑k=0N-1θk(∑i=k-σMk-1‖u(i)‖2+∑i=k-τMk-1‖v(i)‖2).
Here, we note that for τM≥1, σM≥1, we have
(3.28)∑k=0N-1θk(∑i=k-σMk-1‖u(i)‖2+∑i=k-τMk-1‖v(i)‖2)=σM(σM+1)θσMsup⁡-σM≤i≤0‖u(i)‖2+τM(τM+1)θτM×sup⁡-τM≤i≤0‖v(i)‖2+σMθσM∑k=0N-1θk‖u(k)‖2+τMθτM∑k=0N-1θk‖v(k)‖2.
Substituting (3.28) in (3.27) gives
(3.29)θNV(N)≤β3+β4(σMθσM+τMθτM)∑k=0T-1θk(‖u(k)‖2+‖v(k)‖2)+β4[σM(σM+1)θσM+τM(τM+1)θτM](sup⁡-σM≤i≤0‖u(i)‖2+sup⁡-τM≤i≤0‖v(i)‖2)+V(0).
We can observe that
(3.30)V(N)≥{λmin⁡(P1),λmin⁡(P2)}(‖u(N)‖2+‖v(N)‖2).
It follows easily from (3.24) that
(3.31)V(0)≤β1σMsup⁡-σM≤i≤0‖u(i)‖2+β2τMsup⁡-τM≤i≤0‖v(i)‖2.
Then, it follows from (3.25), (3.29), and (3.31) that
(3.32)‖u(N)‖+‖v(N)‖≤νGT(sup⁡-σM≤i≤0‖u(i)‖+sup⁡-τM≤i≤0‖v(i)‖),
where 𝒢=ω-1/2 and
(3.33)ν=β1σM+β2τM+β4[σM(σM+1)θσM+τM(τM+1)θτM]λmin⁡(P1),λmin⁡(P2).
This indicates that the discrete-time stochastic BAM neural network (2.7) is said to be globally exponentially stable. This completes the proof of this theorem.For a deterministic BAM neural network, we have the following system of equations:(3.34)x(k+1)=Ax(k)+Cf(y(k))+Wg(y(k-τ(k)))+M∑M=1+∞μMh(y(k-M))+I,y(k+1)=By(k)+Df̂(x(k))+Vĝ(x(k-σ(k)))+N∑N=1+∞ρNĥ(x(k-N))+J.Then, by Theorem3.1, it is very easy to obtain the following theorem.Theorem 3.2.
Under Assumptions1–4, the discrete-time BAM neural network (3.34) is globally exponentially stable, if there exist diagonal matrices Λ1=diag{λ1(1),λ2(1),…,λn(1)}>0, Λ2=diag{λ1(2),λ2(2),…,λn(2)}>0, Γ1=diag{γ1(1),γ2(1),…,γn(1)}>0, Γ2=diag{γ1(2),γ2(2),…,γn(2)}>0, Ω1=diag{ω1(1),ω2(1),…,ωn(1)}>0 and Ω2=diag{ω1(2),ω2(2),…,ωn(2)}>0 and positive definite matrices P1>0,P2>0,Q1>0,Q2>0R1>0, and R2>0, such that the following LMIs hold:
(3.35)Ξ3=[Ψ110Λ2U2Γ2V20Ω2W20*-Q200000**Π330000***-Γ2000****Π5500*****-Ω20******Π77]<0,Ξ4=[Φ110Λ1L2Γ1M20Ω1N20*-Q100000**Θ330000***-Γ1000****Θ5500*****-Ω10******Θ77]<0,
where
(3.36)Ψ11=ATP1A-2P1-Λ2U1-Γ2V1-Ω2W1+(1+σM-σm)Q2+ρ¯R2,Φ11=BTP2B-2P2-Λ1L1-Γ1M1-Ω1N1+(1+τM-τm)Q1+μ¯R1,
and Π33,Π55,Π77,Θ33,Θ55,andΘ77 are defined in Theorem 3.1.Proof.
Similar to the proof of Theorem3.1, we can derive the stability result. The proof is straightforward and hence omitted.If we neglect the distributed delay term in (2.2), it can be reduced to(3.37)x(k+1)=[Ax(k)+Cf(y(k))+Wg(y(k-τ(k)))+I]+δ(x(k),y(k-τ(k)),k)w1(k),y(k+1)=[By(k)+Df̂(x(k))+Vĝ(x(k-σ(k)))+J]+χ(y(k),x(k-σ(k)),k)w2(k).For system (3.37), we have the following stability result.Corollary 3.3.
Under Assumptions1–5, the discrete-time BAM neural network (3.37) is globally exponentially stable, if there exist diagonal matrices Λ1=diag{λ1(1),λ2(1),…,λn(1)}>0, Λ2=diag{λ1(2),λ2(2),…,λn(2)}>0, Γ1=diag{γ1(1),γ2(1),…,γn(1)}>0, and Γ2=diag{γ1(2),γ2(2),…,γn(2)}>0, and positive definite matrices P1>0,P2>0,Q1>0,and Q2>0, such that the following LMI holds:
(3.38)Ξ5=[Υ110Λ2U2Γ2V20*-Q2000**Π3300***-Γ20****Π55]<0,Ξ6=[Σ110Λ1L2Γ1M20*-Q1000**Θ3300***-Γ10****Θ55]<0,
where
(3.39)Υ11=ATP1A-2P1-Λ2U1-Γ2V1+(1+σM-σm)Q2,Σ11=BTP2B-2P2-Λ1L1-Γ1M1+(1+τM-τm)Q1,
and Π33,Π55,Θ33,andΘ55 are defined in Theorem 3.1.


## 4. Numerical Example

To illustrate the effectiveness of our stability criterion, we give the following numerical example.Example 4.1.
Consider the SBAM neural networks (2.2) with the following parameters:
(4.1)A=[0.30000.30000.4],B=[0.30000.20000.2],C=[0.40.2-0.100.20.3-0.100.2],D=[-0.20.100.20.30.20-0.20.2],W=[-0.20.20.60.30.100-0.2-0.5],V=[0.40.4-0.200.10.2-0.300.3],M=[-0.20.60.10.10.300-0.7-0.5],N=[0.40.5-0.300.10.3-0.400.4],G=[0.20000.30000.3],K=[0.40000.20000.2],τ(k)=3+sin⁡(kπ2),σ(k)=2-cos⁡(kπ2),I=-3sin⁡(kπ2),J=2cos⁡(kπ2),μk=ρk=e-4k,f(y(k))=g(y(k))=h(y(k))=[tanh(-4y1(k))tanh(-4y2(k))tanh(-y3(k))],f̂(y(k))=ĝ(y(k))=ĥ(y(k))=[tanh(-x1(k))tanh(-4x2(k))tanh(-x3(k))].
It can be verified that σm=τm=3, σM=τM=4, l1+=m1+=n1+=2, l1-=m1-=n1-=-2, l2+=m2+=n2+=2, l2-=m2-=n2-=-2, l3+=m3+=n3+=1, l3-=m3-=n3-=-1, u1+=v1+=w1+=1, u1-=v1-=w1-=-1, u2+=v2+=w2+=2, u2-=v2-=w2-=-2, u3+=v3+=w3+=1 and u3-=v3-=w3-=-1 with
(4.2)L1=M1=N1=[-4000-4000-1],L2=M2=N2=[000000000],U1=V1=W1=[-1000-4000-1],U2=V2=W2=[000000000].
By using Matlab LMI toolbox, we solve the LMIs (3.1) in Theorem 3.1 and obtain the feasible solutions as follows:
(4.3)P1=[-3.74770.5120-0.25860.5120-7.5557-0.6515-0.2586-0.6515-9.0264],P2=[-7.67440.25120.01540.2512-6.2020-0.00460.0154-0.0046-8.4207],Q1=[2.84110.0680-0.22370.06803.6480-0.2702-0.2237-0.27022.1788],Q2=[2.53750.5389-0.62480.53893.6343-0.3781-0.6248-0.37812.1497],R1=[1.87380.60691.21000.60692.22121.33031.21001.33032.1414],R2=[-0.5616-0.70362.2034-0.70360.8218-0.47702.2034-0.4770-1.2174],Λ1=[-6.5450000-10.4004000-16.6900],Λ2=[-12.0795000-9.8357000-15.5096],Γ1=[1.70500003.67650002.1403],Γ2=[2.31760003.29880002.0030],Ω1=[1.70500003.67650002.1403],Ω2=[2.31760003.29880002.0030],λ0=1.7593,ϵ0=3.6570.
Then, it follows from Theorem 3.1 that the SBAMNN (2.7) with given parameters is globally exponentially stable in the mean square. Our main purpose in this example is to estimate the maximum allowable upper bound delay σM and τMfor given lower bound σm and τm (Table 1). For instance, if we set σm=τm=2, the allowable time delay upper bound obtained by Gao and Cui [19] is 4. However, in our paper, we obtained that for any time delay satisfying 0<τ(t)≤τM=foranylargefinitevalue, 0<σ(t)≤σM=foranylargefinitevalue. This is much larger than that in [19], which shows the less conservativeness of our developed method (Figure 1).Table 1
Allowable upper bound forσM and τM for given σm and τm.
In [19]σm=τm=2σM=τM=4In this paperσm=τm=2σM=τM=foranylargefinitevalueState trajectories ofx1(k),x2(k),x3(k),y1(k),y2(k),y3(k) for Example 4.1.
(a)(b)(c)(d)(e)(f)


## 5. Conclusion

In this paper, we have considered the stability analysis problem for a class of discrete-time stochastic BAM neural networks with both discrete and distributed delays. Employing a Lyapunov-Krasovskii functional and a Linear matrix inequality approach has been developed to establish sufficient conditions for the SBAMNNs to be globally exponentially stable. It has been shown that the delayed SBAMNNs are globally exponentially stable if some LMIs are solvable and the feasibility of such LMIs can be easily checked by using the numerically efficient LMI toolbox in Matlab. A numerical example has been given to demonstrate the effectiveness of the obtained stability conditions.



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*Source: 153409-2012-01-16.xml*