# Global Exponential Stability Criteria for Bidirectional Associative Memory Neural Networks with Time-Varying Delays

Authors: J. Thipcha; P. Niamsup
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The global exponential stability for bidirectional associative memory neural networks with time-varying delays is studied. In our study, the lower and upper bounds of the activation functions are allowed to be either positive, negative, or zero. By constructing new and improved Lyapunov-Krasovskii functional and introducing free-weighting matrices, a new and improved delay-dependent exponential stability for BAM neural networks with time-varying delays is derived in the form of linear matrix inequality (LMI). Numerical examples are given to demonstrate that the derived condition is less conservative than some existing results given in the literature.


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## Body

## 1. Introduction

A class of neural networks related to bidirectional associative memory (BAM) has been introduced by Kosko [1]. This model generalized the single-layer autoassociative Hebbian correlator to a two-layer pattern-matched heteroassociative circuit. It is an important model with the ability of information memory and information association, which is crucial for various applications such as pattern recognition, solving optimization problems, and automatic control engineering [2–10]. In [1, 11], Kosko investigates the global stability of BAM models and obtains a severe constraint of having a symmetric connection weight matrix. Since it is impossible to maintain an absolutely symmetric connection weight matrix, asymmetric connection has been a focus of this field. Some of these applications require that there should be a well-defined computable solution for all possible initial states. From a mathematical point of view, this means that the equilibrium point of the designed cellular neural networks (CNNs) is globally asymptotically stable (GAS) or globally exponentially stable (GES). Moreover, in biological and artificial neural networks, time delays arise in the process of information transmission; for example, in the electronic implementation of analogue neural networks, time delays occur in the communication and response of neurons owing to the finite switching speed of amplifiers. It is known that they can create an oscillatory or an unstable phenomenon. Therefore, the study of the stability and convergent dynamics of BAM neural networks with delays has raised considerable interest in recent years; see for examples [5, 7, 9, 10, 12–23] and the references cited therein. In [14, 15, 18, 20–22, 24–27], several sufficient conditions on the global exponential stability of BAM neural networks with time-varying delays have been derived. It is worth pointing out that the given criteria in [14, 15, 18, 20–22, 24–27] required the following hypothesis: the time-varying delays are continuously differentiable, the derivative of time-varying delays is smaller than one, and activation functions are bounded and monotonically nondecreasing. The common approach for studying stability of BAM neural networks is Lyapunov stability theory. With a properly designed Lyapunov-Krasovskii functional as well as introducing free-weighting matrices, one may derive stability criteria in term of linear matrix inequality (LMI) which is easily solved by several available algorithms.Based on the above discussion, we propose to study the problem of global exponential stability of BAM neural networks with time-varying delays and generalized activation functions. The main contributions of our works are that the system consists of both memoryless and delayed activation functions, and the lower and upper bounds of the activation functions are allowed to be either positive, negative, or zero which is more general than systems considered in [14, 15, 18, 21, 22, 24–27]. By constructing a new and improved Lyapunov-Krasovskii functional which contains some integral terms of the activation functions, less conservative results are obtained by introducing appropriate free-weighting matrices and by using some improved integral inequality. Finally, two numerical examples are presented to show that our result is less conservative than some existing ones.Notations. Throughout the paper, ℝ denotes the set of all real numbers. * denotes the elements below the main diagonal of a symmetric block matrix. diag{⋯} denotes the diagonal matrix. For symmetric matrices X and Y, the notation X>Y (resp., X≥Y) means that the matrix X-Y is positive definite (resp., nonnegative). λm(·) and λM(·) denote the smallest and largest eigenvalue of given square matrix, respectively.


## 2. Model Description and Preliminaries

Consider the following BAM neural network with time-varying delays of the form(1)u˙i(t)=-ciui(t)+∑j=1m‍aij(1)g~j(vj(t))+∑j=1m‍aij(2)g~j(vj(t-h(t)))+Ii,v˙j(t)=-djvj(t)+∑i=1n‍bji(1)f~i(ui(t))+∑i=1n‍bji(2)f~i(ui(t-d(t)))+Ij,i=1,2,…,n,j=1,2,…,m,
where ui(t) and vj(t) are the state of the ith neurons from the neural field Fu and the jth neurons from the neural field Fv, at time t, respectively; ci and dj denote the neuron charging time constants and passive delay rates, respectively; aij(1) and bji(1) are the synaptic connection strengths; aij(2) and bji(2) are delayed synaptic connection strengths; f~i(·) and g~j(·) denote the activation functions of the ith neurons from the neural field Fu and the jth neurons from the neural field Fv, respectively; Ii and Ij denote the external inputs; and d(t) and h(t) represent the time-varying differentiable functions which satisfy
(2)(i)0≤d(t)≤d,d˙(t)≤τ<1,(ii)0≤h(t)≤h,h˙(t)≤μ<1,
where d,h,μ, and τ are positive scalars.The initial conditions associated with (1) are assumed to be
(3)ui(s)=ϕ~i(s),vj(s)=φ~j(s),s∈[-max{d,h},0],i=1,2,…,n,j=1,2,…,m.Throughout this paper, we make the following assumption on the activation functionf~i(·), g~j(·).(A1)
f~i(·) and g~j(·) are bounded on ℝ.(A2)
For anyα,β∈ℝ,α≠β, there exist four constant matrices E=diag(E1,E2,…,En), F=diag(F1,F2,…,Fn), M=diag(M1,M2,…,Mm), and N=diag(N1,N2,…,Nm) satisfying
(4)Ei≤f~i(α)-f~i(β)α-β≤Fi,i=1,2,…,n,Nj≤g~j(α)-g~j(β)α-β≤Mj,j=1,2,…,m.It is clear that under (A1) and (A2), the system (1) has at least one equilibrium; see [20]. In order to simplify our proof, we shift the equilibrium point u*=(u1*,u2*,…,un*)T, v*=(v1*,v2*,…,vm*)T of system (1) to the origin. Let  xi(t)=ui(t)-ui*, yj(t)=vj(t)-vj*, fi(xi(t))=f~i(xi(t)+ui*)-f~i(ui*), gj(yj(t))=g~j(yj(t)+vj*)-g~j(vj*), fi(xi(t-d(t)))=f~i(xi(t-d(t))+ui*)-f~i(ui*), gj(yj(t-h(t)))=g~j(yj(t-h(t))+vj*)-g~j(vj*), i=1,2,…,n and j=1,2,…,m. Then the system (1) can be transformed to(5)x˙(t)=-cixi(t)+∑j=1m‍aij(1)gj(yj(t))+∑j=1m‍aij(2)gj(yj(t-h(t))),y˙(t)=-djyj(t)+∑i=1n‍bji(1)fi(xi(t))+∑i=1n‍bji(2)fi(xi(t-d(t))).
The activation functions fi(·) and gj(·) satisfy the following properties.(H1)
fi and gj are bounded on ℝ.(H2)
For anyα,β∈ℝ,α≠β, there exist constant matrices E=diag(E1,E2,…,En),F=diag(F1,F2,…,Fn), M=diag(M1,M2,…,Mm), and N=diag(N1,N2,…,Nm) satisfying
(6)Ei≤fi(α)-fi(β)α-β≤Fi,i=1,2,…,n,Nj≤gj(α)-gj(β)α-β≤Mj,j=1,2,…,m.(H3)
fi(0)=0 and gj(0)=0, i=1,2,…,n, j=1,2,…,m.Rewrite the system (7) into the vector form
(7)x˙(t)=-Cx(t)+A1g(y(t))+A2g(y(t-h(t))),y˙(t)=-Dy(t)+B1f(x(t))+B2f(x(t-d(t))).
The initial conditions associated with (7) are assumed to be
(8)xi(s)=ϕi(s),yj(s)=φj(s),s∈[-max{d,h},0],i=1,2,…,n,j=1,2,…,m,
where x(t)=(x1(t),x2(t),…,xn(t))T, y(t)=(y1(t),y2(t),…,ym(t))T, C=diag(c1,c2,…,cn), D=diag(d1,d2,…,dn), A1=(aij(1))m×n, A2=(aij(2))m×n, B1=(bij(1))n×m, B2=(bij(2))n×m, f(·)=(f1(·),f2(·),…,fn(·))T, g(·)=(g1(·), g2(·),…,gm(·))T.Definition 1 (see [14]).
The trivial solution of system (7) is said to be globally exponentially stable if there exist constants k>0 and ρ≥1 such that
(9)∥x(t)∥2+∥y(t)∥2≤ρe-2kt(∥ϕ∥2+∥φ∥2),∀t≥0,
where one denotes
(10)∥ϕ∥2+∥φ∥2=sup-max{d,h}≤s≤0∥ϕ(s)∥2+sup-max{d,h}≤s≤0∥φ(s)∥2.Lemma 2 (see [28]).
If there exist symmetric positive-definite matrixX33>0 and arbitrary matrices X11,X12,X13,X22, and X23 such that
(11)X=[X11X12X13X12TX22X23X13TX23TX33]≥0,
then,
(12)-∫t-h(t)t‍x˙(s)X33x˙(s)ds≤∫t-h(t)t‍[xT(t)xT(t-h(t))x˙T(s)]×[X11X12X13X12TX22X23X13TX23T0][x(t)x(t-h(t))x˙(s)]ds.Lemma 3 (see [25]).
For any real vectorsa, b and any matrix Q>0 with appropriate dimensions, it follows that
(13)2aTb≤aTQa+bTQ-1b.Lemma 4 (see [25]).
Suppose that (H2) holds; then(14)∫vu‍[fi(s)-fi(v)]ds≤[u-v][fi(u)-fi(v)],i=1,2,…,n,∫vu‍[gj(s)-gj(v)]ds≤[u-v][gj(u)-gj(v)],j=1,2,…,m.


## 3. Main Result

In this section, we present a theorem which states the conditions that guarantee the global exponential stability of the system (7) employing the Lyapunov stability theory and linear matrix inequality approach.Theorem 5.
Under the assumptions (H1)–(H3), for given four diagonal matricesE=diag(E1,E2,…,En), F=diag(F1,F2,…,Fn), M=diag(M1,M2,…,Mm), and N=diag(N1,N2,…,Nm) and positive constants d,h,τ,μ,  and k, the system (7) is globally exponentially stable with the convergent rate k, if there exist positive matrices Pi,Wi,Zi, i=1,2, Qj,j=1,2,3,4, positive diagonal matrices K=diag(k1,k2,…,kn), R=diag(r1,r2,…,rm), and positive-definite matrices
(15)S=[S11S12S13S12TS22S23S13TS23TZ1],T=[T11T12T13T12TT22T23T13TT23TZ2],
such that the following LMI holds:
(16)Ξ=[Σ11Σ21Σ31000Σ71Σ81*Σ22000000**Σ33Σ43Σ530Σ73Σ83***Σ44Σ540Σ740****Σ55Σ65Σ750*****Σ6600******Σ77Σ87*******Σ88]<0,
where
(17)Σ11=2kP1-P1C-CTP1-2kETK-2kKTE+ETKC+CTKE+Q1+dCTZ1C+e-2kd(dS11+S13T+S13)+W1,Σ21=e-2kd(dS12+S23T-S13),Σ31=2kK,Σ71=P1A1-ETKA1-dCTZ1A1,Σ81=P2A2-ETKA2-dCTZ1A2,Σ22=e-2kd(dS22-S23T-S23)-e-2kdQ1,Σ33=hB1TZ2B1+Q3-2KCF-1,Σ43=hB1TZ2B2,Σ53=P2B1-NTRB1-hB1TZ2D,Σ73=KA1+RB1,Σ83=KA2,Σ44=hB2TZ2B2-(1-τ)e-2kdQ3-(1-τ)e-2kdF-1W1F-1,Σ54=P2B2-NTRB2-hB2TZ2D,Σ74=RB2,Σ55=2kP2-P2D-DTP2-2kNTR-2kRTN+NTRD+DTRN+Q2+hDTZ2D+e-2kh(hT11+T13T+T13)+W2,Σ65=e-2kh(hT12+T23T-T13),Σ75=2kR,Σ66=e-2kh(hT22-T23T-T23)-e-2khQ2,Σ77=dA1TZ1A1+Q4-2RDM-1,Σ87=dA1TZ1A2,Σ88=dA2TZ1A2-(1-μ)e-2khQ4-(1-μ)e-2khM-1W2M-1.Proof.
Choose the Lyapunov-Krasovskii function candidate for the system (7) to be
(18)V(t)=∑l=15‍Vl(t),
where
(19)V1(t)=e2ktxT(t)P1x(t)+2∑i=1n‍kie2kt∫0xi(t)‍[fi(s)-Eis]ds+e2ktyT(t)P2y(t)+2∑j=1m‍rje2kt∫0yj(t)‍[gj(s)-Njs]ds,V2(t)=∫t-dt‍e2ksxT(s)Q1x(s)ds+∫t-ht‍e2ksyT(s)Q2y(s)ds,V3(t)=∫t-d(t)t‍e2ksfT(x(s))Q3f(x(s))ds+∫t-h(t)t‍e2ksgT(y(s))Q4g(y(s))ds,V4(t)=∫t-d(t)t‍e2ksxT(s)W1x(s)ds+∫t-h(t)t‍e2ksyT(s)W2y(s)ds,V5(t)=∫t-dt‍∫st‍e2kθx˙T(θ)Z1x˙(θ)dθds+∫t-ht‍∫st‍e2kθy˙T(θ)Z2y˙(θ)dθds.
The derivative of V(t) along the trajectories of system (7) is given by
(20)V˙1(t)=2ke2ktxT(t)P1x(t)+2e2ktxT(t)P1x˙(t)+4k∑i=1n‍kie2kt∫0xi(t)‍[fi(s)-Eis]ds+2∑i=1n‍kie2ktx˙i(t)[fi(xi(t))-Eixi(t)]+2ke2ktyT(t)P2y(t)+2e2ktyT(t)P2y˙(t)+4k∑j=1m‍rje2kt∫0yj(t)‍[gj(s)-Njs]ds+2∑j=1m‍rje2kty˙j(t)[gj(yj(t))-Njyj(t)]≤e2kt{[f(x(t))-Ex(t)]T2kxT(t)P1x(t)+2xT(t)×P1[-Cx(t)+A1g(y(t))+A2g(y(t-h(t)))]+4k[f(x(t))-Ex(t)]TKx(t)+2[f(x(t))-Ex(t)]T×K[-Cx(t)+A1g(y(t))+A2g(y(t-h(t)))]+2kyT(t)P2y(t)+2yT(t)P2[-Dy(t)+B1f(x(t))+B2f(x(t-d(t)))]+4k[g(y(t))-Ny(t)]TRy(t)+2[g(y(t))-Ny(t)]TR×[-Dy(t)+B1f(x(t))+B2f(x(t-d(t)))][f(x(t))-Ex(t)]T}.
By (H2), we have
(21)-2fT(x(t))KCx(t)≤-2fT(x(t))KCF-1f(x(t)),-2gT(y(t))RDy(t)≤-2gT(y(t))RDM-1g(y(t)).
Substituting (21) into (20), we obtain
(22)V˙1(t)≤e2kt×{2kxT(t)P1x(t)-2xT(t)P1Cx(t)+2xT(t)P1A1g(y(t))+2xT(t)P1A2×g(y(t-h(t)))+4kfT(x(t))Kx(t)-4kxT(t)ETKx(t)-2fT(x(t))KC×F-1f(x(t))+2fT(x(t))KA1g(y(t))+2fT(x(t))KA2g(y(t-h(t)))+2xT(t)ETKCx(t)-2xT(t)ETKA1g(y(t))-2xT(t)ETKA2g(y(t-h(t)))+2kyT(t)P2y(t)-2yT(t)P2Dy(t)+2yT(t)P2B1f(x(t))+2yT(t)P2B2f(x(t-d(t)))+4kgT(y(t))Ry(t)-4kyT(t)NTRy(t)-2gT(y(t))RDM-1g(y(t))+2gT(y(t))RB1f(x(t))+2gT(y(t))RB2f(x(t-d(t)))+2yT(t)NTRDy(t)-2yT(t)NTRB1f(x(t))-2yT(t)NTRB2f(x(t-d(t)))},V˙2(t)=e2kt{xT(t)Q1x(t)-e-2kdxT(t-d)Q1x(t-d)+yT(t)Q2y(t)-e-2khyT(t-h)Q2y(t-h)},V˙3(t)=e2kt×{fT(x(t))Q3f(x(t))-(1-d˙(t))e-2kd(t)fT(x(t-d(t)))×Q3f(x(t-d(t)))+gT(y(t))Q4g(y(t))-(1-h˙(t))e-2kh(t)×gT(y(t-h(t)))Q4g(y(t-h(t)))}≤e2kt{fT(x(t))Q3f(x(t))-(1-τ)e-2kdfT(x(t-d(t)))×Q3f(x(t-d(t)))+gT(y(t))Q4g(y(t))-(1-μ)e-2khgT(y(t-h(t)))×Q4g(y(t-h(t)))},(23)V˙4(t)=e2kt{xT(t)W1x(t)-(1-d˙(t))e-2kd(t)xT(t-d(t))×W1x(t-d(t))+yT(t)W2y(t)-(1-h˙(t))e-2kh(t)×yT(t-h(t))(s)W2yT(t-h(t))}≤e2kt{xT(t)W1x(t)-(1-τ)e-2kd×xT(t-d(t))W1x(t-d(t))+yT(t)W2y(t)-(1-μ)e-2kh×yT(t-h(t))(s)W2yT(t-h(t))}.
By (H2), we have
(24)-e-2kdxT(t-d(t))W1x(t-d(t))≤-e-2kdfT(x(t-d(t)))F-1W1×F-1f(x(t-d(t))),-e-2khyT(t-h(t))(s)W2yT(t-h(t))≤-e-2khgT(y(t-h(t)))M-1W2×M-1g(y(t-h(t))).
By (24), we conclude that
(25)V˙4(t)≤e2kt{xT(t)W1x(t)-(1-τ)e-2kd×fT(x(t-d(t)))F-1W1F-1×f(x(t-d(t)))+yT(t)W2y(t)-(1-μ)e-2kh×gT(y(t-h(t)))M-1W2M-1×g(y(t-h(t)))}.(26)V˙5(t)=de2ktx˙T(t)Z1x˙(t)-∫t-dt‍e2ksx˙T(s)Z1x˙(s)ds+he2kty˙T(t)Z2y˙(t)-∫t-ht‍e2ksy˙T(s)Z2y˙(s)ds≤e2kt{∫t-ht‍dx˙T(t)Z1x˙(t)-e-2kd∫t-dt‍x˙T(s)Z1x˙(s)ds×hy˙T(t)Z2y˙(t)-e-2kh∫t-ht‍y˙T(s)Z2y˙(s)ds}.
By Lemma 2, we obtain
(27)-∫t-dt‍x˙T(s)Z1x˙(s)ds≤∫t-dt‍[xT(t)xT(t-d)x˙T(s)]×[S11S12S13S12TS22S23S13TS23T0]×[x(t)x(t-d)x˙(s)]ds=∫t-dt‍(xT(t)S11+xT(t-d)S12T+x˙T(s)S13T)x(t)ds+∫t-dt‍(xT(t)S12+xT(t-d)S22+x˙T(s)S23T)×x(t-d)ds+∫t-dt‍(xT(t)S13+xT(t-d)S23)x˙(s)ds=xT(t)dS11x(t)+xT(t-d)dS12Tx(t)+∫t-dt‍x˙T(s)dsS13Tx(t)+xT(t)dS12x(t-d)+xT(t-d)dS22x(t-d)+∫t-dt‍x˙T(s)dsS23Tx(t-d)+xT(t)S13∫t-dt‍x˙(s)ds+xT(t-d)S23∫t-dt‍x˙(s)ds=xT(t)dS11x(t)+xT(t-d)dS12Tx(t)+xT(t)dS12x(t-d)+xT(t-d)dS22x(t-d)+[x(t)-x(t-d)]T×[S13Tx(t)+S23Tx(t-d)]+[xT(t)S13+xT(t-d)S23]×[x(t)-x(t-d)],-∫t-ht‍y˙T(s)Z2y˙(s)ds≤∫t-ht‍[yT(t)yT(t-h)y˙T(s)]×[T11T12T13T12TT22T23T13TT23T0]×[y(t)y(t-h)y˙(s)]ds=∫t-ht‍(yT(t)T11+yT(t-h)T12T+y˙T(s)T13T)y(t)ds+∫t-ht‍(yT(t)T12+yT(t-h)T22+y˙T(s)T23T)×y(t-h)ds+∫t-ht‍(yT(t)T13+yT(t-h)T23)y˙(s)ds=yT(t)hT11y(t)+yT(t-h)hT12Ty(t)+∫t-ht‍y˙T(s)dsT13Ty(t)+yT(t)hT12y(t-h)+yT(t-h)hT22y(t-h)+∫t-ht‍y˙T(s)dsT23Ty(t-h)+yT(t)T13∫t-ht‍y˙(s)ds+yT(t-h)T23∫t-ht‍y˙(s)ds=yT(t)hT11y(t)+yT(t-h)hT12Ty(t)+yT(t)hT12y(t-h)+yT(t-h)hT22y(t-h)+[y(t)-y(t-h)]T×[T13Ty(t)+T23Ty(t-h)]+[yT(t)T13+yT(t-h)T23]×[y(t)-y(t-h)].
Substituting (27) into (26), we have
(28)V˙5(t)≤e2kt{dx˙T(t)Z1x˙(t)+e-2kd[dxT(t)S11x(t)+2dxT(t)S12Tx(t-d)+2[x(t)-x(t-d)]T×[S13Tx(t)+S23Tx(t-d)]+dxT(t-d)S22x(t-d)]+hy˙T(t)Z2y˙(t)+e-2kh[hyT(t)T11y(t)+2hyT(t)T12y(t-h)+2[y(t)-y(t-h)]T×[T13Ty(t)+T23Ty(t-h)]+hyT(t-h)T22y(t-h)]}.
From (22), (25), and (28) we obtain
(29)V˙(t)≤e2ktξT(t)Ξξ(t),
where Ξ is defined as in (16), and ξ(t)=[x(t),x(t-d),f(x(t)),f(x(t-d(t))),y(t),y(t-h),g(y(t)),g(y(t-h(t)))]. Since the matrix Ξ given in Theorem 5 is the negative definite matrix, we have V˙(t)≤0, for all t≥0 which implies that V(t)≤V(0). From the definition of V(t) in (20), we obtain
(30)V(0)=xT(0)P1x(0)+2∑i=1n‍ki∫0xi(0)‍[fi(s)-Eis]ds+∫-d0‍e2ksxT(s)Q1x(s)ds+∫-d(0)0‍e2ksfT(x(s))Q3f(x(s))ds+∫-d(0)0‍e2ksxT(s)W1x(s)ds+yT(0)P2y(0)+2∑j=1m‍rj∫0yj(0)‍[gj(s)-Njs]ds+∫-h0‍e2ksyT(s)Q2y(s)ds+∫-h(0)0‍e2ksgT(y(s))Q4g(y(s))ds+∫-h(0)0‍yT(s)W2y(s)ds+∫-d0‍∫s0‍e2kθx˙T(θ)Z1x˙(θ)dθds+∫-h0‍∫s0‍e2kθy˙T(θ)Z2y˙(θ)dθds≤λM(P1)∥ϕ∥2+2KM(F-E)∥ϕ∥2+[λM(Q1)+λM(Q3)(F-E)+λM(W1)]×∫-d0‍e2ksxT(s)x(s)ds+λM(P2)∥φ∥2+2RM(M-N)∥φ∥2+[λM(Q2)+λM(Q4)(M-N)+λM(W2)]×∫-h0‍e2ksyT(s)y(s)ds+λM(Z1)∫-d0‍∫s0‍x˙T(θ)x˙(θ)dθds+λM(Z2)∫-h0‍∫s0‍y˙T(θ)y˙(θ)dθds,
where KM=max1≤i≤n(ki), F-E=max1≤i≤n(Fi-Ei), RM=max1≤j≤m(rj) and M-N=max1≤j≤n(Mj-Nj).
It follows from Lemma3 that
(31)x˙T(θ)x˙(θ)=[-Cx(θ)+A1g(y(θ))+A2g(y(θ-h(θ)))]T×[-Cx(θ)+A1g(y(θ))+A2g(y(θ-h(θ)))]=xT(θ)CTCx(θ)+gT(y(θ))A1TA1g(y(θ))+gT(y(θ-h(θ)))×A2TA2g(y(θ-h(θ)))-2xT(θ)CTA1g(y(θ))-2xT(θ)CTA2×g(y(θ-h(θ)))+2gT(y(θ))A1TA2g(y(θ-h(θ)))≤3xT(θ)CTCx(θ)+3gT(y(θ))A1TA1g(y(θ))+3gT(y(θ-h(θ)))A2T×A2g(y(θ-h(θ))),y˙T(θ)y˙(θ)=[-Dy(θ)+B1f(x(θ))+B2f(x(θ-d(θ)))]T×[-Dy(θ)+B1f(x(θ))+B2f(x(θ-d(θ)))]=yT(θ)DTDy(θ)+fT(x(θ))B1TB1f(x(θ))+fT(x(θ-d(θ)))×B2TB2f(x(θ-d(θ)))-2yT(θ)DTB1f(x(θ))-2yT(θ)DTB2×f(x(θ-d(θ)))+2fT(x(θ))B1TB2f(x(θ-d(θ)))≤3yT(θ)DTDy(θ)+3fT(x(θ))B1TB1f(x(θ))+3fT(x(θ-d(θ)))B2T×B2g(y(θ-h(θ))).
Substituting (31) into (30), we obtain the bound of V(0) as follows:
(32)V(0)≤λM(P1)∥ϕ∥2+2KM(F-E)∥ϕ∥2+(1-e-2kd2k)×[λM(Q1)+λM(Q3)×(F-E)+λM(W1)]+λM(P2)∥φ∥2+2RM(M-N)∥φ∥2+(1-e-2kh2k)×[λM(Q2)+λM(Q4)(M-N)+λM(W2)]+d22λM(Z1)[3λM(CTC)∥ϕ∥2+3λM(A1TA1)∥φ∥2+3λM(A2TA2)∥φ∥2]+h22λM(Z2)[3λM(DTD)∥φ∥2+3λM(B1TB1)∥ϕ∥2+3λM(B2TB2)∥ϕ∥2].
Thus,
(33)V(0)≤χ1∥ϕ∥2+χ2∥φ∥2,
where
(34)χ1=λM(P1)+2KM(F-E)+(1-e-2kd2k)×[λM(Q1)+λM(Q3)(F-E)+λM(W1)]+3d22λM(Z1)λM(CTC)+h22λM(Z2)[3λM(B1TB1)+3λM(B2TB2)],χ2=λM(P2)+2RM(M-N)+(1-e-2kh2k)×[λM(Q2)+λM(Q4)(M-N)+λM(W2)]+3h22λM(Z2)λM(DTD)+d22λM(Z1)[3λM(A1TA1)+3λM(A2TA2)].
On the other hand, we have
(35)V(t)≥e2kt{λm(P1)∥x(t)∥2+λm(P2)∥y(t)∥2}.
Therefore,
(36)∥x(t)∥2+∥y(t)∥2≤ρe-2kt{∥ϕ∥2+∥φ∥2},
where ρ=max(χ1,χ2)/min(λm(P1),λm(P2))≥1. Therefore, the system (7) is global exponentially stable with the convergent rate k>0. This completes the proof.Remark 6.
In hypothesis (H2), lower boundsEi,Nj and upper bounds Fi, Mj, i=1,2,…,n, j=1,2,…,m, of activation functions are allowed to be either positive, negative, or zero. Clearly, hypothesis (H2) in our paper is more general than those given in [14, 15, 18, 21, 22, 24–27]. Hence, our result is less conservative than some existing results given in the literature.


## 4. Numerical Examples

Example 1. Consider the BAM neural networks in (7) with C=diag(1,1,1), D=diag(2,2,2),A1=B1=0,
(37)A2=[0.050.250.050.10.050.150.150.150.05],B2=[0.750.750.9500.500.150.150.150.05].In this example, the activation function and time delay are given as follows:f1(x)=f2(x)=f3(x)=(1/2)(|x+1|-|x-1|), g1(y)=g2(y)=g3(y)=(1/2)(|y+1|-|y-1|), d(t)=0.5, h(t)=1. It follows that d=0.5, τ=0.3, h=1, and μ=0.3. The assumption (H2) is satisfied with F=diag(1,1,1), M=diag(1,1,1), E=N=0. Let k=0.1. By using the LMI Toolbox in MATLAB, the LMI (16) of Theorem 5 is feasible with k=0.1 and a set of solutions of (16) is given by
(38)P1=[28.63820.68430.75010.684328.20611.09810.75011.098127.7326],P2=[11.76310.56710.41700.567125.82960.79340.41700.793426.2860],Q1=[14.4157-0.0495-0.0610-0.049514.4217-0.0911-0.0610-0.091114.4462],Q2=[12.8030-0.9294-0.9704-0.929416.9982-0.4287-0.9704-0.428717.2419],Q3=[16.77250.44950.25280.449515.06690.59690.25280.596913.1708],Q4=[11.9146-0.2010-0.3870-0.201015.1767-0.6757-0.3870-0.675715.3739],W1=[14.62181.08580.99971.085813.02601.30010.99971.300111.6739],W2=[8.4065-0.0639-0.2523-0.063911.99590.1677-0.25230.167711.6957],Z1=[13.39930.21420.38990.214213.57460.39410.38990.394113.7729],Z2=[2.35110.74680.67360.746811.26270.35000.67360.350011.7724],S11=[12.4617-0.0242-0.0266-0.024212.4732-0.0399-0.0266-0.039912.4949],S12=[-1.1535-0.0211-0.0257-0.0211-1.1444-0.0335-0.0257-0.0335-1.1320],S13=[-0.9075-0.0415-0.0544-0.0415-0.9093-0.0805-0.0544-0.0805-0.9019],S22=[12.2314-0.0115-0.0101-0.011512.2463-0.0163-0.0101-0.016312.2626],S23=[1.63060.01290.02080.01291.64300.02700.02080.02701.6490],T11=[8.6800-0.5650-0.6089-0.565012.0158-0.3052-0.6089-0.305212.2653],T12=[-1.8596-0.2702-0.2795-0.2702-1.2902-0.1294-0.2795-0.1294-1.2317],T13=[-1.7089-0.3795-0.4069-0.3795-0.6642-0.2437-0.4069-0.2437-0.5078],T22=[8.2813-0.2103-0.2276-0.210310.2975-0.1027-0.2276-0.102710.4221],T23=[1.97070.18710.18700.18712.30620.07830.18700.07832.3346],K=[14.942700014.273400013.4467],R=[5.83150008.19630008.3562].
Thus, the system (7) is 0.1-exponentially stable and the value ρ=13.4606. The solution of the closed-loop system satisfies
(39)∥x(t)∥2+∥y(t)∥2≤13.4606e-2(0.1)t{∥ϕ∥2+∥φ∥2},∀t∈ℝ+.By applying Theorem5 and by solving the LMI (16) using MATLAB LMI Toolbox, we obtain the convergence rate k which guarantees that the global exponential stability is 0.998. In Table 1, we give comparison of maximum allowable convergence rate k obtained by Theorem 5 and by other methods in some previous existing results. From Table 1, it is shown that the proposed global exponential stability criterion is less conservative than those obtained in [24, 25, 29].Table 1
Maximum allowable convergence rate.
τ
=
μ
0
0.3
0.5
0.7
0.9
[24]
—
0.459
0.455
0.455
0.455
[29]
—
0.445
0.424
0.408
0.407
[25]
—
0.52
0.47
0.39
0.21
This paper
0.998
0.998
0.998
0.998
0.998Example 2. Consider the BAM neural networks in (7) with
(40)C=[500050005],A1=[-0.72-0.44-0.21-0.72-0.83-0.1-0.010.01-0.04],A2=[-0.01-0.12-0.240.17-0.33-0.43-0.250.33-0.05],D=[500050005],B1=[-0.31-0.310.920.34-0.33-0.780.340.470.25],B2=[-0.83-0.12-0.52-0.650.5-0.14-0.05-0.14-0.65].In this example, the activation function and time-varying delay are given as follows:f1(x)=f2(x)=f3(x)=(1/2)(|x+1|-|x-1|), g1(y)=g2(y)=g3(y)=(1/2)(|y+1|-|y-1|), d(t)=sin2(0.5t), h(t)=0.1cos2(t); the assumption (H2) is satisfied with E=diag(-0.2,-0.25,-0.2), F=diag(0.3,0.4,0.5), N=diag(-0.2,-0.25,-0.2), M=diag(0.3,0.4,0.5). It follows that d=0.5,τ=0.3,h=0.1, and μ=0.3. Let k=0.1. By using the LMI Toolbox in MATLAB, the LMI (16) is feasible with k=0.1 and a set of solutions of (16) is given by
(41)P1=[110.3704-30.1135-5.4152-30.113590.30138.8948-5.41528.8948115.0609],P2=[43.9625-2.30317.3341-2.303138.16714.99697.33414.996962.3183],Q1=[125.4489-1.88020.9399-1.8802127.5495-1.00630.9399-1.0063130.0399],Q2=[100.3079-4.1254-0.7526-4.1254101.84901.0897-0.75261.0897114.9297],Q3=[136.0956-4.291938.6712-4.291990.480819.202238.671219.2022113.1213],Q4=[42.6986-34.8491-15.2241-34.849196.77282.0801-15.22412.0801145.8684],W1=[140.805925.3914-0.935725.3914146.4301-4.4760-0.9357-4.476011.6128],W2=[105.4681-0.14055.1590-0.1405107.441916.76375.159016.7637134.1991],Z1=[69.1695-24.4444-4.5155-24.444448.71777.8545-4.51557.854565.4657],Z2=[99.94901.335735.24701.335780.217910.668235.247010.6682111.8243],S11=[103.3199-2.98950.4043-2.9895103.2019-0.14040.4043-0.1404106.3073],S12=[-2.9235-0.63140.1584-0.6314-2.4889-0.16720.1584-0.1672-1.9572],S13=[1.0569-2.46490.0237-2.46490.40900.21130.02370.21132.4210],S22=[97.0839-2.17720.0865-2.177296.32840.19800.08650.198098.4407],S23=[8.9358-1.7429-0.3501-1.74297.02420.7119-0.35010.71198.3850],T11=[105.0332-0.41930.0331-0.4193105.22660.30430.03310.3043107.5064],T12=[-1.5261-0.6096-0.1149-0.6096-1.24150.0894-0.11490.08940.3340],T13=[-7.6562-3.2414-0.6934-3.2414-5.76500.5527-0.69340.55274.8321],T22=[104.9648-0.5090-0.0471-0.5090105.19400.0964-0.04710.0964106.5444],T23=[2.60720.14200.20770.14202.6111-0.04040.2077-0.04042.2053],K=[106.595700063.39210006.3380],R=[86.005900072.023100071.2919].
Thus, the system (7) is 0.1-exponentially stable and the value ρ=14.1811. The solution of the closed-loop system satisfies
(42)∥x(t)∥2+∥y(t)∥2≤14.1811e-2(0.1)t{∥ϕ∥2+∥φ∥2},∀t∈ℝ+.
The maximum allowable convergence rate k for different values of τ=μ is given in Table 2. The trajectory of solutions of BAM neural networks with time-varying delays is shown in Figure 1, where the initial conditions are chosen as ϕ1=cos(s), ϕ2=sin(s),ϕ3=sin(s)-1, φ1=cos(s)+1,φ2=sin(s)-2, φ3=cos(s)+1, s∈[-0.5,0].Table 2
Maximum allowable convergence rate.
τ
=
μ
0
0.3
0.5
0.7
Convergence ratek
0.81
0.638
0.48
0.221Figure 1
Time responses of state variables.


## 5. Conclusion

This paper has proposed a new sufficient condition guaranteeing the global exponential stability criteria for bidirectional associative memory neural networks with time-varying delays and generalized activation functions. The developed stability condition is in terms of LMI, which can be easily solved by some existing software packages. Furthermore, the proposed stability conditions are less conservative than some works in the literature.



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*Source: 576721-2013-05-30.xml*