# Improved Stability Analysis for Neural Networks with Time-Varying Delay

Authors: Yongming Li; Junkang Tian; Jinzhou Zhao; Liehui Zhang; Tiejun Li
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper concerned the problem of delay-dependent asymptotic stability for neural networks with time-varying delay. A new class of Lyapunov functional dividing the interval delay is constructed to derive some new delay-dependent stability criteria. The obtained criteria are less conservative because free-weighting matrices method, a convex optimization approach, and a mixed dividing delay interval approach are considered. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

In the past few decades, delayed neural networks have been investigated extensively because of their successful applications in various scientific areas, such as pattern recognition, image processing, associative memories, and parallel computation. It is well known that time delay is frequently encountered in neural networks, and it is often a major cause of instability and oscillation. Thus, the stability analysis of delayed neural networks has been widely considered by many research results, delay-independent ones [1–3], and delay-dependent ones [4–37]. Generally speaking, the delay-dependent stability criteria are less conservative than delay-independent ones when the time delay is small. Therefore, much attention has been paid to develop delay-dependent stability conditions. Some less conservative stability criteria were derived in [6] by considering some useful terms and using the free-weighting matrices method. By the fact that the neuron activation functions are sector bounded and nondecreasing, [7] presents an improved method, named the delay-slope-dependent method, for stability analysis of neural networks with time-varying delays. The method includes more information on the slope of neuron activation functions and fewer matrix variables in the constructing Lyapunov functionals. Then some new delay-dependent stability criteria with less conservatism are obtained. Recently, some new Lyapunov functionals based on the idea of decomposing the delay were introduced to investigate the stability of neural networks with time-invariant delay [10–12] and time-varying delay [13–16], which significantly reduced the conservativeness of the derived stability criteria. In [13], different from some previous results, the delay interval [0,d(t)] is divided into some variable subintervals by employing weighting delays. Thus, some new delay-dependent stability criteria for neural networks with time varying delay are derived by applying the weighting-delay method, which are less conservative than the existing results. However, when the delay is time-varying, the information of subinterval is not considered sufficiently. For example, the time-varying delay τ(t) satisfies 0⩽τ(t)⩽h. When the delay interval [0,τ(t)] is divided into some subintervals, the delay interval [0,h] is also divided into some subintervals, in essence. But in the construction of Lyapunov functional in [15], this important information is ignored, which is a major source of conservativeness. Furthermore, the purpose of reducing conservatism is still limited due to the existence of multiple coefficients and the impact of subintervals with uniform size. Thus, it is still a quite difficult task to divide interval [0,τ(t)] in a more reasonable manner, so that the functional with the augmented matrix can easily be constructed to obtain less conservative stability results, which motivates our present study.In this paper, the problem of delay-dependent asymptotic stability criterion for neural networks with time-varying delay has been considered. A new class of Lyapunov functional is constructed to derive some new delay-dependent stability criteria. The obtained criteria are less conservative because a mixed dividing delay interval approach is considered. Finally, the numerical examples are given to indicate significant improvements over some existing results.


## 2. Problem Formulation

Consider the following neural networks with time-varying delay:(2.1)ẋ(t)=-Cx(t)+Ag(x(t))+Bg(x(t-τ(t)))+μ,
where x(t)=[x1(t),x2(t),…,xn(t)]T∈ℛn is the neuron state vector, g(x(·))=[g1(x1(·)),g2(x2(·)),…,gn(xn(·))]T∈ℛn denotes the neuron activation function, and μ=(μ1,μ2,…,μn)T∈ℛn is a constant input vector. A,B∈ℛn×n are the connection weight matrix and the delayed connection weight matrix, respectively. C=diag⁡(C1,C2,…,Cn) with Ci>0,i=1,2,…,n. τ(t) is a time-varying continuous function that satisfies 0⩽τ(t)⩽h, τ̇(t)⩽u, where h and u are constants. In addition, it is assumed that each neuron activation function in (2.1), gi(·),i=1,2,…,n, is bounded and satisfies the following condition:(2.2)0⩽gi(x)-gi(y)x-y⩽ki,∀x,y∈R,x≠y,i=1,2,…,n,
where ki,i=1,2,…,n are positive constants.Assuming thatx*=[x1*,x2*,…,xn*]T is the equilibrium point of (2.1) whose uniqueness has been given in [31] and using the transformation z(·)=x(·)-x*, system (2.1) can be converted to the following system:(2.3)ż(t)=-Cz(t)+Af(z(t))+Bf(z(t-τ(t))),
where z(t)=[z1(t),z2(t),…,zn(t)]T, f(z(·))=[f1(z1(·)),f2(z2(·)),…,fn(zn(·))]T, and fi(zi(·))=gi(zi(·)+xi*)-gi(xi*),i=1,2,…,n. According to the inequality (2.2), one can obtain that(2.4)0⩽fi(zi(t))zi(t)⩽ki,fi(0)=0,i=1,2,…,n.
Thus, under this assumption, the following inequality holds for any diagonal matrix Q>0,(2.5)zT(t)KQKz(t)-fT(z(t))Qf(z(t))⩾0,
where K=diag⁡(k1,k2,…,kn).Lemma 2.1 (see [38]).
For any constant matrixZ∈ℛn×n,Z=ZT>0, scalars h2>h1>0, such that the following integrations are well defined, then
(2.6)-(h2-h1)∫t-h2t-h1xT(s)Zx(s)ds⩽-∫t-h2t-h1xT(s)dsZ∫t-h2t-h1x(s)ds.


## 3. Main Results

In this section, a new Lyapunov functional is constructed, and a new delay-dependent stability criterion is obtained.Theorem 3.1.
For given scalarsK=diag⁡(k1,k2,…,kn), h⩾0,u,and0<α<1, the system (2.3) is globally asymptotically stable if there exist symmetric positive matrices P=[Pij]3×3,Qi(i=1,2,…,6), Ri(i=1,2,…,6), positive diagonal matrices T1,T2,Q,Δ=diag⁡(δ1,δ2,…,δn),Λ=diag⁡(λ1,λ2,…,λn), and any matrices P1,P2,Ni,Mi,Li,Hi,Zi,Si,Ui,Vi,Wi(i=1,2,3) with appropriate dimensions, such that the following LMIs hold:
(3.1)E1=[EαhNα2h22H(1-α2)h22M*-αhR300**-α2h22R50***-(1-α2)h22R6]<0,(3.2)E2=[E(1-α)αhLα2hSα2h22H(1-α2)h22M*-(1-α)αhR3000**-α2hR300***-α2h22R50****-(1-α2)h22R6]<0,(3.3)Φ1=[Φ(1-α)αhU(1-α)hWα2h22H(1-α2)h22M*-(1-α)αhR3000**-(1-α)hR400***-α2h22R50****-(1-α2)h22R6]<0,(3.4)Φ2=[Φ(1-α)αhV(1-α)hZα2h22H(1-α2)h22M*-(1-α)αhR3000**-(1-α)hR400***-α2h22R50****-(1-α2)h22R6]<0,
where
(3.5)E=(E11E12E13E140-P13E17P1A+KT1P1BP22-H1P23-M1*E22E23-N200000-H2-M2**E33-N30000KT2-H3-M3***E440R4(1-α)h000-P22+P23T-P23+P33****-Q5000000*****E66000-P23T-P33******E77P2A+Λ-ΔP2BP12P13*******Q6-2T1-Q000********E9900*********-R10**********-R2),Φ=(Φ11Φ12Φ13Φ14V1+R3α2h-P13-W1Φ17P1A+KT1P1BP22-H1P23-M1*Φ22Φ23-U2+Z2V2-W2000-H2-M2**Φ33-U3+Z3V3-W300KT2-H3-M3***-Q300000-P22+P23T-P23+P33****-Q5-R3α2h000000*****-Q4000-P23T-P33******Φ77P2A+Λ-ΔP2BP12P13*******Q6-2T1-Q000********Φ9900*********-R10**********-R2)E11=P12+P12T+Q1+Q2+Q3+Q4+Q5+α2h2R1+(1-α)2h2R2+S1+S1T-P1C-CP1T+αh(H1+H1T)+(1-α)h(M1+M1T)+KQK,E12=L1+S2T-S1+αhH2T+(1-α)hM2T,E13=N1-L1+S3T+αhH3T+(1-α)hM3T,E14=-P12+P13-N1,E17=P11+KΔ-P1-CP2T,E22=-(1-αu)Q1+L2+L2T-S2-S2T,E23=-L2+L3T-S3T+N2,E33=-(1-u)Q2+N3+N3T-L3-L3T-(1-u)KQK,E44=-Q3-R4(1-α)h,E66=-Q4-R4(1-α)h,E77=αhR3+(1-α)hR4+α2h22R5+(1-α2)h22R6-P2-P2T,E99=-(1-u)Q6-2T2+(1-u)Q,Φ11=P12+P12T+Q1+Q2+Q3+Q4+Q5+α2h2R1+(1-α)2h2R2-P1C-CP1T+αh(H1+H1T)+(1-α)h(M1+M1T)-R3α2h+KQK,Φ12=U1-V1+αhH2T+(1-α)hM2T,Φ13=W1-Z1+αhH3T+(1-α)hM3T,Φ14=-P12+P13-U1+Z1,Φ17=P11+KΔ-P1-CP2T,Φ22=-(1-αu)Q1+U2+U2T-V2-V2T,Φ23=U3T-V3T+W2-Z2,Φ33=-(1-u)Q2+W3+W3T-Z3-Z3T-(1-u)KQK,Φ77=αhR3+(1-α)hR4+α2h22R5+(1-α2)h22R6-P2-P2T,Φ99=-(1-u)Q6-2T2+(1-u)Q,N=(N1TN2TN3T00000000)T,L=(L1TL2TL3T00000000)T,S=(S1TS2TS3T00000000)T,H=(H1TH2TH3T00000000)T,M=(M1TM2TM3T00000000)T,U=(U1TU2TV3T00000000)T,V=(V1TV2TV3T00000000)T,W=(W1TW2TW3T00000000)T,Z=(Z1TZ2TZ3T00000000)T.Proof.
Construct a new class of Lyapunov functional candidate as follows:(3.6)V(z(t))=∑i=16Vi(z(t)),
where
(3.7)V1(z(t))=ξT(t)Pξ(t)+2∑i=1nλi∫0zi(t)fi(s)ds+2∑i=1nδi∫0zi(t)(kis-fi(s))ds,V2(z(t))=∫t-ατ(t)tzT(s)Q1z(s)ds+∫t-τ(t)tzT(s)Q2z(s)ds+∫t-αhtzT(s)Q3z(s)ds+∫t-htzT(s)Q4z(s)ds+∫t-α2htzT(s)Q5z(s)ds+∫t-τ(t)tfT(z(s))Q6f(z(s))ds,V3(z(t))=αh∫-αh0∫t+θtzT(s)R1z(s)dsdθ+(1-α)h∫-h-αh∫t+θtzT(s)R2z(s)dsdθ,V4(z(t))=∫-αh0∫t+θtżT(s)R3ż(s)dsdθ+∫-h-αh∫t+θtżT(s)R4ż(s)dsdθ,V5(z(t))=∫-αh0∫θ0∫t+λtżT(s)R5ż(s)dsdλdθ+∫-h-αh∫θ0∫t+λtżT(s)R6ż(s)dsdλdθ,V6(z(t))=∫t-τ(t)t[zT(s)KQKz(s)-fT(z(s))Qf(z(s))]ds,
where
(3.8)ξT(t)=[zT(t)∫t-αhtzT(s)ds∫t-ht-αhzT(s)ds],0<α<1.Remark 3.2.
Since the term2∑i=1nδi∫0zi(t)(kis-fi(s))ds in V1(z(t)) and V6(z(t))=∫t-τ(t)t[zT(s)KQKz(s)-fT(z(s))Qf(z(s))]ds is taken into account, it is clear that the Lyapunov functional candidate in this paper is more general than that in [5, 6, 8, 9]. So the stability criteria in this paper may be more applicable.The time derivative ofV(z(t)) along the trajectory of system (2.3) is given by(3.9)V̇(z(t))=∑i=16V̇i(z(t)),
where (3.10)V̇1(z(t))=2ξT(t)P[ż(t)z(t)-z(t-αh)z(t-αh)-z(t-h)]+2fT(z(t))Λż(t)+2(Kz(t))-f(z(t))TΔż(t),V̇2(z(t))=zT(t)(Q1+Q2+Q3+Q4+Q5)z(t)+fT(z(t))Q6f(z(t))-(1-αu)zT(t-ατ(t))Q1z(t-ατ(t))-(1-u)zT(t-τ(t))Q2z(t-τ(t))-zT(t-αh)Q3z(t-αh)-zT(t-h)Q4z(t-h)-zT(t-α2h)Q5z(t-α2h)-(1-u)fT(z(t-τ(t)))Q6f(z(t-τ(t))).
Using Lemma 2.1, one can obtain that
(3.11)V̇3(z(t))=α2h2zT(t)R1z(t)+(1-α)2h2zT(t)R2z(t)-αh∫t-αhtzT(s)R1z(s)ds-(1-α)h∫t-ht-αhzT(s)R2z(s)ds⩽α2h2zT(t)R1z(t)+(1-α)2h2zT(t)R2z(t)-(∫t-αhtz(s)ds)TR1(∫t-αhtz(s)ds)-(∫t-ht-αhz(s)ds)TR2(∫t-ht-αhz(s)ds),V̇4(z(t))=żT(t)(αhR3+(1-α)hR4)ż(t)-∫t-αhtżT(s)R3ż(s)ds-∫t-ht-αhżT(s)R4ż(s)ds,V̇5(z(t))=żT(t)(α2h22R5+(1-α2)h22R6)ż(t)-∫-αh0∫t+θtżT(s)R5ż(s)dsdθ-∫-h-αh∫t+θtżT(s)R6ż(s)dsdθ,V̇6(z(t))⩽zT(t)KQKz(t)-fT(z(t))Qf(z(t))-(1-u)zT(t-τ(t))KQKz(t-τ(t))+(1-u)fT(z(t-τ(t)))Qf(z(t-τ(t))).(1)
For the case of0⩽τ(t)⩽αh, then it gets(3.12)-∫t-αhtżT(s)R3ż(s)ds=-∫t-αht-τ(t)żT(s)R3ż(s)ds-∫t-τ(t)t-ατ(t)żT(s)R3ż(s)ds-∫t-ατ(t)tżT(s)R3ż(s)ds.
Similar to [8], the following equalities hold: (3.13)2ζT(t)N[z(t-τ(t))-z(t-αh)-∫t-αht-τ(t)ż(s)ds]=0,(3.14)2ζT(t)L[z(t-ατ(t))-z(t-τ(t))-∫t-τ(t)t-ατ(t)ż(s)ds]=0,(3.15)2ζT(t)S[z(t)-z(t-ατ(t))-∫t-ατ(t)tż(s)ds]=0,(3.16)2[zT(t)P1+żT(t)P2][-ż(t)-Cz(t)+Af(z(t))+Bf(z(t-τ(t)))]=0,(3.17)2ζT(t)H[αhz(t)-∫t-αhtz(s)ds-∫-αh0∫t+θtż(s)dsdθ]=0,(3.18)2ζT(t)M[(1-α)hz(t)-∫t-ht-αhz(s)ds-∫-h-αh∫t+θtż(s)dsdθ]=0,(3.19)ζT(t)=[∫t-αhtzT(t)zT(t-ατ(t))zT(t-τ(t))zT(t-αh)zT(t-α2h)zT(t-h)żT(t)fT(z(t))fT(z(t-τ(t)))∫t-αhtzT(s)ds∫t-ht-αhzT(s)ds].
It is easy to see that (3.20)-2ζT(t)N∫t-αht-τ(t)ż(s)ds⩽(αh-τ(t))ζT(t)NR3-1NTζ(t)+∫t-αht-τ(t)żT(s)R3ż(s)ds,(3.21)-2ζT(t)L∫t-τ(t)t-ατ(t)ż(s)ds⩽(1-α)τ(t)ζT(t)LR3-1LTζ(t)+∫t-τ(t)t-ατ(t)żT(s)R3ż(s)ds,(3.22)-2ζTS∫t-ατ(t)tż(s)ds⩽ατ(t)ζT(t)SR3-1STζ(t)+∫t-ατ(t)tżT(s)R3ż(s)ds,(3.23)-2ζTH∫-αh0∫t+θtż(s)dsdθ⩽α2h22ζT(t)HR5-1HTζ(t)+∫-αh0∫t+θtżT(s)R5ż(s)dsdθ,(3.24)-2ζTM∫-h-αh∫t+θtż(s)dsdθ⩽(1-α2)h22ζT(t)MR6-1MTζ(t)+∫-h-αh∫t+θtżT(s)R6ż(s)dsdθ.
Using Lemma 2.1, it is easy to obtain that (3.25)-∫t-ht-αhżT(s)R4ż(s)ds⩽-[z(t-αh)-z(t-h)]TR4(1-α)h[z(t-αh)-z(t-h)].
Furthermore, there exist positive diagonal matrices T1,T2, such that the following inequalities hold based on (2.4): (3.26)-2fT(z(t))T1f(z(t))+2zT(t)KT1f(z(t))⩾0,(3.27)-2fT(z(t-τ(t)))T2f(z(t-τ(t)))+2zT(t-τ(t))KT2f(z(t-τ(t)))⩾0.
From (3.10)–(3.27), one can obtain that (3.28)V̇(z(t))⩽ζT(t)Σ1ζ(t),
where (3.29)Σ1=E+(αh-τ(t))NR3-1NT+(1-α)τ(t)LR3-1LT+ατ(t)SR3-1ST+α2h22HR5-1HT+(1-α2)h22MR6-1MT.
Note that 0⩽τ(t)⩽αh, (αh-τ(t))NR3-1NT+(1-α)τ(t)LR3-1LT+ατ(t)SR3-1ST can be seen as the convex combination of NR3-1NT,LR3-1LT, and SR3-1ST on τ(t). Therefore, Σ1<0 holds if and only if (3.30)E+αhNR3-1NT+α2h22HR5-1HT+(1-α2)h22MR6-1MT<0,(3.31)E+(1-α)αhLR3-1LT+α2hSR3-1ST+α2h22HR5-1HT+(1-α2)h22MR6-1MT<0.
Applying the Schur complement, the inequalities (3.30) and (3.31) are equivalent to the LMI (3.1) and (3.2), respectively. (2)
Whenαh⩽τ(t)⩽h, then it gets(3.32)-∫t-αhtżT(s)R3ż(s)ds=-∫t-αht-ατ(t)żT(s)R3ż(s)ds-∫t-ατ(t)t-α2hżT(s)R3ż(s)ds-∫t-α2htżT(s)R3ż(s)ds,-∫t-ht-αhżT(s)R4ż(s)ds=-∫t-ht-τ(t)żT(s)R4ż(s)ds-∫t-τ(t)t-αhżT(s)R4ż(s)ds.
Similar to [8], the following equalities hold: (3.33)2ζT(t)U[z(t-ατ(t))-z(t-αh)-∫t-αht-ατ(t)ż(s)ds]=0,2ζT(t)V[z(t-α2h)-z(t-ατ(t))-∫t-ατ(t)t-α2hż(s)ds]=0,2ζT(t)W[z(t-τ(t))-z(t-h)-∫t-ht-τ(t)ż(s)ds]=0,2ζT(t)Z[z(t-αh)-z(t-τ(t))-∫t-τ(t)t-αhż(s)ds]=0.
It is easy to obtain that (3.34)-2ζT(t)U∫t-αht-ατ(t)ż(s)ds⩽α(h-τ(t))ζT(t)UR3-1UTζ(t)+∫t-αht-ατ(t)żT(s)R3ż(s)ds,-2ζT(t)V∫t-ατ(t)t-α2hż(s)ds⩽α(τ(t)-αh)ζT(t)VR3-1VTζ(t)+∫t-ατ(t)t-α2hżT(s)R3ż(s)ds,-2ζT(t)W∫t-ht-τ(t)ż(s)ds⩽(h-τ(t))ζT(t)WR4-1WTζ(t)+∫t-ht-τ(t)żT(s)R4ż(s)ds,-2ζT(t)Z∫t-τ(t)t-αhż(s)ds⩽(τ(t)-αh)ζT(t)ZR4-1ZTζ(t)+∫t-τ(t)t-αhżT(s)R4ż(s)ds.
Using Lemma 2.1, one can obtain that (3.35)-∫t-α2htżT(s)R4ż(s)ds⩽-1α2h[z(t)-z(t-α2h)]TR3[z(t)-z(t-α2h)].
From (3.10)–(3.11), (3)–(3.18), (3.23), (3.24), (3.26), (3.27) and (3.33)–(3.35), one can obtain that (3.36)V̇(z(t))⩽ζT(t)Σ2ζ(t),
where (3.37)Σ2=Φ+α(h-τ(t))UR3-1UT+α(τ(t)-αh)VR3-1VT+(h-τ(t))WR4-1WT+(τ(t)-αh)ZR4-1ZT+α2h22HR5-1HT+(1-α2)h22MR6-1MT.
Note that αh⩽τ(t)⩽h, α(h-τ(t))UR3-1UT+α(τ(t)-αh)VR3-1VT+(h-τ(t))WR4-1WT+(τ(t)-αh)ZR4-1ZT can be seen as the convex combination of UR3-1UT,VR3-1VT,WR4-1WT, and ZR4-1ZT on τ(t). Therefore, Σ2<0 holds if and only if (3.38)Φ+α(1-α)hUR3-1UT+(1-α)hWR4-1WT+α2h22HR5-1HT+(1-α2)h22MR6-1MT<0,(3.39)Φ+α(1-α)hVR3-1VT+(1-α)hZR4-1ZT+α2h22HR5-1HT+(1-α2)h22MR6-1MT<0.
Applying the Schur complement, the inequalities (3.38) and (3.39) are equivalent to the LMI (3.3) and (3.4), respectively. Therefore, if the LMIs (3.1)–(3.4) are satisfied, then the system (2.3) is guaranteed to be asymptotically stable for 0⩽τ(t)⩽h.Remark 3.3.
It is well known that the delay-dividing approach can reduce the conservatism notably. But some previous literature only uses single method to divide the delay interval[0,h]. Unlike [10, 25], the new Lyapunov functional in our paper which not only divides the delay interval [0,h] into two ones [0,αh] and [αh,h] but also divides the delay interval [0,h] into three ones [0,ατ(t)],  [ατ(t),τ(t)], and [τ(t),h] is proposed. Each segment has a different positive matrix, which has the potential to yield less conservative results.Remark 3.4.
In this paper, by taking the statesz(t-αh),z(t-τ(t)),z(t-α2h),z(t-h), and z(t-ατ(t)) as augmented variables, the stability in Theorem 3.1 utilizes more information on state variables. And in deriving upper bounds of integral terms in V̇4(z(t)), different free-weighting matrices are introduced in two different intervals 0⩽τ(t)⩽αh and αh⩽τ(t)⩽h. These methods mentioned above may lead to obtain an improved feasible region for delay-dependent stability criteria.Remark 3.5.
In (3.28), E+(αh-τ(t))NR3-1NT+(1-α)τ(t)LR3-1LT+ατ(t)SR3-1ST+(α2h2/2)HR5-1HT+((1-α2)h2/2)MR6-1MT is not simply guaranteed by E+αhNR3-1NT+(1-α)αhLR3-1LT+α2hSR3-1ST+(α2h2/2)HR5-1HT+((1-α2)h2/2)MR6-1MT but is evaluated by the LMIs (3.1) and (3.2), which can help reduce much more conservatism than the results in [8].Remark 3.6.
In many cases,u is unknown. For this situation, a rate-independent criterion for a delay satisfying 0≤τ(t)≤h is derived as follows by setting Q1=0,Q2=0,Q6=0,  and Q=0 in the proof of Theorem 3.1.Corollary 3.7.
For given scalarsK=diag⁡(k1,k2,…,kn), h⩾0,  and 0<α<1, the system (2.3) is globally asymptotically stable if there exist symmetric positive matrices P=[Pij]3×3,Q3,Q4,Q5,  Ri(i=1,2,…,6), positive diagonal matrices T1,T2,Λ=diag⁡(λ1,λ2,…,λn),Δ=diag⁡(δ1,δ2,…,δn), and any matrices  P1,P2,Ni,Mi,Li,Hi,Zi,Si,Ui,Vi,Wi(i=1,2,3) with appropriate dimensions, such that the following LMIs hold:
(3.40)E̅1=[E̅αhNα2h22H(1-α2)h22M*-αhR300**-α2h22R50***-(1-α2)h22R6]<0,E̅2=[E̅(1-α)αhLα2hSα2h22H(1-α2)h22M*-(1-α)αhR3000**-α2hR300***-α2h22R50****-(1-α2)h22R6]<0,Φ̅1=[Φ̅(1-α)αhU(1-α)hWα2h22H(1-α2)h22M*-(1-α)αhR3000**-(1-α)hR400***-α2h22R50****-(1-α2)h22R6]<0,Φ̅2=[Φ̅(1-α)αhV(1-α)hZα2h22H(1-α2)h22M*-(1-α)αhR3000**-(1-α)hR400***-α2h22R50****-(1-α2)h22R6]<0,
where
(3.41)E̅=(E̅11E12E13E140-P13E17P1A+KT1P1BP22-H1P23-M1*E̅22E23-N200000-H2-M2**E̅33-N30000KT2-H3-M3***E440R4(1-α)h000-P22+P23T-P23+P33****-Q5000000*****E66000-P23T-P33******E77P2A+Λ-ΔP2BP12P13*******-2T1000********-2T200*********-R10**********-R2),Φ̅=(Φ̅11Φ12Φ13Φ14V1+R3α2h-P13-W1Φ17P1A+KT1P1BP22-H1P23-M1*Φ̅22Φ23-U2+Z2V2-W2000-H2-M2**Φ̅33-U3+Z3V3-W300KT2-H3-M3***-Q300000-P22+P23T-P23+P33****-Q5-R3α2h000000*****-Q4000-P23T-P33******Φ77P2A+Λ-ΔP2BP12P13*******-2T1000********-2T200*********-R10**********-R2)E̅11=P12+P12T+Q3+Q4+Q5+α2h2R1+(1-α)2h2R2+S1+S1T-P1C-CP1T+αh(H1+H1T)+(1-α)h(M1+M1T),E̅22=L2+L2T-S2-S2T,E̅33=N3+N3T-L3-L3T,Φ̅11=P12+P12T+Q3+Q4+Q5+α2h2R1+(1-α)2h2R2-P1C-CP1T+αh(H1+H1T)+(1-α)h(M1+M1T)-R3α2h,Φ̅22=U2+U2T-V2-V2T,Φ̅33=W3+W3T-Z3-Z3T.
The other Eij,Φij are defined in Theorem 3.1.


## 4. Numerical Examples

Example 4.1.
Consider the stability of system (2.3) with time-varying delay and
(4.1)C=diag⁡(2,2),A=[11-1-1],B=[0.88111],σ1=0.4,σ2=0.8,γ1=0,γ2=0.
Our purpose is to estimate the allowable upper bounds delay h under different u such that the system (2.3) is globally asymptotically stable. According to Table 1, this example shows that the stability criterion in this paper gives much less conservative results than those in the literature. By using the Matlab LMI toolbox, we solve LMIs (3.1)–(3.4) for the case α=0.4,u=0.8, and  h=2.9144 and obtain
(4.2)P11=[0.0005-0.0002-0.00020.0018],P12=1.0e-003×[-0.03630.00810.1663-0.1945],P13=1.0e-006×[-0.10020.02520.3993-0.2498],P22=1.0e-004×[0.1678-0.2152-0.21520.5047],P23=1.0e-006×[0.0221-0.08360.0468-0.3688],P33=1.0e-006×[0.2614-0.2146-0.21460.5418],Q1=1.0e-006×[0.2368-0.2454-0.24540.4008],Q2=1.0e-005×[0.0174-0.0087-0.00870.1976],Q3=1.0e-003×[0.05260.02960.02960.2541],Q4=1.0e-004×[0.99710.51100.51100.4639],Q5=1.0e-005×[0.0232-0.0561-0.05610.1584],Q6=[0.00210.00350.00350.0057],R1=1.0e-004×[0.4523-0.6020-0.60200.8760],R2=1.0e-006×[0.1049-0.2159-0.21590.5299],R3=1.0e-004×[0.79150.36500.36500.4110],R4=1.0e-004×[0.97090.50710.50710.4212],R5=1.0e-003×[0.1083-0.1872-0.18720.3776],R6=1.0e-006×[0.0698-0.1266-0.12660.2910],Λ=[0.0000000.0014],Δ=1.0e-003×[0.1686000.0027],T1=[0.0037000.0039],T2=[0.0009000.0010],Q=[0.0000000.0038].
Therefore, it follows from Theorem 3.1 that the system (2.3) with given parameters is globally asymptotically stable.Table 1
Allowable upper bound ofh for different u (Example 4.1).
uu=0.8u=0.9Unknownu[4, 5]1.22810.86360.8298[6]1.68311.14931.0880[7]1.73471.1662—[8]2.35341.60501.5103[13] (ρ=0.8)2.54061.7273—[14] (m=2)2.24951.59661.4902[15] (m=2)2.11501.42861.3126[15] (m=3)2.38381.62291.4740This paper(α=0.4)2.91441.90951.7437This paper(α=0.5)2.90631.94431.7769Example 4.2.
Consider the stability of system (2.3) with time-varying delay and
(4.3)C=diag⁡(1.5,0.7),A=[0.05030.04540.09870.2075],B=[0.23810.93200.03880.5062],σ1=0.3,σ2=0.8,γ1=0,γ2=0.Table2 gives the comparison results on the maximum delay bound allowed via the methods in recent paper and our new study. According to Table 2, this example shows that the stability criterion in this paper can lead to less conservative results. By using the Matlab LMI toolbox, we solve LMIs (3.1)–(3.4)  for the case α=0.4,u=0.4, and  h=4.7444 and obtain(4.4)P11=[1.0206-0.7528-0.75281.2378],P12=1.0e-003×[-0.00450.0417-0.0652-0.0137],P13=1.0e-003×[-0.24670.6404-0.1839-0.3921],P22=[0.0073-0.0020-0.00200.0022],P23=1.0e-004×[-0.00860.4616-0.41080.1085],P33=1.0e-003×[0.1904-0.1292-0.12920.1059],Q1=[0.0013-0.0012-0.00120.0012],Q2=[0.02580.05660.05660.1268],Q3=[0.06530.00510.00510.0050],Q4=[0.0114-0.0300-0.03000.0839],Q5=[0.0014-0.0009-0.00090.0006],Q6=[0.03400.03000.03000.0310],R1=[0.0090-0.0066-0.00660.0062],R2=1.0e-003×[0.1943-0.1361-0.13610.1036],R3=[0.2485-0.3846-0.38460.7348],R4=[0.0802-0.2197-0.21970.6122],R5=[0.1553-0.0821-0.08210.0709],R6=1.0e-003×[0.7651-0.4906-0.49060.5825],Λ=[0.0318000.0344],Δ=1.0e-003×[0.0797000.3593],T1=[0.0136000.1968],T2=[0.3056000.3423],Q=[0.4567000.0141].
Therefore, it follows from Theorem 3.1 that the system (2.3) with given parameters is globally asymptotically stable.Table 2
Allowable upper bound ofh for different u (Example 4.2).
uu=0.4u=0.45u=0.5u=0.55[6]3.99723.27603.05942.9814[9]4.38143.60083.33773.2350This paper(α=0.4)4.74443.94823.70003.5667This paper(α=0.5)4.65873.90693.64963.5276


## 5. Conclusions

In this paper, a new delay-dependent asymptotic stability criterion for neural networks with time-varying delay has been proposed. A new class of Lyapunov functional has been introduced to derive some less conservative delay-dependent stability criteria by using the free-weighting matrices method and the technique of dealing with some integral terms. Finally, numerical examples have been given to illustrate the effectiveness of the proposed method.



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*Source: 950269-2012-06-05.xml*