# Stability Analysis for Uncertain Neural Networks of Neutral Type with Time-Varying Delay in the Leakage Term and Distributed Delay

Authors: Qi Zhou; Xueying Shao; Jin Zhu; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The stability problem is investigated for a class of uncertain networks of neutral type
with leakage, time-varying discrete, and distributed delays. Both the parameter uncertainty and the
generalized activation functions are considered in this paper. New stability results are achieved by constructing
an appropriate Lyapunov-Krasovskii functional and employing the free weighting matrices and
the linear matrix inequality (LMI) method. Some numerical examples are given to show the effectiveness
and less conservatism of the proposed results.


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## Body

## 1. Introduction

Neural networks are a complex large-scale power system and have very rich dynamic property. In the past years, neural networks have some applications in areas of associative memory [1], pattern recognition [2], and optimization problems [3–5]. Recently, the stability problem for neural networks has been widely investigated and some results have been reported in [6–14].Time delays are often encountered in various engineering, biological, and economical systems [15–23]. Due to the finite speed of information processing, the existence of time delays frequently causes oscillation, divergence, or instability in neural networks. Recently, the stability of neural networks with time delays has drawn considerable attention, and many results on stability of neural networks with time delays have been reported in the literature [24–30]. In practice, the time-varying delay often belongs to a given interval, and the lower bound of the delays may not be zero. Recently, some papers investigated the stability conditions for neural networks with interval time-varying delay in the literature, for example [31]. Furthermore, due to the presence of parallel pathways of different axonal sizes and lengths, there may exist a spatial extent in neural networks, which may cause distributed time delays [32–35]. Recently, the stability problem of neural networks with neutral-type was studied in [36, 37].More recently, more and more attention has been paid to time delay in the leakage (or “forgetting”) term [38, 39], since there exist some theoretical and technical difficulties when handling the leakage delay [40]. It has been pointed out that the leakage delay has a great impact on the dynamics of neural networks [41, 42]. The authors in [43] showed that time delay in the stabilizing negative feedback term has a tendency to destabilize a system. More recently, the authors in [44] focused on recurrent neural networks with time delay in the leakage term and showed that all the results mentioned about the existence and uniqueness of the equilibrium point are independent of time delays and initial conditions. Then, it can be seen that time delays in leakage terms do not affect the existence and uniqueness of the equilibrium point. Recently, some results about stability analysis for neural networks with leakage delay have been reported in [38, 39, 41]. To mention a few, the work in [14] investigated the problems of delay-dependent stability analysis and strict (Q,S,R)-α-dissipativity analysis for cellular neural networks with distributed delay. However, it should be mentioned that there are few results about stability analysis for uncertain neural networks of neutral type with time-varying delay in the leakage term and distributed delay, which motivates this study.In this paper, the stability problem is investigated for uncertain neural networks of neutral type with time-varying delay in the leakage term and time-varying distributed delay. Firstly, by constructing a new type of Lyapunov functional and developing some novel techniques to handle the delays considered in this paper, some novel robust stability criteria are proposed. Secondly, the proposed conditions can be expressed in terms of linear matrix inequalities (LMIs), which can be easily solved via standard software. Finally, some numerical examples are given to demonstrate the effectiveness and less conservatism of the proposed results.Notation. Throughout this paper, the notations are standard. ℝn and ℝn×m denote the n-dimensional Euclidean space and the set of all n×m real matrices, respectively. In this paper, the superscript T denotes matrix transposition. For real symmetric matrices X and Y, the natation X≥Y (X>Y resp.) means that the X-Y is positive-semidefinite (positive-definite resp.). The notation diag{} stands for a block-diagonal matrix. I is the identity matrix with appropriate dimensions. Matrices if not explicitly stated, where the symbol “*” stands for the symmetric term in a matrix, are assumed to have compatible dimensions.


## 2. Problem Formulation

Consider the following uncertain neural network of neutral type with time delay in the leakage term and distributed delay:(1)x˙(t)=-(W1+ΔW1(t))x(t-δ)+(W2+ΔW2(t))f(x(t))+(W3+ΔW3(t))f(t-τ(t))+(W4+ΔW4(t))∫t-r(t)tf(x(s))ds+(W5+ΔW5(t))x˙(t-h(t)),
where x(t)=[x1(t)x2(t)⋯xn(t)]T∈ℝn is the network state vector at time t; f(t)∶=f(x(t))=[f1(x1(t))f2(x2(t))⋯fn(xn(t))]T∈ℝn denotes the activation function at time t; W1 is a positive diagonal matrix; W1=(W1ij)n×n, W2=(W2ij)n×n, W3=(W3ij)n×n, W4=(W4ij)n×n, and W5=(W5ij)n×n are known constant matrices; ΔW1(t), ΔW2(t), ΔW3(t), ΔW4(t), and ΔW5(t), are unknown matrices; δ is leakage delay, τ(t) is time-varying discrete delay; h(t) is neutral delay; and r(t) is time-varying distributed delay. They satisfy the following conditions:
(2)0≤δ,0<τ1≤τ(t)≤τ2,τ˙(t)≤τd,0<h(t)≤h,h˙(t)≤hd<1,0<r(t)≤r-,
where δ, τ1, τ2, τd, h, and hd are constants. The time-varying parameter uncertainties ΔW1(t), ΔW2(t), ΔW3(t), ΔW4(t), and ΔW5(t) are assumed to be of the form
(3)[ΔW1(t)ΔW2(t)ΔW3(t)ΔW4(t)ΔW5(t)]=HG(t)[E1E2E3E4E5],
where H, E1, E2, E3, E4, E5 are known constant matrices and G(t) is an unknown time-varying matrix satisfying
(4)G(t)TG(t)≤I.
Throughout this paper, we make the following assumption:(
H
1
) for any i=1,2,…,n, fi(0)=0, there exist constants Fi-, Fi+, for all α1, α2∈Rn with α1≠α2 such that
(5)Fi-≤fi(α1)-fi(α2)α1-α2≤Fi+.Lemma 1 (see [45]).
For any positive symmetric constant matricesM∈ℝn×n, scalar τ>0, vector function ω(s)∈ℝn such that the integrations concerned are well defined; then
(6)-∫t-τt‍ωT(s)Mω(s)ds≤-1τ(∫t-τt‍ω(s)ds)TM(∫t-τt‍ω(s)ds).Lemma 2 ([46] Schur complement).
Given constant matricesΩ1, Ω2, and Ω3 with appropriate dimensions, where Ω1T=Ω1 and Ω2T=Ω2>0, then Ω1+Ω3TΩ2-1Ω3<0 if and only if
(7)[Ω1Ω3T*-Ω2]<0or[-Ω2Ω3*Ω1]<0.
In order to present novel stability criteria for neural networks (1), the following notations are defined:
(8)F1=diag(F1-,F2-,…,Fn-),F2=diag(F1+,F2+,…,Fn+),F3=diag(F1-F1+,F2-F2+,…,Fn-Fn+),F4=diag(F1-+F1+2,F2-+F2+2,…,Fn-+Fn+2).


## 3. Main Results

In this section, by considering the delay in the leakage term and distributed delay and using some new techniques, novel stability criteria will be proposed for uncertain neural network of neutral-type with time-varying delay in (1). Firstly, considering neutral-type delay, the delay in the leakage term, and distributed delay, we have the following theorem.Theorem 3.
For given scalars0≤δ, 0<τ1<τ2, 0<h, 0<r-, hd<1, and τd, neural network (1) under Assumption (H1) is robustly asymptotically stable, if there exist matrices Pi>0(i=1,2,…,5), Q=[Qij]2×2>0, K=[Kij]2×2>0, Zi>0(i=1,2,…,7), D=diag{d1,d2,…,dn}>0, L=diag{l1,l2,…,ln}>0, positive diagonal matrices R and S, and appropriately dimensioned matrices Nk, Mk(k=1,2,…,14), such that the following LMIs hold:
(9)[Z3T*Z3]>0,(10)[Ξ+Π+ΠThNMHεET*-Z700**-εI0***-εI]<0,
where(11)Θ12=[0P1-F1TDT+F2TLT0F4R0W1P1W100000F4S00000000000000000000000000000000000-W1P1W10],Ξ=[Θ11Θ12*Θ22],Θ22=[-Z5000000*Ψ990D-L0-P1W10**-(1-hd)Z60000***P5+r-2Z4-R000****-(1-τd)P5-S00*****-P30******-Z4],Θ11=[Ψ110Z20Z1+Q12K12P1W1*Ψ22Z3-TTZ3-T000**P4-Z2-Z3-Q22T-Q12T00***-K22-Z30-K12T0****-Z1+Q22-Q1100*****K22-K110******-P2],Ψ11=-P1W1-W1TP1T+P2+δ2P3+Q11+K11-Z1-Z2-F3R+Z5,Ψ22=-(1-τd)P4-Z3-Z3T-F3S+T+TT,Ψ99=τ124Z1+τ12Z2+τ122Z3+Z6+hZ7,Π=[N,0,0,0,0,0,0,-N,0,0,0,0,0,0]+M[0,0,0,0,0,0,-W1,0,-I,W5,W2,W3,0,W4],N=[N1T,N2T,N3T,N4T,N5T,N6T,N7T,N8T,N9T,N10T,N11T,N12T,N13T,N14T]T,M=[M1T,M2T,M3T,M4T,M5T,M6T,M7T,M8T,M9T,M10T,M11T,M12T,M13T,M14T]T,E=[0,0,0,0,0,0,-E1,0,0,E5,E2,E3,0,E4].Proof.
By using Newton-Leibniz formulation and considering neural network (1), the following equalities hold for appropriately dimensioned matrices N and M:
(12)2ηT(t)N[x(t)-x(t-h(t))-∫t-h(t)t‍x˙(t)ds]=0,2ηT(t)M[∫t-r(t)tf(x(s))ds-(W1+ΔW1(t))x(t-δ)+(W2+ΔW2(t))2ηT(t)M×f(x(t))+(W3+ΔW3(t))f(t-τ(t))2ηT(t)M+(W4+ΔW4(t))∫t-r(t)tf(x(s))ds2ηT(t)M+(W5+ΔW5(t))x˙(t-h(t))-x˙(t)∫t-r(t)tf(x(s))ds]=0,
with
(13)ηT(t)=[(∫t-r(t)t‍f(x(s))ds)TxT(t),xT(t-τ(t)),xT(t-τ1),xT(t-τ2),xT(t-τ12)xT(t-τ22),xT(t-δ),xT(t-h(t)),x˙T(t),x˙T(t-h(t)),fT(x(t)),fT(x(t-τ(t))),(∫t-δt‍x(s)ds)T,(∫t-r(t)t‍f(x(s))ds)T]T.
For positive diagonal matrices R and S, based on Assumption (H1), it can be seen that the following inequalities hold:
(14)[x(t)f(x(t))]T[F3R-F4R-F4RR][x(t)f(x(t))]≤0,[x(t-τ(t))f(x(t-τ(t)))]T[F3S-F4S-F4SS][x(t-τ(t))f(x(t-τ(t)))]≤0.
Now, choosing the following Lyapunov-Krasovskii functional:(15)V(t)=∑i=17‍Vi(t),
where
(16)V1(t)=(x(t)-W1∫t-δt‍x(s)ds)TP1(x(t)-W1∫t-δt‍x(s)ds),V2(t)=2∑i=1n‍di∫0xi(t)‍(fi(s)-Fi-s)ds+2∑i=1n‍li∫0xi(t)‍(Fi+s-fi(s))ds,V3(t)=∫t-δt‍xT(s)P2x(s)ds+δ∫-δ0‍∫t+θt‍xT(s)P3x(s)dsdθ,V4(t)=∫t-(τ1/2)t‍ω1T(s)Qω1(s)ds+∫t-(τ2/2)t‍ω2T(s)Kω2(s)ds+∫t-τ(t)t-τ1‍xT(s)P4x(s)ds+∫t-τ(t)t‍fT(x(s))P5f(x(s))ds,V5(t)=∫-τ1/20‍∫t+θt‍τ12x˙T(s)Z1x˙(s)dsdθ+∫-τ10‍∫t+θt‍τ1x˙T(s)Z2x˙(s)dsdθ+∫-τ2-τ1‍∫t+θt‍τ12x˙T(s)Z3x˙(s)dsdθ,V6(t)=r-∫-r-0‍∫t+θt‍fT(x(s))Z4f(x(s))dsdθ,V7(t)=∫t-h(t)t‍xT(s)Z5x(s)ds+∫t-h(t)t‍x˙T(s)Z6x˙(s)ds+∫-h0‍∫t+θt‍x˙T(s)Z7x˙(s)dsdθ,
where τ12=τ2-τ1, and
(17)ωi(t)=[xT(t)xT(t-τi2)]T,i=1,2.
Then, the derivatives of Vi(t), (i=1,…,7) with time t can be obtained as
(18)V˙1(t)=2(x(t)-W1∫t-δt‍x(s)ds)T×P1(x˙(t)-W1x(t)+W1x(t-δ)),V˙2(t)=2x˙T(t)D(f(x(t))-F1x(t))+2x˙T(t)L(F2x(t)-f(x(t))),V˙3(t)=xT(t)P2x(t)+xT(t-δ)P2x(t-δ)+δ2xT(t)P3x(s)-δ∫t-δt‍xT(s)P3x(s)ds,V˙4(t)=ω1T(t)Qω1(t)-ω1T(t-τ12)Qω1(t-τ12)+ω2T(t)Kω2(t)-ω2T(t-τ22)Kω2(t-τ22)+xT(t-τ1)P4x(t-τ1)-(1-τ˙(t))xT(t-τ(t))P4x(t-τ(t))+fT(x(t))P5f(x(t))-(1-τ˙(t))fT(x(t-τ(t)))P5f(x(t-τ(t))),V˙5(t)=τ124x˙T(t)Z1x˙(t)-∫t-(τ1/2)t‍τ12x˙T(s)Z1x˙(s)ds+τ12x˙T(t)Z2x˙(t)-∫t-τ1t‍τ1x˙T(s)Z2x˙(s)ds+τ122x˙T(t)Z3x˙(t)-∫t-τ2t-τ1‍τ12x˙T(s)Z3x˙(s)ds,V˙6(t)=r-2fT(x(t))Z4f(x(t))-∫t-r-t‍r-fT(x(s))Z4f(x(s))ds,V˙7(t)=xT(t)Z5x(t)-xT(t-h(t))Z5x(t-h(t))+x˙T(t)Z6x˙(t)-(1-h˙(t))x˙T(t-h(t))×Z6x˙(t-h(t))+hx˙T(t)Z7x˙(t)-∫t-ht‍x˙T(s)Z7x˙(s)ds.
By Lemma 1, one can have
(19)-∫t-(τ1/2)t‍τ12x˙T(s)Z1x˙(s)ds≤-[x(t)-x(t-τ12)]TZ1[x(t)-x(t-τ12)],(20)-∫t-τ1t‍τ1x˙T(s)Z2x˙(s)ds≤-[x(t)-x(t-τ1)]TZ2[x(t)-x(t-τ1)],(21)-∫t-τ1t-τ2‍τ12x˙T(s)Z3x˙(s)ds=-∫t-τ(t)t-τ1‍τ12x˙T(s)Z3x˙(s)ds-∫t-τ2t-τ(t)‍τ12x˙T(s)Z3x˙(s)ds≤-τ12τ(t)-τ1ηT(t)(e3-e2)Z3(e3-e2)Tη(t)-τ12τ2-τ(t)ηT(t)(e2-e4)Z3(e2-e4)Tη(t)=-1α1ηT(t)(e3-e2)Z3(e3-e2)Tη(t)-1α2ηT(t)(e2-e4)Z3(e2-e4)Tη(t)=-ηT(t)(e3-e2)Z3(e3-e2)Tη(t)-ηT(t)(e2-e4)Z3(e2-e4)Tη(t)-α2α1ηT(t)(e3-e2)Z3(e3-e2)Tη(t)-α1α2ηT(t)(e2-e4)Z3(e2-e4)Tη(t),
where α1=(τ(t)-τ1)/τ12 and α2=(τ2-τ(t))/τ12.
It can be seen from the condition (9) that
(22)-α2α1ηT(t)(e3-e2)Z3(e3-e2)Tη(t)-α1α2ηT(t)(e2-e4)Z3(e2-e4)Tη(t)+ηT(t)(e3-e2)T(e2-e4)Tη(t)+ηT(t)(e2-e4)TT(e3-e2)Tη(t)=-ηT(t)[α2α1(e3T-e2T)-α1α2(e2T-e4T)]T[Z3TTTZ3]×[α2α1(e3T-e2T)-α1α2(e2T-e4T)]η(t)≤0,
which means
(23)-α2α1ηT(t)(e3-e2)Z3(e3-e2)Tη(t)-α1α2ηT(t)(e2-e4)Z3(e2-e4)Tη(t)≤-ηT(t)(e3-e2)T(e2-e4)Tη(t)-ηT(t)×(e2-e4)TT(e3-e2)Tη(t).
It follows from (21) and (23) that
(24)-∫t-τ1t-τ2‍τ12x˙T(s)Z3x˙(s)ds≤-ηT(t)(e2-e4)TT(e3T-e2T)η(t)-ηT(t)(e3-e2)T(e2T-e4T)η(t)-ηT(t)(e3-e2)Z3(e3-e2)Tη(t)-ηT(t)(e2-e4)Z3(e2-e4)Tη(t)=-ηT(t)[e3T-e2Te2T-e4T]T[Z3TTTZ3]×[e3T-e2Te2T-e4T]η(t).
Therefore, it is straight forward to obtain that
(25)V˙(t)≤2(x(t)-W1∫t-δt‍x(s)ds)T×P1(x˙(t)-W1x(t)+W1x(t-δ))+2x˙T(t)D(f(x(t))-F1x(t))+2x˙T(t)L(F2x(t)-f(x(t)))+xT(t)P2x(t)-xT(t-δ)P2x(t-δ)+δ2xT(t)P3x(s)ds-(∫t-δt‍x(s)ds)TP3∫t-δt‍x(s)ds+ω1T(t)Qω1(t)-ω1T(t-τ12)Qω1(t-τ12)+ω2T(t)Kω2(t)-ω2T(t-τ22)Kω2(t-τ22)+xT(t-τ1)P4x(t-τ1)-(1-τd)xT(t-τ(t))P4x(t-τ(t))+fT(x(t))P5f(x(t))-(1-τd)fT(x(t-τ(t)))P5f(x(t-τ(t)))+τ124x˙T(t)Z1x˙(t)-[x(t)-x(t-τ12)]TZ1[x(t)-x(t-τ12)]+τ12x˙T(t)Z2x˙(t)-[x(t)-x(t-τ1)]TZ2[x(t)-x(t-τ1)]+τ122x˙T(t)Z3x˙(t)-ηT(t)[e3T-e2Te2T-e4T]T[Z3TTTZ3][e3T-e2Te2T-e4T]η(t)+r-2fT(x(t))Z4f(x(t))-∫t-r-t‍r-fT(x(s))Z4f(x(s))ds+xT(t)Z5x(t)-xT(t-h(t))Z5x(t-h(t))+x˙T(t)Z6x˙(t)-(1-hd)x˙T(t-h(t))Z6x˙(t-h(t))+hx˙T(t)Z7x˙(t)-∫t-ht‍x˙T(s)Z7x˙(s)ds-[x(t)f(x(t))]T[F3R-F4R-F4RR][x(t)f(x(t))]-[x(t-τ(t))f(x(t-τ(t)))]T[F3S-F4S-F4SS]×[x(t-τ(t))f(x(t-τ(t)))]+2ηT(t)M[∫t-r(t)t‍-(W1+ΔW1(t))x(t-δ)+2ηT(t)M+(W2+ΔW2(t))f(x(t))+2ηT(t)M+(W3+ΔW3(t))f(t-τ(t))+2ηT(t)M+(W4+ΔW4(t))∫t-r(t)tf(x(s))ds+2ηT(t)M+(W5+ΔW5(t))x˙(t-h(t))-x˙(t)∫t-r(t)t‍]≤ηT(t)(Ξ+Π+ΠT+hNZ7-1NT+ΔΞ)η(t)-∫t-h(t)t‍(NTη(t)+Z7x˙(s))T-∫t-h(t)t‍×Z7-1(NTη(t)+Z7x˙(s))ds≤ηT(t)(Ξ+Π+ΠT+hNZ7-1NT+ΔΞ)η(t)≤ηT(t)(Ξ+Π+ΠT+hNZ7-1NT111111+ε-1MHHTMT+εETE)η(t),
where ΔΞ=MHF(t)E+(MHF(t)E)T. By Schur complement, it can be seen from the condition (10) that V˙(t)<0, which means that neural network (1) under Assumption (H1) is robustly asymptotically stable. This completes this proof.When neural network (1) without uncertainties, we have
(26)x˙(t)=-W1x(t-δ)+W2f(x(t))+W3f(t-τ(t))+W4∫t-r(t)tf(x(s))ds+W5x˙(t-h(t)).The stability condition can be easily obtained from Theorem3 in the following corollary.Corollary 4.
Given scalars0≤δ, 0≤τ1<τ2, 0<h, 0<r-, hd<1, and τd, neural network (26) is asymptotically stable, if there exist matrices Pi>0(i=1,2,…,5), Q=[Qij]2×2>0, K=[Kij]2×2>0, Zi>0(i=1,2,…,7), D=diag{d1,d2,…,dn}>0, L=diag{l1,l2,…,ln}>0, positive diagonal matrices R and S, and appropriately dimensioned matrices Nk, Mk(k=1,2,…,14), such that the following LMIs hold:
(27)[Z3TTTZ3]>0,[Ξ+Π+ΠThN*-Z7]<0,
where Ξ, Π and N have been defined in Theorem 3.Remark 5.
For neural network (1), if time delay in the leakage term is not considered in this paper, the following model can be obtained:
(28)x˙(t)=-W1x(t)+W2f(x(t))+W3f(t-τ(t))+W4∫t-r(t)tf(x(s))ds+W5x˙(t-h(t)).
In order to present stability criterion for neural network (28), we choose the following Lyapunov-Krasovskii functional:
(29)V(t)=xT(t)Px(t)+∑i=27‍Vi(t),
where Vi (i=2,3,4,5,6,7) is defined in (15). From the proof of Theorem 3, the following stability condition for neural network (28) can be obtained.Corollary 6.
For given scalars0≤τ1<τ2, 0<h, 0<r-, hd<1, and τd, neural network (1) under Assumption (H1) is asymptotically stable, if there exist matrices Pi>0(i=1,4,…,5), Q=[Qij]2×2>0, K=[Kij]2×2>0, Zi>0(i=1,2,…,7), D=diag{d1,d2,…,dn}>0, L=diag{l1,l2,…,ln}>0, positive diagonal matrices R and S, and appropriately dimensioned matrices N^k, M^k(k=1,2,…,12), such that the following LMIs hold:
(30)[Z3TTTZ3]>0,[Ξ^+Π^+Π^ThN^*-Z7]<0,
where(31)Ξ^=[Θ^11Θ^12*Θ^22],Θ^12=[0P1-F1TDT+F2TLT0F4R000000F4S0000000000000000000000000],Θ^11=[Ψ^110Z20Z1+Q12K12*Ψ^22Z3-TTZ3-T00**P4-Z2-Z3-Q22T-Q12T0***-K22-Z30-K12T****-Z1+Q22-Q110*****K22-K11],Ψ^11=Q11+K11-Z1-Z2-F3R+Z5,Ψ^22=-(1-τd)P4-Z3-Z3T-F3S+T+TT,Ψ^88=(τ124)Z1+τ12Z2+τ122Z3+Z6+hZ7,Π^=[N^,0,0,0,0,0,-N^,0,0,0,0,0]+M^[-W1,0,0,0,0,0,0,-I,W5,W2,W3,W4],Θ^22=[-Z500000*Ψ^880D-L00**-(1-hd)Z6000***P5+r-2Z4-R00****-(1-τd)P5-S0*****-Z4],N^=[N1T,N2T,N3T,N4T,N5T,N6T,N7T,N8T,N9T,N10T,N11T,N12T]T,M^=[M1T,M2T,M3T,M4T,M5T,M6T,M7T,M8T,M9T,M10T,M11T,M12T]T.Remark 7.
When neural network (1) without time-varying neutral delay h(t), the following model can be obtained:
(32)x˙(t)=-W1x(t-δ)+W2f(x(t))+W3f(t-τ(t))+W4∫t-r(t)tf(x(s))ds.
In order to present stability criterion for neural network (32), we choose the following Lyapunov-Krasovskii functional:
(33)V(t)=∑i=16‍Vi(t),
where Vi (i=1,…,6) are defined in (15). Following the same line of Theorem 3, the following corollary can be presented.Corollary 8.
Given scalars0≤δ, 0≤τ1<τ2, 0<r-, and τd, the neural network (32) under Assumptions (H1)-(H2) is asymptotically stable, if there exist matrices Pi>0(i=1,2,…,5), Q=[Qij]2×2>0, K=[Kij]2×2>0, Zi>0(i=1,2,…,4), D=diag{d1,d2,…,dn}>0,  L=diag{l1,l2,…,ln}>0, positive diagonal matrices R and S, and appropriately dimensioned matrices Mˇk(k=1,2,…,12), such that the following LMIs hold:
(34)[Z3TTTZ3]>0,Ξˇ+Πˇ+ΠˇT<0,
where(35)Ψˇ11=-P1W1-W1TP1T+P2+δ2P3+Q11+K11-Z1-Z2-F3R,Ψˇ22=-(1-τd)P4-Z3-Z3T-F3S+T+TT,Ψˇ88=(τ124)Z1+τ12Z2+τ122Z3,Π^=M^[0,0,0,0,0,0,-W1,-I,W2,W3,0,W4],M^=[M1T,M2T,M3T,M4T,M5T,M6T,M7T,M8T,M9T,M10T,M11T,M12T]T,Θˇ11=[Ψˇ110Z20Z1+Q12K12*Ψˇ22Z3-TTZ3-T00**P4-Z2-Z3-Q22T-Q12T0***-K22-Z30-K12T****-Z1+Q22-Q110*****K22-K11],Θˇ12=[P1W1P1-F1TDT+F2TLTF4R0W1P1W10000F4S00000000000000000000000000],Ξˇ=[Θˇ11Θˇ12*Θˇ22],Θˇ22=[-P2000-W1P1W10*Ψˇ88D-L0-P1W10**P5+r-2Z4-R000***-(1-τd)P5-S00****-P30*****-Z4].
Consider the following neural network with time-varying delay:(36)x˙(t)=-W1x(t)+W2f(x(t))+W3f(t-τ(t)).
By choosing the Lyapunov-Krasovskii functional(37)V(t)=xT(t)Px(t)+V2(t)+V4(t)+V6(t),
where Vi (i=2,4,6) is defined in (15). The novel stability condition for neural network with time-varying delay (36) can be presented from Theorem 3 in the following corollary.Corollary 9.
For given scalars0≤τ1<τ2, and τd, neural network (36) under Assumptions (H1)-(H2) is asymptotically stable, if there exist matrices Pi>0(i=1,4,5), Q=[Qij]2×2>0, K=[Kij]2×2>0, Zi>0(i=1,2,3), D=diag{d1,d2,…,dn}>0,  L=diag{l1,l2,…,ln}>0, positive diagonal matrices R and S, and appropriately dimensioned matrices M-k(k=1,2,…,9), such that the following LMIs hold:
(38)[Z3TTTZ3]>0,Ξ-+Π-+Π-T<0,
where(39)Θ-11=[Ψ-110Z20Z1+Q12*Ψ-22Z3-TTZ3-T0**P4-Z2-Z3-Q22T-Q12T***-K22-Z30****-Z1+Q22-Q11],Θ-12=[K12P1-F1TDT+F2TLTF4R0000F4S0000-K12T0000000],Ψ-77=τ124Z1+τ12Z2+τ122Z3,Θ-22=[K22-K11000*Ψ-77D-L0**P5-R0***-(1-τd)P5-S],Ξ-=[Θ-11Θ-12*Θ-22],Ψ-11=Q11+K11-Z1-Z2-F3R,Ψ-22=-(1-τd)P4-Z3-Z3T-F3S+T+TT,Π-=M-[-W1,0,0,0,0,0,-I,W2,W3],M-=[M1T,M2T,M3T,M4T,M5T,M6T,M7T,M8T,M9T]T.


## 4. Numerical Examples

In this section, two numerical examples are provided to show the effectiveness of the proposed results.Example 1.
Consider uncertain neural network of neutral type with leakage delay time-varying delay, and distributed delay as follows:(40)x˙(t)=-(W1+ΔW1(t))x(t-δ)+(W2+ΔW2(t))f(x(t))+(W3+ΔW3(t))f(t-τ(t))+(W4+ΔW4(t))∫t-r(t)tf(x(s))ds+(W5+ΔW5(t))x˙(t-h(t)),
where
(41)W1=diag{1.6305,1.9221,2.5973,1.3775},W2=[-2.5573-1.38131.9574-1.1398-1.0226-0.88450.5045-0.21111.03781.55320.66451.1902-0.38960.7079-0.3398-2.2543],W3=[0.2853-0.07930.46940.5354-0.59551.3352-0.90360.5529-0.1497-0.6065-0.1641-0.2037-0.4348-1.3474-0.8275-2.2543],W4=[0.02650.11570.0578-0.09300.3186-0.1363-0.08590.07420.2037-0.2049-0.01120.1475-0.3161-0.2469-0.0736-2.2543],W5=[-0.30540.36820.1761-0.0235-0.0546-0.2089-0.07540.26680.45630.00230.14400.6928-0.0115-0.23490.20040.1574],F1=diag{0,0,0,0},F2=diag{1.0275,0.9960,0.3223,0.2113},H=0.1×I,E1=0.1×I,E2=0.2×I,E3=0.3×I,E4=0.4×I,E5=0.5×I.By using Matlab LMI Toolbox, from Theorem3, it can be found that the uncertain neural network (1) under Assumption (H1) is robustly asymptotically stable for 0<δ=τd=hd<0.7. To calculate the maximum allowable τ2=h=r- for different δ=τd=hd in this paper, the numerical results in Table 1 illustrate the effectiveness of the proposed results.Table 1
Allowable upper boundτ2=h=r- for different δ=τd=hd.
Method
τ
1
δ
=
τ
d
=
h
d
=
0.1
δ
=
τ
d
=
h
d
=
0.3
δ
=
τ
d
=
h
d
=
0.5
Theorem3
τ
1
=
0.1
τ
2
=
h
=
r
-
=
0.2200
τ
2
=
h
=
r
-
=
0.1904
τ
2
=
h
=
r
-
=
0.1428
Theorem3
τ
1
=
0.2
τ
2
=
h
=
r
-
=
0.2951
τ
2
=
h
=
r
-
=
0.2687
τ
2
=
h
=
r
-
=
0.2257Example 2.
Consider the neural network with time-varying delay as follows:(42)x˙(t)=-W1x(t)+W2f(x(t))+W3f(t-τ(t)).(43)W1=[2002],W2=[11-1-1],W3=[0.88111],F2=[0.4000.8],F1=diag{0,0}.
By using Matlab LMI Toolbox, it can be found from Corollary9 that the maximum allowable τ2 can be obtained to guarantee the stability of neural network with time-varying delay in (42) for different τd. Compared with the previous results proposed in [28], it is clear that the new stability condition in Corollary 9 is less conservative than the one in [28] (see Table 2).Table 2
Allowable upper boundτ2 for different τd.
Methods
τ
1
τ
d
=
0.1
τ
d
=
0.5
τ
d
=
0.9
[28]
τ
1
=
1
τ
2
=
3.3068
τ
2
=
2.5802
τ
2
=
2.2736
Corollary9
τ
1
=
1
τ
2
=
3.6720
τ
2
=
2.6649
τ
2
=
2.3779
[28]
τ
1
=
2
τ
2
=
3.3125
τ
2
=
2.7500
τ
2
=
2.6468
Corollary9
τ
1
=
2
τ
2
=
3.7871
τ
2
=
2.9085
τ
2
=
2.8008


## 5. Conclusion

In this paper, the problem of stability analysis for neural networks of neutral type with time-varying delay in the leakage term and distributed delay has been studied. By constructing appropriate Lyapunov-Krasovskii functional and employing some advanced methods, some novel stability criteria have been proposed in terms of LMIs, which can be easily solved by standard software. Two examples have been given to illustrate the effectiveness and merit of the proposed results. It should be mentioned that the leakage delay handling method proposed in this paper can also be used to investigate the systems with the delay in the leakage term, for example, the fault-tolerant control systems [47, 48].



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*Source: 517604-2013-12-31.xml*