# Robust Exponential Stability of Impulsive Stochastic Neural Networks with Leakage Time-Varying Delay

Authors: Chunge Lu; Linshan Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper investigates mean-square robust exponential stability of the equilibrium point of stochastic neural networks with leakage time-varying delays and impulsive perturbations. By using Lyapunov functions and Razumikhin techniques, some easy-to-test criteria of the stability are derived. Two examples are provided to illustrate the efficiency of the results.


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## Body

## 1. Introduction

In recent years, stability for neural networks with time delay has been extensively studied due to their great applications in some practical engineering problems such as signal processing, associate memory, and combinatorial optimization (see [1–3]). In particular, the leakage delay, which exists in the negative feedback terms (known as forgetting or leakage terms) of the system, has great impact on the dynamical behavior of neural networks (see [4–10]). Gopalsamy [4] initially discussed the problem of bidirectional associative memory neural networks with constant delays in the leakage term by using model transformation technique. Then, many results of stability of neural networks with delays in the leakage terms are obtained (see, [5–12]).However, besides time delay, neural networks are often subject to impulsive perturbation—the abrupt changes at certain instants, which may be caused by switching phenomenon, frequency change, or other sudden noise (see [13, 14]). The impulsive effect can affect the dynamical behaviors of the system. Now there are many results on stability of the neural networks with time delays in the leakage term and impulsive perturbations under the corresponding delayed neural networks without impulses must be stable themselves (see [5–8]). To best of the authors’ knowledge, this is the first attempt to investigate the stability of the systems under the corresponding delayed neural networks without impulses which are unstable themselves.On the other hand, in real nervous systems, the synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes [15]. It is well known that a neural network could be stabilized or destabilized by certain stochastic inputs. Therefore, noise disturbances have an important effect on the stability of neural networks. Recently, many interesting results on stochastic effect to the stability of neural networks with delays have been reported (see [11, 16, 17]). Moreover, uncertainties are unavoidable in practical implementation of neural networks due to modeling errors and parameter fluctuation, which also cause instability and poor performance [12]. Hence, we can obtain a more perfect model of this situation if we include parameter uncertainties and stochastic effects in neural networks.Motivated by the above, it is of practical and theoretical importance to study the stability problem of impulsive neural networks with time-varying delays in the leakage term. In this paper, we will investigate stability for a class of stochastic neural networks with time-varying delay in the leakage term and impulses. By using Razumikhin techniques [18–21], some new robust mean-square exponential stability criteria will be given under the corresponding delayed neural networks without impulses which are stable and unstable, respectively.


## 2. Problem Formulation

Consider the following uncertain neural networks model with impulses and leakage time-varying delay(1)dx(t)=[-C(t)x(t-σ(t))+A(t)f(x(t))+B(t)f(x(t-τ(t)))]dt+h(x(t),x(t-τ(t)),x(t-σ(t)))dω(t),mmmmmmmmmmmmmmmlt≥t0,t≠tk;Δx(t)=Ik(x(t-),x(t--τ(t)),x(t--σ(t))),mmmmmmmmmmmmmt=tk,k∈N,x(t0+s)=φ(s),s∈[-ρ,0],
where x(t)=(x1(t),x2(t),…,xn(t))T∈Rn is the neuron state vector of the neural networks, C(t)=diag⁡(ci(t))n×n>0, f(x)=(f1(x1),f2(x2),…,fn(xn))T represents the neuron activation function, and A(t),B(t)∈Rn×n are the connection weight matrix. τ(t) is the time-varying delay and σ(t) is the leakage time-varying delay satisfying 0≤τ(t)≤τ, 0≤σ(t)≤σ,  ρ=max⁡(τ,σ). ω(t)=(ω1(t),ω2(t),…,ωn(t))T is a n-dimensional Brownian motion defined on complete probability space (Ω,F,P).    tk∈J={{tk}:0=t0<t1<⋯<tk<⋯,limk→∞tk→∞} is impulsive time. Δx(t)=x(tk)-x(tk-) represents the jump in the state x at tk with Ik determining the size of the jump and x(tk+)=x(tk). The initial conditions φ(t)∈PCF0b([-ρ,0];Rn), where PCF0b([-ρ,0];Rn) denotes the family of all bounded F0 measurable and PC([-ρ,0],Rn) valued random variable φ, satisfying E∥φ∥2=sup-ρ≤s≤0E|φ(s)|2<+∞; E denotes the mathematical expectation. PC(J,Rn)={φ:J→Rnispiecewisecontinuous}.Throughout this paper, symmetric matrixM≥0 (resp., M>0) means that the matrix M is positive semidefinite (resp., positive definite). I denotes an identity matrix. The notation MT represents the transpose of the matrix M. The symmetric terms in asymmetric matrix are denoted by *. λmax⁡(A) and λmin⁡(A) mean the largest and the smallest eigenvalue of A, respectively.In this paper, we assume thatx*=(x1*,x2*,…,xn*)T is the equilibrium point of the system (1). And we have the following assumptions.(A1)
The neuron activation functionfj(x) is continuous on R and satisfies
(2)mmmmmmmmlj≤fj(u)-fj(v)u-v≤Lj,j=1,2,…,n,foranyu,v∈R,u≠v,
where lj,Lj are some real constants and they may be positive, zero, or negative.(A2)
Consider(3)Ik(x(t-),x(t--τ(t)),x(t--σ(t)))=(G1k(t)-I)(x(t-)-x*)+G2k(t)(x(t--τ(t))-x*)+G3k(t)(x(t--σ(t))-x*),
where Gik(t)∈Rn×n, i=1,2,3, and k∈N.(A3)
Considerh(x(t),x(t-τ(t)),x(t-σ(t)))=H1(t)(x(t)-x*)+H2(t)(x(t-τ(t))-x*)+H3(t)(x(t − σ(t))-x*).Lety(t)=x(t)-x*, and system (1) becomes
(4)dy(t)=[+B(t)g(y(t-τ(t)))-C(t)y(t-σ(t))+A(t)g(y(t))+B(t)g(y(t-τ(t)))]dt+[H1(t)y(t)+H2(t)y(t-τ(t))+H3(t)y(t-σ(t))]dω(t),t≥t0,t≠tk;Δy(t)=(G1k(t)-I)y(t-)+G2k(t)y((t-τ(t))-)+G3k(t)y((t-σ(t))-),t=tk,k∈N,y(t0+s)=ϕ(s),s∈[-ρ,0],
where g(y(t))=f(x(t))-f(x*)=(g1(y1(t)),g2(y2(t)),…,gn(yn(t)))T, ϕ(s)=φ(s)-x*.(A4)
We consider the parameter uncertainties expressed as(5)A(t)=A+ΔA(t),B(t)=B+ΔB(t),C(t)=C+ΔC(t),Hi(t)=Hi+ΔHi(t),Gik(t)=Gik+ΔGik(t),mmmmmmmmmmmmmmmmi=1,2,3,k∈N,
where A, B, C, Hi, and Gik are known real constant matrices; ΔA(t), ΔB(t), ΔC(t), ΔHi(t), and ΔGik(t) are unknown matrices representing the parameter uncertainties, which are assumed to be the following form:
(6)[ΔC(t)ΔA(t)ΔB(t)]=E1F1(t)[M1M2M3],[ΔH1(t)ΔH2(t)ΔH3(t)]=E2F2(t)[N1N2N3],[ΔG1k(t)ΔG2k(t)ΔG3k(t)]=E3F2(t)[U1U2U3],
where Ui, Ei, Mi, Ni (i=1,2,3) are known real constant matrices and Fi(t) are unknown real time-varying matrix functions satisfying FiT(t)Fi(t)≤I, i=1,2,3.Remark 1.
Assumptions (A1)–(A4) imply that system (4) satisfies the local Lipschitz condition and linear growth condition. Thus there exists a unique solution of system (4).Definition 2.
The equilibrium pointx* of system (1) is said to be robustly exponentially stable in the mean square, if there exists scalars γ>0 and δ>0 such that for any ε>0 and initial condition φ satisfying E∥φ∥≤δ which implies E|x(t;t0,φ)-x*|<εe-γ(t-t0),   t≥t0.Lemma 3 (see [22]).
Given matricesD, E, and F with FTF≤I and a scalar ε>0, then
(7)DFE+ETFTDT≤εDDT+ε-1ETE.


## 3. Main Results

Theorem 4.
Suppose that assumptions (A1)–(A4) hold and for prescribed scalar Δmax⁡>0, choose positive scalars α,β, and μ1>0, μ2>0, μ3>0, such that μ1+μ2+μ3<1. Then the equilibrium point of system (1) is robustly exponentially stable in the mean square over any impulse time sequences satisfying sup⁡{tk-tk-1∣k=1,2,…}≤Δmax⁡, if there exist positive definite matrix P>0 and definite diagonal matrices W1, W2, and positive constants ε1, ε2, ε3 such that the following LMI holds:(8)[-μ1P+ε1U1TU1ε1U1TU2ε1U1TU3G1kTP0*-μ2P+ε1U2TU2ε1U2TU3G2kTP0**-μ3P+ε1U3TU3G3kTP0***-PPE3****-ε1I]<0,(9)[Ω11ε3N1TN2-PC+ε3N1TN3PA+W1K1PBH1TPPE10*Ω22ε3N2TN30W2K1H2TP00**Ω33-ε2M1TM2-ε2M1TM3H3TP00***-W1+ε2M2TM2ε2M2TM3000****-W2+ε2M3TM3000*****-P0PE2******-ε2I0*******-ε3I]<0,
whereΩ11=[(α+β)/(μ1+μ2+μ3)+ln⁡(μ1+μ2+μ3)/Δmax⁡]P-W1K2+ε3N1TN1, Ω22=-αP-W2K2 + ε3N2TN2, Ω33=-βP+ε2M1TM1+ε3N3TN3, K1=diag⁡(((l1+L1)/2),((l2+L2)/2),…,((ln+Ln)/2)), K2=diag⁡(l1L1,l2L2,…,lnLn).Proof.
Since the matrix inequality (9) holds, we can choose small enough scalars η>0, h>0 satisfying μ1+μ2e2ητ+μ3e2ησ<1 and h<1-μ1-μ2e2ητ-μ3e2ησ, such that(10)[Ω-11ε3N1TN2-PC+ε3N1TN3PA+W1K1PBH1TPPE10*Ω-22ε3N2TN30W2K1H2TP00**Ω-33-ε2M1TM2-ε2M1TM3H3TP00***-W1+ε2M2TM2ε2M2TM3000****-W2+ε2M3TM3000*****-P0PE2******-ε2I0*******-ε3I]<0,
whereΩ-11=[2η+(α+β)/(μ1+μ2e2ητ+μ3e2ησ)+ln⁡(μ1+μ2e2ητ+μ3e2ησ+h)/Δmax⁡]P-W1K2+ε3N1TN1, Ω-22=-αe-2ητP-W2K2+ε3N2TN2, Ω-33=-βe-2ησP+ε2M1TM1+ε3N3TN3.
Letq=1/(μ1+μ2e2ητ+μ3e2ησ), λ=-ln⁡(μ1+μ2e2ητ+μ3e2ησ+h)/Δmax⁡; then λ>0, q>1, eλΔmax⁡<q.
Define Lyapunov function(11)V(t,y(t))=e2ηtyT(t)Py(t).
From the condition (8), applying Schur complement [23] and Lemma 3, we have
(12)[-μ1P000-μ2P000-μ3P]+[G1kT(t)G2kT(t)G3kT(t)]×P[G1k(t)G2k(t)G3k(t)]<0.
Therefore, when t=tk,
(13)V(tk)=e2ηtyT(tk)Py(tk)=e2ηt[y(tk-)y(tk--τ(tk))y(tk--σ(tk))]T[G1kT(t)G2kT(t)G3kT(t)]×P[G1k(t)G2k(t)G3k(t)][y(tk-)y(tk--τ(tk))y(tk--σ(tk))]≤μ1V(tk-)+μ2e2ητV(tk--τ(tk))+μ3e2ησV(tk--σ(tk)).
When t≠tk, applying the Itô formula, we have
(14)LV(t,y(t))=2ηe2ηtyT(t)Py(t)+2e2ηtyT(t)×P[+B(t)g(y(t-τ(t)))-C(t)y(t-σ(t))+A(t)g(y(t))+B(t)g(y(t-τ(t)))]+e2ηt[H1(t)y(t)+H2(t)y(t-τ(t))+H3(t)y(t-σ(t))]T×P[H1(t)y(t)+H2(t)y(t-τ(t))+H3(t)y(t-σ(t))].
Let
(15)W(t)=LV(t)+α(qV(t)-V(t-τ(t)))+β(qV(t)-V(t-σ(t)))-λV(t).
From assumption (A1), the following inequalities hold for any diagonal matrices W1>0, W2>0,
(16)e2ηt[g(y(t))W1g(y(t))-2yT(t)W1K1g(y(t))+yT(t)W1K2y(t)]≤0,e2ηt[g(y(t-τ(t)))W2g(y(t-τ(t)))-2yT(t-τ(t))×W2K1g(y(t-τ(t)))+yT(t-τ(t))×W2K2y(t-τ(t))g(y(t-τ(t)))W2g(y(t-τ(t)))-2yT(t-τ(t))]≤0.
Set
(17)ξT(t)=(gT(y(t-τ(t)))yT(t),yT(t-τ(t)),yT(t-σ(t)),gT(y(t)),gT(y(t-τ(t)))).
Combining (15)–(16) together, we have
(18)W(t)≤e2ηtξT(t)Ψξ(t),
where
(19)Ψ=[Γ110-PC(t)PA(t)+W1K1PB(t)*Γ2200W2K1**-βP00***-W10****-W2]+[H1T(t)PH2T(t)PH3T(t)P00]P-1[H1T(t)PH2T(t)PH3T(t)P00]T,Γ11=2ηP+αqP+βqP-W1K2-λP,Γ22=-αP-W2K2.
Using the similar method for uncertain parameters as above and from (10), we can get Ψ<0. Then we have EW(t)<0; that is,
(20)ELV(t)+α(qEV(t)-EV(t-τ(t)))+β(qEV(t)-EV(t-σ(t)))<λEV(t),mmmmmmmmmmmmmmmmmmmmlt≠tk.
Let λ1=λmax⁡(P), λ0=λmin⁡(P). For any ε>0, there exists a δ>0, such that qλ1δ2<λ0ε2.
In the following, we will prove that when the initial functionϕ satisfies E∥ϕ∥<δ, we have
(21)EV(t)<λ0ε2,t≥t0-ρ.
First, for t∈[t0-ρ,t0],
(22)EV(t)=E(e2ηtyT(t)Py(t))≤λ1E∥ϕ∥2≤λ1δ2<1qλ0ε2<λ0ε2.
Then we will prove that
(23)EV(t)<λ0ε2,t∈[t0,t1).
If the above inequality does not hold, then there exist t*=inf⁡{t∈(t0,t1):EV(t)≥λ0ε2}, such that EV(t*)=λ0ε2. Set t-=sup⁡{t∈[t0,t*):EV(t)≤(1/q)λ0ε2}; then t-∈(t0,t*) and EV(t-)=(1/q)λ0ε2. Hence for all t∈(t-,t*), qEV(t)≥λ0ε2>EV(t+θ),∀θ∈[-ρ,0], which implies that
(24)qEV(t)>EV(t-τ(t)),qEV(t)>EV(t-σ(t))mmmmmmmmmmmmmmmmmmmmfort∈(t-,t*).
It follows from (20) that for any α>0, β>0 and t∈(t-,t*), we have
(25)D+EV(t)=ELV(t)<ELV(t)+α[qEV(t)-EV(t-τ(t))]+β[qEV(t)-EV(t-σ(t))]<λEV(t),
which leads to EV(t*)≤EV(t-)eλ(t*-t-)≤EV(t-)eλΔmax⁡=(1/q)eλΔmax⁡λ0ε2<λ0ε2. This is a contradiction.
Thus (23) holds.
Now we assume that for somem∈N, EV(t)<λ0ε2, t∈[t0-ρ,tm); we will prove that
(26)EV(t)<λ0ε2,t∈[tm,tm+1).
From (13), we have
(27)EV(tm)≤μ1EV(tk-)+μ2e2ητEV(tk--τ(tk))+μ3e2ησEV(tk--σ(tk))<(μ1+μ2e2ητ+μ3e2ησ)λ0ε2=1qλ0ε2<λ0ε2.
Suppose (26) does not hold; then there exists t*=inf⁡{t∈(tm,tm+1):EV(t)≥λ0ε2} and EV(t*)=λ0ε2. Set t-=sup⁡{t∈[tm,t*):EV(t)≤(1/q)λ0ε2}, and then from (27), t-∈(tm,t*) and EV(t-)=(1/q)λ0ε2. In the sequel, the proof is very similar with the proof of (23). Therefore (26) holds. By mathematical induction, inequality (21) holds. This together with EV(t)≥λ0e2ηtE|y(t)|2, we have
(28)E|y(t)|<εe-ηt.
This completes the proof of Theorem 4.Remark 5.
Theorem4 shows that robustly exponential stability of system (1) can be achieved by adjusting suitable impulsive control and appropriate impulsive intervals even if the given networks without impulses may be unstable or chaotic themselves.Remark 6.
In [5], the authors investigated the stability of neural networks with  delayed  leakage term and impulsive perturbations. However, the neural networks without impulses must be stable. Moreover, parameter uncertainties and stochastic effects were not taken into account in the models. Hence, the results in this paper have wider adaptive range.Theorem 7.
Suppose that assumptions (A1)–(A4) hold and for prescribed scalars Δmin⁡>0, choose positive scalars α, β, μ1, μ2, μ3, satisfying μ1+μ2+μ3≥1 and μ1+μ2ecτ+μ3ecσ<ecΔmin⁡. Then the equilibrium point of system (1) is robustly exponentially stable in the mean square over any impulse time sequences satisfying inf⁡{tk-tk-1∣k=1,2,…}≥Δmin⁡, if there exist positive definite matrix P and positive definite diagonal matrices W1, W2 and positive constants ε1, ε2, ε3 such that (8) and the following LMI hold:(29)[Ω11ε3N1TN2-PC+ε3N1TN3PA+W1K1PBH1TPPE10*Ω22ε3N2TN30W2K1H2TP00**Ω33-ε2M1TM2-ε2M1TM3H3TP00***-W1+ε2M2TM2ε2M2TM3000****-W2+ε2M3TM3000*****-P0PE2******-ε2I0*******-ε3I]<0,
whereΩ11=[(α+β)(μ1+μ2ecτ+μ3ecσ)+ln⁡(μ1+μ2ecτ+μ3ecσ)/Δmin⁡]P-W1K2+ε3N1TN1 and Ω22=-αP-W2K2+ε3N2TN2, Ω33=-βP+ε2M1TM1+ε3N3TN3.Proof.
Since the matrix inequality (29) holds, we can choose small enough scalars η>0, h>0 satisfying ln⁡(μ1+μ2ecτ+μ3ecσ+2h)/Δmin⁡+2η≤c, such that the following matrix inequality holds:(30)[Ω-11ε3N1TN2-PC+ε3N1TN3PA+W1K1PBH1TPPE10*Ω-22ε3N2TN30W2K1H2TP00**Ω-33-ε2M1TM2-ε2M1TM3H3TP00***-W1+ε2M2TM2ε2M2TM3000****-W2+ε2M3TM3000*****-P0PE2******-ε2I0*******-ε3I]<0,
whereΩ-11=[2η+(α+β)(μ1+μ2ecτ+μ3ecσ+h)+ln⁡(μ1+μ2ecτ+μ3ecσ+2h)/Δmin⁡]P-W1K2+ε3N1TN1, Ω-22=-αe-2ητP-W2K2+ε3N2TN2, and Ω-33=-βe-2ησP+ε2M1TM1+ε3N3TN3.
Setq=μ1+μ2ecτ+μ3ecσ+h, λ=ln⁡(μ1+μ2ecτ+μ3ecσ+2h)/Δmin⁡; then λ>0, q>1, eλΔmin⁡>q.
Define Lyapunov function(31)V(t,y(t))=e2ηtyT(t)Py(t)
and function
(32)W(t)=LV(t)+α(qV(t)-V(t-τ(t)))+β(qV(t)-V(t-σ(t)))+λV(t).
Similar to the proof Theorem 4, if (8) and (29) hold, we have
(33)V(tk)=e2ηtyT(tk)Py(tk)≤μ1V(tk-)+μ2e2ητV(tk--τ(tk))+μ3e2ησV(tk--σ(tk)),t=tk,
and EW(t)<0 for t≠tk, which implies that
(34)ELV(t)+α(qEV(t)-EV(t-τ(t)))+β(qEV(t)-EV(t-σ(t)))<-λEV(t).
For anyε>0, there exists δ>0, such that qλ1δ2<λ0ε2. In the following, we will prove that when the initial function ϕ∈PCF0b([-ρ,0];Rn) satisfies E∥ϕ∥<δ, we have
(35)EV(t)<λ0ε2,t≥t0-ρ.
First, for t∈[t0-ρ,t0],
(36)EV(t)=Ee2ηtyT(t)Py(t)≤λ1E∥ϕ∥2≤λ1δ2<1qλ0ε2<λ0ε2.
Then we will prove that
(37)EV(t)<λ0ε2,t∈[t0,t1).
If (37) does not hold, there exists t*=inf⁡{t∈[t0,t1):EV(t)≥λ0ε2}. From (36), t*∈(t0,t1) and EV(t*)=λ0ε2. Set t-=sup⁡{t∈[t0,t*):EV(t)≤(1/q)λ0ε2}; then t-∈(t0,t*) and EV(t-)=(1/q)λ0ε2.
So fort∈(t-,t*) and any θ∈[-ρ,0], we have qEV(t)≥λ0ε2>EV(t+θ). It follows from (34) that for any α>0, β>0 and t∈(t-,t*),
(38)D+EV(t)=ELV(t)≤ELV(t)+α[qEV(t)-EV(t-τ(t))]+β[qEV(t)-EV(t-σ(t))]<-λEV(t).
Then we have λ0ε2=EV(t*)<EV(t-)e-λ(t*-t-)≤EV(t-)=(1/q)λ0ε2<λ0ε2. This is a contradiction. Thus (37) holds.
Suppose that for somem∈N, EV(t)<λ0ε2, t∈[t0-ρ,tm). We will prove that
(39)EV(t)<λ0ε2,t∈[tm,tm+1).
We first claim that EV(tm-)≤(1/q)λ0ε2. The proof is very similar to [18, 19], so we omit it here. Next, we show that
(40)EV(tm--σ(tm))≤1qeλσλ0ε2.
Suppose not; then we have EV(tm--σ(tm))>(1/q)eλσλ0ε2. Without loss of generality, we assume tm-σ(tm)∈(tl-1,tl],l∈N,l≤m. There are two cases to be considered.
Case
1. EV(t)>(1/q)eλσλ0ε2 for any t∈[tl-1,tm-σ(tm)). Then we have
(41)qEV(t)>eλσλ0ε2≥λ0ε2>EV(t+θ),m∀θ∈[-ρ,0],t∈[tl-1,tm-σ(tm)).
It follows from (34) that for any α>0, β>0 and t∈(tl-1,tm-σ(tm)) (38) holds, which leads to
(42)EV(tm--σ(tm))<EV(tl-1)e-λ(tm-σ(tm)-tl-1)<e-λΔmin⁡eλσλ0ε2<1qeλσλ0ε2.
This is a contradiction.
Case
2. There exist some t∈[tl-1,tm-σ(tm)), such that EV(t)≤(1/q)eλσλ0ε2.
Sett-=sup⁡{t∈[tl-1,tm-σ(tm)):EV(t)≤(1/q)eλσλ0ε2}. Then we have t-∈[tl-1,tm--σ(tm)) and EV(t-)=(1/q)eλσλ0ε2. Hence for t∈(t-,tm-σ(tm)), we have
(43)qEV(t)≥eλσλ0ε2≥λ0ε2>EV(t+θ),∀θ∈[-ρ,0].
It follows from (34) that for any α>0, β>0 and t∈(t-,tm-σ(tm)) (38) holds, which leads to
(44)EV(tm--σ(tm))<EV(t-)e-λ(tm-σ(tm)-t-)≤EV(t-)=1qeλσλ0ε2.
This is a contradiction.
Therefore (40) holds. By the same methods, we can prove EV(tm--τ(tm))≤(1/q)eλτλ0ε2. Then from (33), we get
(45)EV(tm)≤μ1EV(tm-)+μ2e2ητEV(tm--τ(tm))+μ3e2ησEV(tm--σ(tm))≤(μ1+μ2e(2η+λ)τ+μ3e(2η+λ)σ)1qλ0ε2≤(μ1+μ2ecτ+μ3ecσ)1qλ0ε2<λ0ε2.
Suppose (39) does not hold; then there exist t*=inf⁡{t∈(tm,tm+1):EV(t)≥λ0ε2} and EV(t*)=λ0ε2. If EV(t)>(1/q)λ0ε2 for all t∈[tm,t*), set t-=tm. Otherwise let t-=sup⁡{t∈[tm,t*):EV(t)≤(1/q)λ0ε2}. Then for t∈(t-,t*) and any θ∈[-ρ,0], we have qEV(t)≥λ0ε2>EV(t+θ). It follows from (34) that for any α>0, β>0, and t∈(t-,t*), (38) holds. Then, λ0ε2=EV(t*)<EV(t-)e-λ(t*-t-)≤EV(t-)=(1/q)λ0ε2<λ0ε2.
This is a contradiction. Thus (39) holds.
The next proof is very similar to Theorem4. This completes the proof.Remark 8.
Theorem7 shows that the system will remain exponentially stable on the condition that the impulses, which may destabilize the system, do not occur too frequently.Remark 9.
If there is no leakage delay, that is,σ(t)=0, H3(t)=0, G3k(t)=0, k∈N, t≥0, then the system (4) is the one investigated in [19]. If there is no stochastic perturbation either, then system (4) is the one investigated in [18, 20].


## 4. Examples

In this section, we present some examples to verify the effectiveness of the theoretical results.Example 1.
Consider (4) with two neurons. The uncertain parameters satisfy assumption (A4), where C=[7006], A=[0.6-0.20.70.5], B=[0.5-0.1-1.2-0.9], H1=H2=H3=0.03I, M1=M2=M3=0.03I, G1k=G2k=G3k=0.1I,k∈N, N1=N2=N3=0.3I, U1=U2=U3=0.1I, E1=E2=E3=0.03I. τ<+∞,σ<+∞. g1(y)=g2(y)=|y+1|-|y-1|, and it is obvious that K1=I, K2=0. Choose μ1=0.05, μ2=0.1, μ3=0.1, α=β=0.5, for tk-tk-1≤Δmax⁡=0.01; then the LMIs in Theorem 4 have the following feasible solution via MATLAB LMI toolbox: ε1=1.2054, ε2=11.5508, ε3=0.7167, P=[1.25160.04920.04921.1285], W1=[9.71010010.6592], W2=[0.3534000.2846]. Thus from Theorem 4, the equilibrium (0,0)Tof system (4) is robustly exponentially stable in the mean square.Example 2.
We consider neural network shown in [5] as follows:
(46)x˙(t)=[-900-9]x(t-σ)+[11-11]f(x(t)),t>0,
where f1(s)=f2(s)=tanh(s). As shown in [5], the system (46) is stable when σ=0, and it becomes unstable when σ=0.2. We consider that the system has the following impulsive perturbation at times tk:
(47)Δx(t)=(G1k(t)-I)x(t-),t=tk,k∈N,
where G1k=0.3I, k∈N. It is obvious that K1=0.5I, K2=0. For tk-tk-1≤Δmax⁡=0.01, choosing μ1=0.1, μ2=0.01, μ3=0.05, α=0.5, β=0.5, by using the LMI toolbox in MATLAB, a feasible solution of Theorem 4 is
(48)P=[0.0210000.0210],W1=[0.6321000.6321],W2=[0.0289000.0289],ε1=ε2=ε3=0.6748.
Thus from Theorem 4, the equilibrium (0,0)T of system (46) is robustly exponentially stable in the mean square. It can be seen that the impulses play an important role in inducing the stability of neural network in the leakage time delay.


## 5. Conclusion

Robust exponential stability of stochastic neural networks with time-varying delay in the leakage term under impulsive perturbations is investigated. The leakage delay is time varying and the impulsive perturbations depend not only on the current state of neurons at impulse times but also on the state of neurons in its recent history. Based on Lyapunov functions and Razumikhin techniques, some new criteria are derived. Some examples have been given to demonstrate that, even though the corresponding delayed neural networks without impulses are unstable, impulses may compensate the deviating trend.



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*Source: 831027-2014-07-24.xml*