# Global Exponential Robust Stability of Static Interval Neural Networks with Time Delay in the Leakage Term

Authors: Guiying Chen; Linshan Wang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The stability of a class of static interval neural networks with time delay in the leakage term is investigated. By using the method ofM-matrix and the technique of delay differential inequality, we obtain some sufficient conditions ensuring the global exponential robust stability of the networks. The results in this paper extend the corresponding conclusions without leakage delay. An example is given to illustrate the effectiveness of the obtained results.


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## Body

## 1. Introduction

Recently, neural networks have been widely studied because of their successful applications in different areas, such as pattern recognition, image processing, detection of moving objects, and optimization problems. The stability of the neural networks with time delay, upon which these applications largely depend, has been extensively studied (see [1–10]). However, to the best of our knowledge, there has been very little existing work on neural networks, especially, on static neural networks with time delay in the leakage term [11–18]. This is due to some theoretical and technical difficulties [13]. So, the main purpose of this paper is to study the stability of the static interval neural networks with time delay in the leakage term. By using the properties of M-matrix and delay differential inequality, we obtain some sufficient conditions ensuring the global exponential robust stability. Our results extend the corresponding conclusions without leakage delay.


## 2. Model Description and Preliminaries

In this section, we list all the notations which will be frequently used throughout the paper and give a few definitions, lemmas, and assumptions.Notations. Let R be the set of real number, and let R^n and Rm×n be the space of n-dimensional real vectors and m×n real matrices, separately. E denotes an n×n unit matrix.N
≜
{
1,2
,
…
,
n
}. For A,B∈Rm×n or A,B∈Rn, the notation A≥B (A>B) means that each pair of corresponding elements of A and B satisfies the inequality “≥(>).”  |·| denotes the Euclidean norm. For any u∈R, sgn(u) is the sign function of u.C
[
X
,
Y
] denotes the space of continuous mappings from the topological space X to the topological space Y. Particularly, let C≜C([-τ,0],Rn) denote the family of all continuous Rn-valued function ϕ defined on [-τ,0] with the norm ∥ϕ∥=sup-τ≤s≤0|ϕ(s)|.Forx∈Rn, φ∈C,  we define [x]+=(|x1|,…,|xn|)T, [φ(t)]τ=([φ1(t)]τ,…,[φn(t)]τ)T, [φi(t)]τ=sup-τ≤s≤0{φi(t+s)},  i∈N, and [φ(t)]τ+≜[[φ(t)]+]τ.  D+φ(t) denotes the upper-right-hand derivative of φ(t) at time t.Consider the following interval static neural network model with leakage delay:(1)dxi(t)dt=-ai(λ)xi(t-σ)+fi(∑j=1n∫-τ(λ)0xj(t+θ)dwij(θ,λ)+Ii),aaaaaaaaaaaaaaaaaaaaaaaaaaaaaat≥0,xi(t)=ϕi(t),-r≤t≤0,
where i,j=1,2,…,n, xi and Ii denote the state and the external inputs of the ith neuron, separately. The integer n corresponds to the number of units in a neural network, and fi(·) denotes the signal propagation function of the ith unit.  λ∈Λ⊂R is a parameter, and  ai(λ) represents the rate with which ith neuron will reset its potential to the resting state in isolation when disconnected from the network and external inputs. wij(θ,λ) is nondecreasing bounded variation functions on [-τ(λ),0], and ∫-τ(λ)0xj(t+θ)dwij(θ,λ) is a Lebesgue-Stieltjes integration. There exist positive constants τ, a_i, a-i, and wij* such that for any λ∈Λ,  0<a_i≤ai(λ)≤a-i, 0≤τ(λ)≤τ, and  |∫-τ(λ)0dwij(θ,λ)|≤wij*<∞. σ≥0 represents the leakage delay, r=max{σ,τ}. ϕ(s)=(ϕ1(s),…,ϕn(s))T, where ϕi(s) is derivative on  [-r,0], and [ϕ·i]r+≤mi, i∈N.Definition 1.
The equilibrium pointx* of system (1) is said to be globally exponentially stable if there exist a positive constant γ and a vector M>0 such that
(2)[x(t)-x*]+≤Me-γ(t-t0),t≥t0.Definition 2.
System (1) is said to be globally exponentially robustly stable if its equilibrium point x* is globally exponentially stable for any τ(λ)∈[0,τ] and  ai(λ)∈[a_i,a-i].Definition 3 (see [19]).
Let the matrixD=(dij)n×n with  dii>0 and  dij≤0, i≠j, i,j=1,2,…,n. Then each of the following conditions is equivalent to the statement “D is a nonsingular M-matrix.”(1)
All the leading principle minors ofD are positive.(2)
The diagonal elements ofD are all positive, and there exists a positive vector d such that Dd>0 or DTd>0.Lemma 4 (see [20]).
Leta<b≤+∞, and v(t)∈C[[a,b],Rn] satisfies
(3)D+v(t)≤Pv(t)+Q[v(t)]τ,t∈[a,b),v(a+s)∈PC,s∈[-τ,0],
where P=(pij)n×n,  pij≥0 for i≠j, Q=(qij)n×n≥0, and  J=(J1,…,Jn)T≥0,  i,j=1,2,…,n. Suppose that there exist a scalar λ>0 and a vector z=(z1,z2,…,zn)T>0 such that
(4)[λE+P+Qeλτ]z<0.
If the initial condition satisfies
(5)v(t)≤Mze-λ(t-a)-(P+Q)-1J,kkkkkkkkM≥0,t∈[a-τ,a],
then  v(t)≤Mze-λ(t-a)-(P+Q)-1J for  t∈[a,b).For the model (1), we introduce the following assumptions.(A1)
The signal propagation functionsfi(·) are Lipschitz continuous; that is, there are positive constants ki,  i∈N  such that for all s1,s2∈R(6)|fi(s1)-fi(s2)|≤ki|s1-s2|.(A2)
Let-(U+V) be a nonsingular M-matrix, where
(7)U=(uij)n×n,V=(vij)n×n,uij=0,i≠j,uii=-a_i,vii=σa-i2+(1+σa-i)kiwii*,vij=(1+σa-i)kiwij*,i≠j.


## 3. Main Results

Theorem 5.
Suppose that the conditions (A1) and (A2) hold, and then system (1) has at least one equilibrium point.Proof.
From (A2) and Definition 3, we know there exists a positive vector d=(d1,…,dn)T  such that (U+V)d<0. That is
(8)-a_idi+σa-i2di+(1+σa-i)ki∑j=1nwij*dj<0,kkkkkkkkkkkkkkkkkkkkkkkkki=1,…,n.
From (8) we can get
(9)ki∑j=1nwij*dj<a_i-σa-i21+σa-idi≤a_idi,kkkkkkkkkkkkkkkkki=1,…,n.
Combining with Definition3, we know A_-KW is a nonsingular M-matrix, where A_=diag(a_1,…,a_n), K=diag(k1,…,kn), and W=(wij*)n×n.
In a similar way of proof for the literature [7], by the theory of topological degree and homotopy invariance theorem, the existence of the equilibrium point of system (1) can be proved.        Suppose that x* is an equilibrium point of system (1), let yi(t)=xi(t)-xi*, φi(t)=ϕi(t)-xi*, and then system (1) becomes
(10)dyi(t)dt=-ai(λ)yi(t-σ)+fi(∑j=1n∫-τ(λ)0(yj(t+θ)+xj*)dwij(θ,λ)+Ii)-fi(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii),t≥0,yi(t)=φi(t),-r≤t≤0.
It is clear that the stability of zero solution to system (10) is equivalent to the stability of the equilibrium point  x* of system (1). So we only consider the stability of zero solution to system (10).Theorem 6.
Assume that the conditions (A1) and (A2) are satisfied, and then the zero solution to system (10) is globally exponentially robustly stable.Proof.
From the Middle Value theorem, we obtain(11)-yi(t-σ)=-yi(t)+σy˙i(t-(1-α)σ),i=1,2,…,n,
where  0<α<1. Then from (10) we get
(12)dyi(t)dt=-ai(λ)yi(t)+σai(λ)y˙i(t-(1-α)σ)+fi(∑j=1n∫-τ(λ)0(yj(t+θ)+xj*)dwij(θ,λ)+Ii)-fi(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii).
Case  1. If  0≤t<(1-α)σ, then  y˙i(t-(1-α)σ)=φi˙(t-(1-α)σ).
LetVi(t)=|yi(t)|, i∈N. Then from (A1), we have
(13)D+Vi(t)=sgn(yi(t))y˙i(t)=sgn(yi(t))((∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii)-ai(λ)yi(t)+σai(λ)y˙i(t-(1-α)σ)aaaaaaaaaaaaa+fi(∑j=1n∫-τ(λ)0(yj(t+θ)+xj*)dwij(θ,λ)aaaaaaaaaaaaaaaaaaaaa+Ii∑j=1n∫-τ(λ)0(yj(t+θ)+xj*))aaaaaaaaa-fi(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii))≤-ai(λ)|yi(t)|+σai(λ)|y˙i(t-(1-α)σ)|+ki∑j=1n∫-τ(λ)0|yj(t+θ)|dwij(θ,λ)≤-a_i|yi(t)|+σa-imi+ki∑j=1nwij*[yj(t)]τ+=∑j=1npijVj(t)+∑j=1nqij[Vj(t)]τ+li,
where  pii=-a_i, pij=0, i≠j, qij=kiwij*, and  li=σa-imi.
From (A2) and Definition 3, we know that there exists a vector d>0 such that (U+V)d<0. That is
(14)-a_idi+σa-idi+(1+σa-i)ki∑j=1nwij*dj<0.
From (14), we can get
(15)∑j=1nwij*dj≤dia_i-σa-i21+σa-i≤dia_i.
Hence, -(P+Q)  is a nonsingular M-matrix. Thus, there exists a vector z0>0 such that (P+Q)z0<0. By using the continuity, we know there exists at least one constant  λ0>0 such that
(16)(λ0E+P+Qeλ0τ)z0<0.
Sinceφ∈C[[-r,0],Rn], then, for z0>0, there exists a constant  M≥0 such that
(17)v(t)=[y(t)]+≤Mz0e-λ0t-(P+Q)-1L,kkkkkkkkkkkkkkkkkkkkkkkkt∈[-r,0],
where L=(l1,…,ln)T. Then from (13), (16), (17), and Lemma 4, we get
(18)v(t)=[y(t)]+≤Mz0e-λ0t-(P+Q)-1L,kkkkkkkkkkkkkkkkkkkt∈[0,(1-α)σ].
Case  2. If t≥(1-α)σ, then from (10) we have
(19)y˙i(t-(1-α)σ)=-ai(λ)yi(t-(2-α)σ)+fi(∑j=1n∫-τ(λ)0(+xj*yj(t-(1-α)σ+θ)aaaaaaaaaaaaaaa+xj*yj(t-(1-α)σ+θ))dwij(θ,λ)+Ii∑j=1n∫-τ(λ)0(yj(t-(1-α)σ+θ)+xj*))-fi(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii).
Substituting (19) into (12), we get
(20)dyi(t)dt=-ai(λ)yi(t)+σai(λ)×[(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii)-ai(λ)yi(t-(2-α)σ)aaaaa+fi(∑j=1n∫-τ(λ)0(yj(t-(1-α)σ+θ)aaaaaaaaaaaaaaaaaa+xj*)dwij(θ,λ)+Ii∑j=1n∫-τ(λ)0(yj(t-(1-α)σ+θ)+xj*))aaaaa-fi(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii)]+fi(∑j=1n∫-τ(λ)0(yj(t+θ)+xj*)dwij(θ,λ)+Ii)-fi(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii).
LetVi(t)=|yi(t)|,  i∈N. Then,  from (A1) and (A2), we have
(21)D+Vi(t)=sgn(yi(t))y˙i(t)=sgn(yi(t)){(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii)-ai(λ)yi(t)+σai(λ)AAAAAAAAi×[(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii)-ai(λ)yi(t-(2-α)σ)aaaaaaaaaaaaaaA+fi(∑j=1n∫-τ(λ)0(yj(t-(1-α)σ+θ)aaaaaaaaaaaaaaaaaaaaaaaa+xj*)dwij(θ,λ)+Ii∑j=1n∫-τ(λ)0(yj(t-(1-α)σ+θ)+xj*))aaaaaaaaaaaaaaa-fi(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii)]aaaaaaaaaaaaaaa+fi(∑j=1n∫-τ(λ)0(yj(t+θ)aaaaaaaaaaaaaaaaaaaa+xj*)dwij(θ,λ)+Ii∑j=1n∫-τ(λ)0(yj(t+θ)+xj*)+)aaaaaaaaaaaaaaa-fi(∑j=1n∫-τ(λ)0xj*dwij(θ,λ)+Ii)}≤-ai(λ)|yi(t)|+ai2(λ)σ[yi(t)]r+(1-α)σ++ai(λ)σki∑j=1nwij*[yj(t)]r+(1-α)σ++ki∑j=1nwij*[yj(t)]r+(1-α)σ+≤-a_i|yi(t)|+σa-i2[yi(t)]r+(1-α)σ++(σa-i+1)ki∑j=1nwij*[yj(t)]r+(1-α)σ+=∑j=1nuijVj(t)+∑j=1nvij[Vj(t)]r+(1-α)σ.
Since-(U+V) is a nonsingular M-matrix, there exists a vector z>0 such that (U+V)z<0. By using the continuity, we know there exists at least one constant γ>0 such that
(22)(γE+P+Qeγ(r+σ))z<0.
From (14) and (15), we know y(t) is bounded on [-r,(1-α)σ]. So there exists a vector η^=η(1,…,1)T such that [y(t)]+≤η^,  t∈[-r,(1-α)σ]. Then we can get
(23)vi(t)=|yi(t)|≤η^ze-γ(t-(1-α)σ),kkkkkt∈[-r,(1-α)σ],i∈N.
Then from (21), (22), (23), and Lemma 4 with J=0, we have
(24)vi(t)=|yi(t)|≤η^ze-γ(t-(1-α)σ),kkkkkt∈[(1-α)σ,∞],i∈N.
LetM=zη^eγσ, we get
(25)[x(t)-x*]+≤Me-γt,t≥0.
Thus, the equilibrium x*of system (1) is globally exponentially robustly stable.Remark 7.
Ifσ=0, the system (1) becomes the static interval neural networks without time delay in the leakage term. So, this paper includes the results of Han et al. (2011) as a special case.Remark 8.
Ifσ=0,
(26)wij(θ,λ)={wij,θ=0,0,-τ≤θ<0,τ(λ)=τ,ai(λ)=a,bij=τwij,
then system (1) becomes the following static neural network model:
(27)dxi(t)dt=-aixi(t)+fi(∑j=1nbijxj(t)+Ii),t≥0,xi(t)=ϕi(t),-r≤t≤0.
Ifσ=0 and
(28)wij(θ,λ)={∑k=0mwijk(λ),θ=τ0=0,∑k=1mwijk(λ),-τ1≤θ<0,∑k=2mwijk(λ),-τ2≤θ<-τ1⋮wijm(λ),-τm≤θ<-τm-1,0,-τ≤θ<τm,
where  -τ<-τm<⋯<-τ1<τ0=0, then system (1) becomes a class of static neural network models with discrete time delays as follows:
(29)dxi(t)dt=-ai(λ)xi(t)+fi(∑k=0m∑j=1nwijk(λ)xj(t-τk)+Ii),aaaaaaaaaaaaaaaaaaAAAAAAt≥0,xi(t)=ϕi(t),-r≤t≤0.
Ifσ=0  and  wij(θ,λ)∈C1[-τ,0], then system (1) becomes the following static model with continuous time delays:
(30)dxi(t)dt=-ai(λ)xi(t)+fi(∑j=1n∫-τ(λ)0xj(t+θ)wij′(θ,λ)dθ+Ii),aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaat≥0,xi(t)=ϕi(t),-r≤t≤0.
From (29) and (30), we can see that S-type distributed time delay contains discrete time delays and continuous time delays as two special cases, so our results generalized the results of the related literature [1, 3, 10].


## 4. Example

Consider the following system:(31)dx1(t)dt=-(2+λ)x1(t-110)+sin(∑j=12∫-τ(λ)0xj(t+θ)dwij(θ,λ)+1),dx2(t)dt=-(2+λ)x2(t-110)+cos(∑j=12∫-τ(λ)0xj(t+θ)dwij(θ,λ)+2),xi(t)=ϕi(t),-r≤t≤0,
where λ∈[0,1], |∫-τ(λ)0dw11(θ,λ)|≤1/3, |∫-τ(λ)0dw12(θ,λ)|≤1/2, |∫-τ(λ)0dw21(θ,λ)|≤1/2, and |∫-τ(λ)0dw22(θ,λ)|≤1/4,  k1=k2=1. It can be obtained that
(32)U=(-200-2),V=(43132013204940).
Thus
(33)-(U+V)=-(-2313201320-3140)
is a M-matrix. From Theorem 6, the equilibrium point of system (27) is globally exponentially robust stable.



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*Source: 972608-2014-01-12.xml*