# Exponential Convergence for Cellular Neural Networks with Time-Varying Delays in the Leakage Terms

Authors: Zhibin Chen; Junxia Meng
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

We consider a class of cellular neural networks with time-varying delays in the leakage terms. By applying Lyapunov functional method and differential
inequality techniques, we establish new results to ensure that all solutions of the networks
converge exponentially to zero point.


---

## Body

## 1. Introduction

It is well known that the delayed cellular neural networks (CNNs) have been successfully applied to signal and image processing, pattern recognition, and optimization (see [1]). Hence, they have been the object of intensive analysis by numerous authors in the past decades. In particular, extensive results on the problem of the existence and stability of the equilibrium point for CNNs are given out in many works in the literature. We refer the reader to [2–6] and the references cited therein. Recently, to consider CNNs with the incorporation of time delays in the leakage terms, Gopalsamy [7] and Wang et al. [8] investigated a class of CNNS described by(1.1)xi′(t)=-ci(t)xi(t-ηi(t))+∑j=1n‍aij(t)fj(xj(t-τij(t)))+∑j=1n‍bij(t)∫0∞‍Kij(u)gj(xj(t-u))du+Ii(t),i=1,2,…,n,
in which n corresponds to the number of units in a neural network, xi(t) corresponds to the state vector of the ith unit at the time t, and ci(t) represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at the time t. aij(t) and bij(t) are the connection weights at the time t, ηi(t)≥0 and τij(t)≥0 denote the leakage delay and transmission delay, respectively, Ii(t) denotes the external bias on the ith unit at the time t, fj and gj are activation functions of signal transmission, and i,j=1,2,…,n.Suppose that the following conditions(H0)ci and ηi are constants, where i=1,2,…,n,(H0*) for each j∈{1,2,…,n}, there exists a nonnegative constant L~j such that
(1.2)|fj(u)-fj(v)|≤L~j|u-v|,∀u,v∈R,are satisfied. Avoiding the continuously distributed delay terms, the authors of [7, 8] obtained that all solutions of system (1.1) converge to the equilibrium point or the periodic solution. However, to the best of our knowledge, few authors have considered the convergence behavior for all solutions of system (1.1) without the assumptions (H0) and (H0*). Thus, it is worthwhile to continue to investigate the convergence behavior of system (1.1) in this case.The main purpose of this paper is to give the new criteria for the convergence behavior for all solutions of system (1.1). By applying Lyapunov functional method and differential inequality techniques, without assuming (H0) and (H0*), we derive some new sufficient conditions ensuring that all solutions of system (1.1) converge exponentially to zero point. Moreover, an example is also provided to illustrate the effectiveness of our results.Throughout this paper, fori,j=1,2,…,n, it will be assumed that ci,Ii,ηi,aij,bij,τij:R→R and Kij:[0,+∞)→R are continuous functions, and there exist constants ci+,ηi+,aij+,bij+ and τij+ such that
(1.3)ci+=supt∈Rci(t),ηi+=supt∈Rηi(t),aij+=supt∈R|aij(t)|,bij+=supt∈R|bij(t)|,τij+=supt∈Rτij(t).
We also assume that the following conditions (H1), (H2), and (H3) hold: (H1)  for each i,j∈{1,2,…,n}, there exist nonnegative constants L~j and Lj such that
(1.4)|fj(u)|≤L~j|u|,|gj(u)|≤Lj|u|,∀u∈R,(H2)  for all t>0 and i,j∈{1,2,…,n}, there exist constants η>0,λ>0 and ξi>0 such that
(1.5)∫0∞‍|Kij(u)|eλudu<+∞,-η>-[ci(t)-λe-ληi(t)-ηi(t)ci(t)(λ+ci+eληi+)]eληi(t)ξi+∑j=1n‍L~j(|aij(t)|eλτij(t)+aij+ηi(t)ci(t)eληi(t)eλτij+)ξj+∑j=1n‍Lj∫0∞‍|Kij(u)|eλudu(|bij(t)|+bij+ηi(t)ci(t)eληi(t))ξj;(H3)Ii(t)=O(e-λt)(t→±∞),i=1,2,…,n.The initial conditions associated with system (1.1) are of the form
(1.6)xi(s)=φi(s),s∈(-∞,0],i=1,2,…,n,
where φi(·) denotes real-valued-bounded continuous function defined on (-∞,0].


## 2. Main Results

Theorem 2.1.
Let(H1), (H2), and (H3) hold. Then, for every solution Z(t)=(x1(t),x2(t),…,xn(t))T of system (1.1) with any initial value φ=(φ1(t),φ2(t),…,φn(t))T, there exists a positive constant K such that
(2.1)|xi(t)|≤Kξie-λt∀t>0,i=1,2,…,n.Proof.
LetZ(t)=(x1(t),x2(t),…,xn(t))T be a solution of system (1.1) with any initial value φ=(φ1(t),φ2(t),…,φn(t))T,and let
(2.2)Xi(t)=eλtxi(t),i=1,2,…,n.
In view of (1.1), we have
(2.3)Xi′(t)=λXi(t)+eλt[-ci(t)xi(t-ηi(t))+∑j=1n‍aij(t)fj(xj(t-τij(t)))asdffghhhhhh+∑j=1n‍bij(t)∫0∞‍Kij(u)gj(xj(t-u))du+Ii(t)]=λXi(t)-ci(t)eληi(t)Xi(t-ηi(t))+eλt[∑j=1n‍aij(t)fj(e-λ(t-τij(t))Xj(t-τij(t)))aaaaa+∑j=1n‍bij(t)∫0∞‍Kij(u)gj(e-λ(t-u)Xj(t-u))du+Ii(t)],i=1,2,…,n.
Let(2.4)M=maxi=1,2,…,nsups≤0{eλs|φi(s)|}.
From (1.3), (H2), and (H3), we can choose a positive constant K such that(2.5)Kξi>M,η>[ηi(t)ci(t)eληi(t)+1]supt∈R|Ii(t)eλt|K,∀t>0,i=1,2,…,n.
Then, it is easy to see that(2.6)|Xi(t)|≤M<Kξi∀t≤0,i=1,2,…,n.
We now claim that(2.7)|Xi(t)|<Kξi∀t>0,i=1,2,…,n.If this is not valid, then, one of the following two cases must occur:(1)
there existi∈{1,2,…,n} and t*>0 such that
(2.8)Xi(t*)=Kξi,|Xj(t)|<Kξj∀t<t*,j=1,2,…,n,(2)
there existi∈{1,2,…,n} and t**>0 such that
(2.9)Xi(t**)=-Kξi,|Xj(t)|<Kξj∀t<t**,j=1,2,…,n.Now, we consider two cases.
Case i. If (2.8) holds. Then, from (2.3), (2.5), and (H1)−(H3), we have
(2.10)0≤Xi′(t*)=λXi(t*)-ci(t*)eληi(t*)Xi(t*-ηi(t*))+eλt*[∑j=1n‍aij(t*)fj(e-λ(t*-τij(t*))Xj(t*-τij(t*)))ii+eλt*+∑j=1n‍bij(t*)∫0∞‍Kij(u)gj(e-λ(t*-u)Xj(t*-u))du+Ii(t*)]=λXi(t*)-ci(t*)eληi(t*)Xi(t*)+ci(t*)eληi(t*)[Xi(t*)-Xi(t*-ηi(t*))]+eλt*[∑j=1n‍aij(t*)fj(e-λ(t*-τij(t*))Xj(t*-τij(t*)))ii+eλt*+∑j=1n‍bij(t*)∫0∞‍Kij(u)gj(e-λ(t*-u)Xj(t*-u))du+Ii(t*)]=-[ci(t*)eληi(t*)-λ]Xi(t*)+ci(t*)eληi(t*)∫t*-ηi(t*)t*‍Xi′(s)ds+eλt*[∑j=1n‍aij(t*)fj(e-λ(t*-τij(t*))Xj(t*-τij(t*)))asdasdi+∑j=1n‍bij(t*)∫0∞‍Kij(u)gj(e-λ(t*-u)Xj(t*-u))du+Ii(t*)]=-[ci(t*)eληi(t*)-λ]Xi(t*)+ci(t*)eληi(t*)∫t*-ηi(t*)t*‍[∑j=1n‍λXi(s)-ci(s)eληi(s)Xi(s-ηi(s))+ci(t*)eληi(t*)∫t*-ηi(t*)t*‍ii+eλs(∑j=1n‍aij(s)fj(e-λ(s-τij(s))Xj(s-τij(s)))+ci(t*)ia+ci(t*)eληi(t*)∫t*-ηi(t*)t*‍+∑j=1n‍bij(s)∫0∞‍Kij(u)gj(e-λ(s-u)Xj(s-u))du+Ii(s))]ds+eλt*[∑j=1n‍aij(t*)fj(e-λ(t*-τij(t*))Xj(t*-τij(t*)))ii+eλt*+∑j=1n‍bij(t*)∫0∞‍Kij(u)gj(e-λ(t*-u)Xj(t*-u))du+Ii(t*)]≤-[ci(t*)eληi(t*)-λ-ηi(t*)ci(t*)eληi(t*)(λ+ci+eληi+)]ξiK+∑j=1n‍L~j(|aij(t*)|eλτij(t*)+aij+ηi(t*)ci(t*)eληi(t*)eλτij+)ξjK+∑j=1n‍Lj∫0∞‍|Kij(u)|eλudu(|bij(t*)|+bij+ηi(t*)ci(t*)eληi(t*))ξjK+[ηi(t*)ci(t*)eληi(t*)+1]supt∈R|Ii(t)eλt|={[ηi(t*)ci(t*)eληi(t*)+1]supt∈R|Ii(t)eλt|K-[ci(t*)-λe-ληi(t*)-ηi(t*)ci(t*)(λ+ci+eληi+)]eληi(t*)ξiii+∑j=1n‍L~j(|aij(t*)|eλτij(t*)+aij+ηi(t*)ci(t*)eληi(t*)eλτij+)ξjii+∑j=1n‍Lj∫0∞‍|Kij(u)|eλudu(|bij(t*)|+bij+ηi(t*)ci(t*)eληi(t*))ξja+[ηi(t*)ci(t*)eληi(t*)+1]supt∈R|Ii(t)eλt|K}K<{-η+[ηi(t*)ci(t*)eληi(t*)+1]supt∈R|Ii(t)eλt|K}K<0.
This contradiction implies that (2.8) does not hold.
Case ii. If (2.9) holds. Then, from (2.3), (2.5), and (H1)−(H3), we get
(2.11)0≥Xi′(t**)=-[ci(t**)eληi(t**)-λ]Xi(t**)+ci(t**)eληi(t**)∫t**-ηi(t**)t**‍[∑j=1n‍λXi(s)-ci(s)eληi(s)Xi(s-ηi(s))+ci(t**)eληi(t**)+ci(t**)ais+eλs(∑j=1n‍aij(s)fj(e-λ(s-τij(s))Xj(s-τij(s)))+ci(t**)eληi(t**)∫t**-ηi(t**)t**‍adddddia+∑j=1n‍bij(s)∫0∞‍Kij(u)gj(e-λ(s-u)Xj(s-u))du+Ii(s))]ds+eλt**[∑j=1n‍aij(t**)fj(e-λ(t**-τij(t**))Xj(t**-τij(t**)))+eλt**ii+∑j=1n‍bij(t**)∫0∞‍Kij(u)gj(e-λ(t**-u)Xj(t**-u))du+Ii(t**)]≥-[ci(t**)eληi(t**)-λ-ηi(t**)ci(t**)eληi(t**)(λ+ci+eληi+)](-ξiK)+∑j=1n‍L~j(|aij(t**)|eλτij(t**)+aij+ηi(t**)ci(t**)eληi(t**)eλτij+)(-ξjK)+∑j=1n‍Lj∫0∞‍|Kij(u)|eλudu(|bij(t**)|+bij+ηi(t**)ci(t**)eληi(t**))(-ξjK)-[ηi(t**)ci(t**)eληi(t**)+|Ii(t**)|]eλt**={[ηi(t**)ci(t**)eληi(t**)+1]supt∈R|Ii(t)eλt|K-[ci(t**)-λe-ληi(t**)-ηi(t**)ci(t**)(λ+ci+eληi+)]eληi(t**)ξi'a+∑j=1n‍L~j(|aij(t**)|eλτij(t**)+aij+ηi(t**)ci(t**)eληi(t**)eλτij+)ξj′a+∑j=1n‍Lj∫0∞‍|Kij(u)|eλudu(|bij(t**)|+bij+ηi(t**)ci(t**)eληi(t**))ξjia+[ηi(t**)ci(t**)eληi(t**)+1]supt∈R|Ii(t)eλt|K}(-K)>{-η+[ηi(t**)ci(t**)eληi(t**)+1]supt∈R|Ii(t)eλt|K}(-K)>0,
which is a contradiction and yields that (2.9) does not hold.
Consequently, we can obtain that (2.7) is true. Thus,
(2.12)|xi(t)|≤Kξie-λt∀t>0,i=1,2,…,n.
This implies that the proof of Theorem 2.1 is now completed.


## 3. An Example

Example 3.1.
Consider the following CNNs with time-varying delays in the leakage terms:(3.1)x1′(t)=-(20-(1+|t|)sin2t1+2|t|)x1(t-1+|sint|2000)+|t|3sint1+4|t|3f1(x1(t-2sin2t))+|t|5sint1+36|t|5f2(x2(t-3sin2t))+|t|7sint1+4|t|7∫0∞‍e-ug1(x1(t-u))du+t2sint1+36t2∫0∞‍e-ug2(x2(t-u))du+e-3tsint,x2′(t)=-(40-(1+|t|7)cos2t1+2|t|7)x2(t-1+|cost|2000)+t5cost1+2|t|5f1(x1(t-2sin2t))+tcost1+5|t|f2(x2(t-5sin2t))+|t|3cost1+6|t|3∫0∞‍e-ug1(x1(t-u))du+(1+|t|)cost7+7|t|∫0∞‍e-ug2(x2(t-u))du+e-tsint,
where f1(x)=f2(x)=xcos(x3),g1(x)=g2(x)=xsin(x2).
Noting that(3.2)18≤c1(t)=20-(1+|t|)sin2t1+2|t|≤20,38≤c2(t)=40-(1+|t|7)cos2t1+2|t|7≤40,η1(t)=1+|sint|2000≤11000,η2(t)=1+|cost|2000≤11000,a11(t)=|t|3sint1+4|t|3,b11(t)=|t|7sint1+4|t|7,a12(t)=|t|5sint1+36|t|5,b12(t)=t2sint1+36t2,a21(t)=t5cost1+2|t|5,b21(t)=|t|3cost1+6|t|3,a22(t)=tcost1+5|t|,b22(t)=(1+|t|)cost7+7|t|,τ11(t)=τ21(t)=2sin2t,τ12(t)=3sin2t,τ22(t)=5sin2t,Li=L~i=1,Kij(u)=e-u,i,j=1,2.
Define a continuous function Γi(ω) by setting
(3.3)Γi(ω)=-[ci(t)-ωe-ωηi(t)-ηi(t)ci(t)(ω+ci+eωηi+)]eωηi(t)+∑j=12‍L~j(|aij(t)|eωτij(t)+aij+ηi(t)ci(t)eωηi(t)eωτij+)+∑j=12‍Lj∫0∞‍|Kij(u)|eωudu(|bij(t)|+bij+ηi(t)ci(t)eωηi(t)),wheret>0,i=1,2.
Then, we obtain
(3.4)Γi(0)=-[ci(t)-ηi(t)ci(t)ci+]+∑j=12‍L~j(|aij(t)|+aij+ηi(t)ci(t))+∑j=12‍Lj∫0∞‍|Kij(u)|du(|bij(t)|+bij+ηi(t)ci(t)),i=1,2.
Therefore,
(3.5)Γ1(0)≤-(18-11000×20×20)+2[(14+14×11000×20)+(136+136×11000×20)],<-10,Γ2(0)≤-(38-11000×40×40)+[(12+12×11000×40)+(15+15×11000×40)]+[(16+16×11000×40)+(17+17×11000×40)]<-20,
which, together with the continuity of Γi(ω), implies that we can choose positive constants λ>0 and η>0 such that for all t>0, there holds
(3.6)-η>Γi(λ)=-[ci(t)-λ-ηi(t)ci(t)(λ+ci+eληi+)]eληi(t)ξi+∑j=12‍L~j(|aij(t)|eλτij(t)+aij+ηi(t)ci(t)eληi(t)eλτij+)ξj+∑j=12‍Lj∫0∞‍|Kij(u)|eλudu(|bij(t)|+bij+ηi(t)ci(t)eληi(t))ξj,whereξi=1,i=1,2.
This yields that system (3.1) satisfied (H1), (H2), and (H3). Hence, from Theorem 2.1, all solutions of system (3.1) converge exponentially to the zero point (0,0,…,0)T.Remark 3.2.
Sincef1(x)=f2(x)=xcos(x3),g1(x)=g2(x)=xsin(x2), and CNNs (3.1) are a very simple form of CNNs with time-varying delays in the leakage terms, it is clear that the conditions (H0) and (H0*) are not satisfied. Therefore, all the results in [7–9] and the references therein cannot be applicable to prove that all solutions of system (3.1) converge exponentially to the zero point.



---

*Source: 941063-2012-11-01.xml*