# Global Exponential Stability of Discrete-Time Neural Networks with Time-Varying Delays

Authors: S. Udpin; P. Niamsup
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper presents some global stability criteria of discrete-time neural networks with time-varying delays. Based on a discrete-type inequality, a new global stability condition for nonlinear difference equation is derived. We consider nonlinear discrete systems with time-varying delays and independence of delay time. Numerical examples are given to illustrate the effectiveness of our theoretical results.


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## Body

## 1. Introduction

In recent years, neural networks (NNs) have been investigated extensively due to their broad applications in information processing problems, associative memory, parallel computation, pattern recognition, signal processing, and optimization problems. It is well known that delays are often the sources of instability and oscillation in system. In practical studies, discrete-time systems have been used for a variety of phenomena in electrical networks, genetics, ecological systems, and so forth. Therefore, the stability analysis of discrete-time neural networks (DNNs) with delays has become an important topic of theoretical studies in neural networks; for example, asymptotic stability and exponential stability of neural networks have been studied by many researchers. In [1], the authors have studied robust stability of discrete-time linear-parameter-dependent (LPD) neural networks with time-varying delay. In order to derive stability criteria of discrete-time, one common approach is the use of appropriate inequalities for difference equations. Another approach is the use of Lyapunov stability theory. In [2], the authors have studied global exponential stability of impulsive discrete-time neural networks with time-varying delays, based on some inequality analysis techniques. In [3], the authors have studied new discrete-type inequalities and global stability of nonlinear difference equation. In [4], the authors have studied global exponential stability of discrete-time Hopfield neural networks with variable by using the difference inequality. In [5], the authors have considered the problem of robust stability analysis of generalized neural networks with multiple discrete delays and multiple distributed delays by using the Lyapunov-Krasovskii functional method and the linear matrix inequality technique. In [6], the authors have studied delay-dependent exponential stability criteria for discrete-time nonlinear system with multiple time-varying delays. In this paper, we propose to study global exponential stability of discrete-time neural networks with time-varying delays. In Section 2, we have introduced discrete-time neural networks with time-varying delays and presented some preliminaries. In Section 3, we have derived new discrete-type inequalities; global exponential stability criteria are derived by using new discrete-type inequalities. Finally, numerical examples are given to illustrate the effectiveness of our theoretical results.


## 2. Notations, Definitions, and Preliminaries Results

In this section, we give some notations definitions and preliminaries results which will be used throughout this paper.Letℝ denote the set of all real numbers;ℝ+the set of positive real numbers;ℝ0+the set of nonnegative real numbers; ℤ the set of integers; andℤ+the set of positive integers;ℤ-r={z∈ℤ:z≥-r}. We consider the following discrete-time neural networks with time-varying delays:
(1)u(n+1)=Cu(n)+Af(u(n))+∑i=0r‍Bif(u(n-τi(n)))+b,
whereu(n)=[u1(n),…,up(n)]T∈ℝpis the neuron state vector,b=[b1,…,bp]Tis constant input vector,τi(n), i=0,2,…,r are positive integers denoting the time-varying delays satisfying
(2)0≤τi(n)≤τi,i=0,1,2,…,r,
whereτi≥0are known positive integers,C=diag(cm),(cm∈(0,1)),AandBiare the interconnection matrices, andf(x(·))=[f1(x1(·)),f2(x2(·)),…,fn(xn(·))]Tis the activation function wherefm(·),m=1,…,nsatisfy
(3)|fm(x)-fm(y)|≤Lm|x-y|,∀x,y∈ℝ.
The initial conditions of system (1) are of the form
(4)um(s)=ϕm(s),m=1,2,…,n,
whereϕ(s)=[ϕ1(s),ϕ2(s),…,ϕn(s)]Tis an initial functions,s∈{-τr,-τr+1,…,0}, andτris the maximum value ofτi. Letu*=[u1*,u2*,…,un*]Tbe an equilibrium point of system (1). We shift the equilibrium pointu*to the origin by using the transformationx(·)=u(·)-u*. Then, we obtain the new system
(5)x(n+1)=Cx(n)+Ag(x(n))+∑i=0r‍Big(x(n-τi(n))),
wherex(n)={x1(n),x2(n),…,xp(n)}is the state vector of the transformed system,g(x(·))=[g1(x1(·)),…,gn(xn(·))]T, wheregm(xm(·))=fm(xm(·)+um*)-fm(um*),m=1,2,…,nand the transformed activation functions satisfy the condition
(6)|gm(x)|≤Lm|x|.Definition 1 (see [4]).
The equilibrium point at the origin of system (5) is globally exponentially stable if for any solutionx(n,ϕ)with the initial conditionϕ, there exist constantϵ∈(0,1)andD>0such that
(7)∥x(n)∥≤D∥ϕ∥ϵn∀n≤0,
where∥ϕ∥=maxs∈[-τr,0]{∥ϕ(s)∥}.


## 3. Main Result

Throughout this section, we denotex(n)byxn.In this section, we provide global exponential stability criteria for system (5). First, we introduce new discrete-type inequalities which will be used to derive global exponential stability condition.Lemma 2.
Letqi∈ℝ0+, ki∈ℤ+, i=1,…r; p,qr∈ℝ+, where 0≤k0<k1<⋯<kr and ∑i=0r‍qi(ki+1)<p≤1, and let {xj}j∈ℤ-kr be a sequence of real numbers satisfying the inequality
(8)Δxn≤-pxn+∑i=0r‍qi(ki+1)∑j=0ki‍xn-j,n∈ℤ0,
where Δxn=x(n+1)-x(n). Then, there existsλ0∈(0,1)such that
(9)xn≤max{0,x0,x-1,…,x-kr}λ0n,n∈ℤ0.
Moreover,λ0might be chosen as the smallest positive root of the polynomial
(10)P(λ)=λkr+1-(1-p)λkr-∑i=0r‍qi∑j=0ki‍λkr-j
which lies in the interval(0,1).Proof.
Let{yn}be a solution of the difference equation
(11)Δyn=-pyn+∑i=0r‍qi(ki+1)∑j=0ki‍yn-j,n∈ℤ0.
Fromqi∈ℝ0+and0<p<1, one may follow the proof of Theorem2.1in [3] to show that ifxnsatisfies (8) andxn≤ynforn=-hr,…,0, thenxn≤ynfor alln∈ℤ0. For givenK>0andλ∈(0,1), the sequence{yn}defined byyn=Kλnis a solution of (11) if and only ifλis a root of the polynomial (10). Sincelimλ→0+P(λ)=-qr<0andP(1)=p-∑i=0r‍qi(ki+1)>0, it follows from continuity ofPthat there exists a smallest real numberλ0∈(0,1)such thatP(λ0)=0. Thus, for anyK∈𝕂0+, the sequence {Kλ0n} is a solution of (11). LetK0=max{0,xn,x-1,…,x-kr}. Therefore,yn={K0λ0n}is a solution of (11) and obviously, we havexn≤yn, forn=-kr,…,0. Hence, by using the first part of the proof, we conclude thatxn≤yn=K0λ0n,n∈ℤ0.Now, we will provide a global exponential stability condition for system (5).Theorem 3.
The equilibrium point at the origin of system (5) is globally exponentially stable if
(12)cmax+l∥A∥+l∑i=0r‍(τi+1)∥Bi∥<1,
wherel=maxm(Lm),cmax=maxm(cm), andmaxm(Lm)is maximum of vector.Proof.
Letzn=∥xn∥.Then, from (5), the difference of system is given by
(13)Δz(n)=∥xn+1∥-∥xn∥=∥Cxn+Ag(xn)+∑i=0r‍Big(xn-τi(n))∥-∥xn∥≤∥Cxn∥+∥Ag(xn)∥+∑i=0r‍∥Big(xn-τi(n))∥-∥xn∥≤-(1-cmax-l∥A∥)∥xn∥+l∑i=0r‍∥Bi∥∥xn-τi(n)∥≤-(1-cmax-l∥A∥)∥xn∥+l∑i=0r‍∑j=0τi‍∥Bi∥∥xn-j∥≤-(1-cmax-l∥A∥)∥xn∥+l∑i=0r‍(τi+1)∥Bi∥∑j=0τi‍∥xn-j∥=-pzn+∑i=0r‍qi(ki+1)∑j=0τi‍zn-j,
where
(14)p=1-cmax-l∥A∥,qi=∥Bi∥,ki=τi.
From Lemma 2, it follows assumptions of theorem that there existλ0∈(0,1)such that
(15)zn≤max{0,z0,z-1,…,z-τr}λ0n,n∈ℤ0.
Thus, we obtain
(16)zn=∥xn∥≤max{0,z0,z-1,…,z-τr}λ0n=max{0,∥x0∥,∥x-1∥,…,∥x-τr∥}λ0n≤∥ϕ∥λ0n.
By Definition 1, we conclude that (5) is globally exponentially stable. The proof is complete.Remark 4.
In our main result, we derived global exponential stability criteria for discrete-time neural networks with multiple time-varying delays by using discrete-type inequality. In [4], the global exponential stability criteria of discrete-time Hopfield neural networks are given. Nevertheless, the stability criteria in [4] cannot be applied to discrete-time system with multiple time-varying delays.


## 4. Numerical Example

Example 1.
Consider the NNs (1) withr=2where
(17)C=[0.100-0.15],A=[0.1-0.15-0.20.1],B0=[0.2-0.1-0.10.15],B1=[-0.1-0.150.1-0.1]b=[-3,-1],τ0(n)=1+sin(nπ2),τ1(n)=2+sin(nπ2),f1(s)=tanh(-0.4s)+0.2sins,f2(s)=tanh(0.2s).
Then, ∥A∥1=0.3,∥B0∥1=0.3,∥B1∥1=0.25,τ0=1,τ1=2,cmax=0.15,L1=0.5,L2=0.2, and l=0.5.
The equilibrium point of system in Example1 is
(18)u*=[u1*,u2*]T=[-3.3507,-0.7254]T.
From which it follows that
(19)cmax+0.5∥A∥1+0.5(τ0+1)∥B0∥1+0.5(τ1+1)∥B1∥1=0.975<1.
Therefore, from Theorem 3, it follows that the equilibrium point at the origin of system (1) is globally exponentially stable.
The trajectory of solutions of Example1 with the initial state [ϕ1(s),ϕ2(s)]T=[-3,1]Tfors∈[-1,0] is shown in Figure 1.Figure 1
The trajectory of solutions of system (1) in Example 1.Example 2.
Consider the NNs (1) withr=2where
(20)C=[0.2000.1],A=[-0.10.150.1-0.15],B0=[0.15-0.1-0.20.05],B1=[-0.150.10.1-0.15]b=[2,3],τ0(n)=1+sin(nπ2),τ1(n)=2+sin(nπ2),f1(s)=tanh(-0.4s)+0.2sin(s),f2(s)=tanh(0.2s).
Then∥A∥∞=0.3,∥B0∥∞=0.25,∥B1∥∞=0.25,τ0=1,τ1=2,cmax=0.2, andl=0.5. The equilibrium point of system in Example 2 is
(21)u*=[u1*,u2*]T=[2.2435,3.5554]T.
From which it follows that
(22)cmax+0.5∥A∥∞+0.5(τ0+1)∥B0∥∞+0.5(τ1+1)∥B1∥∞=0.975<1.
Therefore, from Theorem 3, it follows that the equilibrium point at the origin of system (1) is globally exponentially stable.
The trajectory of solutions of Example2 with the initial state[ϕ1(s),ϕ2(s)]T=[1,2]Tfors∈[-1,0] is shown in Figure 2.Figure 2
The trajectory of solutions of system (1) in Example 2.


## 5. Conclusion

In this paper, we obtained sufficient condition for global exponential stability of discrete-time neural networks with multiple time-varying delays. The stability criteria are derived by using a discrete-type inequality. Numerical examples are given to show the effectiveness of our theoretical results.



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