Xiahedan Haliding,JIANG Haijun,WANG Jinling

(School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830046,China)

Abstract: In this paper,the stability of cellular neural networks(CNNs)is investigated.firstly,based on the theories of functional differential equations,stability analysis and Lyapunov functional,we study the dynamical behaviors of continuous-time cellular neural networks with in finite delays and some sufficient conditions are achieved to guarantee its equilibrium point which uniquely exist,and is globally exponentially stable.Secondly,we obtain the discrete-time analogues of the continuous-time systems via the semi-discretization technique and also investigate its dynamic behaviors.finally,the numerical simulation is used to validate the validity of proposed approaches.

Key words:continuous-time CNNs;discrete-time CNNs;in finite delays;lyapunov functional;global exponential stability;global asymptotic stability

0 Introduction

In recent years,since neural network has played a vital role in signal processing,image optimization and other practical applications,a variety of neural networks have been studied[1?7],such as CNNs,Cohen-Grossberg neural networks(CGNNs),associative memory neural networks(AMNNs).

On the one hand,in the implementation of computer simulations,the continuous-time CNNs need to be discretized in some extent.Nowadays,it is worth studying whether the discrete analogues of continuous-time cellular neural networks can maintain dynamic behaviors of the continuous-time CNNs.Mohamad and Gopalsamy have been considered dynamic behaviors of continuous-time and discrete-time cellular neural networks[8],while the delays have been assumed to be constant.

On the other hand,the time-delay phenomenon obviously exists in the process of signal transmission and processing of finite speed.That is to say,it is imperative to study the in fluence of delays on the stability of neural networks.Based on Lyapunov functionals and Halanay-type inequalities,sufficient conditions for the continuous-time and discrete-time cellular neural networks with delays have been obtained to ensure the global exponential stability of the system[9].Li and Liu have been studied stability of discrete-time neural networks with finite delays[10?11].Yao has been investigated a new exponentially stable sliding mode control approach for uncertain discrete systems with time delay[12].However,the results on the stability of continuous-time and discrete-time cellular neural networks with in finite delays are very few.For the rest of the above reasons,we will study the dynamic behaviors of continuous-time and discrete-time CNNs with in finite delays in this paper.

In this paper,by using Lyapunov functionals,we obtain the existence,uniqueness and global exponential stability of the equilibrium of continuous-time cellular neural networks with in finite delays.In the second part,we obtain the discrete-time analogues of the continuous-time systems via the semi-discretization techniques,and ensure the existence,uniqueness and global asymptotical stability of the equilibrium of discrete-time CNNs with in finite delays.finally,we use the example to verify that the conclusion of this paper is valid.

1 Continuous-time Cellular Neural Networks

In this part,we consider a continuous-time CNN with in finite delays

fori∈ ? ={1,2,...,m},xi(t)denotes the potential of the celliat timet;fi(·)denotes a non-linear output function;bij,cij,eijdenote the strengths of connectivity between the cellsjandiat timetandt?τij,respectively;τijcorresponds to the time delay required in processing and transmitting a signal from thejth cell to theith cell;Iidenotes theith component of an external input source introduced from outside the network to the celli;aidenotes the rate with which the celliresets its potential to the resting state when isolated from other cells and inputs;kij(s)denotes delay kernel function andkij(s):[0,∞)→ [0,∞)is continuous.

We denote a vector solution of(1)asx(t)=(x1(t),x2(t),...,xm(t))T,and initial conditions holdxi(s)= ψi(s),s∈(?∞,0].

In this section,the following conditions of system(1)are assumed

(H1)ai>0,bij,cij,eij∈ R,τij≥ 0,fori,j∈ ?,where R=(?∞,∞),the connection weightsbij,cij,eijdo not have to be symmetric;

(H2)Non-linear activation functionfi(·)is bounded and globally Lipschitzian,that is,there exists constantsMi>0,Li>0,such that

whereLidenotes the Lipschitz constant;

(H3)

fori,j∈?,α is constant.

Definition 1The solutionx(t)is said to be globally exponentially stable.That is,there exist constants γ>0 andM≥1,such that

Lemma 1[13](Brouwer’s fixed point theorem)Let B be closed unit sphere in Rn.LetT:B7→ B be a continuous map,thenThas a fixed pointx∈B.

We denote the equilibrium of(1)by a vectorwhere

Theorem 1Let(H1),(H2)and(H3)hold.Suppose that

then there exist constants α >0 and β ≥ 1 such that

Prooffirstly,we establish the existence and uniqueness of the equilibriumx?of(1).Denote a mapping

where

fori∈?.From the assumptions(H1)and(H2),we get that

where It follows that

Based on Brouwer’s fixed point theorem,there exists at least one fixed point.Next,we show the uniqueness of the fixed point.Suppose that there exists another fixed point denoted byy?,then

From(H1)and(H2),we have

In conclusion,it’s easy to check that

From(3)and(5),we acquirefori∈ ?.Then,the equilibriumx?of(1)exists and is unique.

finally,we proceed to establish the global exponential stability of the equilibriumx?of(1).By using(H1)and(H2),we get that

fori∈?,t>0,wheredenotes the upper right derivative.

From the condition(3),note that

for alli∈ ?,where η =>0.

We consider functionsFi(·)de fined by

for αi∈[0,∞),i∈?.

Combining(7)with(8),it means thatFi(0)≥ η≥0,fori∈ ?.According to the continuity ofFi(·)on[0,∞),there exists a constant α>0 such that

Now,we consider the following function

De fine a Lyapunov functionalV(t)as follows

Calculating the rate of change ofV(t)along(11),we obtain

fort>0.Based on(9)and(12),we have˙V(t)≤0 fort>0 implying thatV(t)≤V(0),fort>0.From(11),we derive that

and

where α>0 and

Hence,the proof is complete.

Remark 1Thestabilityofcontinuous-timecellularneuralnetworkswithconstantdelayshasbeeninvestigated[9].However,in the Theorem 1,the stability of continuous-time cellular neural networks with in finite delays is investigated.That is to say,we promoted their conclusions and fill this gap of CNNs with in finite delays.

2 Discrete-time Cellular Neural Networks

In this part,we acquire discrete-time analogue of the continuous-time network(1)via semi-discretization technique.Let Z denote the set of all integers,Z={...,?2,?1,0,1,2,...},Z+={1,2,3,...},={0,1,2,...}.Fora,b∈Z anda≤b,we denote the discrete interval[a,b]Z={a,a+1,...,b?1,b}.Ifb=∞,then[a,∞)Z={a,a+1,a+2,...}.

Based on the semi-discretization technique,we reformulate system(1)with the following approximation

fori∈?,t∈[nh,(n+1)h),wherehis a fixed positive real number denoting a uniform discretization step-size and[r]denotes the integer part of the real numberr.Clearly,fort∈[nh,(n+1)h),we have=ν,and τij≥ 0,we have ‘ij∈For convenience in the following,we use the notationxi(nh)=xi(n).With these preparations,we rewrite(14)as

fori∈?,t∈[nh,(n+1)h),n∈We can integrate(15)over[nh,t),wheret<(n+1)hand by allowingt→(n+1)h(due to continuity of solutions of(1))in the above,we gain that

fori∈ ? ,

It can be veri fied that φi(h)>0,ifai>0,h>0 and φi(h)≈h+o(h2)forh>0.One can show that the discrete-time analogue(16)converges towards the continuous-time network(1)whenh→0+.

In this section,the following conditions of system(16)are assumed:

(H4)ai>0,bij,cij,eij,Ii∈R,‘ij∈Z+0,fori,j∈?;

(H5)Eachfi(·)satis fies(H2);

(H6)and there exists a constant λ such that

A solution of(16)is denoted by a vectorx(n)=(x1(n),x2(n),...,xm(n))Twherexi(n)=xi(n,ψi),forn∈ Z+.Letdenote an equilibrium of(16).From(16),We have to

In the following we establish the global asymptotic stability ofx?of(16).

Theorem 2Leth>0 andlet(H4),(H5)and(H6)hold.Suppose the condition(3)holds,then there existconstants λ>1 and γ≥1 such that

ProofThe existence and uniqueness of the equilibriumx?of(16)can be obtained immediately from Theorem 1.Letx(n)denote an arbitrary solution of(16)forn∈Z+.From(H4),(H5)and(16),we have

fori∈ ?,We consider functionsGi(·)given by

Based on the continuity ofGi(λi)on[1,∞),it follows that there exists a real number λ >1 such that

Now,consider a Lyapunov functionalV(·)as

Calculating the difference 4V(n)=V(n+1)?V(n),we acquire that

From(20)and(22),we assert that 4V(n)≤0 forwhich impliesV(n)≤V(0)forand it follows that

where λ>1 and

We conclude that the equilibriumx?is globally asymptotically stable and hence the proof is complete.

Remark 2The stability of discrete-time CNNs with delays has been investigated[8?9].However,such delays are assumed to be constant or finite.Therefore,the stability of discrete-time systems with in finite delays are acquired in the Theorem 2,and our conclusions are more universal than before.

3 Numerical simulations

In this section,we use the following example to verify that the conclusion of this paper is valid.

ExampleConsider following discrete-time CNNs with in finite delays

where

Further,we verify that the network(24)satis fies the conditions of the Theorem 2.Therefore,the equilibrium of network(24)is globally asymptotically stable(fig 1).

fig 1 The response curves of the system with in finite delays

4 Conclusion

In this paper,the Brower’s fixed point theorem is used to assert the existence and uniqueness of the equilibrium of continuous-time and discrete-time CNNs with in finite delays.Some sufficient conditions are also obtained to guarantee the exponential stability and asymptotic stability of continuous-time and discrete-time CNNs via constructing appropriate Lyapunov functionals.