CHENG Peng,HU Cheng,YU Juan

(School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830017,China)

Abstract: A type of generalized Halanay inequality is established and applied to the asymptotic stability analysis of neural networks with different time-delays.Firstly,by introducing?-class functions and using the method of reduction to absurdity,a differential time-delayed inequality is established to generalize the well-known Halanay inequality,where the time-delay can be bounded or infinite,and can be discrete form or proportional form.As an application of the inequality,the asymptotic stability is investigated for neural networks with different types of delays and some stability criteria are derived.Finally,some numerical simulations are provided to verify the theoretical results.

Key words:Halanay inequality;time delay;asymptotic stability;neural network

0 Introduction

Since neural networks(NNs)can simulate the learning,storage and processing of complex information owned by human brain nervous system,they play an important role in artificial intelligence[1],image processing[2]and other multiple fields.As a theoretical basis of these applications,the stability of NNs has attracted a great deal of attention[3].

In the process of simulating implementation,due to circuit aging and sensor performance differences,transmission delays among neurons are inevitable[4].In addition,it has been shown that time delays may destroy the stability of NNs and lead to oscillation or chaos[5].In the last few decades,various NNs with time delays have been studied extensively[6].At present,time delays can be roughly divided into two categories.The first one is constant delay[7],which means that the delay time of information transmission among neurons is fixed and constant.However,the size and length of axons of neurons are different and so the time of information transmission should be changed.Thus,another type of time delay,time-varying delays have been proposed[8].In recent years,NNs with time-varying delays have attracted the attention of many fields[9].Particularly,as a special class of time-varying delays,proportional delay plays an important role in physics,biology and control theory[10]since it can control the running time of the system by adjusting the proportional delay factor.At present,the stability of NNs with proportional delays has been explored widely[11].

In the investigation of NNs with time delays,in addition to Lyapunov stability theory[12],differential inequality is also a significant tool to discuss the stability of delayed NNs.Halanay’s inequality was first proposed by Halanay[13]to investigate the exponential stability of differential systems.In the work[14],the global exponential stability of a class of high order neural networks with constant delay was discussed by using Halanay inequality.A type of Halanay inequality with unbounded delay was applied to analyse the exponential stability of Hopfield NNs in[15].Halanay inequality with proportional delay was constructed to investigate the global exponential stability of NNs with proportional delays[16].

Although many versions of Halanay inequality have been proposed so far,the forms of time delays are not unified in these related generalization forms of Halanay inequality,so it is urgent and interesting to establish a new form of Halanay inequality to uniform bounded delay,infinite delay,discrete delay and proportional delay.Inspired by the above discussion,a new generalization of Halanay inequality is developed based on?-class functions and the method of contradiction,where the considered time delay can be bounded,infinite,discrete or proportional.Besides,based on the new inequality,the stability of NNs with different forms of delays is addressed.

The rest of this article is arranged as follows.In Sect.1,a generalized Halanay’s inequality is established and its special forms are given.In Sect.2,the stability of a class of neural networks with different time delays is analyzed.In Sect.3,some examples and numerical simulations are given.Finally,the paper is summarized.

Notations In this paper,R=(?∞,+∞)and R+=[0,+∞).For a positive integern,denote→n={1,2,···,n}.D+V(t)is the upper-right Dini derivative.For a time pointt0≥0 and a nonnegative continuous functionτ(t):[t0,+∞)→R+,denotewhich can be finite and can be also infinite.

1 A Generalization of Halanay Inequality

In this section,the classical Halanay inequality will be extended by introducing a new function class.

Definition 1(?-class function)For a functionω(t):R?→R+,it is said thatω(t)belongs to?,ifω(t)=1 fort≤t0,and ω(t)>0 is differentiable and monotonically decreasing to 0 fort∈(t0,+∞).

Theorem 1For a nonnegative continuous functionV(t):[t0?p,+∞)→R+,if there exist constantsα>0,β>0,m≥1 andω(t)∈?,such that

This is the end of the proof.

If the time-varying delayτ(t)is bounded,that is,the supremumpis a finite constant,then we have the well-known Halanay inequality.

Corollary 1For a nonnegative continuous functionV(t):[t0?p,+∞)→R+,if there exist constantsα>0 andβ>0 satisfying?α+β<0,such that

According to Theorem 1,the inequality(7)is obtained.

In the following,we consider a special type of unbounded delays,namely,proportional time-varying delay,which has been widely investigated[9,12].Without loss of generality,the form of the proportional delay is assumed asτ(t)=(1?q)twith 0

wherecis a constant satisfyingc≥max{1,t0},μ?∈(0,α).The following statement is directly obtained from Theorem 1.

Corollary 2For a nonnegative continuous functionV(t):[qt0,+∞)→R+withq∈(0,1),if there existα>0 andβ>0 satisfying?α+β<0,such that

then there exists a constantμ?∈(0,α)such that

Thus the condition(2)is satisfied and the inequality(10)is obtained from Theorem 1.

Remark1 In fact,Theorem 1 generalizes the classical inequality to the more general case of time delays.Ifpis a bounded positive constant,as shown in Corollary 1,Theorem 1 is reduced to Halanay inequality with a bounded delay which has been investigated extensively[14].Ifp=∞,inequality(1)contains unbounded time delay[15].Therefore,Theorem 1 shows a more general result in our paper.

Remark 2From Corollary 1 and Corollary 2,?-class functions satisfying condition(2)in Theorem 1 can be found according to different time delays.Ifω(t)is selected as the form(6),the inequality(10)with unbounded proportional delay is derived in Corollary 1,which includes the result in[14].By Corollary 1,ifω(t)is chosen as the form(8),classical Halanay inequality[15]with bounded time delay can be directly obtained by Theorem 1.

Remark 3In previous related inequalities[13?15],the convergence rateλ?of the system is a solution of a transcendental equation,which results in difficulty to calculate it in practical problems.Different from these works,an effective interval of λ?is given in Corollary 1 and Corollary 2.Obviously our results are more realistic.

2 Application in Delayed Neural Networks

In this section,the above generalized Halanay inequalities are applied to analyze the stability of NNs with different time delays.

Firstly,consider the following delayed NN

wherexi(t)∈R is the state variable of theith neuron at timet,di∈R is the self-inhibition of theith neuron,gj(·)∈R represents the activation function,bi j∈R denotes the connection weight of thejth neuron on theith neuron at timet?τ(t).τ(t):[t0,+∞)→R represents time-varying delay,and denotep=supt∈[t0,+∞)τ(t).In addition,γi(t)∈R is the external input function ofith neuron.

Definition 2Letxi(t)and?xi(t)be two any different solutions of system(11)starting from different initial valuesxi(s)=φi(s)and(s)=(s),wheres∈[t0?p,t0].The delayed NN(11)is said to be globally asymptotically stable if

Assumption 1For anyi∈,there exists a positive constantGisuch that

In addition,the following notations are introduced for simplicity.For anyi∈,

Firstly,if the time delayτ(t)is bounded,i.e.,0≤p<+∞,the following stability criteria will be obtained.

Theorem 2Under Assumption 1,the neural model(11)is globally asymptotically stable if?α+β<0,whereα=mini∈→n{αi},β=maxi∈→n{βi}.

Therefore,the model(11)is globally asymptotically stable.

Next,we will consider the stability of system(11)with the proportional delayτ(t)=(1?q)twith 0

Theorem 3Based on Assumption 1,if?α+β<0,then system(11)with proportional delay is globally asymptotically stable.

ProofSimilar with Theorem 2,choose a Lyapunov function as follows:

The time derivative ofV(t)along(12)is given by

Therefore,system(11)with unbounded proportional delay is globally asymptotically stable.

3 Numerical Simulation

In this section,two examples are given to verify the validity of the theoretical results.

Consider the following neural network with time-varying delay

wheregj(v)=tanh(v),d1=4,d2=3,b11=?1,b12=0.2,b21=0.1,andb22=?1.3.

Firstly,consider the case thatτ(t)=2,γ1=1.By simply computing,we can obtain thatGj=1,α=1.6 andβ=1.5.Obviously,the condition?α+β<0 in Theorem 2 is satisfied.Therefore,system(15)is globally asymptotically stable,which is verified in Fig 1 with different initial values from[?1.5,1.5].

Fig 1 Dynamic evolution of system(15)withτ(t)=2 andγi(t)=1

Fig 2 Dynamic evolution of system(15)withτ(t)=qt andγi(t)=cos t

Next,let’s consider the time-varying proportional delayτ(t)=qtand periodic external input functionγi(t)=costin(15).By computing,we can obtain thatGj=1,α=1.6 andβ=1.5.Evidently,?α+β<0.According to Theorem 3,system(15)is globally asymptotically stable,which is shown in Fig 2 with different initial values in[?1.5,1.5].

4 Conclusion

In this paper,the well-known Halanay inequality is generalized to unify the discrete-time delay and unbounded proportional delay.Based on?functions,a generalization form of Halanay inequality is established.Especially,by choosing different?functions,the result can be applied to different types of time delays.Based on the results,the stability of a class of neural networks with different delays is analyzed.Future work will focus on the generalization of Halanay inequality to impulsive form.