Abstract and Applied Analysis

Volume 2013 (2013), Article ID 638496, 10 pages

http://dx.doi.org/10.1155/2013/638496

Research Article

Asymptotic Stability and Exponential Stability of Impulsive Delayed Hopfield Neural Networks

Jing Chen,¹ Xiaodi Li,2,3 and Dequan Wang⁴

¹Department of Mathematics, Shandong University, Jinan 250100, China

²School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China

³Research Center on Logistics Optimization and Prediction of Engineering Technology, Jinan, Shandong 250014, China

⁴School of Computer Science, Fudan University, Shanghai 200433, China

Received 27 June 2013; Accepted 11 August 2013

Academic Editor: Jinde Cao

Copyright © 2013 Jing Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A criterion for the uniform asymptotic stability of the equilibrium point of impulsive delayed Hopfield neural networks is presented by using Lyapunov functions and linear matrix inequality approach. The criterion is a less restrictive version of a recent result. By means of constructing the extended impulsive Halanay inequality, we also analyze the exponential stability of impulsive delayed Hopfield neural networks. Some new sufficient conditions ensuring exponential stability of the equilibrium point of impulsive delayed Hopfield neural networks are obtained. An example showing the effectiveness of the present criterion is given.

1. Introduction

In the last several years, Hopfield neural networks (HNN) have received especially considerable attention due to their extensive applications in solving optimization problem, traveling salesman problem, and many other subjects in recent years [1–9]. In hardware implementation of neural networks, time delays are inevitably present due to the finite switching speeds of the amplifiers. Hence, it is vital to investigate the stability of delayed HNN. Recently, various results for the stability of delayed HNN are obtained via different approaches. In [3], Rakkiyappan and Balasubramaniam studied the exponential stability for fuzzy impulsive neural networks by utilizing the Lyapunov-Krasovskii functional and the linear matrix inequality approach. In [8], Li studied the global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type based on the similar methods. In [9], Xia et al. derived some sufficient conditions for the synchronization problem of coupled identical Yang-Yang type fuzzy cellular neural networks with time-varying delays based on using the invariance principle of functional differential equations.

On the other hand, impulsive differential equations have attracted a great deal of attention due to its potential applications in biological systems, chemical reactions, and various results are obtained; for instance, see [10–14]. Impulses can make unstable systems stable, and stable systems can become unstable after impulse effects. Hence, the stability properties of impulsive HNN with time delays have become an important topic of theoretical studies and have been investigated by many researchers; see [5, 6, 15–22]. In [5], Zhang and Sun obtained a result for the uniform stability of the equilibrium point of the impulsive HNN systems with time delays by using Lyapunov functions and analysis technique. In [6], global exponential stability of impulsive delay HNN is investigated by applying the piecewise continuous vector Lyapunov function.

The purpose of this paper is to present some sufficient conditions for uniform asymptotic stability and global exponential stability of impulsive HNN with time delays by means of constructing the extended impulsive Halanay inequality which is different from that given in [23], Lyapunov functional methods, and linear matrix inequality approach. The results here are also discussed from the point of view of thier comparison with the earlier results. Our results improve and generalize the earlier results. At last, we discuss an example to illustrate the advantage of the results we obtained.

2. Systems Description and Preliminaries

Let denote the set of real numbers, let denote the set of nonnegative real numbers, and let denote the -dimensional real space equipped with the Euclidean norm .

Consider the following impulsive and delayed HNN model:

where . corresponds to the number of units in a neural network; the impulse times satisfy ; corresponds to the membrane potential of the unit at time ; is positive constant; , denote, respectively, the measures of response or activation to its incoming potentials of the unit at time and ; constant denotes the synaptic connection weight of the unit on the unit at time ; constant denotes the synaptic connection weight of the unit on the unit at time ; is the input of the unit is the transmission delay of the th neuron such that , and .

Assume that system (1) is supplemented with initial conditions of the form

where ,  , and , which is continuous everywhere except at finite number of points , at which and exist and . For any given , , the norm of is defined by . For any, let.

In this paper, we assume that some conditions are satisfied, so that the equilibrium point of (1) without impulse does exist denoted by ; see [2, 5]. Impulsive operator is viewed as perturbation of the equilibrium point of system (1) without impulsive effects. We assume that , , and ,  .

Since is an equilibrium point of (1), one can derive from (1) that the transformation , , transforms system (1) into the following system:

where , .

Clearly, is uniformly asymptotically stable for system (1) if and only if the trivial solution of system (3) is uniformly asymptotically stable. Hence, we only need to prove the stability of the trivial solution of system (3).

Remark 1. If , then we cannot get through the transformation . So some of the results [5] are incorrect.

The following notations will be used throughout the paper. The notation and means the transpose of and the inverse of a square matrix . Let , , ; , , , ; , , . Then system (3) with initial condition becomes

where .

We introduce some definitions as follows.

Definition 2 (see [10]). The function belongs to class if

() is continuous on each of the sets and exists;() is locally Lipschitzian in and .

Definition 3 (see [10]). Let , for any ; the upper right-hand Dini derivative of along the solution of (4) is defined by

Definition 4 (see [11]). Assume that is the solution of (4) through . Then the zero solution of (4) is said to be

(1)uniformly stable, if for any and, there exists a such that implies that ;(2)uniformly asymptotically stable, if it is uniformly stable, and there exists a such that for any, , there is a such that implies that ;(3)globally exponentially stable, if for any , there exist constants , such that

In this paper, we always assume that the following assumption holds:

there exist constants , such that

In addition, we have the following basic lemmas.

Lemma 5 (see [24]). For any vectors , , the inequality

holds, in which is any matrix with .

Lemma 6 (see [25]). Assume that there exist constants , and such that

(i)for , , are constants and satisfy ;(ii)for , ,

where .

Then for ,

where satisfies the following inequality:

3. Main Results

In this section, we will establish some theorems which provide sufficient conditions for uniformly asymptotically stable and global exponential stability of system (1).

Theorem 7. The equilibrium point the system (1) is uniformly asymptotically stable, if there exists symmetric, and positive definite matrix satisfies the following conditions:

(), where is the largest eigenvalue of ;(), where is the smallest eigenvalue of and is the largest eigenvalue of .

Proof. First, we will prove that the zero solution of system (4) is uniformly stable. For any , we may choose a such that , where is the largest eigenvalue of . For any , let be a solution of (4) through (for convenience, that we assume ); then we can prove that .

Consider the following Lyapunov function: ; then we have

By virtue of Lemma 5, we obtain for , ,

First, it is obvious that for ,

Then we can prove that for ,

Suppose that this is not true; then there exists such that.

Set

It is obvious that . Then it follows that

(); (); () for any , there exists such that .

So

In view of condition , from (13), we obtain

which is a contradiction with (). Hence, (15) holds.

Considering

we will prove that for ,

Suppose that this is false; then we can define

Similarly, we can obtain

(); (); () for any , there exists such that .

So

In fact, if , then it is obvious that inequality (22) holds. If , then . So, inequality (22) still holds.

Considering condition , from (13), we obtain

which contradicts (). Hence, (20) holds.

By induction hypothesis, we may prove, in general, that for ,

that is,

Finally, we arrive at

Therefore, we obtain . In view of the choice of , the zero solution of (4) is uniformly stable; that is, the equilibrium point of (1) is uniformly stable.

Next we show the uniformly asymptotical stability. For any given , we find a corresponding such that for any implies that , ; that is, .

For any small , we choose such that

In fact, it is feasible to choose small enough such that in (27) is large enough to satisfy (28).

Since implies that , there exists sufficient large such that

Let

Next we show that there exists such that

Or else, for all ,

Thus, we get

From (13), we have

Integrating the above inequality from to , we have

which is a contradiction. So (31) holds. We may choose .

We next claim that for all ,

Suppose that this is not true; then there exists a such that

and for ,

Suppose that , . We claim that . Otherwise, . Since (31) holds, it is clear that there exists a such that

Furthermore, we note that

From (13), we have

which implies that

This is a contradiction.

Hence, we obtain ; without loss of generality, we may suppose that . Next we first claim that there exists satisfying such that

Suppose that this is false; then for all ,

which implies that in view of (37). Consequently, we have

which implies that

Hence, we get , which contradicts (29). So (43) holds.

Therefore, there are two situations and . Next we discuss them, respectively.

() If , let

We first show that . Suppose on the contrary that ; then

From (13), we have

which implies that

which is a contradiction with the definition of . Thus we obtain that . Suppose that .

We also have two cases.

() If is not impulsive point, that is to say , then considering the definition of , we have

By the same argument as the above mentioned, we obtain that (48) still holds.

Hence, from (13), we get

which implies that

which is a contradiction with (29). So is some impulsive point.

() If is some impulsive point, that is to see , then from the definition of , it is clear that

which implies that

On the other hand, note that inequality (48) still holds; from (13) and (29), we have

which implies that

That means

which contradicts (29).

Hence, the first situation is impossible.

() If , then by the same arguments as in the proof in and (43), we have

Then let

The rest of the arguments are omitted. Finally we can find our desirable contradiction. Hence, (36) holds.

With above mentioned, the same arguments as before, if we replace with , then there exists a such that for ,

Let replace ; then there exists a such that for ,

By induction hypothesis, we may prove, in general, that there exists a such that ,

Therefore, we obtain that . In view of the choice of , , and , the zero solution of (4) is uniformly asymptotically stable; that is, the equilibrium point of (1) is uniformly asymptotically stable. The proof of Theorem 7 is therefore complete.

Let , , , , , in Theorem 7; then we can have the following result.

Corollary 8. The equilibrium point of system (1) is uniformly asymptotically stable, if there exists symmetric and positive definite matrix satisfying

() ; () , where is the smallest eigenvalue of and is the largest eigenvalue of .

Remark 9. For using the less conservative conditions in Theorem 7, our results obviously improve some results established in the earlier references. In [5], condition holds for all ; here note in our Theorem 7 that we only require that the solutions satisfy the hypothesis at impulsive points. In addition, our conditions are without requirement of the range of the largest eigenvalues of on , which are milder than the restrictions in [5].

By utilizing Lemma 6, we will give some sufficient conditions for globally exponential stability of the equilibrium point of system (1).

Theorem 10. Assume that there exists symmetric and positive definite matrix such that is the smallest eigenvalue of , is the largest eigenvalue of , is the largest eigenvalue of , and , and satisfy the following conditions:

() one has

() there exist constants , such that and the following inequality

where satisfies the following inequality:

Then the equilibrium point of the system (1) is globally exponentially stable, and the exponential convergence rate is equal to ( )/2.

Proof. Let be a solution of (4) through , (for convenience, we assume that ). We next will prove that the zero solution of (4) is globally exponentially stable. Construct the Lyapunov functional as in Theorem 7; that is, ; then we obtain the following:

(1), where is the largest eigenvalue of ;(2)for , ,

(3).

From , we have for , ,

where ,   , and .

For any , suppose that , . By Lemma 6 and condition , we obtain

where satisfies inequality (66).

Hence, we obtain for any ,

where satisfies inequality (66).

Therefore, the zero solution of system (4) is globally exponentially stable; that is, the equilibrium point of system (1) is globally exponentially stable. The proof of Theorem 10 is complete.

Remark 11. In Theorem 10, if , then we can choose in condition .

Let , in Theorem 7; then we can have the following result.

Corollary 12. Assume that there exists symmetric and positive definite matrix such that is the smallest eigenvalue of , is the largest eigenvalue of , is the largest eigenvalue of , and condition

Then the equilibrium point of the system (1) is globally exponentially stable, and the exponential convergence rate is equal to , where satisfies the following inequality:

4. Example

In this section, we present a numerical example to illustrate our results.

Example 1. We consider Example  1 in [2] as follows:

with impulses

The delayed feedback matrixes , , and are

Since the activation function in [2] is described by , , then we have . We may choose (unit matrix); then . Note that

then we get , . Matrix s characteristic equation is

By a straightforward calculation, we obtain that the largest characteristic root . By Theorem 7, the equilibrium point of system (73) is uniformly asymptotically stable with impulses (74) for any , which is shown in Figure 1(a).

(a)

(b)

Figure 1: (a) State trajectories of system (73) with impulses (74) and . (b) State trajectories of system (73) with impulses (79) and .

However, the criteria in [5] are invalid here. In fact, condition is not satisfied here. Moreover, because of the impulsive effect, the criteria in [2] are also invalid here. Therefore, our results are less conservative than those given result in [2, 5].

Furthermore, let . Note that , ; then one can choose in Theorem 10. So all conditions of Theorem 10 are satisfied. Therefore, the equilibrium point of system (73) is globally exponentially stable, and the exponential convergence rate is equal to , where satisfies

If , in above example are given as follows:

then we finally get , . So all conditions of Theorem 10 are still satisfied. Therefore, the equilibrium point of system (73) is globally exponentially stable with impulses (79), and the exponential convergence rate is equal to , where satisfies

The simulation is shown in Figure 1(b). However, it is easy to check that the impulsive delayed Halanay inequality in [23, 26] is not feasible here. Thus our results can be applied to the case not covered in [23, 26].

5. Conclusion

The uniform asymptotic stability and global exponential stability of impulsive HNN with time delays are considered in this paper. Some new stability conditions are obtained by means of constructing the extended impulsive Halanay inequality, Lyapunov functional methods, and linear matrix inequality approach. Moreover, our results can be applied to the case not covered in some other existing criteria. Hence, the results extend and improve the earlier publications. An example is given to illustrate the feasibility of the results and the effects of impulses.

Acknowledgments

This work was jointly supported by the Project of Shandong Province Higher Educational Science and Technology Program (J12LI04), Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2012DX039), and National Natural Science Foundation of China (11226136, 11301308).

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