Abstract and Applied Analysis

Volume 2012, Article ID 825643, 17 pages

http://dx.doi.org/10.1155/2012/825643

## Dynamical Analysis for High-Order Delayed Hopfield Neural Networks with Impulses

College of Physics and Electronics, Shandong Normal University, Jinan 250014, China

Received 25 June 2012; Accepted 3 September 2012

Academic Editor: José J. Oliveira

Copyright © 2012 Dengwang Li. 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

The global exponential stability and uniform stability of the equilibrium point for high-order delayed Hopfield neural networks with impulses are studied. By utilizing Lyapunov functional method, the quality of negative definite matrix, and the linear matrix inequality approach, some new stability criteria for such system are derived. The results are related to the size of delays and impulses. Two examples are also given to illustrate the effectiveness of our results.

#### 1. Introduction

In the last several years, Hopfield neural networks (HNNs) have received especially considerable attention due to their extensive applications in solving optimization problem, traveling salesman problem, and many other subjects, see [1–17]. However such neural networks are shown to have limitations such as limited capacity when used in pattern recognition problems, see [2, 3]. This led many researchers to use neural networks with high order connections. The high-order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks. Recently, various results on stability of high-order delayed HNN are obtained, see [11–15]. For example, Lou and Cui [13] studied the global asymptotic stability of high-order HNN with time-varying delays by using Lyapunov method, linear matrix inequality (LMI), and analytic technique as follows: But the authors only obtained some global asymptotic stability criteria for the above high-order HNN. Those results cannot ensure the global exponential stability of the equilibrium point. It is well known that global exponential stability plays an important role in many areas such as designs and applications of neural networks and synchronization in secure communication [5, 17–23]. One purpose of this paper is to improve the results in [13]. We obtain several new criteria on global exponential stability and uniform stability for the above high-order HNN.

On the other hand, it is well known that the artificial electronic networks are subject to instantaneous perturbations and experience change of the state abruptly, that is, do exhibit impulsive effects. Such systems are described by impulsive differential systems which have been used successfully in modeling many practical problems arisen in the fields of natural sciences and technology, see [12, 24–30]. Hence, it is very important and, in fact, necessary to investigate the issue of the stability of high-order delayed HNN with impulses. However, to the best of the authors’ knowledge, there are few results on the stability of high-order delayed HNN with impulses. In [12], Liu et al. obtained some sufficient conditions for ensuring global exponential stability of impulsive high order HNN with time-varying delays by using the method of Lyapunov functions.

The purpose of this paper is to present some new criteria concerning the global exponential stability and uniform stability for a class of high-order delayed HNN with impulses by utilizing Lyapunov functional method, the quality of negative definite matrix, and the linear matrix inequality approach. The conditions on impulses are different from that presented in [12]. The effects of impulses and delays on the solutions are stressed here. As a special case, several new criteria on global exponential stability and uniform stability for the corresponding high-order HNN without impulses (see [13]) are obtained. To illustrate the validity of those results, two examples are given to illustrate the effectiveness of the results obtained.

#### 2. Preliminaries

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

Consider the following high-order delayed HNN model with impulses 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 ; is the second-order synaptic weights of the neural networks; 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 such that and , ; , are constants.

The initial conditions associated with system (2.1) are of the form where , , is continuous everywhere except at finite number of points , at which and exist and . For , the norm of is defined by . For any, let.

Assume that is an equilibrium point of system (2.1). Impulsive operator is viewed as perturbation of the equilibrium point of such system without impulsive effects. We assume that

Since is an equilibrium point of system (2.1), one can derive from system (2.1)-(2.2) that the transformation , transforms such system into the following system (for more details, please see papers [12, 13]): where in which is a real value between and , .

*Remark 2.1. *Obviously, is an equilibrium point of (2.4). Therefore, there exists at least one equilibrium point of system (2.1). So, the stability analysis of the equilibrium point of (2.1) can now be transformed to the stability analysis of the trivial solution of (2.4).

In the following, the notations and mean the transpose of and the inverse of a square matrix . We will use the notation (or , , ) to denote that the matrix is a symmetric and positive definite (negative definite, positive semidefinite, negative semidefinite) matrix. Let , , respectively, denote the largest and smallest eigenvalue of matrix .

Throughout this paper, we assume that there exist constants such that , , , .

We introduce some definitions as follows.

*Definition 2.2 (see [5]). *Leting , for any , the upper right-hand Dini derivative of along the solution of (2.4) is defined by

*Definition 2.3 (see [25]). *Assume is the solution of (2.4) through . Then the zero solution of (2.4) is said to be uniformly stable, if, for any and , there exists some such that implies , .

*Definition 2.4 (see [5]). *The equilibrium point of the system (2.1) is globally exponentially stable, if there exists constant such that, for any initial value ,

Next, in order to obtain our results, we need to establish the following lemma.

Lemma 2.5 (see [13]). *For any vectors , the inequality
**
holds, in which is any matrix with .*

Lemma 2.6 (see [31]). *Let , then
**
for any if is a symmetric matrix.*

#### 3. Main Results

In this section, some sufficient delay-dependent conditions of global exponential stability and uniform stability for system (2.1) are obtained.

Theorem 3.1. * Assume that there exist constants , and symmetric and positive definite matrices , , such that*(i)*
**where ,*(ii)* there exists constant such that
**where is the largest eigenvalue of .** Then the equilibrium point of the system (2.1) is globally exponentially stable and the approximate exponential convergent rate is .*

* Proof. * We only need to prove that the zero solution of system (2.4) is globally exponentially stable. For any , let be a solution of (2.4) through .

Consider the Lyapunov functional as follows:
then we have
By Lemma 2.5, we get
On the other hand, since and , then we have
where .

Thus, we obtain
Now we consider the derivation of along the trajectories of system (2.4), for , ,
Moreover, we note
By simple induction, considering (3.4)–(3.10), we get, for ,
On the other hand, from (3.4), we get
Substituting the above inequality into (3.11), we obtain
In view of condition (ii), we furthermore have
where
Hence, the zero solution of system (2.4) is globally exponentially stable; that is, the equilibrium point of system (2.1) is globally exponentially stable and the approximate exponential convergent rate is . The proof of Theorem 3.1 is therefore complete.

*Remark 3.2. * In Theorem 3.1, we find that condition (i) can be replaced by
Leting in Theorem 3.1, then we have the following.

Corollary 3.3. * Assume that there exist constants , such that*(i)*
**where is the largest eigenvalue of ;*(ii)* there exists constant such that
**The equilibrium point of the system (2.1) is globally exponentially stable and the approximate exponential convergent rate is .*

Furthermore, if in Corollary 3.3, then we have the following result.

Corollary 3.4. * The equilibrium point of the system (2.1) is globally exponentially stable, if , and there exists constant such that
*

*Remark 3.5. * In fact, Theorem 3.1 implies that if , then one may choose . On the other hand, Luo an Cui [13] obtained some results on global asymptotic stability. However, those results cannot ensure the global exponential stability. Let (i.e., in Corollary 3.4, then we can obtain the desirable result as follows.

Corollary 3.6. * The equilibrium point of the system (2.1) without impulses is globally exponentially stable, if there exist symmetric and positive definite matrices , , such that
**
Furthermore, if in Corollary 3.6, then it becomes as follows.*

Corollary 3.7. * The equilibrium point of the system (2.1) without impulses is globally exponentially stable, if the following condition holds:
*

*Remark 3.8. * Corollaries 3.6 and 3.7 imply that if the above inequality holds, then there exists enough small such that all conditions in Corollary 3.4 are satisfied. Hence, Corollaries 3.6 and 3.7 supplied a new criteria for global exponential stability of equilibrium point of the system (2.1) without impulses.

Next we can establish a theorem which provide sufficient conditions for uniform stability of system (2.1) by constructing another Lyapunov functional. Here we shall emphasize the effects of impulses.

Theorem 3.9. * Assume that there exist symmetric and positive definite matrices , , such that the following condition
**
where , is the largest eigenvalue of .** Then the equilibrium point of the system (2.1) is uniformly stable.*

* Proof. * We only prove the zero solution of system (2.4) is uniformly stable. For any , , , let be a solution of (2.4) through , , then we can prove that , ,

where
Consider the following Lyapunov functional
then we have
Applying the same argument as Theorem 3.1, we get
By simple calculation, we can obtain, for ,
Moreover, we know
By simple induction, from (3.27) and (3.28) we may prove that, for ,
Employing the fact (3.25), we obtain
which implies that
Therefore, the zero solution of system (2.4) is uniformly stable, that is, the equilibrium point of system (2.1) is uniformly stable. The proof of Theorem 3.9 is complete.

Corollary 3.10. *The equilibrium point of the system (2.1) is uniformly stable, if there exist symmetric and positive definite matrices , , such that the following condition holds:
**
where is the largest eigenvalue of .*

If in Theorem 3.9, then we have the following.

Corollary 3.11. *The equilibrium point of the system (2.1) is uniformly stable, if the following condition
**
where .*

#### 4. Examples

In this section we give two examples to demonstrate our results.

*Example 4.1. *Consider the following high-order delayed Hopfield-type neural network with impulses
where , , , , , , , ,
In this case, we easily observe that , , , .

For Theorem 3.1, choosing , then from
we may choose , .

On the other hand, we can compute
which implies that .

Also, we note that
implies that
By Corollary 3.3, the equilibrium point of (4.1) is global exponential stable with the approximate convergence rate 0.0488.

However, the criteria in [12] are invalid here.

*Example 4.2. *Consider the high-order delayed Hopfield-type neural network with impulses [13]
and with impulses
where , , , , , , , ,
It is obvious that , . Here we consider . Choose , , .

Note that
On the other hand, we can compute
which implies that .

One can check that
By Corollary 3.3, the equilibrium point of (4.1)–(4.7) is global exponential stable with the approximate convergence rate 0.061.

In fact, for above-given impulsive condition, we only need time-delay which satisfies the following condition:

*Remark 4.3. * In [13], the author obtained that the equilibrium point of (4.7) without impulses is globally asymptotically stable. From the example, we obtain that the equilibrium point of (4.7) without impulses is global exponential stability. In fact, if in (4.7), then we can choose , which implies that, for any given , there exists corresponding such that all conditions in Corollary 3.6 are satisfied.

#### 5. Conclusions

In this paper, a class of high-order delayed HNN with impulses is considered. We obtain some new criteria ensuring global exponential stability and uniform stability of the equilibrium point for such system by using Lyapunov functional method, the quality of negative definite matrix, and the linear matrix inequality. Our results show delays and impulsive effects on the stability of HNN. Two examples are given to illustrate the feasibility of the results.

#### Acknowledgments

This work was jointly supported by the Project of Shandong Province Higher Educational Science and Technology Program (no. J12LN23), Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (no. BS2012DX038), National Science Foundation for Postdoctoral Scientists of China (no. 2012M511538), and National Natural Science Foundation of China (no. 61201441).

#### References

- J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 81, no. 10, pp. 3088–3092, 1984. View at Google Scholar · View at Scopus - J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 79, no. 8, pp. 2554–2558, 1982. View at Publisher · View at Google Scholar - Y. Kamp and M. Hasler,
*Recursive Neural Networks for Associative Memory*, John Wiley & Sons, New York, NY, USA, 1990. - R. L. Wang, Z. Tang, and Q. P. Cao, “A learning method in Hopfield neural network for combinatorial optimization problem,”
*Neurocomputing*, vol. 48, pp. 1021–1024, 2002. View at Publisher · View at Google Scholar · View at Scopus - H. Zhang and G. Wang, “New criteria of global exponential stability for a class of generalized neural networks with time-varying delays,”
*Neurocomputing*, vol. 70, no. 13–15, pp. 2486–2494, 2007. View at Publisher · View at Google Scholar · View at Scopus - Q. Zhang, X. Wei, and J. Xu, “Global asymptotic stability of Hopfield neural networks with transmission delays,”
*Physics Letters A*, vol. 318, no. 4-5, pp. 399–405, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Zhao, “Global asymptotic stability of Hopfield neural network involving distributed delays,”
*Neural Networks*, vol. 17, no. 1, pp. 47–53, 2004. View at Publisher · View at Google Scholar · View at Scopus - Z. H. Guan and G. R. Chen, “On delayed impulsive Hopfield neural networks,”
*Neural Networks*, vol. 12, no. 2, pp. 273–280, 1999. View at Publisher · View at Google Scholar - H. Akça, R. Alassar, V. Covachev, Z. Covacheva, and E. Al-Zahrani, “Continuous-time additive Hopfield-type neural networks with impulses,”
*Journal of Mathematical Analysis and Applications*, vol. 290, no. 2, pp. 436–451, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Wang and D. Xu, “Stability for Hopfield neural networks with time delay,”
*Journal of Vibration and Control*, vol. 8, no. 1, pp. 13–18, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Ren and J. Cao, “Periodic solutions for a class of higher-order Cohen-Grossberg type neural networks with delays,”
*Computers & Mathematics with Applications*, vol. 54, no. 6, pp. 826–839, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Liu, K. L. Teo, and B. Xu, “Exponential stability of impulsive high-order Hopfield-type neural networks with time-varying delays,”
*IEEE Transactions on Neural Networks*, vol. 16, no. 6, pp. 1329–1339, 2005. View at Publisher · View at Google Scholar · View at Scopus - X.-Y. Lou and B.-T. Cui, “Novel global stability criteria for high-order Hopfield-type neural networks with time-varying delays,”
*Journal of Mathematical Analysis and Applications*, vol. 330, no. 1, pp. 144–158, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Ren and J. Cao, “LMI-based criteria for stability of high-order neural networks with time-varying delay,”
*Nonlinear Analysis. Real World Applications*, vol. 7, no. 5, pp. 967–979, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Xu, X. Liu, and X. Liao, “Global asymptotic stability of high-order Hopfield type neural networks with time delays,”
*Computers & Mathematics with Applications*, vol. 45, no. 10-11, pp. 1729–1737, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Guan, D. Sun, and J. Shen, “Qualitative analysis of high-order hopfield neural networks,”
*Acta Electronica Sinica*, vol. 28, no. 3, pp. 77–80, 2000. View at Google Scholar · View at Scopus - J. Cao, “Global exponential stability of Hopfield neural networks,”
*International Journal of Systems Science. Principles and Applications of Systems and Integration*, vol. 32, no. 2, pp. 233–236, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Xu and J. Lam, “A new approach to exponential stability analysis of neural networks with time-varying delays,”
*Neural Networks*, vol. 19, no. 1, pp. 76–83, 2006. View at Publisher · View at Google Scholar · View at Scopus - F. Ren and J. Cao, “Periodic oscillation of higher-order bidirectional associative memory neural networks with periodic coefficients and delays,”
*Nonlinearity*, vol. 20, no. 3, pp. 605–629, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Cao, J. Liang, and J. Lam, “Exponential stability of high-order bidirectional associative memory neural networks with time delays,”
*Physica D*, vol. 199, no. 3-4, pp. 425–436, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Rakkiyappan and P. Balasubramaniam, “On exponential stability results for fuzzy impulsive neural networks,”
*Fuzzy Sets and Systems*, vol. 161, no. 13, pp. 1823–1835, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Balasubramaniam and V. Vembarasan, “Robust stability of uncertain fuzzy BAM neural networks of neutral-type with Markovian jumping parameters and impulses,”
*Computers & Mathematics with Applications*, vol. 62, no. 4, pp. 1838–1861, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Balasubramaniam and V. Vembarasan, “Asymptotic stability of BAM neural networks of neutral-type with impulsive effects and time delay in the leakage term,”
*International Journal of Computer Mathematics*, vol. 88, no. 15, pp. 3271–3291, 2011. View at Publisher · View at Google Scholar - D. D. Baĭnov and P. S. Simeonov,
*Systems with Impulse Effect Stability, Theory and Applications*, Ellis Horwood, New York, NY, USA, 1989. - X. L. Fu, B. Q. Yan, and Y. S. Liu,
*Introduction of Impulsive Differential Systems*, Science Press, Beijing, China, 2005. - C. Li, W. Hu, and S. Wu, “Stochastic stability of impulsive BAM neural networks with time delays,”
*Computers & Mathematics with Applications*, vol. 61, no. 8, pp. 2313–2316, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Xia, Z. Huang, and M. Han, “Existence and globally exponential stability of equilibrium for BAM neural networks with impulses,”
*Chaos, Solitons & Fractals*, vol. 37, no. 2, pp. 588–597, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov,
*Theory of Impulsive Differential Equations*, vol. 6, World Scientific, Singapore, 1989. - X. Li and J. Shen, “LMI approach for stationary oscillation of interval neural networks with discrete and distributed time-varying delays under impulsive perturbations,”
*IEEE Transactions on Neural Networks*, vol. 21, no. 10, pp. 1555–1563, 2010. View at Publisher · View at Google Scholar · View at Scopus - Y. Zhang and J. Sun, “Stability of impulsive neural networks with time delays,”
*Physics Letters, Section A*, vol. 348, no. 1-2, pp. 44–50, 2005. View at Publisher · View at Google Scholar · View at Scopus - A. Berman and R. J. Plemmons,
*Nonnegative Matrices in the Mathematical Sciences*, Academic Press, New York, NY, USA, 1979.