Abstract and Applied Analysis

Volume 2013 (2013), Article ID 409758, 10 pages

http://dx.doi.org/10.1155/2013/409758

## Stability of Impulsive Cohen-Grossberg Neural Networks with Time-Varying Delays and Reaction-Diffusion Terms

^{1}College of Electric and Electronic Engineering, Wuhan Institute of Shipbuilding Technology, Wuhan, Hubei 430050, China^{2}College of Electronic and Information Engineering, Hubei University of Science and Technology, Xianning, Hubei 437100, China

Received 29 December 2012; Accepted 23 February 2013

Academic Editor: Qi Luo

Copyright © 2013 Jinhua Huang 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

This work concerns the stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms as well as Dirichlet boundary condition. By means of Poincaré inequality and Gronwall-Bellman-type impulsive integral inequality, we summarize some new and concise sufficient conditions ensuring the global exponential stability of equilibrium point. The proposed criteria are relevant to the diffusion coefficients and the smallest positive eigenvalue of corresponding Dirichlet Laplacian. In conclusion, two examples are illustrated to demonstrate the effectiveness of our obtained results.

#### 1. Introduction

Cohen-Grossberg neural networks (CGNNs) were introduced by Cohen and Grossberg in 1983 [1] and have been a hot topic due to their important applications in various fields such as parallel computation, associative memory, image processing, and optimization problems.

By reason that time delays are unavoidably encountered for the finite switching speed of neurons and amplifiers in the implementation of neural networks, a more powerful model of delayed Cohen-Grossberg neural networks (DCGNNs) is afterwards proposed. This kind of mathematical models is widely applied in dynamic image processing and pattern recognition problems. It is worth noting that all these applications depend heavily on the dynamical behaviors such as stability, convergence, and oscillatory [2–6]. Meanwhile, stability is an important consideration in the designs and applications of neural networks. The stability of delayed neural networks is a subject of current interest, and therefore considerable theoretical efforts have been put into this topic followed by a large number of stability criteria reported; for example, see [7–12] and the references therein.

In real world, however, many evolutionary processes are characterized by abrupt changes at certain instants which may be caused by switching phenomena, frequency changes, or other sudden noises. As such, in the past few years, scientists have become gradually interested in the influence that impulses may have on the CGNNs and DCGNNs, thus obtaining some related results; for example, see [13–18] and the references therein.

Actually, besides impulsive effects, we have to recognize that diffusion effects are also nonignorable in reality as diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields. On this account, the model of neural networks with both impulses and diffusion should be more effective for describing the evolutionary process of practical systems. Based on this consideration, we wonder what the influence of diffusion on the stability of CGNNs and DCGNNs is.

So far there have appeared a few theoretical achievements [19–29] on the stability of impulsive reaction-diffusion neural networks with or without delays. Particularly, in [21–26], the main research technique is the impulsive differential inequality whereby the authors discussed the stability of equilibrium point and provided a series of sufficient conditions independent of diffusion. From these results, we fail to see the influence of diffusion on the stability of CGNNs and DCGNNs.

Encouragingly, recently there were reported some new results on the stability of CGNNs and DCGNNs in [19, 20, 27]; thereinto, the presented stability criteria derived from the impulsive differential inequality are related to the diffusion terms, and thereby we know the diffusion do contribute to the stability of impulsive neural networks.

In this paper, different from [20, 27], we shall consider the case where the boundary condition is Dirichlet boundary condition rather than Neumann boundary condition. Moreover, unlike [19], we shall utilize the new method of Poincarè inequality to deal with the reaction-diffusion terms, and Gronwall-Bellman-type impulsive integral inequality is also introduced for stability analysis. The obtained results show that not only the reaction-diffusion coefficients but also the first eigenvalue of corresponding Dirichlet Laplacian can affect the stability.

The rest of this paper is structured as follows. In Section 2, the model of impulsive delayed Cohen-Grossberg neural networks with reaction-diffusion terms as well as Dirichlet boundary condition is outlined and some facts and lemmas are introduced for later reference. By the new agencies of Gronwall-Bellman-type impulsive integral inequality and Poincaré inequality, we discuss the global exponential stability of equilibrium point and develop some new and concise algebraic criteria in Section 3. To conclude, two illustrative examples are given in Section 4 to verify the effectiveness of our results.

#### 2. Preliminaries

Let denote the -dimensional Euclidean space, and let be an open bounded domain with smooth boundary and . Let and .

Consider the following impulsive CGNN with time-varying delays and reaction-diffusion terms: where corresponds to the numbers of units in a neural network, , denotes the state of the th neuron at time and in space represents transmission diffusion of the th unit, represents the amplification function, is the appropriate behavior function, activation function stands for the output of the th unit at time and in space and and are constants: indicates the connection strength of the th unit on the th unit at time and in space , while denotes the connection weight of the th unit on the th unit at time and in space , where corresponds to the transmission delay along the axon of the jth unit satisfying and . is the sequence of impulsive moments meeting and and represent the right-hand and left-hand limit of at time and in space , respectively. stands for the abrupt change of at impulsive moment and in space .

Denote by , the solution of systems (1)-(2), satisfying the initial condition and Dirichlet boundary condition where the vector-valued function is such that is bounded on .

The solution of problems (1)–(4) is, for the time variable , a piecewise continuous function with the first kind discontinuity at the points , where it is left-continuous; that is, the following relations are valid:

Throughout this paper, we define the norm of as and make the following assumptions for convenience. is continuous and bounded; that is, there exist constants and such that is continuous and ; moreover, there exists constant such that is continuous and ; furthermore, there exists constant such that is continuous and for and .

In the light of (H1)–(H4), it is easy to see that problems (1)-(2) admit an equilibrium point .

*Definition 1. *The equilibrium point of problems (1)-(2) is said to be globally exponentially stable if there exist constants and such that
where .

Lemma 2 (see [30] (Gronwall-Bellman-type impulsive integral inequality)). *Assume the following.*(A1)* The sequence satisfies , with .*(A2)* and is left-continuous at .*(A3)* and for ,
**where and . Then,
*

Lemma 3 (see [31] (Poincaré inequality)). * Let be a bounded region in , and on the boundary of ; then
**
where is the smallest positive eigenvalue of the following problem:
*

Lemma 4. *If and , then holds for any .*

#### 3. Main Results

Theorem 5. *Assume the following.*(1)* and denote .*(2)*.*(3)*There exists a constant satisfying and , where
**Then, the equilibrium point of systems (1)-(2) is globally exponentially stable with convergence rate .*

*Proof. *Multiplying both sides of (1) by , we get
which yields, after integrating with respect to spatial variable on ,
where ,

By combining Green formula, Dirichlet boundary condition, Lemma 3, and condition (1) of Theorem 5, we obtain

Moreover, it follows from assumptions (H1), (H2), and (H3) that

Consequently, substituting (19)–(22) into (17) produces
for , , .

Now define Lyapunov function as . It is not difficult to see that is a piecewise continuous function with points of discontinuity of the first kind , where it is continuous from the left; that is, (). In addition, for (), we know
as and , supported by condition 2 of Theorem 5.

Put , . It is derived from (23) that

Define function of the form again. From (25), one then reads
where and .

Construct , where satisfies and . Evidently, is also a piecewise continuous function with the first kind discontinuous points , in which it is continuous from the left; that is, . Moreover, at , we find by the use of (24)

Set . By virtue of (26), one has

Choose small enough . Integrating (28) from to gives
which yields, after letting in (29),

Next, we estimate the value of at . For small enough , we put . Now an application of (30) leads to, for ,

If we let in (31), there results

Note that is applicable for . Thus,
holds for . By synthesizing (30) and (33), we then arrive at

This, together with (27), results in
for .

Recalling the assumptions that and , we therefore obtain
Hence,

By induction argument, we reach
Thus,

Since
we claim

According to Lemma 2, we assert that
which reduces to
This completes the proof.

*Remark 6. *According to the conditions of Theorem 5, we see that the reaction-diffusion terms can influence the stability of equilibrium point . Specifically, the acting factors include the reaction-diffusion coefficients and the first eigenvalue of corresponding Dirichlet Laplacian.

*Remark 7. *It is not difficult to see that there must exist constant satisfying condition 3 of Theorem 5 if .

Theorem 8. *Assume the following.*(1)* and denote .*(2)*, , .*(3)*.*(4)*There exists a constant satisfying and , where and .*

Then, the equilibrium point of systems (1)-(2) is globally exponentially stable with convergence rate .

*Proof. *Define Lyapunov function of the form , where . Obviously, is a piecewise continuous function with the first kind discontinuous points , where it is continuous from the left; that is, . Furthermore, for , we derive from condition 2 of Theorem 8 that
Thereby,

Construct function again, where satisfies and . Then, is also a piecewise continuous function with the first kind discontinuous points , where it is continuous from the left; that is, . And for , it follows from (45) that

Set . Following the same procedure as shown in the proof of Theorem 5, we get

The relations (46) and (47) yield
By induction argument, we obtain
Hence,

Introducing as shown in the proof of Theorem 5 into (50), (50) becomes, for ,
It then results from Lemma 2 that, for ,

On the other hand, since , one has . Thereby,
and (52) can be rewritten as
which implies
The proof is completed.

Due to Lemma 4, we know that the following inequalities: hold for any . Thus, in a similar way to the proofs of Theorems 5–8, we can prove the following theorems.

Theorem 9. *Assume the following.*(1)* and denote .*(2)*, .*(3)*There exist constants and such that and , where , and .**
Then, the equilibrium point of systems (1)-(2) is globally exponentially stable with convergence rate .*

*Remark 10. *There must exist constant satisfying condition 3 of Theorem 9 if there are constants such that .

Theorem 11. *Assume the following.*(1)* and denote .*(2)*, , .*(3)*.*(4)*There exist constants and satisfying and , where
**Further, on the condition that , where and , we obtain
**
for . In an identical way with the proof of Theorem 8, we can present the following.*

Theorem 12. *Assume the following.*(1)*Let and denote .*(2)*, where and .*(3)*.*(4)*There exist constants and such that and , where
*

*Remark 13. *Different from Theorems 5–11, the impulsive part in Theorem 12 could be nonlinear, and this will be of more applicability. Actually, Theorems 5–11 can be regarded as the special cases of Theorem 12.

#### 4. Examples

*Example 14. *Consider problems (1)–(4) with ; moreover, , , , , , , , , , , and .

As and , we know . Further, for , , , and , we compute

Let . Since , we therefore conclude from Theorem 5 that the zero solution of this system is globally exponential stable.

*Example 15. *Consider problems (1)–(4) with ; moreover, , , , , , , , , , , , and .

As and , we know . Further, for , , and , we compute

Let , and ; we can find such that

Therefore it is concluded from Theorem 12 that the zero solution of this system is globally exponential stable.

#### References

- M. A. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,”
*IEEE Transactions on Systems, Man, and Cybernetics*, vol. 13, no. 5, pp. 815–826, 1983. View at Publisher · View at Google Scholar · View at MathSciNet - J. Cao and J. Liang, “Boundedness and stability for Cohen-Grossberg neural network with time-varying delays,”
*Journal of Mathematical Analysis and Applications*, vol. 296, no. 2, pp. 665–685, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - J. Cao and J. Wang, “Global exponential stability and periodicity of recurrent neural networks with time delays,”
*IEEE Transactions on Circuits and Systems. I.*, vol. 52, no. 5, pp. 920–931, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - X. Liu, “Stability results for impulsive differential systems with applications to population growth models,”
*Dynamics and Stability of Systems*, vol. 9, no. 2, pp. 163–174, 1994. View at Google Scholar · View at MathSciNet - X. Liu and Q. Wang, “Impulsive stabilization of high-order Hopfield-type neural networks with time-varying delays,”
*IEEE Transactions on Neural Networks*, vol. 19, no. 1, pp. 71–79, 2008. View at Publisher · View at Google Scholar · View at Scopus - Z. Yang and D. Xu, “Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays,”
*Applied Mathematics and Computation*, vol. 177, no. 1, pp. 63–78, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - S. Arik and Z. Orman, “Global stability analysis of Cohen-Grossberg neural networks with time varying delays,”
*Physics Letters A*, vol. 341, no. 5-6, pp. 410–421, 2005. View at Publisher · View at Google Scholar · View at Scopus - T. Huang, C. Li, and G. Chen, “Stability of Cohen-Grossberg neural networks with unbounded distributed delays,”
*Chaos, Solitons & Fractals*, vol. 34, no. 3, pp. 992–996, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - X. Liao, C. Li, and K. W. Wong, “Criteria for exponential stability of Cohen-Grossberg neural networks,”
*Neural Networks*, vol. 17, no. 10, pp. 1401–1414, 2004. View at Publisher · View at Google Scholar · View at Scopus - J. Zhang, Y. Suda, and H. Komine, “Global exponential stability of Cohen-Grossberg neural networks with variable delays,”
*Physics Letters A*, vol. 338, no. 1, pp. 44–50, 2005. View at Publisher · View at Google Scholar · View at Scopus - W. Zhang, Y. Tang, J. A. Fang, and X. Wu, “Stability of delayed neural networks with time-varying impulses,”
*Neural Networks*, vol. 36, pp. 59–63, 2012. View at Google Scholar - X. Lai and Y. Zhang, “Fixed point and asymptotic analysis of cellular neural networks,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 689845, 12 pages, 2012. View at Google Scholar · View at MathSciNet - Z. Chen and J. Ruan, “Global stability analysis of impulsive Cohen' Grossberg neural networks with delay,”
*Physics Letters A*, vol. 345, pp. 101–111, 2005. View at Google Scholar - Z. Chen and J. Ruan, “Global dynamic analysis of general Cohen-Grossberg neural networks with impulse,”
*Chaos, Solitons and Fractals*, vol. 32, no. 5, pp. 1830–1837, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Yang and D. Xu, “Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays,”
*Applied Mathematics and Computation*, vol. 177, no. 1, pp. 63–78, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - Q. Song and J. Zhang, “Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 2, pp. 500–510, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Chen and J. Ruan, “Global dynamic analysis of general Cohen-Grossberg neural networks with impulse,”
*Chaos, Solitons and Fractals*, vol. 32, no. 5, pp. 1830–1837, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Zhang and Q. Luo, “Global exponential stability of impulsive cellular neural networks with time-varying delays via fixed point theory,”
*Advances in Difference Equations*, vol. 2013, article 23, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. Zhang, S. Wu, and K. Li, “Delay-dependent exponential stability for impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 3, pp. 1524–1532, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J. Pan and S. Zhong, “Dynamical behaviors of impulsive reaction-diffusion Cohen-Grossberg neural network with delays,”
*Neurocomputing*, vol. 73, no. 7–9, pp. 1344–1351, 2010. View at Publisher · View at Google Scholar · View at Scopus - K. Li and Q. Song, “Exponential stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms,”
*Neurocomputing*, vol. 72, no. 1–3, pp. 231–240, 2008. View at Publisher · View at Google Scholar · View at Scopus - J. Qiu, “Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms,”
*Neurocomputing*, vol. 70, no. 4–6, pp. 1102–1108, 2007. View at Publisher · View at Google Scholar · View at Scopus - X. Wang and D. Xu, “Global exponential stability of impulsive fuzzy cellular neural networks with mixed delays and reaction-diffusion terms,”
*Chaos, Solitons & Fractals*, vol. 42, no. 5, pp. 2713–2721, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - W. Zhu, “Global exponential stability of impulsive reaction-diffusion equation with variable delays,”
*Applied Mathematics and Computation*, vol. 205, no. 1, pp. 362–369, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Li and K. Li, “Stability analysis of impulsive Cohen-Grossberg neural networks with distributed delays and reaction-diffusion terms,”
*Applied Mathematical Modelling*, vol. 33, no. 3, pp. 1337–1348, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Li and K. Li, “Stability analysis of impulsive fuzzy cellular neural networks with distributed delays and reaction-diffusion terms,”
*Chaos, Solitons and Fractals*, vol. 42, no. 1, pp. 492–499, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - J. Pan, X. Liu, and S. Zhong, “Stability criteria for impulsive reaction-diffusion Cohen-Grossberg neural networks with time-varying delays,”
*Mathematical and Computer Modelling*, vol. 51, no. 9-10, pp. 1037–1050, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Zhang and Q. Luo, “Novel stability criteria for impulsive delayed reaction-diffusion Cohen-Grossberg neural networks via Hardy-Poincarè inequality,”
*Chaos, Solitons & Fractals*, vol. 45, no. 8, pp. 1033–1040, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Yutian and Z. Minhui, “Stability analysis for impulsive reaction-diffusion Cohen-Grossberg neural networks with time-varying delays,”
*Journal of Nanjing University of Information Science and Technology*, vol. 4, no. 3, pp. 213–219, 2012. View at Google Scholar - V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov,
*Theory of Impulsive Differential Equations*, vol. 6 of*Series in Modern Applied Mathematics*, World Scientific Publishing, Singapore, 1989. View at MathSciNet - D. S. Mitrinovic,
*Analytic Inequalities*, Springer, New York, NY, USA, 1970. View at MathSciNet