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Journal of Applied Mathematics

Volume 2014 (2014), Article ID 972608, 6 pages

http://dx.doi.org/10.1155/2014/972608

## Global Exponential Robust Stability of Static Interval Neural Networks with Time Delay in the Leakage Term

^{1}School of Mathematical Science, Ocean University of China, Qingdao 266100, China^{2}School of Mathematical Science, Liaocheng University, Liaocheng 252059, China

Received 24 July 2013; Accepted 12 December 2013; Published 12 January 2014

Academic Editor: Subhas Abel

Copyright © 2014 Guiying Chen and Linshan Wang. 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 stability of a class of static interval neural networks with time delay in the leakage term is investigated. By using the method of -matrix and the technique of delay differential inequality, we obtain some sufficient conditions ensuring the global exponential robust stability of the networks. The results in this paper extend the corresponding conclusions without leakage delay. An example is given to illustrate the effectiveness of the obtained results.

#### 1. Introduction

Recently, neural networks have been widely studied because of their successful applications in different areas, such as pattern recognition, image processing, detection of moving objects, and optimization problems. The stability of the neural networks with time delay, upon which these applications largely depend, has been extensively studied (see [1–10]). However, to the best of our knowledge, there has been very little existing work on neural networks, especially, on static neural networks with time delay in the leakage term [11–18]. This is due to some theoretical and technical difficulties [13]. So, the main purpose of this paper is to study the stability of the static interval neural networks with time delay in the leakage term. By using the properties of -matrix and delay differential inequality, we obtain some sufficient conditions ensuring the global exponential robust stability. Our results extend the corresponding conclusions without leakage delay.

#### 2. Model Description and Preliminaries

In this section, we list all the notations which will be frequently used throughout the paper and give a few definitions, lemmas, and assumptions.

*Notations*. Let be the set of real number, and let and be the space of -dimensional real vectors and real matrices, separately. denotes an unit matrix.

. For or , the notation () means that each pair of corresponding elements of and satisfies the inequality “.” denotes the Euclidean norm. For any , is the sign function of .

denotes the space of continuous mappings from the topological space to the topological space . Particularly, let denote the family of all continuous -valued function defined on with the norm .

For, , we define , , , , and . denotes the upper-right-hand derivative of at time .

Consider the following interval static neural network model with leakage delay: where , and denote the state and the external inputs of the neuron, separately. The integer corresponds to the number of units in a neural network, and denotes the signal propagation function of the unit. is a parameter, and represents the rate with which neuron will reset its potential to the resting state in isolation when disconnected from the network and external inputs. is nondecreasing bounded variation functions on , and is a Lebesgue-Stieltjes integration. There exist positive constants , , , and such that for any , , , and . represents the leakage delay, . , where is derivative on , and , .

*Definition 1. *The equilibrium point of system (1) is said to be globally exponentially stable if there exist a positive constant and a vector such that

*Definition 2. *System (1) is said to be globally exponentially robustly stable if its equilibrium point is globally exponentially stable for any and .

*Definition 3 (see [19]). *Let the matrix with and , , . Then each of the following conditions is equivalent to the statement “ is a nonsingular -matrix.”(1)All the leading principle minors of are positive.(2)The diagonal elements of are all positive, and there exists a positive vector such that or .

Lemma 4 (see [20]). *Let , and satisfies
**
where , for , , and , . Suppose that there exist a scalar and a vector such that
**
If the initial condition satisfies
**
then for .*

For the model (1), we introduce the following assumptions.)The signal propagation functions are Lipschitz continuous; that is, there are positive constants , such that for all ()Let be a nonsingular -matrix, where

#### 3. Main Results

Theorem 5. *Suppose that the conditions (A _{1}) and (A_{2}) hold, and then system (1) has at least one equilibrium point.*

*Proof. *From (A_{2}) and Definition 3, we know there exists a positive vector such that . That is

From (8) we can get

Combining with Definition 3, we know is a nonsingular -matrix, where , , and .

In a similar way of proof for the literature [7], by the theory of topological degree and homotopy invariance theorem, the existence of the equilibrium point of system (1) can be proved. Suppose that is an equilibrium point of system (1), let , , and then system (1) becomes

It is clear that the stability of zero solution to system (10) is equivalent to the stability of the equilibrium point of system (1). So we only consider the stability of zero solution to system (10).

Theorem 6. *Assume that the conditions (A _{1}) and (A_{2}) are satisfied, and then the zero solution to system (10) is globally exponentially robustly stable.*

*Proof. *From the Middle Value theorem, we obtain
where . Then from (10) we get
*Case 1*. If , then .

Let , . Then from (A_{1}), we have
where , , , , and .

From (A_{2}) and Definition 3, we know that there exists a vector such that . That is
From (14), we can get
Hence, is a nonsingular -matrix. Thus, there exists a vector such that . By using the continuity, we know there exists at least one constant such that

Since , then, for , there exists a constant such that
where . Then from (13), (16), (17), and Lemma 4, we get
*Case 2*. If , then from (10) we have
Substituting (19) into (12), we get

Let , . Then, from (A_{1}) and (A_{2}), we have

Since is a nonsingular -matrix, there exists a vector such that . By using the continuity, we know there exists at least one constant such that

From (14) and (15), we know is bounded on . So there exists a vector such that , . Then we can get
Then from (21), (22), (23), and Lemma 4 with , we have

Let , we get
Thus, the equilibrium of system (1) is globally exponentially robustly stable.

*Remark 7. *If , the system (1) becomes the static interval neural networks without time delay in the leakage term. So, this paper includes the results of Han et al. (2011) as a special case.

*Remark 8. *If ,
then system (1) becomes the following static neural network model:

If and
where , then system (1) becomes a class of static neural network models with discrete time delays as follows:

If and , then system (1) becomes the following static model with continuous time delays:

From (29) and (30), we can see that S-type distributed time delay contains discrete time delays and continuous time delays as two special cases, so our results generalized the results of the related literature [1, 3, 10].

#### 4. Example

Consider the following system: where , , , , and , . It can be obtained that Thus is a -matrix. From Theorem 6, the equilibrium point of system (27) is globally exponentially robust stable.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work was supported by the National Natural Science Foundation of China (11171374) and Natural Science Foundation of Shandong Province (ZR2011AZ001).

#### References

- Z. Wu, J. Lam, H. Su, et al., “Stability and dissipativity analysis of static neural networks with time delay,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 23, no. 2, pp. 199–210, 2012. - L. Wang, R. Zhang, and Y. Wang, “Global exponential stability of reaction-diffusion cellular neural networks with S-type distributed time delays,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 2, pp. 1101–1113, 2009. View at Publisher · View at Google Scholar · View at Scopus - Q. Duan, H. Su, and Z. Wu, “${H}_{\infty}$ state estimation of static neural networks with time-varying delay,”
*Neurocomputing*, vol. 97, no. 15, pp. 16–21, 2012. - L. Wan and D. Xu, “Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays,”
*Science in China F*, vol. 46, no. 6, pp. 466–474, 2003. - D. Xu and S. Long, “Attracting and quasi-invariant sets of non-autonomous neural networks with delays,”
*Neurocomputing*, vol. 77, no. 1, pp. 222–228, 2012. View at Publisher · View at Google Scholar · View at Scopus - L. Wang and D. Xu, “Global asymptotic stability of bidirectional associative memory neural networks with S-type distributed delays,”
*International Journal of Systems Science*, vol. 33, no. 11, pp. 869–877, 2002. View at Publisher · View at Google Scholar · View at Scopus - L. Wang,
*Recurrent Neural Networks with Time Delays*, Science Press, Beijing, China, 2008. - C. Huang and J. Cao, “Almost sure exponential stability of stochastic cellular neural networks with unbounded distributed delays,”
*Neurocomputing*, vol. 72, no. 13-15, pp. 3352–3356, 2009. View at Publisher · View at Google Scholar · View at Scopus - Z.-G. Wu, J. H. Park, H. Su, and J. Chu, “New results on exponential passivity of neural networks with time-varying delays,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 4, pp. 1593–1599, 2012. View at Publisher · View at Google Scholar · View at Scopus - W. Han, Y. Kao, and L. Wang, “Global exponential robust stability of static interval neural networks with S-type distributed delays,”
*Journal of the Franklin Institute*, vol. 348, no. 8, pp. 2072–2081, 2011. View at Publisher · View at Google Scholar · View at Scopus - K. Gopalsamy,
*Stability and Oscillations in Delay Differential Equations of Population Dynamics*, vol. 74, Kluwer Academic, Dordrecht, The Netherlands, 1992. View at MathSciNet - X. Li, R. Rakkiyappan, and P. Balasubramaniam, “Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations,”
*Journal of the Franklin Institute*, vol. 348, no. 2, pp. 135–155, 2011. View at Publisher · View at Google Scholar · View at Scopus - K. Gopalsamy, “Leakage delays in BAM,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 2, pp. 1117–1132, 2007. View at Publisher · View at Google Scholar · View at Scopus - C. Li and T. Huang, “On the stability of nonlinear systems with leakage delay,”
*Journal of the Franklin Institute*, vol. 346, no. 4, pp. 366–377, 2009. View at Publisher · View at Google Scholar · View at Scopus - X. Li, X. Fu, P. Balasubramaniam, and R. Rakkiyappan, “Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 5, pp. 4092–4108, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. J. Park, O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay,”
*Journal of the Franklin Institute*, vol. 349, no. 5, pp. 1699–1720, 2012. View at Publisher · View at Google Scholar · View at Scopus - S. Peng, “Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 3, pp. 2141–2151, 2010. View at Publisher · View at Google Scholar · View at Scopus - X. Li, R. Rakkiyappan, and P. Balasubramaniam, “Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations,”
*Journal of the Franklin Institute*, vol. 348, no. 2, pp. 135–155, 2011. 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. View at MathSciNet - D. Xu and Z. Yang, “Attracting and invariant sets for a class of impulsive functional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 329, no. 2, pp. 1036–1044, 2007. View at Publisher · View at Google Scholar · View at Scopus