Journal of Applied Mathematics

Volume 2014 (2014), Article ID 705496, 8 pages

http://dx.doi.org/10.1155/2014/705496

Research Article

Global Exponential Robust Stability of High-Order Hopfield Neural Networks with S-Type Distributed Time Delays

Haiyong Zheng,¹ Bin Wu,¹ Tengda Wei,² Linshan Wang,² and Yangfan Wang³

¹College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China

²School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

³College of Marine Life Science and School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Received 24 March 2014; Accepted 1 June 2014; Published 26 June 2014

Academic Editor: Reinaldo Martinez Palhares

Copyright © 2014 Haiyong Zheng 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.

By employing differential inequality technique and Lyapunov functional method, some criteria of global exponential robust stability for the high-order neural networks with S-type distributed time delays are established, which are easy to be verified with a wider adaptive scope.

1. Introduction

Neural networks and their various generalizations have been successfully employed in many areas such as pattern recognition, cognitive modeling, adaptive control, and combinatorial optimization [1–7]. Hopfield neural networks (HNNs), as some forms of recurrent artificial neural networks, have been widely studied in recent years [8–12]. The earlier HNNs model proposed by Hopfield [13, 14] was based on the theory of analog circuit consisting of capacitors, resistors, and amplifiers and can be formulated as a system of ordinary equations. Time delays are inevitable in the interactions of neurons in biological and artificial neural networks. The existence of delays is frequently a source of instability for neural networks [9, 10, 15–19].

Over the past few decades, the stability of HNNs with time delays has attracted considerable attention in the literature [20–23]. One of the most investigated problems in the study of HNNs is global exponential stability of the equilibrium point. An equilibrium point of HNNs is globally exponentially stable, if the domain of attraction of the equilibrium point is the whole space and the convergence is in real time.

It is worth noting that although the signal propagation is sometimes instantaneous and can be modeled with discrete delays, it may also be distributed during a certain time period so that the distributed delays should be incorporated in the model [24]. Discrete delays and distributed delays attract the attention of many scholars and have been widely studied [17, 19, 25]. To the best of our knowledge, the stability problem for system with both discrete and distributed delays has been a challenging issue, mainly due to the mathematical difficulties in dealing with discrete and distributed delays simultaneously. In 2002, Wang and Xu [26] presented a new neural network model with S-type distributed time delays and demonstrated that S-type distributed time delays include discrete or continuously distributed time delays, but it is not true in the opposite way. In the following years, S-type distributed time delayed neural network models have raised great interest [12, 27–29].

Compared with traditional Hopfield neural networks, the high-order Hopfield type neural networks (HOHNNs) [11, 12, 30–34] have the advantages of stronger approximation properties, faster convergence rate, greater storage capacity, and higher fault tolerance. Therefore, it is of considerable interest to explore the theoretical foundations and practical applications of HOHNNs.

Motivated by the aforementioned discussion, we studied the problem of global exponential robust stability of HOHNNs with S-type distributed time delays. By employing differential inequality technique and a new Lyapunov functional method, some criteria for the global exponential robust stability of the high-order neural networks with S-type distributed time delays have been established, which are easy to be verified with a wider adaptive scope. Meanwhile, the systems in [10, 12, 26, 31] are some special cases of the HOHNNs with S-type distributed time delays.

2. Model Description and Preliminaries

We consider the following HOHNNs with S-type distributed time delays:

where , , , , , and have the same meanings as those in [28], , are the first- and second-order synaptic weights of the system (1) (see [12]), and are Lebesgue-Stieltjes integrable, where and are nondecreasing bounded variation functions which satisfy

In this paper the superscript “” presents the transpose and denotes a set of continuous bounded functions.

For system (1), the initial condition is

If there is an equilibrium point of system (1) with conditions (3), we can rewrite system (1) as the following equivalent form:

From [12], we know that the following system (5) is equivalent to system (4):

where .

Definition 1. The equilibrium point of system (1) is called globally exponentially robustly stable, if, for any , there exist scalars and such that the solution to system (1) with initial condition , , satisfies

Let , , , , , , , = , = , = , = = , = , = = be Hadamard product of two matrices,

We assume throughout that the neuron activation functions , , , satisfy the following conditions.

 () Consider

 () Consider

 () Consider

3. Main Results

Theorem 2. The equilibrium of system (1) is globally exponentially robustly stable, if system (1) satisfies , , and .

Proof. Part 1: Existence of the Equilibrium Point. Let

It is obvious that the solutions to (11) are the equilibrium points of system (1).

Let us define homotopic mapping as follows:

where

By homotopy invariance theorem (see [35]), topological degree theory (see [36]), (), and the proof, which is similar to Theorem 1 in [28], we can conclude that (13) has at least one solution.

That is, system (1) has at least an equilibrium point.

Part 2: Global Existence of the Solutions to System (1). Since is an -matrix, there exists a constant vector such that (see [15]); that is,

Suppose is a solution to system (1) and also satisfies the initial condition , , .

Let us choose a positive constant such that

From (15) we know that

Then, we will show that

If (17) is not true, there must be some positive integer and , such that

From , (15), and (18), we know

So

which leads by contradiction to (18).

Hence, (17) holds. That is, the solutions to system (1) are bounded. So the solutions to system (1) are of global existence.

Part 3: Global Exponential Stability of System (1). From , we know that there exists constant , such that

So, from , we can choose a constant sufficiently small, such that

Let , and

Then, we will show that there exists such that

Define a Lyapunov functional by

Its Dini derivative reads

If , we have

Then, we will prove that

If (28) is not true, there exists and such that

From (29), we have

Because and , for all , , we can obtain

It is obvious that (31) is in contradiction to (30). Hence, (28) holds. That is,

So

where .

If there exists another equilibrium of system (1), we have , , .

From above proof, the system (1) has a unique equilibrium point , which is globally exponentially robustly stable.

The proof of Theorem 2 is completed.

Remark 3. The system (1) includes system with discrete time delays and with continuously distributed delays. Conversely, it is not true.

When and

system (1) becomes a HNNs model with discrete time delays

When and , , , the value of the synaptic connectivity from neuron to is a continuous function on , which means that time delays influence the network continuously, and system (1) belongs to a HNNs model with continuous time delays:

So system (1) is widely representative.

Remark 4. When , , , , and , system (1) becomes the system in [12]. So the systems in [10, 12, 26, 31] are also the special cases of system (1) (see [12]).

4. Example

For the sake of simplicity, we consider given one-dimension HOHNNs with S-type distributed time delays as follows:

In system (37), and , , which satisfy , , and .

Let ; then , , which satisfies ().

The parameters of the system (37) are given as follows:

From and the above parameters, we can easily obtain that is an -matrix.

Therefore, it follows from Theorem 2 that the null solution to system (37) is globally exponentially robustly stable.

5. Conclusion

We have investigated the global exponential robust stability of high-order Hopfield neural networks with S-type distributed time delays, which is of theoretical as well as practical importance for the development of neural networks with time delays. The system (1) considered here is more general compared to the systems in literatures [10, 12, 26, 31]. By employing differential inequality technique and Lyapunov functional method, some criteria of global exponential robust stability for the high-order neural networks with S-type distributed time delays are established, which are easily verifiable and have a wider applicable range. The linear matrix inequality (LMI) approach is also widely used to establish the desired sufficient conditions for stability analysis of delayed neural networks [11, 37]. Wen et al. [17] have done some great work in control and filtering problems for neural systems. In future extension, we will do some research in stability of high-order Hopfield neural networks with S-type distributed time delays using LMI method.

Conflict of Interests

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

Acknowledgments

Haiyong Zheng was partially supported by the National Natural Science Foundation of China (nos. 61271406 and 61301240). Linshan Wang was partially supported by the National Natural Science Foundation of China (no. 11171374) and the Natural Science Foundation of Shandong (no. ZR2011AZ001). Yangfan Wang was partially supported by the National Natural Science Foundation of China (no. 31302182) and Fundamental Research Funds for Central Universities (nos. 201362034 and 201313003).

References

S. Wen, Z. Zeng, T. Huang, and Y. Zhang, “Exponential adaptive lag synchronization of memristive neural networks via fuzzy method and applications in pseudo random number generators,” IEEE Transactions on Fuzzy Systems, 2013.

J. Cao, G. Chen, and P. Li, “Global synchronization in an array of delayed neural networks with hybrid coupling,” IEEE Transactions on Systems, Man, and Cybernetics B: Cybernetics, vol. 38, no. 2, pp. 488–498, 2008.

J. Cao and M. Xiao, “Stability and Hopf bifurcation in a simplified BAM neural network with two time delays,” IEEE Transactions on Neural Networks, vol. 18, no. 2, pp. 416–430, 2007.

S. Wen, G. Bao, Z. Zeng, Y. Chen, and T. Huang, “Global exponential synchronization of memristor-based recurrent networks with time-varying delays,” Neural Networks, vol. 48, pp. 195–203, 2013.

S. Wen, Z. Zeng, and T. Huang, “Associative learning of integrate-and-fire neurons with memristor-based synapses,” Neural Processing Letters, vol. 38, no. 1, pp. 69–80, 2013.

G. Wallis, “Stability criteria for unsupervised temporal association networks,” IEEE Transactions on Neural Networks, vol. 16, no. 2, pp. 301–311, 2005.

C. Wang and D. J. Hill, “Deterministic learning and rapid dynamical pattern recognition,” IEEE Transactions on Neural Networks, vol. 18, no. 3, pp. 617–630, 2007.

X. Liao and Y. Liao, “Stability of Hopfield-type neural networks II,” Science in China A, vol. 40, no. 8, pp. 813–816, 1997.

K. Gopalsamy and X. Z. He, “Stability in asymmetric Hopfield nets with transmission delays,” Physica D: Nonlinear Phenomena, vol. 76, no. 4, pp. 344–358, 1994.

D. Xu, H. Zhao, and H. Zhu, “Global dynamics of hopfield neural networks involving variable delays,” Computers & Mathematics with Applications, vol. 42, no. 1-2, pp. 39–45, 2001.

Y. Wang, P. Lin, and L. Wang, “Exponential stability of reaction-diffusion high-order Markovian jump Hopfield neural networks with time-varying delays,” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1353–1361, 2012.

Y. Wang, C. Lu, G. Ji, and L. Wang, “Global exponential stability of high-order Hopfield-type neural networks with S-type distributed time delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 8, pp. 3319–3325, 2011.

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.

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.

L. Wang, Delayed Recurrent Neural Networks, Science Press, Beijing, China, 2008.

P. P. Civalleri, M. Gilli, and L. Pandolfi, “On stability of cellular neural networks with delay,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 40, no. 3, pp. 157–165, 1993.

S. Wen, Z. Zeng, and T. Huang, “H∞ filtering for neutral systems with mixed delays and multiplicative noises,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 59, no. 11, pp. 820–824, 2012.

S. Wen, Z. Zeng, and T. Huang, “Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays,” Neurocomputing, vol. 97, pp. 233–240, 2012.

S. Wen, Z. Zeng, T. Huang, and C. Li, “Passivity and passification of stochastic impulsive memristor-based piecewise linear system with mixed delays,” International Journal o f Robust and Nonlinear Control, 2013.

X. Liao, Y. Fu, J. Gao, and X. Zhao, “Stability of Hopfield neural networks with reaction-diffusion terms,” Acta Electronica Sinica, vol. 28, no. 1, pp. 78–80, 2000.

Q. Song, Z. Zhao, and Y. Li, “Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms,” Physics Letters A: General, Atomic and Solid State Physics, vol. 335, no. 2-3, pp. 213–225, 2005.

L. Wang and Y. Gao, “Global exponential robust stability of reaction-diffusion interval neural networks with time-varying delays,” Physics Letters. A, vol. 350, no. 5-6, pp. 342–348, 2006.

L. Wang and D. Xu, “Asymptotic behavior of a class of reaction-diffusion equations with delays,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 439–453, 2003.

S. Ruan and R. S. Filfil, “Dynamics of a two-neuron system with discrete and distributed delays,” Physica D: Nonlinear Phenomena, vol. 191, no. 3-4, pp. 323–342, 2004.

J. Cao, K. Yuan, and H. Li, “Global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays,” IEEE Transactions on Neural Networks, vol. 17, no. 6, pp. 1646–1651, 2006.

L. Wang and D. Xu, “Global asymptotic stability of bidirectional associative memory neural networks with -type distributed delays,” International Journal of Systems Science, vol. 33, no. 11, pp. 869–877, 2002.

P. Liu, F. Yi, Q. Guo, J. Yang, and W. Wu, “Analysis on global exponential robust stability of reaction-diffusion neural networks with S-type distributed delays,” Physica D: Nonlinear Phenomena, vol. 237, no. 4, pp. 475–485, 2008.

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.

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.

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.

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.

M. Brucoli, L. Carnimeo, and G. Grassi, “Associative memory design using discrete-time second-order neural networks with local interconnections,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 44, no. 2, pp. 153–158, 1997.

E. B. Kosmatopoulos and M. A. Christodoulou, “Structural properties of gradient recurrent high-order neural networks,” IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 42, no. 9, pp. 592–603, 1995.

A. Dembo, O. Farotimi, and T. Kailath, “High-order absolutely stable neural networks,” IEEE transactions on circuits and systems, vol. 38, no. 1, pp. 57–65, 1991.

S. Hu, Nonlinear Analysis and Methods, Huazhong University of Sci ence and Technology Press, Wuhan, China, 1993.

D. Guo, J. Sun, and Z. Liu, Functional Methods of Nonlinear Ordinary Differential Equations, Shandong Science Press, Jinan, China, 1995.

X. Liao, G. Chen, and E. N. Sanchez, “L{MI}-based approach for asymptotically stability analysis of delayed neural networks,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 49, no. 7, pp. 1033–1039, 2002.