## Journal of Applied Mathematics

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

#### 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.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 705496, 8 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305858

Digital Object Identifier
doi:10.1155/2014/705496

Mathematical Reviews number (MathSciNet)
MR3228139

#### Citation

Zheng, Haiyong; Wu, Bin; Wei, Tengda; Wang, Linshan; Wang, Yangfan. Global Exponential Robust Stability of High-Order Hopfield Neural Networks with S-Type Distributed Time Delays. J. Appl. Math. 2014 (2014), Article ID 705496, 8 pages. doi:10.1155/2014/705496. https://projecteuclid.org/euclid.jam/1425305858

#### 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$\infty$ 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 $S$-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, “LMI-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. \endinput