About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 131836, 10 pages
http://dx.doi.org/10.1155/2013/131836
Research Article

Exponential Stability of Impulsive Delayed Reaction-Diffusion Cellular Neural Networks via Poincaré Integral Inequality

School of Economics & Management, Nanjing University of Information Science & Technology, Nanjing 210044, China

Received 15 November 2012; Accepted 8 February 2013

Academic Editor: Qi Luo

Copyright © 2013 Xianghong Lai and Tianxiang Yao. 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.

Linked References

  1. L. O. Chua and L. Yang, “Cellular neural networks: theory,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1257–1272, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  2. L. O. Chua and L. Yang, “Cellular neural networks: applications,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1273–1290, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  3. J. Cao, “New results concerning exponential stability and periodic solutions of delayed cellular neural networks,” Physics Letters A, vol. 307, no. 2-3, pp. 136–147, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J. Cao, “On stability of cellular neural networks with delay,” IEEE Transactions on Circuits and Systems I, vol. 40, pp. 157–165, 1993.
  5. P. P. Civalleri and M. Gilli, “A set of stability criteria for delayed cellular neural networks,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 48, no. 4, pp. 494–498, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  6. 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, pp. 3088–3092, 1984.
  7. J. Yan and J. Shen, “Impulsive stabilization of functional-differential equations by Lyapunov-Razumikhin functions,” Nonlinear Analysis. Theory, Methods & Applicationss, vol. 37, no. 2, pp. 245–255, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  8. X. Z. Liu and Q. Wang, “Impulsive stabilization of high-order Hopfield-type neural networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 19, pp. 71–79, 2008.
  9. X. Z. 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 MathSciNet
  10. S. Arik and V. Tavsanoglu, “On the global asymptotic stability of delayed cellular neural networks,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 47, no. 4, pp. 571–574, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  11. L. O. Chua and T. Roska, “Stability of a class of nonreciprocal cellular neural networks,” IEEE Transactions on Circuits and Systems I, vol. 37, pp. 1520–1527, 1990.
  12. Z. H. Guan and G. Chen, “On delayed impulsive Hopfield neural networks,” Neural Network, vol. 12, pp. 273–280, 1999.
  13. Q. Zhang, X. Wei, and J. Xu, “On global exponential stability of delayed cellular neural networks with time-varying delays,” Applied Mathematics and Computation, vol. 162, no. 2, pp. 679–686, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  14. D. D. Baĭnov and P. S. Simeonov, Systems with Impulse Effect, Ellis Horwood, Chichester, UK, 1989. View at MathSciNet
  15. I. M. Stamova, Stability Analysis of Impulsive Functional Differential Equations, Walter de Gruyter, Berlin, Germany, 2009.
  16. M. A. Arbib, Brains, Machines, and Mathematics, Springer, New York, NY, USA, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Englewood Cliffs, NJ, USA, 1998.
  18. 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 MathSciNet
  19. G. T. Stamov, “Almost periodic models of impulsive Hopfield neural networks,” Journal of Mathematics of Kyoto University, vol. 49, no. 1, pp. 57–67, 2009. View at MathSciNet
  20. G. T. Stamov and I. M. Stamova, “Almost periodic solutions for impulsive neural networks with delay,” Applied Mathematical Modelling, vol. 31, pp. 1263–1270, 2007.
  21. S. Ahmad and I. M. Stamova, “Global exponential stability for impulsive cellular neural networks with time-varying delays,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 3, pp. 786–795, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  22. X. Liu and K. L. Teo, “Exponential stability of impulsive high-order Hopfield-type neural networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 16, pp. 1329–1339, 2005.
  23. 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
  24. Y. Zhang and M. Zhang, “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.
  25. 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
  26. J. Pan and S. Zhong, “Dynamical behaviors of impulsive reaction-diffusion Cohen-Grossberg neural network with delay,” Neurocomputing, vol. 73, pp. 1344–1351, 2010.
  27. K. Li and Q. Song, “Exponential stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms,” Neurocomputing, vol. 72, pp. 231–240, 2008.
  28. J. Qiu, “Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms,” Neurocomputing, vol. 70, pp. 1102–1108, 2007.
  29. 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
  30. 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
  31. 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
  32. 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
  33. 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
  34. 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
  35. Y. Zhang and Q. Luo, “Global exponential stability of impulsive delayed reaction-diffusion neural networks via Hardy-Poincarè Inequality,” Neurocomputing, vol. 83, pp. 198–204, 2012.
  36. Y. Zhang, “Asymptotic stability of impulsive reaction-diffusion cellular neural networks with time-varying delays,” Journal of Applied Mathematics, vol. 2012, Article ID 501891, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  37. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. View at MathSciNet
  38. W.-S. Cheung, “Some new Poincaré-type inequalities,” Bulletin of the Australian Mathematical Society, vol. 63, no. 2, pp. 321–327, 2001. View at Publisher · View at Google Scholar · View at MathSciNet