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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 785141, 12 pages
http://dx.doi.org/10.1155/2013/785141
Research Article

Stability Analysis for Impulsive Stochastic Reaction-Diffusion Differential System and Its Application to Neural Networks

1Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
2Training Department, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China

Received 26 March 2013; Revised 28 June 2013; Accepted 12 July 2013

Academic Editor: Debasish Roy

Copyright © 2013 Yanke Du 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.

Linked References

  1. J. O. Alzabut and T. Abdeljawad, “On existence of a globally attractive periodic solution of impulsive delay logarithmic population model,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 463–469, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. C. Li, J. Sun, and R. Sun, “Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects,” Journal of the Franklin Institute, vol. 347, no. 7, pp. 1186–1198, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. X. Li and X. Fu, “Synchronization of chaotic delayed neural networks with impulsive and stochastic perturbations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 885–894, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. L. Pan and J. Cao, “Exponential stability of impulsive stochastic functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 382, no. 2, pp. 672–685, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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 Zentralblatt MATH · View at MathSciNet
  6. Q. J. Wu, J. Zhou, and L. Xiang, “Global exponential stability of impulsive differential equations with any time delays,” Applied Mathematics Letters, vol. 23, no. 2, pp. 143–147, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. X. Fu and X. Li, “LMI conditions for stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 435–454, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. C. Li, J. Shi, and J. Sun, “Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 10, pp. 3099–3111, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Q. Gan, “Exponential synchronization of stochastic Cohen-Grossberg neural networks with mixed time-varying and reaction-diffusion via periodially intermittent control,” Neural Networks, vol. 31, no. 1, pp. 12–21, 2012. View at Publisher · View at Google Scholar
  10. C. Hu, H. Jiang, and Z. Teng, “Impulsive control and synchronization for delayed neural networks with reaction-diffusion terms,” IEEE Transactions on Neural Networks, vol. 21, no. 1, pp. 67–81, 2010. View at Publisher · View at Google Scholar
  11. X. Xu, J. Zhang, and W. Zhang, “Mean square exponential stability of stochastic neural networks with reaction-diffusion terms and delays,” Applied Mathematics Letters, vol. 24, no. 1, pp. 5–11, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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
  13. Q. Song and Z. Wang, “Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays,” Physica A, vol. 387, no. 13, pp. 3314–3326, 2008. View at Publisher · View at Google Scholar
  14. I. M. Stamova, R. Ilarionov, and R. Vaneva, “Impulsive control for a class of neural networks with bounded and unbounded delays,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 285–290, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Y. Zhang and Q. G. Wang, “Stationary oscillation for high-order Hopfield neural networks with time delays and impulses,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 473–477, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Q. Song and J. Cao, “Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays,” Journal of Computational and Applied Mathematics, vol. 197, no. 1, pp. 188–203, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. 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 Zentralblatt MATH · View at MathSciNet
  18. K. Li, “Stability analysis for impulsive Cohen-Grossberg neural networks with time-varying delays and distributed delays,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 2784–2798, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. 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 Zentralblatt MATH · View at MathSciNet
  20. 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 Zentralblatt MATH · View at MathSciNet
  21. C. Huang, X. Yang, Y. He, and L. Huang, “Stability of stochastic reaction-diffusion recurrent neural networks with unbounded distributed delays,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 570295, 16 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Z. Li and R. Xu, “Global asymptotic stability of stochastic reaction-diffusion neural networks with time delays in the leakage terms,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1681–1689, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. X. Lou, B. Cui, and W. Wu, “On global exponential stability and existence of periodic solutions for BAM neural networks with distributed delays and reaction-diffusion terms,” Chaos, Solitons & Fractals, vol. 36, no. 4, pp. 1044–1054, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Q. Song and Z. Wang, “Dynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coefficients and delays,” Applied Mathematical Modelling, vol. 33, no. 9, pp. 3533–3545, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Z. Yang and D. Xu, “Global dynamics for non-autonomous reaction-diffusion neural networks with time-varying delays,” Theoretical Computer Science, vol. 403, no. 1, pp. 3–10, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. P. Balasubramaniam and C. Vidhya, “Global asymptotic stability of stochastic BAM neural networks with distributed delays and reaction-diffusion terms,” Journal of Computational and Applied Mathematics, vol. 234, no. 12, pp. 3458–3466, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. C. Wang, Y. Kao, and G. Yang, “Exponential stability of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks with mixed delays,” Neurocomputing, vol. 89, no. 1, pp. 55–63, 2012. View at Publisher · View at Google Scholar
  28. T. Lv and P. Yan, “Dynamical behaviors of reaction-diffusion fuzzy neural networks with mixed delays and general boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 993–1001, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet