Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014, Article ID 187037, 11 pages
http://dx.doi.org/10.1155/2014/187037
Research Article

Exponential Stability of Stochastic Differential Equation with Mixed Delay

School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 24 November 2013; Revised 21 January 2014; Accepted 21 January 2014; Published 4 March 2014

Academic Editor: Oluwole Daniel Makinde

Copyright © 2014 Wenli Zhu 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. S. Cong, “On exponential stability conditions of linear neutral stochastic differential systems with time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 23, no. 11, pp. 1265–1276, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. Li, Z. Lin, and B. Zhou, “An analysis of the exponential stability of linear stochastic neutral delay systems,” International Journal of Robust and Nonlinear Control, 2013. View at Publisher · View at Google Scholar
  3. X. Mao, “A note on the LaSalle-type theorems for stochastic differential delay equations,” Journal of Mathematical Analysis and Applications, vol. 268, no. 1, pp. 125–142, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. W.-H. Chen, Z.-H. Guan, and X. Lu, “Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach,” Systems & Control Letters, vol. 54, no. 6, pp. 547–555, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Y. Sun and J. Cao, “pth moment exponential stability of stochastic recurrent neural networks with time-varying delays,” Nonlinear Analysis: Real World Applications, vol. 8, no. 4, pp. 1171–1185, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. X. Meng, M. Tian, and S. Hu, “Stability analysis of stochastic recurrent neural networks with unbounded time-varying delays,” Neurocomputing, vol. 74, no. 6, pp. 949–953, 2011. View at Publisher · View at Google Scholar · View at Scopus
  7. Y. Hu, F. Wu, and C. Huang, “Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay,” Automatica, vol. 45, no. 11, pp. 2577–2584, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. X. Mao, “Robustness of exponential stability of stochastic differential delay equations,” IEEE Transactions on Automatic Control, vol. 41, no. 3, pp. 442–447, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. X. Mao and A. Shah, “Exponential stability of stochastic differential delay equations,” Stochastics and Stochastics Reports, vol. 60, no. 1-2, pp. 135–153, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. W. Zhu and J. Hu, “Stability analysis of stochastic delayed cellular neural networks by LMI approach,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 171–174, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. W. Zhu and J. Hu, “Exponential stability of stochastic recurrent neural network with time delays,” International Journal of Computational Intelligence Research, vol. 2, no. 1, pp. 52–54, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  12. S. Xie and L. Xie, “Stabilization of a class of uncertain large-scale stochastic systems with time delays,” Automatica, vol. 36, no. 1, pp. 161–167, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. W. Zhu, X. Ruan, Y. Qin, and J. Zhuang, “Exponential stability of stochastic nonlinear dynamical price system with delay,” Mathematical Problems in Engineering, vol. 2013, Article ID 168169, 9 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  14. B. Song, J. H. Park, Z.-G. Wu, and X. Li, “New results on delay-dependent stability analysis and stabilization for stochastic time-delay systems,” International Journal of Robust and Nonlinear, 2013. View at Publisher · View at Google Scholar
  15. Y. Hu and F. Wu, “Stochastic Kolmogorov-type population dynamics with infinite distributed delays,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1407–1428, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. W. Wu, B. T. Cui, and X. Y. Lou, “Global exponential stability of Cohen-Grossberg neural networks with distributed delays,” Mathematical and Computer Modelling, vol. 47, no. 9-10, pp. 868–873, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. L. Yin, Y. Chen, and Y. Zhao, “Global exponential stability for a class of neural networks with continuously distributed delays,” Advances in Dynamical Systems and Applications, vol. 4, no. 2, pp. 221–229, 2009. View at Google Scholar · View at MathSciNet
  18. J. Zhou, S. Li, and Z. Yang, “Global exponential stability of Hopfield neural networks with distributed delays,” Applied Mathematical Modelling, vol. 33, no. 3, pp. 1513–1520, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Q. Zhu and B. Song, “Exponential stability of impulsive nonlinear stochastic differential equations with mixed delays,” Nonlinear Analysis. Real World Applications, vol. 12, no. 5, pp. 2851–2860, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. F. Deng, M. Hua, X. Liu, Y. Peng, and J. Fei, “Robust delay-dependent exponential stability for uncertain stochastic neural networks with mixed delays,” Neurocomputing, vol. 74, no. 10, pp. 1503–1509, 2011. View at Publisher · View at Google Scholar · View at Scopus
  21. L. Xu and D. Xu, “Exponential p-stability of impulsive stochastic neural networks with mixed delays,” Chaos, Solitons & Fractals, vol. 41, no. 1, pp. 263–272, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  22. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  23. L. Chen and Z. Wu, “Maximum principle for the stochastic optimal control problem with delay and application,” Automatica, vol. 46, no. 6, pp. 1074–1080, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. G. Wang, Z. Wu, and J. Xiong, “Maximum principles for forward-backward stochastic control systems with correlated state and observation noises,” SIAM Journal on Control and Optimization, vol. 51, no. 1, pp. 491–524, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet