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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 183729, 21 pages
http://dx.doi.org/10.1155/2012/183729
Research Article

Almost Sure Stability and Stabilization for Hybrid Stochastic Systems with Time-Varying Delays

1School of Information Science and Technology, Donghua University, Shanghai 200051, China
2College of Information Science and Engineering, Shanxi Agricultural University, Taigu 030801, China
3Department of Applied Mathematics, Donghua University, Shanghai 200051, China

Received 21 June 2012; Accepted 1 August 2012

Academic Editor: Bo Shen

Copyright © 2012 Hua Yang 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. Hu, Z. Wang, H. Gao, and L. K. Stergioulas, “Robust sliding mode control for discrete stochastic systems with mixed time delays, randomly occurring uncertainties, and randomly occurring nonlinearities,” IEEE Transactions on Industrial Electronics, vol. 59, no. 7, pp. 3008–3015, 2012.
  2. L. Hu and X. Mao, “Almost sure exponential stabilisation of stochastic systems by state-feedback control,” Automatica, vol. 44, no. 2, pp. 465–471, 2008. View at Publisher · View at Google Scholar · View at Scopus
  3. X. Li and C. E. De Souza, “Criteria for robust stability and stabilization of uncertain linear systems with state delay,” Automatica, vol. 33, no. 9, pp. 1657–1662, 1997. View at Scopus
  4. X. Li and C. E. De Souza, “Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach,” IEEE Transactions on Automatic Control, vol. 42, no. 8, pp. 1144–1148, 1997. View at Scopus
  5. S. Ma, Z. Cheng, and C. Zhang, “Delay-dependent robust stability and stabilisation for uncertain discrete singular systems with time-varying delays,” Control Theory & Applications, vol. 1, no. 4, pp. 1086–1095, 2007. View at Publisher · View at Google Scholar · View at Scopus
  6. B. Shen, Z. Wang, Y. S. Hung, and G. Chesi, “Distributed H filtering for polynomial nonlinear stochastic systems in sensor networks,” IEEE Transactions on Industrial Electronics, vol. 58, no. 5, pp. 1971–1979, 2011. View at Publisher · View at Google Scholar · View at Scopus
  7. D. Yue and Q. L. Han, “Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching,” IEEE Transactions on Automatic Control, vol. 50, no. 2, pp. 217–222, 2005. View at Publisher · View at Google Scholar · View at Scopus
  8. L. Liu, Y. Shen, and F. Jiang, “The almost sure asymptotic stability and th moment asymptotic stability of nonlinear stochastic differential systems with polynomial growth,” IEEE Transactions on Automatic Control, vol. 56, no. 8, pp. 1985–1990, 2011.
  9. G. Wei, Z. Wang, H. Shu, and J. Fang, “Robust H control of stochastic time-delay jumping systems with nonlinear disturbances,” Optimal Control Applications and Methods, vol. 27, no. 5, pp. 255–271, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. G. Wei, Z. Wang, and H. Shu, “Nonlinear H control of stochastic time-delay systems with Markovian switching,” Chaos, Solitons and Fractals, vol. 35, no. 3, pp. 442–451, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. X. Mao, “Stochastic versions of the LaSalle theorem,” Journal of Differential Equations, vol. 153, no. 1, pp. 175–195, 1999. View at Scopus
  12. X. Mao, Y. Shen, and C. Yuan, “Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching,” Stochastic Processes and their Applications, vol. 118, no. 8, pp. 1385–1406, 2008. View at Publisher · View at Google Scholar · View at Scopus
  13. B. Bercu, F. Dufour, and G. G. Yin, “Almost sure stabilization for feedback controls of regime-switching linear systems with a hidden Markov Chain,” IEEE Transactions on Automatic Control, vol. 54, no. 9, pp. 2114–2125, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. Z. Lin, Y. Lin, and W. Zhang, “H filtering for non-linear stochastic Markovian jump systems,” Control Theory & Applications, vol. 4, no. 12, pp. 2743–2756, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. L. Wu, D. W. C. Ho, and C. W. Li, “Stabilisation and performance synthesis for switched stochastic systems,” Control Theory & Applications, vol. 4, no. 10, pp. 1877–1888, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. L. Wu, D. W. C. Ho, and C. W. Li, “Sliding mode control of switched hybrid systems with stochastic perturbation,” Systems and Control Letters, vol. 60, no. 8, pp. 531–539, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. J. Yao, F. Lin, and B. Liu, “H control for stochastic stability and disturbance attenuation in a class of networked hybrid systems,” Control Theory & Applications, vol. 5, no. 15, pp. 1698–1708, 2011.
  18. N. Zeng, Z. Wang, Y. Li, M. Du, and X. Liu, “a Hybrid EKF and switching PSO algorithm for joint state and parameter estimation of lateral flow immunoassay models,” IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 9, no. 2, pp. 321–329, 2012.
  19. X. Mao, “Stability of stochastic differential equations with Markovian switching,” Stochastic Processes and their Applications, vol. 79, no. 1, pp. 45–67, 1999. View at Scopus
  20. C. Yuan and J. Lygeros, “Stabilization of a class of stochastic differential equations with Markovian switching,” Systems and Control Letters, vol. 54, no. 9, pp. 819–833, 2005. View at Publisher · View at Google Scholar · View at Scopus
  21. X. Mao, J. Lam, and L. Huang, “Stabilisation of hybrid stochastic differential equations by delay feedback control,” Systems and Control Letters, vol. 57, no. 11, pp. 927–935, 2008. View at Publisher · View at Google Scholar · View at Scopus
  22. Z. Wang, H. Qiao, and K. J. Burnham, “On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters,” IEEE Transactions on Automatic Control, vol. 47, no. 4, pp. 640–646, 2002. View at Publisher · View at Google Scholar · View at Scopus
  23. Z. Wang, Y. Liu, and X. Liu, “Exponential stabilization of a class of stochastic system with markovian jump parameters and mode-dependent mixed time-delays,” IEEE Transactions on Automatic Control, vol. 55, no. 7, pp. 1656–1662, 2010. View at Publisher · View at Google Scholar · View at Scopus
  24. L. Huang and X. Mao, “On almost sure stability of hybrid stochastic systems with mode-dependent interval delays,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1946–1952, 2010. View at Publisher · View at Google Scholar · View at Scopus
  25. C. Yuan and X. Mao, “Robust stability and controllability of stochastic differential delay equations with Markovian switching,” Automatica, vol. 40, no. 3, pp. 343–354, 2004. View at Publisher · View at Google Scholar · View at Scopus
  26. 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 Scopus
  27. R. S. Lipster and A. N. Shiryayev, Theory of Martingales, Kluwer Academic, Dodrecht, The Netherlands, 1989.