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Robust Exponential Stability of Stochastically Nonlinear Jump Systems with Mixed Time Delays

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Abstract

In this paper, the problem of robust exponential stability is investigated for a class of stochastically nonlinear jump systems with mixed time delays. By applying the Lyapunov–Krasovskii functional and stochastic analysis theory as well as matrix inequality technique, some novel sufficient conditions are derived to ensure the exponential stability of the trivial solution in the mean square. Time delays proposed in this paper comprise both time-varying and distributed delays. Moreover, the derivatives of time-varying delays are not necessarily less than 1. The results obtained in this paper extend and improve those given in the literature. Finally, two numerical examples and their simulations are provided to show the effectiveness of the obtained results.

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References

  1. Krasovskii, N.N., Lidskii, E.A.: Analysis design of controller in systems with random attributes. Part 1. Autom. Remote Control 22, 1021–1025 (1961)

    MathSciNet  Google Scholar 

  2. Krasovskii, N.N., Lidskii, E.A.: Analysis design of controller in systems with random attributes. Part 2. Autom. Remote Control 22, 1141–1146 (1961)

    MathSciNet  Google Scholar 

  3. Benjelloun, K., Boukas, E.K., Costa, O.L.V.: \(\mathcal{H}_{\infty}\) control for linear time-delay systems with Markovian jumping parameters. J. Optim. Theory Appl. 105, 73–95 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boukas, E.K., Yang, H.: Exponential stabilizability of stochastic systems with Markovian jumping parameters. Automatica 35, 1437–1441 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boukas, E.K., Liu, Z.: Robust stability and stability of Markov jump linear uncertain systems with mode dependent time delays. J. Optim. Theory Appl. 109, 587–600 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boukas, E.K.: Stabilization of stochastic singular nonlinear hybrid systems. Nonlinear Anal. 64, 217–228 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cao, Y., Lam, J., Hu, L.: Delay-dependent stochastic stability and \(\mathcal {H}_{\infty}\) analysis for time-delay systems with Markovian jumping parameters. J. Franklin Inst. 340, 423–434 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. de Farias, D.P., Geromel, J.C., do Val, J.B.R., Costa, O.L.V.: Output feedback control of Markov jump linear systems in continuous-time. IEEE Trans. Autom. Control 45, 944–949 (2000)

    Article  MATH  Google Scholar 

  9. Fei, Z., Gao, H., Shi, P.: New results on stabilization of Markovian jump systems with time delay. Automatica 45, 2300–2306 (2009)

    Article  MATH  Google Scholar 

  10. He, Y., Zhang, Y., Wu, M., She, J.: Improved exponential stability for stochastic Markovian jump systems with nonlinearity and time-varying delay. Int. J. Robust Nonlinear Control 20, 16–26 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mao, X., Yuan, C.: Stochastic Differential Delay Equations with Markovian Switching. Imperial College Press, London (2006)

    Google Scholar 

  12. Mariton, M., Bertrand, P.: Output feedback for a class of linear systems with stochastic jump parameters. IEEE Trans. Autom. Control 30, 898–900 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shi, P., Boukas, E.K.: \(\mathcal{H}_{\infty}\) control for Markovian jumping linear systems with parametric uncertainty. J. Optim. Theory Appl. 95, 75–99 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. de Souza, C.E.: Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems. IEEE Trans. Autom. Control 51, 836–841 (2006)

    Article  Google Scholar 

  15. Wang, Y., Zhang, H.: \(\mathcal{H}_{\infty}\) control for uncertain Markovian jump systems with mode dependent mixed delays. Prog. Nat. Sci. 18, 309–314 (2008)

    Article  MathSciNet  Google Scholar 

  16. Wang, Z., Qiao, H., Burnham, K.J.: On stabilization of bilinear uncertain time-delay systems with Markovian jumping parameters. IEEE Trans. Autom. Control 47, 640–646 (2002)

    Article  MathSciNet  Google Scholar 

  17. Wu, J., Chen, T., Wang, L.: Delay-dependent robust stability and \(\mathcal{H}_{\infty}\) control for jump linear systems with delays. Syst. Control Lett. 55, 939–948 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xia, Y., Boukas, E.K., Shi, P., Zhang, J.: Stability and stabilization of continuous-time singular hybrid systems. Automatica 45, 1504–1509 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Xu, S., Chen, T., Lam, J.: Robust \(\mathcal{H}_{\infty}\) filtering for uncertain Markovian jump systems with mode-dependent time delays. IEEE Trans. Autom. Control 48, 900–907 (2003)

    Article  MathSciNet  Google Scholar 

  20. Yue, D., Han, Q.: Delay-dependent exponential stability of stochastic systems with time-varying delay nonlinearity and Markovian switching. IEEE Trans. Autom. Control 50, 217–222 (2005)

    Article  MathSciNet  Google Scholar 

  21. He, S., Liu, F.: Robust peak-to-peak filtering for Markov jump systems. Signal Process. 90, 513–522 (2010)

    Article  MATH  Google Scholar 

  22. Shi, P., Mahmoud, M., Nguang, S.K., Ismail, A.: Robust filtering for jumping systems with mode-dependent delays. Signal Process. 86, 140–152 (2006)

    Article  MATH  Google Scholar 

  23. Boyd, S., Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  24. Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of 39th IEEE Conference on Decision and Control, Sydney, Australia, pp. 2805–2810 (2000)

    Google Scholar 

  25. Zhu, Q., Cao, J.: Adaptive synchronization under almost every initial data for stochastic neural networks with time-varying delays and distributed delays. Commun. Nonlinear Sci. Numer. Simul. 16, 2139–2159 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhu, Q., Cao, J.: pth moment exponential synchronization for stochastic delayed Cohen–Grossberg neural networks with Markovian switching. Nonlinear Dyn. 67, 829–845 (2012)

    Article  Google Scholar 

  27. Zhu, Q., Cao, J.: Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 41, 341–353 (2011)

    Google Scholar 

  28. Zhu, Q., Cao, J.: Robust exponential stability of Markovian jump impulsive stochastic Cohen–Grossberg neural networks with mixed time delays. IEEE Trans. Neural Netw. 21, 1314–1325 (2010)

    Article  Google Scholar 

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Acknowledgements

This work was jointly supported by the National Natural Science Foundation of China (10801056, 11171024), the Natural Science Foundation of Ningbo (2010A610094) and K.C. Wong Magna Fund in Ningbo University. The authors would like to thank the associate editor and anonymous referees for their helpful comments and valuable suggestions regarding this paper.

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Authors and Affiliations

  1. Department of Mathematics, Ningbo University, Ningbo, 315211, Zhejiang, China

    Quanxin Zhu

  2. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, Beijing, China

    Fubao Xi

  3. School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, Shandong, China

    Xiaodi Li

Authors
  1. Quanxin Zhu
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  2. Fubao Xi
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  3. Xiaodi Li
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Correspondence to Quanxin Zhu.

Additional information

Communicated by Negash G. Medhin.

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Zhu, Q., Xi, F. & Li, X. Robust Exponential Stability of Stochastically Nonlinear Jump Systems with Mixed Time Delays. J Optim Theory Appl 154, 154–174 (2012). https://doi.org/10.1007/s10957-012-9997-5

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  • DOI: https://doi.org/10.1007/s10957-012-9997-5

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