Skip to main content
Log in

Exponential estimates for stochastic delay equations with norm-bounded uncertainties

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper presents the sufficient conditions for the exponential stability of linear or semilinear stochastic delay equations with time-varying norm bounded parameter uncertainties. Exponential estimates for the solutions are also obtained by using a modified Lyapunov-Krasovski functional. These conditions can be tested numerically using interior point algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. S. Razumikhin, On stability of systems with a delay, Prikl. Mat. Mekh, 1956, 20(4): 500–512.

    Google Scholar 

  2. N. N. Krasovskii, On the application of the second method of Lyapunov for equations with time delays, Prikl. Mat. Mekh, 1956, 20(3): 315–327.

    MathSciNet  Google Scholar 

  3. X. R. Mao, Stochastic Differential Equations and Their Applications, Chichester, Horwood, 1997.

  4. X. R. Mao, N. Koroleva, and A. Rodkina, Robust stability of uncertain stochastic differential delay equations, Sys. Control Lett., 1998, 35(5): 325–336.

    Article  MATH  MathSciNet  Google Scholar 

  5. V. L. Kharitonov and A. P. Zhabko, Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems, Automatica, 2003, 39(1): 15–20.

    Article  MATH  MathSciNet  Google Scholar 

  6. V. L. Kharitonov and D. Hinrichsen, Exponential estimates for time delay systems, Sys. Control Lett., 2004, 53(5): 395–405.

    Article  MATH  MathSciNet  Google Scholar 

  7. Y. Wang, L. Xie, and C. E. De Souza, Robust control of a class of uncertain nonlinear system, Sys. Control Lett., 1992, 19(2): 139–149.

    Article  Google Scholar 

  8. S. I. Niculescu, H memoryless control with an α-stability constraint for time-delay systems: An LMI approach, IEEE tranc. Auto. Control, 1998, 43(5): 739–743.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Mondié and V. L. Kharitonov, Exponential estimates for retarded time-delay systems: An LMI approach, IEEE trans. Autom. control, 2005, 50(2): 268–272.

    Article  Google Scholar 

  10. X. R. Mao, Robustness of exponential stability of stochastic differential delay equations, IEEE Trans. Automat. Control, 1996, 41(3): 442–447.

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Chen, Z. Guan, and X. Lu, Delay-dependent robust stabilization and H-infinity- control of uncertain stochastic systems with time-varying delay, IMA J. Math. Control Inform., 2004, 21(3): 345–358.

    Article  MathSciNet  Google Scholar 

  12. W. Chen, Z. Guan, and X. Lu, Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: An LMI approach, Sys. Control Lett., 2005, 54(6): 547–555.

    Article  MATH  MathSciNet  Google Scholar 

  13. S. Xu, J. Lam, X. Mao, and Y. Zou, A new LMI condition for delay-dependent robust stability of stochastic time-delay systems, Asian Journal of Control, 2005, 7(4): 419–423.

    MathSciNet  Google Scholar 

  14. D. Yue, Delay-dependent robust stability of stochastic uncertain systems with time delay and Markovian jump parameters, Circuits Systems and Signal Processing, 2003, 22(4): 351–365.

    Article  MATH  MathSciNet  Google Scholar 

  15. D. Yue and Q. Han, Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching, IEEE Trans. Automat. Control, 2005, 50(2): 217–222.

    Article  MathSciNet  Google Scholar 

  16. P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali, LMI Control Toolbox User’s Guide, the math works, Natick, MA, 1995.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hongchu Wang.

Additional information

This work is Supported by the National Natural Science Foundation of China under Grant Nos. 10801056 and 10826095.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, H., Hu, S. Exponential estimates for stochastic delay equations with norm-bounded uncertainties. J Syst Sci Complex 22, 324–332 (2009). https://doi.org/10.1007/s11424-009-9167-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-009-9167-5

Key words

Navigation