Skip to main content

Practical Stability

  • Chapter
  • First Online:
  • 576 Accesses

Abstract

This chapter investigates stochastic systems with Markovian jump parameters and time-varying delays in terms of their practical stability in probability and in the pth mean, and the practical controllability in probability and in the pth mean, respectively.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Delfour, M.C., Mitter, S.K.: Hereditary differential systems with constant delays. II.A class of affine systems and the adjoint problem. J. Differ. Equat. 18(75), 18C28 (1975)

    Google Scholar 

  2. Eller, D.H., Aggarwal, J.K., Banks, H.T.: Optimal control of linear time-delay systems. IEEE Trans. Autom. Control 14(2), 301–305 (1969)

    MathSciNet  Google Scholar 

  3. Ji, Y., Chizeck, H.J.: Controllability, stability, and continuous-time Markovian jump linear quadratic control. IEEE Trans. Autom. Control 35(7), 777–788 (1990)

    Article  MATH  Google Scholar 

  4. Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1990)

    Book  MATH  Google Scholar 

  5. LaSalle, J.P., Lefschetz, S.: Stability by Liapunov’s Direct Method with Applications. Academic Press, New York (1961)

    MATH  Google Scholar 

  6. Luo, J.W., Zou, J.Z., Hou, Z.T.: Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching. Sci. China 46(1), 129–138 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mahmoud, S.M., Shi, Y., Nounou, H.N.: Resilient observer-based control of uncertain time-delay systems. Int. J. Innov. Comput. Inf. Control 3(2), 407–418 (2007)

    Google Scholar 

  8. Malek-Zavarei, M., Jamshidi, M.: Time-Delay Systems: Analysis Optimization and Applications. Elsevier Science, New York (1987)

    MATH  Google Scholar 

  9. Mao, X.: Stability of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 79(1), 45–67 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mao, X.: A note on the Lasalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268(1), 125–142 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mariton, M.: Jump Linear Systems in Automatic Control. Marcel Dekker, New York (1990)

    Google Scholar 

  12. Mariton, M., Bertrand, P.: Output feedback for a class of linear systems with stochastic jump parameters. IEEE Trans. Autom. Control 30(9), 898–900 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Martynyuk, A.: Methods and problems of practical stability of motion theory. Zagadnienia Drgan Nieliniowych 22, 9–46 (1984)

    Google Scholar 

  14. Martynyuk, A., Sun, Z.Q.: Practical Stability and Applications (in Chinese). Science Press, Beijing (2004)

    Google Scholar 

  15. Ross, D.W., Flugge-Lotz, I.: An optimal control problem for systems with differential-difference equation dynamics. SIAM J. Control 7(4), 609–623 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sathananthan, S., Keel, L.H.: Optimal practical stabilization and controllability of systems with Markovian jumps. Nonlinear Anal. 54(6), 1011–1027 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shi, P., Boukas, E.K.: \(H_{\infty }\) control for Markovian jump linear systems with parametric uncertainty. J. Optim. Theory Appl. 95(1), 75–99 (1997)

    Google Scholar 

  18. Shi, P., Boukas, E.K., Agarwal, R.K.: Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay. IEEE Trans. Autom. Control 44(11), 2139–2144 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, Z.D., Lam, J., Liu, X.H.: Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances. IEEE Trans. Circuits Syst. II Express Briefs 51(5), 262–268 (2004)

    Article  Google Scholar 

  20. Wang, Z.D., Qiao, H., Burnham, K.J.: On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters. IEEE Trans. Autom. Control 47(4), 640–646 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. You, B.L.: Complemental Tutorial for Ordinary Differential Equations(in Chinese). Higher Education Press, Beijing (1981)

    Google Scholar 

  22. Yuan, C., Mao, X.: Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 103(2), 277–291 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yu Kang .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd and Science Press, Beijing

About this chapter

Cite this chapter

Kang, Y., Zhao, YB., Zhao, P. (2018). Practical Stability. In: Stability Analysis of Markovian Jump Systems. Springer, Singapore. https://doi.org/10.1007/978-981-10-3860-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-3860-0_5

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-3859-4

  • Online ISBN: 978-981-10-3860-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics