Skip to main content

Suppression of Functional System with Markovian Switching

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7664))

Abstract

In this paper, we will investigate a class of functional system whose coefficient satisfies the local Lipschitz condition and the one-sided polynomial growth condition under Markovian switching. We introduce two appropriate intensity Brownian noise to perturb the system so as to suppress its potential explosion.

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   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Mao, X., Marion, G., Renshaw, E.: Environmental noise suppresses explosion in population dynamics. Stochastic Process Appl. 97, 96–110 (2002)

    Article  MathSciNet  Google Scholar 

  2. Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)

    MathSciNet  MATH  Google Scholar 

  3. Mao, X., Sabanis, S., Renshaw, E.: Asymptotic behaviour of the stochastic Lotka-Volterra model. J. Math. Anal. Appl. 287, 141–156 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Song, Y., Baker, C.T.H.: Qualitative behaviour of numerical approximations to Volterra integro-differential equations. J. Comput. Appl. Math. 172, 101–115 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Wu, F., Hu, S.: Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-types ystem. J. Math. Anal. Appl. 364, 104–118 (2012)

    Article  MathSciNet  Google Scholar 

  6. Xu, Y., Wu, F., Tan, Y.: Stochastic Lotka-Volterra system with infinite delay. J. Comput. Appl. Math. 232, 472–480 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wu, F., Mao, X., Hu, S.: Stochastic suppression and stabilization of functional differential equations. Systems & Control Letters 59, 745–753 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wu, F., Hu, S.: Stochastic functional Kolmogorov-type population dynamics. J. Math. Anal. Appl. 347, 534–548 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Shen, Y., Wang, J.: Almost sure exponential stability of recurrent neural networks with Markovian switching. IEEE Trans. Neural Netw. 20, 840–855 (2009)

    Article  Google Scholar 

  10. Bolzern, P., Colaneri, P., De Nicolao, G.: On almost sure stability of continuous-time Markov jump linear systems. Automatica 42, 983–988 (2006)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, L., Mao, X., Deng, F.: Stability of hybrid stochastic retarded systems. IEEE Trans. Circuits Syst. I. Regul. Pap. 55, 3413–3420 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  15. Mao, X., Yin, G.G., Yuan, C.: Stabilization and destabilization of hybrid systems of stochastic differential equations. Automatica 43, 264–273 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yuan, C., Lygeros, J.: On the exponential stability of switching diffusion processes. IEEE Trans. Automat. Control 50, 1422–1426 (2005)

    Article  MathSciNet  Google Scholar 

  17. Yuan, C., Lygeros, J.: Stabilization of a class of stochastic differential equations with Markovian switching. Systems Control Lett. 54, 819–833 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yuan, C., Yin, G.: Stability of hybrid stochasti delay systems whose discrete components have a large state space: A two-time-scale approach. J. Math. Anal. Appl. 368, 103–119 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yang, Y., Li, J., Chen, G.: Finite-time stability and stabilization of nonlinear stochastic hybrid systems. J. Math. Anal. Appl. 356, 338–345 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhao, L., Xi, F.: Explicit conditions for asymptotic stability of stochastic Linard-type equations with Markovian switching. J. Math. Anal. Appl. 348, 267–273 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, L., Shen, Y.: The asymptotic stability and exponential stability of nonlinear stochastic differential systems with Markovian switching and with polynomial growth. J. Math. Anal. Appl. 391, 323–334 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu, F., Hu, S.: Suppression and stabilisation of noise. Inter. J. Contr. 82, 2150–2157 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Feng, L., Shen, Y., Li, Z. (2012). Suppression of Functional System with Markovian Switching. In: Huang, T., Zeng, Z., Li, C., Leung, C.S. (eds) Neural Information Processing. ICONIP 2012. Lecture Notes in Computer Science, vol 7664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34481-7_56

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-34481-7_56

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-34480-0

  • Online ISBN: 978-3-642-34481-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics