Skip to main content
SpringerLink
Log in

Exponential Stability in Mean Square of Stochastic Functional Differential Equations with Infinite Delay

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

A novel approach to the exponential stability in mean square of stochastic functional differential equations and neutral stochastic functional differential equations with infinite delay is presented. Consequently, some new criteria for the exponential stability in mean square of the considered equations are obtained. Lastly, some examples are investigated to illustrate the theory.

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

Access this article

Log in via an institution

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. Arnolda, L., Schmalfuss, B.: Lyapunov’s second method for random dynamical systems. J. Differ. Equ. 177, 235–265 (2001)

    Article  MathSciNet  Google Scholar 

  2. Hu, G., Wang, K.: Stability in distribution of neutral stochastic functional differential equations with Markovian switching. J. Math. Anal. Appl. 385, 757–769 (2012)

    Article  MathSciNet  Google Scholar 

  3. Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Academic Press, New York (1986)

    Google Scholar 

  4. Li, X.D., Song, S.J., Wu, J.H.: Exponential stability of nonlinear systems with delayed impulses and applications. IEEE Trans. Autom. Control 64, 4024–4034 (2019)

    Article  MathSciNet  Google Scholar 

  5. Liu, R., Mandrekar, V.: Stochastic semilinear evolution equations: Lyapunov function, stability and ultimate boundedness. J. Math. Anal. Appl. 212, 537–553 (1997)

    Article  MathSciNet  Google Scholar 

  6. Liu, B., Marquez, H.J.: Razumikhin-type stability theorems for discrete delay systems. Automatica 43, 1219–1225 (2007)

    Article  MathSciNet  Google Scholar 

  7. Liu, K., Xia, X.: On the exponential stability in mean square of neutral stochastic functional differential equations. Syst. Control Lett. 37, 207–215 (1999)

    Article  MathSciNet  Google Scholar 

  8. Mao, X.: Exponential Stability of Stochastic Differential Equations. Dekker, New York (1994)

    MATH  Google Scholar 

  9. Mao, X.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations. Stoch. Process. Appl. 65, 233–250 (1996)

    Article  MathSciNet  Google Scholar 

  10. Mao, X.: Razumikhin-type theorems on exponential stability of neutral stochastic differential equations. SIAM J. Math. Anal. 28, 389–401 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horvood, Chichester (2007)

    MATH  Google Scholar 

  13. Ngoc, P.H.A.: Novel criteria for exponential stability in mean square of stochastic functional differential equations. Proc. Am. Math. Soc. 148, 3427–3436 (2020)

    Article  MathSciNet  Google Scholar 

  14. Ngoc, P.H.A.: New criteria for mean square exponential stability of stochastic delay differential equations. Int. J. Control (2020). https://doi.org/10.1080/00207179.2020.1770334

    Article  Google Scholar 

  15. Ngoc, P.H.A., Hieu, L.T.: A novel approach to mean square exponential stability of stochastic delay differential equations. IEEE Trans. Autom. Control 66(5), 2351–2356 (2021). https://doi.org/10.1109/TAC.2020.3005587

    Article  MathSciNet  MATH  Google Scholar 

  16. Pavlović, G., Janković, S.: Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay. J. Comput. Appl. Math. 236, 1679–1690 (2012)

    Article  MathSciNet  Google Scholar 

  17. Peng, S., Zhang, Y.: Razumikhin-type theorems on \(p\)-th moment exponential stability of impulsive stochastic delay differential equations. IEEE Trans. Autom. Control 55, 1917–1922 (2010)

    Article  MathSciNet  Google Scholar 

  18. Trigeassoua, J.C., Maamrib, N., Sabatiera, J., Oustaloup, A.: A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91, 437–445 (2011)

    Article  Google Scholar 

  19. Wu, F., Mao, X.: Numerical solutions of neutral stochastic functional differential equations. SIAM J. Numer. Anal. 46, 1821–1841 (2008)

    Article  MathSciNet  Google Scholar 

  20. Xu, B., Liu, Y.: An improved Razumikhin-type theorem and its applications. IEEE Trans. Autom. Control 33, 839–841 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Yang, Z.H., Zhu, E.W., Xu, Y., Tan, Y.M.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations with infinite delay. Acta Appl. Math. 111, 219–231 (2010)

    Article  MathSciNet  Google Scholar 

  22. Zhao, X.Y., Deng, F.Q.: New type of stability criteria for stochastic functional differential equations via Lyapunov functions. SIAM J. Control Optim. 52, 2319–2347 (2014)

    Article  MathSciNet  Google Scholar 

  23. Zhu, Q.: Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching. Int. J. Control 90, 1703–1712 (2017)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, 434023, P.R. China

    Zhi Li & Liping Xu

Authors
  1. Zhi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liping Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi Li.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research is partially supported by the NNSF of China (No.11901058).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Z., Xu, L. Exponential Stability in Mean Square of Stochastic Functional Differential Equations with Infinite Delay. Acta Appl Math 174, 8 (2021). https://doi.org/10.1007/s10440-021-00426-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-021-00426-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation