Skip to main content
SpringerLink
Log in

Stability and Stabilization of Stochastic Neutral-type Markovian Jump Time-delay Systems with Two Delays

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

This article aims to explore the stability and stabilization of a more practical class of stochastic neutral-type Markovian jump time-delay systems (SNMJTDSs) with two delays in the derivative. First, a less restrictive condition is imposed on the associated difference operator to ensure its stability. Second, by constructing novel Lyapunov-Krasovskii functionals, sufficient conditions for exponential stability (in the sense of mean square and almost sure) are established. Then, the SNMJTDS is stabilized by designing a delayed feedback controller. Finally, the less conservativeness of the proposed stability criterion and the validity of the designed controller are checked through two examples.

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. X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chichester, UK, 2007.

    MATH  Google Scholar 

  2. L. R. Huang and X. Mao, “Delay-dependent exponential stability of neutral stochastic delay systems,” IEEE Trans. on Automatic Control, vol. 54, no. 1, pp. 147–152, February 2009.

    Article  MathSciNet  Google Scholar 

  3. L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Heidelberg, 2013.

    Book  Google Scholar 

  4. M. Milošević, “Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama approximation,” Mathematical and Computer Modelling, vol. 57, no. 3–4, pp. 887–899, February 2013.

    Article  MathSciNet  Google Scholar 

  5. B. Song, J. H. Park, Z.-G. Wu, and Y. Zhang, “New results on delay-dependent stability analysis for neutral stochastic delay systems,” Journal of the Franklin Institute, vol. 350, no. 4, pp. 840–852, May 2013.

    Article  MathSciNet  Google Scholar 

  6. Y. B. Yu, X. L. Tang, T. Li, and S. M. Fei, “Mixed-delay-dependent L2 − L∞ filtering for neutral stochastic systems with time-varying delays,” International Journal of Control, Automation, and Systems, vol. 17, no. 11, pp. 2862–2870, November 2019.

    Article  Google Scholar 

  7. M. Tanelli, B. Picasso, P. Bolzern, and P. Colaneri, “Almost sure stabilization of uncertain continuous-time Markov jump linear systems,” IEEE Trans. on Automatic Control, vol. 55, no. 1, pp. 195–201, January 2010.

    Article  MathSciNet  Google Scholar 

  8. A. R. Teel, A. Subbaraman, and A. Sferlazza, “Stability analysis for stochastic hybrid systems: A survey,” Automatica, vol. 50, no. 10, pp. 2435–2456, October 2014.

    Article  MathSciNet  Google Scholar 

  9. X. Zhang, W. N. Zhou, and Y. Q. Sun, “Exponential stability of neural networks with Markovian switching parameters and general noise,” International Journal of Control, Automation, and Systems, vol. 17, no. 4, pp. 966–975, April 2019.

    Article  Google Scholar 

  10. V. Ugrinovskii and H. R. Pota, “Decentralized control of power systems via robust control of uncertain Markov jump parameter systems,” International Journal of Control, vol. 78, no. 9, pp. 662–677, June 2005.

    Article  MathSciNet  Google Scholar 

  11. R. Sakthivel, H. Divya, R. Sakthivel, and Y. Liu, “Robust hybrid control design for stochastic Markovian jump system via fault alarm approach,” IEEE Trans. on Circuits and Systems II: Express Briefs, vol. 67, no. 10, pp. 2004–2008, October 2020.

    Article  Google Scholar 

  12. V. Kolmanovskii, N. Koroleva, T. Maizenberg, X. Mao, and A. Matasov, “Neutral stochastic differential delay equations with Markovian switching,” Stochastic Analysis and Applications, vol. 21, no. 4, pp. 819–847, 2003.

    Article  MathSciNet  Google Scholar 

  13. M. G. Hua, H. S. Tan, J. F. Chen, and J. T. Fei, “Robust delay-range-dependent non-fragile H filtering for uncertain neutral stochastic systems with Markovian switching and mode-dependent time delays,” Journal of the Franklin Institute, vol. 352, no. 3, pp. 1318–1341, March 2015.

    Article  MathSciNet  Google Scholar 

  14. S. Muralisankar, A. Manivannan, and P. Balasubramaniam, “Robust stability criteria for uncertain neutral type stochastic system with Takagi—Sugeno fuzzy model and Markovian jumping parameters,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 10, pp. 3876–3893, October 2012.

    Article  MathSciNet  Google Scholar 

  15. Y. F. Song and Z. G. Zeng, “Razumikhin-type theorems on pth moment boundedness of neutral stochastic functional differential equations with Markovian switching,” Journal of the Franklin Institute, vol. 355, no. 17, pp. 8296–8312, November 2018.

    Article  MathSciNet  Google Scholar 

  16. M. Obradović, “Implicit numerical methods for neutral stochastic differential equations with unbounded delay and Markovian switching,” Applied Mathematics and Computation, vol. 347, pp. 664–687, April 2019.

    Article  MathSciNet  Google Scholar 

  17. R. R. Kumar and K. T. Chong, “The stability of distributed neutral delay differential systems with Markovian switching,” Chaos, Solitons & Fractals, vol. 45, no. 4, pp. 416–425, April 2012.

    Article  MathSciNet  Google Scholar 

  18. H. Chen, P. Shi, and C.-C. Lim, “Stability analysis for neutral stochastic delay systems with Markovian switching,” Systems & Control Letters, vol. 110, pp. 38–48, December 2017.

    Article  MathSciNet  Google Scholar 

  19. W. M. Chen, B. Y. Zhang, and Q. Ma, “Decay-rate-dependent conditions for exponential stability of stochastic neutral systems with Markovian jumping parameters,” Applied Mathematics and Computation, vol. 321, pp. 93–105, March 2018.

    Article  MathSciNet  Google Scholar 

  20. J. Xie, Y. G. Kao, C. H. Wang, and C. C. Gao, “Delay-dependent robust stability of uncertain neutral-type Itô stochastic systems with Markovian jumping parameters,” Applied Mathematics and Computation, vol. 251, pp. 576–585, January 2015.

    Article  MathSciNet  Google Scholar 

  21. Y. Xu and Z. M. He, “Exponential stability of neutral stochastic delay differential equations with Markovian switching,” Applied Mathematics Letters, vol. 52, pp. 64–73, February 2016.

    Article  MathSciNet  Google Scholar 

  22. G. M. Zhuang, S. Y. Xu, J. W. Xia, Q. Ma, and Z. Q. Zhang, “Non-fragile delay feedback control for neutral stochastic Markovian jump systems with time-varying delays,” Applied Mathematics and Computation, vol. 355, pp. 21–32, August 2019.

    Article  MathSciNet  Google Scholar 

  23. J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, New York, USA, 1993.

    Book  Google Scholar 

  24. V. Kharitonov, J. Collado, and S. Mondié, “Exponential estimates for neutral time delay systems with multiple delays,” International Journal of Robust and Nonlinear Control, vol. 16, pp. 71–84, 2006.

    Article  MathSciNet  Google Scholar 

  25. X. Liao, Y. Liu, H. Wang, and T. Huang, “Exponential estimates and exponential stability for neutral-type neural networks with multiple delays,” Neurocomputing, vol. 149, pp. 868–883, February 2015.

    Article  Google Scholar 

  26. R. Villafuerte, B. Saldivar, and S. Mondié, “Practicalstability and stabilization of aclass of nonlinear neutral type time delay systems with multiple delays: BMI’s approaches,” International Journal of Control, Automation, and Systems, vol. 11, no. 5, pp. 859–867, October 2013.

    Article  Google Scholar 

  27. Z.-Y. Li, J. Lam, and Y. Wang, “Stability analysis of linear stochastic neutral-type time-delay systems with two delays,” Automatica, vol. 91, pp. 179–189, May 2018.

    Article  MathSciNet  Google Scholar 

  28. S. A. Karthick, R. Sakthivel, Y. K. Ma, and A. Leelamani, “Observer based guaranteed cost control for Markovian jump stochastic neutral-type neural networks,” Chaos, Solitons & Fractals, vol. 133, pp. 109621, April 2020.

    Article  MathSciNet  Google Scholar 

  29. R. Rakkiyappan, Q. Zhu, and A. Chandrasekar, “Stability of stochastic neural networks of neutral type with Markovian jumping parameters: A delay-fractioning approach,” Journal of the Franklin Institute, vol. 351, pp. 1553–1570, March 2014.

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the National Natural Science Foundation of China under grant number 61573120 and the Natural Science Foundation of Heilongjiang Province under the grant number YQ2020F006.

Author information

Authors and Affiliations

  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Qiuqiu Fan & Zhao-Yan Li

  2. School of Management, Harbin Institute of Technology, Harbin, 150001, China

    Longsuo Li

Authors
  1. Qiuqiu Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhao-Yan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Longsuo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhao-Yan Li.

Additional information

Qiuqiu Fan received her B.S. degree from the Department of Mathematics at Changchun University of Science and Technology, Changchun, China, in 2015, and an M.S. degree in the Department of Mathematics, Harbin Institute of Technology, in 2017. Now, she is a Ph.D. candidate in the School of Mathematics, Harbin Institute of Technology. Her research interests include stability of continuous difference equations and stability and control of stochastic neutral time-delay systems.

Zhao-Yan Li received her Ph.D. from Harbin Institute of Technology in 2010. Since 2011, she has been working in Harbin Institute of Technology. She was appointed as an Associate Professor and Ph.D. Supervisor in the Department of Mathematics, in 2014 and 2017, respectively. At present, she is a Professor in the School of Mathematics. Her research interests mainly focus on the stability theory of stochastic time-delay systems and functional differential equations.

Longsuo Li received his Ph.D. degree from Harbin Institute of Technology in 2004. Currently, he is a professor at the School of Management, Harbin Institute of Technology. His research includes quantum statistics, prediction theory of stationary stochastic process and stochastic systems.

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fan, Q., Li, ZY. & Li, L. Stability and Stabilization of Stochastic Neutral-type Markovian Jump Time-delay Systems with Two Delays. Int. J. Control Autom. Syst. 20, 365–374 (2022). https://doi.org/10.1007/s12555-020-0702-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-020-0702-4

Keywords

Search

Navigation