Skip to main content
SpringerLink
Log in

Stability of nonlinear impulsive stochastic systems with Markovian switching under generalized average dwell time condition

  • Research Paper
  • Published:
Science China Information Sciences Aims and scope Submit manuscript

Abstract

This paper examines the p-th moment globally asymptotic stability in probability and p-th moment stochastic input-to-state stability for a type of impulsive stochastic system withMarkovian switching. By applying the generalized average dwell time approach, some novel sufficient conditions are obtained to ensure these stability properties. Furthermore, the coefficient of the Lyapunov function is allowed to be time-varying, which generalizes and improves the existing results. As corollaries, nonlinear restrictions on the drift and diffusion coefficients are used as substitutes for some previous conditions. Two examples are provided to illustrate the effectiveness of the theoretical results.

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. Sontag E D. Smooth stabilization implies coprime factorization. IEEE Trans Autom Control, 1989, 34: 435–443

    Article  MathSciNet  Google Scholar 

  2. Peng S G, Deng F Q. New criteria on pth moment input-to-state stability of impulsive stochastic delayed differential systems. IEEE Trans Autom Control, 2017, 62: 3573–3579

    Article  Google Scholar 

  3. Wu X T, Tang Y, ZhangWB. Input-to-state stability of impulsive stochastic delayed systems under linear assumptions. Automatica, 2016, 66: 195–204

    Article  MathSciNet  Google Scholar 

  4. Guo R N, Zhang Z Y, Liu X P, et al. Exponential input-to-state stability for complex-valued memristor-based BAM neural networks with multiple time-varying delays. Neurocomputing, 2018, 275: 2041–2054

    Article  Google Scholar 

  5. Pin G, Parisini T. Networked predictive control of uncertain constrained nonlinear systems: recursive feasibility and input-to-state stability analysis. IEEE Trans Autom Control, 2011, 56: 72–87

    Article  MathSciNet  Google Scholar 

  6. Sun F L, Guan Z H, Zhang X H, et al. Exponential-weighted input-to-state stability of hybrid impulsive switched systems. IET Control Theory Appl, 2012, 6: 430–436

    Article  MathSciNet  Google Scholar 

  7. Vu L, Chatterjee D, Liberzon D. Input-to-state stability of switched systems and switching adaptive control. Automatica, 2007, 43: 639–646

    Article  MathSciNet  Google Scholar 

  8. Wang J. A necessary and sufficient condition for input-to-state stability of quantised feedback systems. Int J Control, 2017, 90: 1846–1860

    Article  MathSciNet  Google Scholar 

  9. Liu X M, Yang C Y, Zhou L N. Global asymptotic stability analysis of two-time-scale competitive neural networks with time-varying delays. Neurocomputing, 2018, 273: 357–366

    Article  Google Scholar 

  10. Florchinger P. Global asymptotic stabilisation in probability of nonlinear stochastic systems via passivity. Int J Control, 2016, 89: 1406–1415

    Article  MathSciNet  Google Scholar 

  11. Wang B, Zhu Q X. Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems. Syst Control Lett, 2017, 105: 55–61

    Article  MathSciNet  Google Scholar 

  12. Shen L J, Sun J T. p-th moment exponential stability of stochastic differential equations with impulse effect. Sci China Inf Sci, 2011, 54: 1702–1711

    Article  MathSciNet  Google Scholar 

  13. Peng S G, Zhang Y. Razumikhin-type theorems on pth moment exponential stability of impulsive stochastic delay differential equations. IEEE Trans Autom Control, 2010, 55: 1917–1922

    Article  Google Scholar 

  14. Yao F Q, Deng F Q. Stability of impulsive stochastic functional differential systems in terms of two measures via comparison approach. Sci China Inf Sci, 2012, 55: 1313–1322

    Article  MathSciNet  Google Scholar 

  15. Wang H L, Ding C M. Impulsive control for differential systems with delay. Math Meth Appl Sci, 2013, 36: 967–973

    Article  MathSciNet  Google Scholar 

  16. Jiang F, Yang H, Shen Y. Stability of second-order stochastic neutral partial functional differential equations driven by impulsive noises. Sci China Inf Sci, 2016, 59: 112208

    Article  MathSciNet  Google Scholar 

  17. Li X D, Song S J. Stabilization of delay systems: delay-dependent impulsive control. IEEE Trans Autom Control, 2017, 62: 406–411

    Article  MathSciNet  Google Scholar 

  18. Sakthivel R, Luo J. Asymptotic stability of nonlinear impulsive stochastic differential equations. Stat Probab Lett, 2009, 79: 1219–1223

    Article  MathSciNet  Google Scholar 

  19. Cheng P, Deng F Q, Yao F Q. Exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses. Commun Nonlinear Sci Numer Simul, 2013, 19: 2104-2114

    Article  MathSciNet  Google Scholar 

  20. Hespanha J P, Liberzon D, Teel A R. Lyapunov conditions for input-to-state stability of impulsive systems. Automatica, 2008, 44: 2735–2744

    Article  MathSciNet  Google Scholar 

  21. Dashkovskiy S, Mironchenko A. Input-to-state stability of nonlinear impulsive systems. SIAM J Control Optim, 2013, 51: 1962–1987

    Article  MathSciNet  Google Scholar 

  22. Li X D, Zhang X L, Song S J. Effect of delayed impulses on input-to-state stability of nonlinear systems. Automatica, 2017, 76: 378–382

    Article  MathSciNet  Google Scholar 

  23. Yao F Q, Cao J D, Cheng P, et al. Generalized average dwell time approach to stability and input-to-state stability of hybrid impulsive stochastic differential systems. Nonlinear Anal-Hybrid Syst, 2016, 22: 147–160

    Article  MathSciNet  Google Scholar 

  24. Zhu Q X, Song S Y, Tang T R. Mean square exponential stability of stochastic nonlinear delay systems. Int J Control, 2017, 90: 2384–2393

    Article  MathSciNet  Google Scholar 

  25. Mao X R, Yuan C G. Stochastic Differential Delay Equations with Markovian Switching. London: Imperial College Press, 2006

    Book  Google Scholar 

  26. Zhu Q X, Zhang Q Y. pth moment exponential stabilisation of hybrid stochastic differential equations by feedback controls based on discrete-time state observations with a time delay. IET Control Theory Appl, 2017, 11: 1992–2003

    Article  MathSciNet  Google Scholar 

  27. Zhu Q X. Razumikhin-type theorem for stochastic functional differential equations with L´evy noise and Markov switching. Int J Control, 2017, 90: 1703–1712

    Article  Google Scholar 

  28. Zhu EW, Tian X,Wang Y H. On p-th moment exponential stability of stochastic differential equations with Markovian switching and time-varying delay. J Inequal Appl, 2015, 2015: 137

    Article  MathSciNet  Google Scholar 

  29. Wu X T, Shi P, Tang Y, et al. Input-to-state stability of nonlinear stochastic time-varying systems with impulsive effects. Int J Robust Nonlinear Control, 2017, 27: 1792–1809

    Article  MathSciNet  Google Scholar 

  30. Khasminskii R. Stochastic Stability of Differential Equations. 2nd ed. Berlin: Springer, 2012

    Book  Google Scholar 

  31. Li D Q, Cheng P, He S P. Exponential stability of hybrid stochastic functional differential systems with delayed impulsive effects: average impulsive interval approach. Math Meth Appl Sci, 2017, 40: 4197–4210

    Article  MathSciNet  Google Scholar 

  32. Jiang Z P, Teel A R, Praly L. Small-gain theorem for ISS systems and applications. Math Control Signal Syst, 1994, 7: 95–120

    Article  MathSciNet  Google Scholar 

  33. Liu J, Liu X Z, Xie W C. Impulsive stabilization of stochastic functional differential equations. Appl Math Lett, 2011, 24: 264–269

    Article  MathSciNet  Google Scholar 

  34. Xu L G, Xu D Y. Mean square exponential stability of impulsive control stochastic systems with time-varying delay. Phys Lett A, 2009, 373: 328–333

    Article  Google Scholar 

  35. Liu B. Stability of solutions for stochastic impulsive systems via comparison approach. IEEE Trans Autom Control, 2008, 53: 2128–2133

    Article  MathSciNet  Google Scholar 

  36. Ning C Y, He Y, Wu M, et al. Indefinite Lyapunov functions for input-to-state stability of impulsive systems. Inf Sci, 2018, 436: 343–351

    Article  MathSciNet  Google Scholar 

  37. Lu J Q, Ho D W C, Cao J D. A unified synchronization criterion for impulsive dynamical networks. Automatica, 2010, 46: 1215–1221

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was jointly supported by National Natural Science Foundation of China (Grant Nos. 61773217, 61374080), Natural Science Foundation of Jiangsu Province (Grant No. BK20161552), Qing Lan Project of Jiangsu Province, and Priority Academic Program Development of Jiangsu Higher Education Institutions.

Author information

Authors and Affiliations

  1. Key Laboratory of HPC-SIP (MOE), College of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China

    Xiaozheng Fu & Quanxin Zhu

  2. School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing, 210023, China

    Xiaozheng Fu & Quanxin Zhu

Authors
  1. Xiaozheng Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Quanxin Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quanxin Zhu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, X., Zhu, Q. Stability of nonlinear impulsive stochastic systems with Markovian switching under generalized average dwell time condition. Sci. China Inf. Sci. 61, 112211 (2018). https://doi.org/10.1007/s11432-018-9496-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11432-018-9496-6

Keywords

Search

Navigation