SCIENCE CHINA Information Sciences, Volume 61 , Issue 11 : 112211(2018) https://doi.org/10.1007/s11432-018-9496-6

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

More info
  • ReceivedFeb 12, 2018
  • AcceptedJun 1, 2018
  • PublishedOct 19, 2018


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 with Markovian 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 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.


[1] Sontag E D. Smooth stabilization implies coprime factorization. IEEE Trans Autom Control, 1989, 34: 435-443 CrossRef Google Scholar

[2] Peng S, Deng F. New Criteria on $p$th Moment Input-to-State Stability of Impulsive Stochastic Delayed Differential Systems. IEEE Trans Autom Control, 2017, 62: 3573-3579 CrossRef Google Scholar

[3] Wu X, Tang Y, Zhang W. Input-to-state stability of impulsive stochastic delayed systems under linear assumptions. Automatica, 2016, 66: 195-204 CrossRef Google Scholar

[4] Guo R, Zhang Z, Liu X. Exponential input-to-state stability for complex-valued memristor-based BAM neural networks with multiple time-varying delays. Neurocomputing, 2018, 275: 2041-2054 CrossRef 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 CrossRef Google Scholar

[6] Sun F L, Guan Z H, Zhang X H. Exponential-weighted input-to-state stability of hybrid impulsive switched systems. IET Control Theor Appl, 2012, 6: 430-436 CrossRef 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 CrossRef 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 CrossRef Google Scholar

[9] Liu X, Yang C, Zhou L. Global asymptotic stability analysis of two-time-scale competitive neural networks with time-varying delays. Neurocomputing, 2018, 273: 357-366 CrossRef Google Scholar

[10] Florchinger P. Global asymptotic stabilisation in probability of nonlinear stochastic systems via passivity. Int J Control, 2016, 89: 1406-1415 CrossRef Google Scholar

[11] Wang B, Zhu Q. Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems. Syst Control Lett, 2017, 105: 55-61 CrossRef 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 CrossRef Google Scholar

[13] Peng S G, Zhang Y. Razumikhin-Type Theorems on $p$th Moment Exponential Stability of Impulsive Stochastic Delay Differential Equations. IEEE Trans Autom Control, 2010, 55: 1917-1922 CrossRef 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 CrossRef Google Scholar

[15] Wang H, Ding C. Impulsive control for differential systems with delay. Math Meth Appl Sci, 2013, 36: 967-973 CrossRef ADS 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 CrossRef Google Scholar

[17] Li X, Song S. Stabilization of Delay Systems: Delay-Dependent Impulsive Control. IEEE Trans Autom Control, 2017, 62: 406-411 CrossRef Google Scholar

[18] Sakthivel R, Luo J. Asymptotic stability of nonlinear impulsive stochastic differential equations. Stat Probability Lett, 2009, 79: 1219-1223 CrossRef 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. 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 CrossRef Google Scholar

[21] Dashkovskiy S, Mironchenko A. Input-to-State Stability of Nonlinear Impulsive Systems. SIAM J Control Optim, 2013, 51: 1962-1987 CrossRef Google Scholar

[22] Li X, Zhang X, Song S. Effect of delayed impulses on input-to-state stability of nonlinear systems. Automatica, 2017, 76: 378-382 CrossRef Google Scholar

[23] Yao F, Cao J, Cheng P. 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 CrossRef Google Scholar

[24] Zhu Q, Song S, Tang T. Mean square exponential stability of stochastic nonlinear delay systems. Int J Control, 2017, 90: 2384-2393 CrossRef Google Scholar

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

[26] Zhu Q, Zhang Q. pth moment exponential stabilisation of hybrid stochastic differential equations by feedback controls based on discrete-time state observations with a time delay. IET Contr Theor Appl, 2017, 11: 1992-2003 CrossRef Google Scholar

[27] Zhu Q. Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching. Int J Control, 2017, 90: 1703-1712 CrossRef Google Scholar

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

[29] Wu X, Shi P, Tang Y. Input-to-state stability of nonlinear stochastic time-varying systems with impulsive effects. Int J Robust NOnlinear Control, 2017, 27: 1792-1809 CrossRef Google Scholar

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

[31] Li D, Cheng P, He S. Exponential stability of hybrid stochastic functional differential systems with delayed impulsive effects: average impulsive interval approach. Math Meth Appl Sci, 2017, 40: 4197-4210 CrossRef ADS 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 CrossRef Google Scholar

[33] Liu J, Liu X, Xie W C. Impulsive stabilization of stochastic functional differential equations. Appl Math Lett, 2011, 24: 264-269 CrossRef Google Scholar

[34] Xu L, Xu D. Mean square exponential stability of impulsive control stochastic systems with time-varying delay. Phys Lett A, 2009, 373: 328-333 CrossRef ADS Google Scholar

[35] Liu B. Stability of Solutions for Stochastic Impulsive Systems via Comparison Approach. IEEE Trans Autom Control, 2008, 53: 2128-2133 CrossRef Google Scholar

[36] Ning C, He Y, Wu M. Indefinite Lyapunov functions for input-to-state stability of impulsive systems. Inf Sci, 2018, 436-437: 343-351 CrossRef Google Scholar

[37] Lu J, Ho D W C, Cao J. A unified synchronization criterion for impulsive dynamical networks. Automatica, 2010, 46: 1215-1221 CrossRef Google Scholar

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备17057255号       京公网安备11010102003388号