Asymptotic stability in distribution of stochastic systems with semi-Markovian switching

ABSTRACT This paper investigates the asymptotic stability in distribution of stochastic systems with semi-Markovian switching whose transition rates are time varying, which are more general than stochastic systems with Markovian switching. Some sufficient conditions for the asymptotic stability in distribution of stochastic systems with semi-Markovian switching are established and those conditions are employed to design the state-feedback controllers in terms of linear matrix inequalities. Finally, several examples are given to illustrate the effectiveness of the theoretical results.

[1]  R. Situ.  Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering , 2005 .

[2]  C. Yuan,et al.  Stability in distribution of neutral stochastic differential delay equations with Markovian switching , 2009 .

[3]  Mrinal K. Ghosh,et al.  Risk Minimizing Option Pricing in a Semi-Markov Modulated Market , 2009, SIAM J. Control. Optim..

[4]  John Lygeros,et al.  Stabilization of a class of stochastic differential equations with Markovian switching , 2005, Syst. Control. Lett..

[5]  X. Mao,et al.  Asymptotic stability in distribution of stochastic differential equations with Markovian switching , 2003 .

[6]  Martin Crowder.  Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications by Vlad Stefan Barbu, Nikolaos Limnios , 2009 .

[7]  Jacques Janssen,et al.  Applied Semi-Markov Processes , 2005 .

[8]  X. Mao.  Stability of stochastic differential equations with Markovian switching , 1999 .

[9]  Ji Huang,et al.  Stochastic stability of semi-Markov jump linear systems: An LMI approach , 2011, IEEE Conference on Decision and Control and European Control Conference.

[10]  A. Deshpande.  Asymptotic Stability of Semi-Markov Modulated Jump Diffusions , 2012 .

[11]  Yonggui Kao,et al.  Exponential stability and instability of impulsive stochastic functional differential equations with Markovian switching , 2015, Appl. Math. Comput..

[12]  A. Skorokhod.  Asymptotic Methods in the Theory of Stochastic Differential Equations , 2008 .

[13]  Zhengzhi Han,et al.  Stability analysis of stochastic differential equations with Markovian switching , 2012, Syst. Control. Lett..

[14]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[15]  J. Zhou,et al.  Stochastic stabilization of semi-Markov jump systems with linear fraction uncertainty: State feedback case , 2014, 2014 International Conference on Mechatronics and Control (ICMC).

[16]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[17]  Gopal K. Basak,et al.  Stability of a Random diffusion with linear drift , 1996 .

[18]  Quanxin Zhu,et al.  pth Moment exponential stability of impulsive stochastic functional differential equations with Markovian switching , 2014, J. Frankl. Inst..

[19]  Yang Tang,et al.  Stability analysis of switched stochastic neural networks with time-varying delays , 2014, Neural Networks.

[20]  Quanxin Zhu,et al.  pth moment exponential stabilisation of hybrid stochastic differential equations by feedback controls based on discrete-time state observations with a time delay , 2017 .

[21]  Nguyen Hai Dang,et al.  A note on sufficient conditions for asymptotic stability in distribution of stochastic differential equations with Markovian switching , 2014 .

[22]  Yang Shi,et al.  Stochastic stability and robust stabilization of semi‐Markov jump linear systems , 2013 .

[23]  Hisashi Kobayashi,et al.  Probability, Random Processes, and Statistical Analysis: Moment-generating function and characteristic function , 2011 .

[24]  Debasish Chatterjee,et al.  On Stability of Randomly Switched Nonlinear Systems , 2007, IEEE Transactions on Automatic Control.

[25]  Gang George Yin,et al.  Numerical solutions of regime-switching jump diffusions , 2014, Appl. Math. Comput..

[26]  Quanxin Zhu,et al.  Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching , 2017, Int. J. Control.

[27]  Nikolaos Limnios,et al.  Semi-Markov Chains and Hidden Semi-Markov Models toward Applications: Their Use in Reliability and DNA Analysis , 2008 .

[28]  Hailing Dong,et al.  Asymptotic stability in the distribution of nonlinear stochastic systems with semi-Markovian switching , 2007, The ANZIAM Journal.

[29]  Quanxin Zhu,et al.  Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems , 2017, Syst. Control. Lett..

[30]  Xuerong Mao,et al.  Stochastic Differential Equations With Markovian Switching , 2006 .