STABILITY ANALYSIS OF HIGHLY NONLINEAR HYBRID MULTIPLE-DELAY STOCHASTIC DIFFERENTIAL EQUATIONS

Chen Fei; Weiyin Fei; Xuerong Mao; Mingxuan Shen; Litan Yan

doi:10.11948/2156-907X.20180257

2019 Volume 9 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2019 > 9(3): 1053-1070

Next Article Previous Article

Chen Fei, Weiyin Fei, Xuerong Mao, Mingxuan Shen, Litan Yan. STABILITY ANALYSIS OF HIGHLY NONLINEAR HYBRID MULTIPLE-DELAY STOCHASTIC DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 1053-1070. doi: 10.11948/2156-907X.20180257

Citation:

Chen Fei, Weiyin Fei, Xuerong Mao, Mingxuan Shen, Litan Yan. STABILITY ANALYSIS OF HIGHLY NONLINEAR HYBRID MULTIPLE-DELAY STOCHASTIC DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 1053-1070. doi: 10.11948/2156-907X.20180257

STABILITY ANALYSIS OF HIGHLY NONLINEAR HYBRID MULTIPLE-DELAY STOCHASTIC DIFFERENTIAL EQUATIONS

1.
Glorious Sun School of Business and Management, Donghua University, Shanghai, 200051, China
2.
School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, Anhui, 241000, China
3.
Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, U.K.

The authors would like to thank Liangjian Hu and Wei Liu for their valuable comments and suggestions
Corresponding author: Email address: wyfei@ahpu.edu.cn(W. Fei)
Fund Project: This paper is supported by the Natural Science Foundation of China (71571001) for the financial support

Abstract

Abstract

Stability criteria for stochastic differential delay equations (SDDEs) have been studied intensively for the past few decades. However, most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability of highly nonlinear hybrid stochastic differential equations with a single delay is investigated in [Fei, Hu, Mao and Shen, Automatica, 2017], whose work, in this paper, is extended to highly nonlinear hybrid stochastic differential equations with variable multiple delays. In other words, this paper establishes the stability criteria of highly nonlinear hybrid variable multiple-delay stochastic differential equations. We also discuss an example to illustrate our results.
- Variable multiple-delay stochastic differential equation /
- nonlinear growth condition /
- asymptotic stability /
- Markovian switching /
- Lyapunov functional
MSC: 60H10, 34K20, 93E15

FullText(HTML)

References

[1]	A. Ahlborn and U. Parlitz, Stabilizing unstable steady states using multiple delay feedback control, Phys. Rev. Lett., 2014, 93, 264101. Google Scholar
[2]	C. Briat, Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints, Automatica, 2016, 74, 279-287. doi: 10.1016/j.automatica.2016.08.001 CrossRef Google Scholar
[3]	H. B. Chen, P. Shi and C. Lim, Stability analysis for neutral stochastic delay systems with Markovian switching, Syst. Control Lett., 2017, 110, 38-48. doi: 10.1016/j.sysconle.2017.10.008 CrossRef Google Scholar
[4]	V. Dragan and H. Mukaidani, Exponential stability in mean square of a singularly perturbed linear stochastic system with state-multiplicative white-noise perturbations and Markovian switching, IET Control Theory Appl., 2016, 9, 1040-1051. Google Scholar
[5]	C. Fei, W. Y. Fei and L. T. Yan, Existence and stability of solutions to highly nonlinear stochastic differential delay equations driven by G-Brownian motion, Appl. Math. J. Chinese Univ., to appear. Google Scholar
[6]	C. Fei, M. X. Shen, W. Y. Fei, X. R. Mao and L. T. Yan, Stability of highly nonlinear hybrid stochastic integro-differential delay equations, Nonlinear Anal. Hybrid Syst., 2019, 31, 180-199. doi: 10.1016/j.nahs.2018.09.001 CrossRef Google Scholar
[7]	W. Y. Fei, L. J. Hu, X. M. Mao and M. X. Shen, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 2017, 28, 165-170. Google Scholar
[8]	W. Y. Fei, L. J. Hu, X. R. Mao and M. X. Shen, Structured robust stability and boundedness of nonlinear hybrid delay systems, SIAM J. Control Optim., 2018, 56, 2662-2689. doi: 10.1137/17M1146981 CrossRef Google Scholar
[9]	W. Y. Fei, L. J. Hu, X. R. Mao and M. X. Shen, Generalised criteria on delay dependent stability of highly nonlinear hybrid stochastic systems, Int. J. Robust. Nonlin., 2019, 25, 1201-1215. Google Scholar
[10]	M. Frederic, Stability analysis of time-varying neutral time-delay systems, IEEE Trans. Automat. Control, 2016, 60, 540-546. Google Scholar
[11]	E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Birkhauser, 2014. Google Scholar
[12]	A. Garab, V. Kovács and T. Krisztin Global stability of a price model with multiple delays, Discrete Contin. Dyn. Syst. A., 2016, 36(12), 6855-6871. doi: 10.3934/dcdsa CrossRef Google Scholar
[13]	J. K. Hale and S. M. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. Google Scholar
[14]	L. J. Hu, X. R. Mao and Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Syst. Control Lett., 2013, 62, 178-187. doi: 10.1016/j.sysconle.2012.11.009 CrossRef Google Scholar
[15]	L. J. Hu, X. R. Mao and L. G. Zhang, Robust stability and boundedness of nonlinear hybrid stochastic delay equations, IEEE Trans. Automat Control, 2013, 58(9), 2319-2332. doi: 10.1109/TAC.2013.2256014 CrossRef Google Scholar
[16]	V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, London, 1986. Google Scholar
[17]	M. L. Li and M. C. Huang, Approximate controllability of second-order impulsive stochastic differential equations with state-dependent delay, J. Appl. Anal. Comput., 2018, 8(2), 598-619. Google Scholar
[18]	X. D. Li, Q. X. Zhu and D. O'Reganc, pth Moment exponential stability of impulsive stochastic functional differential equations and application to control problems of NNs, J. Franklin Inst., 2014, 351, 4435-4456. doi: 10.1016/j.jfranklin.2014.04.008 CrossRef Google Scholar
[19]	J. Lei and M. Mackey, Stochastic differential delay equation, moment stability, and application to hematopoitic stem cell regulation systems, SIAM J. Appl. Math., 2007, 67(2), 387-407. doi: 10.1137/060650234 CrossRef Google Scholar
[20]	J. Liu, On asymptotic convergence and boundedness of stochastic systems with time-delay, Automatica, 2012, 48, 3166-3172. doi: 10.1016/j.automatica.2012.08.041 CrossRef Google Scholar
[21]	K. Liu, Almost sure exponential stability sensitive to small time delay of stochastic neutral functional differential equations, Appl. Math. Lett., 2018, 77, 57-63. doi: 10.1016/j.aml.2017.09.008 CrossRef Google Scholar
[22]	X. R. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential eqautions, Stoch. Process. Appl., 1996, 65, 233-250. doi: 10.1016/S0304-4149(96)00109-3 CrossRef Google Scholar
[23]	X. R. Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Trans. Auto. Control, 2002, 47(10), 1604-1612. doi: 10.1109/TAC.2002.803529 CrossRef Google Scholar
[24]	X. R. Mao, Stochastic Differential Equations and Their Applications, 2nd Edition, Chichester: Horwood Pub., 2007. Google Scholar
[25]	X. R. Mao, J. Lam, and L. R. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Syst. Control Lett., 2008, 57, 927-935. doi: 10.1016/j.sysconle.2008.05.002 CrossRef Google Scholar
[26]	X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. Google Scholar
[27]	S. E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1984. Google Scholar
[28]	C. Park, N. Kwon and P. Park, Optimal H_∞ filtering for singular Markovian jump systems, Syst. Control Lett., 2018, 118, 22-28. doi: 10.1016/j.sysconle.2018.05.005 CrossRef Google Scholar
[29]	A. Rathinasamy and M. Balachandran, Mean-square stability of semi-implicit Euler method for linear stochastic differential equations with multiple delays and Markovian switching, Appl. Math. Comput., 2008, 206, 968-979. Google Scholar
[30]	M. X. Shen, C. Fei, W. Y. Fei and X. R. Mao, The boundedness and stability of highly nonlinear hybrid neutral stochastic systems with multiple delays, Sci. China Inf. Sci., DOI: 10.1007/s11432-018-9755-7. Google Scholar
[31]	M. X. Shen, W. Y. Fei, X. R. Mao and S. N. Deng, Exponential stability of highly nonlinear neutral pantograph stochastic differential equations, Asian J. Control, DOI: 10.1002/asjc.1903. Google Scholar
[32]	M. X. Shen, W. Y. Fei, X. R. Mao and Y. Liang, Stability of highly nonlinear neutral stochastic differential delay equations, Syst. Control Lett., 2018, 115, 1-8. doi: 10.1016/j.sysconle.2018.02.013 CrossRef Google Scholar
[33]	S. Y. Xu, J. Lam and X. R. Mao, Delay-dependent H_∞ control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Trans. Circuits Syst. I, 2007, 54(9), 2070-2077. doi: 10.1109/TCSI.2007.904640 CrossRef Google Scholar
[34]	S. R. You, W. Liu, J. Q. Lu, X. R. Mao and J. W. Qiu, Stablization of hybrid systems by feedback control based on discrete-time state observation, SIAM J. Contrl Optim., 2015, 53(2), 905-925. doi: 10.1137/140985779 CrossRef Google Scholar
[35]	D. Yue and Q. Han, Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching, IEEE Trans. Automat. Control, 2005, 50, 217-222. doi: 10.1109/TAC.2004.841935 CrossRef Google Scholar
[36]	Q. X. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Syst. Control Lett., 2018, 118, 62-68. doi: 10.1016/j.sysconle.2018.05.015 CrossRef Google Scholar