
-
Previous Article
Attractors for A sup-cubic weakly damped wave equation in $ \mathbb{R}^{3} $
- DCDS-B Home
- This Issue
-
Next Article
Detecting coupling directions with transcript mutual information: A comparative study
Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations
1. | School of Electronic and Information Engineering, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China |
2. | School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, Hubei 430074, China |
3. | Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, U.K |
Given an unstable hybrid stochastic differential equation (SDE, also known as an SDE with Markovian switching), can we design a delay feedback control to make the controlled hybrid SDE become asymptotically stable? The paper [
References:
[1] |
A. Ahlborn and U. Parlitz, Stabilizing unstable steady states using multiple delay deedback control, Physical Review Letters, 93 (2004), 264101. |
[2] |
A. Bahar and X. Mao,
Stochastic delay population dynamics, Journal of International Applied Mathematics, 11 (2004), 377-400.
|
[3] |
J. Cao, H. X. Li and D. W. C. Ho,
Synchronization criteria of Lur's systems with time-delay feedback control, Chaos, Solitons and Fractals, 23 (2005), 1285-1298.
doi: 10.1016/S0960-0779(04)00380-7. |
[4] |
L. Hu, X. Mao and Y. Shen,
Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Control Letters, 62 (2013), 178-187.
doi: 10.1016/j.sysconle.2012.11.009. |
[5] |
G. S. Ladde and V. Lakshmikantham, Random Differential Inequalities, Academic Press, 1980.
![]() ![]() |
[6] |
Y. Ji and H. J. Chizeck,
Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Transaction on Automatic Control, 35 (1990), 777-788.
doi: 10.1109/9.57016. |
[7] |
V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986.
![]() ![]() |
[8] |
A. L. Lewis, Option Valuation under Stochastic Volatility: with Mathematica Code, Finance Press, 2000.
![]() ![]() |
[9] |
X. Mao, Stability of Stochastic Differential Equations with Respect to Semimartingales, Longman Scientific and Technical, 1991. |
[10] |
X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, 1994. |
[11] |
X. Mao, Stochastic Differential Equations and Their Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2007. |
[12] |
X. Mao,
Stability of stochastic differential equations with Markovian switching, Stochastic Processes and Their Applications, 79 (1999), 45-67.
doi: 10.1016/S0304-4149(98)00070-2. |
[13] |
X. Mao,
Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica, 49 (2013), 3677-3681.
doi: 10.1016/j.automatica.2013.09.005. |
[14] |
X. Mao, J. Lam and L. Huang,
Stabilisation of hybrid stochastic differential equations by delay feedback control, Control Letters, 57 (2008), 927-935.
doi: 10.1016/j.sysconle.2008.05.002. |
[15] |
X. Mao, A. Matasov and A. B. Piunovskiy,
Stochastic differential delay equations with Markovian switching, Bernoulli, 6 (2000), 73-90.
doi: 10.2307/3318634. |
[16] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.
doi: 10.1142/p473.![]() ![]() ![]() |
[17] |
M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker, 1990. |
[18] |
S.-E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1984. |
[19] |
K. Pyragas,
Control of chaos via extended delay feedback, Physics Letters A, 206 (1995), 323-330.
doi: 10.1016/0375-9601(95)00654-L. |
[20] |
L. Shaikhet,
Stability of stochastic hereditary systems with Markov switching, Theory of Stochastic Processes, 2 (1996), 180-184.
|
[21] |
L. Wu, X. Su and P. Shi,
Sliding mode control with bounded $L_2$ gain performance of Markovian jump singular time-delay systems, Automatica, 48 (2012), 1929-1933.
doi: 10.1016/j.automatica.2012.05.064. |
[22] |
S. You, W. Liu, J. Lu, X. Mao and Q. Qiu,
Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM Journal on Control and Optimization, 53 (2015), 905-925.
doi: 10.1137/140985779. |
[23] |
D. Yue and Q. Han,
Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching, IEEE Transaction on Automatic Control, 50 (2005), 217-222.
doi: 10.1109/TAC.2004.841935. |
show all references
References:
[1] |
A. Ahlborn and U. Parlitz, Stabilizing unstable steady states using multiple delay deedback control, Physical Review Letters, 93 (2004), 264101. |
[2] |
A. Bahar and X. Mao,
Stochastic delay population dynamics, Journal of International Applied Mathematics, 11 (2004), 377-400.
|
[3] |
J. Cao, H. X. Li and D. W. C. Ho,
Synchronization criteria of Lur's systems with time-delay feedback control, Chaos, Solitons and Fractals, 23 (2005), 1285-1298.
doi: 10.1016/S0960-0779(04)00380-7. |
[4] |
L. Hu, X. Mao and Y. Shen,
Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Control Letters, 62 (2013), 178-187.
doi: 10.1016/j.sysconle.2012.11.009. |
[5] |
G. S. Ladde and V. Lakshmikantham, Random Differential Inequalities, Academic Press, 1980.
![]() ![]() |
[6] |
Y. Ji and H. J. Chizeck,
Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Transaction on Automatic Control, 35 (1990), 777-788.
doi: 10.1109/9.57016. |
[7] |
V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986.
![]() ![]() |
[8] |
A. L. Lewis, Option Valuation under Stochastic Volatility: with Mathematica Code, Finance Press, 2000.
![]() ![]() |
[9] |
X. Mao, Stability of Stochastic Differential Equations with Respect to Semimartingales, Longman Scientific and Technical, 1991. |
[10] |
X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, 1994. |
[11] |
X. Mao, Stochastic Differential Equations and Their Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2007. |
[12] |
X. Mao,
Stability of stochastic differential equations with Markovian switching, Stochastic Processes and Their Applications, 79 (1999), 45-67.
doi: 10.1016/S0304-4149(98)00070-2. |
[13] |
X. Mao,
Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica, 49 (2013), 3677-3681.
doi: 10.1016/j.automatica.2013.09.005. |
[14] |
X. Mao, J. Lam and L. Huang,
Stabilisation of hybrid stochastic differential equations by delay feedback control, Control Letters, 57 (2008), 927-935.
doi: 10.1016/j.sysconle.2008.05.002. |
[15] |
X. Mao, A. Matasov and A. B. Piunovskiy,
Stochastic differential delay equations with Markovian switching, Bernoulli, 6 (2000), 73-90.
doi: 10.2307/3318634. |
[16] |
X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.
doi: 10.1142/p473.![]() ![]() ![]() |
[17] |
M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker, 1990. |
[18] |
S.-E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1984. |
[19] |
K. Pyragas,
Control of chaos via extended delay feedback, Physics Letters A, 206 (1995), 323-330.
doi: 10.1016/0375-9601(95)00654-L. |
[20] |
L. Shaikhet,
Stability of stochastic hereditary systems with Markov switching, Theory of Stochastic Processes, 2 (1996), 180-184.
|
[21] |
L. Wu, X. Su and P. Shi,
Sliding mode control with bounded $L_2$ gain performance of Markovian jump singular time-delay systems, Automatica, 48 (2012), 1929-1933.
doi: 10.1016/j.automatica.2012.05.064. |
[22] |
S. You, W. Liu, J. Lu, X. Mao and Q. Qiu,
Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM Journal on Control and Optimization, 53 (2015), 905-925.
doi: 10.1137/140985779. |
[23] |
D. Yue and Q. Han,
Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching, IEEE Transaction on Automatic Control, 50 (2005), 217-222.
doi: 10.1109/TAC.2004.841935. |


[1] |
Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281 |
[2] |
Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure and Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397 |
[3] |
Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 |
[4] |
Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325 |
[5] |
Yong Ren, Wensheng Yin, Dongjin Zhu. Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3347-3360. doi: 10.3934/dcdsb.2018248 |
[6] |
Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $. Evolution Equations and Control Theory, 2021, 10 (4) : 921-935. doi: 10.3934/eect.2020096 |
[7] |
Nobuyuki Kato. Linearized stability and asymptotic properties for abstract boundary value functional evolution problems. Conference Publications, 1998, 1998 (Special) : 371-387. doi: 10.3934/proc.1998.1998.371 |
[8] |
Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic and Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001 |
[9] |
Samuel N. Cohen, Lukasz Szpruch. On Markovian solutions to Markov Chain BSDEs. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 257-269. doi: 10.3934/naco.2012.2.257 |
[10] |
Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255 |
[11] |
Dingjun Yao, Rongming Wang, Lin Xu. Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission. Journal of Industrial and Management Optimization, 2015, 11 (2) : 461-478. doi: 10.3934/jimo.2015.11.461 |
[12] |
Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197 |
[13] |
Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257 |
[14] |
Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3725-3757. doi: 10.3934/dcds.2021014 |
[15] |
Zhengyan Lin, Li-Xin Zhang. Convergence to a self-normalized G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 4-. doi: 10.1186/s41546-017-0013-8 |
[16] |
Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations and Control Theory, 2022, 11 (3) : 925-937. doi: 10.3934/eect.2021031 |
[17] |
Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004 |
[18] |
Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005 |
[19] |
Tzong-Yow Lee. Asymptotic results for super-Brownian motions and semilinear differential equations. Electronic Research Announcements, 1998, 4: 56-62. |
[20] |
Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control and Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]