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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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |


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