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August  2019, 24(8): 4099-4116. doi: 10.3934/dcdsb.2019052

Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations

Zhenyu Lu 1, , Junhao Hu 2,, and  Xuerong Mao 3,

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

* Corresponding author: J. Hu

Received  April 2018 Revised  August 2018 Published  August 2019 Early access  February 2019

Fund Project: The authors would like to thank the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the Royal Society of Edinburgh (61294), the EPSRC (EP/K503174/1), the Natural Science Foundation of China (61773220, 61473334, 61876192, 61374085), the Ministry of Education (MOE) of China (MS2014DHDX020) for their financial support.

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 [14] by Mao et al. was the first to study the stabilisation by delay feedback controls for hybrid SDEs, though the stabilization by non-delay feedback controls had been well studied. A critical condition imposed in [14] is that both drift and diffusion coefficients of the given hybrid SDE need to satisfy the linear growth condition. However, many hybrid SDE models in the real world do not fulfill this condition (namely, they are highly nonlinear) and hence there is a need to develop a new theory for these highly nonlinear SDE models. The aim of this paper is to design delay feedback controls in order to stabilise a class of highly nonlinear hybrid SDEs whose coefficients satisfy the polynomial growth condition.

Keywords: Brownian motion, Markov chain, asymptotic stability, Lypunov functional.
Mathematics Subject Classification: Primary: 60H10, 60J10; Secondary: 93D15.
Citation: Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
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.

Figure 4.1.  The computer simulation of the sample paths of the Markov chain and the SDDE (2.4) with control (4.2) and $ \tau = 0.06 $ using the Euler–Maruyama method with step size $ 10^{-4} $.
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Figure 4.2.  The computer simulation of the sample paths of the Markov chain and the SDDE (2.4) with control (4.3) and $ \tau = 0.01 $ using the Euler–Maruyama method with step size $ 10^{-4} $
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