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

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

* Corresponding author: J. Hu

Received  April 2018 Revised  August 2018 Published  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.

Citation: Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & 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. Google Scholar

[2]

A. Bahar and X. Mao, Stochastic delay population dynamics, Journal of International Applied Mathematics, 11 (2004), 377-400.   Google Scholar

[3]

J. CaoH. 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.  Google Scholar

[4]

L. HuX. 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.  Google Scholar

[5] G. S. Ladde and V. Lakshmikantham, Random Differential Inequalities, Academic Press, 1980.   Google Scholar
[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.  Google Scholar

[7] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986.   Google Scholar
[8] A. L. Lewis, Option Valuation under Stochastic Volatility: with Mathematica Code, Finance Press, 2000.   Google Scholar
[9]

X. Mao, Stability of Stochastic Differential Equations with Respect to Semimartingales, Longman Scientific and Technical, 1991.  Google Scholar

[10]

X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

X. MaoJ. 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.  Google Scholar

[15]

X. MaoA. Matasov and A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli, 6 (2000), 73-90.  doi: 10.2307/3318634.  Google Scholar

[16] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.  doi: 10.1142/p473.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[20]

L. Shaikhet, Stability of stochastic hereditary systems with Markov switching, Theory of Stochastic Processes, 2 (1996), 180-184.   Google Scholar

[21]

L. WuX. 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.  Google Scholar

[22]

S. YouW. LiuJ. LuX. 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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[3]

J. CaoH. 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.  Google Scholar

[4]

L. HuX. 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.  Google Scholar

[5] G. S. Ladde and V. Lakshmikantham, Random Differential Inequalities, Academic Press, 1980.   Google Scholar
[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.  Google Scholar

[7] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986.   Google Scholar
[8] A. L. Lewis, Option Valuation under Stochastic Volatility: with Mathematica Code, Finance Press, 2000.   Google Scholar
[9]

X. Mao, Stability of Stochastic Differential Equations with Respect to Semimartingales, Longman Scientific and Technical, 1991.  Google Scholar

[10]

X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

X. MaoJ. 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.  Google Scholar

[15]

X. MaoA. Matasov and A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli, 6 (2000), 73-90.  doi: 10.2307/3318634.  Google Scholar

[16] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.  doi: 10.1142/p473.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[20]

L. Shaikhet, Stability of stochastic hereditary systems with Markov switching, Theory of Stochastic Processes, 2 (1996), 180-184.   Google Scholar

[21]

L. WuX. 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.  Google Scholar

[22]

S. YouW. LiuJ. LuX. 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.  Google Scholar

[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.  Google Scholar

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} $.
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} $
[1]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[2]

Sushil Kumar Dey, Bibhas C. Giri. Coordination of a sustainable reverse supply chain with revenue sharing contract. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020165

[3]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[4]

Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167

[5]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[6]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[7]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[8]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[9]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[10]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[11]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[12]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[13]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[14]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[15]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[16]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (265)
  • HTML views (423)
  • Cited by (1)

Other articles
by authors

[Back to Top]