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doi: 10.3934/dcdsb.2021055
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Stabilization by intermittent control for hybrid stochastic differential delay equations

Wei Mao 1, , Yanan Jiang 2, , Liangjian Hu 3,, and  Xuerong Mao 4,

1. 

School of mathematics and information technology, Jiangsu Second Normal University, Nanjing, 210013, China
2. 

College of Information Sciences and Technology, Donghua University, Shanghai, 201620, China
3. 

Department of Applied Mathematics, Donghua University, Shanghai 201620, China
4. 

Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, U.K

* Corresponding author: Liangjian Hu

Received  August 2020 Revised  January 2021 Early access February 2021

Fund Project: The research of W.Mao was supported by the National Natural Science Foundation of China(11401261), "333 High-level Project" of Jiangsu Province and the Qing Lan Project of Jiangsu Province. The research of L.Hu was supported by the National Natural Science Foundation of China (11471071). The research of X.Mao was supported by the Leverhulme Trust (RF-2015-385), the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the EPSRC (EP/K503174/1)

This paper is concerned with stablization of hybrid differential equations by intermittent control based on delay observations. By M-matrix theory and intermittent control strategy, we establish a sufficient stability criterion on intermittent hybrid stochastic differential equations. Meantime, we show that hybrid differential equations can be stabilized by intermittent control based on delay observations if the delay time $ \tau $ is bounded by $ \tau^* $. Finally, an example is presented to illustrate our theory.

Keywords: Stochastic stabilization, hybrid stochastic differential equations, intermittent control, delay observation.
Mathematics Subject Classification: Primary: 60H10, 93E15.
Citation: Wei Mao, Yanan Jiang, Liangjian Hu, Xuerong Mao. Stabilization by intermittent control for hybrid stochastic differential delay equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021055
References:
[1]

J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control., 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[2]

J. A. D. Appleby and X. Mao, Stochastic stabilization of functional differential equations, Syst. Control. Lett., 54 (2005), 1069-1081.  doi: 10.1016/j.sysconle.2005.03.003.  Google Scholar

[3]

L. Arnold, H. Crauel and V. Wihstutz, Stabilization of linear systems by noise, SIAM J. Control Optim., 21 (1983), 451-461.  doi: 10.1137/0321027.  Google Scholar

[4]

T. Caraballo, M. J. Garrido-Atienza and J. Real, Stochastic stabilization of differential systems with general decay rate, Syst. Control. Lett., 48 (2003), 397-406.  doi: 10.1016/S0167-6911(02)00293-1.  Google Scholar

[5]

W. Chen, S. Xu and Y. Zou, Stabilization of hybrid neutral stochastic differential delay equations by delay feedback control, Syst. Control. Lett., 88 (2016), 1-13.  doi: 10.1016/j.sysconle.2015.04.004.  Google Scholar

[6]

F. Deng, Q. Luo and X. Mao, Stochastic stabilization of hybrid differential equations, Automatica., 48 (2012), 2321-2328.  doi: 10.1016/j.automatica.2012.06.044.  Google Scholar

[7]

R. Z. Has'minskiǐ, Stochastic Stability of Differential Equations, Sithoff Noordhoff, Alphen aan den Rijn, Netherlands., 1980.  Google Scholar

[8]

J. Hu, W. Liu, F. Deng and X. Mao, Advances in stabilization of hybrid stochastic differential equations by delay feedback control, SIAM J. Control Optim., 58 (2020), 735-754.  doi: 10.1137/19M1270240.  Google Scholar

[9]

X. Li and X. Mao, Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control, Automatica., 112 (2020), 108657. doi: 10.1016/j.automatica.2019.108657.  Google Scholar

[10]

L. Liu, M. Perc and J. Cao, Aperiodically intermittent stochastic stabilization via discrete time or delay feedback control, Science in China Information Sciences., 62 (2019), 1-13.  doi: 10.1007/s11432-018-9600-3.  Google Scholar

[11]

L. Liu and Z. Wu, Intermittent stochastic stabilization based on discrete-time observation with time delay, Syst. Control. Lett., 137 (2020), 1-11.  doi: 10.1016/j.sysconle.2020.104626.  Google Scholar

[12]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College, London., (2006). doi: 10.1142/p473.  Google Scholar

[13]

X. Mao, G. G. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica., 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar

[14]

X. Mao, Stochastic stabilisation and destabilisation, Syst. Control. Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[15]

X. Mao, J. Lam and L. Huang, Stabilization of hybrid stochastic differential equations by delay feedback control, Syst. Control. Lett., 57 (2008), 927-935.  doi: 10.1016/j.sysconle.2008.05.002.  Google Scholar

[16]

X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Trans. Automat. Control., 61 (2016), 1619-1624. doi: 10.1109/TAC.2015.2471696.  Google Scholar

[17]

Y. Ren and W. Yin, Quasi sure exponential stabilization of nonlinear systems via intermittent G-Brownian motion, Discret. Contin. Dyn. Syst. Ser. B., 24 (2019), 5871-5883.  doi: 10.3934/dcdsb.2019110.  Google Scholar

[18]

Y. Ren, W. Yin and R. Sakthivel, Stabilization of stochastic differential equations driven by G-Brownian motion with feedback control based on discrete-Time state observation, Automatica., 95 (2018), 146-151.  doi: 10.1016/j.automatica.2018.05.039.  Google Scholar

[19]

M. Scheutzow, Stabilization and destabilization by noise in the plane, Stocha. Anal. Appl., 11 (1993), 97-113.  doi: 10.1080/07362999308809304.  Google Scholar

[20]

F. Wu and S. Hu, Suppression and stabilisation of noise, Int. J. Control., 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108.  Google Scholar

[21]

F. Wu and S. Hu, Stochastic Suppression and stabilization of delay differential systems, Int. J. Robust. Nonlin. Control., 21 (2011), 488-500.  doi: 10.1002/rnc.1606.  Google Scholar

[22]

G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM. J. Appl. Math., 72 (2012), 1361-1382.  doi: 10.1137/110851171.  Google Scholar

[23]

W. Yin, J. Cao and Y. Ren, Quasi-sure exponential stabilization of stochastic systems induced by G-Brownian motion with discrete time feedback control, J. Math, Anal. Appl., 474 (2019), 276-289.  doi: 10.1016/j.jmaa.2019.01.045.  Google Scholar

[24]

W. Yin and J. Cao, Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4493-4513. doi: 10.3934/dcdsb.2020109.  Google Scholar

[25]

C. Yuan and J. Lygeros, Stabilization of a class of stochastic differential equations with Markovian switching, Syst. Control. Lett., 54 (2005), 819-833.  doi: 10.1016/j.sysconle.2005.01.001.  Google Scholar

[26]

B. Zhang, F. Deng, S. Peng and S. Xie, Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system, J. Frankl. Inst., 355 (2018), 3829-3852.  doi: 10.1016/j.jfranklin.2017.12.033.  Google Scholar

[27]

X. Zong, F. Wu and G. Yin, Stochastic regularization and stabilization of hybrid functional differential equations, 2015 54th IEEE Conference on Decision and Control (CDC)., (2015), 1211–1216. doi: 10.1109/CDC.2015.7402376.  Google Scholar

show all references

References:
[1]

J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control., 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[2]

J. A. D. Appleby and X. Mao, Stochastic stabilization of functional differential equations, Syst. Control. Lett., 54 (2005), 1069-1081.  doi: 10.1016/j.sysconle.2005.03.003.  Google Scholar

[3]

L. Arnold, H. Crauel and V. Wihstutz, Stabilization of linear systems by noise, SIAM J. Control Optim., 21 (1983), 451-461.  doi: 10.1137/0321027.  Google Scholar

[4]

T. Caraballo, M. J. Garrido-Atienza and J. Real, Stochastic stabilization of differential systems with general decay rate, Syst. Control. Lett., 48 (2003), 397-406.  doi: 10.1016/S0167-6911(02)00293-1.  Google Scholar

[5]

W. Chen, S. Xu and Y. Zou, Stabilization of hybrid neutral stochastic differential delay equations by delay feedback control, Syst. Control. Lett., 88 (2016), 1-13.  doi: 10.1016/j.sysconle.2015.04.004.  Google Scholar

[6]

F. Deng, Q. Luo and X. Mao, Stochastic stabilization of hybrid differential equations, Automatica., 48 (2012), 2321-2328.  doi: 10.1016/j.automatica.2012.06.044.  Google Scholar

[7]

R. Z. Has'minskiǐ, Stochastic Stability of Differential Equations, Sithoff Noordhoff, Alphen aan den Rijn, Netherlands., 1980.  Google Scholar

[8]

J. Hu, W. Liu, F. Deng and X. Mao, Advances in stabilization of hybrid stochastic differential equations by delay feedback control, SIAM J. Control Optim., 58 (2020), 735-754.  doi: 10.1137/19M1270240.  Google Scholar

[9]

X. Li and X. Mao, Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control, Automatica., 112 (2020), 108657. doi: 10.1016/j.automatica.2019.108657.  Google Scholar

[10]

L. Liu, M. Perc and J. Cao, Aperiodically intermittent stochastic stabilization via discrete time or delay feedback control, Science in China Information Sciences., 62 (2019), 1-13.  doi: 10.1007/s11432-018-9600-3.  Google Scholar

[11]

L. Liu and Z. Wu, Intermittent stochastic stabilization based on discrete-time observation with time delay, Syst. Control. Lett., 137 (2020), 1-11.  doi: 10.1016/j.sysconle.2020.104626.  Google Scholar

[12]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College, London., (2006). doi: 10.1142/p473.  Google Scholar

[13]

X. Mao, G. G. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica., 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar

[14]

X. Mao, Stochastic stabilisation and destabilisation, Syst. Control. Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[15]

X. Mao, J. Lam and L. Huang, Stabilization of hybrid stochastic differential equations by delay feedback control, Syst. Control. Lett., 57 (2008), 927-935.  doi: 10.1016/j.sysconle.2008.05.002.  Google Scholar

[16]

X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Trans. Automat. Control., 61 (2016), 1619-1624. doi: 10.1109/TAC.2015.2471696.  Google Scholar

[17]

Y. Ren and W. Yin, Quasi sure exponential stabilization of nonlinear systems via intermittent G-Brownian motion, Discret. Contin. Dyn. Syst. Ser. B., 24 (2019), 5871-5883.  doi: 10.3934/dcdsb.2019110.  Google Scholar

[18]

Y. Ren, W. Yin and R. Sakthivel, Stabilization of stochastic differential equations driven by G-Brownian motion with feedback control based on discrete-Time state observation, Automatica., 95 (2018), 146-151.  doi: 10.1016/j.automatica.2018.05.039.  Google Scholar

[19]

M. Scheutzow, Stabilization and destabilization by noise in the plane, Stocha. Anal. Appl., 11 (1993), 97-113.  doi: 10.1080/07362999308809304.  Google Scholar

[20]

F. Wu and S. Hu, Suppression and stabilisation of noise, Int. J. Control., 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108.  Google Scholar

[21]

F. Wu and S. Hu, Stochastic Suppression and stabilization of delay differential systems, Int. J. Robust. Nonlin. Control., 21 (2011), 488-500.  doi: 10.1002/rnc.1606.  Google Scholar

[22]

G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM. J. Appl. Math., 72 (2012), 1361-1382.  doi: 10.1137/110851171.  Google Scholar

[23]

W. Yin, J. Cao and Y. Ren, Quasi-sure exponential stabilization of stochastic systems induced by G-Brownian motion with discrete time feedback control, J. Math, Anal. Appl., 474 (2019), 276-289.  doi: 10.1016/j.jmaa.2019.01.045.  Google Scholar

[24]

W. Yin and J. Cao, Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4493-4513. doi: 10.3934/dcdsb.2020109.  Google Scholar

[25]

C. Yuan and J. Lygeros, Stabilization of a class of stochastic differential equations with Markovian switching, Syst. Control. Lett., 54 (2005), 819-833.  doi: 10.1016/j.sysconle.2005.01.001.  Google Scholar

[26]

B. Zhang, F. Deng, S. Peng and S. Xie, Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system, J. Frankl. Inst., 355 (2018), 3829-3852.  doi: 10.1016/j.jfranklin.2017.12.033.  Google Scholar

[27]

X. Zong, F. Wu and G. Yin, Stochastic regularization and stabilization of hybrid functional differential equations, 2015 54th IEEE Conference on Decision and Control (CDC)., (2015), 1211–1216. doi: 10.1109/CDC.2015.7402376.  Google Scholar

Figure 1.  The sample paths of the hybrid differential equations (21)
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Figure 2.  The sample paths of the intermittently hybrid SDEs (22) with $ \theta = 0.95 $
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Figure 3.  The sample paths of the intermittently hybrid SDDEs (23)
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