# American Institute of Mathematical Sciences

January  2022, 27(1): 569-581. doi: 10.3934/dcdsb.2021055

## Stabilization by intermittent control for hybrid stochastic differential delay equations

 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 Published  January 2022 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.

Citation: Wei Mao, Yanan Jiang, Liangjian Hu, Xuerong Mao. Stabilization by intermittent control for hybrid stochastic differential delay equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 569-581. 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. [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. [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. [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. [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. [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. [7] R. Z. Has'minskiǐ, Stochastic Stability of Differential Equations, Sithoff Noordhoff, Alphen aan den Rijn, Netherlands., 1980. [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. [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. [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. [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. [12] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College, London., (2006). doi: 10.1142/p473. [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. [14] X. Mao, Stochastic stabilisation and destabilisation, Syst. Control. Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7. [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. [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. [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. [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. [19] M. Scheutzow, Stabilization and destabilization by noise in the plane, Stocha. Anal. Appl., 11 (1993), 97-113.  doi: 10.1080/07362999308809304. [20] F. Wu and S. Hu, Suppression and stabilisation of noise, Int. J. Control., 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [7] R. Z. Has'minskiǐ, Stochastic Stability of Differential Equations, Sithoff Noordhoff, Alphen aan den Rijn, Netherlands., 1980. [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. [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. [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. [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. [12] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College, London., (2006). doi: 10.1142/p473. [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. [14] X. Mao, Stochastic stabilisation and destabilisation, Syst. Control. Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7. [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. [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. [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. [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. [19] M. Scheutzow, Stabilization and destabilization by noise in the plane, Stocha. Anal. Appl., 11 (1993), 97-113.  doi: 10.1080/07362999308809304. [20] F. Wu and S. Hu, Suppression and stabilisation of noise, Int. J. Control., 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108. [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. [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. [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. [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. [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. [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. [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.
The sample paths of the hybrid differential equations (21)
The sample paths of the intermittently hybrid SDEs (22) with $\theta = 0.95$
The sample paths of the intermittently hybrid SDDEs (23)
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