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doi: 10.3934/dcdsb.2021207
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## Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control

 Department of Mathematics, Anhui Normal University, Wuhu 241000, China

* Corresponding author: Yong Ren

Received  February 2020 Revised  July 2021 Early access August 2021

Fund Project: This work is supported by the NNSF of China (11871076)

In this work, the issue of stabilization for a class of continuous-time hybrid stochastic systems with Lévy noise (HLSDEs, in short) is explored by using periodic intermittent control. As for the unstable HLSDEs, we design a periodic intermittent controller. The main idea is to compare the controlled system with a stabilized one with a periodic intermittent drift coefficient, which enables us to use the existing stability results on the HLSDEs. An illustrative example is proposed to show the feasibility of the obtained result.

Citation: Yong Ren, Qi Zhang. Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021207
##### References:
 [1] K. Ding, Q. Zhu and H. Li, A generalized system approach to intermittent nonfragile control of stochastic neutral time-varying delay systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, on line, (2020). doi: 10.1109/TSMC.2020.2965091.  Google Scholar [2] M. Li and F. Deng, Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with Lévy noise, Nonlinear Anal. Hybrid Syst., 24 (2017), 171-185.  doi: 10.1016/j.nahs.2017.01.001.  Google Scholar [3] C. Liu, X. Xun, Q. Zhang and Y. Li, Dynamical analysis and optimal control in a hybrid stochastic double delayed bioeconomic system with impulsive contaminants emission and Lévy jumps, Appl. Math. Comput., 352 (2019), 99-118.  doi: 10.1016/j.amc.2019.01.045.  Google Scholar [4] D. Liu, W. Wang and J. L. Menaldi, Almost sure asymptotic stabilization of differential equations with time-varying delay by Lévy noise, Nonlinear Dynamics, 79 (2015), 163-172.  doi: 10.1007/s11071-014-1653-1.  Google Scholar [5] M. Liu and Y. Zhu, Stationary distribution and ergodicity of a stochastic hybrid competition model with Lévy jumps, Nonlinear Anal. Hybrid Syst., 30 (2018), 225-239.  doi: 10.1016/j.nahs.2018.05.002.  Google Scholar [6] X. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.  Google Scholar [7] X. Mao, J. Lam and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Systems Control Lett., 57 (2008), 927-935.  doi: 10.1016/j.sysconle.2008.05.002.  Google Scholar [8] X. Mao, G. G Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica J. IFAC, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar [9] Q. Qiu, W. Liu, L. Hu and J. Lu, Stabilisation of hybrid stochastic systems under disrete observation and sample delay, Control. Theorey Appl., 33 (2016), 1023-1030.  doi: 10.7641/CTA.2016.50995.  Google Scholar [10] J. Shao, Stabilization of regime-switching processes by feedback control based on discrete time state observations, SIAM J. Control Optim., 55 (2017), 724-740.  doi: 10.1137/16M1066336.  Google Scholar [11] F. Wu, X. Mao and S. Hu, Stochastic suppression and stabilization of functional differential equations, Systems Control Lett., 59 (2010), 745-753.  doi: 10.1016/j.sysconle.2010.08.011.  Google Scholar [12] Y. Wu, S. Zhuang and W. Li, Periodically intermittent discrete observation control for synchronization of the general stochastic complex network, Automatica J. IFAC, 110 (2019), 108591, 11 pp. doi: 10.1016/j.automatica.2019.108591.  Google Scholar [13] Y. Xu, H. Zhou and W. Li, Stabilisation of stochastic delayed systems with Lévy noise on networks via periodically intermittent control, Internat. J. Control, 93 (2020), 505-518.  doi: 10.1080/00207179.2018.1479538.  Google Scholar [14] 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 [15] W. Yin, J. Cao, Y. Ren and G. Zheng, Improved results on stabilization of $G$-SDEs by feedback control based on discrete-time observations, SIAM J. Control Optim., 59 (2021), 1927-1950.  doi: 10.1137/20M1311028.  Google Scholar [16] 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 J. Control Optim., 53 (2015), 905-925.  doi: 10.1137/140985779.  Google Scholar [17] W. Zhou, J. Yang, X. Yang, A. Dai, H. Liu and J. Fang, $p$th Moment exponential stability of stochastic delayed hybrid systems with Lévy noise, Appl. Math. Model., 39 (2015), 5650-5658.  doi: 10.1016/j.apm.2015.01.025.  Google Scholar [18] Q. Zhu, Asymptotic stability in the $p$th moment for stochastic differential equations with Lévy noise, J. Math. Anal. Appl., 416 (2014), 126-142.  doi: 10.1016/j.jmaa.2014.02.016.  Google Scholar [19] Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with Lêvy noise and Markov switching, Internat. J. Control, 90 (2017), 1703-1712.  doi: 10.1080/00207179.2016.1219069.  Google Scholar [20] Q. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Systems Control Lett., 118 (2018), 62-68.  doi: 10.1016/j.sysconle.2018.05.015.  Google Scholar [21] X. Zong, F. Wu, G. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.  Google Scholar

show all references

##### References:
 [1] K. Ding, Q. Zhu and H. Li, A generalized system approach to intermittent nonfragile control of stochastic neutral time-varying delay systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, on line, (2020). doi: 10.1109/TSMC.2020.2965091.  Google Scholar [2] M. Li and F. Deng, Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with Lévy noise, Nonlinear Anal. Hybrid Syst., 24 (2017), 171-185.  doi: 10.1016/j.nahs.2017.01.001.  Google Scholar [3] C. Liu, X. Xun, Q. Zhang and Y. Li, Dynamical analysis and optimal control in a hybrid stochastic double delayed bioeconomic system with impulsive contaminants emission and Lévy jumps, Appl. Math. Comput., 352 (2019), 99-118.  doi: 10.1016/j.amc.2019.01.045.  Google Scholar [4] D. Liu, W. Wang and J. L. Menaldi, Almost sure asymptotic stabilization of differential equations with time-varying delay by Lévy noise, Nonlinear Dynamics, 79 (2015), 163-172.  doi: 10.1007/s11071-014-1653-1.  Google Scholar [5] M. Liu and Y. Zhu, Stationary distribution and ergodicity of a stochastic hybrid competition model with Lévy jumps, Nonlinear Anal. Hybrid Syst., 30 (2018), 225-239.  doi: 10.1016/j.nahs.2018.05.002.  Google Scholar [6] X. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.  Google Scholar [7] X. Mao, J. Lam and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Systems Control Lett., 57 (2008), 927-935.  doi: 10.1016/j.sysconle.2008.05.002.  Google Scholar [8] X. Mao, G. G Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica J. IFAC, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar [9] Q. Qiu, W. Liu, L. Hu and J. Lu, Stabilisation of hybrid stochastic systems under disrete observation and sample delay, Control. Theorey Appl., 33 (2016), 1023-1030.  doi: 10.7641/CTA.2016.50995.  Google Scholar [10] J. Shao, Stabilization of regime-switching processes by feedback control based on discrete time state observations, SIAM J. Control Optim., 55 (2017), 724-740.  doi: 10.1137/16M1066336.  Google Scholar [11] F. Wu, X. Mao and S. Hu, Stochastic suppression and stabilization of functional differential equations, Systems Control Lett., 59 (2010), 745-753.  doi: 10.1016/j.sysconle.2010.08.011.  Google Scholar [12] Y. Wu, S. Zhuang and W. Li, Periodically intermittent discrete observation control for synchronization of the general stochastic complex network, Automatica J. IFAC, 110 (2019), 108591, 11 pp. doi: 10.1016/j.automatica.2019.108591.  Google Scholar [13] Y. Xu, H. Zhou and W. Li, Stabilisation of stochastic delayed systems with Lévy noise on networks via periodically intermittent control, Internat. J. Control, 93 (2020), 505-518.  doi: 10.1080/00207179.2018.1479538.  Google Scholar [14] 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 [15] W. Yin, J. Cao, Y. Ren and G. Zheng, Improved results on stabilization of $G$-SDEs by feedback control based on discrete-time observations, SIAM J. Control Optim., 59 (2021), 1927-1950.  doi: 10.1137/20M1311028.  Google Scholar [16] 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 J. Control Optim., 53 (2015), 905-925.  doi: 10.1137/140985779.  Google Scholar [17] W. Zhou, J. Yang, X. Yang, A. Dai, H. Liu and J. Fang, $p$th Moment exponential stability of stochastic delayed hybrid systems with Lévy noise, Appl. Math. Model., 39 (2015), 5650-5658.  doi: 10.1016/j.apm.2015.01.025.  Google Scholar [18] Q. Zhu, Asymptotic stability in the $p$th moment for stochastic differential equations with Lévy noise, J. Math. Anal. Appl., 416 (2014), 126-142.  doi: 10.1016/j.jmaa.2014.02.016.  Google Scholar [19] Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with Lêvy noise and Markov switching, Internat. J. Control, 90 (2017), 1703-1712.  doi: 10.1080/00207179.2016.1219069.  Google Scholar [20] Q. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Systems Control Lett., 118 (2018), 62-68.  doi: 10.1016/j.sysconle.2018.05.015.  Google Scholar [21] X. Zong, F. Wu, G. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.  Google Scholar
The paths of the solution $Y(s)$ to the system (39) for state 1
The paths of the solution $Y(s)$ to the system (40) for state 1
The paths of the solution $Y(s)$ to the system (39) for state 2
The paths of the solution $Y(s)$ to the system (40) for state 2
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