Figure 1.
The paths of the solution $ Y(s) $ to the system (39) for state 1
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July
2022, 27(7): 3811-3829.
doi: 10.3934/dcdsb.2021207
Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control
Department of Mathematics, Anhui Normal University, Wuhu 241000, China |
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.
Keywords:
Stabilization,
Lévy noise,
periodically intermittent control,
discrete time state observation.
Mathematics Subject Classification:
Primary: 93D15; Secondary: 60H10, 60J75.
Citation:
Yong Ren, Qi Zhang. Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control. Discrete and Continuous Dynamical Systems - B,
2022, 27
(7)
: 3811-3829.
doi: 10.3934/dcdsb.2021207
![]()
References:
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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. |
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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.
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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. |
| [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. |
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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.
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X. Mao,
Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.
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X. Mao, J. Lam and L. Huang,
Stabilisation of hybrid stochastic differential equations by delay feedback control, Systems Control Lett., 57 (2008), 927-935.
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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. |
| [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. |
| [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. |
| [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. |
| [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.
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| [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.
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| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
Figure 2.
The paths of the solution $ Y(s) $ to the system (40) for state 1
Figure 3.
The paths of the solution $ Y(s) $ to the system (39) for state 2
Figure 4.
The paths of the solution $ Y(s) $ to the system (40) for state 2
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