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A learning-enhanced projection method for solving convex feasibility problems
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 |
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.
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. |
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. |
[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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