
-
Previous Article
Maximal discrete sparsity in parabolic optimal control with measures
- MCRF Home
- This Issue
-
Next Article
The Kato smoothing effect for the nonlinear regularized Schrödinger equation on compact manifolds
Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations
1. | Warwick Manufacturing Group, University of Warwick, Coventry, CV4 7AL, UK |
2. | Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, UK |
In 2013, Mao initiated the study of stabilization of continuous-time hybrid stochastic differential equations (SDEs) by feedback control based on discrete-time state observations. In recent years, this study has been further developed while using a constant observation interval. However, time-varying observation frequencies have not been discussed for this study. Particularly for non-autonomous periodic systems, it's more sensible to consider the time-varying property and observe the system at periodic time-varying frequencies, in terms of control efficiency. This paper introduces a periodic observation interval sequence, and investigates how to stabilize a periodic SDE by feedback control based on periodic observations, in the sense that, the controlled system achieves $ L^p $-stability for $ p>1 $, almost sure asymptotic stability and $ p $th moment asymptotic stability for $ p \ge 2 $. This paper uses the Lyapunov method and inequalities to derive the theory. We also verify the existence of the observation interval sequence and explain how to calculate it. Finally, an illustrative example is given after a useful corollary. By considering the time-varying property of the system, we reduce the observation frequency dramatically and hence reduce the observational cost for control.
References:
[1] |
L. Arnold and C. Tudor,
Stationary and almost periodic solutions of almost periodic affine stochastic differential equations, Stochastics and Stochastic Reports, 64 (1998), 177-193.
doi: 10.1080/17442509808834163. |
[2] |
G. K. Basak, A. Bisi and M. K. Ghosh,
Stability of a random diffusion with linear drift, J. Math. Anal. Appl., 202 (1996), 604-622.
doi: 10.1006/jmaa.1996.0336. |
[3] |
P. H. Bezandry and T. Diagana, Almost Periodic Stochastic Processes, Springer, New York, 2011.
doi: 10.1007/978-1-4419-9476-9. |
[4] |
R. Dong, Stabilization of Stochastic Differential Equations by Feedback Controls Based on Discrete-time Observations, PhD thesis, University of Strathclyde, UK, 2019. Google Scholar |
[5] |
R. Dong,
Almost sure exponential stabilization by stochastic feedback control based on discrete-time observations, Stochastic Analysis and Applications, 36 (2018), 561-583.
doi: 10.1080/07362994.2018.1433046. |
[6] |
R. Dong and X. R. Mao,
On $p$th moment stabilization of hybrid systems by discrete-time feedback control, Stochastic Analysis and Applications, 35 (2017), 803-822.
doi: 10.1080/07362994.2017.1324798. |
[7] |
L. Y. Hu, Y. Ren and T. B. Xu,
$p$-Moment stability of solutions to stochastic differential equations driven by $G$-Brownian motion, Applied Mathematics and Computation, 230 (2014), 231-237.
doi: 10.1016/j.amc.2013.12.111. |
[8] |
C. X. Huang, Y. G. He, L. H. Huang and W. J. Zhu,
$p$th moment stability analysis of stochastic recurrent neural networks with time-varying delays, Information Sciences, 178 (2008), 2194-2203.
doi: 10.1016/j.ins.2008.01.008. |
[9] |
Y. D. Ji and H. J. Chizeck,
Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control, 35 (1990), 777-788.
doi: 10.1109/9.57016. |
[10] |
Y. Y. Li, J. Q. Lu, C. H. Kou, X. R. Mao and J. F. Pan,
Robust stabilization of hybrid uncertain stochastic systems by discrete-time feedback control, Optimal Control Applications and Methods, 38 (2017), 847-859.
doi: 10.1002/oca.2293. |
[11] |
X. Y. Li and X. R. Mao,
A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica J. IFAC, 48 (2012), 2329-2334.
doi: 10.1016/j.automatica.2012.06.045. |
[12] |
J. Q. Lu, Y. Y. Li, X. R. Mao and Q. W. Qiu,
Stabilization of hybrid systems by feedback control based on discrete-time state and mode observations, Asian Journal of Control, 19 (2017), 1943-1953.
doi: 10.1002/asjc.1515. |
[13] |
X. R. Mao,
Stability of stochastic differential equations with Markovian switching, Sto. Proc. Their Appl., 79 (1999), 45-67.
doi: 10.1016/S0304-4149(98)00070-2. |
[14] |
X. R. Mao,
Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Transactions on Automatic Control, 47 (2002), 1604-1612.
doi: 10.1109/TAC.2002.803529. |
[15] |
X. R. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
[16] |
X. R. 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. |
[17] |
X. R. Mao,
Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Transactions on Automatic Control, 61 (2016), 1619-1624.
doi: 10.1109/TAC.2015.2471696. |
[18] |
X. R. Mao, G. G. Yin and C. G. Yuan,
Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43 (2007), 264-273.
doi: 10.1016/j.automatica.2006.09.006. |
[19] |
X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.
doi: 10.1142/p473.![]() ![]() |
[20] |
X. R. Mao, W. Liu, L. J. Hu, Q. Luo and J. Q. Lu,
Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Systems Control Lett., 73 (2014), 88-95.
doi: 10.1016/j.sysconle.2014.08.011. |
[21] |
Y. G. Niu, D. W. C. Ho and J. Lam,
Robust integral sliding mode control for uncertain stochastic systems with time-varying delay, Automatica J. IFAC, 41 (2005), 873-880.
doi: 10.1016/j.automatica.2004.11.035. |
[22] |
R. Rifhat, L. Wang and Z. D. Teng,
Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients, Physica A: Statistical Mechanics and its Applications, 481 (2017), 176-190.
doi: 10.1016/j.physa.2017.04.016. |
[23] |
J. L. Sabo and D. M. Post,
Quantifying periodic, stochastic, and catastrophic environmental variation, Ecological Monographs, 78 (2008), 19-40.
doi: 10.1890/06-1340.1. |
[24] |
J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, New Jersey, 1991. Google Scholar |
[25] |
G. F. Song, B.-C. Zheng and X. R. Mao,
Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time observations of state and mode, IET Control Theory Appl., 11 (2017), 301-307.
doi: 10.1049/iet-cta.2016.0635. |
[26] |
M. H. Sun, J. Lam, S. Y. Xu and Y. Zou,
Robust exponential stabilization for Markovian jump systems with mode-dependent input delay, Automatica J. IFAC, 43 (2007), 1799-1807.
doi: 10.1016/j.automatica.2007.03.005. |
[27] |
I. Tsiakas,
Periodic stochastic volatility and fat tails, Journal of Financial Econometrics, 4 (2006), 90-135.
doi: 10.1093/jjfinec/nbi023. |
[28] |
C. Wang and R. P. Agarwal,
Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model, Applied Mathematics Letters, 70 (2017), 58-65.
doi: 10.1016/j.aml.2017.03.009. |
[29] |
C. Wang, R. P. Agarwal and S. Rathinasamy,
Almost periodic oscillations for delay impulsive stochastic Nicholson's blowflies timescale model, Computational and Applied Mathematics, 37 (2018), 3005-3026.
doi: 10.1007/s40314-017-0495-0. |
[30] |
G. C. Wang, Z. Wu and J. Xiong,
A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information, IEEE Transactions on Automatic Control, 60 (2015), 2904-2916.
doi: 10.1109/TAC.2015.2411871. |
[31] |
Y. Wang and Z. Liu,
Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25 (2012), 2803-2821.
doi: 10.1088/0951-7715/25/10/2803. |
[32] |
L. G. Wu, P. Shi and H. J. Gao,
State estimation and sliding mode control of Markovian jump singular systems, IEEE Transactions on Automatic Control, 55 (2010), 1213-1219.
doi: 10.1109/TAC.2010.2042234. |
[33] |
S. R. You, L. J. Hu, W. Mao and X. R. Mao,
Robustly exponential stabilization of hybrid uncertain systems by feedback controls based on discrete-time observations, Statist. Probab. Lett., 102 (2015), 8-16.
doi: 10.1016/j.spl.2015.03.006. |
[34] |
S. R. You, W. Liu, J. Q. Lu, X. R. Mao and Q. W. 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. |
show all references
References:
[1] |
L. Arnold and C. Tudor,
Stationary and almost periodic solutions of almost periodic affine stochastic differential equations, Stochastics and Stochastic Reports, 64 (1998), 177-193.
doi: 10.1080/17442509808834163. |
[2] |
G. K. Basak, A. Bisi and M. K. Ghosh,
Stability of a random diffusion with linear drift, J. Math. Anal. Appl., 202 (1996), 604-622.
doi: 10.1006/jmaa.1996.0336. |
[3] |
P. H. Bezandry and T. Diagana, Almost Periodic Stochastic Processes, Springer, New York, 2011.
doi: 10.1007/978-1-4419-9476-9. |
[4] |
R. Dong, Stabilization of Stochastic Differential Equations by Feedback Controls Based on Discrete-time Observations, PhD thesis, University of Strathclyde, UK, 2019. Google Scholar |
[5] |
R. Dong,
Almost sure exponential stabilization by stochastic feedback control based on discrete-time observations, Stochastic Analysis and Applications, 36 (2018), 561-583.
doi: 10.1080/07362994.2018.1433046. |
[6] |
R. Dong and X. R. Mao,
On $p$th moment stabilization of hybrid systems by discrete-time feedback control, Stochastic Analysis and Applications, 35 (2017), 803-822.
doi: 10.1080/07362994.2017.1324798. |
[7] |
L. Y. Hu, Y. Ren and T. B. Xu,
$p$-Moment stability of solutions to stochastic differential equations driven by $G$-Brownian motion, Applied Mathematics and Computation, 230 (2014), 231-237.
doi: 10.1016/j.amc.2013.12.111. |
[8] |
C. X. Huang, Y. G. He, L. H. Huang and W. J. Zhu,
$p$th moment stability analysis of stochastic recurrent neural networks with time-varying delays, Information Sciences, 178 (2008), 2194-2203.
doi: 10.1016/j.ins.2008.01.008. |
[9] |
Y. D. Ji and H. J. Chizeck,
Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control, 35 (1990), 777-788.
doi: 10.1109/9.57016. |
[10] |
Y. Y. Li, J. Q. Lu, C. H. Kou, X. R. Mao and J. F. Pan,
Robust stabilization of hybrid uncertain stochastic systems by discrete-time feedback control, Optimal Control Applications and Methods, 38 (2017), 847-859.
doi: 10.1002/oca.2293. |
[11] |
X. Y. Li and X. R. Mao,
A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica J. IFAC, 48 (2012), 2329-2334.
doi: 10.1016/j.automatica.2012.06.045. |
[12] |
J. Q. Lu, Y. Y. Li, X. R. Mao and Q. W. Qiu,
Stabilization of hybrid systems by feedback control based on discrete-time state and mode observations, Asian Journal of Control, 19 (2017), 1943-1953.
doi: 10.1002/asjc.1515. |
[13] |
X. R. Mao,
Stability of stochastic differential equations with Markovian switching, Sto. Proc. Their Appl., 79 (1999), 45-67.
doi: 10.1016/S0304-4149(98)00070-2. |
[14] |
X. R. Mao,
Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Transactions on Automatic Control, 47 (2002), 1604-1612.
doi: 10.1109/TAC.2002.803529. |
[15] |
X. R. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
[16] |
X. R. 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. |
[17] |
X. R. Mao,
Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Transactions on Automatic Control, 61 (2016), 1619-1624.
doi: 10.1109/TAC.2015.2471696. |
[18] |
X. R. Mao, G. G. Yin and C. G. Yuan,
Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43 (2007), 264-273.
doi: 10.1016/j.automatica.2006.09.006. |
[19] |
X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.
doi: 10.1142/p473.![]() ![]() |
[20] |
X. R. Mao, W. Liu, L. J. Hu, Q. Luo and J. Q. Lu,
Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Systems Control Lett., 73 (2014), 88-95.
doi: 10.1016/j.sysconle.2014.08.011. |
[21] |
Y. G. Niu, D. W. C. Ho and J. Lam,
Robust integral sliding mode control for uncertain stochastic systems with time-varying delay, Automatica J. IFAC, 41 (2005), 873-880.
doi: 10.1016/j.automatica.2004.11.035. |
[22] |
R. Rifhat, L. Wang and Z. D. Teng,
Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients, Physica A: Statistical Mechanics and its Applications, 481 (2017), 176-190.
doi: 10.1016/j.physa.2017.04.016. |
[23] |
J. L. Sabo and D. M. Post,
Quantifying periodic, stochastic, and catastrophic environmental variation, Ecological Monographs, 78 (2008), 19-40.
doi: 10.1890/06-1340.1. |
[24] |
J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, New Jersey, 1991. Google Scholar |
[25] |
G. F. Song, B.-C. Zheng and X. R. Mao,
Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time observations of state and mode, IET Control Theory Appl., 11 (2017), 301-307.
doi: 10.1049/iet-cta.2016.0635. |
[26] |
M. H. Sun, J. Lam, S. Y. Xu and Y. Zou,
Robust exponential stabilization for Markovian jump systems with mode-dependent input delay, Automatica J. IFAC, 43 (2007), 1799-1807.
doi: 10.1016/j.automatica.2007.03.005. |
[27] |
I. Tsiakas,
Periodic stochastic volatility and fat tails, Journal of Financial Econometrics, 4 (2006), 90-135.
doi: 10.1093/jjfinec/nbi023. |
[28] |
C. Wang and R. P. Agarwal,
Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model, Applied Mathematics Letters, 70 (2017), 58-65.
doi: 10.1016/j.aml.2017.03.009. |
[29] |
C. Wang, R. P. Agarwal and S. Rathinasamy,
Almost periodic oscillations for delay impulsive stochastic Nicholson's blowflies timescale model, Computational and Applied Mathematics, 37 (2018), 3005-3026.
doi: 10.1007/s40314-017-0495-0. |
[30] |
G. C. Wang, Z. Wu and J. Xiong,
A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information, IEEE Transactions on Automatic Control, 60 (2015), 2904-2916.
doi: 10.1109/TAC.2015.2411871. |
[31] |
Y. Wang and Z. Liu,
Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25 (2012), 2803-2821.
doi: 10.1088/0951-7715/25/10/2803. |
[32] |
L. G. Wu, P. Shi and H. J. Gao,
State estimation and sliding mode control of Markovian jump singular systems, IEEE Transactions on Automatic Control, 55 (2010), 1213-1219.
doi: 10.1109/TAC.2010.2042234. |
[33] |
S. R. You, L. J. Hu, W. Mao and X. R. Mao,
Robustly exponential stabilization of hybrid uncertain systems by feedback controls based on discrete-time observations, Statist. Probab. Lett., 102 (2015), 8-16.
doi: 10.1016/j.spl.2015.03.006. |
[34] |
S. R. You, W. Liu, J. Q. Lu, X. R. Mao and Q. W. 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. |


Subinterval | Observation interval | Observation times |
[0, 0.5) | 0.05556 | 9 |
[0.5, 1) | 0.1 | 5 |
[1, 2.42) | 0.142 | 10 |
[2.42, 3) | 0.19333 | 3 |
[3, 4.27) | 0.21167 | 6 |
[4.27, 5) | 0.10429 | 7 |
[5, 5.48) | 0.06 | 8 |
[5.48, 6.37) | 0.01745 | 51 |
[6.37, 11.28) | 0.00164 | 2988 |
[11.28, 12) | 0.01714 | 42 |
Subinterval | Observation interval | Observation times |
[0, 0.5) | 0.05556 | 9 |
[0.5, 1) | 0.1 | 5 |
[1, 2.42) | 0.142 | 10 |
[2.42, 3) | 0.19333 | 3 |
[3, 4.27) | 0.21167 | 6 |
[4.27, 5) | 0.10429 | 7 |
[5, 5.48) | 0.06 | 8 |
[5.48, 6.37) | 0.01745 | 51 |
[6.37, 11.28) | 0.00164 | 2988 |
[11.28, 12) | 0.01714 | 42 |
[1] |
Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 |
[2] |
Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020317 |
[3] |
Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 |
[4] |
Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020323 |
[5] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[6] |
Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 |
[7] |
Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571 |
[8] |
Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020371 |
[9] |
Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020047 |
[10] |
Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 |
[11] |
Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 |
[12] |
Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170 |
[13] |
Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020398 |
[14] |
Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028 |
[15] |
Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020468 |
[16] |
Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020339 |
[17] |
Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112 |
[18] |
Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264 |
[19] |
Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 |
[20] |
Yi Guan, Michal Fečkan, Jinrong Wang. Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1157-1176. doi: 10.3934/dcds.2020313 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]