# American Institute of Mathematical Sciences

December  2020, 10(4): 715-734. doi: 10.3934/mcrf.2020017

## 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

* Corresponding author: Ran Dong

Received  July 2019 Revised  November 2019 Published  December 2020 Early access  December 2019

Fund Project: The first author was partially supported by the PhD studentship of the University of Strathclyde. The second author is partially supported by the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship) and the EPSRC (EP/K503174/1)

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.

Citation: Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control and Related Fields, 2020, 10 (4) : 715-734. doi: 10.3934/mcrf.2020017
##### 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. [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. [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.

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##### 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. [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. [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.
Sample averages of $|x|^2$ from $500$ simulated paths by the Euler-Maruyama method with step size $1e-5$ and random initial values. Upper plot shows original system (55); lower plot shows controlled system (56) for mean square asymptotically stabilization with corresponding observation frequencies
Plot of parameters $K_1(t)$, $K_2(t)$, $K_3(t)$ and $\lambda(t)$
Plot of observation intervals. The dashed blue line shows the auxiliary function and the solid orange line is observation interval sequence
Period partition, observation interval and observation times in each subinterval
 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
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