Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations

Ran Dong; Xuerong Mao

doi:10.3934/mcrf.2020017

Article Contents

2020, Volume 10, Issue 4: 715-734. Doi: 10.3934/mcrf.2020017

This issue Previous Article The Kato smoothing effect for the nonlinear regularized Schrödinger equation on compact manifolds Next Article Maximal discrete sparsity in parabolic optimal control with measures

Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations

Ran Dong^1, , and
Xuerong Mao^2,

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
^* Corresponding author: Ran Dong

Received: June 30, 2019

Revised: October 31, 2019

Early access: December 2019

Published: October 2020

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 93D15, 93E03; Secondary: 93D20.

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Figure 1. 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

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Figure 2. Plot of parameters $ K_1(t) $, $ K_2(t) $, $ K_3(t) $ and $ \lambda(t) $

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Figure 3. Plot of observation intervals. The dashed blue line shows the auxiliary function and the solid orange line is observation interval sequence

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Table 1. 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

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