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

  • * Corresponding author: Ran Dong

    * Corresponding author: Ran Dong 
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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  • 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.

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

    Citation:

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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

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

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

    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
     | Show Table
    DownLoad: CSV
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