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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-stability for <inline-formula><tex-math id="M2">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula>, almost sure asymptotic stability and <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>th moment asymptotic stability for <inline-formula><tex-math id="M4">\begin{document}$ p \ge 2 $\end{document}</tex-math></inline-formula>. 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.