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Almost sure exponential stabilization by stochastic feedback control based on discrete-time observations. (English) Zbl 1397.93216

Summary: Since Mao initiated the study of stabilization of ordinary differential equations (ODEs) by stochastic feedback controls based on discrete-time state observations in 2016, no more work on this intriguing topic has been reported. This article investigates how to stabilize a given unstable linear non-autonomous ODE by controller \(\sigma (t)x(\delta_t) dB(t)\), and how to stabilize an unstable nonlinear hybrid SDE by controller \(G(r(\delta_t))x(\delta_t)dB(t)\), where \(\delta_t\) represents time points of observation with sufficiently small observation interval, \(B(t)\) is a Brownian motion and \(r(t)\) is the Markov Chain, in the sense of \(p\)th moment \((0<p<1)\) and almost sure exponential stability.

MSC:

93E15 Stochastic stability in control theory
93D15 Stabilization of systems by feedback
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