Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations
The authors investigate the stabilization via feedback control based on periodic observations for two types of \(n\)-dimensional stochastic differential systems, \[dx(t)= [f(x(t),t)+ u(x(\delta_t),t)]\,dt+ g(x(t),t)\,dB(t)\] and \[dx(t)= [f(x(t),r(t),t)+ u(x(\delta_t),r(\delta_t),t)]\,dt+ g(x(t),r(t),t)\,dB(t),\] where \(B(t)\) is \(m\)-dimensional Brownian motion, \(r(t)\) a right-continuous Markov chain with a finite state space, \(u\) the control function, and \[\delta_t= \sum^k_{j=1} \tau_j\le t< \sum^{k+1}_{j=1}\tau_j\] with \(0\), \(\tau_1\), \(\tau_1+\tau_2\), \(\tau_1+ \tau_2+\tau_3,\dots\) as time points of observation. Explicitly time-dependent coefficients are assumed to be periodically so. The aim of the control is \(L^p\)-stability of the systems for \(p>1\), almost sure asymptotic stability and \(p\)th moment asymptotic stability for \(p\ge 2\). The existence of the interval sequence \(\{\tau_j\}_{j\ge 0}\) is verified, a way to calculate it provided, and an illustrative example given. It is shown how to significantly reduce the observation frequency and thus the control costs.
scientific article; zbMATH DE number 7344169
scientific article