Approximation of linear controlled dynamical systems with small random noise and fast periodic sampling

In this paper, we study the dynamics of a linear control system with given state feedback control law in the presence of fast periodic sampling at temporal frequency \begin{document}$ 1/\delta $\end{document} ( \begin{document}$ 0 < \delta \ll 1 $\end{document} ), together with small white noise perturbations of size \begin{document}$ \varepsilon $\end{document} ( \begin{document}$ 0< \varepsilon \ll 1 $\end{document} ) in the state dynamics. For the ensuing continuous-time stochastic process indexed by two small parameters \begin{document}$ \varepsilon,\delta $\end{document} , we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as \begin{document}$ \varepsilon,\delta \searrow 0 $\end{document} . The effective fluctuation process is found to vary, depending on whether \begin{document}$ \delta \searrow 0 $\end{document} faster than/at the same rate as/slower than \begin{document}$ \varepsilon \searrow 0 $\end{document} . The most interesting case is found to be the one where \begin{document}$ \delta, \varepsilon $\end{document} are comparable in size; here, the limiting stochastic differential equation for the fluctuations has both a diffusive term due to the small noise and an effective drift term which captures the cumulative effect of the fast sampling. In this regime, our results yield a time-inhomogeneous Markov process which provides a strong (pathwise) approximation of the original non-Markovian process, together with estimates on the ensuing error. A simple example involving an infinite time horizon linear quadratic regulation problem illustrates the results.