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 $1/\delta$ ($0 < \delta \ll 1$), together with small white noise perturbations of size $\varepsilon$ ($0<\varepsilon \ll 1$) in the state dynamics. For the ensuing continuous-time stochastic process indexed by two small parameters $\varepsilon,\delta$, we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as $\varepsilon,\delta \searrow 0$. The effective fluctuation process is found to vary, depending on whether $\delta \searrow 0$ faster than/at the same rate as/slower than $\varepsilon \searrow 0$. The most interesting case is found to be the one where $\delta,\varepsilon$ 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.