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/δ (0 < δ ≪ 1), together with small white noise perturbations of size ε (0 < ε ≪ 1) in the state dynamics. For the ensuing continuous-time stochastic process indexed by two small parameters ε, δ, we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as ε, δ ↘ 0. The effective fluctuation process is found to vary, depending on whether δ ↘ 0 faster than/at the same rate as/slower than ε ↘ 0. The most interesting case is found to be the one where δ, ε are comparable in size; here, the limiting stochastic differential equation for the fluctuations has both a dif-fusive 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.