  We stabilize a prescribed cycle or an equilibrium of the difference equation
using pulsed stochastic control. Our technique, inspired by the Kolmogorov's
Law of Large Numbers, activates a stabilizing effect of stochastic perturbation
and allows for stabilization using a much wider range for the control parameter
than would be possible in the absence of noise.
  Our main general result applies to both Prediction-Based and Target-Oriented
Controls. This analysis is the first to make use of the stabilizing effects of
noise for Prediction-Based Control; the stochastic version has been examined in
the literature, but only the destabilizing effect of noise was demonstrated. A
stochastic variant of Target-Oriented Control has never been considered, to the
best of our knowledge, and we propose a specific form that uses a point
equilibrium or one point on a cycle as a target. We demonstrate our results
numerically on the logistic, Ricker and Maynard Smith models from population
biology.
