  We consider the influence of stochastic perturbations on stability of a
unique positive equilibrium of a difference equation subject to
prediction-based control. These perturbations may be multiplicative
$$x_{n+1}=f(x_n)-\left( \alpha + l\xi_{n+1} \right) (f(x_n)-x_n), \quad n=0, 1,
\dots$$ if they arise from stochastic variation of the control parameter, or
additive $$x_{n+1}=f(x_n)-\alpha(f(x_n)-x_n) +l\xi_{n+1}, \quad n=0, 1, \dots
$$ if they reflect the presence of systemic noise.
  We begin by relaxing the control parameter in the deterministic equation, and
deriving a range of values for the parameter over which all solutions
eventually enter an invariant interval. Then, by allowing the variation to be
stochastic, we derive sufficient conditions (less restrictive than known ones
for the unperturbed equation) under which the positive equilibrium will be
globally a.s. asymptotically stable:i.e. the presence of noise improves the
known effectiveness of prediction-based control. Finally, we show that systemic
noise has a "blurring" effect on the positive equilibrium, which can be made
arbitrarily small by controlling the noise intensity. Numerical examples
illustrate our results.

Dynamical Systems