  We consider optimal control problems for discrete-time random dynamical
systems, finding unique perturbations that provoke maximal responses of
statistical properties of the system. We treat systems whose transfer operator
has an $L^2$ kernel, and we consider the problems of finding (i) the
infinitesimal perturbation maximising the expectation of a given observable and
(ii) the infinitesimal perturbation maximising the spectral gap, and hence the
exponential mixing rate of the system. Our perturbations are either (a)
perturbations of the kernel or (b) perturbations of a deterministic map
subjected to additive noise. We develop a general setting in which these
optimisation problems have a unique solution and construct explicit formulae
for the unique optimal perturbations. We apply our results to a
Pomeau-Manneville map and an interval exchange map, both subjected to additive
noise, to explicitly compute the perturbations provoking maximal responses.
