  Reduced basis approximations of Optimal Control Problems (OCPs) governed by
steady partial differential equations (PDEs) with random parametric inputs are
analyzed and constructed. Such approximations are based on a Reduced Order
Model, which in this work is constructed using the method of weighted Proper
Orthogonal Decomposition. This Reduced Order Model then is used to efficiently
compute the reduced basis approximation for any outcome of the random
parameter. We demonstrate that such OCPs are well-posed by applying the adjoint
approach, which also works in the presence of admissibility constraints and in
the case of non linear-quadratic OCPs, and thus is more general than the
conventional Lagrangian approach. We also show that a step in the construction
of these Reduced Order Models, known as the aggregation step, is not
fundamental and can in principle be skipped for noncoercive problems, leading
to a cheaper online phase. Numerical applications in three scenarios from
environmental science are considered, in which the governing PDE is steady and
the control is distributed. Various parameter distributions are taken, and
several implementations of the weighted Proper Orthogonal Decomposition are
compared by choosing different quadrature rules.
