  The optimal control of problems that are constrained by partial differential
equations with uncertainties and with uncertain controls is addressed. The
Lagrangian that defines the problem is postulated in terms of stochastic
functions, with the control function possibly decomposed into an unknown
deterministic component and a known zero-mean stochastic component. The extra
freedom provided by the stochastic dimension in defining cost functionals is
explored, demonstrating the scope for controlling statistical aspects of the
system response. One-shot stochastic finite element methods are used to find
approximate solutions to control problems. It is shown that applying the
stochastic collocation finite element to the formulated problem leads to a
coupling between stochastic collocation points when a deterministic optimal
control is considered or when moments are included in the cost functional,
thereby obviating the primary advantage of the collocation method over the
stochastic Galerkin method for the considered problem. The application of the
presented methods is demonstrated through a number of numerical examples. The
presented framework is sufficiently general to also consider a class of inverse
problems, and numerical examples of this type are also presented.
