  Optimal control problems without control costs in general do not possess
solutions due to the lack of coercivity. However, unilateral constraints
together with the assumption of existence of strictly positive solutions of a
pre-adjoint state equation, are sufficient to obtain existence of optimal
solutions in the space of Radon measures. Optimality conditions for these
generalized minimizers can be obtained using Fenchel duality, which requires a
non-standard perturbation approach if the control-to-observation mapping is not
continuous (e.g., for Neumann boundary control in three dimensions). Combining
a conforming discretization of the measure space with a semismooth Newton
method allows the numerical solution of the optimal control problem.
