  This work is concerned with a class of optimal control problems governed by a
partial differential equation that are motivated by an application in
radiotherapy treatment planning, where the primary design objective is to
minimize the volume where a functional of the state violates a prescribed
level, but prescribing these levels in the form of pointwise state constraints
can lead to infeasible problems. We therefore propose an alternative approach
based on $L^1$ penalization of the violation. We establish well-posedness of
the corresponding optimal control problem, derive first-order optimality
conditions, and present a semismooth Newton method for the efficient numerical
solution of these problems. The performance of this method for a model problem
is illustrated and contrasted with the alternative approach based on
(regularized) state constraints.
