L1 penalization of volumetric dose objectives in optimal control of PDEs

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.