We study the effect of total variation regularization on PDE-constrained optimization problems, where the control input functions may only attain finitely many integer values. The regularization helps to avoid undesirable effects such as chattering behavior. In particular, the weak-* compactness of the feasible set in the space of functions of bounded variation allows to derive a meaningful stationarity concept and a trust-region algorithm that produces iterates such that all limits are stationary under mild assumptions on the governing PDE. We sketch a possible approach for discretizing the trust-region subproblems and point out the arising difficulties, in particular for the case that the dimension of the domain is larger than one.

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