  This paper treats a distributed optimal control problem for a tumor growth
model of Cahn-Hilliard type including chemotaxis. The evolution of the tumor
fraction is governed by a variational inequality corresponding to a double
obstacle nonlinearity occurring in the associated potential. In addition, the
control and state variables are nonlinearly coupled and, furthermore, the cost
functional contains a nondifferentiable term like the $L^1$-norm in order to
include sparsity effects which is of utmost relevance, especially time
sparsity, in the context of cancer therapies as applying a control to the
system reflects in exposing the patient to an intensive medical treatment. To
cope with the difficulties originating from the variational inequality in the
state system, we employ the so-called "deep quench approximation" in which the
convex part of the double obstacle potential is approximated by logarithmic
functions. For such functions, first-order necessary conditions of optimality
can be established by invoking recent results. We use these results to derive
corresponding optimality conditions also for the double obstacle case, by
deducing a variational inequality in terms of the associated adjoint state
variables. The resulting variational inequality can be exploited to also obtain
sparsity results for the optimal controls.
