  In this paper, a distributed optimal control problem is studied for a diffuse
interface model of tumor growth which was proposed in [A. Hawkins-Daruud, K.G.
van der Zee, J.T. Oden, Numerical simulation of a thermodynamically consistent
four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng. 28 (2011),
3-24]. The model consists of a Cahn-Hilliard equation for the tumor cell
fraction coupled to a reaction-diffusion equation for a variable representing
the nutrient-rich extracellular water volume fraction. The distributed control
monitors as a right-hand side the reaction-diffusion equation and can be
interpreted as a nutrient supply or a medication, while the cost function,
which is of standard tracking type, is meant to keep the tumor cell fraction
under control during the evolution. We show that the control-to-state operator
is Frechet differentiable between appropriate Banach spaces and derive the
first-order necessary optimality conditions in terms of a variational
inequality involving the adjoint state variables.
