  In this paper, we study a distributed optimal control problem for a diffuse
interface model for tumor growth. The model consists of a Cahn-Hilliard type
equation for the phase field variable coupled to a reaction diffusion equation
for the nutrient and a Brinkman type equation for the velocity. The system is
equipped with homogeneous Neumann boundary conditions for the tumor variable,
the chemical potential and the nutrient as well as a "no-friction" boundary
condition for the velocity. The control acts as a medication by cytotoxic drugs
and enters the phase field equation. The cost functional is of standard
tracking type and is designed to track the phase field variable during the
evolution and at some fixed final time. We prove that the model satisfies the
basics for calculus of variations and we establish first-order and second-order
conditions for local optimality. Moreover, we present a globality condition for
critical controls and we show that the optimal control is unique on small time
intervals.
