  A distributed optimal control problem for a diffuse interface model, which
physical context is that of tumour growth dynamics, is addressed. The system we
deal with comprises a Cahn--Hilliard equation for the tumour fraction coupled
with a reaction-diffusion for a nutrient species surrounding the tumourous
cells. The cost functional to be minimised possesses some objective terms and
it also penalises long treatments time, which may affect harm to the patients,
and big aggregations of tumourous cells. Hence, the optimisation problem leads
to the optimal strategy which reduces the time exposure of the patient to the
medication and at the same time allows the doctors to achieve suitable clinical
goals.
