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A distributed optimal control problem for an extended model of phase fieldtype for tumor growth is addressed. In this model, the chemotaxis effects arealso taken into account. The control is realized by two control variables thatdesign the dispensation of some drugs to the patient. The cost functional is oftracking type, whereas the potential setting has been kept quite general inorder to allow regular and singular potentials to be considered. In thisdirection, some relaxation terms have been introduced in the system. We showthe well-posedness of the state system, the Fr\'echet differentiability of thecontrol-to-state operator in a suitable functional analytic framework, and,lastly, we characterize the first-order necessary conditions of optimality interms of a variational inequality involving the adjoint variables.

Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials