  We study a Cahn-Hilliard-Darcy system with mass sources, which can be
considered as a basic, though simplified, diffuse interface model for the
evolution of tumor growth. This system is equipped with an impermeability
condition for the (volume) averaged velocity $\mathbf{u}$ as well as
homogeneous Neumann boundary conditions for the phase function $\varphi$ and
the chemical potential $\mu$. The source term in the convective Cahn-Hilliard
equation contains a control $R$ that can be thought, for instance, as a drug or
a nutrient in applications. Our goal is to study a distributed optimal control
problem in the two dimensional setting with a cost functional of tracking-type.
In the physically relevant case with unmatched viscosities for the binary fluid
mixtures and a singular potential, we first prove the existence and uniqueness
of a global strong solution with $\varphi$ being strictly separated from the
pure phases $\pm 1$. This well-posedness result enables us to characterize the
control-to-state mapping $\mathcal{S}:R \mapsto \varphi$. Then we obtain the
existence of an optimal control, the Fr\'{e}chet differentiability of
$\mathcal{S}$ and first-order necessary optimality conditions expressed through
a suitable variational inequality for the adjoint variables. Finally, we prove
the differentiability of the control-to-costate operator and establish a
second-order sufficient condition for the strict local optimality.
