  In this paper we study nonlocal-to-local asymptotics for a tumor-growth model
coupling a viscous Cahn-Hilliard equation describing the tumor proportion with
a reaction-diffusion equation for the nutrient phase parameter. First, we prove
that solutions to the nonlocal Cahn-Hilliard system converge, as the
nonlocality parameter tends to zero, to solutions to its local counterpart.
Second, we provide first-order optimality conditions for an optimal control
problem on the local model, accounting also for chemotaxis, and both for
regular or singular potentials, without any additional regularity assumptions
on the solution operator. The proof is based on an approximation of the local
control problem by means of suitable nonlocal ones, and on proving
nonlocal-to-local convergence both for the corresponding dual systems and for
the associated first-order optimality conditions.
