  This paper investigates an optimal control problem associated with a
two-dimensional multi-species Cahn-Hilliard-Keller-Segel tumor growth model,
which incorporates complex biological processes such as species diffusion,
chemotaxis, angiogenesis, and nutrient consumption, resulting in a highly
nonlinear system of nonlinear partial differential equations. The modeling
derivation and corresponding analysis have been addressed in a previous
contribution. Building on this foundation, the scope of this study involves
investigating a distributed control problem with the goal of optimizing a
tracking-type cost functional. This latter aims to minimize the deviation of
tumor cell location from desired target configurations while penalizing the
costs associated with implementing control measures, akin to introducing a
suitable medication. Under appropriate mathematical assumptions, we demonstrate
that sufficiently regular solutions exhibit continuous dependence on the
control variable. Furthermore, we establish the existence of optimal controls
and characterize the first-order necessary optimality conditions through a
suitable variational inequality.
