  In this paper, we address a distributed control problem for a system of
partial differential equations describing the evolution of a tumor that takes
the biological mechanism of chemotaxis into account. The system describing the
evolution is obtained as a nontrivial combination of a Cahn-Hilliard type
system accounting for the segregation between tumor cells and healthy cells,
with a Keller-Segel type equation accounting for the evolution of a nutrient
species and modeling the chemotaxis phenomenon. First, we develop a robust
mathematical background that allows us to analyze an associated optimal control
problem. This analysis forced us to select a source term of logistic type in
the nutrient equation and to restrict the analysis to the case of two space
dimensions. Then, the existence of an optimal control and first-order necessary
conditions for optimality are established.
