  This paper concerns a distributed optimal control problem for a tumor growth
model of Cahn-Hilliard type including chemotaxis with possibly singular
potentials, where the control and state variables are nonlinearly coupled.
First, we discuss the weak well-posedness of the system under very general
assumptions for the potentials, which may be singular and nonsmooth. Then, we
establish the strong well-posedness of the system in a reduced setting, which
however admits the logarithmic potential: this analysis will lay the foundation
for the study of the corresponding optimal control problem. Concerning the
optimization problem, we address the existence of minimizers and establish both
first-order necessary and second-order sufficient conditions for optimality.
The mathematically challenging second-order analysis is completely performed
here, after showing that the solution mapping is twice continuously
differentiable between suitable Banach spaces via the implicit function
theorem. Then, we completely identify the second-order Fr\'echet derivative of
the control-to-state operator and carry out a thorough and detailed
investigation about the related properties.
