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This paper concerns a distributed optimal control problem for a tumor growthmodel of Cahn-Hilliard type including chemotaxis with possibly singularpotentials, where the control and state variables are nonlinearly coupled.First, we discuss the weak well-posedness of the system under very generalassumptions for the potentials, which may be singular and nonsmooth. Then, weestablish the strong well-posedness of the system in a reduced setting, whichhowever admits the logarithmic potential: this analysis will lay the foundationfor the study of the corresponding optimal control problem. Concerning theoptimization problem, we address the existence of minimizers and establish bothfirst-order necessary and second-order sufficient conditions for optimality.The mathematically challenging second-order analysis is completely performedhere, after showing that the solution mapping is twice continuouslydifferentiable between suitable Banach spaces via the implicit functiontheorem. Then, we completely identify the second-order Fr\'echet derivative ofthe control-to-state operator and carry out a thorough and detailedinvestigation about the related properties.

Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis