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

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