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

TL;DR

This paper performs a second-order analysis of an optimal control problem for a complex tumor growth model involving phase field methods, singular potentials, and chemotaxis, establishing well-posedness and optimality conditions.

Contribution

It provides the first comprehensive second-order analysis for a coupled tumor growth control model with singular potentials and chemotaxis, including differentiability and derivative characterization.

Findings

Established weak and strong well-posedness of the system.

Proved existence of optimal controls and derived first- and second-order optimality conditions.

Identified the second-order Fréchet derivative of the control-to-state map.

Abstract

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.