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

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 Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

arXiv commentPlease note that the formula (6.5), in both the arXiv preprint and in the journal article, contains three sign errors: indeed, the term in the second line of (6.5) involving P'' should carry a `plus' sign, while the two terms in the third line should carry `minus' signs. These sign errors, however, do not have an impact on the validity of the results

References in corpus (12)

• A coupled surface-Cahn--Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes
• A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis
• Optimal distributed control of a diffuse interface model of tumor growth
• Long-time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth
• On a phase field model of Cahn-Hilliard type for tumour growth with mechanical effects
• Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources
• Optimal distributed control of an extended model of tumor growth with logarithmic potential
• Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation
• Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach
• On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport
• Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials
• Penalisation of Long Treatment Time and Optimal Control of a Tumour Growth Model of Cahn-Hilliard