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.
Keywords: Optimal control, tumor growth models, singular potentials, optimality conditions, second-order analysis
@article{COCV_2021__27_1_A75_0,
author = {Colli, Pierluigi and Signori, Andrea and Sprekels, J\"urgen},
title = {Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021072},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021072/}
}
TY - JOUR AU - Colli, Pierluigi AU - Signori, Andrea AU - Sprekels, Jürgen TI - Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021072/ DO - 10.1051/cocv/2021072 LA - en ID - COCV_2021__27_1_A75_0 ER -
%0 Journal Article %A Colli, Pierluigi %A Signori, Andrea %A Sprekels, Jürgen %T Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021072/ %R 10.1051/cocv/2021072 %G en %F COCV_2021__27_1_A75_0
Colli, Pierluigi; Signori, Andrea; Sprekels, Jürgen. Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 73. doi: 10.1051/cocv/2021072
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