We investigate a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn–Hilliard type equation for the phase field variable, a reaction diffusion equation for the nutrient concentration and a Brinkman type equation for the velocity field. These PDEs are endowed with homogeneous Neumann boundary conditions for the phase field variable, the chemical potential and the nutrient as well as a “no-friction” boundary condition for the velocity. The control represents a medication by cytotoxic drugs and enters the phase field equation. The aim is to minimize a cost functional of standard tracking type that is designed to track the phase field variable during the time evolution and at some fixed final time. We show that our model satisfies the basics for calculus of variations and we present first-order and second-order conditions for local optimality. Moreover, we present a globality condition for critical controls and we show that the optimal control is unique on small time intervals.
Mots-clés : Optimal control with PDEs, calculus of variations, tumor growth, Cahn–Hilliard equation, Brinkman equation, first-order necessary optimality conditions, second-order sufficient optimality conditions, uniqueness of globally optimal solutions
@article{COCV_2020__26_1_A71_0,
author = {Ebenbeck, Matthias and Knopf, Patrik},
title = {Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019059},
mrnumber = {4155227},
zbl = {1451.35233},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019059/}
}
TY - JOUR AU - Ebenbeck, Matthias AU - Knopf, Patrik TI - Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019059/ DO - 10.1051/cocv/2019059 LA - en ID - COCV_2020__26_1_A71_0 ER -
%0 Journal Article %A Ebenbeck, Matthias %A Knopf, Patrik %T Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019059/ %R 10.1051/cocv/2019059 %G en %F COCV_2020__26_1_A71_0
Ebenbeck, Matthias; Knopf, Patrik. Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 71. doi : 10.1051/cocv/2019059. http://www.numdam.org/articles/10.1051/cocv/2019059/
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