Volume 27, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Number of page(s)||27|
|Published online||01 March 2021|
Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth
Weierstrass Institute for Applied Analysis and Stochastics,
2 Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany.
3 Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany.
* Corresponding author: firstname.lastname@example.org
Accepted: 1 December 2020
In this paper, we study an optimal control problem for a nonlinear system of reaction–diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in H. Garcke, et al. [Math. Model. Methods Appl. Sci. 26 (2016) 1095–1148]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [P. Colli, et al. To appear in: Appl. Math. Optim. (2019)] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L1−norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [P. Colli, et al. To appear in: Appl. Math. Optim. (2019)], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.
Mathematics Subject Classification: 49J20 / 49K20 / 49K40 / 35K57 / 37N25
Key words: Sparse optimal control / tumor growth models / singular potentials / optimality conditions
© EDP Sciences, SMAI 2021
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