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Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials

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A Correction to this article was published on 15 April 2021

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Abstract

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

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Acknowledgements

The authors are very grateful to the anonymous referee for the careful reading of the manuscript and for some useful suggestions. The research of Pierluigi Colli is supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022)—Dept. of Mathematics “F. Casorati”, University of Pavia. In addition, PC gratefully acknowledges some other support from the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI – C.N.R. Pavia, Italy.

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Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 5, 27100, Pavia, Italy

    Pierluigi Colli

  2. Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, via Cozzi 55, 20125, Milan, Italy

    Andrea Signori

  3. Department of Mathematics, Humboldt-Univesität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Jürgen Sprekels

  4. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117, Berlin, Germany

    Jürgen Sprekels

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  1. Pierluigi Colli
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  2. Andrea Signori
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  3. Jürgen Sprekels
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Correspondence to Andrea Signori.

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Colli, P., Signori, A. & Sprekels, J. Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials. Appl Math Optim 83, 2017–2049 (2021). https://doi.org/10.1007/s00245-019-09618-6

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