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A Distributed Control Problem for a Fractional Tumor Growth Model

Author

Listed:
  • Pierluigi Colli

    (Dipartimento di Matematica “F. Casorati”, Università di Pavia and Research Associate at the IMATI—C.N.R. Pavia, via Ferrata 5, 27100 Pavia, Italy)

  • Gianni Gilardi

    (Dipartimento di Matematica “F. Casorati”, Università di Pavia and Research Associate at the IMATI—C.N.R. Pavia, via Ferrata 5, 27100 Pavia, Italy)

  • Jürgen Sprekels

    (Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
    Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany)

Abstract

In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn–Hilliard type phase field system modeling tumor growth that has been proposed by Hawkins–Daarud, van der Zee and Oden. The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in a recent work by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

Suggested Citation

  • Pierluigi Colli & Gianni Gilardi & Jürgen Sprekels, 2019. "A Distributed Control Problem for a Fractional Tumor Growth Model," Mathematics, MDPI, vol. 7(9), pages 1-32, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:792-:d:262765
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    Citations

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    Cited by:

    1. Chaeyoung Lee & Darae Jeong & Junxiang Yang & Junseok Kim, 2020. "Nonlinear Multigrid Implementation for the Two-Dimensional Cahn–Hilliard Equation," Mathematics, MDPI, vol. 8(1), pages 1-23, January.
    2. Pierluigi Colli & Andrea Signori & Jürgen Sprekels, 2022. "Optimal Control Problems with Sparsity for Tumor Growth Models Involving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 25-58, July.

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