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Optimal Control Problems with Sparsity for Tumor Growth Models Involving Variational Inequalities

Author

Listed:
  • Pierluigi Colli

    (Università di Pavia)

  • Andrea Signori

    (Università di Pavia)

  • Jürgen Sprekels

    (Humboldt-Universität zu Berlin
    Weierstrass Institute for Applied Analysis and Stochastics)

Abstract

This paper treats a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the $$L^1$$ L 1 -norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the so-called deep quench approximation in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, first-order necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:194:y:2022:i:1:d:10.1007_s10957-022-02000-7
    DOI: 10.1007/s10957-022-02000-7
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    References listed on IDEAS

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    1. 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.
    2. Roland Herzog & Johannes Obermeier & Gerd Wachsmuth, 2015. "Annular and sectorial sparsity in optimal control of elliptic equations," Computational Optimization and Applications, Springer, vol. 62(1), pages 157-180, September.
    3. Georg Stadler, 2009. "Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices," Computational Optimization and Applications, Springer, vol. 44(2), pages 159-181, November.
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