WIAS Preprint No. 2845, (2021)

Optimal control problems with sparsity for phase field tumor growth models involving variational inequalities


  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Signori, Andrea
    ORCID: 0000-0001-7025-977X
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2020 Mathematics Subject Classification

  • 49J20 49K20 49K40 35K57 37N25


  • Optimal control, tumor growth models, double obstacle potentials, optimality conditions, variational inequality, sparsity




This paper treats a distributed optimal control problem for a tumor growth model of Cahn--Hilliard type including chemotaxis. 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 $L^1$--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.

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