2021/04/20 by Pierluigi Colli, Colli, Pierluigi, Andrea Signori +3
Computer Science · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Mathematical Biology Tumor Growth #Optimization and Control (math.OC) #Solidification and crystal growth phenomena #math.OC
paper · pdf · doi:10.48550/arxiv.2104.09814
arxiv created 2021/04/20 · openalex publication_date 2021/04/20 · arxiv updated 2021/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L1-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.