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

TL;DR

This paper develops an optimal control framework for a tumor growth model involving variational inequalities and sparsity-promoting terms, providing new mathematical conditions for optimality in complex nonlinear systems.

Contribution

It introduces a novel approach using deep quench approximation to derive optimality conditions for a variational inequality-based tumor model with sparsity effects.

Findings

Established first-order necessary optimality conditions.

Derived variational inequalities for the adjoint states.

Demonstrated sparsity in optimal controls.

Abstract

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.