Optimal control of a tumor growth model with hyperbolic relaxation of the chemical potential

TL;DR

This paper investigates the optimal control of a tumor growth phase field model with a hyperbolic relaxation of the chemical potential, covering different potential types, and derives conditions for optimality and sparsity of controls.

Contribution

It introduces a novel hyperbolic relaxation approach in a tumor growth model and establishes differentiability and optimality conditions for control strategies.

Findings

Proves Fréchet differentiability of the control-to-state operator.

Derives first-order necessary optimality conditions.

Establishes sparsity results for optimal controls.

Abstract

In this paper, we study the optimal control of a phase field model for a tumor growth model of Cahn--Hilliard type in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. Both the cases when the double-well potential governing the phase evolution is of either regular or logarithmic type are covered by the analysis. We show the Fr\'echet differentiability of the associated control-to-state operator in suitable Banach spaces and establish first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. The necessary optimality conditions are then used to derive sparsity results for the optimal controls.