Optimal distributed control of a diffuse interface model of tumor growth

Abstract

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed in [A. Hawkins-Daruud, K.G. van der Zee, J.T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng. 28 (2011), 3-24]. The model consists of a Cahn-Hilliard equation for the tumor cell fraction coupled to a reaction-diffusion equation for a variable representing the nutrient-rich extracellular water volume fraction. The distributed control monitors as a right-hand side the reaction-diffusion equation and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Frechet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

A revised version of the paper has been published on Nonlinearity 30 (2017), 2518-2546. Let us point out that in this arXiv:1601.04567 [math.AP] version there is something missing in assumption (H3) at page 6: the first initial value in (H6) must also satisfy a Neumann homogeneous condition at the boundary of the domain

References in corpus (5)

• A coupled surface-Cahn--Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes
• Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth
• Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in 2D
• A rigorous sharp interface limit of a diffuse interface model related to tumor growth
• Analysis of a Mixture Model of Tumor Growth

Cited by in corpus (18)

• Long-time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth
• Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources
• Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth
• Optimal distributed control of an extended model of tumor growth with logarithmic potential
• Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation
• On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport
• Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials
• Optimal control of stochastic phase-field models related to tumor growth
• Numerical analysis for a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport
• Penalisation of Long Treatment Time and Optimal Control of a Tumour Growth Model of Cahn-Hilliard
• Pontryagin's maximum principle and second order optimality condition for optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions
• Mass-conservative and positivity preserving second-order semi-implicit methods for high-order parabolic equations
• Well-posedness and regularity for a fractional tumor growth model
• Second Order Optimality Conditions for Optimal Control Problems Governed by 2D Nonlocal Cahn Hilliard Navier Stokes Equations
• Vanishing parameter for an optimal control problem modeling tumor growth
• Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms
• Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis
• Optimal control problems with sparsity for phase field tumor growth models involving variational inequalities