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Optimal Distributed Control of an Extended Model of Tumor Growth with Logarithmic Potential

Optimal Distributed Control of an Extended Model of Tumor Growth with Logarithmic Potential This paper is intended to tackle the control problem associated with an extended phase field system of Cahn–Hilliard type that is related to a tumor growth model. This system has been investigated in previous contributions from the viewpoint of well-posedness and asymptotic analyses. Here, we aim to extend the mathematical studies around this system by introducing a control variable and handling the corresponding control problem. We try to keep the potential as general as possible, focusing our investigation towards singular potentials, such as the logarithmic one. We establish the existence of optimal control, the Lipschitz continuity of the control-to-state mapping and even its Fréchet differentiability in suitable Banach spaces. Moreover, we derive the first-order necessary conditions that an optimal control has to satisfy. Keywords Distributed optimal control · Tumor growth · Phase field model · Cahn–Hilliard equation · Optimal control · Necessary optimality conditions · Adjoint system Mathematics Subject Classification 35K61 · 35Q92 · 49J20 · 49K20 · 92C50 1 Introduction In this paper, we deal with a distributed optimal control problem for a system of partial differential equations whose physical context is that of tumor growth dynamics. Our aim is to devote this section to explain the general http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Optimal Distributed Control of an Extended Model of Tumor Growth with Logarithmic Potential

Applied Mathematics and Optimization , Volume OnlineFirst – Oct 30, 2018

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References (31)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-018-9538-1
Publisher site
See Article on Publisher Site

Abstract

This paper is intended to tackle the control problem associated with an extended phase field system of Cahn–Hilliard type that is related to a tumor growth model. This system has been investigated in previous contributions from the viewpoint of well-posedness and asymptotic analyses. Here, we aim to extend the mathematical studies around this system by introducing a control variable and handling the corresponding control problem. We try to keep the potential as general as possible, focusing our investigation towards singular potentials, such as the logarithmic one. We establish the existence of optimal control, the Lipschitz continuity of the control-to-state mapping and even its Fréchet differentiability in suitable Banach spaces. Moreover, we derive the first-order necessary conditions that an optimal control has to satisfy. Keywords Distributed optimal control · Tumor growth · Phase field model · Cahn–Hilliard equation · Optimal control · Necessary optimality conditions · Adjoint system Mathematics Subject Classification 35K61 · 35Q92 · 49J20 · 49K20 · 92C50 1 Introduction In this paper, we deal with a distributed optimal control problem for a system of partial differential equations whose physical context is that of tumor growth dynamics. Our aim is to devote this section to explain the general

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 30, 2018

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