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P. Colli, G. Gilardi, E. Rocca, J. Sprekels (2016)
Optimal distributed control of a diffuse interface model of tumor growthNonlinearity, 30
O. Ladyženskaja (1968)
Linear and Quasilinear Equations of Parabolic Type, 23
X. Wu, van Zwieten, van Zee (2014)
Stabilized second‐order convex splitting schemes for Cahn–Hilliard models with application to diffuse‐interface tumor‐growth modelsInternational Journal for Numerical Methods in Biomedical Engineering, 30
P. Colli, G. Gilardi, E. Rocca, J. Sprekels (2015)
Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growtharXiv: Analysis of PDEs
P. Colli, G. Gilardi, J. Sprekels (2017)
Optimal Velocity Control of a Viscous Cahn-Hilliard System with Convection and Dynamic Boundary ConditionsSIAM J. Control. Optim., 56
H. Garcke, K. Lam (2016)
On a Cahn–Hilliard–Darcy System for Tumour Growth with Solution Dependent Source TermsarXiv: Analysis of PDEs
P. Colli, G. Gilardi, G. Marinoschi, E. Rocca (2017)
Optimal control for a conserved phase field system with a possibly singular potentialarXiv: Analysis of PDEs
H. Garcke, K. Lam (2016)
Global weak solutions and asymptotic limits of a Cahn--Hilliard--Darcy system modelling tumour growtharXiv: Analysis of PDEs
H. Garcke, K. Lam (2015)
Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transportEuropean Journal of Applied Mathematics, 28
S. Frigeri, M. Grasselli, E. Rocca (2014)
On a diffuse interface model of tumour growthEuropean Journal of Applied Mathematics, 26
(1987)
Compact sets in the space L p(0, T ; B)
P. Colli, G. Gilardi, J. Sprekels (2014)
On the Cahn–Hilliard equation with dynamic boundary conditions and a dominating boundary potentialJournal of Mathematical Analysis and Applications, 419
P. Colli, J. Sprekels (2012)
Optimal Control of an Allen-Cahn Equation with Singular Potentials and Dynamic Boundary ConditionSIAM J. Control. Optim., 53
P. Colli, G. Gilardi, E. Rocca, J. Sprekels (2015)
Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growtharXiv: Analysis of PDEs
A Hawkins-Daruud, KG Zee, JT Oden (2011)
Numerical simulation of a thermodynamically consistent four-species tumor growth modelInt. J. Numer. Math. Biomed. Eng., 28
H. Garcke, K. Lam (2016)
Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxisarXiv: Analysis of PDEs
G. Gilardi, A. Miranville, G. Schimperna (2009)
On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditionsCommunications on Pure and Applied Analysis, 8
P. Colli, G. Gilardi, J. Sprekels (2014)
A Boundary Control Problem for the Viscous Cahn–Hilliard Equation with Dynamic Boundary ConditionsApplied Mathematics & Optimization, 73
Mimi Dai, E. Feireisl, E. Rocca, G. Schimperna, M. Schonbek (2015)
Analysis of a diffuse interface model of multispecies tumor growthNonlinearity, 30
J Simon (1987)
Compact sets in the space $$L^p(0,T; B)$$ L p ( 0 , T ; B )Ann. Mat. Pura Appl. (4), 146
P. Colli, G. Gilardi, G. Marinoschi, E. Rocca (2014)
Optimal control for a phase field system with a possibly singular potentialarXiv: Analysis of PDEs
H. Garcke, K. Lam, E. Rocca (2016)
Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor GrowthApplied Mathematics & Optimization, 78
S. Frigeri, K. Lam, E. Rocca, G. Schimperna (2017)
On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentialsarXiv: Analysis of PDEs
A. Miranville, S. Zelik (2004)
Robust exponential attractors for Cahn‐Hilliard type equations with singular potentialsMathematical Methods in the Applied Sciences, 27
P. Colli, G. Gilardi, J. Sprekels (2016)
Optimal boundary control of a nonstandard viscous Cahn-Hilliard system with dynamic boundary conditionarXiv: Analysis of PDEs
A. Hawkins-Daarud, Kristoffer Zee, J. Oden (2012)
Numerical simulation of a thermodynamically consistent four‐species tumor growth modelInternational Journal for Numerical Methods in Biomedical Engineering, 28
D. Hilhorst, Johannes Kampmann, Thanh-Nam Nguyen, Kristoffer Zee (2015)
Formal asymptotic limit of a diffuse-interface tumor-growth modelMathematical Models and Methods in Applied Sciences, 25
H. Brezis (1973)
Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert
H Garcke, KF Lam (2018)
Trends on Applications of Mathematics to Mechanics
P. Colli, G. Gilardi, D. Hilhorst (2014)
On a Cahn-Hilliard type phase field system related to tumor growtharXiv: Analysis of PDEs
H. Garcke, K. Lam, Robert Nurnberg, Emanuel Sitka (2017)
A multiphase Cahn--Hilliard--Darcy model for tumour growth with necrosisMathematical Models and Methods in Applied Sciences, 28
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
Applied Mathematics and Optimization – Springer Journals
Published: Oct 30, 2018
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