Asymptotic analysis of an optimal control problem for a phase field tumor growth model with hyperbolic relaxation of the chemical potential
Abstract
In this paper, we study the optimal control of a phase field tumor growth model of Cahn--Hilliard type with possibly singular potential in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. We investigate the asymptotic behavior of the control problem as the relaxation parameter α approaches zero. It is shown that the corresponding state and adjoint state variables of the relaxed problem converge in a suitable sense to their counterparts for the unrelaxed regime where α=0. Also, weak limits of optimal controls for the relaxed problem can be shown to be optimal controls for the unrelaxed case. Moreover, every fixed optimal control u* for the unrelaxed system turns out to be the limit of a family {u*α}α>0 of minimizers of suitable optimal control problems for the relaxed systems, where the original cost functional is replaced by the so-called ``adapted'' cost functional. It is also shown that the variational inequalities satisfied by u*α, which constitute the first-order necessary optimality conditions of the relaxed problems with adapted cost functional, converge to the corresponding variational inequality fulfilled by u* for the unrelaxed problem with the original (nonadapted) cost functional.
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BibTeX
@misc{colli2026asymptotic,
title = {{Asymptotic analysis of an optimal control problem for a phase field tumor growth model with hyperbolic relaxation of the chemical potential}},
author = {Pierluigi Colli and Elisabetta Rocca and Jürgen Sprekels},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04380},
eprint = {2610.04380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04380},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotic analysis of an optimal control problem for a phase field tumor growth model with hyperbolic relaxation of the chemical potential
AU - Pierluigi Colli
AU - Elisabetta Rocca
AU - Jürgen Sprekels
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.04380
UR - https://arxiv.org/abs/2610.04380
AB - In this paper, we study the optimal control of a phase field tumor growth model of Cahn--Hilliard type with possibly singular potential in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. We investigate the asymptotic behavior of the control problem as the relaxation parameter α approaches zero. It is shown that the corresponding state and adjoint state variables of the relaxed problem converge in a suitable sense to their counterparts for the unrelaxed regime where α=0. Also, weak limits of optimal controls for the relaxed problem can be shown to be optimal controls for the unrelaxed case. Moreover, every fixed optimal control u^* for the unrelaxed system turns out to be the limit of a family {u^*_α}_α>0 of minimizers of suitable optimal control problems for the relaxed systems, where the original cost functional is replaced by the so-called ``adapted'' cost functional. It is also shown that the variational inequalities satisfied by u^*_α, which constitute the first-order necessary optimality conditions of the relaxed problems with adapted cost functional, converge to the corresponding variational inequality fulfilled by u^* for the unrelaxed problem with the original (nonadapted) cost functional.
ER -