| Issue |
ESAIM: M2AN
Volume 50, Number 2, March-April 2016
|
|
|---|---|---|
| Page(s) | 381 - 414 | |
| DOI | https://doi.org/10.1051/m2an/2015048 | |
| Published online | 19 February 2016 | |
Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach∗,∗∗
1
Institute for Mathematics and Scientific Computing, University of
Graz, Heinrichstraße
36, 8010
Graz,
Austria
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; This email address is being protected from spambots. You need JavaScript enabled to view it.
2
Radon Institute, Austrian Academy of Sciences, Austria
3
Chair of Optimal Control, Technische Universität München,
Boltzmannstraße 3,
85748
Garching bei München,
Germany
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Received:
1
December
2014
Revised:
6
May
2015
Abstract
Motivated by the termination of undesirable arrhythmia, a time optimal control formulation for the monodomain equations is proposed. It is shown that, under certain conditions, the optimal solutions of this problem steer the system into an appropriate stable neighborhood of the resting state. Towards this goal, some new regularity results and asymptotic properties for the monodomain equations with the Rogers−McCulloch ionic model are obtained. For the numerical realization, a monolithic approach, which simultaneously optimizes for the optimal times and optimal controls, is presented and analyzed. Its practical realization is based on a semismooth Newton method. Numerical examples and comparisons are included.
Mathematics Subject Classification: 35M30 / 49K20 / 49J52 / 90C46
Key words: Time optimal control / monodomain equations / semismooth Newton method / reaction diffusion system / asymptotic behavior
The first and second author gratefully acknowledge support from the International Research Training Group IGDK 1754, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF).
The third author gratefully acknowledges the Austrian Science Fund (FWF) for financial support under SFB F32, “Mathematical Optimization and Applications in Biomedical Sciences”.
© EDP Sciences, SMAI 2016
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