Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach

Kunisch, Karl; Pieper, Konstantin; Rund, Armin

doi:10.1051/m2an/2015048

Kunisch, Karl ^1,² ; Pieper, Konstantin ³ ; Rund, Armin ¹

¹ Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria
² Radon Institute, Austrian Academy of Sciences, Austria
³ Chair of Optimal Control, Technische Universität München, Boltzmannstraße 3, 85748 Garching bei München, Germany

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 381-414.

Résumé

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.

Reçu le : 2014-12-01

MR Zbl

DOI : 10.1051/m2an/2015048

Classification : 35M30, 49K20, 49J52, 90C46
Mots clés : Time optimal control, monodomain equations, semismooth Newton method, reaction diffusion system, asymptotic behavior

Affiliations des auteurs :

Kunisch, Karl ^{1, 2} ; Pieper, Konstantin ³ ; Rund, Armin ¹

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     title = {Time optimal control for a reaction diffusion system arising in cardiac electrophysiology {\textendash} a monolithic approach},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {381--414},
     publisher = {EDP-Sciences},
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     url = {http://archive.numdam.org/articles/10.1051/m2an/2015048/}
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Kunisch, Karl; Pieper, Konstantin; Rund, Armin. Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 381-414. doi : 10.1051/m2an/2015048. http://archive.numdam.org/articles/10.1051/m2an/2015048/

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