Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach∗,∗∗
Institute for Mathematics and Scientific Computing, University of
2 Radon Institute, Austrian Academy of Sciences, Austria
3 Chair of Optimal Control, Technische Universität München, Boltzmannstraße 3, 85748 Garching bei München, Germany
Revised: 6 May 2015
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).
© EDP Sciences, SMAI 2016