Boundary feedback stabilization of the monodomain equations

Tobias Breiten; Karl Kunisch

doi:10.3934/mcrf.2017013

Article Contents

2017, Volume 7, Issue 3: 369-391. Doi: 10.3934/mcrf.2017013

This issue Previous Article Quantification of the unique continuation property for the heat equation Next Article Finite element approximation of sparse parabolic control problems

Boundary feedback stabilization of the monodomain equations

Tobias Breiten^1, , and
Karl Kunisch^1,2,

1.
Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität Graz, Heinrichstr. 36,8010 Graz, Austria
2.
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria

* Corresponding author: Tobias Breiten
* Corresponding author: Tobias Breiten

Received: June 30, 2016

Revised: October 31, 2016

Published: October 2017

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Abstract

Boundary feedback control for a coupled nonlinear PDE-ODE system (in the two and three dimensional cases) is studied. Particular focus is put on the monodomain equations arising in the context of cardiac electrophysiology. Neumann as well as Dirichlet based boundary control laws are obtained by an algebraic operator Riccati equation associated with the linearized system. Local exponential stability of the nonlinear closed loop system is shown by a fixed-point argument. Numerical examples are given for a finite element discretization of the two dimensional monodomain equations.

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Mathematics Subject Classification: Primary: 35K57, 93D15, 93B52.

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Figure 1. Control setup

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Figure 2. Stabilization of perturbed initial state

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Figure 3. Stabilization of a reentry wave

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Figure 4. Stabilization of perturbed initial state

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