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.

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 :
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
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
@article{M2AN_2016__50_2_381_0,
     author = {Kunisch, Karl and Pieper, Konstantin and Rund, Armin},
     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},
     volume = {50},
     number = {2},
     year = {2016},
     doi = {10.1051/m2an/2015048},
     mrnumber = {3482548},
     zbl = {1341.35174},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015048/}
}
TY  - JOUR
AU  - Kunisch, Karl
AU  - Pieper, Konstantin
AU  - Rund, Armin
TI  - Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 381
EP  - 414
VL  - 50
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2015048/
DO  - 10.1051/m2an/2015048
LA  - en
ID  - M2AN_2016__50_2_381_0
ER  - 
%0 Journal Article
%A Kunisch, Karl
%A Pieper, Konstantin
%A Rund, Armin
%T Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 381-414
%V 50
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2015048/
%R 10.1051/m2an/2015048
%G en
%F M2AN_2016__50_2_381_0
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://www.numdam.org/articles/10.1051/m2an/2015048/

H. Amann, Linear and Quasilinear Parabolic Problems: Volume I: Abstract Linear Theory. Birkhäuser, Basel (1995). | MR | Zbl

W. Bangerth, R. Hartmann and G. Kanschat, deal.II – a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33 (2007) 24/1–24/27. | DOI | MR | Zbl

R. Becker, D. Meidner and B. Vexler, Efficient numerical solution of parabolic optimization problems by finite element methods. Optim. Methods Softw. 22 (2007) 813–833. | DOI | MR | Zbl

A. Borzì and R. Griesse, Distributed optimal control of lambda-omega systems. J. Numer. Math. 14 (2006) 17–40. | DOI | MR | Zbl

Y. Bourgault, Y. Coudière and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlin. Anal. Real World Appl. 10 (2009) 458–482. | DOI | MR | Zbl

A.J. Brandão, E. Fernández-Cara, P.M. Magalhães and M.A. Rojas-Medar, Theoretical analysis and control results for the Fitzhugh-Nagumo equation. Electron. J. Differ. Eq. 2008 (2008) 1–20. | MR | Zbl

T. Breiten and K. Kunisch, Compensator design for the monodomain equations with the Fitzhugh-Nagumo model. To appear in ESAIM: COCV (2016). Doi: | DOI | Numdam | MR

E. Casas, C. Ryll and F. Tröltzsch, Sparse optimal control of the Schlögl and FitzHugh–Nagumo systems. Comput. Meth. Appl. Math. 13 (2013) 415–442. | DOI | MR | Zbl

M. Chipot, Elements of Nonlinear Analysis, Adv. Texts Series. Springer (2000). | MR | Zbl

P. Colli Franzone, P. Deuflhard, B. Erdmann, J. Lang and L. Pavarino, Adaptivity in space and time for reaction-diffusion systems in electrocardiology. SIAM J. Sci. Comput. 28 (2006) 942–962. | DOI | MR | Zbl

P. Deuflhard and M. Weiser, Numerische Mathematik 3: Adaptive Lösung partieller Differentialgleichungen, De Gruyter Studium. De Gruyter (2011). | MR | Zbl

J. Dieudonné, Foundations of Modern Analysis. Academic Press (1969). | MR | Zbl

L.C. Evans, Partial Differential Equations. American Mathematical Society (2010). | MR | Zbl

H.O. Fattorini, infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems. North-Holland Math. Stud. Elsevier Science, Amsterdam (2005). | MR | Zbl

J.A. Griepentrog, H.-C. Kaiser and J. Rehberg, Heat kernel and resolvent properties for second order elliptic differential operators with general boundary conditions on L p . Adv. Math. Sci. Appl. 11 (2001) 87–112. | MR | Zbl

J.A. Griepentrog and L. Recke, Linear elliptic boundary value problems with non-smooth data: Normal solvability on Sobolev-Campanato spaces. Math. Nachr. 225 (2001) 39–74. | DOI | MR | Zbl

H. Hermes and J.P. Lasalle, Functional Analysis and Time Optimal Control. Math. Sci. Eng. Academic Press, New York (1969). | MR | Zbl

K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. SIAM (2008). | MR | Zbl

K. Ito and K. Kunisch, Semismooth Newton methods for time-optimal control for a class of ODEs. SIAM J. Control Optim. 48 (2010) 3997–4013. | DOI | MR | Zbl

K. Kunisch and A. Rund, Time optimal control of the monodomain model in cardiac electrophysiology. IMA J. Appl. Math. (2015). | MR

K. Kunisch and D. Wachsmuth, On time optimal control of the wave equation and its numerical realization as parametric optimization problem. SIAM J. Control Optim. 51 (2013) 1232–1262. | DOI | MR | Zbl

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I. Springer (1972). | MR | Zbl

J.D. Murray, Mathematical Biology I. An Introduction. In vol. 17 of Interdisciplinary Applied Mathematics. 3rd edition. Springer, New York (2002). | MR | Zbl

C. Nagaiah, K. Kunisch and G. Plank, Optimal control approach to termination of re-entry waves in cardiac electrophysiology. J. Math. Biol. 67 (2013) 359–388. | DOI | MR | Zbl

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. In vol. 44 of Appl. Math. Sci. Springer-Verlag, New York (1983). | MR | Zbl

K. Pieper, Finite element discretization and efficient numerical solution of elliptic and parabolic sparse control problems. Ph.D. dissertation, Technische Universität München (2015).

J.-P. Raymond and H. Zidani, Pontryagin’s principle for time-optimal problems. J. Optim. Theory Appl. 101 (1999) 375–402. | DOI | MR | Zbl

S.M. Robinson, Normal maps induced by linear transformations. Math. Oper. Res. 17 (1992) 691–714. | DOI | MR | Zbl

F. Schlögl, Chemical reaction models for non-equilibrium phase transitions. Z. Phys. A 253 (1972) 147–161. | DOI

T. Steihaug, The conjugate gradient method and trust regions in large scale optimization. SIAM J. Numer. Anal. 20 (1983) 626–637. | DOI | MR | Zbl

J. Sundnes, G.T. Lines, X. Cai, B.F. Nielsen, K.-A. Mardal and A. Tveito, Computing the Electrical Activity in the Heart. Springer, Berlin, Heidelberg (2006). | MR | Zbl

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. In vol. 18 of North-Holland Math. Library. North-Holland Publ., Amsterdam (1978). | MR | Zbl

F. Tröltzsch, Optimal Control of Partial Differential Equations. In vol. 112 of Grad. Stud. Math. AMS, Providence, Rhode Island (2010). | MR | Zbl

L. Tung, A Bi-domain Model for Describing Ischemic Myocardial D-c Potentials. Ph.D. thesis, Massachusetts Institute of Technology (1978).

M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. MOS-SIAM Series on Optimization. SIAM (2011). | MR | Zbl

W.P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Springer (1989). | MR | Zbl

Cité par Sources :