Skip to main content Accessibility help

Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions

  • Karl Kunisch (a1) and Marcus Wagner (a2)


We consider optimal control problems for the bidomain equations of cardiac electrophysiology together with two-variable ionic models, e.g. the Rogers–McCulloch model. After ensuring the existence of global minimizers, we provide a rigorous proof for the system of first-order necessary optimality conditions. The proof is based on a stability estimate for the primal equations and an existence theorem for weak solutions of the adjoint system.



Hide All
[1] Ainseba, B., Bendahmane, M. and Ruiz-Baier, R., Analysis of an optimal control problem for the tridomain model in cardiac electrophysiology. J. Math. Anal. Appl. 388 (2012) 231247.
[2] Aliev, R.R. and Panfilov, A.V., A simple two-variable model of cardiac excitation. Chaos, Solitons and Fractals 7 (1996) 293301.
[3] M.S. Berger, Nonlinearity and Functional Analysis. Academic Press, New York, San Francisco, London (1977).
[4] Bourgault, Y., Coudière, Y. and Pierre, C., Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlinear Analysis: Real World Appl. 10 (2009) 458482.
[5] A.J.V. Brandaõ, E. Fernández-Cara, P.M.D. Magalhães and M.A. Rojas-Medar, Theoretical analysis and control results for the FitzHugh–Nagumo equation. Electron. J. Differ. Eq. (2008) 1–20.
[6] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer, New York (2008).
[7] L.C. Evans, Partial Differential Equations. Amer. Math. Soc. Providence (1998).
[8] FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1 (1961) 445466.
[9] K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008).
[10] Kunisch, K., Nagaiah, C. and Wagner, M., A parallel Newton-Krylov method for optimal control of the monodomain model in cardiac electrophysiology. Comput. Visualiz. Sci. 14 (2011) [2012], 257269.
[11] Kunisch, K. and Wagner, M., Optimal control of the bidomain system (I): The monodomain approximation with the Rogers–McCulloch model. Nonlinear Anal.: Real World Appl. 13 (2012) 15251550.
[12] K. Kunisch and M. Wagner, Optimal control of the bidomain system (II): Uniqueness and regularity theorems. University of Graz, Institute for Mathematics and Scientific Computing, SFB-Report No. 2011–008 (to appear: Ann. Mat. Pura Appl.)
[13] S. Muzdeka and E. Barbieri, Control theory inspired considerations for the mathematical model defibrillation, in Proc. of the 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference 7416–7421.
[14] Nagaiah, C. and Kunisch, K., Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology. Appl. Num. Math. 61 (2011) 5365.
[15] Nagaiah, C., Kunisch, K. and Plank, G., Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology. Comput. Optim. Appl. 49 (2011) 149178.
[16] C. Nagaiah, K. Kunisch and G. Plank, Optimal control approach to termination of re-entry waves in cardiac electrophysiology. University of Graz, Institute for Mathematics and Scientific Computing, SFB-Report No. 2011–020 (to appear: J. Math. Biol., doi: 10.1007/s00285-012-0557-2)
[17] Nagumo, J., Arimoto, S. and Yoshizawa, S., An active pulse transmission line simulating nerve axon. Proc. Institute of Radio Engineers 50 (1962) 20612070.
[18] Rogers, J.M. and McCulloch, A.D., A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Engrg. 41 (1994) 743757.
[19] S. Rolewicz, Funktionalanalysis und Steuerungstheorie. Springer, Berlin, Heidelberg, New York (1976).
[20] 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 (2006).
[21] L. Tung, A Bi-Domain Model for Describing Ischemic Myocardial D-C Potentials. Ph.D. thesis. Massachusetts Institute of Technology (1978).
[22] Veneroni, M., Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field. Nonlinear Analysis: Real World Appl. 10 (2009) 849868.
[23] J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York, London (1972).
[24] K. Yosida, Functional Analysis. Springer, Berlin (1995) (reprint of the 6th edn. from 1980).


Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions

  • Karl Kunisch (a1) and Marcus Wagner (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed