Abstract
Motivated by the study of related optimal control problems, weak and strong solution concepts for the bidomain system together with two-variable ionic models are analyzed. A key ingredient for the analysis is the bidomain bilinear form. Global existence of weak and strong solutions as well as stability and uniqueness theorems is proven.
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Adams R.A., Fournier J.J.F.: Sobolev Spaces, 2nd ed. Academic Press / Elsevier, Amsterdam etc (2007)
Aliev R.R., Panfilov A.V.: A simple two-variable model of cardiac excitation. Chaos, Solitons Fractals 7, 293–301 (1996)
Boulakia, M., Fernández, M.A., Gerbeau, J.-F., Zemzemi, N.: A coupled system of PDEs and ODEs arising in electrocardiograms modeling. Appl. Math. Res. Express (2008), http://dx.doi.org/10.1093/amrx/abn002 (electronically published)
Bourgault Y., Coudière Y., Pierre C.: Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlinear Anal. Real World Appl. 10, 458–482 (2009)
Colli Franzone, P., Savaré, G.: Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level. In: Lorenzi, A., Ruf, B. (eds.) Evolution Equations, Semigroups and Functional Analysis, pp. 49–78. Birkhäuser, Basel, Boston, Berlin (2002) (Progress in Nonlinear Differential Equations and their Applications, vol. 50)
Colli Franzone P., Guerri L., Tentoni S.: Mathematical modeling of the excitation process in myocardial tissue: influence of fiber rotation on wavefront propagation and potential field. Math. Biosci. 101, 155–235 (1990)
Evans L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)
FitzHugh R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
Ito K., Kunisch K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008)
Kunisch K., Wagner M.: Optimal control of the bidomain system (I): The monodomain approximation with the Rogers-McCulloch model. Nonlinear Anal. Real World Appl. 13, 1525–1550 (2012)
Nagaiah C., Kunisch K., Plank G.: Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology. Comput. Optim. Appl. 49, 149–178 (2011)
Nagumo J., Arimoto S., Yoshizawa S.: An active pulse transmission line simulating nerve axon. Proc. Inst. Radio Eng. 50, 2061–2070 (1962)
Rogers J.M., McCulloch A.D.: A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng. 41, 743–757 (1994)
Schaback R., Wendland H.: Numerische Mathematik, 5th ed. Springer, Berlin (2005)
Showalter R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society, Providence (1997)
Sundnes J., Lines G.T., Cai X., Nielsen B.F., Mardal K.-A., Tveito A.: Computing the Electrical Activity in the Heart. Springer, Berlin (2006)
Tung, L.: A Bi-Domain Model for Describing Ischemic Myocardial D-C Potentials. PhD thesis. Massachusetts Institute of Technology (1978)
Veneroni M.: Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field. Nonlinear Anal. Real World Appl. 10, 849–868 (2009)
Warga J.: Optimal Control of Differential and Functional Equations. Academic Press, New York (1972)
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Kunisch, K., Wagner, M. Optimal control of the bidomain system (II): uniqueness and regularity theorems for weak solutions. Annali di Matematica 192, 951–986 (2013). https://doi.org/10.1007/s10231-012-0254-1
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DOI: https://doi.org/10.1007/s10231-012-0254-1
Keywords
- PDE constrained optimization
- Bidomain equations
- Two-variable models
- Weak solution
- Uniqueness theorem
- Regularity theorem