Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology

Bedr'Eddine Ainseba; Mostafa Bendahmane; Yuan He

doi:10.3934/nhm.2015.10.369

Article Contents

2015, Volume 10, Issue 2: 369-385. Doi: 10.3934/nhm.2015.10.369

This issue Previous Article Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition Next Article Inhomogeneities in 3 dimensional oscillatory media

Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology

1.
Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux, 3 ter Place de la Victoire, 33076 Bordeaux cedex
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received: October 31, 2013

Revised: February 28, 2015

Published: May 2015

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Abstract

In this paper, we study the stability result for the conductivities diffusion coefficients to a strongly reaction-diffusion system modeling electrical activity in the heart. To study the problem, we establish a Carleman estimate for our system. The proof is based on the combination of a Carleman estimate and certain weight energy estimates for parabolic systems.

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Mathematics Subject Classification: Primary: 35K57, 35R30.

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References

[1]	M. Bendahmane and F. W. Chaves-Silva, Controllability of a degenerating reaction-diffusion system in electrocardiology, to appear in SIAM Journal on Control and Optimization,* arXiv:1106.1788.*
[2]	M. Bendahmane and K. H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218.doi: 10.3934/nhm.2006.1.185.
[3]	M. Bendahmane and K. H. Karlsen, Convergence of a finite volume scheme for the bidomain model of cardiac tissue, Appl. Numer. Math., 59 (2009), 2266-2284.doi: 10.1016/j.apnum.2008.12.016.
[4]	M. Bendahmane, R. Bürger and R. Ruiz Baier, A finite volume scheme for cardiac propagation in media with isotropic conductivities, Math. Comp. Simul., 80 (2010), 1821-1840.doi: 10.1016/j.matcom.2009.12.010.
[5]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[6]	A. L. Bukhgeĭm, Carleman estimates for Volterra operators and uniqueness of inverse problems, in Non-classical Problems of Mathematical Physics, Computing Center of Siberian Branch of Soviet Academy of Sciences, Novosibirsk, 1981, 56-64.
[7]	A. L. Bukhgeim, Introduction to the Theory of Inverse Problems, VSP, Utrecht, 2000.
[8]	A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large class of multidimensional inverse problems, (Russian) Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.
[9]	K. C. Chang, Methods in Nonlinear Analysis, Springer-Verlag Berlin Heidelberg, Netherlands, 2005.
[10]	P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, 2002, 49-78.
[11]	M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a $2 \times 2$ reaction-diffusion system using a Carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573.doi: 10.1088/0266-5611/22/5/003.
[12]	A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, RIM Seoul National University, Korea, 1996.
[13]	O.Yu. Imanuvilov, M. Yamamoto, Lipschitz stability in inverse problems by Carleman estimates, Inverse Problems, 14 (1998), 1229-1245.doi: 10.1088/0266-5611/14/5/009.
[14]	O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM, COCV, 11 (2005), 1-56.doi: 10.1051/cocv:2004030.
[15]	V. Isakov, Carleman estimates and applications to inverse problems, Milan J. Math., 72 (2004), 249-271.doi: 10.1007/s00032-004-0033-6.
[16]	M. V. Klibanov, Carleman estimates and inverse problems in the lasrt two decades, in Surveys on Solution Methods for Inverse Problems, Springer, Vienna, 2000, 119-146.
[17]	M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation, Appl. Anal., 85 (2006), 515-538.doi: 10.1080/00036810500474788.
[18]	M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
[19]	G. Lebeau and L. Robbiano, Contrôle exact de l'equation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.doi: 10.1080/03605309508821097.
[20]	J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems, 12 (1996), 995-1002.doi: 10.1088/0266-5611/12/6/013.
[21]	K. Sakthivel, N. Baranibalan, J.-H. Kim and K. Balachandran, Stability of diffusion coefficients in an inverse problem for the Lotka-Volterra competition system, Acta Appl. Math., 111 (2010), 129-147.doi: 10.1007/s10440-009-9455-z.
[22]	Z. Q. Wu, J. X. Yin and C. P. Wang, Elliptic and Parabolic Equations, World Scientific Publishing Co. Pte. Ltd, 2003.doi: 10.1142/6238.
[23]	M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 65-98.doi: 10.1016/S0021-7824(99)80010-5.
[24]	G. Yuan and M. Yamamoto, Lipshitz stability in the determination of the principal part of a parabolic equation, ESAIM: Control Optim. Calc. Var., 15 (2009), 525-554.doi: 10.1051/cocv:2008043.