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June  2015, 10(2): 369-385. doi: 10.3934/nhm.2015.10.369

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

Bedr'Eddine Ainseba 1, , Mostafa Bendahmane 1, and  Yuan He 2,

1. 

Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux, 3 ter Place de la Victoire, 33076 Bordeaux cedex, France, France
2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received  November 2013 Revised  March 2015 Published  April 2015

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.
Keywords: Carleman estimates., cardiac electric field, reaction-diffusion system, Inverse problem, stability.
Mathematics Subject Classification: Primary: 35K57, 35R3.
Citation: Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks and Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369
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, (). 

[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.

show all references

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, (). 

[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.

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