September  2016, 6(3): 407-427. doi: 10.3934/mcrf.2016009

Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary

1. 

Institut de Mathématiques de Marseille, CNRS, UMR 7373, École Centrale, Aix-Marseille Université, 13453 Marseille, France

2. 

Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui Province, 230026, China

3. 

Aix-Marseille Université de Toulon, CNRS, CPT, 13288 Marseille, France

Received  September 2015 Revised  February 2016 Published  August 2016

We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove Hölder stability with the aid of a Carleman estimate specifically designed for hyperbolic waveguides.
Citation: Michel Cristofol, Shumin Li, Eric Soccorsi. Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary. Mathematical Control & Related Fields, 2016, 6 (3) : 407-427. doi: 10.3934/mcrf.2016009
References:
[1]

M. Bellassoued, Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation, Inverse Problems, 20 (2004), 1033-1052. doi: 10.1088/0266-5611/20/4/003.  Google Scholar

[2]

M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Applicable Analysis, 83 (2004), 983-1014. doi: 10.1080/0003681042000221678.  Google Scholar

[3]

M. Bellassoued, M. Cristofol and E. Soccorsi, Inverse boundary value problem for the dynamical heterogeneous Maxwell's system, Inverse Problems, 28 (2012), 095009, 18pp. doi: 10.1088/0266-5611/28/9/095009.  Google Scholar

[4]

M. Bellassoued, M. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494. doi: 10.1016/j.jde.2009.03.024.  Google Scholar

[5]

M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability in in an inverse problem for a hyperbolic equation with a finite set of boundary data, Applicable Analysis, 87 (2008), 1105-1119. doi: 10.1080/00036810802369231.  Google Scholar

[6]

M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation, J. Math. Pures Appl., 85 (2006), 193-224. doi: 10.1016/j.matpur.2005.02.004.  Google Scholar

[7]

M. Bellassoued and M. Yamamoto, Determination of a coefficient in the wave equation with a single measurement, Applicable Analysis, 87 (2008), 901-920. doi: 10.1080/00036810802369249.  Google Scholar

[8]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimentional inverse problems, Soviet Math. Dokl., 24 (1981), 244-247. Google Scholar

[9]

L. Cardoulis, M. Cristofol and P. Gaitan, Inverse problem for the Schrödinger operator in an unbounded strip, C. R. Math. Acad. Sci. Paris, 346 (2008), 635-640. doi: 10.1016/j.crma.2008.04.004.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[11]

O. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems, 17 (2001), 717-728. doi: 10.1088/0266-5611/17/4/310.  Google Scholar

[12]

O. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with single measurement, Inverse Problems, 19 (2003), 157-171. doi: 10.1088/0266-5611/19/1/309.  Google Scholar

[13]

V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206. doi: 10.1088/0266-5611/8/2/003.  Google Scholar

[14]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, Berlin, 2006.  Google Scholar

[15]

Y. Kian, Q. S. Phan and E. Soccorsi, Carleman estimate for infinite cylindrical quantum domains and application to inverse problems, Inverse Problems, 30 (2014), 055016, 16pp. doi: 10.1088/0266-5611/30/5/055016.  Google Scholar

[16]

Y. Kian, Q. S. Phan and E. Soccorsi, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, Journal of Mathematical Analysis and Applications, 426 (2015), 194-210. doi: 10.1016/j.jmaa.2015.01.028.  Google Scholar

[17]

M. V. Klibanov and J. Malinsky, Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time dependent data, Inverse Problems, 7 (1991), 577-596. doi: 10.1088/0266-5611/7/4/007.  Google Scholar

[18]

M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an accoustic equation, Applicable Analysis, 85 (2006), 515-538. doi: 10.1080/00036810500474788.  Google Scholar

[19]

J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, vol. 1, Dunod, 1968.  Google Scholar

[20]

P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358. doi: 10.1006/jfan.1997.3188.  Google Scholar

[21]

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.  Google Scholar

show all references

References:
[1]

M. Bellassoued, Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation, Inverse Problems, 20 (2004), 1033-1052. doi: 10.1088/0266-5611/20/4/003.  Google Scholar

[2]

M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Applicable Analysis, 83 (2004), 983-1014. doi: 10.1080/0003681042000221678.  Google Scholar

[3]

M. Bellassoued, M. Cristofol and E. Soccorsi, Inverse boundary value problem for the dynamical heterogeneous Maxwell's system, Inverse Problems, 28 (2012), 095009, 18pp. doi: 10.1088/0266-5611/28/9/095009.  Google Scholar

[4]

M. Bellassoued, M. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494. doi: 10.1016/j.jde.2009.03.024.  Google Scholar

[5]

M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability in in an inverse problem for a hyperbolic equation with a finite set of boundary data, Applicable Analysis, 87 (2008), 1105-1119. doi: 10.1080/00036810802369231.  Google Scholar

[6]

M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation, J. Math. Pures Appl., 85 (2006), 193-224. doi: 10.1016/j.matpur.2005.02.004.  Google Scholar

[7]

M. Bellassoued and M. Yamamoto, Determination of a coefficient in the wave equation with a single measurement, Applicable Analysis, 87 (2008), 901-920. doi: 10.1080/00036810802369249.  Google Scholar

[8]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimentional inverse problems, Soviet Math. Dokl., 24 (1981), 244-247. Google Scholar

[9]

L. Cardoulis, M. Cristofol and P. Gaitan, Inverse problem for the Schrödinger operator in an unbounded strip, C. R. Math. Acad. Sci. Paris, 346 (2008), 635-640. doi: 10.1016/j.crma.2008.04.004.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[11]

O. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems, 17 (2001), 717-728. doi: 10.1088/0266-5611/17/4/310.  Google Scholar

[12]

O. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with single measurement, Inverse Problems, 19 (2003), 157-171. doi: 10.1088/0266-5611/19/1/309.  Google Scholar

[13]

V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206. doi: 10.1088/0266-5611/8/2/003.  Google Scholar

[14]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, Berlin, 2006.  Google Scholar

[15]

Y. Kian, Q. S. Phan and E. Soccorsi, Carleman estimate for infinite cylindrical quantum domains and application to inverse problems, Inverse Problems, 30 (2014), 055016, 16pp. doi: 10.1088/0266-5611/30/5/055016.  Google Scholar

[16]

Y. Kian, Q. S. Phan and E. Soccorsi, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, Journal of Mathematical Analysis and Applications, 426 (2015), 194-210. doi: 10.1016/j.jmaa.2015.01.028.  Google Scholar

[17]

M. V. Klibanov and J. Malinsky, Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time dependent data, Inverse Problems, 7 (1991), 577-596. doi: 10.1088/0266-5611/7/4/007.  Google Scholar

[18]

M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an accoustic equation, Applicable Analysis, 85 (2006), 515-538. doi: 10.1080/00036810500474788.  Google Scholar

[19]

J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, vol. 1, Dunod, 1968.  Google Scholar

[20]

P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358. doi: 10.1006/jfan.1997.3188.  Google Scholar

[21]

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.  Google Scholar

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