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. doi: 10.1088/0266-5611/20/4/003.

[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. doi: 10.1080/0003681042000221678.

[3]

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

[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. doi: 10.1016/j.jde.2009.03.024.

[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. doi: 10.1080/00036810802369231.

[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. doi: 10.1016/j.matpur.2005.02.004.

[7]

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

[8]

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

[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. doi: 10.1016/j.crma.2008.04.004.

[10]

L. C. Evans, Partial Differential Equations,, Amer. Math. Soc., (2010). doi: 10.1090/gsm/019.

[11]

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

[12]

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

[13]

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

[14]

V. Isakov, Inverse Problems for Partial Differential Equations,, Springer-Verlag, (2006).

[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). doi: 10.1088/0266-5611/30/5/055016.

[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. doi: 10.1016/j.jmaa.2015.01.028.

[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. doi: 10.1088/0266-5611/7/4/007.

[18]

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

[19]

J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications,, vol. 1, (1968).

[20]

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

[21]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl., 78 (1999), 65. doi: 10.1016/S0021-7824(99)80010-5.

show all references

References:
[1]

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

[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. doi: 10.1080/0003681042000221678.

[3]

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

[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. doi: 10.1016/j.jde.2009.03.024.

[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. doi: 10.1080/00036810802369231.

[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. doi: 10.1016/j.matpur.2005.02.004.

[7]

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

[8]

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

[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. doi: 10.1016/j.crma.2008.04.004.

[10]

L. C. Evans, Partial Differential Equations,, Amer. Math. Soc., (2010). doi: 10.1090/gsm/019.

[11]

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

[12]

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

[13]

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

[14]

V. Isakov, Inverse Problems for Partial Differential Equations,, Springer-Verlag, (2006).

[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). doi: 10.1088/0266-5611/30/5/055016.

[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. doi: 10.1016/j.jmaa.2015.01.028.

[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. doi: 10.1088/0266-5611/7/4/007.

[18]

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

[19]

J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications,, vol. 1, (1968).

[20]

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

[21]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl., 78 (1999), 65. doi: 10.1016/S0021-7824(99)80010-5.

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