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On a hyperbolic coefficient inverse problem via partial dynamic boundary measurements. (English) Zbl 1200.35321
Summary: This paper is devoted to the identification of the unknown smooth coefficient $c$ entering the hyperbolic equation $c(x)\partial_t^2u -\Delta u=0$ in a bounded smooth domain in $\Bbb R^d$ from partial (on part of the boundary) dynamic boundary measurements. In this paper, we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset $\Gamma $ of the boundary determines explicitly the coefficient $c$ provided that $c$ is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula for calculating the coefficient $c$ from the knowledge of the difference between the local Dirichlet-to-Neumann maps.

MSC:
35R30Inverse problems for PDE
49J20Optimal control problems with PDE (existence)
WorldCat.org
Full Text: DOI EuDML
References:
[1] H. Ammari, “Identification of small amplitude perturbations in the electromagnetic parameters from partial dynamic boundary measurements,” Journal of Mathematical Analysis and Applications, vol. 282, no. 2, pp. 479-494, 2003. · Zbl 1082.78006 · doi:10.1016/S0022-247X(02)00709-6
[2] L. Beilina and M. V. Klibanov, “A globally convergent numerical method for a coefficient inverse problem,” SIAM Journal on Scientific Computing, vol. 31, no. 1, pp. 478-509, 2008. · Zbl 1185.65175 · doi:10.1137/070711414
[3] M. I. Belishev, “Dynamical inverse problem for the equation - \Delta u - \nabla ln\rho .\nabla u=0 (the BC method),” Cubo, vol. 10, no. 2, pp. 15-30, 2008. · Zbl 1155.35483
[4] M. I. Belishev, “The Caldéron problem for two-dimensional manifolds by the BC-method,” SIAM Journal on Mathematical Analysis, vol. 35, no. 1, pp. 172-182, 2003. · Zbl 1048.58019 · doi:10.1137/S0036141002413919
[5] S. I. Kabanikhin, Projection-Difference Methods for Determining the Coeffcients of Hyperbolic Equations, Nauka, Novosibirsk, Russia, 1988. · Zbl 0658.65131
[6] M. V. Klibanov, “Inverse problems and Carleman estimates,” Inverse Problems, vol. 8, no. 4, pp. 575-596, 1992. · Zbl 0755.35151 · doi:10.1088/0266-5611/8/4/009
[7] A. I. Nachman, “Global uniqueness for a two-dimensional inverse boundary value problem,” Annals of Mathematics, vol. 143, no. 1, pp. 71-96, 1996. · Zbl 0857.35135 · doi:10.2307/2118653
[8] A. G. Ramm, Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, NY, USA, 2005. · Zbl 1083.35002
[9] A. G. Ramm and Rakesh, “Property C and an inverse problem for a hyperbolic equation,” Journal of Mathematical Analysis and Applications, vol. 156, no. 1, pp. 209-219, 1991. · Zbl 0729.35147 · doi:10.1016/0022-247X(91)90391-C
[10] M. Yamamoto, “Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method,” Inverse Problems, vol. 11, no. 2, pp. 481-496, 1995. · Zbl 0822.35154 · doi:10.1088/0266-5611/11/2/013
[11] M. Yamamoto, “Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method,” Inverse Problems, vol. 11, no. 2, pp. 481-496, 1995. · Zbl 0822.35154 · doi:10.1088/0266-5611/11/2/013
[12] L. Beilina and M. V. Klibanov, “A globally convergent numerical method and the adaptivity technique for a hyperbolic coefficient inverse problem. Part I: analytical study,” preprint, 2009.
[13] M. V. Klibanov and A. Timonov, “Numerical studies on the globally convergent convexification algorithm in 2D,” Inverse Problems, vol. 23, no. 1, pp. 123-138, 2007. · Zbl 1115.65111 · doi:10.1088/0266-5611/23/1/006
[14] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse and Ill-posed Problems Series, VSP, Utrecht, The Netherlands, 2004. · Zbl 1069.65106
[15] J. Xin and M. V. Klibanov, “Comparative studies of the globally convergent convexification algorithm with application to imaging of antipersonnel land mines,” Applicable Analysis, vol. 86, no. 9, pp. 1147-1176, 2007. · Zbl 1131.65086 · doi:10.1080/00036810701609869
[16] L. Päivärinta and V. Serov, “Recovery of jumps and singularities in the multidimensional Schrödinger operator from limited data,” Inverse Problems and Imaging, vol. 1, no. 3, pp. 525-535, 2007. · Zbl 1151.35103 · doi:10.3934/ipi.2007.1.525
[17] V. Serov and L. Päivärinta, “Inverse scattering problem for two-dimensional Schrödinger operator,” Journal of Inverse and Ill-Posed Problems, vol. 14, no. 3, pp. 295-305, 2006. · Zbl 1111.35126 · doi:10.1163/156939406777340946
[18] Yu. Chen, “Inverse scattering via Heisenberg’s uncertainty principle,” Inverse Problems, vol. 13, no. 2, pp. 253-282, 1997. · Zbl 0872.35123 · doi:10.1088/0266-5611/13/2/005
[19] C. Daveau, A. Khelifi, and A. Sushchenko, “Reconstruction of closely spaced small inhomogeneities via boundary measurements for the full time-dependent Maxwell’s equations,” Applied Mathematical Modelling, vol. 33, no. 3, pp. 1719-1728, 2009. · Zbl 1168.78305 · doi:10.1016/j.apm.2008.03.012
[20] D. J. Cedio-Fengya, S. Moskow, and M. S. Vogelius, “Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction,” Inverse Problems, vol. 14, no. 3, pp. 553-595, 1998. · Zbl 0916.35132 · doi:10.1088/0266-5611/14/3/011
[21] A. Friedman and M. Vogelius, “Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence,” Archive for Rational Mechanics and Analysis, vol. 105, no. 4, pp. 299-326, 1989. · Zbl 0684.35087 · doi:10.1007/BF00281494
[22] J. Sylvester and G. Uhlmann, “A global uniqueness theorem for an inverse boundary value problem,” Annals of Mathematics, vol. 125, no. 1, pp. 153-169, 1987. · Zbl 0625.35078 · doi:10.2307/1971291
[23] H. Brezis, Analyse Fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, France, 1983.
[24] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1998. · Zbl 0902.35002
[25] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, Contrôlabilité Exacte, vol. 8 of Recherches en Mathématiques Appliquées, Masson, Paris, France, 1988. · Zbl 0653.93002
[26] M. I. Belishev and Y. V. Kurylev, “Boundary control, wave field continuation and inverse problems for the wave equation,” Computers & Mathematics with Applications, vol. 22, no. 4-5, pp. 27-52, 1991. · Zbl 0768.35077 · doi:10.1016/0898-1221(91)90130-V
[27] G. Bruckner and M. Yamamoto, “Determination of point wave sources by pointwise observations: stability and reconstruction,” Inverse Problems, vol. 16, no. 3, pp. 723-748, 2000. · Zbl 0962.35184 · doi:10.1088/0266-5611/16/3/312
[28] J.-P. Puel and M. Yamamoto, “Applications de la contrôlabilité exacte à quelques problèmes inverses hyperboliques,” Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, vol. 320, no. 10, pp. 1171-1176, 1995. · Zbl 0829.93019
[29] J.-P. Puel and M. Yamamoto, “On a global estimate in a linear inverse hyperbolic problem,” Inverse Problems, vol. 12, no. 6, pp. 995-1002, 1996. · Zbl 0862.35141 · doi:10.1088/0266-5611/12/6/013
[30] Rakesh and W. W. Symes, “Uniqueness for an inverse problem for the wave equation,” Communications in Partial Differential Equations, vol. 13, no. 1, pp. 87-96, 1988. · Zbl 0667.35071 · doi:10.1080/03605308808820539
[31] Z. Q. Sun, “On continuous dependence for an inverse initial-boundary value problem for the wave equation,” Journal of Mathematical Analysis and Applications, vol. 150, no. 1, pp. 188-204, 1990. · Zbl 0733.35107 · doi:10.1016/0022-247X(90)90207-V
[32] C. Bardos, G. Lebeau, and J. Rauch, “Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,” SIAM Journal on Control and Optimization, vol. 30, no. 5, pp. 1024-1065, 1992. · Zbl 0786.93009 · doi:10.1137/0330055
[33] M. Yamamoto, “On an inverse problem of determining source terms in Maxwell’s equations with a single measurement,” in Inverse Problems, Tomography, and Image Processing (Newark, DE, 1997), pp. 241-256, Plenum, New York, NY, USA, 1998. · Zbl 0910.35144
[34] N. U. Ahmed and T. Wan, “Exact boundary controllability of electromagnetic fields in general regions,” Dynamic Systems and Applications, vol. 5, no. 2, pp. 229-243, 1996. · Zbl 0858.93010