On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements

This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation in a bounded smooth domain in 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 of the boundary determines explicitly the coefficient provided that 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 from the knowledge of the difference between the local Dirichlet-to-Neumann maps.

[1]  A. Nachman,et al.  Global uniqueness for a two-dimensional inverse boundary value problem , 1996 .

[2]  Mikhail I. Belishev,et al.  The Calderon Problem for Two-Dimensional Manifolds by the BC-Method , 2003, SIAM J. Math. Anal..

[3]  Masahiro Yamamoto,et al.  ON AN INVERSE PROBLEM OF DETERMINING SOURCE TERMS IN MAXWELL'S EQUATIONS WITH A SINGLE MEASUREMENT , 1998 .

[4]  Avner Friedman,et al.  Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence , 1989 .

[5]  J.-P. Puel,et al.  Applications de la contrôlabilité exacte à quelques problèmes inverses hyperboliques , 1995 .

[6]  A. Khelifi,et al.  Reconstruction of closely spaced small inhomogeneities via boundary measurements for the full time-dependent Maxwell’s equations , 2009, 0912.1122.

[7]  Michael V. Klibanov,et al.  Inverse Problems and Carleman Estimates , 1992 .

[8]  Stability , 1973 .

[9]  Yaroslav Kurylev,et al.  Boundary control, wave field continuation and inverse problems for the wave equation , 1991 .

[10]  Michael V. Klibanov,et al.  A Globally Convergent Numerical Method for a Coefficient Inverse Problem , 2008, SIAM J. Sci. Comput..

[11]  Masahiro Yamamoto,et al.  On a global estimate in a linear inverse hyperbolic problem , 1996 .

[12]  D. L. Russell.  Review: J.-L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués , 1990 .

[13]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[14]  Michael Vogelius,et al.  Identification of conductivity imperfections of small diameter by boundary measurements. Continuous , 1998 .

[15]  Ziqi Sun,et al.  On continuous dependence for an inverse initial boundary value problem for the wave equation , 1990 .

[16]  Rakesh,et al.  Property C and an inverse problem for a hyperbolic equation , 1991 .

[17]  Rakesh,et al.  Uniqueness for an inverse problem for the wave equation , 1988 .

[18]  Masahiro Yamamoto,et al.  Determination of point wave sources by pointwise observations: stability and reconstruction , 2000 .

[19]  C. Bardos,et al.  Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , 1992 .

[20]  Michael V. Klibanov,et al.  Carleman estimates for coefficient inverse problems and numerical applications , 2004 .

[21]  Masahiro Yamamoto,et al.  Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method , 1995 .

[22]  H. Ammari.  Identification of small amplitude perturbations in the electromagnetic parameters from partial dynamic boundary measurements , 2003 .

[23]  J. Xin,et al.  Comparative studies of the globally convergent convexification algorithm with application to imaging of antipersonnel land mines , 2007 .

[24]  Michael V. Klibanov,et al.  Numerical studies on the globally convergent convexification algorithm in 2D , 2007 .

[25]  J. Sylvester,et al.  A global uniqueness theorem for an inverse boundary value problem , 1987 .

[26]  A. Ramm,et al.  Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering , 2004 .

[27]  Yu Chen,et al.  Inverse scattering via Heisenberg's uncertainty principle , 1996 .