Equations of the Boundary Control method for the inverse source problem
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 42, Tome 409 (2012), pp. 121-129 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the dynamical inverse source problem for the abstract nonselfadjoint operator in the Hilbert space and derive the equation of the Boundary Control method for this problem. It is shown that the solution of this equation crucially depends on the property of certain exponential family. We provide the applications of this equation to inverse source problem and to the problem of the extension of the inverse data.
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A. S. Mikhaylov; V. S. Mikhaylov. Equations of the Boundary Control method for the inverse source problem. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 42, Tome 409 (2012), pp. 121-129. http://geodesic.mathdoc.fr/item/ZNSL_2012_409_a7/

[1] C. Alves, A. L. Silvestre, T. Takahashi, M. Tucsnak, “Solving inverse source problems using observability. Applications to the Euler–Bernoulli plate equation”, SIAM J. Control Optim., 48:3 (2009), 1632–1659 | DOI | MR | Zbl

[2] S. A. Avdonin, A. S. Bulanova, “Boundary control approach to the spectral estimation problem. The case of multiple poles”, Math. Contr. Sign. Syst., 22:3 (2011), 245–265 | DOI | MR | Zbl

[3] S. A. Avdonin, A. S. Bulanova, D. Nicolsky, “Boundary control approach to the spectral estimation problem. The case of simple poles”, Sampling Theory in Signal and Image Processing, 8:3 (2009), 225–248 | MR | Zbl

[4] S. Avdonin, F. Gesztesy, K. A. Makarov, “Spectral estimation and inverse initial boundary value problems”, Inverse Probl. Imaging, 4:1 (2010), 1–9 | DOI | MR | Zbl

[5] S. Avdonin, S. Ivanov, Families of Exponentials, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[6] S. Avdonin, V. Mikhaylov, “The boundary control approach to inverse spectral theory”, Inverse Problems, 26:4 (2010), 045009, 19 pp. | DOI | MR | Zbl

[7] S. Avdonin, V. Mikhaylov, “Inverse source problem for the 1-D Schrödinger equation”, Zap. Nauchn. Semin. POMI, 393, 2011, 5–11 | MR

[8] S. Avdonin, V. Mikhaylov, K. Ramdani, Reconstructing the potential for the 1D Schrödinger equation from boundary measurements, submitted

[9] L. Baudouin, J.-P. Puel, “Uniqueness and stability in an inverse problem for the Schrödinger equation”, Inverse Problems, 18:6 (2002), 1537–1554 | DOI | MR | Zbl

[10] M. Bellassoued, M. Choulli, “Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation”, J. Math. Pures Appl., 91 (2009), 233–255 | DOI | MR | Zbl

[11] M. Belishev, “Dynamical systems with boundary control: models and characterization of inverse data”, Inverse Problems, 17:4 (2001), 659–682 | DOI | MR | Zbl

[12] M. Belishev, “On a relation between data of dynamic and spectral inverse problems”, Zap. Nauchn. Semin. POMI, 297, 2003, 30–48 | MR | Zbl

[13] M. Belishev, “Recent progress in the boundary control method”, Inverse Problems, 23:5 (2007), R1–R67 | DOI | MR | Zbl

[14] A. Mercado, A. A. Osses, L. Rosier, “Carleman inequalities and inverse problems for the Schrödinger equation”, C. R. Acad. Sci. Paris, Ser. I, 346 (2008), 53–58 | DOI | MR | Zbl

[15] K. Ramdani, M. Tucsnak, G. Weiss, “Recovering the initial state of an infinite-dimensional system using observers”, Automatica, 46 (2010), 1616–1625 | DOI | MR | Zbl

[16] M. Tucsnak, G. Weiss, Observation and control for operator semigroups, Birkhaüser Advanced Texts, Basler Lehrbucher [Birkhaüser Advanced Texts, Basel Textbooks], Birkhaüser Verlag, Basel, 2009, xii+483 pp. | MR | Zbl

[17] M. Yamamoto, “Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method”, Inverse Problems, 11:2 (1995), 481–496 | DOI | MR | Zbl