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Characterization of the inverse problem data for one-dimensional two-velocity dynamical system


Authors: M. I. Belishev and A. L. Pestov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 26 (2014), nomer 3.
Journal: St. Petersburg Math. J. 26 (2015), 411-440
MSC (2010): Primary 35R30
DOI: https://doi.org/10.1090/S1061-0022-2015-01344-7
Published electronically: March 20, 2015
MathSciNet review: 3289178
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Abstract | References | Similar Articles | Additional Information

Abstract: The evolution of the dynamical system in question is described by the wave equation $ \rho u_{tt}-(\gamma u_{x}) _{x}+Au_{x}+Bu=0$, $ x>0$, $ t>0$, with the zero Cauchy data at $ t=0$ and the Dirichlet boundary control at $ x=0$. Here $ \rho $, $ \gamma $, $ A$, $ B$ are smooth real $ 2\times 2$-matrix-valued functions of $ x$; $ \rho =\mathrm {diag}\{\rho _1, \rho _2\}$ and $ \gamma =\mathrm {diag}\{\gamma _1, \gamma _2\}$ are matrices with positive entries; and $ u=u(x,t)$ is a solution (an $ {\mathbb{R}}^2$-valued function). For $ x\geq 0$, it is assumed that $ \sqrt {\frac {\gamma _{2}}{\rho _{2}}}< \sqrt {\frac {\gamma _{1}}{\rho _{1}}}$ and $ A^{\mathrm {tr}} =-A$, $ A_x =B -B^{\mathrm {tr}}$. The ``input-output'' correspondence is realized by the response operator $ R\colon u(0,t) \mapsto \gamma (0)u_x(0,t)$, $ t\geq 0$, which plays the role of inverse problem data in applications. In the paper, a constructive characterization is given for the response operators of the systems of this type.


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Additional Information

M. I. Belishev
Affiliation: St. Petersburg Branch, Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023; Physics Department, St. Petersburg State University, Russia
Email: belishev@pdmi.ras.ru

A. L. Pestov
Affiliation: St. Petersburg Branch, Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023; Physics Department, St. Petersburg State University, Russia
Email: pestov@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-2015-01344-7
Keywords: Two-velocity dynamical system with boundary control, characterization of the inverse problem data
Received by editor(s): August 22, 2013
Published electronically: March 20, 2015
Additional Notes: The author were supported by RFBR (grants nos. 14-01-00535A and 12-01-31446mol-a) and by the grants NSh-1771.2014.1 and SPbGU 6.38.670.2013
Dedicated: To the 75th anniversary of A. S. Blagoveshchenskiĭ
Article copyright: © Copyright 2015 American Mathematical Society