Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

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
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • 1. J. D. Achenbach, Wave propagation in elastic solids,
    North-Holland Publ. Co., Amsterdam, 1973.
  • 2. E. I. Grigolyuk and I. J. Selezov, Nonclassical theories of vibration of rods, plates and shells, Mechanics of Solids, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesouz. Inst. Nauch. i Tekhn. Inform., Moscow, 1973, pp. 4-303. (Russian)
  • 3. L. P. Nizhnik, Inverse scattering problems for hyperbolic equations, Naukova Dumka, Kiev, 1991. (Russian) MR 1146436 (93i:35156)
  • 4. A. S. Blagoveshchenskiĭ, An inverse axisymmetric Lamb problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI) 203 (1992), 51-67; English transl., J. Math. Sci. (N.Y.) 79 (1996), no. 4, 1191-1202. MR 1193678 (93k:35174)
  • 5. M. I. Belishev, A. S. Blagovestchenskiĭ, and S. A. Ivanov, Erratum to ``The two-velocity dynamical system: boundary control of waves and inverse problems'' [Wave Motion 25 (1997), 83-107], Wave Motion 26 (1997), no. 1, 99. MR 1431888 (98g:73015a)
  • 6. M. I. Belishev and S. A. Ivanov, Characterization of data in dynamical inverse problem for a two-velocity system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 259 (1999), 19-45; English transl., J. Math. Sci. (N.Y.) 109 (2002), no. 5, 1814-1834. MR 1754356 (2001i:35283)
  • 7. -, Uniqueness in the small in a dynamic inverse problem for a two-velocity system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 275 (2001), 41-54; English transl., J. Math. Sci. (N.Y.) 117 (2003), no. 2, 3910-3917. MR 1854499 (2002h:35326)
  • 8. -, Reconstraction of the parametres of a system of connected teams from dynamic boundary measurements, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 324 (2005), 20-42; English transl., J. Math. Sci. (N. Y.) 138 (2006), no. 2, 5491-5502. MR 2159346 (2007g:35262)
  • 9. A. Morassi, G. Nakamura, and M. Sini, An inverse dynamical problem for connected beams, European J. Appl. Math. 16 (2005), no. 1, 83-109. MR 2148682 (2006a:74035)
  • 10. P. S. Rakesh, Stability for an inverse problem for a two speed hyperbolic PDE in one space dimension, Inverse Problems 26 (2010), no. 2, 025005, 20 pp. MR 2575362 (2011a:35579)
  • 11. V. G. Romanov, On the problem of determining the parameter of a layered elastic medium and an impulse source, Sibirsk. Mat. Zh. 49 (2008), no. 5, 1157-1183; English transl., Sib. Math. J. 49 (2008), no. 5, 919-943. MR 2469061 (2009i:76022)
  • 12. M. G. Krein, On a method of effective solution of inverse boundary problem, Dokl. Akad. Nauk SSSR 94 (1954), no. 6, 767-770. (Russian) MR 0062904 (16:38h)
  • 13. A. S. Blagoveshchenskiĭ, The local method of solution of the nonstationary inverse problem for an inhomogeneous string, Tr. Mat. Inst. Steklov. 115 (1971), 28-38; English transl., Proc. Steklov. Inst. Math. 115 (1971), 30-41. MR 0307558 (46:6678)
  • 14. V. G. Romanov, Inverse problems of mathematical physics, Nauka, Novosibirsk, 1984. (Russian) MR 759893 (86g:35205)
  • 15. M. I. Belishev, Boundary control method in dynamical inverse problems -- an introductory course, Dynamical Inverse Problems: Theory and Application, Springer, New York, 2011, pp. 85-150. MR 3050418
  • 16. M. I. Belishev and A. V. Zurov, Effects associated with the coincidence of velocities in a two-velocity dynamical system, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 264 (2000), 44-65; English transl., J. Math. Sci. (N.Y.) 111 (2002), no. 4, 3645-3656. MR 1796997 (2002e:47018)
  • 17. M. I. Belishev and A. L. Pestov, The direct dynamic problem for the Timoshenko beam, Zap. Nauchn. Sem. S.-Peterburg. Mat. Inst. Steklov. (POMI) 369 (2009), 16-47; English transl., J. Math. Sci. (N.Y.) 167 (2010) no. 5, 603-621. MR 2749199 (2011m:74080)
  • 18. M. I. Belishev and A. S. Blagoveshchenskiĭ, Dynamic inverse problems of the theory of waves, St.Petersburg Univ., St.Petersburg, 1999. (Russian)
  • 19. A. S. Blagoveshchenskiĭ, The nonselfadjoint inverse matrix boundary problem for a hyperbolic differential equation, Probl. Math. Phys., No. 5, Leningrad. Univ., 1971, pp. 38-61. (Russian) MR 0303124 (46:2262)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35R30

Retrieve articles in all journals with MSC (2010): 35R30


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

American Mathematical Society