Journal of Mathematical Sciences

, Volume 142, Issue 6, pp 2528–2539 | Cite as

On a control problem for the wave equation in R3

  • M. I. Belishev
  • A. F. Vakulenko


Solutions of the wave equation (waves) initiated by infinitely distant sources (controls) are considered, and the L2-completeness of reachable sets consisting of such waves is stidued. This problem is a natural analog of the control problem for a bounded domain where the completeness (local approximate controllability) in subdomains filled with waves generated by boundary controls occurs. It is shown that, in contrast to the latter case, the reachable sets formed by waves incoming from infinity are not complete in filled subdomains and describe the associated defect. Next, extending the class of controls to a set of special polynomials, the completeness is gained. A transform defined by jumps that arise in projecting functions to reachable sets is introduced. Its relevance to the Radon transform is clarified. Bibliography: 7 titles.


Radon Control Problem Wave Equation Boundary Control Distant Source 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. A. Avdonin, M. I. Belishev, and S. A. Ivanov, “Controllability in a filled domain for a multidimensional wave equation with a singular boundary control,” Zap. Nauchn. Semin. POMI, 210, 3–14 (1994).Google Scholar
  2. 2.
    M. I. Belishev, “Boundary control in reconstruction of manifolds and metrics (the BC-method),” Inverse Problems, 13, R1–R45 (1997).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. I. Belishev and A. K. Glasman, “Dynamical inverse problem for the Maxwell system: recovering the velocity in a regular zone (the BC-method),” Algebra Analiz, 12, No. 2, 131–182 (2000).MathSciNetGoogle Scholar
  4. 4.
    M. I. Belishev, “On a unitary transform in the space \(\overrightarrow L _2 (\Omega ;R^3 )\) connected with the Weyl decomposition,” Zap. Nauchn. Semin. POMI, 275, 25–40 (2001).Google Scholar
  5. 5.
    R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, Inc. (1982).Google Scholar
  6. 6.
    P. Lax and R. Phillips, Scattering Theory, Academic Press, New York-London (1967).MATHGoogle Scholar
  7. 7.
    D. L. Russell, “Controllability and stabilizability theory for linear partial differential equations,” SIAM Review, 20, No. 4, 639–739 (1978).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • M. I. Belishev
    • 1
  • A. F. Vakulenko
    • 1
  1. 1.St.Petersburg Department of the Steklov Mathematical InstituteSt.PetersburgRussia

Personalised recommendations