Advertisement

Local Boundary Controllability in Classes of Differentiable Functions for the Wave Equation

  • M. I. BelishevEmail author
Article
  • 1 Downloads

The well-known fact following from the Holmgren-John-Tataru uniqueness theorem is a local approximate boundary L2-controllability of the dynamical system governed by the wave equation. Generalizing this result, we establish the controllability in certain classes of differentiable functions in the domains filled up with waves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. I. Belishev, “Boundary control in reconstruction of manifolds and metrics (the BC method),” Inverse Problems, 13(5): R1–R45 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M. I. Belishev, “Recent progress in the boundary control method,” Inverse Problems, 23, No. 5, R1–R67 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. I. Belishev and A. N. Dolgoborodov, “Local boundary controllability in smooth classes of functions for the wave equation,” PDMI Preprint, 1/1997, 1–9 (1997).Google Scholar
  4. 4.
    M. S. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D.Reidel Publishing Comp. (1987).Google Scholar
  5. 5.
    M. Ikawa, Hyperbolic PDEs and Wave Phenomena, Translations of Mathematical Monographs, v. 189, AMS; Providence, Rhode Island (1997).Google Scholar
  6. 6.
    I. Lasiecka, J-L. Lions, and R. Triggiani, “Nonhomogeneous boundary value problems for second order hyperbolic operators,” J. Math. Pures Appl., 65, No. 3, 142–192 (1986).MathSciNetzbMATHGoogle Scholar
  7. 7.
    I. Lasiecka and R. Triggiani, “Recent advances in regularity of second-order hyperbolic mixed problems, and applications,” in: Christopher K. R. T. (ed.) et al., Jones, editor, Dynamics reported. Expositions in dynamical systems, V. 3, Springer-Verlag, Berlin (1994), pp. 104–162.Google Scholar
  8. 8.
    J-L. Lions and E. Magenes, Problémes aux Limites Non Homogenes et Applications, Vols. 1–3, Dunod, Paris (1968).zbMATHGoogle Scholar
  9. 9.
    D. L. Russell, “Boundary value control theory of the higher-dimensional wave equation,” SIAM J. Control, 9, 29–42 (1971).MathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Tataru, “Unique continuation for solutions to PDE’s: between Hormander’s and Holmgren’s theorem,” Comm. PDE, 20, 855–884 (1995).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St.Petersburg Department of the Steklov Mathematical Institute, RASSt.Petersburg State UniversitySt.PetersburgRussia

Personalised recommendations