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Spectral Controllability of Some Singular Hyperbolic Equations on Networks

Journal of Dynamical and Control Systems volume 23, pages 459–480 (2017)Cite this article

Abstract

The purpose of this paper is to address the question of well-posedness and spectral controllability of the wave equation perturbed by potential on networks which may contain unbounded potentials in the external edges. It has been shown before that in the absence of any potential, there exists an optimal time T ∗ (which turns out to be simply twice the sum of all length of the strings of the network) that describes the spectral controllability of the system. We will show that this holds in our case too, i.e., the potentials have no effect on the value of the optimal time T ∗. The proof is based on the famous Beurling-Malliavin’s Theorem on the completeness interval of real exponentials and on a result by Redheffer who had shown that under some simple condition the completeness interval of two complex sequences are the same.

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References

  1. Baras P, Goldstein J. Remarks on the inverse square potential in quantum mechanics, Differential equations, 31–35, North-Holland Math. Stud., 92. Amsterdam: North-Holland; 1984.

    Google Scholar 

  2. Baudouin L, Crepeau E, Valein J. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control and Related Fields, AIMS 2011;3:307–330.

    MathSciNet  Article  MATH  Google Scholar 

  3. Bebernes J, Eberly D, Vol. 83. Mathematical problems from combustion theory, Math. Sci. New York: Springer; 1989.

  4. Beurling A, Malliavin P. On the closure of characters and zeros of entire functions. Acta Math 1967;118:79–93.

    MathSciNet  Article  MATH  Google Scholar 

  5. Brezis H, Vazquez JL. Blow-up solutions of some nonlinear elliptic equations. Rev Mat Complut 1997;10:443–469.

    MATH  Google Scholar 

  6. Dager R, Zuazua E. Wave propagation, observation and control in 1-d flexible multi-structures: Springer; 2006.

  7. de Castro AS. Bound states of the Dirac equation for a class of effective quadratic plus inversely quadratic potentials. Ann Phys 2004;311:170–181.

    MathSciNet  Article  MATH  Google Scholar 

  8. Dold JW, Galaktionov VA, Lacey AA, Vazquez JL. Rate of approach to a singular steady state in quasilinear reaction-diffusion equations. Ann. Sc. Norm. Super Pisa Cl. Sci. 1998;4(26):663–687.

    MathSciNet  MATH  Google Scholar 

  9. Duyckaerts T, Zhang X, Zuazua E. On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Annales de l Institut Henri Poincare (C) Analyse Non Lineaire 2008;25(1):1–41.

    MathSciNet  Article  MATH  Google Scholar 

  10. Galaktionov V, Vazquez JL. Continuation of blow-up solutions of nonlinear heat equations in several space dimensions. Comm. Pure Appl. Math. 1997;1:1–67.

    Article  MATH  Google Scholar 

  11. Gohberg IC, Krein MG. Introduction to the theory of linear nonselfadjoint operators. Trans. Math. Monographs 1969;18. AMS.

  12. Haraux A, Jaffard S. Pointwise and spectral control of plate vibrations. Revista Matematica Iberoamericana 1991;7:1–24.

    MathSciNet  Article  MATH  Google Scholar 

  13. Mehmeti FA. A charactrisation of generalized C ∞ notions on nets. Int Eq Oper Theory 1986;9:753–766.

    Article  MATH  Google Scholar 

  14. Redheffer R. Two consequences of the Beurling- Malliavin theory. Proc Am Math Soc 1972;36(1):116–122.

    MathSciNet  Article  MATH  Google Scholar 

  15. Reed M, Simon B, Vol. II. Methods of modern mathematical physics. New York: Academic; 1979.

  16. Watson GN. A treatise on the theory of bessel functions, 2nd ed. London: Cambridge Univ. Press; 1944.

    MATH  Google Scholar 

  17. Yosida K. Functional analysis, 6th ed.: Springer; 1980.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Morteza Fotouhi

  2. Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran

    Leila Salimi

Authors
  1. Morteza Fotouhi
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  2. Leila Salimi
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Correspondence to Leila Salimi.

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Fotouhi, M., Salimi, L. Spectral Controllability of Some Singular Hyperbolic Equations on Networks. J Dyn Control Syst 23, 459–480 (2017). https://doi.org/10.1007/s10883-016-9330-y

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  • DOI: https://doi.org/10.1007/s10883-016-9330-y

Keywords

  • Network
  • Spectral controllability
  • Approximate controllability
  • Spectrum of an operator

Mathematics Subject Classification (2010)

  • 35K65
  • 93B05
  • 93B07