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Uniform controllability for the beam equation with vanishing structural damping

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Abstract

This paper is devoted to studying the effects of a vanishing structural damping on the controllability properties of the one dimensional linear beam equation. The vanishing term depends on a small parameter ɛ ∈ (0, 1). We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls \({v_\varepsilon }\) as ɛ goes to zero. It is shown that for any time T sufficiently large but independent of ɛ and for each initial data in a suitable space there exists a uniformly bounded family of controls \({({v_\varepsilon })_\varepsilon }\) in L 2(0, T) acting on the extremity x = π. Any weak limit of this family is a control for the beam equation. This analysis is based on Fourier expansion and explicit construction and evaluation of biorthogonal sequences. This method allows us to measure the magnitude of the control needed for each eigenfrequency and to show their uniform boundedness when the structural damping tends to zero.

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References

  1. S. A. Avdonin, S. A. Ivanov: Families of Exponentials. The method of moments in controllability problems for distributed parameter systems. Cambridge University Press, Cambridge, 1995.

    MATH  Google Scholar 

  2. J. M. Ball, M. Slemrod: Nonharmonic Fourier series and the stabilization of distributed semilinear control systems. Commun. Pure Appl. Math. 32 (1979), 555–587.

    Article  MathSciNet  Google Scholar 

  3. R. J. DiPerna: Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82 (1983), 27–70.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Edward: Ingham-type inequalities for complex frequencies and applications to control theory. J. Math. Anal. Appl. 324 (2006), 941–954.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Edward, L. Tebou: Internal null-controllability for a structurally damped beam equation. Asymptotic Anal. 47 (2006), 55–83.

    MATH  MathSciNet  Google Scholar 

  6. H. O. Fattorini, D. L. Russell: Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Q. Appl. Math. 32 (1974), 45–69.

    MATH  MathSciNet  Google Scholar 

  7. H. O. Fattorini, D. L. Russell: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43 (1971), 272–292.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. I. Ignat, E. Zuazua: Dispersive properties of numerical schemes for nonlinear Schrödinger equations. Foundations of Computational Mathematics (L. M. Pardo et al., eds.). Santander, Spain, 2005, London Math. Soc. Lecture Note Ser. 331, Cambridge University Press, Cambridge, 2006, pp. 181–207.

    Google Scholar 

  9. L. I. Ignat, E. Zuazua: Dispersive properties of a viscous numerical scheme for the Schrödinger equation. C. R., Math., Acad. Sci. Paris 340 (2005), 529–534.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. E. Ingham: Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936), 367–379.

    Article  MathSciNet  Google Scholar 

  11. V. Komornik, P. Loreti: Fourier Series in Control Theory. Springer Monographs in Mathematics, Springer, New York, 2005.

    MATH  Google Scholar 

  12. G. Lebeau, E. Zuazua: Null-controllability of a system of linear thermoelasticity. Arch. Ration. Mech. Anal. 141 (1998), 297–329.

    Article  MATH  MathSciNet  Google Scholar 

  13. J.-L. Lions: Exact controllability, perturbations and stabilization of distributed systems. Volume 1: Exact controllability. Research in Applied Mathematics 8, Masson, Paris, 1988. (In French.)

    Google Scholar 

  14. S. Micu, I. Rovenţa: Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity. ESAIM, Control Optim. Calc. Var. 18 (2012), 277–293.

    Article  MATH  MathSciNet  Google Scholar 

  15. T. I. Seidman, S.A. Avdonin, S.A. Ivanov: The ‘window problem’ for series of complex exponentials. J. Fourier Anal. Appl. 6 (2000), 233–254.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. M. Young: An Introduction to Nonharmonic Fourier Series. Pure and Applied Mathematics 93, Academic Press, New York, 1980.

    MATH  Google Scholar 

  17. J. Zabczyk: Mathematical Control Theory: An Introduction. Systems & Control: Foundations & Applications, Birkhäuser, Boston, 1992.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Exact Sciences, Department of Computer Science, University of Craiova, Strada Alexandru Ioan Cuza 13, 200 585, Craiova, Romania

    Ioan Florin Bugariu

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  1. Ioan Florin Bugariu
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Correspondence to Ioan Florin Bugariu.

Additional information

This work was supported by the strategic grant POSDRU/CPP107/DMI1.5/S/78421, Project ID 78421 (2010), co-financed by the European Social Fund—Investing in People, within the Sectoral Operational Programme Human Resources Development 2007–2013 and by a grant of the Romanian National Authority for Scientific Research, CNCS—UEFISCDI, project number PN-II-ID-PCE-2011-3-0257.

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Bugariu, I.F. Uniform controllability for the beam equation with vanishing structural damping. Czech Math J 64, 869–881 (2014). https://doi.org/10.1007/s10587-014-0140-7

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  • DOI: https://doi.org/10.1007/s10587-014-0140-7

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