Uniform controllability for the beam equation with vanishing structural damping

Ioan Florin Bugariu

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 4, page 869-881
  • ISSN: 0011-4642

Abstract

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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 ε ∈ ( 0 , 1 ) . We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls v ε as ε goes to zero. It is shown that for any time T sufficiently large but independent of ε and for each initial data in a suitable space there exists a uniformly bounded family of controls ( v ε ) ε 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.

How to cite

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Bugariu, Ioan Florin. "Uniform controllability for the beam equation with vanishing structural damping." Czechoslovak Mathematical Journal 64.4 (2014): 869-881. <http://eudml.org/doc/269867>.

@article{Bugariu2014,
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 $\varepsilon \in (0,1)$. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls $v_\{\varepsilon \}$ as $\varepsilon $ goes to zero. It is shown that for any time $T$ sufficiently large but independent of $\varepsilon $ 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 = \pi $. 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.},
author = {Bugariu, Ioan Florin},
journal = {Czechoslovak Mathematical Journal},
keywords = {beam equation; null-controllability; structural damping; moment problem; biorthogonals; beam equation; null-controllability; structural damping; moment problem; biorthogonals},
language = {eng},
number = {4},
pages = {869-881},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Uniform controllability for the beam equation with vanishing structural damping},
url = {http://eudml.org/doc/269867},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Bugariu, Ioan Florin
TI - Uniform controllability for the beam equation with vanishing structural damping
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 4
SP - 869
EP - 881
AB - 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 $\varepsilon \in (0,1)$. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls $v_{\varepsilon }$ as $\varepsilon $ goes to zero. It is shown that for any time $T$ sufficiently large but independent of $\varepsilon $ 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 = \pi $. 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.
LA - eng
KW - beam equation; null-controllability; structural damping; moment problem; biorthogonals; beam equation; null-controllability; structural damping; moment problem; biorthogonals
UR - http://eudml.org/doc/269867
ER -

References

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  1. Avdonin, S. A., Ivanov, S. A., Families of Exponentials. The method of moments in controllability problems for distributed parameter systems, Cambridge University Press Cambridge (1995). (1995) Zbl0866.93001MR1366650
  2. Ball, J. M., Slemrod, M., 10.1002/cpa.3160320405, Commun. Pure Appl. Math. 32 (1979), 555-587. (1979) MR0528632DOI10.1002/cpa.3160320405
  3. DiPerna, R. J., 10.1007/BF00251724, Arch. Ration. Mech. Anal. 82 (1983), 27-70. (1983) Zbl0519.35054MR0684413DOI10.1007/BF00251724
  4. Edward, J., 10.1016/j.jmaa.2005.12.074, J. Math. Anal. Appl. 324 (2006), 941-954. (2006) Zbl1108.26019MR2265092DOI10.1016/j.jmaa.2005.12.074
  5. Edward, J., Tebou, L., Internal null-controllability for a structurally damped beam equation, Asymptotic Anal. 47 (2006), 55-83. (2006) Zbl1098.35096MR2224406
  6. Fattorini, H. O., Russell, D. L., 10.1090/qam/510972, Q. Appl. Math. 32 (1974), 45-69. (1974) Zbl0281.35009MR0510972DOI10.1090/qam/510972
  7. Fattorini, H. O., Russell, D. L., 10.1007/BF00250466, Arch. Ration. Mech. Anal. 43 (1971), 272-292. (1971) Zbl0231.93003MR0335014DOI10.1007/BF00250466
  8. Ignat, L. I., Zuazua, E., Dispersive properties of numerical schemes for nonlinear Schrödinger equations, Foundations of Computational Mathematics; Santander, Spain, 2005, London Math. Soc. Lecture Note Ser. 331 Cambridge University Press, Cambridge (2006), 181-207 L. M. Pardo et al. (2006) Zbl1106.65321MR2277106
  9. Ignat, L. I., Zuazua, E., 10.1016/j.crma.2005.02.024, C. R., Math., Acad. Sci. Paris 340 (2005), 529-534. (2005) Zbl1063.35016MR2135236DOI10.1016/j.crma.2005.02.024
  10. Ingham, A. E., 10.1007/BF01180426, Math. Z. 41 (1936), 367-379. (1936) Zbl0014.21503MR1545625DOI10.1007/BF01180426
  11. Komornik, V., Loreti, P., Fourier Series in Control Theory, Springer Monographs in Mathematics Springer, New York (2005). (2005) Zbl1094.49002MR2114325
  12. Lebeau, G., Zuazua, E., 10.1007/s002050050078, Arch. Ration. Mech. Anal. 141 (1998), 297-329. (1998) Zbl1064.93501MR1620510DOI10.1007/s002050050078
  13. Lions, J.-L., Exact controllability, perturbations and stabilization of distributed systems. Volume 1: Exact controllability, Research in Applied Mathematics 8 Masson, Paris French (1988). (1988) MR0963060
  14. Micu, S., RovenĹŁa, I., 10.1051/cocv/2010055, ESAIM, Control Optim. Calc. Var. 18 (2012), 277-293. (2012) MR2887936DOI10.1051/cocv/2010055
  15. Seidman, T. I., Avdonin, S. A., Ivanov, S. A., 10.1007/BF02511154, J. Fourier Anal. Appl. 6 (2000), 233-254. (2000) Zbl0960.42012MR1755142DOI10.1007/BF02511154
  16. Young, R. M., An Introduction to Nonharmonic Fourier Series, Pure and Applied Mathematics 93 Academic Press, New York (1980). (1980) Zbl0493.42001MR0591684
  17. Zabczyk, J., Mathematical Control Theory: An Introduction, Systems & Control: Foundations & Applications Birkhäuser, Boston (1992). (1992) Zbl1071.93500MR1193920

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