Analytic controllability of the wave equation over a cylinder

Brice Allibert

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 4, page 177-207
  • ISSN: 1292-8119

Abstract

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We analyze the controllability of the wave equation on a cylinder when the control acts on the boundary, that does not satisfy the classical geometric control condition. We obtain precise estimates on the analyticity of reachable functions. As the control time increases, the degree of analyticity that is required for a function to be reachable decreases as an inverse power of time. We conclude that any analytic function can be reached if that control time is large enough. In the C∞ class, a precise description of all reachable functions is given.

How to cite

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Allibert, Brice. "Analytic controllability of the wave equation over a cylinder." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 177-207. <http://eudml.org/doc/197336>.

@article{Allibert2010,
abstract = { We analyze the controllability of the wave equation on a cylinder when the control acts on the boundary, that does not satisfy the classical geometric control condition. We obtain precise estimates on the analyticity of reachable functions. As the control time increases, the degree of analyticity that is required for a function to be reachable decreases as an inverse power of time. We conclude that any analytic function can be reached if that control time is large enough. In the C∞ class, a precise description of all reachable functions is given. },
author = {Allibert, Brice},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Wave equation; attainable set; analyticity; boundary control.; boundary control; wave equation; controllability; observability},
language = {eng},
month = {3},
pages = {177-207},
publisher = {EDP Sciences},
title = {Analytic controllability of the wave equation over a cylinder},
url = {http://eudml.org/doc/197336},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Allibert, Brice
TI - Analytic controllability of the wave equation over a cylinder
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 177
EP - 207
AB - We analyze the controllability of the wave equation on a cylinder when the control acts on the boundary, that does not satisfy the classical geometric control condition. We obtain precise estimates on the analyticity of reachable functions. As the control time increases, the degree of analyticity that is required for a function to be reachable decreases as an inverse power of time. We conclude that any analytic function can be reached if that control time is large enough. In the C∞ class, a precise description of all reachable functions is given.
LA - eng
KW - Wave equation; attainable set; analyticity; boundary control.; boundary control; wave equation; controllability; observability
UR - http://eudml.org/doc/197336
ER -

References

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  1. B. Allibert, Contrôle analytique de l'équation des ondes sur des surfaces de révolution. C. R. Acad. Sci. Paris322 (1996) 835-838.  Zbl0852.35021
  2. B. Allibert, Contrôle analytique de l'équation des ondes sur des surfaces de révolution. Thèse de doctorat de l'École Polytechnique (1997).  
  3. A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl.68 (1989) 457-465.  Zbl0685.93039
  4. G. Lebeau, Control for hyperbolic equations. Journées ``équations aux dérivées partielles'' Saint Jean de Monts, 1992. École Polytechnique, Palaiseau (1992).  
  5. G. Lebeau, Fonctions harmoniques et spectre singulier. Ann. Sci. École Norm. Sup.13 (1980) 269-291.  Zbl0446.46035
  6. J.-L. Lions, Contrôlabilité exacte, perturbation et stabilisation des systèmes distribués. Rech. Math. Appl.8-9 (Masson, Paris, 1988).  Zbl0653.93002
  7. F. Treves, Introduction to pseudodifferential and Fourier integral operators, New York and London Plenum Press Vol. 2, 25 cm (The university series in mathematics, 1980).  Zbl0453.47027
  8. A. Zygmund, Trigonometric Series, Cambridge Univ. Press (1968).  Zbl0157.38204

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