Simultaneous controllability in sharp time for two elastic strings

Sergei Avdonin; Marius Tucsnak

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

  • Volume: 6, page 259-273
  • ISSN: 1292-8119

Abstract

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We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time.

How to cite

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Avdonin, Sergei, and Tucsnak, Marius. "Simultaneous controllability in sharp time for two elastic strings." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 259-273. <http://eudml.org/doc/197323>.

@article{Avdonin2010,
abstract = { We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time. },
author = {Avdonin, Sergei, Tucsnak, Marius},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Exact controllability; spectral controllability; approximate controllability; simultaneous controllability; string equation; boundary control; Riesz basis.; simultaneous controllability; Riesz basis; simultaneous reachable space},
language = {eng},
month = {3},
pages = {259-273},
publisher = {EDP Sciences},
title = {Simultaneous controllability in sharp time for two elastic strings},
url = {http://eudml.org/doc/197323},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Avdonin, Sergei
AU - Tucsnak, Marius
TI - Simultaneous controllability in sharp time for two elastic strings
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 259
EP - 273
AB - We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time.
LA - eng
KW - Exact controllability; spectral controllability; approximate controllability; simultaneous controllability; string equation; boundary control; Riesz basis.; simultaneous controllability; Riesz basis; simultaneous reachable space
UR - http://eudml.org/doc/197323
ER -

References

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  1. S.A. Avdonin, Simultaneous controllability of several elastic strings, in Proc. CD of the Fourteenth International Symposium on Mathematical Theory of Networks and Systems. Perpignan, France, June 19-23 (2000).  
  2. S.A. Avdonin and S.A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, New York (1995).  Zbl0866.93001
  3. C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory. Boll. Un. Mat. Ital. B2 (1999) 33-63.  Zbl0924.42022
  4. C. Baiocchi, V. Komornik and P. Loreti, Généralisation d'un théorème de Beurling et application à la théorie du contrôle. C. R. Acad. Sci. Paris Sér. I Math.330 (2000) 281-286.  Zbl0964.42019
  5. J.W.S. Cassels, An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1965).  Zbl0077.04801
  6. S. Dolecki and D.L. Russell, A general theory of observation and control. SIAM J. Control Optim.15 (1977) 185-220.  Zbl0353.93012
  7. S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. J. Differential Equations145 (1998) 184-215.  Zbl0920.35029
  8. S. Jaffard, M. Tucsnak and E. Zuazua, On a theorem of Ingham. J. Fourier Anal. Appl.3 (1997) 577-582.  Zbl0939.42004
  9. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. John Wiley & Sons, New York (1974).  Zbl0281.10001
  10. J.E. Lagnese and J.L. Lions, Modelling, Analysis and Control of Thin Plates. Masson, Paris (1988).  
  11. S. Lang, Introduction to Diophantine Approximations. Addison Wesley, New York (1966).  Zbl0144.04005
  12. J.-L. Lions, Controlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués, Volume 1. Masson, Paris (1988).  Zbl0653.93002
  13. N.K. Nikol'skii, A Treatise on the Shift Operator. Moscow, Nauka, 1980 (Russian); Engl. Transl., Springer, Berlin (1986).  
  14. B.S. Pavlov, Basicity of an exponential systems and Muckenhoupt's condition. Dokl. Akad. Nauk SSSR 247 (1979) 37-40 (Russian); English transl. in Soviet Math. Dokl. 20 (1979) 655-659.  Zbl0429.30004
  15. W. Rudin, Real and complex analysis. McGraw-Hill, New York (1987).  Zbl0925.00005
  16. D.L. Russell, The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region. SIAM J. Control Optim.24 (1986) 199-229.  Zbl0594.49026
  17. M. Tucsnak and G. Weiss, Simultaneous exact controllability and some applications. SIAM J. Control Optim.38 (2000) 1408-1427.  Zbl0982.93021
  18. R. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980).  Zbl0493.42001

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