Controllability of a slowly rotating Timoshenko beam

Martin Gugat

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 333-360
  • ISSN: 1292-8119

Abstract

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Consider a Timoshenko beam that is clamped to an axis perpendicular to the axis of the beam. We study the problem to move the beam from a given initial state to a position of rest, where the movement is controlled by the angular acceleration of the axis to which the beam is clamped. We show that this problem of controllability is solvable if the time of rotation is long enough and a certain parameter that describes the material of the beam is a rational number that has an even numerator and an odd denominator or vice versa.

How to cite

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Gugat, Martin. "Controllability of a slowly rotating Timoshenko beam." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 333-360. <http://eudml.org/doc/90597>.

@article{Gugat2001,
abstract = {Consider a Timoshenko beam that is clamped to an axis perpendicular to the axis of the beam. We study the problem to move the beam from a given initial state to a position of rest, where the movement is controlled by the angular acceleration of the axis to which the beam is clamped. We show that this problem of controllability is solvable if the time of rotation is long enough and a certain parameter that describes the material of the beam is a rational number that has an even numerator and an odd denominator or vice versa.},
author = {Gugat, Martin},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {rotating Timoshenko beam; exact controllability; eigenvalues; moment problem; spectral properties; fractional order equations},
language = {eng},
pages = {333-360},
publisher = {EDP-Sciences},
title = {Controllability of a slowly rotating Timoshenko beam},
url = {http://eudml.org/doc/90597},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Gugat, Martin
TI - Controllability of a slowly rotating Timoshenko beam
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 333
EP - 360
AB - Consider a Timoshenko beam that is clamped to an axis perpendicular to the axis of the beam. We study the problem to move the beam from a given initial state to a position of rest, where the movement is controlled by the angular acceleration of the axis to which the beam is clamped. We show that this problem of controllability is solvable if the time of rotation is long enough and a certain parameter that describes the material of the beam is a rational number that has an even numerator and an odd denominator or vice versa.
LA - eng
KW - rotating Timoshenko beam; exact controllability; eigenvalues; moment problem; spectral properties; fractional order equations
UR - http://eudml.org/doc/90597
ER -

References

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  1. [1] S.A. Avdonin and S.S. Ivanov, Families of Exponentials. Cambridge University Press (1995). Zbl0866.93001MR1366650
  2. [2] M.C. Delfour, M. Kern, L. Passeron and B. Sevenne, Modelling of a rotating flexible beam, in Control of Distributed Parameter Systems, edited by H.E. Rauch. Pergamon Press, Los Angeles (1986) 383-387. 
  3. [3] K.F. Graff, Wave Motion in Elastic Solids. Dover Publications, New York (1991). Zbl0314.73022
  4. [4] M. Gugat, A Newton method for the computation of time-optimal boundary controls of one-dimensional vibrating systems. J. Comput. Appl. Math. 114 (2000) 103-119. Zbl0953.93042MR1739846
  5. [5] J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25 (1987) 1417-1429. Zbl0632.93057MR912448
  6. [6] W. Krabs, On moment theory and contollability of one-dimensional vibrating systems and heating processes. Springer-Verlag, Heidelberg, Lecture Notes in Control and Informat. Sci. 173 (1992). Zbl0955.93501MR1162111
  7. [7] W. Krabs, Controllability of a rotating beam. Springer-Verlag, Lecture Notes in Control and Inform. Sci. 185 (1993) 447-458. Zbl0800.93169MR1208288
  8. [8] W. Krabs and G.M. Sklyar, On the controllability of a slowly rotating Timoshenko beam. J. Anal. Appl. 18 (1999) 437-448. Zbl0959.93005MR1701363
  9. [9] M.A. Moreles, A classical approach to uniform null controllability of elastic beams. SIAM J. Control Optim. 36 (1998) 1073-1085. Zbl0915.93005MR1613913
  10. [10] D.L. Russel, Nonharmonic Fourier series in the control theory of distributed parameter systems. J. Math. Anal. Appl. 18 (1967) 542-560. Zbl0158.10201MR211044
  11. [11] M.A. Shubov, Spectral operators generated by Timoshenko beam model. Systems Control Lett. 38 (1999). Zbl0985.93012MR1754907
  12. [12] S.P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. (1921) xli. 

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