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Nonlinear feedback stabilization of a rotating body-beam without damping

Published online by Cambridge University Press:  15 August 2002

Boumediène CHENTOUF  and
Jean-François COUCHOURON
Boumediène CHENTOUF
Affiliation:
INRIA-Lorraine (CONGE project) & UPRES A 7035, ISGMP, bâtiment A, Université de Metz, Ile de Saulcy, 57045 Metz cedex 01, France.
Jean-François COUCHOURON
Affiliation:
INRIA-Lorraine (CONGE project) & UPRES A 7035, ISGMP, bâtiment A, Université de Metz, Ile de Saulcy, 57045 Metz cedex 01, France.
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Abstract

This paper deals with nonlinear feedback stabilization problem of a flexible beam clamped at a rigid body and free at the other end. We assume that there is no damping and the feedback law proposed here consists of a nonlinear control torque applied to the rigid body and either a boundary control moment or a nonlinear boundary control force or both of them applied to the free end of the beam. This nonlinear feedback, which insures the exponential decay of the beam vibrations, extends the linear case studied by Laousy et al. to a more general class of controls.

Keywords

Rotating body-beamnon linear controlexponential stability.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 515 - 535
Copyright
© EDP Sciences, SMAI, 1999

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References

J. Ackermann, Sampled-data control system: Analysis and synthesis, robust system design, Springer-Verlag (1985).
J. Baillieul and M. Levi, Rotational elastic dynamics. Physica D, 27 (1987) 43-62 .
P. Bénilan, Équations d'évolution dans un espace de Banach quelconque et applications, Thèse, Paris XI, Orsay (1972).
P. Bénilan, M.G. Crandal and A. Pazy, Nonlinear evolution equations in Banach spaces, monograph in preparation.
A.M. Bloch and E.S. Titi, On the dynamics of rotating elastic beams, in Proc. Conf. New Trends Syst. theory, Genoa, Italy, July 9-11, 1990, Conte, Perdon, and Wyman, eds., Cambridge, MA: Birkhäuser (1990).
H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, London (1973).
H. Brezis, Analyse Fonctionnelle. Théorie et applications, Masson (1983).
F. Conrad and B. Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback. Asymptotic Analysis, 7 (1993) 159-177.
Coron, J.-M. and d'Andréa-Novel, B., Stabilization of a rotating body-beam without damping. IEEE Trans. Automat. Contr. , 43 (1998) 608-618. CrossRef
Crandall, M.G., Nonlinear semigroups and evolution governed by accretive operators. Pro. Sympo. in pure Math. 45 (1986) 305-337. CrossRef
C.M. Dafermos and M. Slemrod, Asymptotic behaviour of non linear contractions semi-groups, J. Func. Anal . 14 (1973) 97-106.
A. Haraux, Systèms Dynamique Dissipatifs et Applications. Collection RMA (17) Masson, Paris (1991).
V. Jurdjevic and J. P. Quin, Controllability and stability, J. Differential Equations, 28 (1978) 381-389.
Komornik, V. and Zuazua, E., A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. , 69 (1990) 33-54.
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson and John Wiley (1994).
Laousy, H., Xu, C.Z. and Sallet, G., Boundary feedback stabilization of a rotating body-beam system. IEEE Trans. Automat. Contr. , 41 (1996) 241-245. CrossRef
Morgül, O., Orientation and stabilization of a flexible beam attached to a rigid body: Planar motion. IEEE Trans. Automat. Contr. , 36 (1991) 953-963. CrossRef
O. Morgül, Constant angular velocity control of a rotating flexible structure, in Proc. 2nd Conf., ECC'93., Groningen, Netherlands (1993) 299-302.
Morgül, O., Control of a rotating flexible structure. IEEE Trans. Automat. Contr. 39 (1994) 351-356. CrossRef
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Verlag, New York (1983).
Pierre, M., Perturbations localement lipschitziennes et continues d'opérateurs m-accretifs. Proc. Amer. Math. Soc. , 58 (1976) 124-128.
Rao, B., Decay estimate of solution for hybrid system of flexible structures. Euro. J. Appl. Math. 4 (1993) 303-319 . CrossRef
Xu, C.Z. and Baillieul, J., Stabilizability and stabilization of a rotating body-beam system with torque control. IEEE Trans. Automat. Contr. 38 (1993) 1754-1765. CrossRef
C.Z. Xu and G. Sallet, Boundary stabilization of a rotating flexible system. Lecture Notes in Control and Information Sciences 185, R.F. Curtain, A. Bensoussan and J.L. Lions, eds., Springer Verlag, New York (1992) 347-365.
A. Zeidler, Non linear functional analysis and its applications, Vol. 2, Springer Verlag, New York (1986).