Journal of Applied Mathematics

Dynamic boundary controls of a rotating body-beam system with time-varying angular velocity

Boumediène Chentouf

Full-text: Open access


This paper deals with feedback stabilization of a flexible beam clamped at a rigid body and free at the other end. We assume that there is no damping and the rigid body rotates with a nonconstant angular velocity. To stabilize this system, we propose a feedback law which consists of a control torque applied on the rigid body and either a dynamic boundary control moment or a dynamic boundary control force or both of them applied at the free end of the beam. Then it is shown that the closed loop system is well posed and exponentially stable provided that the actuators, which generate the boundary controls, satisfy some classical assumptions and the angular velocity is smaller than a critical one.

Article information

J. Appl. Math., Volume 2004, Number 2 (2004), 107-126.

First available in Project Euclid: 7 July 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B37 35M10
Secondary: 93D15 93D30


Chentouf, Boumediène. Dynamic boundary controls of a rotating body-beam system with time-varying angular velocity. J. Appl. Math. 2004 (2004), no. 2, 107--126. doi:10.1155/S1110757X04312027.

Export citation


  • J. Ackermann, Sampled-Data Control System: Analysis and Synthesis, Robust System Design, Springer-Verlag, Berlin, 1985.
  • J. Baillieul and M. Levi, Rotational elastic dynamics, Phys. D 27 (1987), no. 1-2, 43--62.
  • in [3?] according to the MathSciNet database.} of rotating elastic beams}, New Trends in Systems Theory (Genoa, 1990) (G. Conte, A. Perdon, and B. Wyman, eds.), Progr. Systems Control Theory, vol. 7, Birkhäuser Boston, Massachusetts, 1991, pp. 128--135. \CMP1+125+1011 125 101
  • [4?] according to the MathSciNet database.} Théorie et Applications}, Collection Mathématiques Appliquées pour la Maî trise, Masson, Paris, 1983.
  • G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne, and H. H. West, The Euler-Bernoulli beam equation with boundary energy dissipation, Operator Methods for Optimal Control Problems (New Orleans, La, 1986) (S. J. Lee, ed.), Lecture Notes in Pure and Appl. Math., vol. 108, Marcel Dekker, New York, 1987, pp. 67--96.
  • B. Chentouf, Boundary feedback stabilization of a variant of the SCOLE model, J. Dynam. Control Systems 9 (2003), no. 2, 201--232.
  • B. Chentouf and J.-F. Couchouron, Nonlinear feedback stabilization of a rotating body-beam without damping, ESAIM Control Optim. Calc. Var. 4 (1999), 515--535.
  • B. Chentouf, C. Z. Xu, and G. Sallet, On the stabilization of a vibrating equation, Nonlinear Anal. Ser. A: Theory and Methods 39 (2000), no. 5, 537--558.
  • J.-M. Coron and B. d'Andréa Novel, Stabilization of a rotating body beam without damping, IEEE Trans. Automat. Control 43 (1998), no. 5, 608--618.
  • A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées, vol. 17, Masson, Paris, 1991.
  • F. L. Huang, Characteristic conditions for exponential stability to the MathSciNet database. of linear dynamical systems in Hilbert spaces}, Ann. Differential Equations 1 (1985), no. 1, 43--56.
  • A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 (1936), 367--379.
  • T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1976.
  • H. K. Khalil, Nonlinear Systems, Prentice-Hall, New Jersey, 2002.
  • in [15?].} exponential sums and integrals}, Bull. Amer. Math. Soc. (1931), 213--239.
  • H. Laousy, C. Z. Xu, and G. Sallet, Boundary feedback stabilization of a rotating body-beam system, IEEE Trans. Automat. Control 41 (1996), no. 2, 241--245. \CMP1+375+7591 375 759
  • Ö. Morgül, Constant angular velocity control of a rotating flexible structure, Proc. 2nd European Control Conference (Groningen, 1993), pp. 299--302.
  • --------, Orientation and stabilization of a flexible beam attached to a MathSciNet database. rigid body: planar motion}, IEEE Trans. Automat. Control 36 (1991), no. 8, 953--962.
  • --------, Dynamic boundary control of an Euler-Bernoulli beam, IEEE Trans. Automat. Control 37 (1992), no. 5, 639--642.
  • --------, Control and stabilization of a rotating flexible structure, Automatica J. IFAC 30 (1994), no. 2, 351--356. \CMP1+261+7121 261 712
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.
  • R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc. 105 (1989), no. 2, 375--383.
  • M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, New Jersey, 1978.
  • C.-Z. Xu and J. Baillieul, Stabilizability and stabilization of a rotating body-beam system with torque control, IEEE Trans. Automat. Control 38 (1993), no. 12, 1754--1765.
  • C.-Z. Xu and G. Sallet, Boundary stabilization of rotating flexible systems, Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems (Sophia-Antipolis, 1992) (R. F. Curtain, A. Bensoussan, and J. L. Lions, eds.), Lecture Notes in Control and Inform. Sci., vol. 185, Springer-Verlag, Berlin, 1993, pp. 347--365. \CMP1+208+2801 208 280