608 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 5, MAY 1998 Stabilization of a Rotating Body Beam Without Damping Jean-Michel Coron and Brigitte d’Andr´ ea-Novel Abstract—This paper deals with the stabilization of a rotating body-beam system with torque control. The system we consider is the one studied by Baillieul and Levi in [1]. In [12] it has been proved by Xu and Baillieul that, for any constant angular velocity smaller than a critical one, this system can be stabilized by means of a feedback torque control law if there is damping. We prove that this result also holds if there is no damping. Index Terms—Distributed control system, elastic beam, hybrid system, nonlinear control, stabilization. I. INTRODUCTION T HE GOAL of this paper is to study the stabilization of a system, already considered in [1], consisting of a disk with a beam attached to its center and perpendicular to the disk’s plane. The beam is confined to another plane which is perpendicular to the disk and rotates with the disk; see Fig. 1. The dynamics of motion is (see [1] and [2]) (1) (2) where is the length of the beam, is the mass per unit length of the beam, is the flexural rigidity per unit length of the beam, is the angular velocity of the disk at time is the disk’s moment of inertia, is the beam’s displacement in the rotating plane at time with respect to the spatial variable is the damping term, and is the torque control variable applied to the disk at time (see Fig. 1). If there is no damping , and therefore (1) reads (3) Two types of damping are considered in [12]. 1) Viscous damping: with . 2) Structural damping: with . Manuscript received January 24, 1997. Recommended by Associate Editor, C.-Z. Xu. This work was supported by DRET under Grant 951170 and by the PRC-GDR “Automatique” of the CNRS and the MST. J.-M. Coron is with the Universit´ e de Paris-Sud, Analyze Num´ erique et EDP, 91405 Orsay Cedex, France (e-mail: Jean-Michel.Coron@math.u- psud.fr). B. d’Andr´ ea-Novel is with the Centre de Robotique, ´ Ecole des Mines de Paris, 75272 Paris Cedex 06, France. Publisher Item Identifier S 0018-9286(98)03589-2. Fig. 1. The body-beam structure. The asymptotic behavior of the solutions of (1) and (2) when there is no control (i.e., ) has been studied by Baillieul and Levi in [1] and by Bloch and Titi in [4]. For both types of damping, Xu and Baillieul have con- structed in [12] a feedback torque control law which globally asymptotically stabilizes the equilibrium point with (4) where is an explicit critical angular velocity. This critical angular velocity is given by (5) where is the first eigenvalue of the unbounded linear operator in with domain (6) They also prove that is optimal: if , they prove that there is no feedback law which asymptotically stabilizes . The asymptotically stabilizing feedback law constructed in [12] is linear, and the stabilization is strong and exponential. In [8] Laousy et al. have constructed a globally asymptotically stabilizing feedback in the case where there is no damping but when there is a control also on the free boundary of the beam ( ). The goal of this paper is to investigate the stabilization problem when there is no damping and no control on the free boundary. We construct in this case a (nonlinear) feedback torque control law which globally asymptotically stabilizes the equilibrium point , provided that (4) holds. Our paper is organized as follows: in Section II we introduce some notations and construct stabilizing feedback laws. In 0018–9286/98$10.00  1998 IEEE