Exact controllability of the SCOLE model

This paper investigates the exact controllability of the SCOLE (NASA Spacecraft Control Laboratory Experiment) model. This model describes the movement in one plane of a beam clamped at one end and connected to a rigid body at the other end. The mass and moment of inertia of the rigid body are not negligible. The inputs are the force and the torque acting on the rigid body. The importance of this model stems from it being used to model the vibrations of a flexible mast holding an antenna on a spacecraft. More recently, it is being used also to model the vibrations of the tower of a wind-turbine. By exact controllability space we actually mean the reachable space by L inputs. The controllability of the SCOLE model has been investigated by several researchers. The exact controllability of the uniform SCOLE model in a smooth state space of type H6 × H4 (with boundary conditions) was proved in Littman and Markus [3] based on the theory of semi-infinite beams. Using a constructive cutoff approach, they proved the existence of smooth torque and force controls for the finite beam. Guo and Ivanov [2] proved the exact controllability of the non-uniform SCOLE model in the domain of the generator with the graph norm, denoted X1, using only one input (torque or force) by the Riesz basis approach. The exact controllability of the SCOLE model in the usual energy space X cannot be achieved using L inputs as the control operator is compact from the input space to X . The space X1 is the biggest space whose reachability using L2 controls has been proved so far. Using the Hilbert Uniqueness Method, Rao [6] obtained the exact controllability of the uniform SCOLE model in the energy space by means of a singular control (the torque component is not in L2). Rao also proved the exact controllability of the SCOLE model in the energy space by only one singular control (the