Control and stabilization of a flexible beam attached to a rigid body: planar motion

The authors consider a flexible spacecraft modeled as a rigid body constrained to rotate around its principal axis D₁, fixed in inertial space; a light flexible beam is clamped to the rigid body at one end and is free at the other. The equations of motion are obtained by using free-body diagrams. It is shown that suitable boundary controls applied to the free end of the beam and a control torque applied to the rigid body stabilize the system. The proof is obtained by using the energy of the system as a Lyapunov functional