Well-posedness, regularity and exact controllability of the SCOLE model

The SCOLE model is a coupled system consisting of a flexible beam (modelled as an Euler–Bernoulli equation) with one end clamped and the other end linked to a rigid body. Its inputs are the force and the torque acting on the rigid body. It is well-known that the SCOLE model is not exactly controllable with L ² input signals in the natural energy state space H ^c , because the control operator is bounded from the input space mathbbC²{mathbb{C}^2} to H ^c , and hence compact. We regard the velocity and the angular velocity of the rigid body as the output signals of this system. Using the theory of coupled linear systems (one infinite-dimensional and one finite-dimensional) developed by us recently in another paper, we show that the SCOLE model is well-posed, regular and exactly controllable in arbitrarily short time when using a certain smoother state space X ì H^c{mathcal{X}subset H^c}.