Well-posedness and controllability of a class of coupled linear systems

We consider coupled systems consisting of an infinite-dimensional part and a finite-dimensional part connected in feedback, such as, for example, the well-known SCOLE system (a beam with a rigid body attached at one end). The external world interacts with the coupled system via the finite-dimensional part, which receives the external input and sends out the output. The infinite-dimensional part is assumed to be such that it becomes well-posed and strictly proper when connected in cascade with an integrator. Under several assumptions, we derive well-posedness and exact controllability results for such coupled systems. The first main result concerns the case when the input signal of the finite-dimensional part is the difference between the external input and the feedback signal. The second main result allows a more general structure for the finite-dimensional part. We also prove a result for the approximate controllability of coupled systems.