Exact Controllability of Coupled PDE-ODE Systems

We consider coupled systems consisting of an infinite-dimensional part (a PDE system) and a finite-dimensional part (an ODE system) 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 a system node. 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 the coupled system.