Well-posedness and generic exact controllability of coupled systems

We consider coupled systems consisting of a well-posed and strictly proper subsystem and a finite-dimensional subsystem connected in feedback. The external world interacts with the coupled system via the finite-dimensional part, which receives the external input and sends out the output. We allow the external input and the feedback signal from the infinite-dimensional part to be two separate inputs of the finite-dimensional part. We consider the output of the infinite-dimensional part as an additional output. Under several assumptions, we derive well-posedness, regularity and generic exact controllability results for such coupled systems.

I. Introduction

Coupled infinite-dimensional systems have attracted much interest in recent years, we have developed in [7] a theory for the well-posedness, regularity and controllability of coupled systems consisting of an infinite-dimensional subsystem and a finite-dimensional subsystem connected in feedback. In [7] we assume that the feedthrough matrix of is zero and that is such that it becomes well-posed and strictly proper when connected in cascade with an integrator (we call such systems SPI systems). We have shown that such coupled systems are well-posed and regular, when we use as state space a certain subspace of the product of the state spaces of the subsystems. Moreover, under certain assumptions, we have derived exact and approximate controllability results for this class of coupled systems. Fig. 1. A coupled system consisting of a well-posed and strictly proper system and a finite-dimensional system , connected in feedback.