Behavioral controllability and coprimeness for a class of infinite-dimensional systems

Behavioral system theory has become a successful framework in providing a viewpoint that does not depend on a priori notions of inputs/outputs. In particular, this theory provides notions as controllability, without an explicit reference to the state space formalism. One also obtains several interesting consequences of controllability, for example, direct sum decomposition of the signal space with a controllable behavior B as a direct summand. While there are some attempts to extend this theory to infinite-dimensional systems, for example, delay systems, the overall picture remains incomplete. This article extends this theory, particularly the notion of controllability, to a well-behaved class of infinite-dimensional systems, called pseudorational. A crucial notion in this context is the Bezout identity, and we relate a recent result to the context of behavioral controllability. We establish its relationships with notions as image representation and direct sum decompositions.