Pseudorational Behaviors and Bezoutians

Behavioral system theory has been successful in providing a viewpoint that does not depend on a priori notions of inputs/outputs. While there are some attempts to extend this theory to infinite-dimensional systems, for example, delay systems, the overall picture seems to remain still incomplete. The first author has studied a class of infinite-dimensional systems called pseudorational. This class allows a compact fractional representation for systems having bounded-time memory. It is particularly appropriate for extending the behavioral framework to infinite-dimensional context. We have recently studied several attempts to extend this framework to a behavioral context. Among them are characterizations of behavioral controllability, particularly involving a coprimeness condition over an algebra of distributions, and some stability tests involving Lyapunov functions derived from Bézoutians. This article gives a brief overview of pseudorational transfer functions, controllability issues and related criteria, path integrals, and finally the connection with Lyapunov functions derived from Bézoutians.