$H^2 $-Functions and Infinite-Dimensional Realization Theory

Abstract

In this paper the realization question for infinite-dimensional linear systems is examined for both bounded and unbounded operators. In addition to obtaining realizability criteria covering the basic cases, we discuss the relationship between canonical realizations of the same system. What one finds is that the set of transfer functions which are realizable by triples $(A,b,c)$ with A bounded is related in a close way to the space of complex functions analytic and square integrable on the disk $| {s < 1} |$, and that the set of transfer functions which are realizable by triples $(A,b,c)$ with A unbounded but generating a strongly continuous semigroup is related in a close way to functions analytic and square integrable on a half-plane. This relation makes possible a deeper study between the transfer function and the models which realize it. Some examples illustrate the results and their applications.