A Review on Realization Theory for Infinite-Dimensional Systems

TL;DR

This paper reviews the realization theory for infinite-dimensional systems, showing how any bounded analytic function can be represented using operators related to semigroup generators, clarifying decades of research.

Contribution

It provides a comprehensive overview and clarification of the realization theory for infinite-dimensional systems, connecting classical results with modern operator theory.

Findings

Existence of operator realizations for bounded analytic functions

Connection between functions and semigroup generators on Hilbert spaces

Clarification of historical results in realization theory

Abstract

We give an introduction to the realisation theory for infinite-dimensional systems. That is, we show that for any function $G$, analytic and bounded in the right half of the complex plane, there exists operators $A,B,C$ such that $G(s_1)-G(s_2) = (s_2-s_1) C(s_1 I-A)^{-1}(s_2 I-A)^{-1}B$. Here $A$ is the infinitesimal generator of a strongly continuous semigroup on a Hilbert space, and $B$ and $C$ are admissible input and output operators, respectively. Our results summarise and clarify the results as found in the literature, starting more than 40 years ago.