A Review on Realization Theory for Infinite-Dimensional Systems

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.