Transfer functions of regular linear systems Part II: The system operator and the Lax–Phillips semigroup

Abstract:

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as “Part I”. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by $\dot x=Ax+Bu$, $y=Cx+Du$ would be the $s$-dependent matrix $S_\Sigma (s)= \left [ {}^{A-sI}_{\ \; C} {\ } ^{B}_{D} \right ]$. In the general case, $S_\Sigma (s)$ is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks $A-sI$ and $B$, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of $S_\Sigma (s)$ where the right lower block is the feedthrough operator of the system. Using $S_\Sigma (0)$, we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the “initial time” is $-\infty$. We also introduce the Lax-Phillips semigroup $\boldsymbol {\mathfrak {T}}$ induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ${\omega }\in {\mathbb R}$ which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of $A$ and also the points where $S_\Sigma (s)$ is not invertible, in terms of the spectrum of the generator of $\boldsymbol {\mathfrak {T}}$ (for various values of ${\omega }$). The system $\Sigma$ is dissipative if and only if $\boldsymbol {\mathfrak {T}}$ (with index zero) is a contraction semigroup.