Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Transfer functions of regular linear systems Part II: The system operator and the Lax–Phillips semigroupHTML articles powered by AMS MathViewer

by Olof Staffans and George Weiss
Trans. Amer. Math. Soc. 354 (2002), 3229-3262 Request permission

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.
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• Olof Staffans
• Affiliation: Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland
• Email: Olof.Staffans@abo.fi
• George Weiss
• Affiliation: Department of Electrical & Electronic Engineering, Imperial College of Science & Technology, Exhibition Road, London SW7 2BT, United Kingdom
• Email: G.Weiss@ic.ac.uk
• Received by editor(s): February 23, 2001
• Received by editor(s) in revised form: November 16, 2001
• Published electronically: April 3, 2002