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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Olof Staffans and George Weiss
Journal: Trans. Amer. Math. Soc. 354 (2002), 3229-3262
MSC (2000): Primary 93C25; Secondary 34L25, 37L99, 47D06
Published electronically: April 3, 2002
MathSciNet review: 1897398
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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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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02976-8
PII: S 0002-9947(02)02976-8
Keywords: Well-posed linear system, (weakly) regular linear system, operator semigroup, system operator, generating operators, well-posed transfer function, scattering theory, Lax-Phillips semigroup, dissipative system
Received by editor(s): February 23, 2001
Received by editor(s) in revised form: November 16, 2001
Published electronically: April 3, 2002
Article copyright: © Copyright 2002 American Mathematical Society