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

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- by Olof Staffans and George Weiss PDF
- 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.## References

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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
- Received by editor(s): February 23, 2001
- Received by editor(s) in revised form: November 16, 2001
- Published electronically: April 3, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**354**(2002), 3229-3262 - MSC (2000): Primary 93C25; Secondary 34L25, 37L99, 47D06
- DOI: https://doi.org/10.1090/S0002-9947-02-02976-8
- MathSciNet review: 1897398