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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On scattering passive system nodes and maximal scattering dissipative operators


Author: Olof J. Staffans
Journal: Proc. Amer. Math. Soc. 141 (2013), 1377-1383
MSC (2010): Primary 47B44, 93A05, 93C25
Published electronically: September 5, 2012
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Abstract: There is an extensive literature on a class of linear time-invariant dynamical systems called ``well-posed scattering passive systems''. Such a system is generated by an operator $ S$ which is called a scattering passive system node. In the existing literature such a node is typically introduced by first giving a list of assumptions which imply that $ S$ is a system node and then adding an inequality which forces this system node to be scattering passive. Here we proceed in the opposite direction: we start by requiring that $ S$ satisfies the passivity inequality and then ask the question, what additional conditions are needed in order for $ S$ to be a system node? The answer is surprisingly simple: A necessary and sufficient condition for an operator $ S$ to be a scattering passive system node is that $ S$ is closed and maximal within the class of operators that satisfy the passivity inequality. In the absence of external inputs and outputs, this condition is identical to the standard condition which characterizes the class of operators which generate contraction semigroups on Hilbert spaces.


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Additional Information

Olof J. Staffans
Affiliation: Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11887-8
PII: S 0002-9939(2012)11887-8
Keywords: Scattering passive, system node, maximally dissipative operator, well-posed linear system
Received by editor(s): August 20, 2011
Published electronically: September 5, 2012
Communicated by: Richard Rochberg
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.