Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



When is a linear system conservative?

Authors: Jarmo Malinen, Olof J. Staffans and George Weiss
Journal: Quart. Appl. Math. 64 (2006), 61-91
MSC (2000): Primary 47A48, 47N70, 93B28, 93C25
DOI: https://doi.org/10.1090/S0033-569X-06-00994-7
Published electronically: January 24, 2006
MathSciNet review: 2211378
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Abstract: We consider infinite-dimensional linear systems without a-priori well-posedness assumptions, in a framework based on the works of M. Livšic, M. S. Brodski{\u{\i\/}}\kern.15em, Y. L. Smuljan, and others. We define the energy in the system as the norm of the state squared (other, possibly indefinite quadratic forms will also be considered). We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, well posed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control operator is one-to-one and whose observation operator has dense range, the equivalent conditions for being conservative become simpler, and reduce to three algebraic equations.

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Additional Information

Jarmo Malinen
Affiliation: Institute of Mathematics, P.O. Box 1100, Helsinki University of Technology, FIN-02015 HUT, Finland
Email: Jarmo.Malinen@hut.fi

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

George Weiss
Affiliation: Department of Electric and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2BT, United Kingdom
Email: G.Weiss@imperial.ac.uk

DOI: https://doi.org/10.1090/S0033-569X-06-00994-7
Keywords: Conservative system, energy preserving system, well-posed linear system, regular linear system, operator node, Cayley transform
Received by editor(s): January 6, 2005
Published electronically: January 24, 2006
Additional Notes: This research was supported by grants from the European Research Network on Systems Identification (ERNSI), the Academy of Finland under grant 203991, the Mittag–Leffler Institute (Sweden) and EPSRC (UK), under the Portfolio Partnership grant GR/S61256/01.
Article copyright: © Copyright 2006 Brown University

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