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

DOI:
https://doi.org/10.1090/S0002-9939-2012-11887-8

Published electronically:
September 5, 2012

MathSciNet review:
3008884

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Abstract | References | Similar Articles | Additional Information

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 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 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 satisfies the passivity inequality and then ask the question, what additional conditions are needed in order for to be a system node? The answer is surprisingly simple: A necessary and sufficient condition for an operator to be a scattering passive system node is that 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.

**[AN96]**D. Z. Arov and M. A. Nudelman,*Passive linear stationary dynamical scattering systems with continuous time*, Integral Equations Operator Theory**24**(1996), no. 1, 1–45. MR**1366539**, https://doi.org/10.1007/BF01195483**[MS06]**Jarmo Malinen and Olof J. Staffans,*Conservative boundary control systems*, J. Differential Equations**231**(2006), no. 1, 290–312. MR**2287888**, https://doi.org/10.1016/j.jde.2006.05.012**[MSW06]**Jarmo Malinen, Olof J. Staffans, and George Weiss,*When is a linear system conservative?*, Quart. Appl. Math.**64**(2006), no. 1, 61–91. MR**2211378**, https://doi.org/10.1090/S0033-569X-06-00994-7**[Phi59]**R. S. Phillips,*Dissipative operators and hyperbolic systems of partial differential equations*, Trans. Amer. Math. Soc.**90**(1959), 193–254. MR**0104919**, https://doi.org/10.1090/S0002-9947-1959-0104919-1**[Sta01]**Olof J. Staffans,*𝐽-energy preserving well-posed linear systems*, Int. J. Appl. Math. Comput. Sci.**11**(2001), no. 6, 1361–1378. Infinite-dimensional systems theory and operator theory (Perpignan, 2000). MR**1885509****[Sta02a]**Olof J. Staffans,*Passive and conservative continuous-time impedance and scattering systems. I. Well-posed systems*, Math. Control Signals Systems**15**(2002), no. 4, 291–315. MR**1942089**, https://doi.org/10.1007/s004980200012**[Sta02b]**Olof J. Staffans,*Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view)*, Mathematical systems theory in biology, communications, computation, and finance (Notre Dame, IN, 2002) IMA Vol. Math. Appl., vol. 134, Springer, New York, 2003, pp. 375–413. MR**2043247**, https://doi.org/10.1007/978-0-387-21696-6_14**[Sta05]**Olof Staffans,*Well-posed linear systems*, Encyclopedia of Mathematics and its Applications, vol. 103, Cambridge University Press, Cambridge, 2005. MR**2154892****[SW10]**Roland Schnaubelt and George Weiss,*Two classes of passive time-varying well-posed linear systems*, Math. Control Signals Systems**21**(2010), no. 4, 265–301. MR**2672788**, https://doi.org/10.1007/s00498-010-0049-0**[SW12]**Olof J. Staffans and George Weiss,*A physically motivated class of scattering passive linear systems*, submitted to SIAM J. Control Optim., http://users.abo.fi/staffans, 2012.**[WST01]**George Weiss, Olof J. Staffans, and Marius Tucsnak,*Well-posed linear systems—a survey with emphasis on conservative systems*, Int. J. Appl. Math. Comput. Sci.**11**(2001), no. 1, 7–33. Mathematical theory of networks and systems (Perpignan, 2000). MR**1835146**

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Additional Information

**Olof J. Staffans**

Affiliation:
Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland

DOI:
https://doi.org/10.1090/S0002-9939-2012-11887-8

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.