Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
MathSciNet review: 3008884
Full-text PDF

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 $ 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.

References [Enhancements On Off] (What's this?)

  • [AN96] Damir Z. Arov and Mark A. Nudelman, Passive linear stationary dynamical scattering systems with continuous time, Integral Equations Operator Theory 24 (1996), 1-45. MR 1366539 (96k:47016)
  • [MS06] Jarmo Malinen and Olof J. Staffans, Conservative boundary control systems, J. Differential Equations 231 (2006), 290-312. MR 2287888 (2008a:93010)
  • [MSW06] Jarmo Malinen, Olof J. Staffans, and George Weiss, When is a linear system conservative?, Quart. Appl. Math. 64 (2006), 61-91. MR 2211378 (2007m:47022)
  • [Phi59] Ralph S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 193-254. MR 0104919 (21:3669)
  • [Sta01] Olof J. Staffans, $ J$-energy preserving well-posed linear systems, Internat. J. Appl. Math. Comput. Sci. 11 (2001), 1361-1378. MR 1885509 (2002k:93045)
  • [Sta02a] -, Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems, Math. Control Signals Systems 15 (2002), 291-315. MR 1942089 (2003i:93024)
  • [Sta02b] -, Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view), Mathematical Systems Theory in Biology, Communication, Computation, and Finance (New York), IMA Volumes in Mathematics and its Applications, vol. 134, Springer-Verlag, 2002, pp. 375-414. MR 2043247 (2004m:93072)
  • [Sta05] -, Well-posed linear systems, Cambridge University Press, Cambridge and New York, 2005. MR 2154892 (2006k:93003)
  • [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 (2011h:93018)
  • [SW12] Olof J. Staffans and George Weiss, A physically motivated class of scattering passive linear systems, submitted to SIAM J. Control Optim.,, 2012.
  • [WST01] George Weiss, Olof J. Staffans, and Marius Tucsnak, Well-posed linear systems - a survey with emphasis on conservative systems, Internat. J. Appl. Math. Comput. Sci. 11 (2001), 7-34. MR 1835146 (2002f:93068)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47B44, 93A05, 93C25

Retrieve articles in all journals with MSC (2010): 47B44, 93A05, 93C25

Additional Information

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

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.

American Mathematical Society