On scattering passive system nodes and maximal scattering dissipative operators

Staffans, Olof

doi:10.1090/S0002-9939-2012-11887-8

On scattering passive system nodes and maximal scattering dissipative operators
HTML articles powered by AMS MathViewer

by Olof J. Staffans PDF

Proc. Amer. Math. Soc. 141 (2013), 1377-1383 Request permission

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

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, DOI 10.1007/BF01195483
Jarmo Malinen and Olof J. Staffans, Conservative boundary control systems, J. Differential Equations 231 (2006), no. 1, 290–312. MR 2287888, DOI 10.1016/j.jde.2006.05.012
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, DOI 10.1090/S0033-569X-06-00994-7
R. S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 193–254. MR 104919, DOI 10.1090/S0002-9947-1959-0104919-1
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
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, DOI 10.1007/s004980200012
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, DOI 10.1007/978-0-387-21696-6_{1}4
Olof Staffans, Well-posed linear systems, Encyclopedia of Mathematics and its Applications, vol. 103, Cambridge University Press, Cambridge, 2005. MR 2154892, DOI 10.1017/CBO9780511543197
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, DOI 10.1007/s00498-010-0049-0
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.
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