Passive and conservative continuous-time impedance and scattering systems. I: Well-posed systems.
type: Item
instance of: scholarly article
title: Passive and conservative continuous-time impedance and scattering systems. I: Well-posed systems.
zbMATH Open document ID: 1158.93321
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 2003-06-22T00:00:00Z
review text: In many applications of systems theory various metric constraints imposed upon a system are used. For example, for the case of continuous time \(t\in R\), the states \(x(t)\), the input signals \(u(t)\), and the output signals \(y(t)\) of such systems have the following metric relations: (\(*\)) \((d/dt)\| x(t)\| ^2\leq\Phi(u(t),y(t))\), where \(\Phi(\cdot,\cdot)\) is the supply rate quadratic form, and the state, input and output spaces are finite or infinite-dimensional Hilbert spaces. Inequality (\(*\)) has a physical sense of dissipation of energy in a system, thus such systems are called dissipative (e.g., see \textit{J. C. Willems} [Arch. Ration. Mech. Anal. 45, 321--351 (1972; Zbl 0252.93002); ibid. 45, 352--393 (1972; Zbl 0252.93003)]), or passive (e.g., see \textit{M. R. Wohlers} [Lumped and distributed passive networks, Academic Press, New York (1969; Zbl 0217.57302)]; \textit{D. Z. Arov} [Sib. Mat. Zh. 20, 211--228 (1979; Zbl 0414.93014)]; \textit{D. Z. Arov} and \textit{M. A. Nudel'man}, Integral Equations Oper. Theory 24, No. 1, 1--45 (1996; Zbl 0838.47004)] ). In the special case of equalities in (\(*\)) valid for all values of \(t\), a system is called energy-preserving, and if the analogous equalities hold also for the dual system then a system is called conservative. In the terminology originating from Arov [ op. cit.] , particular cases of \(\Phi\) correspond to scattering systems (\(\Phi(u,y)=\| u\| ^2-\| y\| ^2\)), resistance, or impedance, systems (\(\Phi(u,y)=2 \text{ Re}\langle u,y\rangle\)), and transmission systems (\(\Phi(u,y)=[ u,u] _{\text{ in}}-[y,y] _{\text{out}}\) where \([ \cdot, \cdot] \) denotes indefinite inner products in the input and output spaces which turn them into Krein spaces). The problem of well-posedness of continuous-time linear systems whose coefficients are unbounded linear operators in infinite-dimensional Hilbert spaces was raised independently by \textit{Yu. L. Shmul'yan} [ ''Invariant subspaces of semigroups and t…
Mathematics Subject Classification ID: 93B29
Mathematics Subject Classification ID: 47N70
Mathematics Subject Classification ID: 93D15
zbMATH DE Number: 1890502
zbMATH Keywords: dissipative
zbMATH Keywords: passive
zbMATH Keywords: impedance
zbMATH Keywords: scattering
zbMATH Keywords: energy preserving
zbMATH Keywords: conservative systems
zbMATH Keywords: positive real
zbMATH Keywords: bounded real lemma
zbMATH Keywords: Kalman-Yakubovich-Popov lemma
author: Olof Staffans
reviewed by: Dmitry Kaliuzhnyi-Verbovetskyi
MaRDI profile type: Publication
full work available at URL: s004980200012
OpenAlex ID: W2038599078
DOI: 10.1007/S004980200012
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description: scientific article; zbMATH DE number 1890502