Passive and conservative continuous-time impedance and scattering systems. I: Well-posed systems.
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 the Lax-Phillips scheme,'' Dep. at VINITI, No. 8009-B86, Odessa, 49 pp. (1986); RZhMat 1987:3B1080] and \textit{D. Salamon} [Math. Syst. Theory 21, No. 3, 147--164 (1989; Zbl 0668.93018)] , and subsequently investigated also by R. Curtain, O. J. Staffans, G. Weiss and other authors. Continuous-time passive scattering systems (which are automatically well-posed) were studied by Arov and Nudel'man [loc. cit.]. In the paper under review, continuous-time passive, energy-preserving, and conservative impedance and scattering systems are investigated. A criterion for the well-posedness of a passive impedance system is established in terms of its transfer function, namely the transfer function should be bounded on some (or, equivalently, on any) vertical line in the open right half-plane. It is shown that the so-called diagonal transformation of input-output pairs which establishes one-to-one correspondence between passive (energy-preserving, conservative) impedance systems and passive (energy-preserving, conservative) scattering systems, which is well-known for the case of finite-dimensional state space, works as well for the case of infinite-dimensional state space, if it is applied to a well-posed passive impedance system. A feedback interpretation of the diagonal transformation is given: it says that if one applies negative feedback to a well-posed passive impedance system then the resulting closed-loop system is energy stable.
scientific article; zbMATH DE number 1890502