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Passive and conservative scattering systems using none toget none with web,windows applicationcode 128 generating 11.3 Semi-lossless and lossless systems Beaware of Malicious QR Codes In an energy preserving s ystem no energy is lost, but it may be rst transferred from the input to the state, and then trapped in the state space forever, so that it can no longer be retrieved from the outside. Thus, from the point of view of an external observer, a conservative system may be lossy . To speci cally exclude this case we need another notion, that we shall refer to as losslessness.

De nition 11.3.1 (i) An operator D TIC2 (U ; Y ) (where U and Y are Hilbert spaces) is (scattering) semi-lossless if it is isometric, i.

e.,. (D + u)(s). 2 ds = Y 2 . u(s). U ds for all u L 2 (R+ ; U ) . It is (scattering) co-lossless if the causal dual operator Dd = RD R is semi-lossless, and it is (scattering) lossless if it is both semi-lossless and co-lossless. (ii) By a semi-lossless, co-lossless, or lossless system we mean an L 2 -well-posed linear system on three Hilbert spaces (Y, X, U ) whose input/output map is semi-lossless or lossless.

Thus, semi-losslessness, co-losslessness, and losslessness can be interpreted as the input/output versions of energy preservation, co-energy preservation, or conservativity, respectively. Semi-losslessness of an operator D TIC2 (U ; Y ) can alternatively be interpreted as a property of the transfer function D of D: Proposition 11.3.

2 An operator D TIC2 (U ; Y ) (where U and Y are Hilbert spaces) is semi-lossless if and only if its transfer function D is left-inner in the following sense: D is a contractive analytic function on C+ , the restriction of D to every separable subspace of U has a strong limit from the right a.e. at the imaginary axis, and this limit is isometric a.

e.1 Proof By Proposition 10.3.

5, D TIC2 (U ; Y ) if and only if D H (U ; Y ), and D is a contractive analytic function on C+ whenever D is a contraction (in particular, if it is isometric). Without loss of generality, we may assume that U and Y are separable, because any function u L 2 (R+ ; U ) to which we may apply D is almost separable-valued, and so is D + u. The separable case is.

The limit from the right none none at the imaginary axis of Du exists almost everywhere for all u U even if U is nonseparable. By restricting D to a separable subspace of U we can ensure that the limiting operator is de ned almost everywhere. The continuity of D in C+ implies that the values of D lie in a separable subspace of Y in this case.

. 11.3 Semi-lossless and lossless systems well-known and found in m none for none any places (in slightly different settings); see, e.g., Duren (1970, pp.

187 192), Hoffman (1988), or Sz.-Nagy and Foia (1970, s Section V.2, pp.

186 192). Note that, if D is the input/output map of an L 2 -well-posed system with main operator A, then the transfer function D has an analytic extension to all of (A). In particular, if (A) jR = , then at every point (A) jR, D( ) will be isometric if D is semi-lossless and unitary if D is lossless (since D is continuous at every (A)).

As the following theorem says, if a system is controllable and passive and semi-lossless, then it must be energy preserving and output normalized. Theorem 11.3.

3 A controllable semi-lossless passive system = A B on C D (Y, X, U ) is necessarily energy preserving and strongly stable, and C C = 1. In particular, is minimal and output normalized, and it is unitarily similar to the restricted exactly observable shift realization of D given in Proposition 9.5.

2(iv). Proof We begin by showing that the state trajectory x and the output function y of with initial time zero, initial state zero, and input function u satisfy . x(t). 2 + X
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