Passive and Conservative Continuous-Time Impedance and Scattering Systems. Part I: Well-Posed Systems

Abstract

Let U be a Hilbert space. By an ℒ (U)-valued positive analytic function on the open right half-plane we mean an analytic function which satisfies the condition . This function need not be proper, i.e., it need not be bounded on any right half-plane. We study the question under what conditions such a function can be realized as the transfer function of an impedance passive system. By this we mean a continuous-time state space system whose control and observation operators are not more unbounded than the (main) semigroup generator of the system, and, in addition, there is a certain energy inequality relating the absorbed energy and the internal energy. The system is (impedance) energy preserving if this energy inequality is an equality, and it is conservative if both the system and its dual are energy preserving. A typical example of an impedance conservative system is a system of hyperbolic type with collocated sensors and actuators. We give several equivalent sets of conditions which characterize when a system is impedance passive, energy preserving, or conservative. We prove that a impedance passive system is well-posed if and only if it is proper. We furthermore show that the so-called diagonal transform (which may be regarded as a slightly modified feedback transform) maps a proper impedance passive (or energy preserving or conservative) system into a (well-posed) scattering passive (or energy preserving or conservative) system. This implies that, just as in the finite-dimensional case, if we apply negative output feedback to a proper impedance passive system, then the resulting system is (energy) stable. Finally, we show that every proper positive analytic function on the right half-plane has a (essentially unique) well-posed impedance conservative realization, and it also has a minimal impedance passive realization.