Shift Invariant Subspaces, Passivity, Reproducing Kernels and H∞-Optimization

Various notions of passivity are introduced for a lossless circuit, or equivalently, for a rational matrix function θ which is J-unitary on the unit circle. These notions, as well as how they are related to each other, are analyzed from several points of view: energy bookkeeping in the circuit, analytic conditions on θ and on the associated scattering matrix U, geometry of shift invariant subspaces, positive definiteness conditions on associated reproducing kernel functions, connections with classical interpolation problems, and state space representations. This gives a circuit theoretic interpretation for several modern approaches to interpolation such as the geometric one of Ball-Helton.