New results on stable multidimensional polynomials. III. State-space interpretations

Lyapunov characterization of scattering Schur, reactance Schur, and immittance Schur properties of multidimensional polynomials are given. Since these polynomials arise in the description of passive systems, energy-like functions can be conveniently defined for such systems, thus making Lyapunov-type characterization readily available. It is shown that the notion of modal observability and its further variants for multidimensional systems are necessary ingredients of such considerations. Two approaches are followed. One is essentially passive synthesis based, whereas the other is that of viewing multidimensional systems as a parametric family of one-dimensional systems. A nontrivial example is included to illustrate the former method