Multiparametric Dissipative Linear Stationary Dynamical Scattering Systems: Discrete Case

TL;DR

This paper generalizes linear stationary dynamical systems to multiparametric cases with discrete time, using Lax-Phillips semigroups, and characterizes their transfer functions within a holomorphic operator-valued function class.

Contribution

It introduces a new multiparametric framework for dissipative systems, extending classical models and providing a characterization theorem for their transfer functions.

Findings

Defined multiparametric passive and conservative scattering systems.

Established a representation theorem for transfer functions as holomorphic operator-valued functions.

Connected system properties with operator colligations and semigroup theory.

Abstract

We propose the new generalization of linear stationary dynamical systems with discrete time $t\in\mathbb{Z}$ to the case $t\in\nspace{Z}{N}$. The dynamics of such a system can be reproduced by means of its associated multiparametric Lax-Phillips semigroup. We define multiparametric passive, and conservative scattering systems and interpret them in terms of operator colligations, of the associated semigroup and of "energetic" relations for system data. We prove the Agler's type theorem describing the class of holomorphic operator-valued functions on the polydisc $\nspace{D}{N}$ that are the transfer functions of multiparametric conservative scattering systems. Keywords: Passive systems, multiparametric Lax-Phillips semigroup, generalized Schur class, conservative realizations