# Passive discrete-time systems with a Pontryagin state space

## TL;DR

This paper investigates passive discrete-time systems within Pontryagin spaces, providing geometric characterizations, factorization methods, and criteria for system properties, expanding the theory of generalized Schur functions and their realizations.

## Contribution

It introduces new geometric and factorization characterizations for passive systems with Pontryagin space states, and develops criteria for system controllability and observability based on reproducing kernel spaces.

## Key findings

- Characterization of transfer function index and state space negative index equivalence.
- Product representation of isometric systems via Krein-Langer factorization.
- New criteria for controllability and observability preservation in system products.

## Abstract

Passive discrete-time systems with Hilbert spaces as an incoming and outgoing space and a Pontryagin space as a state space are investigated. A geometric characterization when the index of the transfer function coincides with the negative index of the state space is given. In this case, an isometric (co-isometric) system has a product representation corresponding to the left (right) Krein-Langer factorization of the transfer function. A new criterion, based on the inclusion of reproducing kernel spaces, when a product of two isometric (co-isometric) systems preserves controllability (observability), is obtained. The concept of the defect function is expanded for generalized Schur functions, and realizations of generalized Schur functions with zero defect functions are studied.