Periodic behaviors

TL;DR

This paper explores behaviors on a torus, establishing foundational properties similar to classical results, and connects these behaviors to algebraic geometry concepts like integral points on varieties.

Contribution

It extends classical behavior theory to periodic functions on a torus, linking it with algebraic geometry and establishing a Nullstellensatz for these behaviors.

Findings

Established properties of behaviors on a torus analogous to classical results

Connected behaviors to algebraic geometry concepts such as integral points

Proved a Nullstellensatz describing the Willems closure for periodic behaviors

Abstract

This paper studies behaviors that are defined on a torus, or equivalently, behaviors defined in spaces of periodic functions, and establishes their basic properties analogous to classical results of Malgrange, Palamodov, Oberst et al. for behaviors on R^n. These properties - in particular the Nullstellensatz describing the Willems closure - are closely related to integral and rational points on affine algebraic varieties.