Feedback classification of linear systems over von Neumann regular rings

TL;DR

This paper develops a method to classify linear systems over von Neumann regular rings by reducing the problem to manageable parts and provides an explicit algorithm for computing canonical forms, enhancing understanding of system equivalences.

Contribution

It introduces a reduction technique for feedback classification over von Neumann regular rings and offers an explicit algorithm for computing canonical forms, including an example over a finite ring.

Findings

Feedback classification reduces to finite family of systems.

Reachable part in Brunovski-type canonical form.

Explicit algorithm for canonical form computation.

Abstract

It is proved that feedback classification of a linear system over a commutative von Neumann regular ring R can be reduced to the classification of a finite family of systems, each of which is properly split into a reachable and a non-reachable part, where the reachable part is in a Brunovski-type canonical form, while the non-reachable part can only be altered by similarity. If a canonical form is known for similarity of matrices over R, then it can be used to construct a canonical form for feedback equivalence. An explicit algorithm is given to obtain the canonical form in a computable context together with an example over a finite ring.