Pole Placement with Fields of Positive Characteristic

TL;DR

This paper extends classical pole placement results from real and complex fields to arbitrary algebraically closed fields, demonstrating that key controllability conditions hold universally across these fields.

Contribution

It generalizes the pole placement theorem to fields of positive characteristic, providing new proofs and confirming the classical results in a broader algebraic context.

Findings

Pole placement conditions hold over any algebraically closed field.

The number of solutions matches classical intersection numbers.

Classical controllability criteria are field-independent.

Abstract

The pole placement problem belongs to the classical problems of linear systems theory. It is often assumed that the ground field is the real numbers R or the complex numbers C. The major result over the complex numbers derived in 1981 by Brockett and Byrnes states that arbitrary static pole placement is possible for a generic set of m-inputs, p-outputs and McMillan degree n system as soon as mp>=n. Moreover the number of solutions in the situation mp=n is an intersection number first computed by Hermann Schubert in the 19th century. In this paper we show that the same result with slightly different proofs holds over any algebraically closed field.