The Hopf algebra of Fliess operators and its dual pre-Lie algebra

TL;DR

This paper explores the algebraic structures underlying Fliess operators in control theory, revealing a Hopf algebra and dual pre-Lie algebra with compatible structures, advancing the mathematical framework of control systems.

Contribution

It introduces a graded, finite-dimensional Hopf algebra of Fliess operators and characterizes its dual as a Com-pre-Lie algebra, providing new algebraic insights.

Findings

The Hopf algebra of Fliess operators is graded and finite-dimensional.

The dual space forms a Com-pre-Lie algebra with compatible structures.

The dual algebra admits a presentation as a Com-pre-Lie and pre-Lie algebra.

Abstract

We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finte-dimensional, connected gradation. Dually, the vector space IR is both a pre-Lie algebra for the pre-Lie product dual of the coproduct of H, and an associative, commutative algebra for the shuffle product. These two structures admit a compatibility which makes IR a Com-pre-Lie algebra. We give a presentation of this object as a Com-pre-Lie algebra and as a pre-Lie algebra.