Center problem, Abel equation and the Faa di Bruno Hopf algebra for output feedback

TL;DR

This paper explores the algebraic structures underlying the classical center-focus problem, linking combinatorics, Hopf algebras, and control theory through the Abel equation and output feedback.

Contribution

It reveals the connection between Devlin's polynomials, Hopf algebras, and control theory, identifying a Faa di Bruno Hopf algebra as fundamental in the center problem.

Findings

Identifies a Hopf algebra structure related to Devlin's polynomials.

Links the Abel equation to output feedback control systems.

Distinguishes the Faa di Bruno Hopf algebra from the shuffle algebra.

Abstract

A combinatorial interpretation is given of Devlin's word problem underlying the classical center-focus problem of Poincare for non-autonomous differential equations. It turns out that the canonical polynomials of Devlin are from the point of view of connected graded Hopf algebras intimately related to the graded components of a Hopf algebra antipode applied to the formal power series of Ferfera. The link is made by passing through control theory since the Abel equation, which describes a center, is equivalent to an output feedback equation, and the Hopf algebra of output feedback is derived from the composition of iterated integrals rather than just the products of iterated integrals, which yields the shuffle algebra. This means that the primary algebraic structure at play in Devlin's approach is actually not the shuffle algebra, but a Faa di Bruno type Hopf algebra, which is defined in terms of the shuffle product but is a distinct algebraic structure.