The Hopf Algebra of Fliess Operators and Its Dual 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, finite-dimensional, connected grading. Dually, the vector space ℝ ⟨ x 0, x 1 ⟩ 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 ℝ ⟨ x 0, x 1 ⟩ a Com-Pre-Lie algebra. We give a presentation of this object as a pre-Lie algebra.