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.

Key Words:

• Fliess operators
• Hopf algebras
• Pre-Lie algebras

2010 Mathematics Subject Classification:

• 16T05
• 17B60
• 93B25
• 05C05

Notes

Communicated by J. Alev.