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We study the Hopf algebra H of Fliess operators coming from Control Theory inthe 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 forthe pre-Lie product dual of the coproduct of H, and an associative, commutativealgebra for the shuffle product. These two structures admit a compatibilitywhich makes IR a Com-pre-Lie algebra. We give a presentation of this object asa Com-pre-Lie algebra and as a pre-Lie algebra.

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