Vers une notion de dérivation fonctionnelle causale. (Towards a notion of causal functional derivatives) (Q1082306)

The input output behaviour of an important class of nonlinear \(C^{\infty}\) control systems can be characterized by non-commutative generating power series. The objective of the present note is to show that these non-commutative power series are genuine Taylor expansions if a new kind of derivative is used. This derivative was introduced by \textit{R. Ree} [Ann. Math., II. Ser. 68, 210-220 (1958; Zbl 0083.254)] in the context of his work on shuffle algebras. Analytically, it is based on Chen's iterated path-integrals [see \textit{K. T. Chen}, Bull. Am. Math. Soc. 83, 831-879 (1977; Zbl 0389.58001)]. The note gives a concise introduction to this concept of ''causal'' derivative.