The formal Laplace-Borel transform, Fliess operators and the composition product

In this paper, the formal Laplace-Borel transform of an analytic nonlinear input-output system is defined, specifically, an input-output system that can be represented as a Fliess operator. Using this concept and the composition product, an explicit relationship is then derived between the formal Laplace-Borel transforms of the input and output signals. This provides an alternative interpretation of the symbolic calculus introduced by Fliess to compute the output response of such systems. Finally, it is shown that the formal Laplace-Borel transform provides an isomorphism between the semigroup of all well defined Fliess operators under composition and the semigroup of all locally convergent formal power series under the composition product.