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International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 34217, 14 pages

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

1Department of Electrical and Computer Engineering, University of Memphis, Memphis 38152, TN, USA
2Department of Electrical and Computer Engineering, Old Dominion University, Norfolk 23529, VA, USA

Received 28 July 2005; Revised 14 April 2006; Accepted 25 April 2006

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The formal Laplace-Borel transform of an analytic integral operator, known as a Fliess operator, is defined and developed. Then, in conjunction with the composition product over formal power series, the formal Laplace-Borel transform is shown to provide an isomorphism between the semigroup of all Fliess operators under operator composition and the semigroup of all locally convergent formal power series under the composition product. Finally, the formal Laplace-Borel transform is applied in a systems theory setting to explicitly derive the relationship between the formal Laplace transform of the input and output functions of a Fliess operator. This gives a compact interpretation of the operational calculus of Fliess for computing the output response of an analytic nonlinear system.