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

Li, Yaqin; Gray, W. Steven

doi:https://doi.org/10.1155/IJMMS/2006/34217

International Journal of Mathematics and Mathematical Sciences

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Volume 2006 | Article ID 034217 | https://doi.org/10.1155/IJMMS/2006/34217

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

Yaqin Li¹and W. Steven Gray²

Received28 Jul 2005

Revised14 Apr 2006

Accepted25 Apr 2006

Published06 Jul 2006

Abstract

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.

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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.

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