International Journal of Mathematics and Mathematical Sciences
Volume 2006 (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 Yaqin Li and W. Steven Gray. 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.