Functional Derivatives of Chen-Fliess Series with Applications to Optimal Control

Functional optimization problems, such as those appearing in optimal control, are often stated in terms of finding the critical points of a variational derivative. The first goal of this paper is to describe the Frechet derivative of a Chen-Fliess series and to provide an algebraic framework for computing it. The second goal is to show how to characterize and compute critical points of this Frechet derivative both analytically and numerically. The former requires a certain shuffle separability property of the generating series for the Frechet derivative and employs the concept of a nullable series. Finally, some simple examples are provided to show how these ideas can be applied to solve quadratic optimal control problems entirely in the context of Chen-Fliess series.