Abstracts Mathematics

Chen-Fliess Series for Linear Distributed Systems

by Natalie T. Pham

Distributed systems like fluid flow and heat transfer are modeled by partial differential equations (PDEs). In control theory, distributed systems are generally reformulated in terms of a linear state space realization, where the state space is an infinite dimensional Banach space or Hilbert space. In the finite dimension case, the input-output map can always be written in terms of a Chen-Fliess functional series, that is, a weighted sum of iterated integrals of the components of the input function. The Chen-Fliess functional series has been used to describe interconnected nonlinear systems, to solve system inversion and tracking problems, and to design predictive and adaptive controllers. The main goal of this thesis is to show that there is a generalized notion of a Chen-Fliess series for linear distributed systems where the weights are now linear operators acting on the iterated integrals. Sufficient conditions for convergence are developed. The method is compared against classical PDE theory using a number of first-order and second-order examples.