Chen-Fliess Series for Linear Distributed Systems with One Spatial Dimension

Given a smooth finite-dimensional state space system which is linear in the input, its input-output map can be represented by a Chen-Fliess series, namely, a weighted sum of iterated integrals of the input's component functions. The objective of this paper is to propose a generalized notion of a Chen- Fliess series for infinite-dimensional systems. The basic idea is to replace the real field of series coefficients with a ring of linear operators which act on the iterated integrals. The specific goals are to provide sufficient conditions for convergence of this generalized series and to exercise the theory on two specific examples: the transport equation and second-order hyperbolic partial differential equations. It will be shown in these examples that the generalized Chen-Fliess series under suitable conditions yields solutions that converge pointwise to the known classical solutions.