Parameter Dependent Chen--Fliess Series and Their Nonrecursive Interconnections

TL;DR

This paper introduces a new class of parameter-dependent Chen--Fliess series with coefficients in a noncommutative ring, useful for modeling distributed control systems and analyzing their interconnections.

Contribution

It defines a novel class of Chen--Fliess series with noncommutative coefficients and explores their properties under nonrecursive interconnections, extending their applicability.

Findings

Series represent solutions to PDE initial value problems.

Class is nearly closed under nonrecursive interconnections.

Examples include transport and wave equations.

Abstract

A class of parameter dependent Chen--Fliess series is introduced where the series coefficients are taken from a noncommutative ring of multivariable differential operators. Such series are shown in the linear case to represent formal solutions to Cauchy initial value problems for nonhomogeneous PDEs and thus are useful for characterizing the input-output maps of distributed control systems. It is also shown that this class of functional series is almost closed under the set of nonrecursive interconnections, that is, any finite combination of parallel and series interconnections without a closed-loop. Some sufficient conditions are needed for the series interconnection. Specific examples are given involving the transport equation and the wave equation.