Parameter Dependent Chen--Fliess Series and Their Nonrecursive Interconnections

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.