Parameter Dependent Chen–Fliess Series: Convergence and Continuity

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 can represent solutions to Cauchy initial value problems for nonhomogeneous nonlinear PDEs and thus are useful for characterizing the input-output maps of distributed control systems. Sufficient conditions for convergence of these series and absolute continuity of the output are given for a number of special cases. Specific examples involving transport equations are presented.