Distributional Generating Series in Nonlinear Control Theory

In linear system theory, the commutative algebra of integrable functions under convolution does not have a unit unless one includes the Dirac delta or impulse function. In the broader context of nonlinear systems, such generalized functions or distributions have played a more limited role. One exception to this situation is the class of nonlinear input-output systems that have a Chen–Fliess series representation. In this context, a convolutional unit in the form of a distributional generating series was introduced in the literature to characterize the algebraic structures induced by system interconnections. What is missing in this earlier work, however, is a clear description of what a distributional generating series is in an analytic framework. The goal of this paper is to describe this object precisely and show how it interacts with more standard concepts in nonlinear control theory.