Continuity of Formal Power Series Products in Nonlinear Control Theory

TL;DR

This paper investigates the mathematical properties of formal power series products in nonlinear control, proving their continuity and analyticity, which underpin advanced control and learning system techniques.

Contribution

It establishes the continuity and analyticity of formal power series products and shows that a key transformation group forms an analytic Lie group with regularity properties.

Findings

Proved continuity of formal power series products in various topologies.

Established analyticity of these products, supporting control theory applications.

Demonstrated that the output feedback transformation group is an analytic Lie group.

Abstract

Formal power series products appear in nonlinear control theory when systems modeled by Chen-Fliess series are interconnected to form new systems. In fields like adaptive control and learning systems, the coefficients of these formal power series are estimated sequentially with real-time data. The main goal of the present article is to prove the continuity and analyticity of such products with respect to several natural (locally convex) topologies on spaces of locally convergent formal power series in order to establish foundational properties behind these technologies. In addition, it is shown that a transformation group central to describing the output feedback connection is in fact an analytic Lie group in this setting with certain regularity properties.