On symmetries in analytic input-output systems

Abstract The earliest form of symmetry for analytic input-output systems is due to Fliess who introduced systems described by an exchangeable generating series. In this case, one is able to write the output as a memoryless analytic function of the integral of each input. The first goal is to describe two new types of symmetry for such Chen–Fliess input-output systems, namely, coefficient reversible symmetry and palindromic symmetry. Each concept is then related to the notion of an exchangeable series. The second goal is to provide an in-depth analysis of Chen–Fliess input-output systems whose generating series are linear time-varying, palindromic, and have generating series coefficients growing at a maximal rate while ensuring some type of convergence. It is shown that such series have an infinite Hankel rank and Lie rank, have a certain infinite-dimensional state space realization and a description of their relative degree and zero dynamics is given.