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Local Convergence of the Feedback Product via the Asymptotics of the Catalan Numbers

Given two analytic nonlinear input-output systems represented as Fliess operators, Fcand Fd, their feedback connection y = Fc[u + Fd[y]] can be described in terms of a feedback product of their corresponding generating series c and d, namely y = Fc@d[u]. In this paper, sufficient conditions are given under which Fc@dis always well-defined on a closed ball in a suitable input signal space and over a nonzero interval of time. In the process of establishing this result, a connection is derived between the radius of convergence and the asymptotic behavior of the sequence of Catalan numbers or, more specifically, the binomial transform of the sequence of Catalan numbers. This suggests a deeper connection between feedback structures involving analytic systems and algebraic combinatorics on words.