Fliess operators on Lp spaces: convergence and continuity

Abstract

Fliess operators as input–output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory. In the classical analysis of these operators, certain growth conditions on the coefficients in their series representations insure uniform and absolute convergence, provided every input is uniformly bounded by some fixed upperbound. In some emerging applications, however, it is more natural to consider other classes of inputs. In this paper, L p function spaces are considered. In particular, it is shown that the classic growth conditions also provide sufficient conditions for convergence and continuity when the admissible inputs are from a ball in L p [ t 0, t 0+ T], where T is bounded and p⩾1. In addition, stronger global growth conditions are given that apply even for the case where T is unbounded. When the coefficients of a Fliess operator have a state space representation, it is shown that the state space model will always locally realize the corresponding input–output map on L p [ t 0, t 0+ T] for sufficiently small T>0. If certain well-posedness conditions are satisfied then the state space model will globally realized the input–output mapping for unbounded T when the coefficients satisfy the global growth condition.