A gap metric perspective of well-posedness for nonlinear feedback interconnections

Abstract

A differential geometric approach based on the gap metric is taken to examine the uniqueness of solutions of the equations describing a feedback interconnection. It is shown that under sufficiently small perturbations on the Fréchet derivative of a nonlinear plant as measured by the gap metric, the uniqueness property is preserved if solutions exist given exogenous signals. The results developed relate the uniqueness of solutions for a nominal feedback interconnection and that involving the derivative of the plant. Causality of closed-loop operators is also investigated. It is established that if a certain open-loop mapping has an inverse over signals with arbitrary start time (i.e. zero before some initial time), then the closed-loop operator is causal provided the latter is weakly additive.