When is the stability of a nonlinear input-output system robust?

We consider a general nonlinear input-output system governed by operator equations that relate the system's input, state, and output, all of which are in extended spaces. It is assumed that the system variables are separated. Our results give conditions under which the stability of the nominal system is robust; i.e., it is not destroyed by any sufficiently small admissible perturbation of the system. Theorem 1 deals with the case when by stability we mean theincremental stability. Theorem 3 concerns the*-stability; i.e., the case when the stability is essentially the boundedness of the transmission operator. Moreover, in Theorem 2 it is shown that, under certain conditions, the incremental stability of the nominal system implies insensitivity. Basically, our results show that if the operators describing the nominal system are well behaved, and the transition from the nominal system to the perturbed system is not abrupt, then the nominal system stability is robust. The applications of the results are illustrated by several examples.