SENSITIVITY AND ROBUST STABILITY OF GENERAL INPUT-OUTPUT SYSTEMS

We consider a general model of an input-output system that is governed by nonlinear operator equations which relate the input, the state, and the output of the system. This model encompasses feedback systems as a special case. Assuming that the governing equations depend on a parameter A which is allowed to vary in a neighborhood of a nominal value A(o) in a linear space, we study the dependence of the system behavior on A. We call a system insensitive, if for any fixed input, the output depends continuously on A. Similarly, we say that the system is robust, if it is stable for each A in a neighborhood of A(o). By stability, we essentially mean an appropriate continuity of the input-output operator. Our results give various sufficient conditions for insensitivity and robustness. The Theorems 1, 2, 5, and 6 represent the main results. As examples illustrating the applications of our theory, we discuss 1) estimating the difference of operator inverses, and the insensitivity and robust stability of a 2) Hilbert network, 3) feedback-feedforward system, 4) traditional feedback system, and of a 5) time-varying dynamical system described by a linear vector differential equation on [0, infinity).