ROBUST STABILITY AND SENSITIVITY OF INPUT-OUTPUT SYSTEMS OVER EXTENDED SPACES .1. ROBUST STABILITY

In Part I of this paper we consider a general model of an input-output system governed by nonlinear operator equations that relate the system's input, state, and output, all of which are elements in extended spaces. This model encompasses feedback systems as a special case. Assuming that the equations governing the system depend on a parameter A that is allowed to vary in a neighborhood N(r)(A0) of a nominal value A0 in a linear space, we study conditions under which the system is stable for every A is-an-element-of N(r)(A0), i.e., when the system exhibits robust stability. By stability we essentially mean that the input-output operator is continuous. Depending on the type of continuity of a map between two extended spaces, four concepts of robustness are introduced. The main results, Theorems 1 and 2, furnish sufficient conditions for a system to be robust in the respective sense. Basically, they show that if the nominal system satisfies a certain condition guaranteeing its stability, and the operators appearing in the governing equations depend continuously on the parameter A, then we have robust stability. As examples illustrating the applications of our results we discuss (1) a feedback-feedforward system, in particular the case when the extended space consists of locally square-integrable functions or functions continuous on [O, infinity), and (2) a time-varying dynamical system described by a linear vector differential equation, whose variables are continuous functions on [O, infinity) which decrease exponentially to zero as t --> infinity. At the end of the paper some modifications of the presented theory are discussed.