Stability, Extended Spaces and Numerical Ranges

Abstract

This paper looks at a generalized version of the ideas of passivity and positivity used in studying stability of nonlinear feedback systems. We generalize these ideas to normed spaces that may not be Hilbert spaces by using the concept of the numerical range and extending this to nonlinear operators. The stability theorem derived by this method includes the usual results for passive and positive operators but also allows one to obtain different results by applying it to non-Hilbert spaces. Before coming to the stability theorem, it is necessary to examine more carefully the way in which extended spaces are constructed. A certain technical property, needed in the transition between extended and original spaces in stability proofs, holds automatically in Hilbert spaces: it need not hold in general but we show how it can always be achieved by an initial enlargement of the original normed space.