Criteria for the Global Existence of Functional Expansions for Input/Output Maps*

Abstract

Much has been learned in recent years about the existence, determination, and properties of power-series-like expansions for expressing a nonlinear system's outputs in terms of its inputs. In particular, the existence and local convergence of expansions, and of certain “associated expansions,” for important large classes of systems are now well established. While the focus of attention has been on questions such that the size of the inputs for which convergence is guaranteed is not the main issue, some related material has appeared that bears on the problem of determining the extent of the region of convergence. The result most closely related to this paper is a recent theorem that gives necessary and sufficient conditions under which f−1 has a generalized power-series expansion when f is an invertible locally-Lipshitz map between certain general subsets of two complex Banach spaces. In applications involving nonlinear models, ordinarily only real spaces of inputs and outputs are of direct interest. A “complexification” involving a certain solvability condition in complex spaces has to be able to be carried out to use the theorem referred to above. This paper reports on pertinent general results concerning invertible maps between subsets of real Banach spaces, with their complex extensions, and with generalized power-series expansions in both real and complex spaces. It focuses on questions concerning expansions for inverses of maps defined in real spaces. The results show that for a very large class of systems that have input/output maps, the ability to complexify is not just a useful sufficient condition for expandability, but is in fact the key condition for an input/output map to be representable by a generalized power-series expansion.