Representation of hysteresis operators for vector-valued continuous monotaffine input functions by functions on strings

Abstract

In Brokate-Sprekels-1996, it was shown that scalar-valued hysteresis operators for scalar-valued continuous piecewise monotone input functions can be uniquely represented by functionals defined on the set of all finite alternating strings of real numbers. Using this representation, various properties of these hysteresis operators were investigated. In this work, it is shown that a similar representation result can be derived for hysteresis operators dealing with inputs in a general topological linear vector space. Introducing a new class of functions, the so-called emphmonotaffine functions, which can be considered as a vector generalization of monotone scalar functions, and the convexity triple free strings on a vector space as a generalization of the alternating strings allows to formulate the corresponding representation result. As an example for the application of the representation result, a vectorial formulation of the second and third Madelung rule are discussed.