2012, Klein, Olaf

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.

2013, Klein, Olaf

In Brokate-Sprekels 1996, it is shown that hysteresis operators acting on scalar-valued, continuous, piecewise monotone input functions can be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be continuous and piecewise monotaffine, i.e., being piecewise the composition of two functions such that the output of a monotone increasing function is used as input for an affine function. In the current paper, a representation result is formulated for hysteresis operators dealing with input functions being left-continuous and piecewise monotaffine and continuous. The operators are generated by functions acting on an admissible subset of the set of all strings of pairs of elements of the vector space.