On the representation of hysteresis operators acting on vector-valued,      left-continuous and piecewise  monotaffine and continuous functions

Olaf Klein

doi:10.3934/dcds.2015.35.2591

Article Contents

2015, Volume 35, Issue 6: 2591-2614. Doi: 10.3934/dcds.2015.35.2591

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On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions

Olaf Klein^1,

1.
Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin

Received: November 30, 2013

Revised: May 31, 2014

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 47J40.

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