Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions

Abstract

It is well known that the Prandtl-Ishlinskii hysteresis operator is locally Lipschitz continuous in the space of continuous functions provided its primary response curve is convex or concave. This property can easily be extended to any absolutely continuous primary response curve with derivative of locally bounded variation. Under the same condition, the Prandtl-Ishlinskii operator in the Kurzweil integral setting is locally Lipschitz continuous also in the space of regulated functions. This paper shows that the Prandtl-Ishlinskii operator is still continuous if the primary response curve is only monotone and continuous, and that it may not even be locally Holder continuous for continuously differentiable primary response curves.