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

  • * Corresponding author: Wei Liu

    * Corresponding author: Wei Liu 
This work was supported by the Program of High-end Foreign Experts of the SAFEA (No. GDW20163200216). The work of the second author was partially supported by the GACR GrantČ 15-12227S and RVO: 67985840.
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  • 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 Hölder continuous for continuously differentiable primary response curves.

    Mathematics Subject Classification: Primary:34C55;Secondary:58C07, 49J40.


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  • Figure 1.  The memory curves $\lambda(r)$ (the bold solid line) and $\hat \lambda(r)$ (the thin solid line)

    Figure 2.  The memory curves $\lambda(r)$ (the solid line) and $\hat \lambda(r)$ (the {dashed} line)

    Figure 3.  The primary response curve $\psi_1$ (the bold solid line), its derivative $\psi_1'$ (the bold dashed line), and the piecewise linear regularization $\psi_2'$ of $\psi_1'$ (the thin solid line)

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