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hysteresis operators; Preisach operator; Ishlinskii operator
We prove that the classical Prandtl, Ishlinskii and Preisach hysteresis operators are continuous in Sobolev spaces $W^{1,p}(0,T)$ for $1\leq p < +\infty$, (localy) Lipschitz continuous in $W^{1,1}(0,T)$ and discontinuous in $W^{1,\infty}(0,T)$ for arbitrary $T>0$. Examples show that this result is optimal.
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