Weak differentiability of the control-to-state mapping in a parabolic equation with hysteresis

We consider the heat equation on a bounded domain subject to an inhomogeneous forcing in terms of a rate-independent (hysteresis) operator and a control variable. The aim of the paper is to establish a functional analytical setting which allows to prove weak differentiability properties of the control-to-state mapping. Using results of Brokate and Krejčí (DCDS 35(6):2405–2421, 2015) and Brokate (Newton and Bouligand derivatives of the scalar play and stop operator, arXiv:1607.07344, version 2, 2019) on the weak differentiability of scalar rate-independent operators, we prove Bouligand and Newton differentiability in suitable Bochner spaces of the control-to-state mapping in a parabolic problem.