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

  • Martin BrokateEmail author
  • Klemens Fellner
  • Matthias Lang-Batsching


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.


Heat equation Rate independence Hysteresis operator Optimal control Weak differentiability 

Mathematics Subject Classification

47J40 35K10 34K35 



The first author thanks Pavel Gurevich for some helpful suggestions and the university of Graz for several opportunities to visit. The second author acknowledges helpful discussions with Joachim Rehberg and the kind hospitality of the Technical University of Munich. The third author has been supported by the International Research Training Group IGDK 1754 “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures”, funded by the German Research Council (DFG) and the Austrian Science Fund (FWF).


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Center for Mathematical SciencesTechnical University MunichGarching b. MunichGermany
  2. 2.Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria

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