Abstract
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.
Original language | English |
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Article number | 46 |
Journal | Nonlinear Differential Equations and Applications |
Volume | 26 |
Issue number | 6 |
DOIs | |
State | Published - 1 Dec 2019 |
Keywords
- Heat equation
- Hysteresis operator
- Optimal control
- Rate independence
- Weak differentiability