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

2019/05/06 by Martin Brokate, Brokate, Martin, Klemens Fellner +3 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Stability and Controllability of Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1905.01863

Abstract

We consider the heat equation on a bounded domain subject to an inhomogeneous\nforcing in terms of a rate-independent (hysteresis) operator and a control\nvariable. The aim of the paper is to establish a functional analytical setting\nwhich allows to prove weak differentiability properties of the control-to-state\nmapping. Using results of [BK] and [B] on the weak differentiability of scalar\nrate-independent operators, we prove Bouligand and Newton differentiability in\nsuitable Bochner spaces of the control-to-state mapping in a parabolic problem.\n

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• Parabolic hysteresis problems revisited: Finite element error analysis and convergent Newton-type solvers

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