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

## TL;DR

This paper investigates the weak differentiability of the control-to-state mapping in a heat equation influenced by hysteresis, establishing a functional analytical framework and proving Bouligand and Newton differentiability.

## Contribution

It introduces a new functional analytical setting for analyzing weak differentiability of control-to-state mappings in parabolic equations with hysteresis.

## Key findings

- Proves Bouligand differentiability of the control-to-state map.
- Establishes Newton differentiability in Bochner spaces.
- Provides a framework for weak differentiability in hysteresis-influenced PDEs.

## 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 [BK] and [B] 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.