This paper is concerned with the optimal control of hysteresis-reaction-diffusion systems. We study a control problem with two sorts of controls, namely distributed control functions, or controls which act on a part of the boundary of the domain. The state equation is given by a reaction-diffusion system with the additional challenge that the reaction term includes a scalar stop operator. We choose a variational inequality to represent the hysteresis. In this paper, we prove first order necessary optimality conditions. In particular, under certain regularity assumptions, we derive results about the continuity properties of the adjoint system. For the case of distributed controls, we improve the optimality conditions and show uniqueness of the adjoint variables. We employ the optimality system to prove higher regularity of the optimal solutions of our problem. The specific feature of rate-independent hysteresis in the state equation leads to difficulties concerning the analysis of the solution operator. Non-locality in time of the Hadamard derivative of the control-to-state operator complicates the derivation of an adjoint system. This work is motivated by its academic challenge, as well as by its possible potential for applications such as in economic modeling.

Accepted:

DOI: 10.1051/cocv/2018025

Keywords: Optimal control, reaction-diffusion, semilinear parabolic evolution problem, hysteresis operator, stop operator, global existence, solution operator, Hadamard differentiability, optimality conditions, adjoint system

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@article{COCV_2018__24_4_1453_0, author = {M\"unch, Christian}, title = {Optimal control of reaction-diffusion systems with hysteresis}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1453--1488}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2018025}, zbl = {1414.49002}, mrnumber = {3922446}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2018025/} }

TY - JOUR AU - Münch, Christian TI - Optimal control of reaction-diffusion systems with hysteresis JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1453 EP - 1488 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2018025/ DO - 10.1051/cocv/2018025 LA - en ID - COCV_2018__24_4_1453_0 ER -

%0 Journal Article %A Münch, Christian %T Optimal control of reaction-diffusion systems with hysteresis %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1453-1488 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2018025/ %R 10.1051/cocv/2018025 %G en %F COCV_2018__24_4_1453_0

Münch, Christian. Optimal control of reaction-diffusion systems with hysteresis. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1453-1488. doi : 10.1051/cocv/2018025. http://archive.numdam.org/articles/10.1051/cocv/2018025/

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