Volume 24, Number 4, October–December 2018
|Page(s)||1453 - 1488|
|Published online||18 October 2018|
Optimal control of reaction-diffusion systems with hysteresis★
Department of Mathematics – M6, Technical University of Munich,
* Corresponding author: email@example.com
Accepted: 6 April 2018
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.
Mathematics Subject Classification: 49J20 / 47J40 / 35K51
Key words: Optimal control / reaction-diffusion / semilinear parabolic evolution problem / hysteresis operator / stop operator / global existence / solution operator / Hadamard differentiability / optimality conditions / adjoint system
The author is supported by the DFG through the International Research Training Group IGDK 1754 “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures”. The author would like to thank Prof. Brokate from the Technical University of Munich and Prof. Fellner from the Karl-Franzens University of Graz for thoroughly proofreading the manuscript.
© EDP Sciences, SMAI 2018
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