Introduction to Part I Constrained Optimization, Identification and Control

Abstract

In the article Optimal Control of Allen-Cahn Systems the authors Luise Blank, M. Hassan Farshbaf-Shaker, Claudia Hecht, Josef Michl and Christoph Rupprecht report results of their project regarding a control problem for a multi-component Allen-Cahn equation that incorporates elastic effects. This gives rise to a coupled elliptic-parabolic system. Distributed control of the Allen-Cahn equation and Neumann boundary for the stress tensor on a part of the boundary are considered. Existence results and optimality conditions are given for multi-component systems and smooth potential without distributed control. Concerning obstacle potentials, first-order conditions for the limiting systems of approximating problems are given for the boundary control using a penalization approach and for distributed control by a relaxation technique. Finally, there are some numerical results obtained with Trust-Region-Newton-Steihaug-CG method.