Article Contents

2012, Volume 5, Issue 1: 159-181. Doi: 10.3934/dcdss.2012.5.159

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Optimal control problem for Allen-Cahn type equation associated with total variation energy

1.
Division of Mathematical Sciences, Graduate School of Engineering, Gunma University, 4-2 Aramaki-cho, Maebashi, 371-8510
2.
Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522
3.
Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, 221-8686

Received: March 2009

Revised: December 2009

Published: February 2012

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Abstract

In this paper we study an optimal control problem for a singular diffusion equation associated with total variation energy. The singular diffusion equation is derived as an Allen-Cahn type equation, and then the observing optimal control problem corresponds to a temperature control problem in the solid-liquid phase transition. We show the existence of an optimal control for our singular diffusion equation by applying the abstract theory. Next we consider our optimal control problem from the view-point of numerical analysis. In fact we consider the approximating problem of our equation, and we show the relationship between the original control problem and its approximating one. Moreover we show the necessary condition of an approximating optimal pair, and give a numerical experiment of our approximating control problem.

Keywords:

Mathematics Subject Classification: Primary: 49J20, 35K55; Secondary: 35R35.

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