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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 26, Issue 2

Issues

Frechet differentiability in Besov spaces in the optimal control of parabolic free boundary problems

Ugur G. Abdulla
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  • Mathematical Sciences, Florida Institute of Technology, 150 W. University Blvd., Melbourne, FL 32901, USA
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/ Jonathan M. GoldfarbORCID iD: http://orcid.org/0000-0002-6716-3336
Published Online: 2017-09-20 | DOI: https://doi.org/10.1515/jiip-2017-0014

Abstract

We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and the density of the sources are missing and must be found along with the temperature and the free boundary. We pursue the optimal control framework analyzed in [1, 2], where the boundary heat flux, the density of the sources, and the free boundary are components of the control vector. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov spaces for the numerical solution of the inverse Stefan problem.

Keywords: Inverse Stefan problem; optimal control of parabolic PDE; parabolic free boundary problem; Frechet differentiability; Besov spaces; embedding theorems; trace embeddings

MSC 2010: 35R30; 35R35; 35K20

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About the article

Received: 2017-01-26

Revised: 2017-05-22

Accepted: 2017-08-11

Published Online: 2017-09-20

Published in Print: 2018-04-01


Funding Source: National Science Foundation

Award identifier / Grant number: 1359074

This research was funded by National Science Foundation (grant no. 1359074), REU Site: Partial Differential Equations and Dynamical Systems at Florida Institute of Technology.


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 26, Issue 2, Pages 211–227, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2017-0014.

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