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

Ugur G. Abdulla EMAIL logo and Jonathan M. Goldfarb ORCID logo

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.

MSC 2010: 35R30; 35R35; 35K20

Award Identifier / Grant number: 1359074

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

Acknowledgements

Two REU students, Jessica Pillow and Dylanger Pittman, worked on part of the project restricted to heuristic derivation of the Frechet gradient with the intention to implement the gradient type method for numerical analysis.

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Received: 2017-1-26
Revised: 2017-5-22
Accepted: 2017-8-11
Published Online: 2017-9-20
Published in Print: 2018-4-1

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