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

Abstract

AbstractWe 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.