  We consider the inverse Stefan type free boundary problem, where information
on the boundary heat flux and density of the sources are missing and must be
found along with the temperature and the free boundary. We pursue optimal
control framework where boundary heat flux, density of sources, and free
boundary are components of the control vector. The optimality criteria consists
of the minimization of the $L_2$-norm declinations of the temperature
measurements at the final moment, phase transition temperature, and final
position of the free boundary. 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.
