| We consider an inverse Stefan type free boundary problem (ISP), where information on the boundary heat flux and the density of heat sources are missing and must be found along with the temperature and free boundary.;In this work, new variational formulation introduced in U.G. Abdulla, Inverse Problems and Imaging, 7, 2(2013), 307-340, is extended to a new setting and an optimal control problem is pursued where boundary heat flux, density of sources and the free boundary are components of the control vector, and the optimality criteria consist of the minimization of the L2-norm declinations of the temperature measurement at the final moment, phase transition temperature, and final position of the free boundary.;Existence of the optimal control and the Frechet differentiability of the cost functional in Besov-Hilbert-Sobolev spaces is proved under the minimal regularity assumptions on the data. The adjoint PDE problem is introduced and the explicit formula for the Frechet differential is derived. This result opens a way for the application of the iterative gradient type numerical methods of least computational cost in Hilbert spaces framework.;Discretization through finite differences is pursued and the convergence of the sequence of discrete optimal control problems to the continuous optimal control problem both with respect to functional and control is proved. The major tool is the proof of two energy estimates in discrete Sobolev spaces for the semi-discrete PDE problem. The new approach allows the tackling of situations in which the known density of sources is a measure given through the distributional derivative of an integrable function. |
