  We consider the inverse multiphase Stefan problem with homogeneous Dirichlet
boundary condition on a bounded Lipschitz domain, where the density of the heat
source is unknown in addition to the temperature and the phase transition
boundaries. The variational formulation is pursued in the optimal control
framework, where the density of the heat source is a control parameter, and the
criteria for optimality is the minimization of the $L_2-$norm declination of
the trace of the solution to the Stefan problem from a temperature measurement
on the whole domain at the final time. The state vector solves the multiphase
Stefan problem in a weak formulation, which is equivalent to Dirichlet problem
for the quasilinear parabolic PDE with discontinuous coefficient. The optimal
control problem is fully discretized using the method of finite differences. We
prove the existence of the optimal control and the convergence of the discrete
optimal control problems to the original problem both with respect to cost
functional and control. In particular, the convergence of the method of finite
differences for the weak solution of the multidimensional multiphase Stefan
problem is proved. The proofs are based on achieving a uniform $L_{\infty}$
bound and $W_2^{1,1}$ energy estimate for the discrete multiphase Stefan
problem.
