Optimal Stefan problem

Abstract

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 L2−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∞ bound and W 1,1 2 energy estimate for the discrete multiphase Stefan problem.

Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations

“…with bounded measurable coefficients a, b, c and (1. 7) a(x, t) ≥ a 0 > 0, a.e. (x, t) ∈ D = {0 < x < ℓ, 0 < t < T }.…”

“…In [5] existence of the optimal control and convergence of the sequence of discretized optimal control problems via method of finite differences is proved. In [7], this framework is extended to the multidimensional IMSP.…”