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 $L_2-$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.

35 pages

References in corpus (6)

• On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. I.Well-posedness and Convergence of the Method of Lines
• On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. II.Convergence of the Method of Finite Differences
• Optimal Control of the Multiphase Stefan Problem
• Gradient Method in Hilbert-Besov Spaces for the Optimal Control of Parabolic Free Boundary Problems
• Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems
• Optimal control of multiphase free boundary problems for nonlinear parabolic equations

Cited by in corpus (1)

• Discretization and Convergence of the EIT Optimal Control Problem in 2D and 3D Domains