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Optimal Stefan problem
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-03-04 , DOI: 10.1007/s00526-020-1712-z
Ugur G. Abdulla , Bruno Poggi

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 difference 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.



中文翻译:

最优斯蒂芬问题

摘要

我们考虑有界Lipschitz域上具有齐次Dirichlet边界条件的反多相Stefan问题,在该区域中,除了温度和相变边界外,热源的密度未知。在最优控制框架中采用变分公式,其中热源的密度是一个控制参数,而最优性的标准是\(L_2 \)的最小值-在最后一次对整个域进行温度测量时,对Stefan问题的解的迹线的标准差。状态向量以弱公式解决了多相Stefan问题,这与具有不连续系数的拟线性抛物线PDE的Dirichlet问题等效。最优控制问题使用有限差分法完全离散化。我们证明了最优控制的存在以及离散最优控制问题与原始问题在成本函数和控制方面的收敛性。特别地,证明了多维多相Stefan问题的弱解的有限差分方法的收敛性。证明是基于实现统一的\(L _ {\ infty} \)约束和离散多相Stefan问题的(W_2 ^ {1,1} \)能量估计。

更新日期:2020-03-20
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