One proves that the moving interface of a two-phase Stefan problem on $\Omega \subset {\mathbb{R}}^d$, $d=1,2,3,$ is controllable at the end time $T$ by a Neumann boundary controller $u$. The phase-transition region is a mushy region $\{{\sigma }_t^u;0\le t\le T\}$ of a modified Stefan problem and the main result amounts to saying that, for each Lebesgue measurable set ${\Omega }^{\ast }$ with positive measure, there is $u\in L^2((0,T)\times \partial \Omega )$ such that ${\Omega }^{\ast }\subset {\sigma }_T^u.$ To this aim, one uses an optimal control approach combined with Carleman’s inequality and the Kakutani fixed point theorem.

中文翻译：

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两相斯特凡问题的相变区的边界可控性

一个证明了两阶段Stefan问题的运动接口 $\Omega \subset {\mathbb{[R}}^d$， $d=\mathrm{1个}，\mathrm{2个}，3，$ 在结束时间是可控的 $Ť$ 由Neumann边界控制器 $ü$。相变区域是糊状区域$\{{\sigma }_{Ť}^{ü};0\le Ť\le Ť\}$ 修改后的Stefan问题的结论，主要结果是说，对于每个Lebesgue可测集 ${\Omega }^{\ast }$ 用积极的措施，有 $ü\in {\mathrm{大号}}^{\mathrm{2个}}（（0，Ť）\times \partial \Omega ）$ 这样 ${\Omega }^{\ast }\subset {\sigma }_{Ť}^{ü}。$ 为此，人们采用了一种最优控制方法，并结合了Carleman不等式和Kakutani不动点定理。