论文标题
一阶段Stefan问题的轨迹的确切可控性
Exact controllability to the trajectories of the one-phase Stefan problem
论文作者
论文摘要
本文介绍了一个空间维度的一个轨迹的轨迹的确切可控性。这是一个自由边缘的问题，可以建模固化和熔化过程。假定物理结构域是由状态在左侧为液体的介质填充的，右侧是固体，右温度为恒定。在两者之间，我们找到了一个自由晶体（将液体与固体分开的界面）。在液体结构域中，必须通过温度来满足以初始条件和边界条件完成的抛物线方程。在接口上，提出了一个额外的自由界要求，称为{\ it Stefan条件，}。我们证明了（平滑）轨迹的局部确切可控性。为此，我们首先将问题重新制定为具有分布式控件的耦合PDE-ODE系统的局部无效可控性。然后，介绍了线性化PDE系统的伴随的新卡尔曼不等式，并通过空间和内存项中的非局部在边界上进行了耦合。这导致适当的线性系统的无效可控性。最后，通过使用{\ it Liusternik-Graves的定理}获得局部结果。作为我们方法的副产品，我们发现某些包含在边界上定位的内存项的抛物线方程是可控制的。
This paper deals with the exact controllability to the trajectories of the one--phase Stefan problem in one spatial dimension. This is a free-boundary problem that models solidification and melting processes. It is assumed that the physical domain is filled by a medium whose state is liquid on the left and solid, with constant temperature, on the right. In between we find a free-boundary (the interface that separates the liquid from the solid). In the liquid domain, a parabolic equation completed with initial and boundary conditions must be satisfied by the temperature. On the interface, an additional free-boundary requirement, called the {\it Stefan condition,} is imposed. We prove the local exact controllability to the (smooth) trajectories. To this purpose, we first reformulate the problem as the local null controllability of a coupled PDE-ODE system with distributed controls. Then, a new Carleman inequality for the adjoint of the linearized PDE-ODE system, coupled on the boundary through nonlocal in space and memory terms, is presented. This leads to the null controllability of an appropriate linear system. Finally, a local result is obtained via local inversion, by using {\it Liusternik-Graves' Theorem}. As a byproduct of our approach, we find that some parabolic equations which contains memory terms localized on the boundary are null-controllable.
