论文标题
通过域的缓慢变形控制Schrödinger方程
Control of the Schrödinger equation by slow deformations of the domain
论文作者
论文摘要
这项工作的目的是研究schrödinger方程的可控性\ begin {qore} \ label {eq_abstract} i \ partial_t u（t）= - δu（t）~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $ω（t）\ subset \ mathbb {r}^n $是一个时变域。我们通过绝热变形$ω（t）\ subset \ mathbb {r} $（in [0，t] $）证明了$ l^2（ω）$中\ eqref {eq_abstract}的全局近似可控性，以便$ω（0）=ω（0）=ω=ω=ω=ω$。该控制强烈基于[18]提供的\ eqref {eq_abstract}的哈密顿结构，该结构能够使用绝热运动。我们还讨论了我们在矩形域的特定框架中执行的几个明确的有趣控件。
The aim of this work is to study the controllability of the Schrödinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-Δu(t)~~~~~\text{ on }Ω(t) \tag{$\ast$} \end{equation} with Dirichlet boundary conditions, where $Ω(t)\subset\mathbb{R}^N$ is a time-varying domain. We prove the global approximate controllability of \eqref{eq_abstract} in $L^2(Ω)$, via an adiabatic deformation $Ω(t)\subset\mathbb{R}$ ($t\in[0,T]$) such that $Ω(0)=Ω(T)=Ω$. This control is strongly based on the Hamiltonian structure of \eqref{eq_abstract} provided by [18], which enables the use of adiabatic motions. We also discuss several explicit interesting controls that we perform in the specific framework of rectangular domains.
