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引用次数: 4
摘要
本文的目的是研究具有Dirichlet边界条件的Schrödinger方程\begin{equation}\label{eq_abstract} i\partial_t u(t)=-\Delta u(t)~~~~~\text{ on }\Omega(t) \tag{$\ast$} \end{equation}的可控性,其中$\Omega(t)\subset\mathbb{R}^N$为时变域。我们通过一个绝热变形$\Omega(t)\subset\mathbb{R}$ ($t\in[0,T]$)证明了$L^2(\Omega)$中\eqref{eq_abstract}的全局近似可控性,使得$\Omega(0)=\Omega(T)=\Omega$。这种控制强烈地基于[18]提供的\eqref{eq_abstract}哈密顿结构,这使得使用绝热运动成为可能。我们还讨论了我们在矩形域的特定框架中执行的几个显式的有趣控制。
Control of the Schrödinger equation by slow deformations of the domain
The aim of this work is to study the controllability of the Schr\"odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-\Delta u(t)~~~~~\text{ on }\Omega(t) \tag{$\ast$} \end{equation} with Dirichlet boundary conditions, where $\Omega(t)\subset\mathbb{R}^N$ is a time-varying domain. We prove the global approximate controllability of \eqref{eq_abstract} in $L^2(\Omega)$, via an adiabatic deformation $\Omega(t)\subset\mathbb{R}$ ($t\in[0,T]$) such that $\Omega(0)=\Omega(T)=\Omega$. 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.
期刊介绍:
The Nonlinear Analysis section of the Annales de l''Institut Henri Poincaré is an international journal created in 1983 which publishes original and high quality research articles. It concentrates on all domains concerned with nonlinear analysis, specially applicable to PDE, mechanics, physics, economy, without overlooking the numerical aspects.