论文标题
在Dirichlet边界条件的情况下，1D非线性Schrödinger方程的局部精确可控性
Local exact controllability of the 1D nonlinear Schrödinger equation in the case of Dirichlet boundary conditions
论文作者
论文摘要
我们考虑具有双线性对照的1D非线性Schrödinger方程。在Neumann边界条件的情况下，Beauchard和Laurent在Arxiv中证明了该方程在基态附近的局部确切可控性：1001.3288。在本文中，我们研究了Dirichlet边界条件的情况。为了建立线性化方程的可控性，我们使用双线性控制通过四个方向作用：三个傅立叶模式和一个通用方向。适当地选择了傅立叶模式，以便满足饱和属性。这些模式允许控制大致线性化的schrödinger方程。我们表明，线性化方程的可触及设置已关闭。这是通过将解析运算符表示为两个线性连续映射的总和来实现的：一个是冲销的（此处使用了通用方向的控制），另一个是紧凑的。具有密集和闭合图像的映射是汇总的，因此线性化的schrödinger方程是完全可控制的。然后，使用逆映射定理得出了非线性方程的局部精确可控性。
We consider the 1D nonlinear Schrödinger equation with bilinear control. In the case of Neumann boundary conditions, local exact controllability of this equation near the ground state has been proved by Beauchard and Laurent in arXiv:1001.3288. In this paper, we study the case of Dirichlet boundary conditions. To establish the controllability of the linearised equation, we use a bilinear control acting through four directions: three Fourier modes and one generic direction. The Fourier modes are appropriately chosen so that they satisfy a saturation property. These modes allow to control approximately the linearised Schrödinger equation. We show that the reachable set for the linearised equation is closed. This is achieved by representing the resolving operator as a sum of two linear continuous mappings: one is surjective (here the control in generic direction is used) and the other is compact. A mapping with dense and closed image is surjective, so the linearised Schrödinger equation is exactly controllable. Then local exact controllability of the nonlinear equation is derived using the inverse mapping theorem.
