具有非局部项的Schroedinger方程的能控性
Controllability of Schroedinger equation with a nonlocal term
摘要：
本文研究量子半导体模型中一维薛定谔方程$i\,u_t(x,t)=-u_{xx}+α(x)\,u+m(u)\,u,$的内部分布控制问题。这里$m(u)$是一个非局部Hartree型非线性源于与一维泊松方程的耦合，和$α(x)$是一个经常性的功能与线性增长的无穷大，包括恒定的电场。通过Hilbert唯一性方法和Schauder不动点定理，证明了对于初始状态和目标状态属于原点的一个适当的小邻域，以及对于分布控制支持在一个固定的紧区间之外，模型方程是可控的。此外，它表明，分布控制与紧支持，精确可控性问题是不可能的。
英文摘要：
This paper is concerned with the internal distributed control problem for the 1D Schroedinger equation, $i\,u_t(x,t)=-u_{xx}+α(x)\,u+m(u)\,u,$ that arises in quantum semiconductor models. Here $m(u)$ is a non local Hartree--type nonlinearity stemming from the coupling with the 1D Poisson equation, and $α(x)$ is a regular function with linear growth at infinity, including constant electric fields. By means of both the Hilbert Uniqueness Method and the Schauder's fixed point theorem it is shown that for initial and target states belonging to a suitable small neighborhood of the origin, and for distributed controls supported outside of a fixed compact interval, the model equation is controllable. Moreover, it is shown that, for distributed controls with compact support, the exact controllability problem is not possible.
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