Local Exact Controllability and Stabilizability of the Nonlinear Schrödinger Equation on a Bounded Interval

Abstract

This paper studies the exact controllability and the stabilization of the cubic Schrödinger equation posed on a bounded interval. Both internal and boundary controls are considered, and the results are given first in a periodic setting, and next with Dirichlet (resp., Neumann) boundary conditions. It is shown that the systems with either an internal control or a boundary control are locally exactly controllable in the classical Sobolev space $H^s$ for any $s\geq0$. It is also shown that the systems with an internal stabilization are locally exponentially stabilizable in $H^s$ for any $s\geq0$.