Small Time Uniform Controllability of the Linear One-Dimensional Schrödinger Equation with Vanishing Viscosity

This article considers the linear 1-d Schrödinger equation in (0, π) perturbed by a vanishing viscosity term depending on a small parameter ε > 0. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls vε as ε goes to zero. It is shown that, for any time T sufficiently large but independent of ε and for each initial datum in H−1(0, π), there exists a uniformly bounded family of controls (vε)ε in L (0, T ) acting on the extremity x = π. Any weak limit of this family is a control for the Schrödinger equation. Mathematics Subject Classification. 93B05, 30E05, 35Q41. Received July 14, 2010. Revised October 25, 2010. Published online January 19, 2011.