Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity

Facultatea de Matematica si Informatica, Universitatea din Craiova, 200585 Craiova, Romania
sd_micu@yahoo.com, roventaionel@yahoo.com

Received: 14 July 2010
Revised: 25 October 2010

Abstract

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 v_ε 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 (v_ε)_ε in L²(0, T) acting on the extremity x = π. Any weak limit of this family is a control for the Schrödinger equation.