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

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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 L 2(0, T) acting on the extremity x = π. Any weak limit of this family is a control for the Schrödinger equation.

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