Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity
Revised: 25 October 2010
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 L2(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
Key words: Null-controllability / Schrödinger equation / complex Ginzburg-Landau equation / moment problem / biorthogonal / vanishing viscosity
© EDP Sciences, SMAI, 2010