Geodesic
Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 277-293

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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.

DOI : 10.1051/cocv/2010055
Classification : 93B05, 30E05, 35Q41
Keywords: null-controllability, Schrödinger equation, complex Ginzburg-Landau equation, moment problem, biorthogonal, vanishing viscosity 
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     author = {Micu, Sorin and Roven\c{t}a, Ionel},
     title = {Uniform controllability of the linear one dimensional {Schr\"odinger} equation with vanishing viscosity},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {277--293},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {1},
     year = {2012},
     doi = {10.1051/cocv/2010055},
     mrnumber = {2887936},
     zbl = {1242.93019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010055/}
}
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Micu, Sorin; Rovenţa, Ionel. Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 277-293. doi: 10.1051/cocv/2010055

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