Abstract
This paper addresses the exact controllability problem of the linear one-dimensional Schrödinger equation perturbed by a vanishing viscosity term depending on a strictly positive parameter. It is shown that, for any time and for each initial datum in a suitable space, there exists a uniformly bounded family of boundary controls. Any weak limit of this family is a control for the linear Schrödinger equation.
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Acknowledgements
The first author was supported by strategic grant POSDRU/CPP107/ DMI1.5/S/78421, Project ID 78421 (2010), co-financed by the European Social Fund—Investing in People, within the Sectorial Operational Programme Human Resources Development 2007-2013. The second author was supported by Grant PN-II-ID-PCE-2011-3-0257 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI. The authors are grateful to Sorin Micu for several interesting suggestions and comments. We also thank the referees for the constructive and helpful comments and suggestions on the manuscript.
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Communicated by Enrique Zuazua.
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Bugariu, I.F., Rovenţa, I. Small Time Uniform Controllability of the Linear One-Dimensional Schrödinger Equation with Vanishing Viscosity. J Optim Theory Appl 160, 949–965 (2014). https://doi.org/10.1007/s10957-013-0387-4
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DOI: https://doi.org/10.1007/s10957-013-0387-4