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Limitations on the control of Schrödinger equations

Published online by Cambridge University Press:  11 October 2006

Reinhard Illner ,
Horst Lange  and
Holger Teismann
Reinhard Illner
Affiliation:
Department of Mathematics and Statistics, University of Victoria, PO Box 3045, Victoria, B.C., V8W 3P4 Canada; rillner@math.uvic.ca
Horst Lange
Affiliation:
Mathematisches Institut, Universität Köln, Weyertal 86-90, 50931 Köln, Germany; lange@mathematik.uni-koeln.de
Holger Teismann
Affiliation:
Department of Mathematics and Statistics, Acadia University, Wolfville, N.S., B4P 1R6 Canada; hteisman@acadiau.ca
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Abstract

We give the definitions of exact and approximate controllability for linear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No-go” result for evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllability for the linear Schrödinger equation and distributed additive control, and we show that the Hartree equation of quantum chemistry with bilinear control $(E(t)\cdot x) u$ is not controllable in finite or infinite time. Finally, in Section 3, we give criteria for additive controllability of linear Schrödinger equations, and we give a distributed additive controllability result for the nonlinear Schrödinger equation if the data are small.

Keywords

Schrödinger equationsexact and approximate controlquantum control.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 4 , October 2006 , pp. 615 - 635
Copyright
© EDP Sciences, SMAI, 2006

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