Limitations on the control of Schrödinger equations

Reinhard Illner; Horst Lange; Holger Teismann

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 4, page 615-635
  • ISSN: 1292-8119

Abstract

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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 ) · 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.

How to cite

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Illner, Reinhard, Lange, Horst, and Teismann, Holger. "Limitations on the control of Schrödinger equations." ESAIM: Control, Optimisation and Calculus of Variations 12.4 (2006): 615-635. <http://eudml.org/doc/249670>.

@article{Illner2006,
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. },
author = {Illner, Reinhard, Lange, Horst, Teismann, Holger},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Schrödinger equations; exact and approximate control; quantum control.; quantum control},
language = {eng},
month = {10},
number = {4},
pages = {615-635},
publisher = {EDP Sciences},
title = {Limitations on the control of Schrödinger equations},
url = {http://eudml.org/doc/249670},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Illner, Reinhard
AU - Lange, Horst
AU - Teismann, Holger
TI - Limitations on the control of Schrödinger equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/10//
PB - EDP Sciences
VL - 12
IS - 4
SP - 615
EP - 635
AB - 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.
LA - eng
KW - Schrödinger equations; exact and approximate control; quantum control.; quantum control
UR - http://eudml.org/doc/249670
ER -

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