Issue |
ESAIM: COCV
Volume 12, Number 4, October 2006
|
|
---|---|---|
Page(s) | 615 - 635 | |
DOI | https://doi.org/10.1051/cocv:2006014 | |
Published online | 11 October 2006 |
Limitations on the control of Schrödinger equations
1
Department of Mathematics and Statistics,
University of Victoria, PO Box 3045, Victoria, B.C.,
V8W 3P4 Canada; rillner@math.uvic.ca
2
Mathematisches Institut,
Universität Köln, Weyertal 86-90,
50931 Köln, Germany;
lange@mathematik.uni-koeln.de
3
Department of Mathematics and Statistics,
Acadia University, Wolfville, N.S., B4P 1R6 Canada; hteisman@acadiau.ca
Received:
30
September
2004
Revised:
25
March
2005
Revised:
3
May
2005
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 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.
Mathematics Subject Classification: 35Q40 / 35Q55 / 81Q99 / 93B05
Key words: Schrödinger equations / exact and approximate control / quantum control.
© EDP Sciences, SMAI, 2006
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