Local controllability of 1D Schrödinger equations with bilinear control and minimal time

Karine Beauchard; Morgan Morancey

doi:10.3934/mcrf.2014.4.125

Article Contents

2014, Volume 4, Issue 2: 125-160. Doi: 10.3934/mcrf.2014.4.125

This issue Previous Article Next Article Internal control of the Schrödinger equation

Local controllability of 1D Schrödinger equations with bilinear control and minimal time

Karine Beauchard^1, and
Morgan Morancey^1,

1.
CMLS, Ecole Polytechnique, 91 128 Palaiseau cedex

Received: June 30, 2012

Revised: December 31, 2012

Published: May 2014

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Abstract

We consider a linear Schrödinger equation, on a bounded interval, with bilinear control.
In [10], Beauchard and Laurent prove that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. In [18], Coron proves that a positive minimal time is required for this controllability result, on a particular degenerate example.
In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution.

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Mathematics Subject Classification: 93B05, 93C20, 81Q93.

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References

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