Mathematical Control and Related Fields (MCRF)

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

Pages: 125 - 160, Volume 4, Issue 2, June 2014      doi:10.3934/mcrf.2014.4.125

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Karine Beauchard - CMLS, Ecole Polytechnique, 91 128 Palaiseau cedex, France (email)
Morgan Morancey - CMLS, Ecole Polytechnique, 91 128 Palaiseau cedex, France (email)

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.

Keywords:  Exact controllability, Schrödinger equation, bilinear control, minimal time, power series expansion.
Mathematics Subject Classification:  93B05, 93C20, 81Q93.

Received: July 2012;      Revised: January 2013;      Available Online: February 2014.