Local controllability of 1D Schrödinger equations with bilinear control and minimal time
Pages: 125 - 160,
Karine Beauchard - CMLS, Ecole Polytechnique, 91 128 Palaiseau cedex, France (email)
Morgan Morancey - CMLS, Ecole Polytechnique, 91 128 Palaiseau cedex, France (email)
We consider a linear Schrödinger equation, on a bounded interval, with bilinear control.
In , Beauchard and Laurent prove that, under an appropriate non degeneracy assumption,
this system is controllable, locally around the ground state, in arbitrary time.
In , 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;
Available Online: February 2014.