Internal control of the Schrödinger equation

Camille Laurent

doi:10.3934/mcrf.2014.4.161

Article Contents

2014, Volume 4, Issue 2: 161-186. Doi: 10.3934/mcrf.2014.4.161

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Internal control of the Schrödinger equation

Camille Laurent^1,

1.
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris

Received: November 2012

Revised: September 2013

Published: June 2014

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Abstract

In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schrödinger equations.
After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension $1$ using some propagation results. We then discuss how to obtain some similar results on a compact manifold where the zone of control satisfies the Geometric Control Condition. We also discuss some known results and open questions when this condition is not satisfied. Then, we present the links between the controllability and some resolvent estimates. Finally, we discuss the additional difficulties when we consider the nonlinear Schrödinger equation.

Keywords:

Mathematics Subject Classification: Primary: 93B05, 35Q41; Secondary: 35Q55.

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