Mathematical Control and Related Fields (MCRF)

Internal control of the Schrödinger equation

Pages: 161 - 186, Volume 4, Issue 2, June 2014      doi:10.3934/mcrf.2014.4.161

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Camille Laurent - CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France (email)

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:  Controllability, linear Schrödinger equation, nonlinear Schrödinger equation.
Mathematics Subject Classification:  Primary: 93B05, 35Q41; Secondary: 35Q55.

Received: November 2012;      Revised: September 2013;      Available Online: February 2014.