A Weak Spectral Condition for the Controllability of the Bilinear Schrödinger Equation with Application to the Control of a Rotating Planar Molecule

In this paper we prove an approximate controllability result for the bilinear Schrödinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrödinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the L1 norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.

[1]  Karine Beauchard,et al.  Semi-global weak stabilization of bilinear Schrödinger equations , 2010 .

[2]  Kazufumi Ito,et al.  Optimal Bilinear Control of an Abstract Schrödinger Equation , 2007, SIAM J. Control. Optim..

[3]  Mazyar Mirrahimi,et al.  Practical Stabilization of a Quantum Particle in a One-Dimensional Infinite Square Potential Well , 2009, SIAM J. Control. Optim..

[4]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[5]  P. H. Müller,et al.  T. Kato, Perturbation theory for linear operators. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 132) XX + 592 S. m. 3 Fig. Berlin/Heidelberg/New York Springer-Verlag. Preis geb. DM 79,20 , 1967 .

[6]  T. Seideman,et al.  Nonadiabatic Alignment by Intense Pulses. Concepts, Theory, and Directions , 2005 .

[7]  Mario Sigalotti,et al.  Controllability of the discrete-spectrum Schrödinger equation driven by an external field , 2008, 0801.4893.

[8]  M. Slemrod,et al.  Controllability of distributed bilinear systems , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[9]  Andrei A. Agrachev,et al.  An estimation of the controllability time for single-input systems on compact Lie Groups , 2006 .

[10]  Law,et al.  Arbitrary control of a quantum electromagnetic field. , 1996, Physical review letters.

[11]  Mario Sigalotti,et al.  Erratum of “The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent” , 2010 .

[12]  Karine Beauchard,et al.  Local controllability of 1D linear and nonlinear Schr , 2010, 1001.3288.

[13]  Karine Beauchard,et al.  Controllability of a quantum particle in a moving potential well , 2006 .

[14]  Hayk Nersisyan,et al.  Global exact controllability in infinite time of Schrödinger equation: multidimensional case , 2012, 1201.3445.

[15]  Michael Spanner,et al.  Coherent control of rotational wave-packet dynamics via fractional revivals. , 2004, Physical review letters.

[16]  Gabriel Turinici,et al.  On the controllability of bilinear quantum systems , 2000 .

[17]  Vahagn Nersesyan,et al.  Global approximate controllability for Schr\"odinger equation in higher Sobolev norms and applications , 2009, 0905.2438.

[18]  Yu. L. Sachkov,et al.  Controllability of invariant systems on lie groups and homogeneous spaces , 2000 .

[19]  D. D’Alessandro.  Introduction to Quantum Control and Dynamics , 2007 .

[20]  Bronis law Jakubczyk.  Introduction to Geometric Nonlinear Control ; Controllability and Lie Bracket , 2007 .

[21]  A. Agrachev,et al.  Control Theory from the Geometric Viewpoint , 2004 .

[22]  Mazyar Mirrahimi,et al.  Lyapunov control of a quantum particle in a decaying potential , 2008, 0805.0910.

[23]  Karine Beauchard,et al.  Local controllability of a 1-D Schrödinger equation , 2005 .

[24]  M. Zelikin,et al.  Control theory and optimization I , 1999 .

[25]  Thomas Chambrion,et al.  Locomotion and Control of a Self-Propelled Shape-Changing Body in a Fluid , 2009, J. Nonlinear Sci..

[26]  V. Nersesyan.  Growth of Sobolev Norms and Controllability of the Schrödinger Equation , 2008, 0804.3982.

[27]  Roger W. Brockett,et al.  Finite Controllability of Infinite-Dimensional Quantum Systems , 2010, IEEE Transactions on Automatic Control.

[28]  Mario Sigalotti,et al.  Simultaneous approximate tracking of density matrices for a system of Schrödinger equations , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[29]  Mario Sigalotti,et al.  Generic controllability properties for the bilinear Schrödinger equation , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[30]  Henrik Stapelfeldt,et al.  Colloquium: Aligning molecules with strong laser pulses , 2003 .

[31]  Sylvain Ervedoza,et al.  Approximate Controllability for a System of Schrödinger Equations Modeling a Single Trapped Ion , 2009 .

[32]  Mario Sigalotti,et al.  The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent , 2008 .