A sufficient condition for partial ensemble controllability of bilinear schrödinger equations with bounded coupling terms

This note presents a sufficient condition for partial approximate ensemble controllability of a set of bilinear conservative systems in an infinite dimensional Hilbert space. The proof relies on classical geometric and averaging control techniques applied on finite dimensional approximation of the infinite dimensional system. The results are illustrated with the planar rotation of a linear molecule.

[1]  Jr-Shin Li,et al.  Ensemble Control of Bloch Equations , 2009, IEEE Transactions on Automatic Control.

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

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

[4]  Jerrold E. Marsden,et al.  Controllability for Distributed Bilinear Systems , 1982 .

[5]  Nabile Boussaid,et al.  Approximate controllability of the Schrödinger equation with a polarizability term , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[6]  Nabile Boussaid,et al.  Periodic control laws for bilinear quantum systems with discrete spectrum , 2012, 2012 American Control Conference (ACC).

[7]  Mazyar Mirrahimi,et al.  Lyapunov control of bilinear Schrödinger equations , 2005, Autom..

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

[9]  Pierre Rouchon,et al.  Controllability Issues for Continuous-Spectrum Systems and Ensemble Controllability of Bloch Equations , 2009, 0903.2720.

[10]  Mazyar Mirrahimi,et al.  Implicit Lyapunov control of finite dimensional Schrödinger equations , 2007, Syst. Control. Lett..

[11]  Anatoly Zlotnik,et al.  Iterative ensemble control synthesis for bilinear systems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

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

[13]  Morgan Morancey Explicit approximate controllability of the Schrödinger equation with a polarizability term , 2013, Math. Control. Signals Syst..

[14]  Mario Sigalotti,et al.  Controllability of the bilinear Schrödinger equation with several controls and application to a 3D molecule , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[15]  Pierre Rouchon,et al.  Adiabatic passage and ensemble control of quantum systems , 2010, 1012.5429.

[16]  Jr-Shin Li,et al.  Ensemble Controllability of the Bloch Equations , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[17]  Mario Sigalotti,et al.  A Weak Spectral Condition for the Controllability of the Bilinear Schrödinger Equation with Application to the Control of a Rotating Planar Molecule , 2011, ArXiv.

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

[19]  Mario Sigalotti,et al.  Controllability of the rotation of a quantum planar molecule , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[20]  Pierre Rouchon,et al.  Stabilization for an ensemble of half-spin systems , 2010, Autom..

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

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

[23]  Thomas Chambrion,et al.  Periodic excitations of bilinear quantum systems , 2011, Autom..

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

[25]  N. Khaneja,et al.  Control of inhomogeneous quantum ensembles , 2006 .

[26]  M. Mirrahimi Lyapunov control of a particle in a finite quantum potential well , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.