Controllability of the Discrete-Spectrum Schrödinger Equation Driven by an External Field
Annales de l'I.H.P. Analyse non linéaire, Volume 26 (2009) no. 1, pp. 329-349.
@article{AIHPC_2009__26_1_329_0,
     author = {Chambrion, Thomas and Mason, Paolo and Sigalotti, Mario and Boscain, Ugo},
     title = {Controllability of the {Discrete-Spectrum} {Schr\"odinger} {Equation} {Driven} by an {External} {Field}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {329--349},
     publisher = {Elsevier},
     volume = {26},
     number = {1},
     year = {2009},
     doi = {10.1016/j.anihpc.2008.05.001},
     mrnumber = {2483824},
     zbl = {1161.35049},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2008.05.001/}
}
TY  - JOUR
AU  - Chambrion, Thomas
AU  - Mason, Paolo
AU  - Sigalotti, Mario
AU  - Boscain, Ugo
TI  - Controllability of the Discrete-Spectrum Schrödinger Equation Driven by an External Field
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2009
SP  - 329
EP  - 349
VL  - 26
IS  - 1
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2008.05.001/
DO  - 10.1016/j.anihpc.2008.05.001
LA  - en
ID  - AIHPC_2009__26_1_329_0
ER  - 
%0 Journal Article
%A Chambrion, Thomas
%A Mason, Paolo
%A Sigalotti, Mario
%A Boscain, Ugo
%T Controllability of the Discrete-Spectrum Schrödinger Equation Driven by an External Field
%J Annales de l'I.H.P. Analyse non linéaire
%D 2009
%P 329-349
%V 26
%N 1
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2008.05.001/
%R 10.1016/j.anihpc.2008.05.001
%G en
%F AIHPC_2009__26_1_329_0
Chambrion, Thomas; Mason, Paolo; Sigalotti, Mario; Boscain, Ugo. Controllability of the Discrete-Spectrum Schrödinger Equation Driven by an External Field. Annales de l'I.H.P. Analyse non linéaire, Volume 26 (2009) no. 1, pp. 329-349. doi : 10.1016/j.anihpc.2008.05.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2008.05.001/

[1] R. Adami, U. Boscain, Controllability of the Schrödinger equation via intersection of eigenvalues, in: Proceedings of the 44th IEEE Conference on Decision and Control, December 12-15, 2005, pp. 1080-1085.

[2] Agrachev A., Chambrion T., An Estimation of the Controllability Time for Single-Input Systems on Compact Lie Groups, ESAIM Control Optim. Calc. Var. 12 (3) (2006) 409-441. | Numdam | MR | Zbl

[3] Agrachev A., Kuksin S., Sarychev A., Shirikyan A., On Finite-Dimensional Projections of Distributions for Solutions of Randomly Forced 2D Navier-Stokes Equations, Ann. Inst. H. Poincaré Probab. Statist. 43 (4) (2007) 399-415. | Numdam | MR | Zbl

[4] Agrachev A. A., Sachkov Y. L., Control Theory From the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, vol. 87, Springer-Verlag, Berlin, 2004, Control Theory and Optimization, II. | MR | Zbl

[5] Agrachev A. A., Sarychev A. V., Controllability of 2D Euler and Navier-Stokes Equations by Degenerate Forcing, Commun. Math. Phys. 265 (3) (2006) 673-697. | MR | Zbl

[6] Albert J. H., Genericity of Simple Eigenvalues for Elliptic PDE's, Proc. Amer. Math. Soc. 48 (1975) 413-418. | MR | Zbl

[7] Albertini F., D'Alessandro D., Notions of Controllability for Bilinear Multilevel Quantum Systems, IEEE Trans. Automat. Control 48 (8) (2003) 1399-1403. | MR

[8] Altafini C., Controllability of Quantum Mechanical Systems by Root Space Decomposition of su N, J. Math. Phys. 43 (5) (2002) 2051-2062. | MR | Zbl

[9] Altafini C., Controllability Properties for Finite Dimensional Quantum Markovian Master Equations, J. Math. Phys. 44 (6) (2003) 2357-2372. | MR | Zbl

[10] Ball J. M., Marsden J. E., Slemrod M., Controllability for Distributed Bilinear Systems, SIAM J. Control Optim. 20 (4) (1982) 575-597. | MR | Zbl

[11] Baudouin L., Kavian O., Puel J.-P., Regularity for a Schrödinger Equation With Singular Potentials and Application to Bilinear Optimal Control, J. Differential Equations 216 (1) (2005) 188-222. | MR | Zbl

[12] Beauchard K., Local Controllability of a 1-D Schrödinger Equation, J. Math. Pures Appl. (9) 84 (7) (2005) 851-956. | MR | Zbl

[13] Beauchard K., Coron J.-M., Controllability of a Quantum Particle in a Moving Potential Well, J. Funct. Anal. 232 (2) (2006) 328-389. | MR | Zbl

[14] Borzì A., Decker E., Analysis of a Leap-Frog Pseudospectral Scheme for the Schrödinger Equation, J. Comput. Appl. Math. 193 (1) (2006) 65-88. | MR | Zbl

[15] Boscain U., Chambrion T., Charlot G., Nonisotropic 3-Level Quantum Systems: Complete Solutions for Minimum Time and Minimum Energy, Discrete Contin. Dyn. Syst. Ser. B 5 (4) (2005) 957-990, (electronic). | MR | Zbl

[16] Boscain U., Charlot G., Resonance of Minimizers for N-Level Quantum Systems With an Arbitrary Cost, ESAIM Control Optim. Calc. Var. 10 (4) (2004) 593-614, (electronic). | EuDML | Numdam | MR | Zbl

[17] Boscain U., Mason P., Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field, J. Math. Phys. 47 (6) (2006) 29, 062101. | MR | Zbl

[18] Coron J.-M., Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007. | MR | Zbl

[19] D'Alessandro D., Introduction to Quantum Control and Dynamics, Applied Mathematics and Nonlinear Science Series, Chapman, Hall/CRC, Boca Raton, FL, 2008. | MR | Zbl

[20] Davies E. B., Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. | MR | Zbl

[21] Henrot A., Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. | MR | Zbl

[22] Hübler P., Bargon J., Glaser S. J., Nuclear Magnetic Resonance Quantum Computing Exploiting the Pure Spin State of Para Hydrogen, J. Chem. Phys. 113 (6) (2000) 2056-2059.

[23] Ito K., Kunisch K., Optimal Bilinear Control of an Abstract Schrödinger Equation, SIAM J. Control Optim. 46 (1) (2007) 274-287, (electronic). | MR | Zbl

[24] Jurdjevic V., Sussmann H. J., Control Systems on Lie Groups, J. Differential Equations 12 (1972) 313-329. | MR | Zbl

[25] Kato T., Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag, New York, 1966. | MR | Zbl

[26] Khaneja N., Glaser S. J., Brockett R., Sub-Riemannian Geometry and Time Optimal Control of Three Spin Systems: Quantum Gates and Coherence Transfer, Phys. Rev. A 65 (3) (2002) 11, 032301. | MR

[27] M. Mirrahimi, Lyapunov control of a particle in a finite quantum potential well, in: Proceedings of the 45th IEEE Conference on Decision and Control, December 13-15, 2006.

[28] Mirrahimi M., Rouchon P., Controllability of Quantum Harmonic Oscillators, IEEE Trans. Automat. Control 49 (5) (2004) 745-747. | MR

[29] Peirce A., Dahleh M., Rabitz H., Optimal Control of Quantum Mechanical Systems: Existence, Numerical Approximations, and Applications, Phys. Rev. A 37 (1988) 4950-4964. | MR

[30] Pierfelice V., Strichartz Estimates for the Schrödinger and Heat Equations Perturbed With Singular and Time Dependent Potentials, Asymptotic Anal. 47 (1-2) (2006) 1-18. | MR | Zbl

[31] Rabitz H., De Vivie-Riedle H., Motzkus R., Kompa K., Wither the Future of Controlling Quantum Phenomena?, Science 288 (2000) 824-828.

[32] Reed M., Simon B., Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press (Harcourt Brace Jovanovich Publishers), New York, 1978. | MR | Zbl

[33] Rellich F., Perturbation Theory of Eigenvalue Problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon Breach Science Publishers, New York, 1969. | MR | Zbl

[34] Rodnianski I., Schlag W., Time Decay for Solutions of Schrödinger Equations With Rough and Time-Dependent Potentials, Invent. Math. 155 (3) (2004) 451-513. | MR | Zbl

[35] Rodrigues S. S., Navier-Stokes Equation on the Rectangle Controllability by Means of Low Mode Forcing, J. Dynam. Control Syst. 12 (4) (2006) 517-562. | MR | Zbl

[36] P. Rouchon, Control of a quantum particle in a moving potential well, in: Lagrangian and Hamiltonian Methods for Nonlinear Control 2003, IFAC, Laxenburg, 2003, pp. 287-290. | MR

[37] Sachkov Y. L., Controllability of Invariant Systems on Lie Groups and Homogeneous Spaces, Dynamical systems, 8, J. Math. Sci. (New York) 100 (4) (2000) 2355-2427. | MR | Zbl

[38] Shapiro M., Brumer P., Principles of the Quantum Control of Molecular Processes, Wiley-VCH, 2003, pp. 250.

[39] G. Tenenbaum, M. Tucsnak, K. Ramdani, T. Takahashi, A spectral approach for the exact observability of infinite dimensional systems with skew-adjoint generator, J. Funct. Anal., 2007. | MR | Zbl

[40] Turinici G., On the Controllability of Bilinear Quantum Systems, in: Defranceschi M., Le Bris C. (Eds.), Mathematical Models and Methods for Ab Initio Quantum Chemistry, Lecture Notes in Chemistry, vol. 74, Springer, 2000. | MR | Zbl

[41] Zuazua E., Remarks on the Controllability of the Schrödinger Equation, in: Quantum Control: Mathematical and Numerical Challenges, CRM Proc. Lecture Notes, vol. 33, Amer. Math. Soc., Providence, RI, 2003, pp. 193-211. | MR

Cited by Sources: