Controllability of the discrete-spectrum Schrödinger equation driven by an external field

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 26, Issue 1, 2009, Pages 329-349

  Chambrion, Thomas ^a   Mason, Paolo ^a,b   Sigalotti, Mario ^a   Boscain, Ugo ^c  

^a UNIVERSITÉ HENRI POINCARÉ NANCY 1   (France)

^b CNR   (Italy)

^c UNIVERSITÉ DE BOURGOGNE   (France)

Author keywords

Approximate controllability; Bilinear Schr dinger equation; Control of PDE; Density matrix; Galerkin approximation; Quantum control

Indexed keywords

AIRCRAFT ENGINES; CONTROL THEORY; GALERKIN METHODS; OSCILLATORS (ELECTRONIC); PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIAL APPROXIMATION; QUANTUM THEORY;

APPROXIMATE CONTROLLABILITY; CONTROL POTENTIALS; DENSITY MATRIX; EXTERNAL FIELDS; GALERKIN APPROXIMATION; HARMONIC OSCILLATORS; QUANTUM CONTROL; RESONANT SPECTRUMS; UNBOUNDED DOMAINS;

CONTROLLABILITY;

EID: 58249137155     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2008.05.001     Document Type: Article

References (41)

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., and 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
3. Agrachev A., Kuksin S., Sarychev A., and 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
4. Agrachev A.A., and Sachkov Y.L. Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences vol. 87 (2004), Springer-Verlag, Berlin Control Theory and Optimization, II
5. Agrachev A.A., and Sarychev A.V. Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing. Commun. Math. Phys. 265 3 (2006) 673-697
6. Albert J.H. Genericity of simple eigenvalues for elliptic PDE's. Proc. Amer. Math. Soc. 48 (1975) 413-418
7. Albertini F., and D'Alessandro D. Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Automat. Control 48 8 (2003) 1399-1403
8. Altafini C. Controllability of quantum mechanical systems by root space decomposition of su (N). J. Math. Phys. 43 5 (2002) 2051-2062
9. Altafini C. Controllability properties for finite dimensional quantum Markovian master equations. J. Math. Phys. 44 6 (2003) 2357-2372
10. Ball J.M., Marsden J.E., and Slemrod M. Controllability for distributed bilinear systems. SIAM J. Control Optim. 20 4 (1982) 575-597
11. Baudouin L., Kavian O., and 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
12. Beauchard K. Local controllability of a 1-D Schrödinger equation. J. Math. Pures Appl. (9) 84 7 (2005) 851-956
13. Beauchard K., and Coron J.-M. Controllability of a quantum particle in a moving potential well. J. Funct. Anal. 232 2 (2006) 328-389
14. Borzì A., and Decker E. Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation. J. Comput. Appl. Math. 193 1 (2006) 65-88
15. Boscain U., Chambrion T., and 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)
16. Boscain U., and 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)
17. Boscain U., and Mason P. Time minimal trajectories for a spin 1/2 particle in a magnetic field. J. Math. Phys. 47 6 (2006) 29 062101
18. Coron J.-M. Control and Nonlinearity. Mathematical Surveys and Monographs vol. 136 (2007), American Mathematical Society, Providence, RI
19. D'Alessandro D. Introduction to Quantum Control and Dynamics. Applied Mathematics and Nonlinear Science Series (2008), Chapman, Hall/CRC, Boca Raton, FL
20. Davies E.B. Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics vol. 42 (1995), Cambridge University Press, Cambridge
21. Henrot A. Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics (2006), Birkhäuser Verlag, Basel
22. Hübler P., Bargon J., and 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., and Kunisch K. Optimal bilinear control of an abstract Schrödinger equation. SIAM J. Control Optim. 46 1 (2007) 274-287 (electronic)
24. Jurdjevic V., and Sussmann H.J. Control systems on Lie groups. J. Differential Equations 12 (1972) 313-329
25. Kato T. Perturbation Theory for Linear Operators. Die Grundlehren der mathematischen Wissenschaften Band 132 (1966), Springer-Verlag, New York
26. Khaneja N., Glaser S.J., and 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
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., and Rouchon P. Controllability of quantum harmonic oscillators. IEEE Trans. Automat. Control 49 5 (2004) 745-747
29. Peirce A., Dahleh M., and Rabitz H. Optimal control of quantum mechanical systems: Existence, numerical approximations, and applications. Phys. Rev. A 37 (1988) 4950-4964
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
31. Rabitz H., de Vivie-Riedle H., Motzkus R., and Kompa K. Wither the future of controlling quantum phenomena?. Science 288 (2000) 824-828
32. Reed M., and Simon B. Methods of Modern Mathematical Physics. IV. Analysis of Operators (1978), Academic Press (Harcourt Brace Jovanovich Publishers), New York
33. Rellich F. Perturbation Theory of Eigenvalue Problems. Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz (1969), Gordon Breach Science Publishers, New York
34. Rodnianski I., and Schlag W. Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155 3 (2004) 451-513
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
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
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
38. Shapiro M., and Brumer P. Principles of the Quantum Control of Molecular Processes (2003), Wiley-VCH 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
40. Turinici G. On the controllability of bilinear quantum systems. In: Defranceschi M., and Le Bris C. (Eds). Mathematical Models and Methods for ab initio Quantum Chemistry. Lecture Notes in Chemistry vol. 74 (2000), Springer
41. Zuazua E. Remarks on the controllability of the Schrödinger equation. Quantum Control: Mathematical and Numerical Challenges. CRM Proc. Lecture Notes vol. 33 (2003), Amer. Math. Soc., Providence, RI 193-211