On the K + P problem for a three-level quantum system: Optimality implies resonance

Journal of Dynamical and Control Systems

Volumn 8, Issue 4, 2002, Pages 547-572

  Boscain, Ugo ^a   Chambrion, T ^a   Gauthier, J P ^a  

^a UNIVERSITÉ DE BOURGOGNE   (France)

Author keywords

Lie groups; Quantum control; Subriemannian geometry

Indexed keywords

HAMILTONIANS; INTEGRAL EQUATIONS; LASER APPLICATIONS; MAXIMUM PRINCIPLE; QUANTUM THEORY; RESONANCE; TRAJECTORIES;

LIE GROUPS;

OPTIMAL CONTROL SYSTEMS;

EID: 0036803005     PISSN: 10792724     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1020767419671     Document Type: Article

References (23)

1. A. Agrachev and J.P. Gauthier, On the Dido problem and plane isoperimetric problems. Acta Appl. Math. 57 (1999), p. 287.
2. _, On the subanalyticity of Carnot-Carathéodory distances. Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2002), No. 3, 359-382.
3. A. Agrachev, Exponential mappings for contact subriemannian structures. J. Dynam. Control Systems 2 (1996), No. 3, 321-358.
4. A. Agrachev and A.V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 635-690.
5. C. Altafini, Controllability of quantum mechanical systems by root space decomposition of su(N). J. Math. Phys. 43 (2002), 2051-2062.
6. D. D'Alessandro and M. Dahleh, Optimal control of two-level quantum systems. IEEE Trans. Automat. Control 46 (2001), No. 6, 866-876.
7. A. Bellaiche, The tangent space in sub-Riemannian geometry. Sub-Riemannian geometry. Progr. Math., Birkhäuser, Basel (1996), 1-78.
8. A.O. Barut and R. Raczka, Theory of group representations and applications. World Scientific, 1986.
9. U. Boscain, Grégoire Charlot, Jean-Paul Gauthier, Stéphane Guérin, and Hans-Rudolf Jauslin, Optimal control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys. 43 (2002), 2107-2132.
10. A. Borel and J. De Siebenthal, Les sous-groupes fermes de rang maximum des groupes de Lie clos. Comment. Math. Helv. 23 (1949), 200-221.
11. 3. J. Dynam. Control Systems 2 (1996), No. 3, 359-421.
12. M.A. Daleh, A.M. Peirce, and H. Rabitz, Optimal control of quantummechanical systems: existence, numerical approximation, and applications. Phys. Rev. A 37 (1988).
13. M. Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry. In: Progr. Math. 144, Birkhäuser, Basel (1996) 79-323.
14. S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Pure Appl. Math. 80 (1978).
15. R. Hermann, Lie groups for physicists. Benjamin-Cummings, 1966.
16. V. Jurdjevic, Geometric control theory. Cambridge Univ. Press, 1997.
17. _, Optimal control, geometry, and mechanics. Mathematical control teory. (J. Bailleu, J.C. Willems (Eds.)), Springer (1999) 227-267.
18. _, Hamiltonian point of view on non-Euclidean geometry and elliptic functions. System Control Lett. 43 (2001), 25-41.
19. V. Jurdjevic and H.J. Sussmann, Control systems on Lie groups. J. Differential Equations 12 (1972), 313-329.
20. N. Khaneja, R. Brockett, and S.J. Glaser, Time optimal control in spin systems. Phys. Rev. A 63 (2001).
21. R. Montgomery, A tour of subriemannian geometry. Amer. Math. Soc., Math. Surv. and Monogr., 2002.
22. L.S. Pontryagin, V. Boltianski, R. Gamkrelidze, and E. Mishchenko, The mathematical theory of optimal processes. John Wiley and Sons, Inc. 1961.
23. V. Ramakrishna, K.L. Flores, H. Rabitz, and R. Ober, Quantum control by decomposition of su(2). Phys. Rev. A 62 (2000).