Lyapunov control of a quantum particle in a decaying potential

Mazyar Mirrahimi

Annales de l'I.H.P. Analyse non linéaire (2009)

  • Volume: 26, Issue: 5, page 1743-1765
  • ISSN: 0294-1449

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Mirrahimi, Mazyar. "Lyapunov control of a quantum particle in a decaying potential." Annales de l'I.H.P. Analyse non linéaire 26.5 (2009): 1743-1765. <http://eudml.org/doc/78911>.

@article{Mirrahimi2009,
author = {Mirrahimi, Mazyar},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {nonlinear control of PDEs; approximate stabilization; Lyapunov techniques; dispersive estimates; pre-compactness},
language = {eng},
number = {5},
pages = {1743-1765},
publisher = {Elsevier},
title = {Lyapunov control of a quantum particle in a decaying potential},
url = {http://eudml.org/doc/78911},
volume = {26},
year = {2009},
}

TY - JOUR
AU - Mirrahimi, Mazyar
TI - Lyapunov control of a quantum particle in a decaying potential
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2009
PB - Elsevier
VL - 26
IS - 5
SP - 1743
EP - 1765
LA - eng
KW - nonlinear control of PDEs; approximate stabilization; Lyapunov techniques; dispersive estimates; pre-compactness
UR - http://eudml.org/doc/78911
ER -

References

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  1. [1] Agmon S., Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci.4 (2) (1975) 151-218. Zbl0315.47007MR397194
  2. [2] Agmon S., Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schrödinger Operators, Mathematical Notes, vol. 29, Princeton University Press, 1982. Zbl0503.35001MR745286
  3. [3] Albertini F., D'Alessandro D., Notions of controllability for bilinear multilevel quantum systems, IEEE Trans. Automat. Control48 (8) (2003) 1399-1403. MR2004373
  4. [4] Altafini C., Controllability of quantum mechanical systems by root space decomposition of su(n), J. Math. Phys.43 (5) (2002) 2051-2062. Zbl1059.93016MR1893660
  5. [5] Avron J.E., Elgart A., Adiabatic theorem without a gap condition, Comm. Math. Phys.203 (1999) 445-463. Zbl0936.47047MR1697605
  6. [6] Baudouin L., Puel J.P., Kavian O., Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. Differential Equations216 (2005) 188-222. Zbl1109.35094MR2158922
  7. [7] Baudouin L., Salomon J., Constructive solutions of a bilinear control problem, C. R. Acad. Sci. Paris, Ser. I342 (2) (2006) 119-124. Zbl1079.49021MR2193658
  8. [8] Beauchard K., Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl.84 (2005) 851-956. Zbl1124.93009MR2144647
  9. [9] Beauchard K., Coron J.-M., Mirrahimi M., Rouchon P., Implicit Lyapunov control of finite dimensional Schrödinger equations, Systems Control Lett.56 (2007) 388-395. Zbl1110.81063MR2311201
  10. [10] Beauchard K., Coron J.M., Controllability of a quantum particle in a moving potential well, J. Funct. Anal.232 (2006) 328-389. Zbl1188.93017MR2200740
  11. [11] Beauchard K., Mirrahimi M., Practical stabilization of a quantum particle in a 1D infinite square potential well, SIAM J. Control Optim., in press, preliminary version:, arXiv:0801.1522v1. Zbl1194.93176MR2491595
  12. [12] Chambrion T., Mason P., Sigalotti M., Boscain U., Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. I. H. Poincaré – AN26 (2009) 329-349. Zbl1161.35049MR2483824
  13. [13] Chen Y., Gross P., Ramakrishna V., Rabitz H., Mease K., Competitive tracking of molecular objectives described by quantum mechanics, J. Chem. Phys.102 (1995) 8001-8010. 
  14. [14] Coron J.-M., d'Andréa Novel B., Stabilization of a rotating body-beam without damping, IEEE Trans. Automat. Control43 (5) (1998) 608-618. Zbl0908.93055MR1618052
  15. [15] Coron J.-M., d'Andrá Novel B., Bastin G., A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control52 (1) (2007) 2-11. MR2286756
  16. [16] Coron J.M., Global stabilization for controllable systems without drift, Math. Control Signals Systems5 (1992) 295-312. Zbl0760.93067MR1164379
  17. [17] Coron J.M., On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain, SIAM J. Control Optim.37 (1999) 1874-1896. Zbl0954.76010MR1720143
  18. [18] Coron J.M., Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematica Society, USA, 2007. Zbl1140.93002MR2302744
  19. [19] Glass O., Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: The multiconnected case, SIAM J. Control Optim.44 (3) (2005) 1105-1147. Zbl1130.93403MR2178059
  20. [20] O. Glass, Controllability and asymptotic stabilization of the Camassa–Holm equation, preprint, 2007. Zbl1186.35185MR2436454
  21. [21] Goldberg M., Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials, Geom. Funct. Anal.16 (3) (2006) 517-536. Zbl1158.35408MR2238943
  22. [22] Goldberg M., Schlag W., Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys.251 (2004) 157-178. Zbl1086.81077MR2096737
  23. [23] Goldberg M., Schlag W., A limiting absorption principle for the three-dimensional Schrödinger equation with L p potentials, Int. Math. Res. Notices75 (2004) 4049-4071. Zbl1069.35063MR2112327
  24. [24] Goldberg M., Visan M., A counterexample to dispersive estimates for Schrödinger operators in higher dimensions, Comm. Math. Phys.266 (1) (2006) 211-238. Zbl1110.35073MR2231971
  25. [25] Van Handel R., Stockton J.K., Mabuchi H., Modeling and feedback control design for quantum state preparation, J. Opt. B: Quant. Semiclass. Opt.7 (2005) S179-S197, Special issue on quantum control. MR2183700
  26. [26] Haroche S., Contrôle de la décohérence: théorie et expériences, 2004, Notes de cours, Collège de France, http://www.lkb.ens.fr/recherche/qedcav/college/college.html. 
  27. [27] Jensen A., Kato T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J.46 (3) (1979) 583-611. Zbl0448.35080MR544248
  28. [28] Jensen A., Yajima K., A remark on L p -boundedness of wave operators for two-dimensional Schrödinger operators, Comm. Math. Phys.225 (3) (2002) 633-637. Zbl1057.47011MR1888876
  29. [29] Journé J.-L., Soffer A., Sogge C.D., Decay estimates for Schrödinger operators, Comm. Pure Appl. Math.44 (1991). Zbl0743.35008MR1105875
  30. [30] Kato T., Perturbation Theory for Linear Operators, Springer, 1966. Zbl0148.12601
  31. [31] Keel M., Tao T., Endpoint Strichartz estimates, Amer. J. Math.5 (1998) 955-980. Zbl0922.35028MR1646048
  32. [32] Li B., Turinici G., Ramakrishna V., Rabitz H., Optimal dynamic discrimination of similar molecules through quantum learning control, J. Phys. Chem. B106 (33) (2002) 8125-8131. 
  33. [33] M. Mirrahimi, Lyapunov control of a particle in a finite quantum potential well, in: CDC, San Diego, 2006. 
  34. [34] Mirrahimi M., Rouchon P., Turinici G., Lyapunov control of bilinear Schrödinger equations, Automatica41 (2005) 1987-1994. Zbl1125.93466MR2168664
  35. [35] Mirrahimi M., Turinici G., Rouchon P., Reference trajectory tracking for locally designed coherent quantum controls, J. Phys. Chem. A109 (2005) 2631-2637. 
  36. [36] Mirrahimi M., Van Handel R., Stabilizing feedback controls for quantum systems, SIAM J. Control Optim.46 (2) (2007) 445-467. Zbl1136.81342MR2309036
  37. [37] Ramakrishna V., Salapaka M., Dahleh M., Rabitz H., Controllability of molecular systems, Phys. Rev. A51 (2) (1995) 960-966. 
  38. [38] Rauch J., Local decay of scattering solutions to Schrödinger's equation, Comm. Math. Phys.61 (2) (1978) 149-168. Zbl0381.35023MR495958
  39. [39] Reed M., Simon B., Methods of Modern Mathematical Physics, vol. IV: Analysis of Operators, Academic Press, New York, 1978. Zbl0401.47001MR493421
  40. [40] Rodnianski I., Schlag W., Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math.155 (2004) 451-513. Zbl1063.35035MR2038194
  41. [41] Schlag W., Dispersive estimates for Schrödinger operators in two dimensions, Comm. Math. Phys.257 (1) (2005) 87-117. Zbl1134.35321MR2163570
  42. [42] Shi S., Woody A., Rabitz H., Optimal control of selective vibrational excitation in harmonic linear chain molecules, J. Chem. Phys.88 (11) (1988) 6870-6883. MR943930
  43. [43] Stoiciu M., An estimate for the number of bound states of the Schrödinger operator in two dimensions, Proc. Amer. Math. Soc.132 (4) (2004) 1143-1151. Zbl1039.35071MR2045431
  44. [44] Strichartz R., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J.44 (3) (1977) 705-714. Zbl0372.35001MR512086
  45. [45] Sugawara M., General formulation of locally designed coherent control theory for quantum systems, J. Chem. Phys.118 (15) (2003) 6784-6800. 
  46. [46] Sussmann H.J., Jurdjevic V., Controllability of nonlinear systems, J. Differential Equations12 (1972) 95-116. Zbl0242.49040MR338882
  47. [47] T.J. Tarn, J.W. Clark, D.G. Lucarelli, Controllability of quantum mechanical systems with continuous spectra, in: CDC, Sydney, 2000. 
  48. [48] G. Turinici, Controllable quantities for bilinear quantum systems, in: Proceedings of the 39th IEEE Conference on Decision and Control, 2000, pp. 1364–1369. 
  49. [49] Turinici G., Rabitz H., Wavefunction controllability in quantum systems, J. Phys. A36 (2003) 2565-2576. Zbl1064.81558MR1967518
  50. [50] Weder R., L p - L p ' estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Anal.170 (1) (2000) 37-68. Zbl0943.34070MR1736195
  51. [51] Yajima K., The W k , p -continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan47 (3) (1995) 551-581. Zbl0837.35039MR1331331
  52. [52] Yajima K., L p -boundedness of wave operators for two-dimensional Schrödinger operators, Comm. Math. Phys.208 (1) (1999) 125-152. Zbl0961.47004MR1729881

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