Lyapunov Control of a Quantum Particle in a Decaying Potential
Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 5, p. 1743-1765
@article{AIHPC_2009__26_5_1743_0,
     author = {Mirrahimi, Mazyar},
     title = {Lyapunov Control of a Quantum Particle in a Decaying Potential},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {26},
     number = {5},
     year = {2009},
     pages = {1743-1765},
     doi = {10.1016/j.anihpc.2008.09.006},
     zbl = {1176.35169},
     mrnumber = {2566708},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2009__26_5_1743_0}
}
Mirrahimi, Mazyar. Lyapunov Control of a Quantum Particle in a Decaying Potential. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 5, pp. 1743-1765. doi : 10.1016/j.anihpc.2008.09.006. http://www.numdam.org/item/AIHPC_2009__26_5_1743_0/

[1] Agmon S., Spectral Properties of Schrödinger Operators and Scattering Theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (2) (1975) 151-218. | Numdam | MR 397194 | Zbl 0315.47007

[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. | MR 745286 | Zbl 0503.35001

[3] Albertini F., D'Alessandro D., Notions of Controllability for Bilinear Multilevel Quantum Systems, IEEE Trans. Automat. Control 48 (8) (2003) 1399-1403. | MR 2004373

[4] Altafini C., Controllability of Quantum Mechanical Systems by Root Space Decomposition of Su(N), J. Math. Phys. 43 (5) (2002) 2051-2062. | MR 1893660 | Zbl 1059.93016

[5] Avron J. E., Elgart A., Adiabatic Theorem Without a Gap Condition, Comm. Math. Phys. 203 (1999) 445-463. | MR 1697605 | Zbl 0936.47047

[6] Baudouin L., Puel J. P., Kavian O., Regularity for a Schrödinger Equation With Singular Potentials and Application to Bilinear Optimal Control, J. Differential Equations 216 (2005) 188-222. | MR 2158922 | Zbl 1109.35094

[7] Baudouin L., Salomon J., Constructive Solutions of a Bilinear Control Problem, C. R. Acad. Sci. Paris, Ser. I 342 (2) (2006) 119-124. | MR 2193658 | Zbl 1079.49021

[8] Beauchard K., Local Controllability of a 1-D Schrödinger Equation, J. Math. Pures Appl. 84 (2005) 851-956. | MR 2144647 | Zbl 1124.93009

[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. | MR 2311201 | Zbl 1110.81063

[10] Beauchard K., Coron J. M., Controllability of a Quantum Particle in a Moving Potential Well, J. Funct. Anal. 232 (2006) 328-389. | MR 2200740 | Zbl 1188.93017 | Zbl pre05017416

[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. | MR 2491595 | Zbl 1194.93176

[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 - AN 26 (2009) 329-349. | Numdam | MR 2483824 | Zbl 1161.35049

[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] Coron J.-M., D'Andréa Novel B., Stabilization of a Rotating Body-Beam Without Damping, IEEE Trans. Automat. Control 43 (5) (1998) 608-618. | MR 1618052 | Zbl 0908.93055

[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. Control 52 (1) (2007) 2-11. | MR 2286756

[16] Coron J. M., Global Stabilization for Controllable Systems Without Drift, Math. Control Signals Systems 5 (1992) 295-312. | MR 1164379 | Zbl 0760.93067

[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. | MR 1720143 | Zbl 0954.76010

[18] Coron J. M., Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematica Society, USA, 2007. | MR 2302744 | Zbl 1140.93002

[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. | MR 2178059 | Zbl 1130.93403

[20] O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, preprint, 2007. | MR 2436454 | Zbl 1186.35185 | Zbl pre05344300

[21] Goldberg M., Dispersive Bounds for the Three-Dimensional Schrödinger Equation With Almost Critical Potentials, Geom. Funct. Anal. 16 (3) (2006) 517-536. | MR 2238943 | Zbl 1158.35408

[22] Goldberg M., Schlag W., Dispersive Estimates for Schrödinger Operators in Dimensions One and Three, Comm. Math. Phys. 251 (2004) 157-178. | MR 2096737 | Zbl 1086.81077

[23] Goldberg M., Schlag W., A Limiting Absorption Principle for the Three-Dimensional Schrödinger Equation With L p Potentials, Int. Math. Res. Notices 75 (2004) 4049-4071. | MR 2112327 | Zbl 1069.35063

[24] Goldberg M., Visan M., A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions, Comm. Math. Phys. 266 (1) (2006) 211-238. | MR 2231971 | Zbl 1110.35073

[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. | MR 2183700

[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] Jensen A., Kato T., Spectral Properties of Schrödinger Operators and Time-Decay of the Wave Functions, Duke Math. J. 46 (3) (1979) 583-611. | MR 544248 | Zbl 0448.35080

[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. | MR 1888876 | Zbl 1057.47011

[29] Journé J.-L., Soffer A., Sogge C. D., Decay Estimates for Schrödinger Operators, Comm. Pure Appl. Math. 44 (1991). | MR 1105875 | Zbl 0743.35008

[30] Kato T., Perturbation Theory for Linear Operators, Springer, 1966. | MR 203473 | Zbl 0148.12601

[31] Keel M., Tao T., Endpoint Strichartz Estimates, Amer. J. Math. 5 (1998) 955-980. | MR 1646048 | Zbl 0922.35028

[32] Li B., Turinici G., Ramakrishna V., Rabitz H., Optimal Dynamic Discrimination of Similar Molecules Through Quantum Learning Control, J. Phys. Chem. B 106 (33) (2002) 8125-8131.

[33] M. Mirrahimi, Lyapunov control of a particle in a finite quantum potential well, in: CDC, San Diego, 2006.

[34] Mirrahimi M., Rouchon P., Turinici G., Lyapunov Control of Bilinear Schrödinger Equations, Automatica 41 (2005) 1987-1994. | MR 2168664 | Zbl 1125.93466

[35] Mirrahimi M., Turinici G., Rouchon P., Reference Trajectory Tracking for Locally Designed Coherent Quantum Controls, J. Phys. Chem. A 109 (2005) 2631-2637.

[36] Mirrahimi M., Van Handel R., Stabilizing Feedback Controls for Quantum Systems, SIAM J. Control Optim. 46 (2) (2007) 445-467. | MR 2309036 | Zbl 1136.81342

[37] Ramakrishna V., Salapaka M., Dahleh M., Rabitz H., Controllability of Molecular Systems, Phys. Rev. A 51 (2) (1995) 960-966.

[38] Rauch J., Local Decay of Scattering Solutions to Schrödinger's Equation, Comm. Math. Phys. 61 (2) (1978) 149-168. | MR 495958 | Zbl 0381.35023

[39] Reed M., Simon B., Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators, Academic Press, New York, 1978. | MR 493421 | Zbl 0401.47001

[40] Rodnianski I., Schlag W., Time Decay for Solutions of Schrödinger Equations With Rough and Time-Dependent Potentials, Invent. Math. 155 (2004) 451-513. | MR 2038194 | Zbl 1063.35035

[41] Schlag W., Dispersive Estimates for Schrödinger Operators in Two Dimensions, Comm. Math. Phys. 257 (1) (2005) 87-117. | MR 2163570 | Zbl 1134.35321

[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. | MR 943930

[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. | MR 2045431 | Zbl 1039.35071

[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. | MR 512086 | Zbl 0372.35001

[45] Sugawara M., General Formulation of Locally Designed Coherent Control Theory for Quantum Systems, J. Chem. Phys. 118 (15) (2003) 6784-6800.

[46] Sussmann H. J., Jurdjevic V., Controllability of Nonlinear Systems, J. Differential Equations 12 (1972) 95-116. | MR 338882 | Zbl 0242.49040

[47] T.J. Tarn, J.W. Clark, D.G. Lucarelli, Controllability of quantum mechanical systems with continuous spectra, in: CDC, Sydney, 2000.

[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] Turinici G., Rabitz H., Wavefunction Controllability in Quantum Systems, J. Phys. A 36 (2003) 2565-2576. | MR 1967518 | Zbl 1064.81558

[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. | MR 1736195 | Zbl 0943.34070

[51] Yajima K., The W k,p -Continuity of Wave Operators for Schrödinger Operators, J. Math. Soc. Japan 47 (3) (1995) 551-581. | MR 1331331 | Zbl 0837.35039

[52] Yajima K., L p -Boundedness of Wave Operators for Two-Dimensional Schrödinger Operators, Comm. Math. Phys. 208 (1) (1999) 125-152. | MR 1729881 | Zbl 0961.47004