@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}, pages = {1743--1765}, publisher = {Elsevier}, volume = {26}, number = {5}, year = {2009}, doi = {10.1016/j.anihpc.2008.09.006}, mrnumber = {2566708}, zbl = {1176.35169}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2008.09.006/} }
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 SP - 1743 EP - 1765 VL - 26 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2008.09.006/ DO - 10.1016/j.anihpc.2008.09.006 LA - en ID - AIHPC_2009__26_5_1743_0 ER -
%0 Journal Article %A Mirrahimi, Mazyar %T Lyapunov Control of a Quantum Particle in a Decaying Potential %J Annales de l'I.H.P. Analyse non linéaire %D 2009 %P 1743-1765 %V 26 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2008.09.006/ %R 10.1016/j.anihpc.2008.09.006 %G en %F 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, Volume 26 (2009) no. 5, pp. 1743-1765. doi : 10.1016/j.anihpc.2008.09.006. http://www.numdam.org/articles/10.1016/j.anihpc.2008.09.006/
[1] Spectral Properties of Schrödinger Operators and Scattering Theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (2) (1975) 151-218. | Numdam | MR | Zbl
,[2] 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 | Zbl
,[3] Notions of Controllability for Bilinear Multilevel Quantum Systems, IEEE Trans. Automat. Control 48 (8) (2003) 1399-1403. | MR
, ,[4] Controllability of Quantum Mechanical Systems by Root Space Decomposition of Su(N), J. Math. Phys. 43 (5) (2002) 2051-2062. | MR | Zbl
,[5] Adiabatic Theorem Without a Gap Condition, Comm. Math. Phys. 203 (1999) 445-463. | MR | Zbl
, ,[6] Regularity for a Schrödinger Equation With Singular Potentials and Application to Bilinear Optimal Control, J. Differential Equations 216 (2005) 188-222. | MR | Zbl
, , ,[7] Constructive Solutions of a Bilinear Control Problem, C. R. Acad. Sci. Paris, Ser. I 342 (2) (2006) 119-124. | MR | Zbl
, ,[8] Local Controllability of a 1-D Schrödinger Equation, J. Math. Pures Appl. 84 (2005) 851-956. | MR | Zbl
,[9] Implicit Lyapunov Control of Finite Dimensional Schrödinger Equations, Systems Control Lett. 56 (2007) 388-395. | MR | Zbl
, , , ,[10] Controllability of a Quantum Particle in a Moving Potential Well, J. Funct. Anal. 232 (2006) 328-389. | MR | Zbl
, ,[11] 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 | Zbl
, ,[12] Controllability of the Discrete-Spectrum Schrödinger Equation Driven by an External Field, Ann. I. H. Poincaré - AN 26 (2009) 329-349. | EuDML | Numdam | MR | Zbl
, , , ,[13] Competitive Tracking of Molecular Objectives Described by Quantum Mechanics, J. Chem. Phys. 102 (1995) 8001-8010.
, , , , ,[14] Stabilization of a Rotating Body-Beam Without Damping, IEEE Trans. Automat. Control 43 (5) (1998) 608-618. | MR | Zbl
, ,[15] A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws, IEEE Trans. Automat. Control 52 (1) (2007) 2-11. | MR
, , ,[16] Global Stabilization for Controllable Systems Without Drift, Math. Control Signals Systems 5 (1992) 295-312. | MR | Zbl
,[17] 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 | Zbl
,[18] Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematica Society, USA, 2007. | MR | Zbl
,[19] Asymptotic Stabilizability by Stationary Feedback of the Two-Dimensional Euler Equation: the Multiconnected Case, SIAM J. Control Optim. 44 (3) (2005) 1105-1147. | MR | Zbl
,[20] O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, preprint, 2007. | MR | Zbl
[21] Dispersive Bounds for the Three-Dimensional Schrödinger Equation With Almost Critical Potentials, Geom. Funct. Anal. 16 (3) (2006) 517-536. | MR | Zbl
,[22] Dispersive Estimates for Schrödinger Operators in Dimensions One and Three, Comm. Math. Phys. 251 (2004) 157-178. | MR | Zbl
, ,[23] A Limiting Absorption Principle for the Three-Dimensional Schrödinger Equation With Potentials, Int. Math. Res. Notices 75 (2004) 4049-4071. | MR | Zbl
, ,[24] A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions, Comm. Math. Phys. 266 (1) (2006) 211-238. | MR | Zbl
, ,[25] 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
, , ,[26] 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] Spectral Properties of Schrödinger Operators and Time-Decay of the Wave Functions, Duke Math. J. 46 (3) (1979) 583-611. | MR | Zbl
, ,[28] A Remark on -Boundedness of Wave Operators for Two-Dimensional Schrödinger Operators, Comm. Math. Phys. 225 (3) (2002) 633-637. | MR | Zbl
, ,[29] Decay Estimates for Schrödinger Operators, Comm. Pure Appl. Math. 44 (1991). | MR | Zbl
, , ,[30] Perturbation Theory for Linear Operators, Springer, 1966. | MR | Zbl
,[31] Endpoint Strichartz Estimates, Amer. J. Math. 5 (1998) 955-980. | MR | Zbl
, ,[32] 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] Lyapunov Control of Bilinear Schrödinger Equations, Automatica 41 (2005) 1987-1994. | MR | Zbl
, , ,[35] Reference Trajectory Tracking for Locally Designed Coherent Quantum Controls, J. Phys. Chem. A 109 (2005) 2631-2637.
, , ,[36] Stabilizing Feedback Controls for Quantum Systems, SIAM J. Control Optim. 46 (2) (2007) 445-467. | MR | Zbl
, ,[37] Controllability of Molecular Systems, Phys. Rev. A 51 (2) (1995) 960-966.
, , , ,[38] Local Decay of Scattering Solutions to Schrödinger's Equation, Comm. Math. Phys. 61 (2) (1978) 149-168. | MR | Zbl
,[39] Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators, Academic Press, New York, 1978. | MR | Zbl
, ,[40] Time Decay for Solutions of Schrödinger Equations With Rough and Time-Dependent Potentials, Invent. Math. 155 (2004) 451-513. | MR | Zbl
, ,[41] Dispersive Estimates for Schrödinger Operators in Two Dimensions, Comm. Math. Phys. 257 (1) (2005) 87-117. | MR | Zbl
,[42] Optimal Control of Selective Vibrational Excitation in Harmonic Linear Chain Molecules, J. Chem. Phys. 88 (11) (1988) 6870-6883. | MR
, , ,[43] 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 | Zbl
,[44] Restrictions of Fourier Transforms to Quadratic Surfaces and Decay of Solutions of Wave Equations, Duke Math. J. 44 (3) (1977) 705-714. | MR | Zbl
,[45] General Formulation of Locally Designed Coherent Control Theory for Quantum Systems, J. Chem. Phys. 118 (15) (2003) 6784-6800.
,[46] Controllability of Nonlinear Systems, J. Differential Equations 12 (1972) 95-116. | MR | Zbl
, ,[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] Wavefunction Controllability in Quantum Systems, J. Phys. A 36 (2003) 2565-2576. | MR | Zbl
, ,[50] - 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 | Zbl
,[51] The -Continuity of Wave Operators for Schrödinger Operators, J. Math. Soc. Japan 47 (3) (1995) 551-581. | MR | Zbl
,[52] -Boundedness of Wave Operators for Two-Dimensional Schrödinger Operators, Comm. Math. Phys. 208 (1) (1999) 125-152. | MR | Zbl
,Cited by Sources: