Chambrion, Thomas; Mason, Paolo; Sigalotti, Mario; Boscain, Ugo
Controllability of the Discrete-Spectrum Schrödinger Equation Driven by an External Field
Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 1 , p. 329-349
Zbl 1161.35049 | MR 2483824 | 6 citations dans Numdam
doi : 10.1016/j.anihpc.2008.05.001
URL stable : http://www.numdam.org/item?id=AIHPC_2009__26_1_329_0

Bibliographie

[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., 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. Numdam | MR 2224821 | Zbl 1106.93006

[3] Agrachev A., Kuksin S., Sarychev A., 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. Numdam | MR 2329509 | Zbl 1177.60059

[4] Agrachev A. A., Sachkov Y. L., Control Theory From the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, vol. 87, Springer-Verlag, Berlin, 2004, Control Theory and Optimization, II. MR 2062547 | Zbl 1062.93001

[5] Agrachev A. A., Sarychev A. V., Controllability of 2D Euler and Navier-Stokes Equations by Degenerate Forcing, Commun. Math. Phys. 265 (3) (2006) 673-697. MR 2231685 | Zbl 1105.93008

[6] Albert J. H., Genericity of Simple Eigenvalues for Elliptic PDE's, Proc. Amer. Math. Soc. 48 (1975) 413-418. MR 385934 | Zbl 0302.35071

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

[8] 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

[9] Altafini C., Controllability Properties for Finite Dimensional Quantum Markovian Master Equations, J. Math. Phys. 44 (6) (2003) 2357-2372. MR 1979090 | Zbl 1062.82033

[10] Ball J. M., Marsden J. E., Slemrod M., Controllability for Distributed Bilinear Systems, SIAM J. Control Optim. 20 (4) (1982) 575-597. MR 661034 | Zbl 0485.93015

[11] Baudouin L., Kavian O., 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. MR 2158922 | Zbl 1109.35094

[12] Beauchard K., Local Controllability of a 1-D Schrödinger Equation, J. Math. Pures Appl. (9) 84 (7) (2005) 851-956. MR 2144647 | Zbl 1124.93009

[13] Beauchard K., Coron J.-M., Controllability of a Quantum Particle in a Moving Potential Well, J. Funct. Anal. 232 (2) (2006) 328-389. MR 2200740 | Zbl pre05017416

[14] Borzì A., Decker E., Analysis of a Leap-Frog Pseudospectral Scheme for the Schrödinger Equation, J. Comput. Appl. Math. 193 (1) (2006) 65-88. MR 2228707 | Zbl 1118.65107

[15] Boscain U., Chambrion T., 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). MR 2170218 | Zbl 1084.81083

[16] Boscain U., 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). Numdam | MR 2111082 | Zbl 1072.49002

[17] Boscain U., Mason P., Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field, J. Math. Phys. 47 (6) (2006) 29, 062101. MR 2239948 | Zbl 1112.81098

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

[19] D'Alessandro D., Introduction to Quantum Control and Dynamics, Applied Mathematics and Nonlinear Science Series, Chapman, Hall/CRC, Boca Raton, FL, 2008. MR 2357229 | Zbl 1139.81001

[20] Davies E. B., Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825 | Zbl 0893.47004

[21] Henrot A., Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. MR 2251558 | Zbl 1109.35081

[22] Hübler P., Bargon J., 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., Kunisch K., Optimal Bilinear Control of an Abstract Schrödinger Equation, SIAM J. Control Optim. 46 (1) (2007) 274-287, (electronic). MR 2299629 | Zbl 1136.35089

[24] Jurdjevic V., Sussmann H. J., Control Systems on Lie Groups, J. Differential Equations 12 (1972) 313-329. MR 331185 | Zbl 0237.93027

[25] Kato T., Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag, New York, 1966. MR 203473 | Zbl 0148.12601

[26] Khaneja N., Glaser S. J., 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. MR 1891763

[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., Rouchon P., Controllability of Quantum Harmonic Oscillators, IEEE Trans. Automat. Control 49 (5) (2004) 745-747. MR 2057808

[29] Peirce A., Dahleh M., Rabitz H., Optimal Control of Quantum Mechanical Systems: Existence, Numerical Approximations, and Applications, Phys. Rev. A 37 (1988) 4950-4964. MR 949169

[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. MR 2224403 | Zbl 1100.35020

[31] Rabitz H., De Vivie-Riedle H., Motzkus R., Kompa K., Wither the Future of Controlling Quantum Phenomena?, Science 288 (2000) 824-828.

[32] Reed M., Simon B., Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press (Harcourt Brace Jovanovich Publishers), New York, 1978. MR 493421 | Zbl 0401.47001

[33] Rellich F., Perturbation Theory of Eigenvalue Problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon Breach Science Publishers, New York, 1969. MR 240668 | Zbl 0181.42002

[34] Rodnianski I., Schlag W., Time Decay for Solutions of Schrödinger Equations With Rough and Time-Dependent Potentials, Invent. Math. 155 (3) (2004) 451-513. MR 2038194 | Zbl 1063.35035

[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. MR 2253360 | Zbl 1105.35085

[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. MR 2082989

[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. MR 1776551 | Zbl 1073.93511

[38] Shapiro M., Brumer P., Principles of the Quantum Control of Molecular Processes, Wiley-VCH, 2003, 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. MR 2158180 | Zbl 1140.93395

[40] Turinici G., On the Controllability of Bilinear Quantum Systems, in: Defranceschi M., Le Bris C. (Eds.), Mathematical Models and Methods for Ab Initio Quantum Chemistry, Lecture Notes in Chemistry, vol. 74, Springer, 2000. MR 1857459 | Zbl 1007.81019

[41] Zuazua E., Remarks on the Controllability of the Schrödinger Equation, in: Quantum Control: Mathematical and Numerical Challenges, CRM Proc. Lecture Notes, vol. 33, Amer. Math. Soc., Providence, RI, 2003, pp. 193-211. MR 2043529