We give the definitions of exact and approximate controllability for linear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No-go” result for evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllability for the linear Schrödinger equation and distributed additive control, and we show that the Hartree equation of quantum chemistry with bilinear control is not controllable in finite or infinite time. Finally, in Section 3, we give criteria for additive controllability of linear Schrödinger equations, and we give a distributed additive controllability result for the nonlinear Schrödinger equation if the data are small.
Mots clés : Schrödinger equations, exact and approximate control, quantum control
@article{COCV_2006__12_4_615_0, author = {Illner, Reinhard and Lange, Horst and Teismann, Holger}, title = {Limitations on the control of {Schr\"odinger} equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {615--635}, publisher = {EDP-Sciences}, volume = {12}, number = {4}, year = {2006}, doi = {10.1051/cocv:2006014}, mrnumber = {2266811}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2006014/} }
TY - JOUR AU - Illner, Reinhard AU - Lange, Horst AU - Teismann, Holger TI - Limitations on the control of Schrödinger equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 615 EP - 635 VL - 12 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2006014/ DO - 10.1051/cocv:2006014 LA - en ID - COCV_2006__12_4_615_0 ER -
%0 Journal Article %A Illner, Reinhard %A Lange, Horst %A Teismann, Holger %T Limitations on the control of Schrödinger equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 615-635 %V 12 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2006014/ %R 10.1051/cocv:2006014 %G en %F COCV_2006__12_4_615_0
Illner, Reinhard; Lange, Horst; Teismann, Holger. Limitations on the control of Schrödinger equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 615-635. doi : 10.1051/cocv:2006014. http://www.numdam.org/articles/10.1051/cocv:2006014/
[1] Collective oscillations of one-dimensional Bose-Einstein gas under varying in time trap potential and atomic scattering length. Phys. Rev. A 70 (2004) 053604.
and ,[2] Functional Analysis. Academic Press, N.Y. (1966). | MR | Zbl
and ,[3] Controllability for distributed bilinear systems. SIAM J. Contr. Opt. 20 (1982) 575-597. | MR | Zbl
, and ,[4] Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Contr. Opt. 30 (1992) 1024-1065. | MR | Zbl
, and ,[5] A bilinear optimal control problem applied to a time dependent Hartree-Fock equation coupled with classical nuclear dynamics. Portugaliae Mat. (To appear). | MR | Zbl
,[6] Existence and regularity of the solution of a time dependent Hartree-Fock equation coupled with a classical nuclear dynamics. Rev. Mat. Complut. 18 (2005) 285-314. | MR
,[7] Bilinear optimal control problem on a Schrödinger equation with singular potentials. Preprint (2004).
and ,[8] Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. 84 (2005) 851-956. | MR | Zbl
,[9] Controllability of a quantum particle in a moving potential well. J. Funct. Anal. 232 (2006) 328-389. | MR
and ,[10] Principles of the Quantum Control of Molecular Processes. Wiley-VCH, Berlin (2003).
and ,[11] Linear vs. nonlinear effects for nonlinear Schrödinger equations with potential. Commun. Contemp. Math. 7(4) (2005) 483-508. | MR | Zbl
,[12] On the time-dependent Hartree-Fock equations coupled with classical nuclear dynamics. Math. Mod. Meth. Appl. Sci. 9 (1999) 963-990. | MR | Zbl
and ,[13] Contrôle optimale bilinéaire d'une équation de Schrödinger. C. R. Acad. Sci. Paris, Sér. 1 330 (2000) 567-571. | Zbl
, and ,[14] Control of quantum systems. Int. J. Mod. Phys. B 17 (2003) 5397-5412. | Zbl
, and ,[15] Résultats de contrôlabilité exacte interne pour l'équation de Schrödinger at leurs limites asymptotiques, Application à certaines équations de plaques vibrantes. Asymptotic Analysis 5 (1992) 343-379. | Zbl
,[16] Harmonic Analysis. Addison-Wesley, Reading (1983). | MR | Zbl
,[17] Coherent control of self-trapping transition. Eur. Phys. J. B 20 (2001) 451-467.
and ,[18] On the controllability of quantum-mechanical systems. J. Math. Phys. 24 (1983) 2608-2618. | MR | Zbl
, and ,[19] Miscellanea in elementary quantum mechanics II. Prog. Theor. Phys. 9 (1953) 381-402. | MR | Zbl
,[20] A note on the exact internal control of nonlinear Schrödinger equations. CRM Proc. Lecture Notes 33 (2003) 127-137. | MR
, and ,[21] Some trigonometric inequalities with applications to the theory of series. Math. Z. 41 (1936) 367. | MR | Zbl
,[22] Decay estimates for Schrödinger operators. Commun. Pure Appl. Math. 44 (1991) 573-604. | MR | Zbl
, and ,[23] Note on the forced and damped oscillator in quantum mechanics. Can. J. Phys. 36 (1958) 371-377. | Zbl
,[24] Analytic controllability of time-dependent quantum control systems. J. Math. Phys. 46 (2005) 052102 | MR | Zbl
, , and ,[25] Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet controls. Differ. Int. Equ. 5 (1992) 571-535. | MR | Zbl
and ,[26] Control theory for partial differential equations, continuous and approximation theories. I & II. Cambridge University Press, Cambridge (2000). | MR | Zbl
and ,[27] Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates 12 (2004) 43-123. | MR | Zbl
, and ,[28] Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates 12 (2004) 183-231. | MR | Zbl
, and ,[29] Contrôle de l'équation de Schrödinger. Jour. Math. Pures Appl. 71 (1992) 267-291. | Zbl
,[30] Control theory applied to quantum chemistry, some tracks, in Conf. Int. contrôle des systèmes gouvernés par des équations aux derivées partielles. ESAIM Proc. 8 (2000) 77-94. | MR | Zbl
,[31] Computational Chemistry, in Handbook of Numerical Analysis, C. LeBris, Ph.G. Ciarlet Eds. North-Holland, Amsterdam (2003). | MR | Zbl
,[32] Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1 & 2. Masson, Paris (1988). | MR | Zbl
,[33] Exact controllability for the Schrödinger equation. SIAM J. Contr. Opt. 32 (1994) 24-34. | MR | Zbl
,[34] Stabilization of the Schrödinger equation. Portugaliae Mat. 51 (1994) 243-256. | MR | Zbl
and ,[35] Controllability of quantum harmonic oscillators. IEEE Trans. Automatic Control 49 (2004) 745-747. | MR
and ,[36] Observability and control of Schrödinger equations. SIAM J. Contr. Opt. 40 (2001) 211-230. | MR | Zbl
,[37] Optical Control of Molecular Dynamics. John Wiley & Sons, New York (2000).
and ,[38] Controllability and stabilizability theory for linear partial differential equations, recent progress and open questions. SIAM Rev. (1978) 20 639-739. | MR | Zbl
,[39] Degrees of controllability for quantum systems and application to atomic systems. J. Phys. A 35 (2002) 4125-4141. | MR | Zbl
, and ,[40] Coherent states and energy spectrum of the anharmonic osciallator. J. Phys. A 11 (1978) 1771-1780.
,[41] Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1974). | MR | Zbl
,[42] Analyse de méthodes numériques de simulation et contrôle en chimie quantique. Ph.D. Thesis, Univ. Paris VI (2000).
,[43] Controllable quantities for bilinear quantum systems, in Proc. of the 39th IEEE Conference on Decision and Control, Sydney, Australia (2000) 1364-1369.
,[44] An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980). | MR | Zbl
,[45] Introduction to Control Theory. Birkhäuser, Basel (1994). | MR
,[46] Remarks on the controllability of the Schrödinger equation. CRM Proc. Lecture Notes 33 (2003) 193-211. | MR
,Cité par Sources :