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Monotonic time-discretized schemes in quantum control. (English) Zbl 1095.65058

The authors consider the optimal control problem in quantum control of the form \[ J(\varepsilon)= \langle\psi(T)|O|\psi(T)\rangle- \alpha \int^T_0 \varepsilon^2(t)\,dt\to\max, \]
\[ i{\partial\over\partial t} w(x,t)= H(x)\psi(x,t)- \mu(x) \varepsilon(t)\psi(x,t),\quad \psi(x,t= 0)= \psi_0(x). \] For this problem the authors present a stable time and space discretisation which preserves the monotonic properties of the monotonic algorithms of the continuous version. Numerical results are presented.

MSC:

65K10 Numerical optimization and variational techniques
49M25 Discrete approximations in optimal control
49J20 Existence theories for optimal control problems involving partial differential equations
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[1] Rabitz, Science, 299, 525 (2003) · doi:10.1126/science.1080683
[2] Rabitz, H., Turinici, G., Brown, E.: Control of quantum dynamics: Concepts, procedures and future prospects. In: Ph. G. Ciarlet, editor, Computational Chemistry, Special Volume (C. Le Bris Editor) of Handbook of Numerical Analysis, vol X, 833-887. Elsevier Science B.V. (2003) · Zbl 1066.81015
[3] Shi, J. Chem. Phys.,, 88, 6870 (1988) · doi:10.1063/1.454384
[4] Salomon, J.: Limit points of the monotonic schemes for quantum control, to appear in the Proceedings of the 44th IEEE Conference on decision and Control, Sevilla, Spain, December 12-15 (2005)
[5] Elliott, D.L.: Bilinear systems, University of Maryland. J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering Online Copyright © 1999 by John Wiley & Sons
[6] Judson, Phys. Rev. Lett., 68, 1500 (1992) · doi:10.1103/PhysRevLett.68.1500
[7] Levis, Science, 292, 709 (2001) · doi:10.1126/science.1059133
[8] Assion, Science, 282, 919 (1998) · doi:10.1126/science.282.5390.919
[9] Bergt, J. Phys. Chem. A., 103, 10381 (1999) · doi:10.1021/jp992541k
[10] Weinacht, Nature, 397, 233 (1999) · doi:10.1038/16654
[11] Bardeen, Chem. Phys. Lett., 280, 151 (1997) · doi:10.1016/S0009-2614(97)01081-6
[12] Bardeen, J. Am. Chem. Soc., 120, 13023 (1998) · doi:10.1021/ja9824627
[13] Strang, Arch. Rat. Mech. and An., 12, 392 (1963) · Zbl 0113.32303 · doi:10.1007/BF00281235
[14] Hornung, J. Chem.Phys., 115, 3105 (2001) · doi:10.1063/1.1378817
[15] Tannor, D., Kazakov, V., Orlov, V.: Control of photochemical branching: Novel procedures for finding optimal pulses and global upper bounds. In Time Dependent Quantum Molecular Dynamics, edited by Broeckhove J. and Lathouwers L. Plenum, 347-360 (1992)
[16] Zhu, J. Chem. Phys., 109, 385 (1998) · doi:10.1063/1.476575
[17] Maday, J. Chem. Phys., 118, 8191 (2003) · doi:10.1063/1.1564043
[18] Bandrauk, J. Chem. Phys., 99, 1185 (1993) · doi:10.1063/1.465362
[19] Salomon, J.: Phd. thesis. in progress
[20] Truong, J. Chem. Phys., 96, 2077 (1992) · doi:10.1063/1.462870
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