# zbMATH — the first resource for mathematics

Constructive solution of a bilinear optimal control problem for a Schrödinger equation. (English) Zbl 1153.49023
Summary: Often considered in numerical simulations related to the control of quantum systems, the so-called monotonic schemes have not been so far much studied from the functional analysis point of view. Yet, these procedures provide an efficient constructive method for solving a certain class of optimal control problems. This paper aims both at extending the results already available about these algorithms in the finite-dimensional case (i.e., the time-discretized case) and at completing those of the continuous case. This paper starts with some results about the regularity of a functional related to a wide class of models in quantum chemistry. These enable us to extend an inequality due to Łojasiewicz to the infinite-dimensional case. Finally, some inequalities proving the Cauchy character of the monotonic sequence are obtained, followed by an estimation of the rate of convergence.

##### MSC:
 49K40 Sensitivity, stability, well-posedness 49N90 Applications of optimal control and differential games 35J10 Schrödinger operator, Schrödinger equation
Full Text:
##### References:
 [1] Assion, A.; Baumert, T.; Bergt, M.; Brixner, T.; Kiefer, B.; Seyfried, V.; Strehle, M.; Gerber, G., Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses, Science, 282, 919-922, (1998) [2] Ball, J.M.; Marsden, J.E.; Slemrod, M., Controlability for distributed bilinear systems, SIAM J. contol optim., 20, 575-597, (1982) · Zbl 0485.93015 [3] 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, 188-222, (2005) · Zbl 1109.35094 [4] Baudouin, L., A bilinear optimal control problem applied to a time dependent hartree – fock equation coupled with classical nuclear dynamics, Port. math. (N.S.), 63, 1, 293-325, (2006) · Zbl 1109.49003 [5] Baudouin, L., Existence and regularity of the solution of a time dependent hartree – fock equation coupled with a classical nuclear dynamics, Rev. mat. complut., 18, 2, 285-314, (2005) · Zbl 1162.35450 [6] Beauchard, K., Local controllability of a 1D Schrödinger equation, J. math. pures appl., 84, 7, 851-956, (2005) · Zbl 1124.93009 [7] Brixner, T.; Damrauer, N.H.; Niklaus, P.; Gerber, G., Photoselective adaptive femtosecond quantum control in the liquid phase, Nature, 414, 57-60, (2001) [8] Brown, E.; Rabitz, H., Some mathematical and algorithmic challenges in the control of quantum dynamics phenomena, J. math. chem., 31, 17-63, (2002) · Zbl 0996.81001 [9] Cancès, E.; Le Bris, C.; Pilot, M., Optimal bilinear control for a Schrödinger equation, C. R. acad. sci. Paris, 330, Série 1, 567-571, (2000) · Zbl 0953.49005 [10] Cancès, E.; Le Bris, C.; Maday, Y.; Turinici, G., Mathematical foundations of molecular modelling, (2007), Oxford Univ. Press Oxford [11] T. Cazenave, An introduction to nonlinear Schrödinger equation, third ed., in: Textos de Métodos Matemáticos, vol. 26, Rio de Janeiro, 1996 [12] Kasparian, J.; Rodriguez, M.; Méjean, G.; Yu, J.; Salmon, E.; Wille, H.; Bourayou, R.; Frey, S.; Andr, Y.-B.; Mysyrowicz, A.; Sauerbrey, R.; Wolf, J.-P.; Woste, L., White-light filaments for atmospheric analysis, Science, 301, 61-64, (2003) [13] Haraux, A.; Jendoubi, M.A.; Kavian, O., Rate of decay to equilibrium in some semilinear parabolic equations, J. evol. equ., 3, 463-484, (2003) · Zbl 1036.35035 [14] Ito, K.; Kunisch, K., Optimal bilinear control of an abstract Schrödinger equation, SIAM J. control optim., 46, 274, (2007) · Zbl 1136.35089 [15] Jendoubi, M.A., A simple unified approach to some convergence theorems of L. Simon, J. funct. anal., 153, 187-202, (1998) · Zbl 0895.35012 [16] Le Bris, C.; Ciarlet, Ph.G., Computational chemistry, handbook of numerical analysis, vol. X, (2003), North-Holland [17] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in: Colloques internationaux du CNRS. Les équations aux dérivées partielles, 117, 1963 · Zbl 0234.57007 [18] Łojasiewicz, S., Sur la géométrie semi- et sous-analytique, Ann. inst. Fourier, 43, 1575-1595, (1993) · Zbl 0803.32002 [19] Maday, Y.; Salomon, J.; Turinici, G., Monotonic time-discretized schemes in quantum control, Numer. math., 103, 2, 323-338, (2006) · Zbl 1095.65058 [20] Maday, Y.; Turinici, G., New formulations of monotonically convergent quantum control algorithms, J. chem. phys, 118, 18, (2003) [21] Rabitz, H.; de Vivie-Riedle, R.; Motzkus, M.; Kompa, K., Whither the future of controlling quantum phenomena?, Science, 288, 824-828, (2000) [22] Rabitz, H.; Turinici, G.; Brown, E., Control of quantum dynamics: concepts, procedures and future prospects, (), (special volume) · Zbl 1066.81015 [23] Reed, M.; Simon, B., Methods of modern mathematical physics, II, Fourier analysis, self-adjointness, (1975), Academic Press · Zbl 0308.47002 [24] Salomon, J., Convergence of the time-discretized monotonic schemes, M2an, 41, 1, 77-93, (2007) · Zbl 1124.65059 [25] J. Salomon, Limit points of the monotonic schemes in quantum control, in: Proceedings of the 44th IEEE Conference on Decision and Control, Sevilla, 2005 [26] J. Salomon, Contrôle en chimie quantique: conception et analyse de schémas d’optimisation, Thèse de l’Université Pierre et Marie Curie, 2005 [27] Shi, S.; Woody, A.; Rabitz, H., Optimal control of selective vibrational excitation in harmonic linear chain molecules, J. chem. phys., 88, 6870-6883, (1988) [28] Simon, L., Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of math., 118, 525-571, (1983) · Zbl 0549.35071 [29] Tannor, D.; Kazakov, V.; Orlov, V., Control of photochemical branching: novel procedures for finding optimal pulses and global upper bounds, (), 347-360 [30] G. Turinici, Monotonically convergent algorithms for bounded quantum controls, in: Proceedings of the LHMNLC03 IFAC Conference, 2003, pp. 263-266 [31] Vogt, G.; Krampert, G.; Niklaus, P.; Nuernberger, P.; Gerber, G., Optimal control of photoisomerization, Phys. rev. lett., 94, 68305, (2005) [32] Weinacht, T.; Ahn, J.; Bucksbaum, P., Controlling the shape of a quantum wavefunction, Nature, 397, 233-235, (1999) [33] Zeidler, E., Nonlinear functional analysis and its applications, tome 1, (1985), Springer-Verlag Berlin, New York [34] Zhu, W.; Rabitz, H., A rapid monotonically convergent algorithm for quantum optimal control over the expectation value of a definite operator, J. chem. phys., 109, 385-391, (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.