×

zbMATH — the first resource for mathematics

Lyapunov control of a quantum particle in a decaying potential. (English) Zbl 1176.35169
Summary: A Lyapunov-based approach for the trajectory generation of an \(N\)-dimensional Schrödinger equation in whole \(\mathbb R^N\) is proposed. For the case of a quantum particle in an \(N\)-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
35P05 General topics in linear spectral theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
93C20 Control/observation systems governed by partial differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML arXiv
References:
[1] Agmon, S., Spectral properties of Schrödinger operators and scattering theory, Ann. scuola norm. sup. Pisa cl. sci., 4, 2, 151-218, (1975) · Zbl 0315.47007
[2] Agmon, S., Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators, Mathematical notes, vol. 29, (1982), Princeton University Press · Zbl 0503.35001
[3] Albertini, F.; D’Alessandro, D., Notions of controllability for bilinear multilevel quantum systems, IEEE trans. automat. control, 48, 8, 1399-1403, (2003) · Zbl 1364.93059
[4] Altafini, C., Controllability of quantum mechanical systems by root space decomposition of su(n), J. math. phys., 43, 5, 2051-2062, (2002) · Zbl 1059.93016
[5] Avron, J.E.; Elgart, A., Adiabatic theorem without a gap condition, Comm. math. phys., 203, 445-463, (1999) · Zbl 0936.47047
[6] Baudouin, L.; Puel, J.P.; Kavian, O., Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. differential equations, 216, 188-222, (2005) · Zbl 1109.35094
[7] Baudouin, L.; Salomon, J., Constructive solutions of a bilinear control problem, C. R. acad. sci. Paris, ser. I, 342, 2, 119-124, (2006) · Zbl 1079.49021
[8] Beauchard, K., Local controllability of a 1-D Schrödinger equation, J. math. pures appl., 84, 851-956, (2005) · Zbl 1124.93009
[9] Beauchard, K.; Coron, J.-M.; Mirrahimi, M.; Rouchon, P., Implicit Lyapunov control of finite dimensional Schrödinger equations, Systems control lett., 56, 388-395, (2007) · Zbl 1110.81063
[10] Beauchard, K.; Coron, J.M., Controllability of a quantum particle in a moving potential well, J. funct. anal., 232, 328-389, (2006) · Zbl 1188.93017
[11] Beauchard, K.; Mirrahimi, M., Practical stabilization of a quantum particle in a 1D infinite square potential well, SIAM J. Control Optim., in press, preliminary version: · Zbl 1194.93176
[12] Chambrion, T.; Mason, P.; Sigalotti, M.; Boscain, U., Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. I. H. Poincaré - AN, 26, 329-349, (2009) · Zbl 1161.35049
[13] Chen, Y.; Gross, P.; Ramakrishna, V.; Rabitz, H.; Mease, K., Competitive tracking of molecular objectives described by quantum mechanics, J. chem. phys., 102, 8001-8010, (1995)
[14] Coron, J.-M.; d’Andréa Novel, B., Stabilization of a rotating body-beam without damping, IEEE trans. automat. control, 43, 5, 608-618, (1998) · Zbl 0908.93055
[15] Coron, J.-M.; d’Andrá Novel, B.; Bastin, G., A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE trans. automat. control, 52, 1, 2-11, (2007) · Zbl 1366.93481
[16] Coron, J.M., Global stabilization for controllable systems without drift, Math. control signals systems, 5, 295-312, (1992) · Zbl 0760.93067
[17] Coron, J.M., On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain, SIAM J. control optim., 37, 1874-1896, (1999) · Zbl 0954.76010
[18] Coron, J.M., Control and nonlinearity, Mathematical surveys and monographs, vol. 136, (2007), American Mathematica Society USA
[19] Glass, O., Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: the multiconnected case, SIAM J. control optim., 44, 3, 1105-1147, (2005) · Zbl 1130.93403
[20] O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, preprint, 2007
[21] Goldberg, M., Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials, Geom. funct. anal., 16, 3, 517-536, (2006) · Zbl 1158.35408
[22] Goldberg, M.; Schlag, W., Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. math. phys., 251, 157-178, (2004) · Zbl 1086.81077
[23] Goldberg, M.; Schlag, W., A limiting absorption principle for the three-dimensional Schrödinger equation with \(L^p\) potentials, Int. math. res. notices, 75, 4049-4071, (2004) · Zbl 1069.35063
[24] Goldberg, M.; Visan, M., A counterexample to dispersive estimates for Schrödinger operators in higher dimensions, Comm. math. phys., 266, 1, 211-238, (2006) · Zbl 1110.35073
[25] Van Handel, R.; Stockton, J.K.; Mabuchi, H., Modeling and feedback control design for quantum state preparation, J. opt. B: quant. semiclass. opt., 7, S179-S197, (2005), Special issue on quantum control
[26] Haroche, S., Contrôle de la décohérence: théorie et expériences, 2004, Notes de cours, Collège de France
[27] Jensen, A.; Kato, T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke math. J., 46, 3, 583-611, (1979) · Zbl 0448.35080
[28] Jensen, A.; Yajima, K., A remark on \(L^p\)-boundedness of wave operators for two-dimensional Schrödinger operators, Comm. math. phys., 225, 3, 633-637, (2002) · Zbl 1057.47011
[29] Journé, J.-L.; Soffer, A.; Sogge, C.D., Decay estimates for Schrödinger operators, Comm. pure appl. math., 44, (1991) · Zbl 0743.35008
[30] Kato, T., Perturbation theory for linear operators, (1966), Springer · Zbl 0148.12601
[31] Keel, M.; Tao, T., Endpoint Strichartz estimates, Amer. J. math., 5, 955-980, (1998) · Zbl 0922.35028
[32] Li, B.; Turinici, G.; Ramakrishna, V.; Rabitz, H., Optimal dynamic discrimination of similar molecules through quantum learning control, J. phys. chem. B, 106, 33, 8125-8131, (2002)
[33] M. Mirrahimi, Lyapunov control of a particle in a finite quantum potential well, in: CDC, San Diego, 2006
[34] Mirrahimi, M.; Rouchon, P.; Turinici, G., Lyapunov control of bilinear Schrödinger equations, Automatica, 41, 1987-1994, (2005) · Zbl 1125.93466
[35] Mirrahimi, M.; Turinici, G.; Rouchon, P., Reference trajectory tracking for locally designed coherent quantum controls, J. phys. chem. A, 109, 2631-2637, (2005)
[36] Mirrahimi, M.; Van Handel, R., Stabilizing feedback controls for quantum systems, SIAM J. control optim., 46, 2, 445-467, (2007) · Zbl 1136.81342
[37] Ramakrishna, V.; Salapaka, M.; Dahleh, M.; Rabitz, H., Controllability of molecular systems, Phys. rev. A, 51, 2, 960-966, (1995)
[38] Rauch, J., Local decay of scattering solutions to Schrödinger’s equation, Comm. math. phys., 61, 2, 149-168, (1978) · Zbl 0381.35023
[39] Reed, M.; Simon, B., Methods of modern mathematical physics, vol. IV: analysis of operators, (1978), Academic Press New York · Zbl 0401.47001
[40] Rodnianski, I.; Schlag, W., Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. math., 155, 451-513, (2004) · Zbl 1063.35035
[41] Schlag, W., Dispersive estimates for Schrödinger operators in two dimensions, Comm. math. phys., 257, 1, 87-117, (2005) · Zbl 1134.35321
[42] Shi, S.; Woody, A.; Rabitz, H., Optimal control of selective vibrational excitation in harmonic linear chain molecules, J. chem. phys., 88, 11, 6870-6883, (1988)
[43] Stoiciu, M., An estimate for the number of bound states of the Schrödinger operator in two dimensions, Proc. amer. math. soc., 132, 4, 1143-1151, (2004) · Zbl 1039.35071
[44] Strichartz, R., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke math. J., 44, 3, 705-714, (1977) · Zbl 0372.35001
[45] Sugawara, M., General formulation of locally designed coherent control theory for quantum systems, J. chem. phys., 118, 15, 6784-6800, (2003)
[46] Sussmann, H.J.; Jurdjevic, V., Controllability of nonlinear systems, J. differential equations, 12, 95-116, (1972) · Zbl 0242.49040
[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] Turinici, G.; Rabitz, H., Wavefunction controllability in quantum systems, J. phys. A, 36, 2565-2576, (2003) · Zbl 1064.81558
[50] Weder, R., \(L^p\)-\(L^{p^\prime}\) estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. funct. anal., 170, 1, 37-68, (2000) · Zbl 0943.34070
[51] Yajima, K., The \(W^{k, p}\)-continuity of wave operators for Schrödinger operators, J. math. soc. Japan, 47, 3, 551-581, (1995) · Zbl 0837.35039
[52] Yajima, K., \(L^p\)-boundedness of wave operators for two-dimensional Schrödinger operators, Comm. math. phys., 208, 1, 125-152, (1999) · Zbl 0961.47004
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.