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Approximate controllability for a system of Schrödinger equations modeling a single trapped ion. (English) Zbl 1180.35437
Summary: We analyze the approximate controllability properties for a system of Schrödinger equations modeling a single trapped ion. The control we use has a special form, which takes its origin from practical limitations. Our approach is based on the controllability of an approximate finite dimensional system for which one can design explicitly exact controls. We then justify the approximations which link up the complete and approximate systems. This yields approximate controllability results in the natural space \((L^2(\mathbb R))^2\) and also in stronger spaces corresponding to the domains of powers of the harmonic operator.

MSC:
35Q41 Time-dependent Schrödinger equations and Dirac equations
93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
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References:
[1] R. Adami, U. Boscain, Controllability of the Schroedinger equation via intersection of eigenvalues, in: Proc. of the 44rd IEEE Conf. on Decision and Control, 2005
[2] Ball, J.M.; Marsden, J.E.; Slemrod, M., Controllability for distributed bilinear systems, SIAM J. control optim., 20, 4, 575-597, (1982) · Zbl 0485.93015
[3] Baudouin, L., A bilinear optimal control problem applied to a time dependent hartree – fock equation coupled with classical nuclear dynamics, Portugal math. (N.S.), 63, 3, 293-325, (2006) · Zbl 1109.49003
[4] 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, 188-222, (2005) · Zbl 1109.35094
[5] Baudouin, L.; Salomon, J., Constructive solution of a bilinear control problem, C. R. math. acad. sci. Paris, ser. I, 342, 2, 119-124, (2006) · Zbl 1079.49021
[6] Beauchard, K., Local controllability of a 1-D Schrödinger equation, J. math. pures appl. (9), 84, 7, 851-956, (2005) · Zbl 1124.93009
[7] Beauchard, K., Controllability of a quantum particle in a 1D variable domain, ESAIM control optim. calc. var., 14, 1, 105-147, (2008) · Zbl 1132.35446
[8] Beauchard, K.; Coron, J.-M., Controllability of a quantum particle in a moving potential well, J. funct. anal., 232, 2, 328-389, (2006) · Zbl 1188.93017
[9] Beauchard, K.; Coron, J.-M.; Mirrahimi, M.; Rouchon, P., Implicit Lyapunov control of finite dimensional Schrödinger equations, Systems control lett., 56, 5, 388-395, (2007) · Zbl 1110.81063
[10] K. Beauchard, M. Mirrahimi, Approximate stabilization of a quantum particle in a 1D infinite potential well, in: IFAC World Congress, Seoul, 2008 · Zbl 1194.93176
[11] A.M. Bloch, R.W. Brockett, C. Rangan, The controllability of infinite quantum systems and closed subspace criteria, IEEE Trans. Automat. Control, submitted for publication · Zbl 1368.81086
[12] Brezis, H., Analyse fonctionnelle, Collection mathématiques appliquées pour la maîtrise, (1983), Masson Paris, Théorie et applications (Theory and applications) · Zbl 0511.46001
[13] Chambrion, T.; Mason, P.; Sigalotti, M.; Boscain, U., Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. inst. H. Poincaré anal. non linéaire, 26, 1, 329-349, (2009) · Zbl 1161.35049
[14] Coron, J.-M., On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well, C. R. acad. sci. Paris, ser. I, 342, 2, 103-108, (2006) · Zbl 1082.93002
[15] Coron, J.-M., Control and nonlinearity, Mathematical surveys and monographs, vol. 136, (2007), American Mathematical Society Providence, RI
[16] Ito, K.; Kunisch, K., Optimal bilinear control of an abstract Schrödinger equation, SIAM J. control optim., 46, 1, 274-287, (2007), (electronic) · Zbl 1136.35089
[17] Law, C.K.; Eberly, J.H., Arbitrary control of a quantum electromagnetic field, Phys. rev. lett., 76, 7, 1055-1058, (1996)
[18] Mirrahimi, M., Lyapunov control of a quantum particle in a decaying potential, Ann. inst. H. Poincaré anal. non linéaire, 26, 5, 1743-1765, (2009) · Zbl 1176.35169
[19] M. Mirrahimi, Lyapunov control of a particle in a finite quantum potential well, in: IEEE Conf. on Decision and Control, 2006
[20] Mirrahimi, M.; Rouchon, P., Controllability of quantum harmonic oscillators, IEEE trans. automat. control, 49, 5, 745-747, (2004) · Zbl 1365.81065
[21] Mirrahimi, M.; Rouchon, P.; Turinici, G., Lyapunov control of bilinear Schrödinger equations, Automatica J. IFAC, 41, 11, 1987-1994, (2005) · Zbl 1125.93466
[22] V. Nersesyan, Growth of Sobolev norms and controllability of Schrödinger equation, Preprint, 2008
[23] Rangan, C.; Bloch, A.M., Control of finite-dimensional quantum systems: application to a spin-\(\frac{1}{2}\) particle coupled with a finite quantum harmonic oscillator, J. math. phys., 46, 3, 032106, (2005) · Zbl 1076.81010
[24] Reed, M.; Simon, B., Methods of modern mathematical physics. I. functional analysis, (1980), Academic Press Inc. (Harcourt Brace Jovanovich Publishers) New York
[25] Rouchon, P., Quantum systems and control, Arima, 9, 325-357, (2008)
[26] Turinici, G., On the controllability of bilinear quantum systems, (), 75-92
[27] Turinici, G.; Rabitz, H., Wavefunction controllability for finite-dimensional bilinear quantum systems, J. phys. A, 36, 10, 2565-2576, (2003) · Zbl 1064.81558
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