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Local exact controllability of a one-dimensional nonlinear Schrödinger equation. (English) Zbl 1326.35330
The purpose of this article is to study a one-dimensional nonlinear Schrödinger equation. This is equivalent to a control system in which the state is the wave function \(\Psi\) and the control is the length of the box. First new variables are introduced, which impose a constraint on the control \(u\). This implies that a certain map is surjective. The well-posedeness, Cauchy problem and existence and uniqueness of positive solutions are studied, as well as asymptotic estimates. The Banach fixed point theorem, Galiardo-Nirenberg, Cauchy-Schwartz and Gronvalls inequalities are used in the proofs.

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q93 PDEs in connection with control and optimization
93C20 Control/observation systems governed by partial differential equations
35B09 Positive solutions to PDEs
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[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, Mineola, NY, 1965. · Zbl 0171.38503
[2] R. Adami and U. Boscain, Controllability of the Schroedinger equation via intersection of eigenvalues, Proceedings of the 44rd IEEE Conference on Decision and Control, Seville, Spain, 2005.
[3] J. M. Ball, J. E. Marsden, and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20 (1982), pp. 575–597. · Zbl 0485.93015
[4] Y. Band, B. Malomed, and M. Trippenbach, Adiabaticity in nonlinear quantum dynamics: Bose-Einstein condensate in a time-varying box, Phys. Rev. A, 65 (2002), 033607.
[5] L. Baudouin, A bilinear optimal control problem applied to a time dependent Hartree-Fock equation coupled with classical nuclear dynamics, Port. Math. (N.S.), 63 (2006), pp. 293–325. · Zbl 1109.49003
[6] L. Baudouin, O. Kavian, and J.-P. Puel, Regularity for a Schrödinger equation with singular potential and application to bilinear optimal control, J. Differential Equations, 216 (2005), pp. 188–222. · Zbl 1109.35094
[7] L. Baudouin and J. Salomon, Constructive solutions of a bilinear control problem for a Schrödinger equation, Systems Control Lett., 57 (2008), pp. 453–464. · Zbl 1153.49023
[8] K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl., 84 (2005), pp. 851–956. · Zbl 1124.93009
[9] K. Beauchard, Controllability of a quantum particle in a 1D variable domain, ESAIM Control Optim. Calc. Var., 14 (2008), pp. 105–147. · Zbl 1132.35446
[10] K. Beauchard, Local controllability and non controllability for a 1D wave equation with bilinear control, J. Differential Equations, 250 (2010), pp. 2064–2098. · Zbl 1221.35221
[11] K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal., 232 (2006), pp. 328–389. · Zbl 1188.93017
[12] K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl., 94 (2010), pp. 520–554. · Zbl 1202.35332
[13] K. Beauchard and M. Mirrahimi, Practical stabilization of a quantum particle in a one-dimensional infinite square potential well, SIAM J. Control Optim., 48 (2009), pp. 1179–1205. · Zbl 1194.93176
[14] K. Beauchard and M. Morancey, Local controllability of 1D Schrödinger equations with bilinear control and minimal time, Math. Control Rel. Fields, 4 (2014). · Zbl 1281.93016
[15] A. M. Bloch, R. W. Brockett, and C. Rangan, Finite controllability of infinite-dimensional quantum systems, IEEE Trans. Automat. Control, 55 (2010), pp. 1797–1805. · Zbl 1368.81086
[16] U. Boscain, M. Caponigro, T. Chambrion, and M. Sigalotti, A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule, Commun. Math. Phys., 311 (2012), pp. 423–455. · Zbl 1267.35177
[17] N. Boussaid, M. Caponigro, and T. Chambrion, Weakly-coupled systems in quantum control, IEEE Trans. Automat. Control, 58 (2013), pp. 2205–2216. · Zbl 1369.81042
[18] H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Dunod, Paris, 1999.
[19] C. Brif, R. Chakrabarti, and H. Rabitz, Control of quantum phenomena, Adv. Chem. Phys., 148 (2012), pp. 1–76.
[20] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity, Phys. Rev. A, 62 (2010), 063610.
[21] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II, Case of attractive nonlinearity, Phys. Rev. A, 62 (2000), 063611.
[22] R. Castelli and H. Teismann, Rigorous Numerics for NLS: Bound States, Spectra, and Controllability, preprint, 2013. · Zbl 1417.65193
[23] T. Chambrion, P. Mason, M. Sigalotti, and M. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), pp. 329–349. · Zbl 1161.35049
[24] J.-M. Coron, On the small-time local controllability of a quantum particule in a moving one-dimensional infinite square potential well, C. R. Acad. Sci. Paris Ser. I, 342 (2006), pp. 103–108.
[25] J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr. 136, AMS, Providence, RI, 2007.
[26] A. del Campo and M. G. Boshier, Shortcuts to adiabaticity in a time-dependent box, Scientific Reports, 2 (2012), 648.
[27] E. Cances, C. Le Bris, and M. Pilot, Contrôle optimal bilinéaire d’une équation de Schrödinger, C. R. Acad. Sci. Paris, 330 (2000), pp. 567–571. · Zbl 0953.49005
[28] S. Ervedoza and J.-P. Puel, Approximate controllability for a system of Schrödinger equations modeling a single trapped ion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 2009. · Zbl 1180.35437
[29] B. Feng, J. Liu, and J. Zheng, Optimal bilinear control of nonlinear Hartree equation in \(R ^3\), Electron. J. Differential Equations, 130 (2013), pp. 1–14. · Zbl 1288.35430
[30] A. Gaunt, T. Schmidutz, I. Gotlibovych, R. Smith, and Z. Hadzibabic, Bose-Einstein condensation of atoms in a uniform potential, Phys. Rev. Lett., 110 (2013), 200406.
[31] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), pp. 457–465. · Zbl 0685.93039
[32] M. Hintermüller, D. Marahrens, P. Markowich, and C. Sparber, Optimal bilinear control of Gross–Pitaevskii equations, SIAM J. Control Optim., 51 (2013), pp. 2509–2543. · Zbl 1277.49005
[33] U. Hohenester, OCTBEC—A Matlab toolbox for optimal quantum control of Bose–Einstein condensates, Comput. Phys. Comm., 185 (2014), pp. 194–216.
[34] R. Hryniv and P. Lancaster, On the perturbation of analytic matrix functions, Integral Equations Oper. Theory, 34 (1999), pp. 325–338. · Zbl 0940.47008
[35] G. M. Huang, T. J. Tarn, and J. W. Clark, On the controllability of quantum-mechanical systems, J. Math. Phys., 24 (1983), pp. 2608–2618. · Zbl 0539.93003
[36] R. Illner, H. Lange, and H. Teismann, Limitations on the control of Schrödinger equations, ESAIM Control Optim. Calc. Var., 12 (2006), pp. 615–635. · Zbl 1162.93316
[37] T. Kato, Perturbation Theory for Linear Operator, Springer, Berlin, 1980.
[38] A. Y. Khapalov, Bilinear controllability properties of a vibrating string with variable axial load and damping gain, Dyn. Contin. Discrete Impuls. Syst. Ser A Math Anal., 10 (2003), pp. 721–743. · Zbl 1035.35015
[39] A. Y. Khapalov, Controllability properties of a vibrating string with variable axial load, Discrete Contin. Dyn. Syst., 11 (2004), pp. 311–324. · Zbl 1175.93032
[40] A. Y. Khapalov, Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping, ESAIM Control Optim. Calc. Var., 12 (2006), pp. 231–252. · Zbl 1105.93011
[41] H. Lange and H. Teismann, Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state, Math. Methods Appl. Sci., 30 (2007), pp. 1483–1505. · Zbl 1134.93008
[42] T. P. Meyrath, F. Schreck, J. L. Hanssen, C. S. Chuu, and M. G. Raizen, Bose-Einstein condensate in a box, Phys. Rev. A, 71 (2005), 041604.
[43] M. Mirrahimi, Lyapunov control of a quantum particle in a decaying potential, Ann. Inst. H. Poincaré Anal Non Linéare, 26 (2009), pp. 1743–1765. · Zbl 1176.35169
[44] M. Mirrahimi and P. Rouchon, Controllability of quantum harmonic oscillators, IEEE Trans. Automat. Control, 49 (2004), pp. 745–747. · Zbl 1365.81065
[45] K. Nakamura, Z. Sobirov, D. Matrasulov, and S. Avazbaev, Bernoulli’s formula and poisson’s equations for a confined quantum gas: Effects due to a moving piston, Phys. Rev. E, 86 (2012), 061128.
[46] V. Nersesyan, Growth of Sobolev norms and controllability of Schrödinger equation, Comm. Math. Phys., 290 (2009), pp. 371–387. · Zbl 1180.93017
[47] V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. Inst. H. Poincaré Anal Non Linéare, 27 (2010), pp. 901–915. · Zbl 1191.35257
[48] V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation, J. Math. Pures Appl., 97 (2012), pp. 295–317. · Zbl 1263.93038
[49] F. Pinsker, Computing with soliton trains in Bose-Einstein condensates, Internat. J. Modern Phys. C, 26 (2013).
[50] M. W. Ray, E. Ruokokoski, S. Kandel, M. Möttönen, and D. S. Hall, Observation of Dirac monopoles in a synthetic magnetic field, Nature, 505 (2014), pp. 657–660.
[51] I. Rodnianski, W. Schlag, and A. Soffer, Asymptotic stability of N-soliton states of NLS, arXiv:math/0309114, 2005.
[52] L. Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var., 10 (2004), pp. 346–380. · Zbl 1094.93014
[53] C. Sabín, D. Bruschi, M. Ahmadi, and I. Fuentes, Phonon creation by gravitational waves, New J. Phys., 16 (2014), 085003.
[54] T. Schmidutz, I. Gotlibovych, A. Gaunt, R. Smith, N. Navon, and Z. Hadzibabic, Quantum Joule-Thomson effect in a saturated homogeneous Bose gas, Phys. Rev. Lett., 112 (2014), 040403.
[55] D. Stefanatos, Optimal shortcuts to adiabaticity for a quantum piston, Automatica, 49 (2013), pp. 3079–3083. · Zbl 1320.81064
[56] D. Stefanatos and J. S. Li, Frictionless decompression in minimum time of Bose-Einstein condensates in the Thomas-Fermi regime, Phys. Rev. A, 86 (2012), 063602.
[57] H. Teismann, Generalized coherent states and the control of quantum systems, J. Math. Phys., 46 (2005), 122106. · Zbl 1111.81315
[58] S. Theodorakis and C. Psaroudaki, Oscillations of a Bose–Einstein condensate in a rapidly contracting circular box, Phys. Lett. A, 373 (2009), pp. 441–447. · Zbl 1227.82051
[59] E. Torrontegui, S. Ibán͂ez, S. Martínez-Garaot, M. Modugno, A. Del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, J. G. Muga, et al., Shortcuts to adiabaticity, Adv. Atomic Molecular Optical Phys., 62 (2013), pp. 117–169.
[60] G. Turinici, On the controllability of bilinear quantum systems, in Mathematical Models and Methods for Ab Initio Quantum Chemistry, C. Le Bris and M. Defranceschi, eds., Lecture Notes in Chemistry 74, Springer, New York, 2000.
[61] S. van Frank, A. Negretti, T. Berrada, R. Bücker, S. Montangero, J.-F. Schaff, T. Schumm, T. Calarco, and J. Schmiedmayer, Interferometry with non-classical motional states of a Bose–Einstein condensate, Nat. Commun., 5 (2014).
[62] S. Woo, S. Choi, and N. Bigelow, Controlling quasiparticle excitations in a trapped Bose-Einstein condensate, Phys. Rev. A, 72 (2005), 021605.
[63] E. Zuazua, Remarks on the controllability of the Schrödinger equation, CRM Proc. Lecture Notes, 33 (2003), pp. 193–211.
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