Simultaneous local exact controllability of 1D bilinear Schrödinger equations
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, pp. 501-529.

We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schrödinger equations on a bounded interval. This is a bilinear control system in which the state is the N-tuple of wave functions. The control is the real amplitude of the laser field. For N=1, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time if N2. Still, for N=2, we prove that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. This is proved using Coron's return method. We also prove that for N3, local controllability does not hold in small time even up to a global phase. Finally, for N=3, we prove that local controllability holds up to a global phase and a global delay.

DOI: 10.1016/j.anihpc.2013.05.001
Keywords: Bilinear control, Schrödinger equation, Simultaneous control, Return method, Non-controllability
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     title = {Simultaneous local exact controllability of {1D} bilinear {Schr\"odinger} equations},
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Morancey, Morgan. Simultaneous local exact controllability of 1D bilinear Schrödinger equations. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, pp. 501-529. doi : 10.1016/j.anihpc.2013.05.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2013.05.001/

[1] S.A. Avdonin, W. Moran, Simultaneous control problems for systems of elastic strings and beams, Systems Control Lett. 44 no. 2 (2001), 147 -155 | MR | Zbl

[2] S.A. Avdonin, M. Tucsnak, Simultaneous controllability in sharp time for two elastic strings, ESAIM Control Optim. Calc. Var. 6 (2001), 259 -273 | EuDML | Numdam | MR | Zbl

[3] J.M. Ball, J.E. Marsden, M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim. 20 no. 4 (1982), 575 -597 | MR | Zbl

[4] K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. (9) 84 no. 7 (2005), 851 -956 | MR | Zbl

[5] K. Beauchard, J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal. 232 no. 2 (2006), 328 -389 | MR | Zbl

[6] K. Beauchard, C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl. (9) 94 no. 5 (2010), 520 -554 | MR | Zbl

[7] K. Beauchard, M. Mirrahimi, Practical stabilization of a quantum particle in a one-dimensional infinite square potential well, SIAM J. Control Optim. 48 no. 2 (2009), 1179 -1205 | MR | Zbl

[8] K. Beauchard, M. Morancey, Local controllability of 1d Schrödinger equations with bilinear control and minimal time, arXiv:1208.5393 (2012) | MR | Zbl

[9] K. Beauchard, V. Nersesyan, Semi-global weak stabilization of bilinear Schrödinger equations, C. R. Math. Acad. Sci. Paris 348 no. 19–20 (2010), 1073 -1078 | MR | Zbl

[10] R.P. Boas, A general moment problem, Amer. J. Math. 63 (1941), 361 -370 | MR | Zbl

[11] J.F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Ser. Oper. Res. , Springer-Verlag, New York (2000) | MR | Zbl

[12] N. Boussaid, M. Caponigro, T. Chambrion, Weakly-coupled systems in quantum control, IEEE Trans. Automat. Control (2013) | MR

[13] E. Cerpa, E. Crépeau, Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 2 (2009), 457 -475 | EuDML | Numdam | MR | Zbl

[14] T. Chambrion, P. Mason, M. Sigalotti, U. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 1 (2009), 329 -349 | EuDML | Numdam | MR | Zbl

[15] T. Chambrion, M. Sigalotti, Simultaneous approximate tracking of density matrices for a system of Schrödinger equations, in: Proceedings of the 48th IEEE Conference on Decision and Control, Shangai, 2009, pp. 357–362.

[16] M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim. 48 no. 3 (2009), 1567 -1599 | MR | Zbl

[17] M. Chapouly, On the global null controllability of a Navier–Stokes system with Navier slip boundary conditions, J. Differential Equations 247 no. 7 (2009), 2094 -2123 | MR | Zbl

[18] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems 5 no. 3 (1992), 295 -312 | MR | Zbl

[19] J.-M. Coron, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels, C. R. Acad. Sci. Paris Sér. I Math. 317 no. 3 (1993), 271 -276 | MR

[20] J.-M. Coron, On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var. 1 (1996), 35 -75 | EuDML | Numdam | MR | Zbl

[21] J.-M. Coron, Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations, ESAIM Control Optim. Calc. Var. 8 (2002), 513 -554 | EuDML | Numdam | MR | Zbl

[22] J.-M. Coron, On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well, C. R. Math. Acad. Sci. Paris 342 no. 2 (2006), 103 -108 | Zbl

[23] J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr. vol. 136 , American Mathematical Society, Providence, RI (2007) | MR

[24] J.-M. Coron, S. Guerrero, Local null controllability of the two-dimensional Navier–Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl. (9) 92 no. 5 (2009), 528 -545 | MR | Zbl

[25] J.-M. Coron, S. Guerrero, L. Rosier, Null controllability of a parabolic system with a cubic coupling term, SIAM J. Control Optim. 48 no. 8 (2010), 5629 -5653 | MR | Zbl

[26] S. Ervedoza, J.-P. Puel, Approximate controllability for a system of Schrödinger equations modelling a single trapped ion, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 6 (2009), 2111 -2136 | EuDML | Numdam | MR | Zbl

[27] A.V. Fursikov, O.Y. Imanuvilov, Exact controllability of the Navier–Stokes and Boussinesq equations, Uspekhi Mat. Nauk 54 no. 3 (327) (1999), 93 -146 | MR | Zbl

[28] O. Glass, Exact boundary controllability of 3-D Euler equation, ESAIM Control Optim. Calc. Var. 5 (2000), 1 -44 | EuDML | Numdam | MR | Zbl

[29] O. Glass, On the controllability of the Vlasov–Poisson system, J. Differential Equations 195 no. 2 (2003), 332 -379 | MR | Zbl

[30] O. Glass, On the controllability of the 1-D isentropic Euler equation, J. Eur. Math. Soc. (JEMS) 9 no. 3 (2007), 427 -486 | EuDML | MR | Zbl

[31] O. Glass, Controllability and asymptotic stabilization of the Camassa–Holm equation, J. Differential Equations 245 no. 6 (2008), 1584 -1615 | MR | Zbl

[32] O. Glass, S. Guerrero, On the uniform controllability of the Burgers equation, SIAM J. Control Optim. 46 no. 4 (2007), 1211 -1238 | MR | Zbl

[33] M.R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific J. Math. 1 (1951), 525 -581 | MR | Zbl

[34] T. Horsin, On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var. 3 (1998), 83 -95 | EuDML | Numdam | MR | Zbl

[35] B. Kapitonov, Simultaneous exact controllability for a class of evolution systems, Comput. Appl. Math. 18 no. 2 (1999), 149 -161 | MR | Zbl

[36] V. Komornik, P. Loreti, Fourier Series in Control Theory, Springer Monogr. Math. , Springer-Verlag, New York (2005) | MR | Zbl

[37] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Rech. Math. Appl. vol. 8 , Masson, Paris (1988) | MR | Zbl

[38] M. Mirrahimi, Lyapunov control of a quantum particle in a decaying potential, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 5 (2009), 1743 -1765 | EuDML | Numdam | MR | Zbl

[39] M. Mirrahimi, P. Rouchon, G. Turinici, Lyapunov control of bilinear Schrödinger equations, Automatica J. IFAC 41 no. 11 (2005), 1987 -1994 | MR | Zbl

[40] M. Morancey, Explicit approximate controllability of the Schrödinger equation with a polarizability term, Math. Control Signals Systems (2012), 1 -26 , http://dx.doi.org/10.1007/s00498-012-0102-2 | MR

[41] V. Nersesyan, Growth of Sobolev norms and controllability of the Schrödinger equation, Comm. Math. Phys. 290 no. 1 (2009), 371 -387 | MR | Zbl

[42] V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 no. 3 (2010), 901 -915 | Numdam | MR | Zbl

[43] V. Nersesyan, H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation: multidimensional case, 2012, preprint, hal-00660478. | MR

[44] P. Rouchon, Control of a quantum particle in a moving potential well, Lagrangian and Hamiltonian Methods for Nonlinear Control 2003, IFAC, Laxenburg (2003), 287 -290 | MR

[45] G. Turinici, On the controllability of bilinear quantum systems, Mathematical Models and Methods for Ab Initio Quantum Chemistry, Lecture Notes in Chem. vol. 74 , Springer-Verlag, Berlin (2000), 75 -92 | MR

[46] G. Turinici, H. Rabitz, Optimally controlling the internal dynamics of a randomly oriented ensemble of molecules, Phys. Rev. A 70 (2004), 063412

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