Simultaneous local exact controllability of 1D bilinear Schrödinger equations

Morancey, Morgan

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 501-529 / Harvested from Numdam

Résumé

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 $ N⩾2$. 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 $ N⩾3$, 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.

Détail

Publié le : 2014-01-01

MR 3208452 | Zbl 1295.93015

DOI : https://doi.org/10.1016/j.anihpc.2013.05.001

Bibliographie

[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 2007587 | Zbl 0986.93037

[2] S.A. Avdonin, M. Tucsnak, Simultaneous controllability in sharp time for two elastic strings, ESAIM Control Optim. Calc. Var. 6 (2001), 259 -273 | Numdam | MR 1816075 | Zbl 0995.93036

[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 661034 | Zbl 0485.93015

[4] K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. (9) 84 no. 7 (2005), 851 -956 | MR 2144647 | Zbl 1124.93009

[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 2200740 | Zbl 1188.93017

[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 2732927 | Zbl 1202.35332

[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 2491595 | Zbl 1194.93176

[8] K. Beauchard, M. Morancey, Local controllability of 1d Schrödinger equations with bilinear control and minimal time, arXiv:1208.5393 (2012) | MR 3167929 | Zbl 1281.93016

[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 2735010 | Zbl 1205.35051

[10] R.P. Boas, A general moment problem, Amer. J. Math. 63 (1941), 361 -370 | MR 3848 | Zbl 0025.25404

[11] J.F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Ser. Oper. Res. , Springer-Verlag, New York (2000) | MR 1756264 | Zbl 0766.90076

[12] N. Boussaid, M. Caponigro, T. Chambrion, Weakly-coupled systems in quantum control, IEEE Trans. Automat. Control (2013) | MR 3101605

[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 | Numdam | MR 2504039 | Zbl 1158.93006

[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 | Numdam | MR 2483824 | Zbl 1161.35049

[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 2516179 | Zbl 1282.93050

[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 2560050 | Zbl 1178.35285

[18] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems 5 no. 3 (1992), 295 -312 | MR 1164379 | Zbl 0760.93067

[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 1233425

[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 | Numdam | MR 1393067 | Zbl 0872.93040

[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 | Numdam | MR 1932962 | Zbl 1071.76012

[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 1082.93002

[23] J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr. vol. 136 , American Mathematical Society, Providence, RI (2007) | MR 2302744

[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 2558423 | Zbl 1179.35209

[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 2745788 | Zbl 1213.35247

[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 | Numdam | MR 2569888 | Zbl 1180.35437

[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 1728643 | Zbl 0970.35116

[28] O. Glass, Exact boundary controllability of 3-D Euler equation, ESAIM Control Optim. Calc. Var. 5 (2000), 1 -44 | Numdam | MR 1745685 | Zbl 0940.93012

[29] O. Glass, On the controllability of the Vlasov–Poisson system, J. Differential Equations 195 no. 2 (2003), 332 -379 | MR 2016816 | Zbl 1109.93007

[30] O. Glass, On the controllability of the 1-D isentropic Euler equation, J. Eur. Math. Soc. (JEMS) 9 no. 3 (2007), 427 -486 | MR 2314104 | Zbl 1139.35014

[31] O. Glass, Controllability and asymptotic stabilization of the Camassa–Holm equation, J. Differential Equations 245 no. 6 (2008), 1584 -1615 | MR 2436454 | Zbl 1186.35185

[32] O. Glass, S. Guerrero, On the uniform controllability of the Burgers equation, SIAM J. Control Optim. 46 no. 4 (2007), 1211 -1238 | MR 2346380 | Zbl 1140.93013

[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 46590 | Zbl 0045.20806

[34] T. Horsin, On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var. 3 (1998), 83 -95 | Numdam | MR 1612027 | Zbl 0897.93034

[35] B. Kapitonov, Simultaneous exact controllability for a class of evolution systems, Comput. Appl. Math. 18 no. 2 (1999), 149 -161 | MR 2098345 | Zbl 1125.93314

[36] V. Komornik, P. Loreti, Fourier Series in Control Theory, Springer Monogr. Math. , Springer-Verlag, New York (2005) | MR 2114325 | Zbl 1094.49002

[37] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Rech. Math. Appl. vol. 8 , Masson, Paris (1988) | MR 963060 | Zbl 0653.93002

[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 | Numdam | MR 2566708 | Zbl 1176.35169

[39] M. Mirrahimi, P. Rouchon, G. Turinici, Lyapunov control of bilinear Schrödinger equations, Automatica J. IFAC 41 no. 11 (2005), 1987 -1994 | MR 2168664 | Zbl 1125.93466

[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 3084375

[41] V. Nersesyan, Growth of Sobolev norms and controllability of the Schrödinger equation, Comm. Math. Phys. 290 no. 1 (2009), 371 -387 | MR 2520519 | Zbl 1180.93017

[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 2629885 | Zbl 1191.35257

[43] V. Nersesyan, H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation: multidimensional case, 2012, preprint, hal-00660478. | MR 2899810

[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 2082989

[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 1855575

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