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June  2014, 4(2): 125-160. doi: 10.3934/mcrf.2014.4.125

Local controllability of 1D Schrödinger equations with bilinear control and minimal time

Karine Beauchard 1, and  Morgan Morancey 1,

1. 

CMLS, Ecole Polytechnique, 91 128 Palaiseau cedex, France, France

Received  July 2012 Revised  January 2013 Published  February 2014

We consider a linear Schrödinger equation, on a bounded interval, with bilinear control.
    In [10], Beauchard and Laurent prove that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. In [18], Coron proves that a positive minimal time is required for this controllability result, on a particular degenerate example.
    In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution.
Keywords: minimal time, Exact controllability, Schrödinger equation, bilinear control, power series expansion..
Mathematics Subject Classification: 93B05, 93C20, 81Q9.
Citation: Karine Beauchard, Morgan Morancey. Local controllability of 1D Schrödinger equations with bilinear control and minimal time. Mathematical Control & Related Fields, 2014, 4 (2) : 125-160. doi: 10.3934/mcrf.2014.4.125
References:
[1]

R. Adami and U. Boscain, Controllability of the Schroedinger Equation via Intersection of Eigenvalues, Proceedings of the 44rd IEEE Conference on Decision and Control December 12-15, 2005, Seville, (Spain). Also on 'Control Systems: Theory, Numerics and Applications, Roma, Italia 30 Mar - 1 Apr 2005, POS, Proceeding of science. Google Scholar

[2]

J. Ball, J. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim., 20 (1982), 575-597. doi: 10.1137/0320042.  Google Scholar

[3]

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), 293-325.  Google Scholar

[4]

L. Baudouin, O. Kavian and J.-P. Puel, Regularity for a Schrödinger equation with singular potential and application to bilinear optimal control, J. of Differential Equations, 216 (2005), 188-222. doi: 10.1016/j.jde.2005.04.006.  Google Scholar

[5]

L. Baudouin and J. Salomon, Constructive solutions of a bilinear control problem for a Schrödinger equation, Systems and Control Letters, 57 (2008), 453-464. doi: 10.1016/j.sysconle.2007.11.002.  Google Scholar

[6]

K. Beauchard, Local Controllability of a 1-D Schrödinger equation, J. Math. Pures et Appl., 84 (2005), 851-956. doi: 10.1016/j.matpur.2005.02.005.  Google Scholar

[7]

K. Beauchard, Controllability of a quantum particle in a 1D variable domain, ESAIM:COCV, 14 (2008), 105-147. doi: 10.1051/cocv:2007047.  Google Scholar

[8]

K. Beauchard, Local controllability and non controllability for a 1D wave equation with bilinear control, J. Diff. Eq., 250 (2010), 2064-2098. doi: 10.1016/j.jde.2010.10.008.  Google Scholar

[9]

K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Functional Analysis, 232 (2006), 328-389. doi: 10.1016/j.jfa.2005.03.021.  Google Scholar

[10]

K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl., 94 (2010), 520-554. doi: 10.1016/j.matpur.2010.04.001.  Google Scholar

[11]

K. Beauchard and M. Mirrahimi, Practical stabilization of a quantum particle in a one-dimensional infinite square potential well, SIAM J. Contr. Optim., 48 (2009), 1179-1205. doi: 10.1137/070704204.  Google Scholar

[12]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000.  Google Scholar

[13]

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, Communications on Mathematical Physics, 311 (2012), 423-455. doi: 10.1007/s00220-012-1441-z.  Google Scholar

[14]

N. Boussaïd, M. Caponigro and T. Chambrion, Weakly-coupled systems in quantum control, IEEE Transactions on Automatic Control, 58 (2013), 2205-2216. arXiv:1109.1900. doi: 10.1109/TAC.2013.2255948.  Google Scholar

[15]

E. Cancès, C. L. Bris and M. Pilot, Contrôle optimal bilinéaire d'une équation de Schrödinger, CRAS Paris, 330 (2000), 567-571. doi: 10.1016/S0764-4442(00)00227-5.  Google Scholar

[16]

E. Cerpa and E. Crépeau, Boundary controlability for the non linear korteweg-de vries equation on any critical domain, Ann. IHP Analyse Non Linéaire, 26 (2009), 457-475. doi: 10.1016/j.anihpc.2007.11.003.  Google Scholar

[17]

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), 329-349. doi: 10.1016/j.anihpc.2008.05.001.  Google Scholar

[18]

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. Sciences Paris, Ser. I, 342 (2006), 103-108. doi: 10.1016/j.crma.2005.11.004.  Google Scholar

[19]

J.-M. Coron, Control and Nonlinearity, vol. 136, Mathematical Surveys and Monographs, 2007.  Google Scholar

[20]

S. Ervedoza and J.-P. Puel, Approximate controllability for a system of schrödinger equations modeling a single trapped ion, Ann.IHP: Nonlinear Analysis, 26 (2009), 2111-2136. doi: 10.1016/j.anihpc.2009.01.005.  Google Scholar

[21]

R. Ilner, H. Lange and H. Teismann, Limitations on the control of schrödinger equations, ESAIM:COCV, 12 (2006), 615-635. doi: 10.1051/cocv:2006014.  Google Scholar

[22]

A. Y. Khapalov, Bilinear controllability properties of a vibrating string with variable axial load and damping gain, Dyn. Contin. Impuls. Syst. Ser A Math Anal., 10 (2003), 721-743.  Google Scholar

[23]

A. Y. Khapalov, Controllability properties of a vibrating string with variable axial load, Discrete Contin. Dyn. Syst., 11 (2004), 311-324. doi: 10.3934/dcds.2004.11.311.  Google Scholar

[24]

A. Y. Khapalov, Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping, ESAIM:COCV, 12 (2006), 231-252. doi: 10.1051/cocv:2006001.  Google Scholar

[25]

M. Mirrahimi, Lyapunov control of a quantum particle in a decaying potential, Ann. IHP: Nonlinear Analysis, 26 (2009), 1743-1765. doi: 10.1016/j.anihpc.2008.09.006.  Google Scholar

[26]

V. Nersesyan, Growth of Sobolev norms and controllability of Schrödinger equation, Comm. Math. Phys., 290 (2009), 371-387. doi: 10.1007/s00220-009-0842-0.  Google Scholar

[27]

V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. IHP Nonlinear Analysis, 27 (2010), 901-915. doi: 10.1016/j.anihpc.2010.01.004.  Google Scholar

[28]

V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation, J. Math. Pures et Appl., 97 (2012), 295-317. doi: 10.1016/j.matpur.2011.11.005.  Google Scholar

[29]

V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation: multidimensional case,, (preprint)., ().   Google Scholar

[30]

G. Turinici, On the controllability of bilinear quantum systems, In C. Le Bris and M. Defranceschi, editors, Mathematical Models and Methods for Ab Initio Quantum Chemistry, 75-92, Lecture Notes in Chem., 74, Springer, Berlin, 2000. doi: 10.1007/978-3-642-57237-1_4.  Google Scholar

show all references

References:
[1]

R. Adami and U. Boscain, Controllability of the Schroedinger Equation via Intersection of Eigenvalues, Proceedings of the 44rd IEEE Conference on Decision and Control December 12-15, 2005, Seville, (Spain). Also on 'Control Systems: Theory, Numerics and Applications, Roma, Italia 30 Mar - 1 Apr 2005, POS, Proceeding of science. Google Scholar

[2]

J. Ball, J. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim., 20 (1982), 575-597. doi: 10.1137/0320042.  Google Scholar

[3]

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), 293-325.  Google Scholar

[4]

L. Baudouin, O. Kavian and J.-P. Puel, Regularity for a Schrödinger equation with singular potential and application to bilinear optimal control, J. of Differential Equations, 216 (2005), 188-222. doi: 10.1016/j.jde.2005.04.006.  Google Scholar

[5]

L. Baudouin and J. Salomon, Constructive solutions of a bilinear control problem for a Schrödinger equation, Systems and Control Letters, 57 (2008), 453-464. doi: 10.1016/j.sysconle.2007.11.002.  Google Scholar

[6]

K. Beauchard, Local Controllability of a 1-D Schrödinger equation, J. Math. Pures et Appl., 84 (2005), 851-956. doi: 10.1016/j.matpur.2005.02.005.  Google Scholar

[7]

K. Beauchard, Controllability of a quantum particle in a 1D variable domain, ESAIM:COCV, 14 (2008), 105-147. doi: 10.1051/cocv:2007047.  Google Scholar

[8]

K. Beauchard, Local controllability and non controllability for a 1D wave equation with bilinear control, J. Diff. Eq., 250 (2010), 2064-2098. doi: 10.1016/j.jde.2010.10.008.  Google Scholar

[9]

K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Functional Analysis, 232 (2006), 328-389. doi: 10.1016/j.jfa.2005.03.021.  Google Scholar

[10]

K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl., 94 (2010), 520-554. doi: 10.1016/j.matpur.2010.04.001.  Google Scholar

[11]

K. Beauchard and M. Mirrahimi, Practical stabilization of a quantum particle in a one-dimensional infinite square potential well, SIAM J. Contr. Optim., 48 (2009), 1179-1205. doi: 10.1137/070704204.  Google Scholar

[12]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000.  Google Scholar

[13]

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, Communications on Mathematical Physics, 311 (2012), 423-455. doi: 10.1007/s00220-012-1441-z.  Google Scholar

[14]

N. Boussaïd, M. Caponigro and T. Chambrion, Weakly-coupled systems in quantum control, IEEE Transactions on Automatic Control, 58 (2013), 2205-2216. arXiv:1109.1900. doi: 10.1109/TAC.2013.2255948.  Google Scholar

[15]

E. Cancès, C. L. Bris and M. Pilot, Contrôle optimal bilinéaire d'une équation de Schrödinger, CRAS Paris, 330 (2000), 567-571. doi: 10.1016/S0764-4442(00)00227-5.  Google Scholar

[16]

E. Cerpa and E. Crépeau, Boundary controlability for the non linear korteweg-de vries equation on any critical domain, Ann. IHP Analyse Non Linéaire, 26 (2009), 457-475. doi: 10.1016/j.anihpc.2007.11.003.  Google Scholar

[17]

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), 329-349. doi: 10.1016/j.anihpc.2008.05.001.  Google Scholar

[18]

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. Sciences Paris, Ser. I, 342 (2006), 103-108. doi: 10.1016/j.crma.2005.11.004.  Google Scholar

[19]

J.-M. Coron, Control and Nonlinearity, vol. 136, Mathematical Surveys and Monographs, 2007.  Google Scholar

[20]

S. Ervedoza and J.-P. Puel, Approximate controllability for a system of schrödinger equations modeling a single trapped ion, Ann.IHP: Nonlinear Analysis, 26 (2009), 2111-2136. doi: 10.1016/j.anihpc.2009.01.005.  Google Scholar

[21]

R. Ilner, H. Lange and H. Teismann, Limitations on the control of schrödinger equations, ESAIM:COCV, 12 (2006), 615-635. doi: 10.1051/cocv:2006014.  Google Scholar

[22]

A. Y. Khapalov, Bilinear controllability properties of a vibrating string with variable axial load and damping gain, Dyn. Contin. Impuls. Syst. Ser A Math Anal., 10 (2003), 721-743.  Google Scholar

[23]

A. Y. Khapalov, Controllability properties of a vibrating string with variable axial load, Discrete Contin. Dyn. Syst., 11 (2004), 311-324. doi: 10.3934/dcds.2004.11.311.  Google Scholar

[24]

A. Y. Khapalov, Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping, ESAIM:COCV, 12 (2006), 231-252. doi: 10.1051/cocv:2006001.  Google Scholar

[25]

M. Mirrahimi, Lyapunov control of a quantum particle in a decaying potential, Ann. IHP: Nonlinear Analysis, 26 (2009), 1743-1765. doi: 10.1016/j.anihpc.2008.09.006.  Google Scholar

[26]

V. Nersesyan, Growth of Sobolev norms and controllability of Schrödinger equation, Comm. Math. Phys., 290 (2009), 371-387. doi: 10.1007/s00220-009-0842-0.  Google Scholar

[27]

V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. IHP Nonlinear Analysis, 27 (2010), 901-915. doi: 10.1016/j.anihpc.2010.01.004.  Google Scholar

[28]

V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation, J. Math. Pures et Appl., 97 (2012), 295-317. doi: 10.1016/j.matpur.2011.11.005.  Google Scholar

[29]

V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation: multidimensional case,, (preprint)., ().   Google Scholar

[30]

G. Turinici, On the controllability of bilinear quantum systems, In C. Le Bris and M. Defranceschi, editors, Mathematical Models and Methods for Ab Initio Quantum Chemistry, 75-92, Lecture Notes in Chem., 74, Springer, Berlin, 2000. doi: 10.1007/978-3-642-57237-1_4.  Google Scholar

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