-
Previous Article
Internal control of the Schrödinger equation
- MCRF Home
- This Issue
- Next Article
Local controllability of 1D Schrödinger equations with bilinear control and minimal time
1. | CMLS, Ecole Polytechnique, 91 128 Palaiseau cedex, France, France |
  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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
K. Beauchard, Controllability of a quantum particle in a 1D variable domain, ESAIM:COCV, 14 (2008), 105-147.
doi: 10.1051/cocv:2007047. |
[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. |
[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. |
[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. |
[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. |
[12] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
J.-M. Coron, Control and Nonlinearity, vol. 136, Mathematical Surveys and Monographs, 2007. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation: multidimensional case,, (preprint)., ().
|
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
K. Beauchard, Controllability of a quantum particle in a 1D variable domain, ESAIM:COCV, 14 (2008), 105-147.
doi: 10.1051/cocv:2007047. |
[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. |
[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. |
[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. |
[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. |
[12] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
J.-M. Coron, Control and Nonlinearity, vol. 136, Mathematical Surveys and Monographs, 2007. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation: multidimensional case,, (preprint)., ().
|
[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. |
[1] |
Piermarco Cannarsa, Alessandro Duca, Cristina Urbani. Exact controllability to eigensolutions of the bilinear heat equation on compact networks. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1377-1401. doi: 10.3934/dcdss.2022011 |
[2] |
Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143 |
[3] |
Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113 |
[4] |
Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure and Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034 |
[5] |
Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control and Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161 |
[6] |
Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188 |
[7] |
Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901 |
[8] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[9] |
Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations and Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016 |
[10] |
Shuya Kanagawa, Ben T. Nohara. The nonlinear Schrödinger equation created by the vibrations of an elastic plate and its dimensional expansion. Conference Publications, 2013, 2013 (special) : 415-426. doi: 10.3934/proc.2013.2013.415 |
[11] |
Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1 |
[12] |
Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure and Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311 |
[13] |
Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks and Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014 |
[14] |
Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control and Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031 |
[15] |
Dan-Andrei Geba, Evan Witz. Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations. Electronic Research Archive, 2020, 28 (2) : 627-649. doi: 10.3934/era.2020033 |
[16] |
Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085 |
[17] |
J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445 |
[18] |
José R. Quintero, Alex M. Montes. On the exact controllability and the stabilization for the Benney-Luke equation. Mathematical Control and Related Fields, 2020, 10 (2) : 275-304. doi: 10.3934/mcrf.2019039 |
[19] |
Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations and Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020 |
[20] |
Mo Chen, Lionel Rosier. Exact controllability of the linear Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3889-3916. doi: 10.3934/dcdsb.2020080 |
2020 Impact Factor: 1.284
Tools
Metrics
Other articles
by authors
[Back to Top]