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doi: 10.3934/mcrf.2022027
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Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates

Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Received  July 2021 Revised  February 2022 Early access June 2022

Fund Project: This work benefits from the support of ANR project TRECOS, grant ANR-20-CE40-0009

We consider the linear Schrödinger equation, in 1D, on a bounded interval, with Dirichlet boundary conditions and bilinear scalar control. The small-time local exact controllability around the ground state was proved in [5], under an appropriate nondegeneracy assumption. Here, we work under a weaker nondegeneracy assumption and we prove the small-time local exact controllability in projection, around the ground state, with estimates on the control (depending linearly on the target) simultaneously in several spaces. These estimates are obtained at the level of the linearized system, thanks to a new result about trigonometric moment problems. Then, they are transported to the nonlinear system by the inverse mapping theorem, thanks to appropriate estimates of the error between the nonlinear and the linearized dynamics.

Citation: Mégane Bournissou. Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022027
References:
[1]

S. A. Avdonin and S. A. Ivanov, Families of Exponentials, Cambridge University Press, Cambridge, 1995. The method of moments in controllability problems for distributed parameter systems, Translated from the Russian and revised by the authors.

[2]

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

[3]

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

[4]

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

[5]

K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl. (9), 94 (2010), 520-554.  doi: 10.1016/j.matpur.2010.04.001.

[6]

K. Beauchard and F. Marbach, Quadratic obstructions to small-time local controllability for scalar-input systems, J. Differential Equations, 264 (2018), 3704-3774.  doi: 10.1016/j.jde.2017.11.028.

[7]

K. Beauchard and F. Marbach, Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations, J. Math. Pures Appl. (9), 136 (2020), 22-91.  doi: 10.1016/j.matpur.2020.02.001.

[8]

A. BenabdallahF. BoyerM. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680.

[9]

A. BenabdallahF. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Ann. H. Lebesgue, 3 (2020), 717-793.  doi: 10.5802/ahl.45.

[10]

I. Beschastnyi, U. Boscain and M. Sigalotti, An obstruction to small-time controllability of the bilinear Schrödinger equation, J. Math. Phys., 62 (2021), Paper No. 032103, 14 pp. doi: 10.1063/5.0003524.

[11]

U. BoscainM. CaponigroT. 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, Comm. Math. Phys., 311 (2012), 423-455.  doi: 10.1007/s00220-012-1441-z.

[12]

U. BoscainM. Caponigro and M. Sigalotti, Multi-input Schrödinger equation: Controllability, tracking, and application to the quantum angular momentum, J. Differential Equations, 256 (2014), 3524-3551.  doi: 10.1016/j.jde.2014.02.004.

[13]

U. V. BoscainF. ChittaroP. Mason and M. Sigalotti, Adiabatic control of the Schrödinger equation via conical intersections of the eigenvalues, IEEE Trans. Automat. Control, 57 (2012), 1970-1983.  doi: 10.1109/TAC.2012.2195862.

[14]

U. BoscainJ.-P. GauthierF. Rossi and M. Sigalotti, Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems, Comm. Math. Phys., 333 (2015), 1225-1239.  doi: 10.1007/s00220-014-2195-6.

[15]

M. Bournissou, Small-time local controllability of the bilinear Schrödinger equation, despite a quadratic obstruction, thanks to a cubic term, preprint, 2022, arXiv: 2203.03955v2.

[16]

N. BoussaïdM. Caponigro and T. Chambrion, Weakly coupled systems in quantum control, IEEE Trans. Automat. Control, 58 (2013), 2205-2216.  doi: 10.1109/TAC.2013.2255948.

[17]

N. Boussaïd, M. Caponigro and T. Chambrion, Regular propagators of bilinear quantum systems, J. Funct. Anal., 278 (2020), 108412, 66 pp. doi: 10.1016/j.jfa.2019.108412.

[18]

E. BurmanA. Feizmohammadi and L. Oksanen, A fully discrete numerical control method for the wave equation, SIAM J. Control Optim., 58 (2020), 1519-1546.  doi: 10.1137/19M1249424.

[19]

T. ChambrionP. MasonM. Sigalotti and U. 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.

[20]

B. Dehman and G. Lebeau, Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time, SIAM J. Control Optim., 48 (2009), 521-550.  doi: 10.1137/070712067.

[21]

A. Duca, Controllability of bilinear quantum systems in explicit times via explicit control fields, Internat. J. Control, 94 (2021), 724-734.  doi: 10.1080/00207179.2019.1616224.

[22]

A. Duca, Simultaneous global exact controllability in projection of infinite 1D bilinear Schrödinger equations, Dyn. Partial Differ. Equ., 17 (2020), 275-306.  doi: 10.4310/DPDE.2020.v17.n3.a4.

[23]

A. Duca, R. Joly and D. Turaev, Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall, J. Math. Phys., 61 (2020), 101511, 15 pp. doi: 10.1063/5.0005399.

[24]

A. Duca, R. Joly and D. Turaev, Control of the Schrödinger equation by slow deformations of the domain, preprint, 2022, arXiv: 2203.00486.

[25]

A. Duca and V. Nersesyan, Bilinear control and growth of sobolev norms for the nonlinear Schrödinger equation, preprint, 2021, arXiv: 2101.12103.

[26]

A. Duca and V. Nersesyan, Local exact controllability of the 1D nonlinear Schrödinger equation in the case of Dirichlet boundary conditions, preprint, 2022, arXiv: 2202.08723.

[27]

S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1375-1401.  doi: 10.3934/dcdsb.2010.14.1375.

[28]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.

[29]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.

[30]

M. González-Burgos and L. Ouaili, Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions, preprint, 2021.

[31]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl. (9), 68 (1989), 457-465. 

[32]

A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.

[33]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.

[34]

W. Krabs, On Moment Theory and Controllability of One-Dimensional Vibrating Systems and Heating Processes, Lecture Notes in Control and Information Sciences, volume 173, Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0039513.

[35]

P. Lissy, The cost of the control in the case of a minimal time of control: The example of the one-dimensional heat equation, J. Math. Anal. Appl., 451 (2017), 497-507.  doi: 10.1016/j.jmaa.2017.01.096.

[36]

M. Mirrahimi, Lyapunov control of a quantum particle in a decaying potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1743-1765.  doi: 10.1016/j.anihpc.2008.09.006.

[37]

M. Morancey, Simultaneous local exact controllability of 1D bilinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 501-529.  doi: 10.1016/j.anihpc.2013.05.001.

[38]

M. Morancey and V. Nersesyan, Global exact controllability of 1D Schrödinger equations with a polarizability term, C. R. Math. Acad. Sci. Paris, 352 (2014), 425-429.  doi: 10.1016/j.crma.2014.03.013.

[39]

M. Morancey and V. Nersesyan, Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations, J. Math. Pures Appl. (9), 103 (2015), 228-254.  doi: 10.1016/j.matpur.2014.04.002.

[40]

V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 901-915.  doi: 10.1016/j.anihpc.2010.01.004.

[41]

V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation, J. Math. Pures Appl. (9), 97 (2012), 295-317.  doi: 10.1016/j.matpur.2011.11.005.

[42]

V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation: Multidimensional case, J. Math. Pures Appl. (9), 97 (2012), 295-317.  doi: 10.1016/j.matpur.2011.11.005.

[43]

J.-P. Puel, Local exact bilinear control of the Schrödinger equation, ESAIM Control Optim. Calc. Var., 22 (2016), 1264-1281.  doi: 10.1051/cocv/2016049.

[44]

G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, 243 (2007), 70-100.  doi: 10.1016/j.jde.2007.06.019.

[45]

G. Turinici, On the controllability of bilinear quantum systems, Mathematical Models and Methods for ab Initio Quantum Chemistry, volume 74 of Lecture Notes in Chem., pages 75–92. Springer, Berlin, 2000. doi: 10.1007/978-3-642-57237-1_4.

show all references

References:
[1]

S. A. Avdonin and S. A. Ivanov, Families of Exponentials, Cambridge University Press, Cambridge, 1995. The method of moments in controllability problems for distributed parameter systems, Translated from the Russian and revised by the authors.

[2]

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

[3]

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

[4]

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

[5]

K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl. (9), 94 (2010), 520-554.  doi: 10.1016/j.matpur.2010.04.001.

[6]

K. Beauchard and F. Marbach, Quadratic obstructions to small-time local controllability for scalar-input systems, J. Differential Equations, 264 (2018), 3704-3774.  doi: 10.1016/j.jde.2017.11.028.

[7]

K. Beauchard and F. Marbach, Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations, J. Math. Pures Appl. (9), 136 (2020), 22-91.  doi: 10.1016/j.matpur.2020.02.001.

[8]

A. BenabdallahF. BoyerM. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680.

[9]

A. BenabdallahF. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Ann. H. Lebesgue, 3 (2020), 717-793.  doi: 10.5802/ahl.45.

[10]

I. Beschastnyi, U. Boscain and M. Sigalotti, An obstruction to small-time controllability of the bilinear Schrödinger equation, J. Math. Phys., 62 (2021), Paper No. 032103, 14 pp. doi: 10.1063/5.0003524.

[11]

U. BoscainM. CaponigroT. 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, Comm. Math. Phys., 311 (2012), 423-455.  doi: 10.1007/s00220-012-1441-z.

[12]

U. BoscainM. Caponigro and M. Sigalotti, Multi-input Schrödinger equation: Controllability, tracking, and application to the quantum angular momentum, J. Differential Equations, 256 (2014), 3524-3551.  doi: 10.1016/j.jde.2014.02.004.

[13]

U. V. BoscainF. ChittaroP. Mason and M. Sigalotti, Adiabatic control of the Schrödinger equation via conical intersections of the eigenvalues, IEEE Trans. Automat. Control, 57 (2012), 1970-1983.  doi: 10.1109/TAC.2012.2195862.

[14]

U. BoscainJ.-P. GauthierF. Rossi and M. Sigalotti, Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems, Comm. Math. Phys., 333 (2015), 1225-1239.  doi: 10.1007/s00220-014-2195-6.

[15]

M. Bournissou, Small-time local controllability of the bilinear Schrödinger equation, despite a quadratic obstruction, thanks to a cubic term, preprint, 2022, arXiv: 2203.03955v2.

[16]

N. BoussaïdM. Caponigro and T. Chambrion, Weakly coupled systems in quantum control, IEEE Trans. Automat. Control, 58 (2013), 2205-2216.  doi: 10.1109/TAC.2013.2255948.

[17]

N. Boussaïd, M. Caponigro and T. Chambrion, Regular propagators of bilinear quantum systems, J. Funct. Anal., 278 (2020), 108412, 66 pp. doi: 10.1016/j.jfa.2019.108412.

[18]

E. BurmanA. Feizmohammadi and L. Oksanen, A fully discrete numerical control method for the wave equation, SIAM J. Control Optim., 58 (2020), 1519-1546.  doi: 10.1137/19M1249424.

[19]

T. ChambrionP. MasonM. Sigalotti and U. 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.

[20]

B. Dehman and G. Lebeau, Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time, SIAM J. Control Optim., 48 (2009), 521-550.  doi: 10.1137/070712067.

[21]

A. Duca, Controllability of bilinear quantum systems in explicit times via explicit control fields, Internat. J. Control, 94 (2021), 724-734.  doi: 10.1080/00207179.2019.1616224.

[22]

A. Duca, Simultaneous global exact controllability in projection of infinite 1D bilinear Schrödinger equations, Dyn. Partial Differ. Equ., 17 (2020), 275-306.  doi: 10.4310/DPDE.2020.v17.n3.a4.

[23]

A. Duca, R. Joly and D. Turaev, Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall, J. Math. Phys., 61 (2020), 101511, 15 pp. doi: 10.1063/5.0005399.

[24]

A. Duca, R. Joly and D. Turaev, Control of the Schrödinger equation by slow deformations of the domain, preprint, 2022, arXiv: 2203.00486.

[25]

A. Duca and V. Nersesyan, Bilinear control and growth of sobolev norms for the nonlinear Schrödinger equation, preprint, 2021, arXiv: 2101.12103.

[26]

A. Duca and V. Nersesyan, Local exact controllability of the 1D nonlinear Schrödinger equation in the case of Dirichlet boundary conditions, preprint, 2022, arXiv: 2202.08723.

[27]

S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1375-1401.  doi: 10.3934/dcdsb.2010.14.1375.

[28]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.

[29]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.

[30]

M. González-Burgos and L. Ouaili, Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions, preprint, 2021.

[31]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl. (9), 68 (1989), 457-465. 

[32]

A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.

[33]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.

[34]

W. Krabs, On Moment Theory and Controllability of One-Dimensional Vibrating Systems and Heating Processes, Lecture Notes in Control and Information Sciences, volume 173, Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0039513.

[35]

P. Lissy, The cost of the control in the case of a minimal time of control: The example of the one-dimensional heat equation, J. Math. Anal. Appl., 451 (2017), 497-507.  doi: 10.1016/j.jmaa.2017.01.096.

[36]

M. Mirrahimi, Lyapunov control of a quantum particle in a decaying potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1743-1765.  doi: 10.1016/j.anihpc.2008.09.006.

[37]

M. Morancey, Simultaneous local exact controllability of 1D bilinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 501-529.  doi: 10.1016/j.anihpc.2013.05.001.

[38]

M. Morancey and V. Nersesyan, Global exact controllability of 1D Schrödinger equations with a polarizability term, C. R. Math. Acad. Sci. Paris, 352 (2014), 425-429.  doi: 10.1016/j.crma.2014.03.013.

[39]

M. Morancey and V. Nersesyan, Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations, J. Math. Pures Appl. (9), 103 (2015), 228-254.  doi: 10.1016/j.matpur.2014.04.002.

[40]

V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 901-915.  doi: 10.1016/j.anihpc.2010.01.004.

[41]

V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation, J. Math. Pures Appl. (9), 97 (2012), 295-317.  doi: 10.1016/j.matpur.2011.11.005.

[42]

V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation: Multidimensional case, J. Math. Pures Appl. (9), 97 (2012), 295-317.  doi: 10.1016/j.matpur.2011.11.005.

[43]

J.-P. Puel, Local exact bilinear control of the Schrödinger equation, ESAIM Control Optim. Calc. Var., 22 (2016), 1264-1281.  doi: 10.1051/cocv/2016049.

[44]

G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, 243 (2007), 70-100.  doi: 10.1016/j.jde.2007.06.019.

[45]

G. Turinici, On the controllability of bilinear quantum systems, Mathematical Models and Methods for ab Initio Quantum Chemistry, volume 74 of Lecture Notes in Chem., pages 75–92. Springer, Berlin, 2000. doi: 10.1007/978-3-642-57237-1_4.

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