American Institute of Mathematical Sciences

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June  2018, 7(2): 317-334. doi: 10.3934/eect.2018016

Optimal nonlinearity control of Schrödinger equation

Kai Wang 1, , Dun Zhao 1,, and  Binhua Feng 2,

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
2. 

Department of Mathematic, Northwest Normal University, Lanzhou 730070, China

* Corresponding author: Dun Zhao

Received  June 2017 Revised  February 2018 Published  May 2018

Fund Project: This work is supported by the NSFC under grants No. 11475073, No. 11325417 and No. 11601435.

We study the optimal nonlinearity control problem for the nonlinear Schrödinger equation $iu_{t} = -\triangle u+V(x)u+h(t)|u|^α u$, which is originated from the Fechbach resonance management in Bose-Einstein condensates and the nonlinearity management in nonlinear optics. Based on the global well-posedness of the equation for $0<α<\frac{4}{N}$, we show the existence of the optimal control. The Fréchet differentiability of the objective functional is proved, and the first order optimality system for $N≤ 3$ is presented.

Keywords: Nonlinear Schrödinger equation, nonlinearity control, objective functional, Fréchet-differentiability, optimal condition.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 49J20.
Citation: Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016
References:
[1]

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

[2]

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, P. L. Christiansen and Y. B. Gaididei, Self-guiding light in layered nonlinear media, Opt. Lett., 25 (2000), 1037-1039.  doi: 10.1364/OL.25.001037.  Google Scholar

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R. Carles, Nonlinear Schrödinger Equations with time dependent potentials, Commun. Math. Sci., 9 (2011), 937-964.  doi: 10.4310/CMS.2011.v9.n4.a1.  Google Scholar

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T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

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T. Cazenave and M. Scialom, A Schrödinger equation with time-oscillating nonlinearity, Rev. Mat. Complut., 23 (2010), 321-339.  doi: 10.1007/s13163-009-0018-7.  Google Scholar

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M. Centurion, M. A. Porter, P. G. Kevrekidis and D. Psaltis, Nonlinearity management in optics: Experiment, theory and simulation Phys. Rev. Lett., 97 (2006), 033903. doi: 10.1103/PhysRevLett.97.033903.  Google Scholar

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S. Choi and N. P. Bigelow, Quantum control of Bose-Einstein condensates using Feshbach resonance, J. Modern Optics, 52 (2005), 1081-1087.  doi: 10.1080/09500340512331323475.  Google Scholar

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J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, 2007. doi: 10.1090/surv/136.  Google Scholar

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S. Cuccagna, E. Kirr and D. Pelinovsky, Parametric resonance of ground states in the nonlinear Schrödinger equation, J. Differential Equations, 220 (2006), 85-120.  doi: 10.1016/j.jde.2005.07.009.  Google Scholar

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I. Damergi and O. Goubet, Blow-up solutions to the nonlinear Schrödinger equation with oscillating nonlinearities, J. Math. Anal. Appl., 352 (2009), 336-344.  doi: 10.1016/j.jmaa.2008.07.079.  Google Scholar

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D. Y. Fang and Z. Han, A Schrödinger equation with time-oscillating critical nonlinearity, Nonlinear Analysis, 74 (2011), 4698-4708.  doi: 10.1016/j.na.2011.04.035.  Google Scholar

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B. Feng, J. Liu and J. Zheng, Optimal bilinear control of nonlinear Hartree equation in ${{\mathbb{R}}^{3}}$, Electronic. J. Differential Equations, 2013 (2013), 1-14.   Google Scholar

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B. Feng and K. Wang, Optimal bilinear control of nonlinear Hartree equations with singular potentials, J. Optim. Theory Appl., 170 (2016), 756-771.  doi: 10.1007/s10957-016-0976-0.  Google Scholar

[14]

B. Feng, D. Zhao and P. Y. Chen, Optimal bilinear control of nonlinear Schrödinger equations with singular potentials, Nonlinear Analysis, 107 (2014), 12-21.  doi: 10.1016/j.na.2014.04.017.  Google Scholar

[15]

B. Feng and D. Zhao, Optimal bilinear control of Gross-Pitaevskii equations with Coulombian potentials, J. Differential Equations, 260 (2016), 2973-2993.  doi: 10.1016/j.jde.2015.10.026.  Google Scholar

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U. Hohenester, P. K. Rekdel, A. Borzi and J. Schmiedmayer, Optimal quantum control of Bose Einstein condensates in magnetic microtraps Phys. Rev. A, 75 (2007), 023602. doi: 10.1103/PhysRevA.75.023602.  Google Scholar

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S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn and W. Ketterle, Observation of Feshbach resonances in a Bose-Einstein condensate, Nature, 392 (1998), 151-154.  doi: 10.1038/32354.  Google Scholar

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K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation, SIAM J. Control Optim., 46 (2007), 274-287.  doi: 10.1137/05064254X.  Google Scholar

[20]

P. G. Kevrekidis, D. E. Pelinovsky and A. Stefanov, Nonlinearity management in higher dimensions, J. Phys. A: Math. Gen., 39 (2006), 479-488.  doi: 10.1088/0305-4470/39/3/002.  Google Scholar

[21]

P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis and Boris A. Malomed, Feshbach resonance management for Bose-Einstein condensates Phys. Rev. Lett., 90 (2003), 230401. doi: 10.1103/PhysRevLett.90.230401.  Google Scholar

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V. V. Konotop and P. Pacciani, Collapse of solutions of the nonlinear Schrödinger equation with a time-dependent nonlinearity: application to Bose-Einstein condensates Phys. Rev. Lett., 94 2005), 240405. doi: 10.1103/PhysRevLett.94.240405.  Google Scholar

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T. Özsari, Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities, arXiv: 1705.03965, [math. AP] (2017). Google Scholar

[24]

D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Averaging for solitons with nonlinearity management Phys. Rev. Lett., 91 (2003), 240201. doi: 10.1103/PhysRevLett.91.240201.  Google Scholar

[25]

J. Simon, Compact sets in the space $L^{p}(0,T;B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[26]

J. Stenger, S. Inouye, M. R. Andrews, H.-J. Miesner, D. M. Stamper-Kurn and W. Ketterle, Strongly Enhanced Inelastic Collisions in a Bose-Einstein Condensate near Feshbach Resonances, Phys. Rev. Lett., 82 (1999), 2422-2425.  doi: 10.1103/PhysRevLett.82.2422.  Google Scholar

[27]

I. Towers and B. A. Malomed, Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity, J. Opt. Soc. Amer. B Opt. Phys., 19 (2002), 537-543.  doi: 10.1364/JOSAB.19.000537.  Google Scholar

[28]

J. Werschnik and E. Gross, Quantum optimal control theory, J. Phys. B, 40 (2007), 175-211.  doi: 10.1088/0953-4075/40/18/R01.  Google Scholar

[29]

J. Zhang and S. Zhu, Blow-up profile to solutions of NLS with oscillating nonlinearities, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 219-234.  doi: 10.1007/s00030-011-0125-2.  Google Scholar

show all references

References:
[1]

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

[2]

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, P. L. Christiansen and Y. B. Gaididei, Self-guiding light in layered nonlinear media, Opt. Lett., 25 (2000), 1037-1039.  doi: 10.1364/OL.25.001037.  Google Scholar

[3]

R. Carles, Nonlinear Schrödinger Equations with time dependent potentials, Commun. Math. Sci., 9 (2011), 937-964.  doi: 10.4310/CMS.2011.v9.n4.a1.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[5]

T. Cazenave and M. Scialom, A Schrödinger equation with time-oscillating nonlinearity, Rev. Mat. Complut., 23 (2010), 321-339.  doi: 10.1007/s13163-009-0018-7.  Google Scholar

[6]

M. Centurion, M. A. Porter, P. G. Kevrekidis and D. Psaltis, Nonlinearity management in optics: Experiment, theory and simulation Phys. Rev. Lett., 97 (2006), 033903. doi: 10.1103/PhysRevLett.97.033903.  Google Scholar

[7]

S. Choi and N. P. Bigelow, Quantum control of Bose-Einstein condensates using Feshbach resonance, J. Modern Optics, 52 (2005), 1081-1087.  doi: 10.1080/09500340512331323475.  Google Scholar

[8]

J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, 2007. doi: 10.1090/surv/136.  Google Scholar

[9]

S. Cuccagna, E. Kirr and D. Pelinovsky, Parametric resonance of ground states in the nonlinear Schrödinger equation, J. Differential Equations, 220 (2006), 85-120.  doi: 10.1016/j.jde.2005.07.009.  Google Scholar

[10]

I. Damergi and O. Goubet, Blow-up solutions to the nonlinear Schrödinger equation with oscillating nonlinearities, J. Math. Anal. Appl., 352 (2009), 336-344.  doi: 10.1016/j.jmaa.2008.07.079.  Google Scholar

[11]

D. Y. Fang and Z. Han, A Schrödinger equation with time-oscillating critical nonlinearity, Nonlinear Analysis, 74 (2011), 4698-4708.  doi: 10.1016/j.na.2011.04.035.  Google Scholar

[12]

B. Feng, J. Liu and J. Zheng, Optimal bilinear control of nonlinear Hartree equation in ${{\mathbb{R}}^{3}}$, Electronic. J. Differential Equations, 2013 (2013), 1-14.   Google Scholar

[13]

B. Feng and K. Wang, Optimal bilinear control of nonlinear Hartree equations with singular potentials, J. Optim. Theory Appl., 170 (2016), 756-771.  doi: 10.1007/s10957-016-0976-0.  Google Scholar

[14]

B. Feng, D. Zhao and P. Y. Chen, Optimal bilinear control of nonlinear Schrödinger equations with singular potentials, Nonlinear Analysis, 107 (2014), 12-21.  doi: 10.1016/j.na.2014.04.017.  Google Scholar

[15]

B. Feng and D. Zhao, Optimal bilinear control of Gross-Pitaevskii equations with Coulombian potentials, J. Differential Equations, 260 (2016), 2973-2993.  doi: 10.1016/j.jde.2015.10.026.  Google Scholar

[16]

M. Hintermüller, D. Marahrens, P. A. Markowich and C. Sparber, Optimal bilinear control of Gross-Pitaevskii equations, SIAM J. Control Optim., 51 (2013), 2509-2543.  doi: 10.1137/120866233.  Google Scholar

[17]

U. Hohenester, P. K. Rekdel, A. Borzi and J. Schmiedmayer, Optimal quantum control of Bose Einstein condensates in magnetic microtraps Phys. Rev. A, 75 (2007), 023602. doi: 10.1103/PhysRevA.75.023602.  Google Scholar

[18]

S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn and W. Ketterle, Observation of Feshbach resonances in a Bose-Einstein condensate, Nature, 392 (1998), 151-154.  doi: 10.1038/32354.  Google Scholar

[19]

K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation, SIAM J. Control Optim., 46 (2007), 274-287.  doi: 10.1137/05064254X.  Google Scholar

[20]

P. G. Kevrekidis, D. E. Pelinovsky and A. Stefanov, Nonlinearity management in higher dimensions, J. Phys. A: Math. Gen., 39 (2006), 479-488.  doi: 10.1088/0305-4470/39/3/002.  Google Scholar

[21]

P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis and Boris A. Malomed, Feshbach resonance management for Bose-Einstein condensates Phys. Rev. Lett., 90 (2003), 230401. doi: 10.1103/PhysRevLett.90.230401.  Google Scholar

[22]

V. V. Konotop and P. Pacciani, Collapse of solutions of the nonlinear Schrödinger equation with a time-dependent nonlinearity: application to Bose-Einstein condensates Phys. Rev. Lett., 94 2005), 240405. doi: 10.1103/PhysRevLett.94.240405.  Google Scholar

[23]

T. Özsari, Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities, arXiv: 1705.03965, [math. AP] (2017). Google Scholar

[24]

D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Averaging for solitons with nonlinearity management Phys. Rev. Lett., 91 (2003), 240201. doi: 10.1103/PhysRevLett.91.240201.  Google Scholar

[25]

J. Simon, Compact sets in the space $L^{p}(0,T;B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[26]

J. Stenger, S. Inouye, M. R. Andrews, H.-J. Miesner, D. M. Stamper-Kurn and W. Ketterle, Strongly Enhanced Inelastic Collisions in a Bose-Einstein Condensate near Feshbach Resonances, Phys. Rev. Lett., 82 (1999), 2422-2425.  doi: 10.1103/PhysRevLett.82.2422.  Google Scholar

[27]

I. Towers and B. A. Malomed, Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity, J. Opt. Soc. Amer. B Opt. Phys., 19 (2002), 537-543.  doi: 10.1364/JOSAB.19.000537.  Google Scholar

[28]

J. Werschnik and E. Gross, Quantum optimal control theory, J. Phys. B, 40 (2007), 175-211.  doi: 10.1088/0953-4075/40/18/R01.  Google Scholar

[29]

J. Zhang and S. Zhu, Blow-up profile to solutions of NLS with oscillating nonlinearities, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 219-234.  doi: 10.1007/s00030-011-0125-2.  Google Scholar

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