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Asymptotic behavior of a hierarchical size-structured population model
Optimal nonlinearity control of Schrödinger equation
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China |
2. | Department of Mathematic, Northwest Normal University, Lanzhou 730070, China |
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.
References:
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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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, 2007.
doi: 10.1090/surv/136. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn and W. Ketterle,
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doi: 10.1038/32354. |
[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. |
[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. |
[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. |
[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. |
[23] |
T. Özsari, Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities, arXiv: 1705.03965, [math. AP] (2017). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, 2007.
doi: 10.1090/surv/136. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
T. Özsari, Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities, arXiv: 1705.03965, [math. AP] (2017). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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