- Previous Article
- MCRF Home
- This Issue
-
Next Article
Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I
Nonlinear Schrödinger equations on a finite interval with point dissipation
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA USA |
$ iu_t+u_{xx}+f(u) = 0 , \;\;\;\; u ( x, 0 ) = w_0 (x) $ |
$ x\in [0, L] $ |
$ L^2 $ |
$ u(0, t) = \beta u(L, t), \beta u_x(0, t)-u_x(L, t) = i\alpha u(0, t), $ |
$ L>0 $ |
$ \alpha, \beta $ |
$ \alpha\beta<0 $ |
$ \beta\neq \pm 1 $ |
$ f(u) $ |
$ \mathbb{C} $ |
$ \mathbb{C} $ |
$ s \in \left ( \frac12, 1\right ] $ |
$ w_0 (x) \in H^s(0, L ) $ |
$ u \in C([0, T]; H^s (0, L )) $ |
$ t \rightarrow + \infty $ |
References:
[1] |
J. L. Bona, S. M. Sun and B.-Y. Zhang,
Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, J. Math. Pures Appl., 109 (2018), 1-66.
doi: 10.1016/j.matpur.2017.11.001. |
[2] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part Ⅰ: Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.
doi: 10.1007/BF01896020. |
[3] |
J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Colloqium Publication, Vol. 46, American Mathematical Society, Providence, RI, 1999.
doi: 10.1090/coll/046. |
[4] |
H. Brézis and T. Gallouet,
Nonlinear Schrödinger evolution equation, Nonlinear Anal. TMA, 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[5] |
C. Bu,
An initial-boundary value problem of the nonlinear Schrödinger equation, Appl. Anal., 53 (1994), 241-254.
doi: 10.1080/00036819408840260. |
[6] |
C. Bu,
Nonlinear Schrödinger equation on the semi-infinite line, Chinese Annals of Math., 21 (2000), 209-222.
|
[7] |
C. Bu, K. Tsutaya and C Zhang, Nonlinear Schrödinger equation with inhomogebeous Dirichlet boundary data, J. Math. Phys., 46 (2005), 083504, 6pp.
doi: 10.1063/1.1914730. |
[8] |
T. Cazenave, Semilinear Schrödinger Equations, American Math. Soc., Providence, RI, 2003.
doi: 10.1090/cln/010. |
[9] |
T. Cazenave, D. Fang and Z. Han,
Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 28 (2011), 135-147.
doi: 10.1016/j.anihpc.2010.11.005. |
[10] |
T. Cazenave and F. B. Weissler,
The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. TMA, 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[11] |
A. Chabchoub, N. Hoffmann and N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett., 106 (2011), 204502.
doi: 10.1103/PhysRevLett.106.204502. |
[12] |
N. Dunford and J. T. Schwartz, Linear Operators, Part III, Wiley-Interscience, New York, 1971. |
[13] |
M. Fujii and R. Nakamoto,
Simultaneous Extensions of Selberg inequality and Heinz-Kato-Furuta inequality, Nihonkai Math., 9 (1998), 219-225.
|
[14] |
G. Gao and S. M. Sun,
A Korteweg-de Vries type of fifth-order equations on a finite domain with point dissipation, J. Math. Anal. Appl., 438 (2016), 200-239.
doi: 10.1016/j.jmaa.2016.01.050. |
[15] |
J. Ginibre and G. Velo,
On a class of nonlinear Schrödinger equations. Ⅰ. The Cauchy problem, general case, J. Functional Anal., 32 (1979), 1-32.
doi: 10.1016/0022-1236(79)90076-4. |
[16] |
J. Ginibre and G. Velo,
On a class of nonlinear Schrödinger equations. Ⅱ. Scattering theory, general case, J. Functinal Anal., 32 (1979), 33-71.
doi: 10.1016/0022-1236(79)90077-6. |
[17] |
L. F. Ho and D. L. Russell,
Admissible input elements for systems in Hillbert space and Carleson measure criterion, SIAM J. Control. Optim., 21 (1983), 614-640.
doi: 10.1137/0321037. |
[18] |
J. Holmer,
The initial-boundary value problem for the $1$-$d$ nonlinear Schrödinger equation on the half-line, Diff. Integral Equations, 18 (2005), 647-668.
|
[19] |
F.-L. Huang,
Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
|
[20] |
R. Illner, H. Lange and H. Teismann,
A note on the exact internal control of nonlinear Schrödinger equations, CRM Proc. Lecture Notes, 33 (2003), 127-137.
|
[21] |
R. Illner, H. Lange and H. Teismann,
Limitations on the control of Schrödinger equations, ESAIM Control Optim. Calc. Var., 12 (2006), 615-635.
doi: 10.1051/cocv:2006014. |
[22] |
T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Theor., 46 (1987), 113--129. |
[23] |
T. Kato,
On nonlinear Scrhödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. d'Analyse Math., 67 (1995), 281-306.
doi: 10.1007/BF02787794. |
[24] |
S. Kamvissis, Semiclassical nonlinear Schrödinger on the half line, J. Math. Phys., 44 (2003), 5849--5868.
doi: 10.1063/1.1624091. |
[25] |
V. Komornik, A generalization of Ingham's inequality, in Colloq. Math. Soc. $J\grave{a}nos$ Bolyai, Differential Equations Applications, 62 (1991), 213--217. |
[26] |
H. Lange and H. Teismann,
Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state, Math. Methods Appl. Sci., 30 (2007), 1483-1505.
doi: 10.1002/mma.849. |
[27] |
G. Lumer and R. S. Phillips,
Dissipative operators in a Banach space, Pacific J. Math., 11 (1961), 679-698.
doi: 10.2140/pjm.1961.11.679. |
[28] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[29] |
D. Peregrine,
Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. B, 25 (1983), 16-43.
doi: 10.1017/S0334270000003891. |
[30] |
L. Rosier and B.-Y. Zhang,
Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval, SIAM J. Control Optim., 48 (2009), 972-992.
doi: 10.1137/070709578. |
[31] |
D. L. Russell,
Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[32] |
D. L. Russell and B. Y. Zhang,
Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim., 31 (1993), 659-676.
doi: 10.1137/0331030. |
[33] |
D. L. Russell and B. Y. Zhang,
Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation, J. Math. Anal. Appl., 190 (1995), 449-488.
doi: 10.1006/jmaa.1995.1087. |
[34] |
W. Strauss and C. Bu,
Inhomogeneous boundary value problem for a nonlinear Schrödinger equation, J. Diff. Equations, 173 (2001), 79-91.
doi: 10.1006/jdeq.2000.3871. |
[35] |
S. M. Sun,
The Korteweg-de Vries equation on a periodic domain with singular-point dissipation, SIAM J. Control and Optimization, 34 (1996), 892-912.
doi: 10.1137/S0363012994269491. |
[36] |
Y. Tsutsumi,
$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funk. Ekva., 30 (1987), 115-125.
|
[37] |
V. E. Zakharov and S. V. Manakov,
On the complete integrability of a nonlinear Schrödinger equation, J. Theore. and Math. Phys., 19 (1974), 551-559.
|
[38] |
V. E. Zakharov and A. B. Shabat,
Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, J. Experi. and Theore. Phys., 34 (1972), 62-69.
|
show all references
References:
[1] |
J. L. Bona, S. M. Sun and B.-Y. Zhang,
Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, J. Math. Pures Appl., 109 (2018), 1-66.
doi: 10.1016/j.matpur.2017.11.001. |
[2] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part Ⅰ: Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.
doi: 10.1007/BF01896020. |
[3] |
J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Colloqium Publication, Vol. 46, American Mathematical Society, Providence, RI, 1999.
doi: 10.1090/coll/046. |
[4] |
H. Brézis and T. Gallouet,
Nonlinear Schrödinger evolution equation, Nonlinear Anal. TMA, 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[5] |
C. Bu,
An initial-boundary value problem of the nonlinear Schrödinger equation, Appl. Anal., 53 (1994), 241-254.
doi: 10.1080/00036819408840260. |
[6] |
C. Bu,
Nonlinear Schrödinger equation on the semi-infinite line, Chinese Annals of Math., 21 (2000), 209-222.
|
[7] |
C. Bu, K. Tsutaya and C Zhang, Nonlinear Schrödinger equation with inhomogebeous Dirichlet boundary data, J. Math. Phys., 46 (2005), 083504, 6pp.
doi: 10.1063/1.1914730. |
[8] |
T. Cazenave, Semilinear Schrödinger Equations, American Math. Soc., Providence, RI, 2003.
doi: 10.1090/cln/010. |
[9] |
T. Cazenave, D. Fang and Z. Han,
Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 28 (2011), 135-147.
doi: 10.1016/j.anihpc.2010.11.005. |
[10] |
T. Cazenave and F. B. Weissler,
The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. TMA, 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[11] |
A. Chabchoub, N. Hoffmann and N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett., 106 (2011), 204502.
doi: 10.1103/PhysRevLett.106.204502. |
[12] |
N. Dunford and J. T. Schwartz, Linear Operators, Part III, Wiley-Interscience, New York, 1971. |
[13] |
M. Fujii and R. Nakamoto,
Simultaneous Extensions of Selberg inequality and Heinz-Kato-Furuta inequality, Nihonkai Math., 9 (1998), 219-225.
|
[14] |
G. Gao and S. M. Sun,
A Korteweg-de Vries type of fifth-order equations on a finite domain with point dissipation, J. Math. Anal. Appl., 438 (2016), 200-239.
doi: 10.1016/j.jmaa.2016.01.050. |
[15] |
J. Ginibre and G. Velo,
On a class of nonlinear Schrödinger equations. Ⅰ. The Cauchy problem, general case, J. Functional Anal., 32 (1979), 1-32.
doi: 10.1016/0022-1236(79)90076-4. |
[16] |
J. Ginibre and G. Velo,
On a class of nonlinear Schrödinger equations. Ⅱ. Scattering theory, general case, J. Functinal Anal., 32 (1979), 33-71.
doi: 10.1016/0022-1236(79)90077-6. |
[17] |
L. F. Ho and D. L. Russell,
Admissible input elements for systems in Hillbert space and Carleson measure criterion, SIAM J. Control. Optim., 21 (1983), 614-640.
doi: 10.1137/0321037. |
[18] |
J. Holmer,
The initial-boundary value problem for the $1$-$d$ nonlinear Schrödinger equation on the half-line, Diff. Integral Equations, 18 (2005), 647-668.
|
[19] |
F.-L. Huang,
Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
|
[20] |
R. Illner, H. Lange and H. Teismann,
A note on the exact internal control of nonlinear Schrödinger equations, CRM Proc. Lecture Notes, 33 (2003), 127-137.
|
[21] |
R. Illner, H. Lange and H. Teismann,
Limitations on the control of Schrödinger equations, ESAIM Control Optim. Calc. Var., 12 (2006), 615-635.
doi: 10.1051/cocv:2006014. |
[22] |
T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Theor., 46 (1987), 113--129. |
[23] |
T. Kato,
On nonlinear Scrhödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. d'Analyse Math., 67 (1995), 281-306.
doi: 10.1007/BF02787794. |
[24] |
S. Kamvissis, Semiclassical nonlinear Schrödinger on the half line, J. Math. Phys., 44 (2003), 5849--5868.
doi: 10.1063/1.1624091. |
[25] |
V. Komornik, A generalization of Ingham's inequality, in Colloq. Math. Soc. $J\grave{a}nos$ Bolyai, Differential Equations Applications, 62 (1991), 213--217. |
[26] |
H. Lange and H. Teismann,
Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state, Math. Methods Appl. Sci., 30 (2007), 1483-1505.
doi: 10.1002/mma.849. |
[27] |
G. Lumer and R. S. Phillips,
Dissipative operators in a Banach space, Pacific J. Math., 11 (1961), 679-698.
doi: 10.2140/pjm.1961.11.679. |
[28] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[29] |
D. Peregrine,
Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. B, 25 (1983), 16-43.
doi: 10.1017/S0334270000003891. |
[30] |
L. Rosier and B.-Y. Zhang,
Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval, SIAM J. Control Optim., 48 (2009), 972-992.
doi: 10.1137/070709578. |
[31] |
D. L. Russell,
Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[32] |
D. L. Russell and B. Y. Zhang,
Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim., 31 (1993), 659-676.
doi: 10.1137/0331030. |
[33] |
D. L. Russell and B. Y. Zhang,
Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation, J. Math. Anal. Appl., 190 (1995), 449-488.
doi: 10.1006/jmaa.1995.1087. |
[34] |
W. Strauss and C. Bu,
Inhomogeneous boundary value problem for a nonlinear Schrödinger equation, J. Diff. Equations, 173 (2001), 79-91.
doi: 10.1006/jdeq.2000.3871. |
[35] |
S. M. Sun,
The Korteweg-de Vries equation on a periodic domain with singular-point dissipation, SIAM J. Control and Optimization, 34 (1996), 892-912.
doi: 10.1137/S0363012994269491. |
[36] |
Y. Tsutsumi,
$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funk. Ekva., 30 (1987), 115-125.
|
[37] |
V. E. Zakharov and S. V. Manakov,
On the complete integrability of a nonlinear Schrödinger equation, J. Theore. and Math. Phys., 19 (1974), 551-559.
|
[38] |
V. E. Zakharov and A. B. Shabat,
Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, J. Experi. and Theore. Phys., 34 (1972), 62-69.
|
[1] |
Pasquale Palumbo, Pierdomenico Pepe, Simona Panunzi, Andrea De Gaetano. Robust closed-loop control of plasma glycemia: A discrete-delay model approach. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 455-468. doi: 10.3934/dcdsb.2009.12.455 |
[2] |
Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037 |
[3] |
Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117 |
[4] |
Justine Yasappan, Ángela Jiménez-Casas, Mario Castro. Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3267-3299. doi: 10.3934/dcdsb.2015.20.3267 |
[5] |
Riccardo Adami, Diego Noja, Nicola Visciglia. Constrained energy minimization and ground states for NLS with point defects. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1155-1188. doi: 10.3934/dcdsb.2013.18.1155 |
[6] |
Xiaochen Sun, Fei Hu, Yancong Zhou, Cheng-Chew Lim. Optimal acquisition, inventory and production decisions for a closed-loop manufacturing system with legislation constraint. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1355-1373. doi: 10.3934/jimo.2015.11.1355 |
[7] |
Yi Jing, Wenchuan Li. Integrated recycling-integrated production - distribution planning for decentralized closed-loop supply chain. Journal of Industrial & Management Optimization, 2018, 14 (2) : 511-539. doi: 10.3934/jimo.2017058 |
[8] |
Wenbin Wang, Peng Zhang, Junfei Ding, Jian Li, Hao Sun, Lingyun He. Closed-loop supply chain network equilibrium model with retailer-collection under legislation. Journal of Industrial & Management Optimization, 2019, 15 (1) : 199-219. doi: 10.3934/jimo.2018039 |
[9] |
Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187 |
[10] |
Masoud Mohammadzadeh, Alireza Arshadi Khamseh, Mohammad Mohammadi. A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1041-1064. doi: 10.3934/jimo.2016061 |
[11] |
Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215 |
[12] |
Zhijian Yang, Pengyan Ding, Xiaobin Liu. Attractors and their stability on Boussinesq type equations with gentle dissipation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 911-930. doi: 10.3934/cpaa.2019044 |
[13] |
Salvatore A. Marano, Sunra J. N. Mosconi. Multiple solutions to elliptic inclusions via critical point theory on closed convex sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3087-3102. doi: 10.3934/dcds.2015.35.3087 |
[14] |
Alexandre N. Carvalho, Jan W. Cholewa. NLS-like equations in bounded domains: Parabolic approximation procedure. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 57-77. doi: 10.3934/dcdsb.2018005 |
[15] |
Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547 |
[16] |
Bo-Qing Dong, Jiahong Wu, Xiaojing Xu, Zhuan Ye. Global regularity for the 2D micropolar equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4133-4162. doi: 10.3934/dcds.2018180 |
[17] |
Wei Sun. On uniform estimate of complex elliptic equations on closed Hermitian manifolds. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1553-1570. doi: 10.3934/cpaa.2017074 |
[18] |
Jibin Li. Family of nonlinear wave equations which yield loop solutions and solitary wave solutions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 897-907. doi: 10.3934/dcds.2009.24.897 |
[19] |
Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 |
[20] |
Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57 |
2017 Impact Factor: 0.631
Tools
Metrics
Other articles
by authors
[Back to Top]