# American Institute of Mathematical Sciences

• Previous Article
Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I
• MCRF Home
• This Issue
• Next Article
The generalised singular perturbation approximation for bounded real and positive real control systems
June  2019, 9(2): 351-384. doi: 10.3934/mcrf.2019017

## Nonlinear Schrödinger equations on a finite interval with point dissipation

 Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA USA

* Corresponding author: Shu-Ming Sun

Received  July 2017 Revised  July 2018 Published  December 2018

Fund Project: The research was partially supported by the National Science Foundation under grant No. DMS-1210979

The paper considers the initial value problem of a general type of nonlinear Schrödinger equations
 $iu_t+u_{xx}+f(u) = 0 , \;\;\;\; u ( x, 0 ) = w_0 (x)$
posed on a finite domain
 $x\in [0, L]$
with an
 $L^2$
-stabilizing feedback control law
 $u(0, t) = \beta u(L, t), \beta u_x(0, t)-u_x(L, t) = i\alpha u(0, t),$
where
 $L>0$
,
 $\alpha, \beta$
are real constants with
 $\alpha\beta<0$
and
 $\beta\neq \pm 1$
, and
 $f(u)$
is a smooth function from
 $\mathbb{C}$
to
 $\mathbb{C}$
satisfying some growth conditions. It is shown that for
 $s \in \left ( \frac12, 1\right ]$
and
 $w_0 (x) \in H^s(0, L )$
with the boundary conditions described above, the problem is locally well-posed for
 $u \in C([0, T]; H^s (0, L ))$
. Moreover, the solution with small initial condition exists globally and approaches to 0 as
 $t \rightarrow + \infty$
.
Citation: Jing Cui, Shu-Ming Sun. Nonlinear Schrödinger equations on a finite interval with point dissipation. Mathematical Control & Related Fields, 2019, 9 (2) : 351-384. doi: 10.3934/mcrf.2019017
##### 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. Google Scholar [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. Google Scholar [3] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Colloqium Publication, Vol. 46, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/coll/046. Google Scholar [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. Google Scholar [5] C. Bu, An initial-boundary value problem of the nonlinear Schrödinger equation, Appl. Anal., 53 (1994), 241-254. doi: 10.1080/00036819408840260. Google Scholar [6] C. Bu, Nonlinear Schrödinger equation on the semi-infinite line, Chinese Annals of Math., 21 (2000), 209-222. Google Scholar [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. Google Scholar [8] T. Cazenave, Semilinear Schrödinger Equations, American Math. Soc., Providence, RI, 2003. doi: 10.1090/cln/010. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Wiley-Interscience, New York, 1971. Google Scholar [13] M. Fujii and R. Nakamoto, Simultaneous Extensions of Selberg inequality and Heinz-Kato-Furuta inequality, Nihonkai Math., 9 (1998), 219-225. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] F.-L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. Google Scholar [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. Google Scholar [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. Google Scholar [22] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Theor., 46 (1987), 113--129. Google Scholar [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. Google Scholar [24] S. Kamvissis, Semiclassical nonlinear Schrödinger on the half line, J. Math. Phys., 44 (2003), 5849--5868. doi: 10.1063/1.1624091. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] D. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. B, 25 (1983), 16-43. doi: 10.1017/S0334270000003891. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funk. Ekva., 30 (1987), 115-125. Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [3] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Colloqium Publication, Vol. 46, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/coll/046. Google Scholar [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. Google Scholar [5] C. Bu, An initial-boundary value problem of the nonlinear Schrödinger equation, Appl. Anal., 53 (1994), 241-254. doi: 10.1080/00036819408840260. Google Scholar [6] C. Bu, Nonlinear Schrödinger equation on the semi-infinite line, Chinese Annals of Math., 21 (2000), 209-222. Google Scholar [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. Google Scholar [8] T. Cazenave, Semilinear Schrödinger Equations, American Math. Soc., Providence, RI, 2003. doi: 10.1090/cln/010. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Wiley-Interscience, New York, 1971. Google Scholar [13] M. Fujii and R. Nakamoto, Simultaneous Extensions of Selberg inequality and Heinz-Kato-Furuta inequality, Nihonkai Math., 9 (1998), 219-225. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] F.-L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. Google Scholar [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. Google Scholar [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. Google Scholar [22] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Theor., 46 (1987), 113--129. Google Scholar [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. Google Scholar [24] S. Kamvissis, Semiclassical nonlinear Schrödinger on the half line, J. Math. Phys., 44 (2003), 5849--5868. doi: 10.1063/1.1624091. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] D. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. B, 25 (1983), 16-43. doi: 10.1017/S0334270000003891. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funk. Ekva., 30 (1987), 115-125. Google Scholar [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. Google Scholar [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. Google Scholar
 [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] Xiaohong Chen, Kui Li, Fuqiang Wang, Xihua Li. Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2019008 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289 [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] 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 [19] 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 [20] 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

2018 Impact Factor: 1.292

Article outline