
-
Previous Article
$ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations
- EECT Home
- This Issue
-
Next Article
Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity
Stabilization of higher order Schrödinger equations on a finite interval: Part I
1. | Department of Mathematics, İzmir Institute of Technology, Urla, İzmir, 35430 Turkey |
2. | Department of Mathematics, Bilkent University, Bilkent, Ankara, 06800 Turkey |
We study the backstepping stabilization of higher order linear and nonlinear Schrödinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a prescribed rate of decay. The construction of the backstepping kernel is based on a challenging successive approximation analysis. This contrasts with the case of second order pdes. Second, we consider the case where the full state of the system cannot be measured at all times but some partial information such as measurements of a boundary trace are available. For this problem, we simultaneously construct an observer and the associated backstepping controller which is capable of stabilizing the original plant. Wellposedness and regularity results are provided for all pde models. Although the linear part of the model is similar to the KdV equation, the power type nonlinearity brings additional difficulties. We give two examples of boundary conditions and partial measurements. We also present numerical algorithms and simulations verifying our theoretical results to the fullest extent. Our numerical approach is novel in the sense that we solve the target systems first and obtain the solution to the feedback system by using the bounded invertibility of the backstepping transformation.
References:
[1] |
G. P. Agrawal, Nonlinear Fiber Optics, Nonlinear Science at the Dawn of the 21$^{st}$ Century, Springer, Berlin, Heidelberg, 2000,195–211.
doi: 10.1007/3-540-46629-0_9. |
[2] |
A. Balogh and M. Krstic,
Boundary control of the Korteweg-de Vries-Burgers equation: Further results on stabilization and well-posedness, with numerical demonstration, IEEE Trans. Automat. Control, 45 (2000), 1739-1745.
doi: 10.1109/9.880639. |
[3] |
A. Batal and T. Özsarı, Output feedback stabilization of the linearized Korteweg-de Vries equation with right endpoint controllers, Automatica, 109 (2019), 108531, 8 pp.
doi: 10.1016/j.automatica.2019.108531. |
[4] |
E. Bisognin, V. Bisognin and O. P. Vera Villagrán, Stabilization of solutions to higher-order nonlinear Schrödinger equation with localized damping, Electron. J. Differential Equations, (2007), No. 6, 18 pp. |
[5] |
X. Carvajal and F. Linares,
A higher-order nonlinear Schrödinger equation with variable coefficients, Differential Integral Equations, 16 (2003), 1111-1130.
|
[6] |
X. Carvajal, Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices, Electron. J. Differential Equations, (2004), No. 13, 10 pp. |
[7] |
X. Carvajal,
Sharp global well-posedness for a higher order Schrödinger equation, J. Fourier Anal. Appl., 12 (2006), 53-70.
doi: 10.1007/s00041-005-5028-3. |
[8] |
M. M. Cavalcanti, W. J. Correa, M. A. Sepulveda and R. V. Asem,
Finite difference scheme for a high order nonlinear Schrödinger equation with localized damping, Stud. Univ. Babes-Bolyai Math., 64 (2019), 161-172.
doi: 10.24193/subbmath.2019.2.03. |
[9] |
J. C. Ceballos V., R. Pavez F. and O. P. Vera Villagrán, Exact boundary controllability for higher order nonlinear Schrödinger equations with constant coefficients, Electron. J. Differential Equations, (2005), No. 122, 31 pp. |
[10] |
E. Cerpa and J.-M. Coron,
Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Automat. Control, 58 (2013), 1688-1695.
doi: 10.1109/TAC.2013.2241479. |
[11] |
M. Chen, Stabilization of the higher order nonlinear Schrödinger equation with constant coefficients, Proc. Indian Acad. Sci. Math. Sci., 128 (2018), No. 3, Paper No. 39, 15 pp.
doi: 10.1007/s12044-018-0410-7. |
[12] |
J.-M. Coron and Q. Lü,
Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right, J. Math. Pures Appl. (9), 102 (2014), 1080-1120.
doi: 10.1016/j.matpur.2014.03.004. |
[13] |
C. Jia,
Boundary feedback stabilization of the Korteweg–de Vries–Burgers equation posed on a finite interval, J. Math. Anal. Appl., 444 (2016), 624-647.
doi: 10.1016/j.jmaa.2016.06.063. |
[14] |
Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE Journal of Quantum Electronics, 23 (1987), 510-524. Google Scholar |
[15] |
Y. Kodama,
Optical solitons in a monomode fiber, J. Stat. Phys., 39 (1985), 597-614.
doi: 10.1007/BF01008354. |
[16] |
M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008.
doi: 10.1137/1.9780898718607. |
[17] |
C. Laurey,
The Cauchy problem for a third order nonlinear Schrödinger equation, Nonlinear Anal., 29 (1997), 121-158.
doi: 10.1016/S0362-546X(96)00081-8. |
[18] |
W.-J. Liu and M. Krstić,
Global boundary stabilization of the Korteweg-de Vries-Burgers equation, Comput. Appl. Math., 21 (2002), 315-354.
|
[19] |
W. Liu,
Boundary feedback stabilization of an unstable heat equation, SIAM J. Control Optim., 42 (2003), 1033-1043.
doi: 10.1137/S0363012902402414. |
[20] |
S. Marx and E. Cerpa,
Output feedback stabilization of the Korteweg–de Vries equation, Automatica J. IFAC, 87 (2018), 210-217.
doi: 10.1016/j.automatica.2017.07.057. |
[21] |
T. Özsarı and E. Arabacı,
Boosting the decay of solutions of the linearised Korteweg–de Vries–Burgers equation to a predetermined rate from the boundary, Internat. J. Control, 92 (2019), 1753-1763.
doi: 10.1080/00207179.2017.1408923. |
[22] |
T. Özsarı and A. Batal,
Pseudo-backstepping and its application to the control of Korteweg–de Vries equation from the right endpoint on a finite domain, SIAM J. Control Optim., 57 (2019), 1255-1283.
doi: 10.1137/18M1211933. |
[23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[24] |
H. Takaoka,
Well-posedness for the higher order nonlinear Schrödinger equations, Adv. Math. Sci. Appl., 10 (2000), 149-171.
doi: 10.1016/j.na.2006.06.020. |
[25] |
S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries systems with anti-diffusion, in IEEE 2013 American Control Conference, Washington, DC, 2013, 3302–3307. Google Scholar |
[26] |
S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries with anti-diffusion by boundary feedback with non-collocated observation, in IEEE 2015 American Control Conference, Chicago, IL, 2015.
doi: 10.1109/ACC.2015.7171020. |
show all references
References:
[1] |
G. P. Agrawal, Nonlinear Fiber Optics, Nonlinear Science at the Dawn of the 21$^{st}$ Century, Springer, Berlin, Heidelberg, 2000,195–211.
doi: 10.1007/3-540-46629-0_9. |
[2] |
A. Balogh and M. Krstic,
Boundary control of the Korteweg-de Vries-Burgers equation: Further results on stabilization and well-posedness, with numerical demonstration, IEEE Trans. Automat. Control, 45 (2000), 1739-1745.
doi: 10.1109/9.880639. |
[3] |
A. Batal and T. Özsarı, Output feedback stabilization of the linearized Korteweg-de Vries equation with right endpoint controllers, Automatica, 109 (2019), 108531, 8 pp.
doi: 10.1016/j.automatica.2019.108531. |
[4] |
E. Bisognin, V. Bisognin and O. P. Vera Villagrán, Stabilization of solutions to higher-order nonlinear Schrödinger equation with localized damping, Electron. J. Differential Equations, (2007), No. 6, 18 pp. |
[5] |
X. Carvajal and F. Linares,
A higher-order nonlinear Schrödinger equation with variable coefficients, Differential Integral Equations, 16 (2003), 1111-1130.
|
[6] |
X. Carvajal, Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices, Electron. J. Differential Equations, (2004), No. 13, 10 pp. |
[7] |
X. Carvajal,
Sharp global well-posedness for a higher order Schrödinger equation, J. Fourier Anal. Appl., 12 (2006), 53-70.
doi: 10.1007/s00041-005-5028-3. |
[8] |
M. M. Cavalcanti, W. J. Correa, M. A. Sepulveda and R. V. Asem,
Finite difference scheme for a high order nonlinear Schrödinger equation with localized damping, Stud. Univ. Babes-Bolyai Math., 64 (2019), 161-172.
doi: 10.24193/subbmath.2019.2.03. |
[9] |
J. C. Ceballos V., R. Pavez F. and O. P. Vera Villagrán, Exact boundary controllability for higher order nonlinear Schrödinger equations with constant coefficients, Electron. J. Differential Equations, (2005), No. 122, 31 pp. |
[10] |
E. Cerpa and J.-M. Coron,
Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Automat. Control, 58 (2013), 1688-1695.
doi: 10.1109/TAC.2013.2241479. |
[11] |
M. Chen, Stabilization of the higher order nonlinear Schrödinger equation with constant coefficients, Proc. Indian Acad. Sci. Math. Sci., 128 (2018), No. 3, Paper No. 39, 15 pp.
doi: 10.1007/s12044-018-0410-7. |
[12] |
J.-M. Coron and Q. Lü,
Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right, J. Math. Pures Appl. (9), 102 (2014), 1080-1120.
doi: 10.1016/j.matpur.2014.03.004. |
[13] |
C. Jia,
Boundary feedback stabilization of the Korteweg–de Vries–Burgers equation posed on a finite interval, J. Math. Anal. Appl., 444 (2016), 624-647.
doi: 10.1016/j.jmaa.2016.06.063. |
[14] |
Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE Journal of Quantum Electronics, 23 (1987), 510-524. Google Scholar |
[15] |
Y. Kodama,
Optical solitons in a monomode fiber, J. Stat. Phys., 39 (1985), 597-614.
doi: 10.1007/BF01008354. |
[16] |
M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008.
doi: 10.1137/1.9780898718607. |
[17] |
C. Laurey,
The Cauchy problem for a third order nonlinear Schrödinger equation, Nonlinear Anal., 29 (1997), 121-158.
doi: 10.1016/S0362-546X(96)00081-8. |
[18] |
W.-J. Liu and M. Krstić,
Global boundary stabilization of the Korteweg-de Vries-Burgers equation, Comput. Appl. Math., 21 (2002), 315-354.
|
[19] |
W. Liu,
Boundary feedback stabilization of an unstable heat equation, SIAM J. Control Optim., 42 (2003), 1033-1043.
doi: 10.1137/S0363012902402414. |
[20] |
S. Marx and E. Cerpa,
Output feedback stabilization of the Korteweg–de Vries equation, Automatica J. IFAC, 87 (2018), 210-217.
doi: 10.1016/j.automatica.2017.07.057. |
[21] |
T. Özsarı and E. Arabacı,
Boosting the decay of solutions of the linearised Korteweg–de Vries–Burgers equation to a predetermined rate from the boundary, Internat. J. Control, 92 (2019), 1753-1763.
doi: 10.1080/00207179.2017.1408923. |
[22] |
T. Özsarı and A. Batal,
Pseudo-backstepping and its application to the control of Korteweg–de Vries equation from the right endpoint on a finite domain, SIAM J. Control Optim., 57 (2019), 1255-1283.
doi: 10.1137/18M1211933. |
[23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[24] |
H. Takaoka,
Well-posedness for the higher order nonlinear Schrödinger equations, Adv. Math. Sci. Appl., 10 (2000), 149-171.
doi: 10.1016/j.na.2006.06.020. |
[25] |
S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries systems with anti-diffusion, in IEEE 2013 American Control Conference, Washington, DC, 2013, 3302–3307. Google Scholar |
[26] |
S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries with anti-diffusion by boundary feedback with non-collocated observation, in IEEE 2015 American Control Conference, Chicago, IL, 2015.
doi: 10.1109/ACC.2015.7171020. |

















[1] |
Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021022 |
[2] |
Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 |
[3] |
Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284 |
[4] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020450 |
[5] |
Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 |
[6] |
Guangbin CAI, Yang Zhao, Wanzhen Quan, Xiusheng Zhang. Design of LPV fault-tolerant controller for hypersonic vehicle based on state observer. Journal of Industrial & Management Optimization, 2021, 17 (1) : 447-465. doi: 10.3934/jimo.2019120 |
[7] |
Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021004 |
[8] |
José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020376 |
[9] |
Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 |
[10] |
Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 |
[11] |
Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 |
[12] |
Tomasz Szostok. Inequalities of Hermite-Hadamard type for higher order convex functions, revisited. Communications on Pure & Applied Analysis, 2021, 20 (2) : 903-914. doi: 10.3934/cpaa.2020296 |
[13] |
Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 |
[14] |
Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020461 |
[15] |
Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 |
[16] |
Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020108 |
[17] |
Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121 |
[18] |
Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002 |
[19] |
Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021008 |
[20] |
Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 |
2019 Impact Factor: 0.953
Tools
Article outline
Figures and Tables
[Back to Top]