# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020095

## 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

* Corresponding author: Türker Özsarı

Received  May 2020 Revised  July 2020 Published  September 2020

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.

Citation: Ahmet Batal, Türker Özsarı, Kemal Cem Yılmaz. Stabilization of higher order Schrödinger equations on a finite interval: Part I. Evolution Equations & Control Theory, doi: 10.3934/eect.2020095
##### References:
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Differential Equations, (2005), No. 122, 31 pp.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] Y. Kodama and A. 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Control Optim., 42 (2003), 1033-1043.  doi: 10.1137/S0363012902402414.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] X. Carvajal and F. Linares, A higher-order nonlinear Schrödinger equation with variable coefficients, Differential Integral Equations, 16 (2003), 1111-1130.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] W.-J. Liu and M. Krstić, Global boundary stabilization of the Korteweg-de Vries-Burgers equation, Comput. Appl. Math., 21 (2002), 315-354.   Google Scholar [19] W. Liu, Boundary feedback stabilization of an unstable heat equation, SIAM J. Control Optim., 42 (2003), 1033-1043.  doi: 10.1137/S0363012902402414.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
Triangular regions
Backstepping kernel on $\Delta_{x,y}$ for $L = \pi$, $\beta = 1$, $\alpha = 2$, $\delta = 8$ and $r = 1$
Left: Contour plot of $|p(x,y)|$ on $\Delta_{x,y}$ for $L = \pi$, $\beta = 0.5$, $\alpha = 1$, $\delta = 0.5$ and $r = 0.2$. Right: Real and imaginary parts of $p_1(x) = -i \beta p(x,L)$
Uncontrolled solution for linear case
Numerical results for the linear controller case. Left: Time evolution of $|u(x,t)|$. Right: Contour plot of $|u(x,t)|$
Left: Time evolution of $|u(\cdot,t)|_2$ for different values of $r$. Right: Control gain $|k(0,y)|$ for different values of $r$
Uncontrolled solution for nonlinear case, $p \geq 1$
Numerical results of the controlled nonlinear model for $p \geq 1$
Numerical results of the controlled nonlinear model for $0 < p < 1$
Numerical results for the observer case. Left: Time evolution of $|u(x,t)|$. Right: Contour plot of $|u(x,t)|$
Time evolution of $L^2$ norms
Uncontrolled solution for the linear case
Numerical results of the controlled linear model
Uncontrolled solution for nonlinear case, $p \geq 1$
Numerical results of the controlled nonlinear model, $p \geq 1$
Uncontrolled solution for the nonlinear case $0 < p < 1$
Numerical results of the controlled nonlinear model for $0 < p < 1$
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