Stabilization of higher order Schrödinger equations on a finite interval: Part I

Ahmet Batal; Türker Özsarı; Kemal Cem Yılmaz

doi:10.3934/eect.2020095

Article Contents

2021, Volume 10, Issue 4: 861-919. Doi: 10.3934/eect.2020095

This issue Previous Article Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions Next Article Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than

$ 1/2 $

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
^* Corresponding author: Türker Özsarı

Received: April 30, 2020

Revised: June 30, 2020

Early access: September 2020

Published: December 2021

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35Q93, 93B52, 93C20, 93D15, 93D20, 93D23, Secondary: 35A01, 35A02, 35Q55, 35Q60.

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Figure 1. Triangular regions

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Figure 2. Backstepping kernel on $ \Delta_{x,y} $ for $ L = \pi $, $ \beta = 1 $, $ \alpha = 2 $, $ \delta = 8 $ and $ r = 1 $

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Figure 3. 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) $

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Figure 4. Uncontrolled solution for linear case

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Figure 5. Numerical results for the linear controller case. Left: Time evolution of $ |u(x,t)| $. Right: Contour plot of $ |u(x,t)| $

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Figure 6. Left: Time evolution of $ |u(\cdot,t)|_2 $ for different values of $ r $. Right: Control gain $ |k(0,y)| $ for different values of $ r $

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Figure 7. Uncontrolled solution for nonlinear case, $ p \geq 1 $

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Figure 8. Numerical results of the controlled nonlinear model for $ p \geq 1 $

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Figure 9. Numerical results of the controlled nonlinear model for $ 0 < p < 1 $

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Figure 10. Numerical results for the observer case. Left: Time evolution of $ |u(x,t)| $. Right: Contour plot of $ |u(x,t)| $

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Figure 11. Time evolution of $ L^2 $ norms

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Figure 12. Uncontrolled solution for the linear case

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Figure 13. Numerical results of the controlled linear model

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Figure 14. Uncontrolled solution for nonlinear case, $ p \geq 1 $

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Figure 15. Numerical results of the controlled nonlinear model, $ p \geq 1 $

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Figure 16. Uncontrolled solution for the nonlinear case $ 0 < p < 1 $

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Figure 17. Numerical results of the controlled nonlinear model for $ 0 < p < 1 $

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