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

  • * Corresponding author: Türker Özsarı

    * Corresponding author: Türker Özsarı 
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  • 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.

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

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  • Figure 1.  Triangular regions

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

    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) $

    Figure 4.  Uncontrolled solution for linear case

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

    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 $

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

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

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

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

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

    Figure 12.  Uncontrolled solution for the linear case

    Figure 13.  Numerical results of the controlled linear model

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

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

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

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

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