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

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.

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

“…as in Part I (see [3]) with additional assumptions on the coefficients, but this topic is omitted here considering the volume of current text and postponed to a future paper. In addition, it is also possible to consider other sets of boundary conditions here as in Part I that involves second order traces such as u(0, t) = 0, u x (L, t) = h 0 (t), u xx (L, t) = h 1 (t), but this will also be discussed in another place.…”

“…A numerical study of this problem was given in [9]. From the controllability and stabilization perspective, we refer the reader to [10] for exact boundary controllability, [4] and [12] for internal feedback stabilization and [3] for boundary feedback stabilization.…”

“…In order to solve this problem, we use backstepping method (see [20] for a general discussion on the backstepping method), which is a well studied method for the second order evolutionary partial differential equations [19,22,27,28,29]. In recent years, researchers studied backstepping stabilization of several higher order evolutionary equations that include third order dispersion term [3,11,23,24,32,33]. In these studies on KdV type equations, a single boundary feedback control is located at one endpoint and the number of boundary conditions located at the opposite endpoint are two.…”

“…In these studies on KdV type equations, a single boundary feedback control is located at one endpoint and the number of boundary conditions located at the opposite endpoint are two. In particular, Part I of this study [3] assumes a control input acting from the left endpoint, and there are two homogeneous boundary conditions that are imposed from the right endpoint. Conversely, if there are two boundary controllers acting from the right endpoint and a single homogeneous boundary condition imposed at the left endpoint, which is the subject of the present paper, the situation becomes mathematically very different as we explain below.…”

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Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [3] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain.

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