Stabilization of higher order Schr\"odinger equations on a finite interval: Part I

TL;DR

This paper develops a backstepping stabilization method for higher order Schrödinger equations on finite intervals, including observer design for partial measurements, with theoretical analysis and numerical verification.

Contribution

It introduces a novel backstepping approach for higher order Schrödinger equations, handling nonlinearities and partial measurements, with rigorous well-posedness and numerical algorithms.

Findings

Successfully stabilizes higher order Schrödinger equations with prescribed decay rates.

Designs observers for systems with partial boundary measurements.

Provides numerical algorithms and simulations confirming theoretical results.

Abstract

We study the backstepping stabilization of higher order linear and nonlinear Schr\"odinger 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.