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

TL;DR

This paper extends the control and observer design for higher order Schr"odinger equations to the right endpoint boundary control case, addressing overdetermined kernel problems with an imperfect kernel approach and analyzing stability and well-posedness.

Contribution

It introduces a novel boundary control strategy for Schr"odinger equations at the right endpoint, overcoming kernel overdetermination issues with an imperfect kernel method.

Findings

Exponential decay of the $L^2$-norm is achieved with boundary control at the right endpoint.

The imperfect kernel approach allows stabilization despite overdetermined boundary value problems.

Numerical simulations confirm the theoretical stability and control results.

Abstract

Backstepping based controller and observer models were designed for higher order linear and nonlinear Schr\"odinger equations on a finite interval in Part I of this study 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. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the $L^2$-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.