Output feedback stabilization of the linearized Korteweg-de Vries equation with right endpoint controllers

TL;DR

This paper develops an output feedback stabilization method for the linearized Korteweg-de Vries equation on a finite domain, using boundary measurements and observers, overcoming challenges in right endpoint controller design.

Contribution

It introduces a novel observer-based boundary feedback control strategy for the linearized KdV equation with right endpoint control, addressing mathematical difficulties in kernel model construction.

Findings

Exponential stabilization of the system in the L^2 sense.

Observer convergence in higher Sobolev norms.

Reduction of controllers and measurements to a single boundary measurement.

Abstract

In this paper, we prove the output feedback stabilization for the linearized Korteweg-de Vries (KdV) equation posed on a finite domain in the case the full state of the system cannot be measured. We assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution. This allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable. Stabilization of the original system is proved in the $L^2$-sense, while the convergence of the observer system to the original plant is also proved in higher order Sobolev norms. The standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models. In order to deal with this difficulty we use the method of [18] which is based on using modified target systems involving extra trace terms. In addition, we show that the number of controllers and boundary measurements can be reduced to one, with the cost of a slightly lower exponential rate of decay. We provide numerical simulations illustrating the efficacy of our controllers.