Rapid Exponential Stabilization of a Boussinesq System of KdV--KdV Type

TL;DR

This paper demonstrates that a Boussinesq system of KdV--KdV type can be exponentially stabilized using a single scalar control input, with decay rate adjustable arbitrarily high, through spectral analysis and Gramian-based feedback design.

Contribution

It introduces a novel method for exponential stabilization of a coupled Boussinesq system with only one scalar input, expanding control possibilities for such systems.

Findings

Solutions decay uniformly to zero under feedback control

Decay rate can be arbitrarily large

Stabilization achieved with a single scalar input

Abstract

This paper studies the exponential stabilization of a Boussinesq system describing the two-way propagation of small amplitude gravity waves on the surface of an ideal fluid, the so-called Boussinesq system of the Korteweg-de Vries type. We use a Gramian-based method introduced by Urquiza to design our feedback control. By means of spectral analysis and Fourier expansion, we show that the solutions of the linearized system decay uniformly to zero when the feedback control is applied. The decay rate can be chosen as large as we want. The main novelty of our work is that we can exponentially stabilize this system of two coupled equations using only one scalar input.