Asymptotic behavior of Boussinesq system of KdV-KdV type

TL;DR

This paper develops a feedback control law for a Boussinesq system of KdV-KdV type, achieving exponential stabilization of small amplitude long waves on a finite domain with boundary controls.

Contribution

It introduces integral transformations to design a feedback control law that ensures exponential decay of solutions in the Boussinesq KdV-KdV system.

Findings

Solutions decay exponentially in the L2 norm

Decay rate can be arbitrarily increased for small initial data

Boundary controls effectively stabilize the system

Abstract

This work deals with the local rapid exponential stabilization for a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Our goal is to build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the closed-loop system decays exponentially to zero in the $L^2(0,L)$--norm and the decay rate can be tuned to be as large as desired if the initial data is small enough.