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arXiv · 1709.06192

Asymptotic behavior of Boussinesq system of KdV-KdV type

Roberto A. Capistrano-Filho·Fernando A. Gallego
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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.

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Roberto A. Capistrano-Filho, Fernando A. Gallego. 2017-09-18. Asymptotic behavior of Boussinesq system of KdV-KdV type. https://doi.org/10.1016/j.jde.2018.04.034

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