Asymptotic behavior of Boussinesq system of KdV–KdV type

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.2010 Mathematics Subject Classification. Primary: 93B05, 93D15, 35Q53.

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

“…Backstepping with two scalar controls has already been implemented in [13] for a coupled KdV-KdV system with two boundary controls, for which, because of the coupling nature, it is natural to introduce two control terms. More recently, it has been improved in [12], where the authors showed that only one control is sufficient to stabilize the system.…”

On the well-posedness and large-time behavior of higher order Boussinesq system

“…Remark 1. Note that the right hand side of (3.4) is well defined for all τ ∈ [0, T ], since ψ xx (0, ·) and ϕ xx (L, ·) belong to H The next result borrowed from [9], with minor changes, gives us the existence and uniqueness of solution for system (3.3). Its proof is presented here for the sake of completeness.…”

“…In the present work, we address the problems described in the previous subsection and our main results provide a partial positive answer for the Problems A and B. In order to give an answer for Problem B, we apply the ideas suggested in [9,10], therefore, let us consider…”

Stabilization of the coupled ODE-linearized KdV system

“…To prove the stability of the closed-loop system (1) with the control law (6), we need to show that transformation (2) is bounded and invertible.…”

“…Now it has become an important model. People have carried out a lot of research on the KdV equation, see [1][2] and reference therein. For changing the state of KdV for satisfying our purpose, its control problem is introduced, for example, kinds of stability and controllability results for KdV can be found in [3][4][5][6][7] and reference therein.…”