$L^{2}-$ Well-posedness and Bounded Controllability of KdV-B equation

TL;DR

This paper investigates the well-posedness and bounded controllability of the KdV-Burgers equation on a half-plane, establishing foundational results for control theory applications in nonlinear PDEs.

Contribution

It provides the first analysis of well-posedness in $L^2$ and controllability with boundary controls for the KdV-Burgers equation on a half-line.

Findings

Established $L^2$ well-posedness of the IBVP.

Proved bounded controllability with boundary controls.

Analyzed control effects at boundary conditions.

Abstract

In this paper, the initial boundary value problem of the Korteweg-de Vries Burger equation on the negative half-plane is analyzed. Initially, the well-posedness on $H^s(\R^-)$ for $s\geq 0$ of the IBVP is established to concentrate on the $L^2(\R^-)$ controllability problem when the controls are in the Dirichlet and Newmann conditions at $x=0$.