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

Ivonne Rivas, Liliana Esquivel

TL;DR

The paper analyzes the Korteweg-de Vries–Burgers equation on the negative half-line, proving local well-posedness in $H^s(\mathbb{R}^-)$ for $s>0$ and establishing a comprehensive controllability theory. It develops a linear theory with explicit solution representations and Bourgain-space estimates, then handles the nonlinear problem via a fixed-point argument using boundary-corrected linear propagators. A global Carleman estimate is derived to obtain observability-type inequalities, which are instrumental in proving both negative null-controllability for $L^2$-energy-bounded solutions and positive boundary controllability through an approximation- and forcing-based approach. The analysis connects spectral properties of a bounded KdV–B operator to controllability on the half-line, providing a rigorous framework for boundary control in dispersive-dissipative IBVPs with unbounded spatial domains.

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 $x=0$.

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

TL;DR

The paper analyzes the Korteweg-de Vries–Burgers equation on the negative half-line, proving local well-posedness in for and establishing a comprehensive controllability theory. It develops a linear theory with explicit solution representations and Bourgain-space estimates, then handles the nonlinear problem via a fixed-point argument using boundary-corrected linear propagators. A global Carleman estimate is derived to obtain observability-type inequalities, which are instrumental in proving both negative null-controllability for -energy-bounded solutions and positive boundary controllability through an approximation- and forcing-based approach. The analysis connects spectral properties of a bounded KdV–B operator to controllability on the half-line, providing a rigorous framework for boundary control in dispersive-dissipative IBVPs with unbounded spatial domains.

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 for of the IBVP is established to concentrate on the controllability problem when the controls are in the Dirichlet and Newmann conditions at .

Paper Structure

This paper contains 16 sections, 17 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Notation and Main result
  3. Analysis of the linear Korteweg de Vries-Burgers equation
  4. Solution formulas for linear problems
  5. Initial value problem in $\mathbb R$
  6. Boundary value problem with homogeneous initial data
  7. Initial value problem in $\mathbb{R}^+$
  8. Well-posedness of the non-linear KdV-B
  9. Carleman's Estimate
  10. Controllability
  11. Proof of first main result of controllability Theorem \ref{['nonullcontrol']}
  12. Proof of the second main control result:
  13. Spectral analysis of the bounded KdV-B operator
  14. Useful Results
  15. Lemmas for the no-controllability
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $s\in \mathbb{R}$, $0<b\le 1$ and $0 < \delta < \frac{1}{2}$ be given. For $u_0 \in C^{\infty}_0(\mathbb{R})$ (smooth function with compact support):

Theorems & Definitions (39)

  • Definition 2.1: Mild solution
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Remark 3.4
  • Proposition 3.5
  • Proposition 3.6: Bilinear estimate
  • Remark 3.7
  • Lemma 3.8
  • proof
  • ...and 29 more