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Boundary controllability of the Korteweg-de Vries equation: The Neumann case

R. de A. Capistrano-Filho, J. S. da Silva

TL;DR

This work analyzes the Korteweg–de Vries equation on $(0,L)$ with Neumann boundary control at $x=L$ and a single input, focusing on the so-called critical length set ${\mathcal R}_c$. The authors prove local exact controllability in $L^2(0,L)$ when $L\in {\mathcal R}_c$ by combining the return method with a fixed-point argument, and show that linearization around a steady state is not exactly controllable at these lengths while nearby equilibria can be steered. They further establish that for small perturbations of the equilibrium $c$, there exists a neighborhood where $L\notin {\mathcal R}_d$, enabling controllability around those equilibria. The results extend the boundary controllability theory for Neumann controls to the critical-length regime and provide a framework for analyzing controllability costs and short-time behavior. Overall, the paper advances understanding of the critical-length phenomenon for dispersive PDE control and outlines open challenges for HUM-based approaches and non-Neumann boundary conditions.

Abstract

This article gives a necessary first step to understanding the critical set phenomenon for the Korteweg-de Vries (KdV) equation posed on interval $[0,L]$ considering the Neumann boundary conditions with only one control input. We showed that the KdV equation is controllable in the critical case, i.e., when the spatial domain $L$ belongs to the set $\mathcal{R}_c$, where $c\neq-1$ and $$ \mathcal{R}_c:=\left\{\frac{2π}{\sqrt{3(c+1)}}\sqrt{m^2+ml+m^2};\ m,l\in \mathbb{N}^*\right\}\cup\left\{\frac{mπ}{\sqrt{c+1}};\ m\in \mathbb{N}^*\right\}, $$ the KdV equation is exactly controllable in $L^2(0,L)$. The result is achieved using the return method together with a fixed point argument.

Boundary controllability of the Korteweg-de Vries equation: The Neumann case

TL;DR

This work analyzes the Korteweg–de Vries equation on with Neumann boundary control at and a single input, focusing on the so-called critical length set . The authors prove local exact controllability in when by combining the return method with a fixed-point argument, and show that linearization around a steady state is not exactly controllable at these lengths while nearby equilibria can be steered. They further establish that for small perturbations of the equilibrium , there exists a neighborhood where , enabling controllability around those equilibria. The results extend the boundary controllability theory for Neumann controls to the critical-length regime and provide a framework for analyzing controllability costs and short-time behavior. Overall, the paper advances understanding of the critical-length phenomenon for dispersive PDE control and outlines open challenges for HUM-based approaches and non-Neumann boundary conditions.

Abstract

This article gives a necessary first step to understanding the critical set phenomenon for the Korteweg-de Vries (KdV) equation posed on interval considering the Neumann boundary conditions with only one control input. We showed that the KdV equation is controllable in the critical case, i.e., when the spatial domain belongs to the set , where and the KdV equation is exactly controllable in . The result is achieved using the return method together with a fixed point argument.
Paper Structure (12 sections, 13 theorems, 172 equations, 1 figure)

This paper contains 12 sections, 13 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem set
  3. Main results
  4. Heuristic and paper's outline
  5. Overview of the well-posedness theory
  6. Boundary controllability in the critical length
  7. An auxiliary result
  8. Construction of the trajectories
  9. Controllability on $\mathcal{R}_c$
  10. Conclusion and open issues
  11. Control cost and small-time behavior
  12. Neumann versus Dirichlet/mixed controls

Key Result

theorem 1

Let $T>0$, $c\ne -1$ and $L \notin \mathcal{R}_c$. There exists $\delta>0$ such that for any $u_0,u_T\in L^2(0,L)$ with one can find $h \in L^2(0,T)$ such that the system NKdV-2 admits a unique solution satisfying control-r.

Figures (1)

  • Figure 1: Solutions driving states close to 0 to constants and vice versa.

Theorems & Definitions (24)

  • definition 1
  • theorem 1: Caicedo, Capistrano--Filho, Zhang CaCaZh
  • theorem 2: Caicedo, Capistrano--Filho, Zhang CaCaZh
  • theorem 3
  • theorem 4
  • theorem 5
  • proposition 1: Caicedo, Capistrano-Filho, Zhang CaCaZh
  • remark 1
  • lemma 1: Kramer, Zhang KZ
  • proposition 2
  • ...and 14 more