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Stability of a Korteweg--de Vries equation close to critical lengths

Jingrui Niu, Shengquan Xiang

Abstract

In this paper, we investigate the quantitative exponential stability of the Korteweg-de Vries equation on a finite interval with its length close to the critical set. Sharp decay estimates are obtained via a constructive PDE control framework. We first introduce a novel transition-stabilization approach, combining the Lebeau--Robbiano strategy with the moment method, to establish constructive null controllability for the KdV equation. This approach is then coupled with precise spectral analysis and invariant manifold theory to characterize the asymptotic behavior of the decay rate as the length of the interval approaches the set of critical lengths. Building on our classification of the critical lengths, we show that the KdV equation exhibits distinct asymptotic behaviors in neighborhoods of different types of critical lengths.

Stability of a Korteweg--de Vries equation close to critical lengths

Abstract

In this paper, we investigate the quantitative exponential stability of the Korteweg-de Vries equation on a finite interval with its length close to the critical set. Sharp decay estimates are obtained via a constructive PDE control framework. We first introduce a novel transition-stabilization approach, combining the Lebeau--Robbiano strategy with the moment method, to establish constructive null controllability for the KdV equation. This approach is then coupled with precise spectral analysis and invariant manifold theory to characterize the asymptotic behavior of the decay rate as the length of the interval approaches the set of critical lengths. Building on our classification of the critical lengths, we show that the KdV equation exhibits distinct asymptotic behaviors in neighborhoods of different types of critical lengths.

Paper Structure

This paper contains 74 sections, 77 theorems, 519 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Review of the literature
  3. Motivations
  4. The constructive control approach
  5. The fast control cost
  6. The limiting stability problem
  7. Statement of the results
  8. Stability results
  9. Control results
  10. Organization of the paper
  11. Strategy of the proof
  12. Outlines
  13. Preliminary
  14. Spectral analysis of the stationary operator
  15. Eigenvalues and eigenfunctions for ${\mathcal{B}}$
  16. ...and 59 more sections

Key Result

Theorem 1.4

Let $L_0\in \mathcal{N}$ be a fixed critical length. Let $I=[L_0- \delta,L_0+ \delta]$ with $\delta= \frac{\pi^2}{3L_0^2}$. For every $L\in I\setminus\{L_0\}$, the state space $L^2(0,L)$ can be decomposed as $H_{{\mathcal{A}}}(L)\oplus M_{{\mathcal{A}}}(L)$ (see Section sec: Projections and state sp

Figures (3)

  • Figure 1: Relations among different notations
  • Figure 7: Spectral Division of ${\mathcal{B}}$
  • Figure 8: Relations among spaces

Theorems & Definitions (171)

  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Proposition 3.1
  • Remark 3.2
  • Definition 3.3: Symmetric finite co-dimensional projector
  • Definition 3.4: Projective null controllability
  • ...and 161 more