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

TL;DR

This paper studies the exponential stability of the Korteweg-de Vries equation on finite intervals near critical lengths, introducing a novel stabilization method and analyzing decay rates.

Contribution

It presents a new transition-stabilization approach combining PDE control strategies to analyze stability near critical lengths.

Findings

Sharp decay estimates for the KdV equation near critical lengths.

Distinct asymptotic behaviors depending on the type of critical length.

Constructive null controllability established for the KdV equation.

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.