Small-time local controllability of a KdV system for all critical lengths

TL;DR

This paper completes the classification of small-time local controllability for the KdV system at all critical lengths, resolving a longstanding open problem by analyzing the controllability properties based on length parameters.

Contribution

It establishes that the KdV system is not small-time locally controllable at certain critical lengths where $2k+l otin 3\mathbb{N}^*$ and $k \neq l$, completing previous partial results.

Findings

System is not small-time controllable for critical lengths with $2k+l \in 3\mathbb{N}^*$ and $k \neq l$.

Confirms controllability at all other critical lengths, resolving the open problem.

Provides a complete characterization of small-time local controllability for the KdV system at critical lengths.

Abstract

In this paper, we consider the small-time local controllability problem for the KdV system on an interval with a Neumann boundary control. In 1997, Rosier discovered that the linearized system is uncontrollable if and only if the length is critical, namely $L=2\pi\sqrt{(k^2+ kl+ l^2)/3}$ for some integers $k$ and $l$. Coron and Cr\'epeau (2003) proved that the nonlinear system is small-time locally controllable even if the linearized system is not, provided that $k= l$ is the only solution pair. Later, Cerpa and Crepeau showed that the system is large-time locally controllable for all critical lengths. In 2020, Coron, Koenig, and Nguyen found that the system is not small-time locally controllable if $2k+l\not \in 3\mathbb{N}^*$. We demonstrate that if the critical length satisfies $2k+l \in 3\mathbb{N}^*$ with $k\neq l$, then the system is not small-time locally controllable. This paper, together with the above results, gives a complete answer to the longstanding open problem on the small-time local controllability of KdV on all critical lengths since the pioneer work by Rosier