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

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 $L=2π\sqrt{(k^2+ kl+ l^2)/3}$ for some integers $k$ and $l$. Coron and Crépeau (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

💡 Research Summary

The paper addresses the long‑standing open problem of small‑time local controllability (STLC) for the Korteweg‑de Vries (KdV) equation on a bounded interval with a Neumann boundary control at the right endpoint. The equation under study is

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