Small-time local controllability of a KdV system for all critical lengths
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914204537061376 |
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| author | Niu, Jingrui Xiang, Shengquan |
| author_facet | Niu, Jingrui Xiang, Shengquan |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13640 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small-time local controllability of a KdV system for all critical lengths Niu, Jingrui Xiang, Shengquan Analysis of PDEs Optimization and Control 35Q53, 76B15, 93C20 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 |
| title | Small-time local controllability of a KdV system for all critical lengths |
| topic | Analysis of PDEs Optimization and Control 35Q53, 76B15, 93C20 |
| url | https://arxiv.org/abs/2501.13640 |