2025/01/23 by Jingrui Niu, Shengquan Xiang, Niu, Jingrui +1 · 2 citations
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Physics Problems
paper · pdf · doi:10.48550/arxiv.2501.13640
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√((k2+ kl+ l2)/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 ∈ 3ℕ^*. We demonstrate that if the critical length satisfies 2k+l ∈ 3ℕ^* with k≠ 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