Small-time local controllability of a KdV system for all 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=2π√(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 Crépeau 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∉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.