Local controllability of the Korteweg-de Vries equation with the right Dirichlet control

Abstract

The Korteweg-de Vries (KdV) equation with the right Dirichlet control is small time, locally, exactly controllable for all non-critical lengths and its linearized system is not controllable for all critical lengths. In this paper, we give a definitive picture of the local controllability properties of this control problem for all critical lengths. In particular, we show that the unreachable space of the linearized system is always of dimension 1 and the KdV system with the right Dirichlet control is not locally null controllable in small time for any critical length. We also give a criterion to determine whether the system is locally exactly controllable in finite time or not locally null controllable in any positive time for all critical lengths. Consequently, we show that there exist critical lengths such that the system is not locally null controllable in small time but is locally exactly controllable in finite time.