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The Korteweg-de Vries (KdV) equation with the right Dirichlet control wasinitially investigated more than twenty years ago. It was shown that thissystem is small time, locally, exactly controllable for all non-criticallengths and its linearized system is not controllable for {\it all} criticallengths. Even though the controllability of the KdV system has been studiedextensively in the last two decades, the local controllability of this systemfor critical lengths remains an open question. In this paper, we give adefinitive answer to this question. First, we characterize all critical lengthsand the corresponding unreachable space for the linearized system. Inparticular, we show that the unreachable space is always of dimension 1.Second, we prove that the KdV system with the right Dirichlet control is notlocally null controllable in small time. Third, we give a criterion todetermine whether the system is locally exactly controllable in finite time or{\it not} locally null controllable in any positive time for {\it all} criticallengths. Consequently, we show that there exist critical lengths such that thesystem is not locally null controllable in small time but is locally exactlycontrollable in finite time. These facts are surprising and distinct incomparison with related known results. First, it is known that thecorresponding KdV system with the right zero Dirichlet is locally exactlycontrollable in small time using internal controls. Second, the unreachablespace of the linearized system of the corresponding KdV system with the rightNeumann control might be of arbitrary dimension. Third, the KdV system with theright Neumann control is locally exactly controllable in small time if thecorresponding unreachable space is of dimension 1.

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