The paper studies the exact controllability and stabilizability problem of the KdV equation:
with periodic boundary conditions: , , where denotes a distributed control input such that . The exact controllability problem with finite time is sought first for the linear equation: within the framework of the moment problem: It is solved by introducing an associated Riesz basis (eigenfunctions) and the dual Riesz basis. Then the problem for the original KdV equation is solved by interpreting the term as a control via a Fredholm operator. As to the stabilizability problem, the control is chosen as a feedback of the state which reduces monotonically. By establishing a discrete decay inequality first, an exponential decay estimate is finally obtained.