This paper considers the Korteweg-de Vries (KdV) equation u(t) + uu(x) + u(xxx) = 0, 0 < x < 1, t > 0, u(0, x) - u(0)(x), and the periodic boundary conditions u(t, 1) = u(t, 0), u(xx)(t, 0) = u(xx)(t, 1) with an L(2)-stabilizing control input implemented by a feedback mechanism u(x)(t, 1) = aux(t, 0) and alpha < 1. It can be shown that the solutions conserve the volume [u] = integral(0)(1) u(t, x)dx and the constant state [u(0)] possesses the smallest energy among solutions with same volume. It has been proved that the solution of the system exists and approaches [u(0)] as t --> +infinity when alpha not equal -1/2. This paper studies the case for alpha = -1/2 and gives a proof of the existence and exponential decay of the solutions by deriving estimates of the corresponding Green’s function and using semigroup theory. The method used here also works for the other cases with alpha < 1.4