  We prove that the KdV equation on the circle remains exactly controllable in
arbitrary time with localized control, for sufficiently small data, also in
presence of quasi-linear perturbations, namely nonlinearities containing up to
three space derivatives, having a Hamiltonian structure at the highest orders.
We use a procedure of reduction to constant coefficients up to order zero,
classical Ingham inequality and HUM method to prove the controllability of the
linearized operator. Then we prove and apply a modified version of the
Nash-Moser implicit function theorems by H\"ormander.
