We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in the presence of quasilinear 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 (adapting a result of Baldi, Berti and Montalto (2014)), the classical Ingham inequality and the Hilbert uniqueness 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 Hörmander (1976, 1985).
"Exact controllability for quasilinear perturbations of KdV." Anal. PDE 10 (2) 281 - 322, 2017. https://doi.org/10.2140/apde.2017.10.281