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).
Keywords
control of PDEs, exact controllability, internal
controllability, KdV equation, quasilinear PDEs,
observability of PDEs, HUM, Nash–Moser theorem