Controllability of the Hirota equation with quasi-linear Hamiltonian perturbations

Considered herein is the exact controllability of the Hirota equation with quasi-linear Hamiltonian perturbations. Firstly, by employing some diffeomorphism of the circle T, we reduce the linearized operator to a time-dependent variable coefficients operator with a bounded remainder. The main challenge during the reduction is the off-diagonal matrices of the lower-order term coefficients and their coupling with the highest-order term. The method used involves finding some perturbations to diagonalize the coefficient matrices of the lower-order terms. Then we establish the existence of the right inverse for the linearized operator by investigating the associated linear control problem. Finally, utilizing the Nash-Moser implicit function theorem, we prove the exact controllability of the Hirota equation under the influence of quasi-linear Hamiltonian perturbations.