Considered herein is the exact controllability of the Hirota equation with quasi-linear Hamiltonian perturbations. Firstly, by employing some diffeomorphism of the circle \({\mathbb {T}},\) 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.
