Exact Controllability for the Quasilinear Perturbations of the Kawahara Equation

This paper is devoted to studying the exact controllability for the Kawahara equation under the influence of quasilinear perturbations for sufficiently small data on the circle with localized control, the nonlinearities containing up to five space derivatives and having a Hamiltonian structure at the space derivatives of the highest order. Firstly, we conjugate the associated linearized operator to a time-dependent variable coefficient operator up to a bounded remainder. The major difficulties come from five space derivatives and the coupling of the coefficient of the highest order term with the coefficients of other terms. The strategy adopted is to look for appropriate transformations, which are reversible and satisfy the sharp bounds for the reducibility. Then, from the observability and controllability of the corresponding linear control problem, the existence of the right inverse for the linearized operator is verified. Finally, the application of the Nash–Moser–Hörmander theorem implies the exact controllability for the Kawahara equation with the quasilinear perturbations.