Exact controllability for quasi-linear perturbations of KdV

TL;DR

This paper demonstrates the exact controllability of the KdV equation on the circle with localized control, even under quasi-linear perturbations, by combining reduction techniques, classical inequalities, and advanced implicit function theorems.

Contribution

It extends controllability results of KdV to quasi-linear perturbations with Hamiltonian structure using a novel combination of reduction, inequalities, and Nash-Moser methods.

Findings

Exact controllability for small data under quasi-linear perturbations

Reduction to constant coefficients up to order zero achieved

Application of Nash-Moser theorem for nonlinear analysis

Abstract

We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in presence of quasi-linear 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, classical Ingham inequality and HUM 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 H\"ormander.