Controllability of quasi-linear Hamiltonian NLS equations

TL;DR

This paper demonstrates that small data quasi-linear Hamiltonian NLS equations on the circle can be controlled internally within any time frame, using a combination of reduction techniques, HUM method, and Nash-Moser-Hörmander theorem.

Contribution

It introduces a novel approach combining reduction to constant coefficients and advanced functional analysis to establish controllability for this class of equations.

Findings

Proves internal controllability in arbitrary time for small data

Reduces quasi-linear equations to constant coefficients

Applies Nash-Moser-Hörmander theorem for nonlinear problem

Abstract

We prove internal controllability in arbitrary time, for small data, for quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of reduction to constant coefficients up to order zero and HUM method to prove the controllability of the linearized problem. Then we apply a Nash-Moser-H\"ormander implicit function theorem as a black box.