Observability of the linear Zakharov--Kuznetsov equation

TL;DR

This paper investigates the observability, controllability, and stabilization of the linear Zakharov--Kuznetsov equation in two dimensions, introducing new methods and insights applicable to dispersive PDEs.

Contribution

It provides new internal observability theorems and control strategies for the linear Zakharov--Kuznetsov equation using nonharmonic Fourier series and duality principles.

Findings

Established internal observability theorems

Proved exact controllability results

Achieved rapid uniform stabilization

Abstract

We study the linear Zakharov--Kuznetsov equation with periodic boundary conditions. Employing some tools from the nonharmonic Fourier series we obtain several internal observability theorems. Then we prove various exact controllability and rapid uniform stabilization results by applying a duality principle and a general feedback construction. The method presented here introduces a new insight into the control of dispersive equations in two-dimensional cases and may be adapted to more general equations.