Exact controllability and stabilization for linear dispersive PDE's on the two-dimensional torus

TL;DR

This paper proves the exact controllability and exponential stabilization of a broad class of linear dispersive PDEs on the two-dimensional torus, using the moment method and feedback control laws.

Contribution

It extends controllability and stabilization results to bidimensional dispersive PDEs, including models like Benjamin-Ono and Korteweg-de Vries equations, on the 2D torus.

Findings

Exact controllability of bidimensional dispersive PDEs on the torus.

Exponential stabilizability with arbitrary decay rate.

Applicability to models like Benjamin-Ono and Korteweg-de Vries equations.

Abstract

The moment method is used to prove the exact controllability of a wide class of bidimensional linear dispersive PDE's posed on the two-dimensional torus $\mathbb{T}^{2}.$ The control function is considered to be acting on a small vertical and horizontal strip of the torus. Our results apply to several well-known models including some bidimesional extensions of the Benajamin-Ono and Korteweg-de Vries equations. As a by product, the exponential stabilizability with any given decay rate is also established in $H^{s}_{p}(\mathbb{T}^{2}),$ with $s\geq 0,$ by constructing an appropriated feedback control law.