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

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 $H_p^s({\mathbb{T}}^2),$ with $s\ge 0,$ by constructing an appropriated feedback control law.