Two simple criterion to obtain exact controllability and stabilization of a linear family of Dispersive PDE's on a periodic domain

TL;DR

This paper introduces simple criteria using the moment method to determine exact controllability and stabilization of linear dispersive PDEs on periodic domains, with applications to several important equations.

Contribution

It provides a practical criterion for controllability and stabilization of linear dispersive PDEs, extending to various equations like Smith, Benjamin-Ono, and higher-order Schrödinger.

Findings

Criteria applicable to multiple dispersive PDEs.

Achieved controllability and stabilization with arbitrary decay rates.

Validated results for several key linearized equations.

Abstract

In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in $H^{s}_{p}(\mathbb{T})$ with $s\in \mathbb{R}$. We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schr\"odinger equation, and the Higher-order Schr\"odinger equations are exactly controllable and exponentially stabilizable with any given decay rate in $H^{s}_{p}(\mathbb{T})$ with $s\in \mathbb{R}$.