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

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 \begin{document}$ H_{p}^{s}(\mathbb{T}) $\end{document} with \begin{document}$ s\in \mathbb{R}. $\end{document} We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given decay rate in \begin{document}$ H_{p}^{s}(\mathbb{T}) $\end{document} with \begin{document}$ s\in \mathbb{R}. $\end{document}