  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 $s\in \mathbb{R}$.
