  The aim of this paper is to study the controllability and stabilization for
the Benjamin equation on a periodic domain $\mathbb{T}$. We show that the
Benjamin equation is globally exactly controllable and globally exponentially
stabilizable in $H_{p}^{s}(\mathbb{T}),$ with $s\geq 0.$ First we prove
propagation of compactness, propagation of regularity of solution in Bourgain's
spaces and unique continuation property, and use them to obtain the global
exponential stabilizability corresponding to a natural feedback law. Combining
the global exponential stability and the local controllability result we prove
the global controllability as well. Also, we prove that the closed-loop system
with a different feedback control law is locally exponentially stable with an
arbitrary decay rate. Finally, a time-varying feedback law is designed to
guarantee a global exponential stability with an arbitrary decay rate. The
results obtained here extend the ones we proved for the linearized Benjamin
equation in \cite{Vielma and Panthee}.
