On the controllability and stabilization of the Benjamin equation on a periodic domain

The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain $T$. We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in $H_p^s\left(T\right)$, with $s\ge 0$. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in [32].