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 $s\ge 0$. 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].

本文的目的是研究本杰明方程在周期域上的可控性和稳定性 $\mathbb{吨}$. 我们证明了 Benjamin 方程是全局精确可控和全局指数稳定的$H_{磷}^{秒}(\mathbb{吨})$， 和 $秒\ge 0$. 与自然反馈定律相对应的全局指数稳定性首先是借助解的某些性质建立的，即在 Bourgain 空间中紧致性的传播和规律性的传播。系统的全局指数稳定性与局部可控性结果相结合，也产生了全局可控性。使用不同的反馈定律，所得闭环系统显示为具有任意大衰减率的局部指数稳定。进一步设计了随时间变化的反馈定律，以确保具有任意大衰减率的全局指数稳定性。这里获得的结果扩展了我们在 [32] 中为线性化 Benjamin 方程证明的结果。