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 s p (T), with s ≥ 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 [32].1991 Mathematics Subject Classification. 93B05, 93D15, 35Q53.
This paper is concerned with controllability and stabilization properties of the dispersion generalized Benjamin equation on the periodic domain $\mathbb{T}.$ First, by assuming the control input acts on all the domain, the system is proved to be globally exactly controllable in the Sobolev space $H^{s}_{p}(\mathbb{T}),$ with $s\geq 0.$ Second, by providing a locally-damped term added to the equation as a feedback law, it is shown that the resulting equation is globally well-posed and locally exponentially stabilizable in the space $L^{2}_{p}(\mathbb{T}).$ The main ingredient to prove the global well-posedness is the introduction of the dissipation-normalized Bourgain spaces which allows one to gain smoothing properties simultaneously from the dissipation and dispersion present in the equation. Finally, the local exponential stabilizability result is accomplished taking into account the decay of the associated semigroup combined with the fixed point argument.
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