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arXiv · 1904.03492

On the controllability and Stabilization of the Benjamin Equation

M. Panthee·F. Vielma Leal
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Abstract

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}.

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M. Panthee, F. Vielma Leal. 2019-04-06. On the controllability and Stabilization of the Benjamin Equation. https://arxiv.org/abs/1904.03492

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