On the controllability and Stabilization of the Benjamin Equation

TL;DR

This paper investigates the controllability and stabilization of the Benjamin equation on a periodic domain, establishing global controllability and exponential stabilization using various feedback laws and extending previous linearized results.

Contribution

It introduces new methods for global controllability and stabilization of the nonlinear Benjamin equation, including time-varying feedback laws, extending prior linearized analyses.

Findings

The Benjamin equation is globally exactly controllable.

The Benjamin equation can be exponentially stabilized with specific feedback laws.

A time-varying feedback law guarantees global exponential stability with arbitrary decay rate.

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}.