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

## TL;DR

This paper proves that the linearized Benjamin equation on a periodic domain can be controlled and stabilized exponentially, enabling precise management of wave propagation in fluid systems with capillarity effects.

## Contribution

It establishes exact controllability and exponential stabilization results for the linearized Benjamin equation with periodic boundary conditions.

## Key findings

- Exact controllability in $H_{p}^{s}( ext{T})$ for $s\u22650$
- Exponential stabilization with arbitrary decay rate
- Applicable to modeling long wave propagation in fluid systems

## Abstract

In this work we study the controllability and stabilization of the linearized Benjamin equation which models the unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. We show that the linearized Benjamin equation with periodic boundary conditions is exactly controllable and exponentially stabilizable with any given decay rate in $H_{p}^{s}(\mathbb{T})$ with $s\geq0$.