Weak damping for the Korteweg-de Vries equation

TL;DR

This paper introduces a weak damping mechanism for the Korteweg-de Vries equation that achieves global exponential stability with lower control costs, advancing control methods for this classical PDE.

Contribution

It proposes a novel, cost-effective weak forcing approach to stabilize the Korteweg-de Vries equation globally, improving upon existing control strategies.

Findings

Achieves global exponential stability with weaker damping.

Reduces control cost compared to previous methods.

Validates effectiveness through mathematical analysis.

Abstract

For more than 20 years, the Korteweg-de Vries equation has been intensively explored from the mathematical point of view. Regarding control theory, when adding an internal force term in this equation, it is well known that the Korteweg-de Vries equation is exponentially stable in a bounded domain. In this work, we propose a weak forcing mechanism, with a lower cost than that already existing in the literature, to achieve the result of the global exponential stability to the Korteweg-de Vries equation.