Singular perturbation analysis for a coupled KdV-ODE system

TL;DR

This paper applies singular perturbation analysis to a coupled KdV-ODE system, deriving stability results and Tikhonov-type theorems to handle different timescales in infinite-dimensional systems.

Contribution

It extends singular perturbation methods to coupled KdV-ODE systems, providing new stability conditions and theoretical results in an infinite-dimensional context.

Findings

Established stability results for the coupled system.

Derived Tikhonov-type theorems for the system.

Validated the approach for systems with different timescales.

Abstract

Asymptotic stability is with no doubts an essential property to be studied for any system. This analysis often becomes very difficult for coupled systems and even harder when different timescales appear. The singular perturbation method allows to decouple a full system into what are called the reduced order system and the boundary layer system, to get simpler stability conditions for the original system. In the infinite-dimensional setting, we do not have a general result making sure this strategy works. This papers is devoted to this analysis for some systems coupling the Korteweg-the Vries equation and an ordinary differential equation with different timescales. More precisely, We obtain stability results and Tikhonov-type theorems.