Singular Perturbation Analysis for a Coupled KdV–ODE System

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 time-scales 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 article is devoted to this analysis for some systems coupling the Korteweg-de Vries equation and an ordinary differential equation with different time scales. More precisely, we obtain stability results and Tikhonov-type theorems.