Fredholm transform and local rapid stabilization for a Kuramoto–Sivashinsky equation

This paper is devoted to the study of the local rapid exponential stabilization problem for a controlled Kuramoto–Sivashinsky equation on a bounded interval. We build a feedback control law to force the solution of the closed-loop system to decay exponentially to zero with arbitrarily prescribed decay rates, provided that the initial datum is small enough. Our approach uses a method we introduced for the rapid stabilization of a Korteweg–de Vries equation. It relies on the construction of a suitable integral transform and can be applied to many other equations.