Rapid exponential stabilisation of linear-KdV equation with long input delay on the left boundary

Abstract

In this paper, we consider the one-dimensional linear-KdV (Korteweg–de Vries) equation posed in bounded interval with delay boundary control acting on the left boundary by Dirichlet condition where the delay is constant and known, but it can be of arbitrary length. We reformulate the considered problem as an undelayed coupled KdV-Transport system and we use the so-called infinite dimensional backstepping method to derive an explicit feedback control law that transforms system under consideration to a target system. For this target system, we prove its well posedness through the use of semigroup theory and its rapid exponential stability in a suitable functional space by a series of propositions based on the Young's inequality, Wirtinger's inequality and Agmon's inequality. Then, by invertibility of such design, we prove the well posedness and the rapid exponential stabilisation of the original plant.