On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

International audienceThe aim of this work is to study the exponential stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed internal feedback. We first consider the case where the weight of the feedback with delay is smaller than the weight of the feedback without delay and prove the local exponential stability result by two methods: the first one by a Lyapunov method (which holds for restrictive length of the domain but allow to have an estimation on the decay rate) and the second one by an observability inequality for any length (without estimation of the decay rate). We also prove a semiglobal stabilization result for any length. Secondly we study the case where the support of the feedback without delay is not included in the feedback with delay and give a local exponential stability result if the weight of the delayed feedback is small enough. Some numerical simulations are given to illustrate these results