Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback

This article concerns the nonlinear Korteweg-de Vries equation with boundary timedelay feedback. Under appropriate assumption on the coefficients of the feedbacks (delayed or not), we first prove that this nonlinear infinite dimensional system is well-posed for small initial data. The main results of our study are two theorems stating the exponential stability of the nonlinear time delay system. Two different methods are employed: a Lyapunov functional approach (allowing to have an estimation on the decay rate, but with a restrictive assumption on the length of the spatial domain of the KdV equation) and an observability inequality approach, with a contradiction argument (for any non critical lengths but without estimation on the decay rate). Some numerical simulations are given to illustrate the results. Keyword: Korteweg-de Vries non-linear equation, exponential stability, time-delay, Lyapunov functional.