  This article concerns the nonlinear Korteweg-de Vries equation with boundary
time-delay 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.
