Two Approaches for the Stabilization of Nonlinear KdV Equation With Boundary Time-Delay Feedback

This paper 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 function 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 noncritical lengths but without estimation on the decay rate). Some numerical simulations are given to illustrate the results.