Boundary stabilization of the korteweg-de vries equation and the Korteweg-de Vries-Burgers equation

摘要

In this article, we continue our study of a system described by a class of initial boundary value problem (IBVP) of the Korteweg-de Vries (KdV) equation and the KdV Burgers (KdVB) equation posed on a finite interval with nonhomogeneous boundary conditions. While the system is known to be locally well-posed (Kramer et al. arXiv:1012.1057, [2010]; Rivas et al. in Math. Control Relat. Fields 1:61-81, [2011]) and its small amplitude solutions are known to exist globally, it is not clear whether its large amplitude solutions would blow up in finite time or not. This problem is addressed in this article from control theory point of view: look for some appropriate feedback control laws (with boundary value functions as control inputs) to ensure that the finite time blow-up phenomena would never occur. In this article, a simple, but nonlinear, feedback control law is proposed and the resulting closed-loop system is shown not only to be globally well-posed, but also to be locally exponentially stable for the KdV equation and globally exponentially stable for the KdVB equation.