Local exponential stabilization for a class of Korteweg–de Vries equations by means of time-varying feedback laws

Abstract

We study the exponential stabilization problem for a nonlinear Korteweg-de Vries equation on a bounded interval in cases where the linearized control system is not controllable. The system has Dirichlet boundary conditions at the end-points of the interval and a Neumann nonhomogeneous boundary condition at the right end-point, which is the control. We build a class of time-varying feedback laws for which the solutions of the closed-loop systems with small initial data decay exponentially to $0$. We present also results on the well-posedness of the closed-loop systems for general time-varying feedback laws.