Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

Abstract

In [38], Russell and Zhang showed that the Korteweg-de Vries equation posed on a periodic domain 𝕋 with an internal control is locally exactly controllable and locally exponentially stabilizable when the control acts on an arbitrary nonempty subdomain of 𝕋. In this paper, we show that the system is in fact globally exactly controllable and globally exponentially stabilizable. The global exponential stabilizability is established with the aid of certain properties of propagation of compactness and regularity in Bourgain spaces for the solutions of the associated linear system. With Slemrod's feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrarily large decay rate.