Stabilization of the degasperis-procesi equation with periodic boundary condition

In this paper, we shall study the stabilizability of the Degasperis-Procesi (D-P) equation with periodic boundary condition, namely, by linear distributed feedbacks. First, by using the multiplier technique, we show that the solution to the D-P equation with the distributed feedback control is asymptotically stable. Secondly, by using T. Katopsilas theorem, we show the closed-loop system under a distributed feedback control is locally well-posed. Finally, by taking account into the energy estimates, we draw a conclusion that the closed-loop system is in fact globally well-posed. The proof is based on a remarkable property of this equation: the existence of an infinite sequence of conservation laws corresponding to an infinite sequence of useful multipliers.