Stabilization of a new periodic integrable equation: The Degasperis–Procesi equation

In this paper, we shall study, following Degasperis and Procesi [A. Degasperis, M. Procesi, Asymptotic integrability, in: Symmetry an Perturbation Theory, World Scientific, Singapore, 1999, pp. 23–37], Constantin [A. Constantin, W.A. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603–610. [3]] and Yin [Zhaoyang Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math. 47(3) 2003 649–666. Fall], the stabilizability of a new periodic integrable equation, namely the Degasperis–Procesi (D–P) equation, by linear distributed feedbacks. First, by using the multiplier technique, we show that the solution to the D–P equation with distributed feedback control is asymptotically stable. Secondly, by using T. Kato’s theorem, we show that the closed-loop system under a distributed feedback control of the form −k(u−[u])(k>0)−k(u−[u])(k>0) is locally well-posed. Finally, by taking into account the energy estimates, we draw the 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.