有限区间上的Korteweg-de Vries方程的快速镇定

The rapid stabilization of the Korteweg-de Vries equation on a finite interval

Coron和L(2014)研究了区间(0,L)上的Korteweg-de Vries方程所描述的一类控制系统的快速镇定问题.对λ∈(0,∞),他们设计出了适当的状态反馈并证明了相应的闭环系统△_(λ,L)在L^2(0,L)中局部适定,闭环系统△_(λ,L)的状态轨线在L^2(0,L)中呈λ/2型指数衰减.本文进一步证明对_s∈[0,∞)\{j+1/2;j∈N_0}:(i)闭环系统△_(λ,L)在H^s(0,L)中整体适定;(ii)闭环系统△_(λ,L)的状态轨线在H^s(0,L)中仍然呈局部的λ/2型指数衰减.

Coron and Lu（2014） have recently studied the rapid stabilization problem of a control system which describes the Korteweg-de Vries equation posed on a finite interval（0,L） with the control acting as the right-end Neumann boundary value.For any given,λ ∈（0,∞）,a feedback control law is carefully designed by Coron and Lu so that the resulted close-loop system（denoted by △λ,L） possesses the following properties：（1） △λ,L is locally well-posed in the space L2（0,L） and（2） it is locally exponentially stable in the space L2（0,L） with decay rateλ/2.In this paper we continue the work of Coron and Lii to study △λ,L and show that it is（i） globally well-posed in the space Hs（0,L） and（ii） locally exponentially stable in the space Hs（0,L） with the decay rate λ/2 for any_s∈[0,∞）／{j＋1/2;j∈N0}.