Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:254

Article

Rosier, Lionel¹  Zhang, Bing-Yu^2,3 

[1] UHP CNRS INRIA, Inst Elie Carton, UMR 7502, F-54506 Vandoeuvre Les Nancy, France

[2] Univ Cincinnati, Dept Math Sci, Cincinnati, OH 45221 USA

[3] Sichuan Univ, Yangtze Ctr Math, Chengdu 610064, Peoples R China

关键词: Unique continuation property;    Benjamin-Bona-Mahony equation;    Korteweg-de Vries equation;    Moving point control;    Exact controllability;    Stabilization;   

DOI  :  10.1016/j.jde.2012.08.014

来源: Elsevier

【 摘 要 】

We consider the Benjamin-Bona-Mahony (BBM) equation on the one-dimensional torus T = R/(2 pi Z). We prove a Unique Continuation Property (UCP) for small data in H-1(T) with nonnegative zero means. Next we extend the UCP to certain BBM-like equations, including the equal width wave equation and the KdV-BBM equation. Applications to the stabilization of the above equations are given. In particular, we show that when an internal control acting on a moving interval is applied in the BBM equation, then a semiglobal exponential stabilization can be derived in H-s(T) for any s >= 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H-s(T) for any s >= 0 and globally exactly controllable in H-s(T) for any s >= 1 in a sufficiently large time depending on the H-s-norms of the initial and terminal states. (c) 2012 Elsevier Inc. All rights reserved.

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