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

作者：Rosier Lionel^*; Zhang Bing Yu

来源：Journal of Differential Equations, 2013, 254(1): 141-178.

DOI：10.1016/j.jde.2012.08.014

• Unique continuation property
• Benjamin-Bona-Mahony equation
• Korteweg-de Vries equation
• Moving point control
• Exact controllability
• Stabilization

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 %26gt;= 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H-s(T) for any s %26gt;= 0 and globally exactly controllable in H-s(T) for any s %26gt;= 1 in a sufficiently large time depending on the H-s-norms of the initial and terminal states.

• 出版日期2013-1-1
• 单位四川大学; INRIA