Unique Continuation Property and Control for the Benjamin-bona-mahony Equation on the Torus

We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional torus T = R/(2πZ). We prove a Unique Continuation Property (UCP) for small data in H(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 BBM equation, then a semiglobal exponential stabilization can be derived in H(T) for any s ≥ 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H(T) for any s ≥ 0 and globally exactly controllable in H(T) for any s ≥ 1.