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

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 H1(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 Hs(T) for any s⩾1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in Hs(T) for any s⩾0 and globally exactly controllable in Hs(T) for any s⩾1 in a sufficiently large time depending on the Hs-norms of the initial and terminal states.