Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus

TL;DR

This paper establishes unique continuation properties and control results for the Benjamin-Bona-Mahony (BBM) equation on a torus, including stabilization and controllability, with extensions to related equations like KdV-BBM.

Contribution

It proves a unique continuation property for the BBM equation and extends it to similar equations, also demonstrating stabilization and controllability results with moving internal controls.

Findings

Unique continuation property for small data in H^1(T).

Semiglobal exponential stabilization with internal moving control.

Local and global exact controllability results for the BBM equation.

Abstract

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 BBM equation, then a semiglobal exponential stabilization can be derived in H^s(T) for any s \geq 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H^s(T) for any s \geq 0 and globally exactly controllable in H s (T) for any s \geq 1.