Control and stabilization for the Benjamin-Bona-Mahony equation on the one-dimensional torus, 2025

Abstract

In the work Unique continuation property and control for the Benjamin–Bona–Mahony equation on a periodic domain, Journal of Differential Equations, (254), no. 1, 2013 by Lionel Rosier and Bing-Yu Zhang, the authors studied the Benjamin-Bona-Mahony (BBM) equation, a fundamental model for the propagation of long waves with small amplitude in nonlinear dispersive systems, on the one-dimensional torus T = R/(2πZ). First, the authors showed that the initial-value problem associated with the BBM equation is globally well-posed in Hs

(T), for s ⩾ 0. Moreover, the mapping associating the solution to a given initial data is smooth and the solution is analytic in time. Subsequently, they establish a unique continuation property (UCP) for small data in H1

(T) with nonnegative zero means. This result is further extended to certain BBM-like equations, including the equal width wave equation and the KdV-BBM equation, where, for the latter, some Carleman estimates are derived. Applications to stabilization are developed, showing that semiglobal exponential stabilization can be achieved in Hs

(T) for any s ⩾ 1 when an internal control acting on a moving interval is applied. Furthermore, they prove that the BBM equation with a moving control is locally exactly controllable in Hs

(T) for s ⩾ 0 and globally exactly

controllable in Hs

(T) for s ⩾ 1 over sufficiently large times, depending on the Hs -norms of the initial and terminal states. The results of this article are explored in detail in this master’s thesis.