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

• Ano de defesa: 2025
• Autor(a) principal: OLIVEIRA, Matheus Luiz da Silva
• Orientador(a): GONZALEZ MARTINEZ, Victor Hugo
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de Pernambuco
• Programa de Pós-Graduação: Programa de Pos Graduacao em Matematica
• País: Brasil
• Palavras-chave em Português: Benjamin–Bona–Mahony equation; unique continuation property; exact con- trollability; stabilization; moving point control; Korteweg–de Vries equation
• Link de acesso: https://repositorio.ufpe.br/handle/123456789/62106

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.