Control and stabilization for the Benjamin-Bona-Mahony equation on the one-dimensional torus, 2025
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 propagat...
| Autor: | |
|---|---|
| Tipo de documento: | dissertação |
| Estado: | Versão publicada |
| Data de publicação: | 2025 |
| País: | Brasil |
| Recursos: | Universidade Federal de Pernambuco (UFPE) |
| Repositório: | Repositório Institucional da UFPE |
| Idioma: | inglês |
| OAI Identifier: | oai:repositorio.ufpe.br:123456789/62106 |
| Acesso em linha: | https://repositorio.ufpe.br/handle/123456789/62106 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Benjamin–Bona–Mahony equation unique continuation property exact con- trollability stabilization moving point control Korteweg–de Vries equation |
| Resumo: | 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. |
|---|