On the controllability and stabilization of the Benjamin equation and its generalization on a periodic domain

• Grant number: 18/18883-0
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: January 31, 2019
• End date: January 30, 2020
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Mahendra Prasad Panthee
• Grantee: Francisco Javier Vielma Leal
• Supervisor: Lionel Rosier
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: ParisTech, France
• Associated to the scholarship: 15/06131-5 - Study of solutions to some non-linear evolution equations of dispersive type, BP.DR

Abstract

The Benjamin equation is an integro-differential equation that serves as a generic model forunidirectional propagations of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. During the doctoral programme, we have been studying the Controllability and Stabilization of the Benjamin equation posed on a periodic domain.We have already obtained some important results in this direction for Benjamin equation, by proving a small data control and linear exponential stabilization. Furthermore, we also proved that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable.The global exponential stabilizability corresponding to a natural feedback law was first established with the aid of certain properties of propagation of compactness and propagation of regularity in Bourgain spaces for solutions of the associated linear system. Our aim, during the BEPE Fellowship, is to improve the result on stabilization by constructing a smooth time varyingfeedback law ensuring a semiglobal stabilization with an arbitrary large decay rate and to extend our results to the Benjamin equation in dimension 2.Also, in this project we will study two new important problems: The controllability and the stabilization for the generalized Benjamin equation, and the exact boundary controllability for the Benjamin equation on a bounded domain. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

PANTHEE, MAHENDRA; VIELMA LEAL, FRANCISCO J.. On the controllability and stabilization of the linearized Benjamin equation on a periodic domain. NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, v. 51, FEB 2020. (15/06131-5, 18/18883-0)