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

Author(s)

Panthee, M. ^[1] ; Leal, F. Vielma ^[1]

Affiliation

^[1] Campinas Univ, Dept Math Stat & Comp Sci, Sao Paulo - Brazil

• Document type: Journal article
• Source: ANNALES DE L' INSTITUT HENRI POINCARÉ-ANALYSE NON LINÉAIRE; v. 38, n. 5, p. 1605-1652, SEP-OCT 2021.
• Web of Science Citations: 0

Abstract

The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain T. We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in H-p(s)(T), with s >= 0. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in {[}32]. (C) 2020 L'Association Publications de l'Institut Henri Poincare. Published by Elsevier B.V. All rights reserved. (AU)

• FAPESP's process: 16/25864-6 - Nonlinear Evolution Equations of Dispersive Type
• Grantee: Mahendra Prasad Panthee
• Support Opportunities: Regular Research Grants
• FAPESP's process: 15/06131-5 - Study of solutions to some non-linear evolution equations of dispersive type
• Grantee: Francisco Javier Vielma Leal
• Support Opportunities: Scholarships in Brazil - Doctorate