On the controllability and stabilization of the linearized Benjamin-Ono equation

¹ IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brasil; linares@impa.br
² Universidad de Chile, Facultad de Ciencias Físicas y Matemáticas. Departamento de Ingeniería Matemática, Casilla 170/3, Correo 3, Santiago, Chile.
³ Departamento de Ciencias Básicas, Universidad del Bío-Bío, Avda. Andrés Bello s/n, Casilla 447, Chillán, Chile; jortega@dim.uchile.cl

Received: 18 April 2003
Revised: 11 March 2004
Revised: 2 April 2004

Abstract

In this work we are interested in the study of controllability and stabilization of the linearized Benjamin-Ono equation with periodic boundary conditions, which is a generic model for the study of weakly nonlinear waves with nonlocal dispersion. It is well known that the Benjamin-Ono equation has infinite number of conserved quantities, thus we consider only controls acting in the equation such that the volume of the solution is conserved. We study also the stabilization with a feedback law which gives us an exponential decay of the solutions.