Internal controllability of the korteweg–de vries equation on a bounded domain

¹ Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária, Ilha do Fundão, 21941-909 Rio de Janeiro (RJ), Brazil.
ademir@im.ufrj.br
² Institut Elie Cartan, UMR 7502 UHP/CNRS/INRIA, BP 70239, 54506 Vandœuvre-les-Nancy cedex, France.
capistrano@im.ufrj.br
³ Centre Automatique et Systèmes, MINES ParisTech, PSL Research University, 60 boulevard Saint-Michel, 75272 Paris cedex 06, France.
lionel.rosier@mines-paristech.fr

Received: 27 January 2014
Revised: 23 October 2014

Abstract

This paper is concerned with the control properties of the Korteweg–de Vries (KdV) equation posed on a bounded interval (0,L) with a distributed control. When the control region is an arbitrary open subdomain (l₁,l₂), we prove the null controllability of the KdV equation by means of a new Carleman inequality. As a consequence, we obtain a regional controllability result, which roughly tells us that any target function arbitrarily chosen on (0,l₁) and null on (l₂,L) is reachable. Finally, when the control region is a neighborhood of the right endpoint, an exact controllability result in a weighted L²-space is also established.