Internal controllability of the korteweg–de vries equation on a bounded domain
Instituto de Matemática, Universidade Federal do Rio de Janeiro,
C.P. 68530, Cidade Universitária, Ilha do Fundão, 21941-909
Rio de Janeiro (RJ),
2 Institut Elie Cartan, UMR 7502 UHP/CNRS/INRIA, BP 70239, 54506 Vandœuvre-les-Nancy cedex, France.
3 Centre Automatique et Systèmes, MINES ParisTech, PSL Research University, 60 boulevard Saint-Michel, 75272 Paris cedex 06, France.
Received: 27 January 2014
Revised: 23 October 2014
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.
Mathematics Subject Classification: 35Q53 / 37K10 / 93B05 / 93D15
Key words: KdV equation / Carleman estimate / null controllability / exact controllability
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