Stabilization of the wave equation on d

Niveau: Secondaire, Lycée, Terminale
Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedba ks Serge Ni aise, Julie Valein Université de Valen iennes et du Hainaut Cambrésis LAMAV, FR CNRS 2956 Institut des S ien es et Te hniques of Valen iennes F-59313 - Valen iennes Cedex 9 Fran e Julie.Valein,Serge.Ni aiseuniv-valen iennes.fr O tober 24, 2006 Abstra t In this paper we onsider the wave equation on 1-d networks with a delay term in the boundary and/or transmission onditions. We rst show the well posedness of the problem and the de ay of an appropriate energy. We give a ne essary and su ient ondition that guarantees the de ay to zero of the energy. We further give su ient onditions that lead to exponential or polynomial stability of the solution. Some examples are also given. 1 Introdu tion/Notations Time delay ee ts arise in many pra ti al problems, see for instan e [20, 28, 1? for biologi al, ele tri al engineering, or me hani al appli ations. Furthermore it is well known that they an indu e some instabilities [17, 18, 19, 30, 25?, or on the ontrary improve the performan e of the system [28, 1?.

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