Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings

Kais Ammari (1) , Denis Mercier (2) , Virginie Régnier (2) , Julie Valein (3, 4)

1 FSM - Faculté des Sciences de Monastir
2 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015
3 CORIDA - Robust control of infinite dimensional systems and applications
4 IECN - Institut Élie Cartan de Nancy

We consider N Euler-Bernoulli beams and N strings alternatively connected to one another and forming a particular network which is a chain beginning with a string. We study two stabilization problems on the same network and the spectrum of the corresponding conservative system: the characteristic equation as well as its asymptotic behavior are given. We prove that the energy of the solution of the rst dissipative system tends to zero when the time tends to in nity under some irrationality assumptions on the length of the strings and beams. On another hand we prove a polynomial decay result of the energy of the second system, independently of the length of the strings and beams, for all regular initial data. Our technique is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

Keywords: feedback stabilization; resolvent method; spectrum; Euler-Bernoulli beam equation; wave equation; Network

Domains: Analysis of PDEs [math.AP]

Kais Ammari, Denis Mercier, Virginie Régnier, Julie Valein. Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Communications on Pure and Applied Analysis, 2012, 11 (2), pp.785--807. ⟨10.3934/cpaa.2012.11.785⟩. ⟨hal-00644058⟩