2010/05/17 by Kaïs Ammari, K. Ammari, D. Mercier +9 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1005.2916
arxiv created 2010/05/17 · openalex publication_date 2010/05/17 · arxiv updated 2010/05/18 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
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 solutions of the first dissipative system tends to zero when the time tends to infinity under some irrationality assumptions of 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.