Asymptotic behavior of some elastic planar networks of Bernoulli–Euler beams

Abstract

We study the energy decay of a network of elastic Bernoulli–Euler beams in star-shaped and tree-shaped network configurations. We show that the vibrations of star-shaped or tree-shaped network of beams are not exponentially stable in the energy space. So, we give explicit polynomial decay estimates valid for regular initial data. These estimates depend on the diophantine approximations properties.

Keywords:

• Decay rates
• Planar networks of Bernoulli–Euler beams

AMS 2000 Subject Classifications::

• 35B37
• 93B07
• 93D15

Acknowledgement

The author would like to thank V. Komornik for assistance in the proof of Lemma 4.1.