Remark on stabilization of tree-shaped networks of strings

Kaïs Ammari; Mohamed Jellouli

Applications of Mathematics (2007)

  • Volume: 52, Issue: 4, page 327-343
  • ISSN: 0862-7940

Abstract

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We consider a tree-shaped network of vibrating elastic strings, with feedback acting on the root of the tree. Using the d’Alembert representation formula, we show that the input-output map is bounded, i.e. this system is a well-posed system in the sense of G. Weiss (Trans. Am. Math. Soc. 342 (1994), 827–854). As a consequence we prove that the strings networks are not exponentially stable in the energy space. Moreover, we give explicit polynomial decay estimates valid for regular initial data.

How to cite

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Ammari, Kaïs, and Jellouli, Mohamed. "Remark on stabilization of tree-shaped networks of strings." Applications of Mathematics 52.4 (2007): 327-343. <http://eudml.org/doc/33292>.

@article{Ammari2007,
abstract = {We consider a tree-shaped network of vibrating elastic strings, with feedback acting on the root of the tree. Using the d’Alembert representation formula, we show that the input-output map is bounded, i.e. this system is a well-posed system in the sense of G. Weiss (Trans. Am. Math. Soc. 342 (1994), 827–854). As a consequence we prove that the strings networks are not exponentially stable in the energy space. Moreover, we give explicit polynomial decay estimates valid for regular initial data.},
author = {Ammari, Kaïs, Jellouli, Mohamed},
journal = {Applications of Mathematics},
keywords = {networks of strings; input-output map; well-posed system; networks of strings; input-output map; well-posed system},
language = {eng},
number = {4},
pages = {327-343},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remark on stabilization of tree-shaped networks of strings},
url = {http://eudml.org/doc/33292},
volume = {52},
year = {2007},
}

TY - JOUR
AU - Ammari, Kaïs
AU - Jellouli, Mohamed
TI - Remark on stabilization of tree-shaped networks of strings
JO - Applications of Mathematics
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 4
SP - 327
EP - 343
AB - We consider a tree-shaped network of vibrating elastic strings, with feedback acting on the root of the tree. Using the d’Alembert representation formula, we show that the input-output map is bounded, i.e. this system is a well-posed system in the sense of G. Weiss (Trans. Am. Math. Soc. 342 (1994), 827–854). As a consequence we prove that the strings networks are not exponentially stable in the energy space. Moreover, we give explicit polynomial decay estimates valid for regular initial data.
LA - eng
KW - networks of strings; input-output map; well-posed system; networks of strings; input-output map; well-posed system
UR - http://eudml.org/doc/33292
ER -

References

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