Remark on stabilization of tree-shaped networks of strings
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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.
Keywordsnetworks of strings input-output map well-posed system
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- K. Ammari, M. Jellouli: Stabilization of star-shaped networks of strings. Differ. Integral Equations 17 (2004), 1395–1410.Google Scholar
- J. von Below: Classical solvability of linear parabolic equations in networks. J. Differ. Equations 52 (1988), 316–337.Google Scholar
- R. Dáger, E. Zuazua: Wave propagation, observation and control in 1-d flexible multi-structures. Mathématiques et Applications, Vol. 50. Springer-Verlag, Berlin, 2006.Google Scholar
- I. Lasiecka, J.-L. Lions, and R. Triggiani: Nonhomogeneous boundary value problems for second-order hyperbolic generators. J. Math. Pures Appl. 65 (1986), 92–149.Google Scholar