K. Ammari, M. Jellouli, Department of Mathematics, Faculty of Sciences of Monastir, 5019 Monastir, Tunisia, e-mails: kais.ammari@fsm.rnu.tn, mohamed.jellouli@fsm.rnu.tn
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.
Keywords: networks of strings, input-output map, well-posed system
Classification (MSC 2000): 35B37, 93B07, 93D15
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