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Remark on stabilization of tree-shaped networks of strings. (English) Zbl 1164.93315
Summary: 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., the considered system is well-posed in the sense of G. Weiss [Trans. Am. Math. Soc. 342, 827–854 (1994; Zbl 0798.93036)]. As a consequence, we prove that networks of strings are not exponentially stable in the energy space. Moreover, we give explicit polynomial decay estimates valid for regular initial data.

93B07 Observability
93D15 Stabilization of systems by feedback
35B37 PDE in connection with control problems (MSC2000)
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