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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.

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