Abstract
We consider a star-shaped networkconsisting of a single node with $N\geq 3$ connected arcs. The dynamics on each arc is governed by the wave equation.The arcs are coupled at the node and each arc is controlled at the other end.Without assumptions on the lengths of the arcs, we show thatif the feedback control is activeat all exterior ends,the system velocity vanishes in finite time. In order to achieve exponential decay to zero of the system velocity,it is not necessary that the system is controlled at all $N$ exterior ends, but stabilization isstill possible if, from time to time, one of the feedback controllers breaksdown.We give sufficient conditions that guarantee that such a switching feedback stabilizationwhere not all controls are necessarily active ateach timeis successful.
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