Stars of vibrating strings: Switching boundary feedback stabilization

We consider a star-shaped network consisting 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 that if the feedback control is active at 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 is still possible if, from time to time, one of the feedback controllers breaks down. We give sufficient conditions that guarantee that such a switching feedback stabilization where not all controls are necessarily active at each time is successful.