Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks

Two exact formulae for the eigenvalues of one-dimensional wave equations on general feedback controlled networks are presented. Especially, by them together with the exponential polynomial theory and the graph theory, it is shown that the oscillation of tree-shaped networks with one fixed vertex can rest in finite time with appropriate dampers. In the end, the two concrete networks are given and their eigenvalues are calculated to demonstrate our theoretical results.