A further note on a theorem of Ingham and simultaneous observability in critical time

Applying a recent generalization of Parseval’s formula we improve some former simultaneous observability results for star-shaped networks of strings whose equations also contain lower order terms. These results hold only if the lengths of the strings are pairwise incommensurable. Under the somewhat opposite assumption that the lengths of the strings are pairwise commensurable, we solve a weaker inverse problem: a so-called spectral observability problem. This last result will also be established in the critical time. The proof is based on a generalization of a theorem of Ingham.