This book is devoted to analyze the vibrations of simpli?ed 1? d models of multi-body
structures consisting of a ?nite number of ?exible strings d- tributed along planar graphs.
We?rstdiscussissueson existence and uniquenessof solutions that can be solved by standard
methods (energy arguments semigroup theory separation ofvariables transposition
...).Thenweanalyzehowsolutionspropagatealong the graph as the time evolves addressing the
problem of the observation of waves. Roughly the question of observability can be formulated
as follows: Can we obtain complete information on the vibrations by making measu- ments in one
single extreme of the network? This formulation is relevant both in the context of control and
inverse problems. UsingtheFourierdevelopmentofsolutionsandtechniquesofNonharmonic Fourier
Analysis we give spectral conditions that guarantee the observability property to hold in any
time larger than twice the total length of the network in a suitable Hilbert space that can be
characterized in terms of Fourier series by means of properly chosen weights. When the network
graph is a tree we characterize these weights in terms of the eigenvalues of the corresponding
elliptic problem. The resulting weighted observability inequality allows id- tifying the
observable energy in Sobolev terms in some particular cases. That is the case for instance
when the network is star-shaped and the ratios of the lengths of its strings are algebraic
irrational numbers.
