In this talk we will mainly analyze the vibrations of a simplified 1-d model for a multi-body structure consisting of a finite number of flexible strings distributed along a planar graph. In particular we shall analyze how solutions propagate along the graph as time evolves. The problem of the observation of waves is a natural framework to analyze this issue. Roughly, the question can be formulated as follows: Can we obtain complete information on the vibrations by making measurements in one single extreme of the network? This formulation is relevant both in the context of control and inverse problems.
Using the Fourier development of solutions and techniques of Nonharmonic Fourier Analysis, we give spectral conditions that guarantee the observability property to hold in any time larger than twice the total lengths of the network in a suitable Hilbert that can be characterized in terms of Fourier series by means of properly chosen weights. When the network graph is a tree these weights can be identified.
Once this is done these results can be transferred to other models as the Schroedinger, heat or beam-type equations.
This lecture is based on results obtained in collaboration with Rene Dager.