By René Dáger
This publication is dedicated to investigate the vibrations of simpli?ed 1? d types of multi-body constructions along with a ?nite variety of ?exible strings d- tributed alongside planar graphs. We?rstdiscussissueson life and uniquenessof options that may be solved through ordinary equipment (energy arguments, semigroup conception, separation ofvariables,transposition,...).Thenweanalyzehowsolutionspropagatealong the graph because the time evolves, addressing the matter of the commentary of waves. approximately, the query of observability may be formulated as follows: do we receive entire info at the vibrations by means of making measu- ments in a single unmarried severe of the community? This formula is appropriate either within the context of regulate and inverse difficulties. UsingtheFourierdevelopmentofsolutionsandtechniquesofNonharmonic Fourier research, we supply spectral stipulations that warrantly the observability estate to carry in any time higher than two times the whole size of the community in an appropriate Hilbert area that may be characterised when it comes to Fourier sequence through thoroughly selected weights. while the community graph is a tree, we symbolize those weights by way of the eigenvalues of the corresponding elliptic challenge. The ensuing weighted observability inequality permits identification- tifying the observable strength in Sobolev phrases in a few specific instances. that's the case, for example, whilst the community is star-shaped and the ratios of the lengths of its strings are algebraic irrational numbers.