Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures PDF

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.