Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures
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.
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according to Proposition allows approximately controllable assertion boundary conditions Chapter coefficients coincides consequence constructed continuous fraction control problem controllable initial controlled node Corollary corresponding d’Alembert formula deﬁned denote density diﬀerent Diophantine approximation Dirichlet Dirichlet boundary conditions edges eigenfunctions eigenvalues energy equal equivalent exactly controllable exists a constant exterior nodes fact ﬁnite finite sequence ﬁrst Fourier Fourier series function graph guarantees heat equation Hilbert space holds homogeneous system implies initial data irrational numbers Lebesgue measure Let us observe linear combinations method of moments networks of strings norm observability inequality obtain particular planar graph positive numbers problem of moments Proof prove ratios real numbers Remark Riesz basis satisfies Section sequence vn simultaneous control single string space of controllable spectrally controllable star-shaped network sub-trees subspaces of controllable suﬃcient Theorem three string network tree unique continuation property values verified vertices wave equation
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