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Exponential stability of a network of serially connected Euler–Bernoulli beams

Abstract : The aim is to prove the exponential stability of a system modelling the vibrations of a network of N Euler–Bernoulli beams serially connected. Using a result given by K. Ammari and M. Tucsnak, the problem is reduced to the estimate of a transfer function and the obtention of an observability inequality. The solution is then expressed in terms of Fourier series so that one of the sufficient conditions for both the estimate of the transfer function and the observability inequality is that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds. It is proved using the exterior matrix method due to W. H. Paulsen. Two more asymptotic estimates involving the eigenfunctions are required. They are established using an adequate basis.
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Denis Mercier, Virginie Régnier. Exponential stability of a network of serially connected Euler–Bernoulli beams. International Journal of Control, Taylor & Francis, 2014, 87 (6), pp.1266-1281. ⟨10.1080/00207179.2013.874597⟩. ⟨hal-03163679⟩

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