December  2009, 4(4): 709-730. doi: 10.3934/nhm.2009.4.709

Spectrum analysis of a serially connected Euler-Bernoulli beams problem

1. 

LAMAV, FR CNRS 2956, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313 VALENCIENNES Cedex 9, France

Received  April 2009 Revised  August 2009 Published  October 2009

In this article we analyse the eigenfrequencies of a hyperbolic system which corresponds to a chain of Euler-Bernoulli beams. More precisely we show 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 as soon as the roots of a function denoted by $f_{\infty}$ (and giving the asymptotic behaviour of the eigenvalues) are all simple. For a chain of $N$ different beams, this assumption on the multiplicity of the roots of $f_{\infty}$ is proved to be satisfied. A direct consequence of this result is that we obtain the exact controllability of an associated boundary controllability problem. It is well-known that the spectral gap is a important key point in order to get the exact controllabilty of these one-dimensional problem and we think that the new method developed in this paper could be applied in other related problems.
Citation: Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks & Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709
[1]

Bruno Colbois, Alexandre Girouard. The spectral gap of graphs and Steklov eigenvalues on surfaces. Electronic Research Announcements, 2014, 21: 19-27. doi: 10.3934/era.2014.21.19

[2]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917

[3]

Yassine El Gantouh, Said Hadd, Abdelaziz Rhandi. Approximate controllability of network systems. Evolution Equations & Control Theory, 2021, 10 (4) : 749-766. doi: 10.3934/eect.2020091

[4]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205

[5]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239

[6]

Soña Pavlíková, Daniel Ševčovič. On construction of upper and lower bounds for the HOMO-LUMO spectral gap. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 53-69. doi: 10.3934/naco.2019005

[7]

Shuang Chen, Jun Shen. Large spectral gap induced by small delay and its application to reduction. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4533-4564. doi: 10.3934/dcds.2020190

[8]

Kaïs Ammari, Denis Mercier, Virginie Régnier, Julie Valein. Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Communications on Pure & Applied Analysis, 2012, 11 (2) : 785-807. doi: 10.3934/cpaa.2012.11.785

[9]

Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521

[10]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723

[11]

Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042

[12]

Nicolas Augier, Ugo Boscain, Mario Sigalotti. Semi-conical eigenvalue intersections and the ensemble controllability problem for quantum systems. Mathematical Control & Related Fields, 2020, 10 (4) : 877-911. doi: 10.3934/mcrf.2020023

[13]

Andrea Bondesan, Laurent Boudin, Marc Briant, Bérénice Grec. Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2549-2573. doi: 10.3934/cpaa.2020112

[14]

Stefano Galatolo, Rafael Lucena. Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1309-1360. doi: 10.3934/dcds.2020079

[15]

Boguslaw Twarog, Robert Pekala, Jacek Bartman, Zbigniew Gomolka. The changes of air gap in inductive engines as vibration indicator aided by mathematical model and artificial neural network. Conference Publications, 2007, 2007 (Special) : 1005-1012. doi: 10.3934/proc.2007.2007.1005

[16]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016

[17]

Luciano Pandolfi. Riesz systems, spectral controllability and a source identification problem for heat equations with memory. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 745-759. doi: 10.3934/dcdss.2011.4.745

[18]

Yassine El Gantouh, Said Hadd. Well-posedness and approximate controllability of neutral network systems. Networks & Heterogeneous Media, 2021, 16 (4) : 569-589. doi: 10.3934/nhm.2021018

[19]

Ya Li, ShouQiang Du, YuanYuan Chen. Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020147

[20]

Eugenia Pérez. On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 859-883. doi: 10.3934/dcdsb.2007.7.859

2020 Impact Factor: 1.213

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]