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Optimal Internal Stabilization of the Linear System of Elasticity

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Abstract

We address the non-linear optimal design problem which consists in finding the best position and shape of a feedback damping mechanism for the stabilization of the linear system of elasticity. Non-existence of classical designs are related to the over-damping phenomenon. Therefore, by means of Young measures, a relaxation of the original problem is proposed. Due to the vector character of the elasticity system, the relaxation is carried out through div-curl Young measures which let the analysis be direct and the dimension independent. Finally, the relaxed problem is solved numerically, and a penalization technique to recover quasi-optimal classical designs from the relaxed ones is implemented in several numerical experiments.

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Authors and Affiliations

  1. Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, UMR CNRS 6623, 16, route de Gray, 25030, Besançon, France

    Arnaud Münch

  2. Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071, Ciudad Real, Spain

    Pablo Pedregal

  3. Departamento de Matemática Aplicada y Estadística, ETSI Industriales, Universidad Politécnica de Cartagena, 30203, Cartagena, Spain

    Francisco Periago

Authors
  1. Arnaud Münch
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  2. Pablo Pedregal
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  3. Francisco Periago
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Corresponding author

Correspondence to Pablo Pedregal.

Additional information

Communicated by S. S. Antman

A. Münch was partially supported by grants ANR-05-JC-0182-01 and ANR-07- JC-183284.

P. Pedregal was supported by project MTM2004-07114 from Ministerio de Educación y Ciencia (Spain), and PAI05-029 from JCCM (Castilla-La Mancha).

F. Periago was supported by projects MTM2004-07114 from Ministerio de Educación y Ciencia (Spain) and 00675/PI/04 from Fundación Séneca (Gobierno Regional de Murcia, Spain).

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Münch, A., Pedregal, P. & Periago, F. Optimal Internal Stabilization of the Linear System of Elasticity. Arch Rational Mech Anal 193, 171–193 (2009). https://doi.org/10.1007/s00205-008-0187-4

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  • DOI: https://doi.org/10.1007/s00205-008-0187-4

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