Optimal Internal Stabilization of the Linear System of Elasticity

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. 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).