no. 6
Partial Differential Equations/Probability Theory
Stochastic variational inequalities for elasto-plastic oscillators
[Inéquation variationnelle stochastique modélisant un oscillateur elasto-plastique]

Présenté par : Philippe G. Ciarlet

Bensoussan, Alain1 ; Turi, Janos2
1 International Center for Decision and Risk Analysis, School of Management, P.O.Box 830688, SM 30, University of Texas at Dallas, Richardson, TX 75083-0688, USA
2 Programs in Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75083-0688, USA
Comptes Rendus. Mathématique, Tome 343 (2006) no. 6, pp. 399-406.

On montre que les modèles représentant l'effet d'hystérésis pour les oscillateurs non-linéaires elasto-plastiques sont équivalents à une inéquation variationnelle stochastique. Cette technique puissante permet d'étudier complétement les propriétés ergodiques du processus de Markov relatif au déplacement. On caractérise complétement la mesure invariante par une équation aux dérivées partielles elliptique partiellement dégénerée, avec des conditions de Dirichlet nouvelles.

The purpose of this Note is to show that models used in the literature for the hysteresis effect of non-linear elasto-plastic oscillators submitted to random vibrations are equivalent to stochastic variational inequalities. This powerful tool allows to study the ergodic properties of the Markov process related to the displacement. We characterize completely the invariant measure by a partially degenerate elliptic partial differential equation, with new Dirichlet coupling conditions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.08.008
Bensoussan, Alain 1 ; Turi, Janos 2

1 International Center for Decision and Risk Analysis, School of Management, P.O.Box 830688, SM 30, University of Texas at Dallas, Richardson, TX 75083-0688, USA
2 Programs in Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75083-0688, USA 
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Bensoussan, Alain; Turi, Janos. Stochastic variational inequalities for elasto-plastic oscillators. Comptes Rendus. Mathématique, Tome 343 (2006) no. 6, pp. 399-406. doi : 10.1016/j.crma.2006.08.008. http://www.numdam.org/articles/10.1016/j.crma.2006.08.008/

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⁎ This research was partially supported by a grant from CEA, Commissariat à l'énergie atomique.