No abstract

In this paper we study we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control A(x) belongs to L 2 -space (rather than L ∞ ). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissible controls, is well-posed and admits a nonempty set of solutions [9]. However, the optimal solutions to such problem may have a singular character. We show that some of optimal solutions can be attainable by solutions of special optimal control problems in perforated domains with fictitious boundary controls on the holes.

A discussion and an improvement of the Neumann–Kelvin's model are suggested in this paper. This model is used in the simulation of progressive wave phenomenon. As mentioned by several authors ([7, 19, 23, 29]), this model is ill posed unless a capillary energy is introduced. The mathematical explanation is that a compactness inversions' property occurs if the capillary forces are omitted. Theoretical arguments and numerical simulations are used in the following which aims at giving an explanation of what happens from a mechanical point of view.

International audience Lorsqu'un milieu présente une vitesse d'onde inférieure à celles de ceux qui l'entourent, un mécanisme de localisation d'énergie vibratoire peut apparaître. Cela provient du fait qu'une partie plus importante de l'énergie est réfléchie du côté du milieu le plus souple. Or la transition conduit à des contraintes d'interface très fortes qui sont susceptibles de créer un endommagement local. Notre ambition est de proposer une stratégie d'évaluation de l'énergie de ces surcontraintes sous forme d'un taux de restitution dynamique de l'énergie totale du système qui ne nécessite pas un calcul très précis au voisinage de l'interface. At the interface between two media with different wave velocities, local stationary waves can appear. The mechanical explanation is that the reflection of the waves is more important inwards the softest media. The overstressing which appears at the interface between the two media can be at the origin of a damage mechanism. Our goal is to suggest an energetical method in order to be able to compute accurately the energy release rate due to this overstressing without being obliged to use a very refined mesh in the neighbourhood of the interface.