Quadratic Programming in Mechanics: Dynamics of One-Sided Constraints

International audienceLet $S$ be a frictionless mechanical system with $n$ degrees of freedom; we denote by $q_1,q_2,\ldots,q_n$ the generalized coordinates, representing the point $q$ of a configuration space. A finite family of one-sided constraints is imposed on the system; the kinematic effect of these constraints is expressed by the conditions (assumed compatible) $f_{\alpha}(q, t) \geq 0$, $\alpha \in I$, finite set of indexes. For instance, some solid parts of the system may be in contact or become detached but they can never overlap. These constraints are frictionless, i.e., as long as the equalities hold in the expression above, the motion of the system is governed by Lagrange's equations with multipliers $\lambda_{\alpha}$, $\alpha\in I$

Mathematical comparison of three alternative laws for linear viscosity

Article dans "Séminaire d'analyse convexe", Montpellier, exposé n°

Inf-convolution, sous-additivité, convexité des fonctions numériques

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Bounded Variation in Time

International audienceDescribing a motion consists in defining the state or position $q$ of the investigated system as a function of the real variable $t$, the time. Commonly, $q$ takes its values in some set $Q$, suitably structured for the velocity $u$ to be introduced as the derivative of $t\to q$, when it exists. This, in fact, makes sense if $Q$ is a topological linear space or, more generally, a differential manifold modelled on such a space.For smooth situations, classical dynamics rests, in turn, on the consideration of the acceleration. This is the derivative of $t \to u$, if it exists in the sense of the topological linear structure of $Q$, or, when $Q$ is a manifold, in the sense of some connection. But, from its early stages, classical dynamics has also had to face shocks, i.e. velocity jumps. For isolated shocks, one traditionally resorts to the equations of the dynamics of percussions. Even in the absence of impact, it has been known for a long time that systems submitted to such nonsmooth effects as dry friction may exhibit time discontinuity of the velocity. Furthermore, nonsmooth mechanical constraints may also prevent $t\to u$ from admitting a derivative. In all these cases, the laws governing the motion can no longer be formulated in terms of acceleration

Rétraction d'une multiapplication

Article dans "Séminaire d'analyse convexe", Montpellier, exposé n°1

Champs et distributions de tenseurs déformation sur un ouvert de connexité quelconque

Article dans "Séminaire d'analyse convexe", Montpellier, exposé n°

Rafle par un convexe variable (Deuxième partie)

Article dans "Séminaire d'analyse convexe", Montpellier, exposé n°

Intersection de deux convexes mobiles

Article dans "Séminaire d'analyse convexe", Montpellier, exposé n°

Sur les mesures différentielles de fonctions vectorielles et certains problèmes d'évolution

International audienceSoient $X$, $Y$, $Z$ trois espaces de Banach, $\Phi: X \times Y \to Z$ bilinéaire continue, $I$, intervalle réel, $x : I\to X$ et $y : I\to Y$ à variation localement bornée; on calcule la mesure différentielle de $t \mapsto \Phi( (x (t), y (t))$. Inégalités concernant le cas où $\Phi : X \times X \to \mathbb R$ est symétrique et engendre une forme quadratique $\geqslant 0$. Application aux solutions continues à droite d'un processus de rafle

Solutions du processus de rafle au sens des mesures différentielles

Article dans "Séminaire d'analyse convexe", Montpellier, exposé n°