International audienceIn this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset $C(t)$, supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set $C(t)$. This class of problems subsumes as a particular case, the evolution variational inequalities. Assuming that the moving subset $C(t)$ has a continuous variation for every $t \in [0, T ]$ with $C (0)$ bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model, to the planning procedure in mathematical economy, and to nonregular electrical circuits containing nonsmooth electronic devices like diodes. The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau’s seminal work

Given a smooth function f : R n → R and a convex function Φ : R n → R, we consider the following differential inclusion:t≥ 0, where ∂Φ denotes the subdifferential of Φ. The term ∂Φ(ẋ) is strongly related with the notion of friction in unilateral mechanics. The trajectories of (S) are shown to converge toward a stationary solution of (S). Under the additional assumption that 0 ∈ int ∂Φ(0) (case of a dry friction), we prove that the limit is achieved in a finite time. This result may have interesting consequences in optimization.

A LaSalle's Invariance Theory for a class of first-order evolution variational inequalities is developed. Using this approach, stability and asymptotic properties of important classes of secondorder dynamic systems are studied. The theoretical results of the paper are supported by examples in nonsmooth Mechanics and some numerical simulations. Résumé Dans cet article, la théorie d'invariance de LaSalle est généralisée pour une classe d'inéquations variationnelles d'évolution du premier ordre. Des résultats de stabilité (au sens de Lyapunov) et d'attractivité sont ensuite obtenus pour des systèmes dynamiques du second ordre non réguliers. Une extension du théorème de Lagrange aux systèmes conservatifs non réguliers est également proposée. Enfin, quelques exemples et simulations numériques illustrent les principaux résultats théoriques.

International audienceResults on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super)linear convergence of a Newton- type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton’s method, the nonsmooth Newton’s method for semismooth functions, the inexact proximal point algorithm, etc. Moreover, we also cover a forward-backward splitting algorithm for finding a zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton’s method is discussed

In the present paper, we study a perturbed iterative method for solving a general class of variational inclusions. An existence result which generalizes some known results in this field, a convergence result, and a new iterative method are given. We also prove the continuity of the perturbed solution to a parametric variational inclusion problem. Several special cases are discussed. ᮊ