Manuel D. P. Monteiro Marques

An Introduction to Moreau’s Sweeping Process

Starting from an elementary level, this rewiew paper summarizes some results and applications concerning first and second order sweeping processes.

Sweeping by a continuous prox-regular set☆

Abstract The evolution problem known as sweeping process is considered for a class of nonconvex sets called prox-regular (or ϕ -convex). Assuming, essentially, that such sets contain in the interior a suitable subset and move continuously (w.r.t. the Hausdorff distance), we prove local and global existence as well as uniqueness of solutions, which are continuous functions with bounded variation. Some examples are presented.

BV periodic solutions of an evolution problem associated with continuous moving convex sets

This paper is concerned with BV periodic solutions for multivalued perturbations of an evolution equation governed by the sweeping process (or Moreaus process). The perturbed equation has the form −Du∈NC(t)(u(t))+F(t,u(t)), whereC is a closed convex valued continuousT-periodic multifunction from [0,T] to ℝd,NC(t)(u(t)) is the normal cone ofC(t) atu(t),F: [0,T]×ℝd→ℝd is a compact convex valued multifunction and Du is the differential measure of the periodic BV solutionu. Several existence results for this differential inclusion are stated under various assumptions on the perturbationF.

Nonconvex second-order differential inclusions with memory

We prove several existence theorems for the second-order differential inclusion of the form(dot x(t) in G(x(t)), ddot x(t) in - N_{G(x(t))} dot x(t) + F(t,T(t)x)) in the case whenF or bothG andF are maps with nonconvex values in an Euclidean or Hilbert space andF(t, T(t)x) is a memory term ([T(t)x](θ)=x(t+θ)).AbstractWe prove several existence theorems for the second-order differential inclusion of the formn

An Existence Result in Non-Smooth Dynamics

dot x(t) in G(x(t)), ddot x(t) in - N_{G(x(t))} dot x(t) + F(t,T(t)x)

A sweeping process approach to inelastic contact problems with general inertia operators

n in the case whenF or bothG andF are maps with nonconvex values in an Euclidean or Hilbert space andF(t, T(t)x) is a memory term ([T(t)x](θ)=x(t+θ)).

A convergence result for a vibro-impact problem with a general inertia operator

New results on weak invariance in Hilbert spaces, both in autonomous and nonautonomous case, are given. Applications to weakly decreasing systems and to strong invariance are presented.

Viability for Nonautonomous Semilinear Differential Equations

Starting in 1983, Jean Jacques Moreau gave remarkable new formulations of the dynamics of mechanical systems submitted to inelastic or frictional contact as measure-differential inclusions. This paved the way to proofs of existence of solutions, by the first author, in 1985 for the case of standard inelastic shocks and in 1988 for frictional dynamics, both for one contact problems. These mathematical theoretical results left out many other situations, that could be handled numerically by J.J. Moreau and collaborators in a very effective way, as well as the Painleve example. Here we present a generalization of such existence results to variable mass matrices and non isotropic friction.