Evolution equations governed by the sweeping process

Abstract

This paper is concerned with variants of the sweeping process introduced by J.J. Moreau in 1971. In Section 4, perturbations of the sweeping process are studied. The equation has the formX′(t) ∈ -NC(t) (X(t)) +F(t, X(t)). The dimension is finite andF is a bounded closed convex valued multifunction. WhenC(t) is the complementary of a convex set,F is globally measurable andF(t, ·) is upper semicontinuous, existence is proved (Th. 4.1). The Lipschitz constants of the solutions receive particular attention. This point is also examined for the perturbed version of the classical convex sweeping process in Th. 4.1′. In Sections 5 and 6, a second-order sweeping process is considered:X″ (t) ∈ -NC(X(t)) (X′(t)). HereC is a bounded Lipschitzean closed convex valued multifunction defined on an open subset of a Hilbert space. Existence is proved whenC is dissipative (Th. 5.1) or when allC(x) are contained in a compact setK (Th. 5.2). In Section 6, the second-order sweeping process is solved in finite dimension whenC is continuous.