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Evolution equations governed by the sweeping process

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 form X′(t) ∈ -NC(t) (X(t)) +F(t, X(t)). The dimension is finite and F is a bounded closed convex valued multifunction. When C(t) is the complementary of a convex set, F is globally measurable and F(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)). Here C is a bounded Lipschitzean closed convex valued multifunction defined on an open subset of a Hilbert space. Existence is proved when C is dissipative (Th. 5.1) or when all C(x) are contained in a compact set K (Th. 5.2). In Section 6, the second-order sweeping process is solved in finite dimension when C is continuous. © 1993 Kluwer Academic Publishers.

 Castaing C., Duc Ha T.X., Valadier M.
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