Sweeping Processes by Convex Sets with Nonempty Interior

In this Chapter, we shall deal with the sweeping process (Definition 1.1.1) by a moving convex set t→C(t) with nonempty interior. Sometimes the convex set C(t) may be decomposed in the form $$ C(t) = \upsilon (t) + \Gamma (t), $$ where v is a function taking values in a separable Hilbert space H and Γ is a multifunction with closed convex values having nonempty interior in H. If v is continuous and Γ is Lipschitz-continuous in the sense of Hausdorff distance, then the first proof of existence of a solution is due to Castaing ([Cas 1] Th. 6) . It generalizes a previous statement by Tanaka [Taan], where Γ is constant and H is finite-dimensional. Since C need not have a bounded retraction, the assumption on the interior of the convex set is essential. For instance, if we take Γ(t) ≡ {0}, then only the function v could be a solution to the sweeping process, but v may have unboundded variation.