On a Sweeping Process with the Cone of Limiting Normals

Abstract

We present two positive results on the existence of solutions of a sweeping process with the cone of limiting normals. The assumptions on the sets involve o-minimal structures.

Projection Process with Definable Right-Hand Side

“…Since C is Lipschitz (with respect to the Hausdorff distance), it can easily be obtained that K is compact. Lemmata 3.6 and 3.7 from [11] yield that there exists a setT ⊂ [0, T ] , which is at most countable and such that N C(t) (x) ≡ P r(N K (x, t)) for every x ∈ C(t) and t ∈T . This means that the existence of a solution x(t), t ∈ [0, T ] to (7) is equivalent to the existence of a solution to…”

“…Under the definability hypothesis an existence result for (4) has been obtained in the paper [11] provided the perturbation is single-valued and continuous. In the same paper the existence of solution to (1) is proved if the multimapping C(t) is definable and Lipschitz (with respect to the Hausdorff distance).…”

“…Let us point out that now the right-hand sides of (3) and (4) are upper semicontinuous mappings with possibly nonconvex values which makes their investigation a lot harder. Proposition 6.7 on p.219 from [17] (see also Lemma 3.8 from [11]) shows that the righthand side of (4) contains the right-hand side of (3). We do not know whether it is true that (4) has a solution for arbitrary closed set C even in the case when the perturbation F (x) is single-valued and constant.…”

“…In fact, the definability assumption on C (and continuity and single-valuedness of the perturbation) yields the existence of solution of the projection process (6) (and therefore of (3)) as well. The existence of solution to (3) has been announced in [11] in a remark, but the proof there is not complete. Thus, Theorem 4.4 is new (with respect to [11]), even in the stationary case.…”

“…The existence of solution to (3) has been announced in [11] in a remark, but the proof there is not complete. Thus, Theorem 4.4 is new (with respect to [11]), even in the stationary case.…”