Abstract
We prove that the sweeping process on a regular class of convex sets is equicontinuous. Classes of polyhedral sets with a given finite set of normal vectors are regular, as well as classes of uniformly strictly convex sets. Regularity is invariant to certain operations on classes of convex sets such as intersection, finite union, arithmetic sum and affine transformation.
Highlights
The sweeping process is an input-output operator, where the input is a variable closed convex set Z(t) ⊆ Rn, t ∈ I = [0, T ]), and the output is the position of a “lazy” point x(t) ∈ Rn that must remain within Z(t) but tries to minimize the distance passed
We study the problem of equicontinuity of the sweeping process, that is, of the uniform continuity of the output as the function of the initial point of the output and of the set-valued input in the L∞-metric on I
If the variable convex set takes values in a “regular” class of closed convex sets, the sweeping process is equicontinuous
Summary
ALEXANDER VLADIMIROV defined as the limit of the outputs of discrete-time processes that approximate Z(t) The existence of this limit is not guaranteed even if Z(t) is Hausdorff-continuous. Each finite partition F = {0 = t1 < · · · < tm = T } of I generates a discrete time approximation of the process Z(·) as follows. In what follows we will prove the equicontinuity property of “regular” classes Z of closed convex sets. Let us give an example of Hausdorff-continuous input Z(t) for which there is no solution of sweeping process. A class Z of convex closed sets in Rn is regular if, for any h ∈ Rn, there exists an ε = ε(h) > 0 such that Z ∈ Z, x ∈ Z and d(x + h, Z) ≤ ε imply x + h/2 ∈ Z.
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