Optimal Set Motions from Nonconvex Initial Regions

Definitions and mathematical statement

Let $\Omega_0$ denote an initial region and let $t\mapsto\Omega(t)$, $t\in[0,T]$, denote a set motion with $\Omega(0)=\Omega_0$. The supplied excerpt refers to admissible and optimal motions but does not specify their full definitions, the objective functional, or the admissible controls. Thus the questions below are conditional on recovering the underlying optimal set-motion model from the cited report; they do not define a complete optimization problem by themselves.

A subset of a topological space is connected if it cannot be expressed as the union of two disjoint, nonempty subsets that are open in its relative topology. A subset $A\subseteq{\mathbb{R}}^n$ is convex if \[ (1-\lambda)x+\lambda y\in A \qquad\text{for every }x,y\in A\text{ and every }\lambda\in[0,1]. \] No regularity or boundedness assumptions on $\Omega_0$ or $\Omega(t)$ are supplied for the principal questions.

1. (Q2) For the optimal set-motion problem referred to in the report, characterize optimal strategies when $\Omega_0$ is nonconvex. Determine conditions under which optimal motions satisfy \[ \Omega(t)\text{ is connected for every }t\in[0,T]. \] The excerpt does not specify whether this property is required of every optimal motion or whether existence of an optimal motion with this property is sought. In particular, nonconvexity does not itself specify whether $\Omega_0$ is connected, and a condition including $t=0$ necessarily requires connectedness of $\Omega_0$.
2. (Q3) Formulate the referenced planar problems in ${\mathbb{R}}^n$ and on an $n$-dimensional Riemannian manifold, and determine which conclusions of the planar theory extend. In particular, does the convexity conclusion of the report's Theorem 1 remain valid for optimal set motion in ${\mathbb{R}}^3$, under the corresponding hypotheses? Theorem 1 and its hypotheses are absent from the supplied record. Consequently, this question cannot be replaced by an unconditional assertion that optimal motions preserve convexity. The planar results to be extended, the manifold hypotheses, and any relevant notion of convexity on a manifold are also unspecified.

The original excerpt additionally describes a geographically constrained eradication problem. Let $V\subseteq{\mathbb{R}}^2$ be a bounded open set representing an island. Within $V$, contamination is assumed to expand at unit speed in all directions, while control can clean an area $M$ per unit time. The stated task is to find an admissible motion $t\mapsto\Omega(t)\subseteq V$ and a finite time $T>0$ such that \[ \Omega(0)=V,\qquad \Omega(T)=\varnothing, \] and the control expenditure satisfies $E(t)\leq M$ for every $t\in[0,T]$. The extracted expenditure formula is typographically damaged. A possible reading is \[ E(t)=\int_{\partial\Omega(t)\cap V}\left(1+\beta(t,x)\right)\,{\,\mathrm{d}}\sigma\leq M, \qquad t\in[0,T], \] but this reconstruction is not confirmed by the supplied text. The quantity $\beta$, its sign convention and admissible range, the measure $\sigma$, and the required boundary regularity are not defined. The excerpt supplies no further boundary rule at $\partial V$. It also mentions two geographical problems but includes only the eradication problem. This additional setting is retained as source context; the record does not establish whether it is an additional target of the present entry or contextual material following (Q2) and (Q3).

Short English statement

The source asks how optimal motions behave for nonconvex initial regions, when they stay connected throughout the prescribed time interval, and which planar results extend to higher dimensions or Riemannian manifolds, especially the referenced convexity result in ${\mathbb{R}}^3$. It also describes eradication inside a bounded planar island with unit-speed contamination spread and a cleaning budget. The optimization model, the cited theorem, and parts of the geographical constraint are missing or unclear, so a fully specified mathematical formulation requires source review.

Sources

• Local UnsolvedMath registry, permanent ID $30005733$, title Optimal Set Motions from Nonconvex Initial Regions, source problem code catalogue entry. The supplied record attributes the questions to the Oberwolfach Reports collection, gives proposed year 2023, and identifies no individual proposer.
• Supplied authoritative report citation: Control Methods in Hyperbolic PDEs (2024), Oberwolfach Reports, pp. 2941--3011, DOI 10.4171/owr/2023/52. https://doi.org/10.4171/owr/2023/52. The literature-source list also labels this reference Oberwolfach Report 52/2023. The supplied excerpt contains questions (Q2) and (Q3), a reference to Theorem 1, and the geographical eradication passage.
• Supplied literature reference: Optimally Controlled Moving Sets with Geographical Constraints (2025). https://doi.org/10.1007/s00032-025-00419-x.
• Supplied literature assessment dated exactly 2026-08-22: the registry labels the entry partially solved and reports existence and optimality results for constrained motions, including three-dimensional finite-perimeter settings, while describing the extracted nonconvex connectivity and general manifold or ${\mathbb{R}}^n$ convexity questions as not fully settled. This is supplied dated curation, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. That verification claim is supplied metadata; neither the report nor the cited literature has been independently consulted for this draft.