An Equivalence Theorem on Minimum Sheltering Speed for Non-convex Habitats

The paper is devoted to a new family of variational problems for differential inclusions, motivated by the protection of human and wildlife habitats when an invasive environmental disaster occurs. Indeed, the mathematical model consists of a differential inclusion describing the expansion of invaded region over time, and an artificial barrier that cannot be penetrated by the invasive agent is erected to shield the habitat, which serves as the control strategy and is characterized as a one-dimensional rectifiable set. We consider an isotropic case that the disaster spreads with uniform speed in all directions, and develop an equivalence result on the minimum construction speed required for implementing a sheltering strategy determined by a rectifiable Jordan curve. By measuring the invaded portion of barriers over time, it is shown that each connected non-convex habitat admits an equivalent habitat that contains the given habitat and their minimum speeds are equal. In particular, the boundary of an equivalent habitat can be partitioned into two arcs: an arc is locally convex, and the other is part of the boundary of illuminated area. This leads to a corollary on the existence of admissible sheltering strategies for non-convex habitats.