Dynamic Sheltering Problem and An Existence Theorem for Admissible Strategies

Abstract

This paper is devoted to a new class of dynamic problems, originally motivated by the protection issues when a spreading disaster occurs. In the propagation model, the invaded region is described by the reachable set of differential inclusions. To protect an assigned habitat against the disaster, an artificial barrier is constructed to shield the habitat in real time, where the barrier is characterized as one-dimensional rectifiable set in mathematical terms and a dynamic strategy is determined by the barrier constructed over time. In the paper, we start by showing the motivation of dynamic sheltering problems and then describe the mathematical models. Afterwards, we restrict the analysis to an isotropic case, and derive the minimum speed that ensures the existence of an admissible sheltering strategy. As the main conclusion, it is proved that, given a convex habitat, among all barriers each consisting of a Jordan curve, there exists an admissible sheltering strategy, if and only if the construction speed is greater than that determined by the boundary curve of the habitat. After stating the main theorem, an illustrative example is carefully examined, which provides insights into the existence theorem and clarifies several common misconceptions.