On the Regularity of Optimal Dynamic Blocking Strategies

TL;DR

This paper investigates the structure of optimal dynamic barriers for fire containment, proving that such barriers are nowhere dense and providing new estimates on fire spread and barrier trajectories.

Contribution

It establishes that optimal barriers are nowhere dense and introduces new estimates on reachable sets and fire trajectories in a dynamic blocking problem.

Findings

Optimal barriers are nowhere dense.

New estimates on reachable sets and fire trajectories.

Existence of optimal barriers proven.

Abstract

The paper studies a dynamic blocking problem, motivated by a model of optimal fire confinement. While the fire can expand with unit speed in all directions, barriers are constructed in real time. An optimal strategy is sought, minimizing the total value of the burned region, plus a construction cost. It is well known that optimal barriers exists. In general, they are a countable union of compact, connected, rectifiable sets. The main result of the present paper shows that optimal barriers are nowhere dense. The proof relies on new estimates on the reachable sets and on optimal trajectories for the fire, solving a minimum time problem in the presence of obstacles.