Optimality Conditions for a Blocking Strategy Involving Delaying Arcs

In this paper, we study a class of optimization problems, originally motivated by models of confinement of wild fires. The burned region is described by the reachable set for a differential inclusion. To block its spreading, we assume that barriers can be constructed in real time. In mathematical terms, a barrier is a one‐dimensional rectifiable set, which cannot be crossed by trajectories of the differential inclusion. Relying on a classification of blocking arcs, we derive global necessary conditions for an optimal strategy involving ‘delaying arcs,’ which slow down the advancement of the fire. These new optimality conditions take the form of ODE’s describing the delaying arcs, together with tangency conditions on the initial points of such arcs.