Equivalent formulation and numerical analysis of a fire confinement problem

Abstract

We consider a class of variationalproblems for differential inclusions, related to thecontrol of wild fires. The area burned by the fire at time t> 0is modelled as the reachable set fora differential inclusion $\dot x$ ∈F(x), starting froman initial set R ₀. To block the fire, a barrier can be constructedprogressively in time. For each t> 0, the portion of the wall constructedwithin time t is described by a rectifiable setγ(t) ⊂ $\mathbb{R}^2$ . In this paperwe show that the searchfor blocking strategies and for optimal strategies can be reduced toa problem involving one single admissible rectifiable set Γ⊂ $\mathbb{R}^2$ ,rather than the multifunction t $\mapsto$ γ(t) ⊂ $\mathbb{R}^2$ .Relying on this result, we then developa numerical algorithm for the computation ofoptimal strategies, minimizing the total area burned by the fire.