Equivalent formulation and numerical analysis of a fire confinement problem

Abstract

We consider a class of variational problems for differential inclusions, related to the control of wild fires. The area burned by the fire at time t> 0 is modelled as the reachable set for a differential inclusion ˙ x ∈ F (x), starting from an initial set R0. To block the fire, a barrier can be constructed progressively in time. For each t> 0, the portion of the wall constructed within time t is described by a rectifiable set γ(t) ⊂ R 2 . In this paper we show that the search for blocking strategies and for optimal strategies can be reduced to a problem involving one single admissible rectifiable set Γ ⊂ R 2 , rather than the multifunction t � γ(t) ⊂ R 2 . Relying on this result, we then develop a numerical algorithm for the computation of optimal strategies, minimizing the total area burned by the fire. Mathematics Subject Classification. 49Q20, 34A60, 49J24, 93B03.