Equivalent formulation and numerical analysis of a fire confinement problem

Abstract

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 inclusionẋ ∈ 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.

Dynamic Blocking Problems for a Model of Fire Propagation

“…As proved in [9], under the assumptions (A1)-(A3) the two formulations are equivalent. In particular, if t → γ(t) is a complete, optimal strategy for (OP1), then the rectifiable set…”

“…Indeed, for each t ≥ 0 one needs to describe the portion of the wall constructed within time t. The subsequent paper [9] showed that the both the blocking problem and the optimization problem can be reformulated in a simpler way, where an admissible strategy is determined by one single rectifiable set Γ ⊂ IR 2 . This approach is particularly useful for the numerical computation of optimal strategies.…”

“…A first algorithm for the computation of optimal barriers was developed in [9]. This construction relies on two basic assumptions about the optimal barrier, namely:…”

“…Aim of this paper is to survey the main concepts and results in the theory of dynamic blocking problems, developed in [5,6,7,9,10,11,12,15,23], and discuss various open questions.…”

“…Section 5 reviews the classification of arcs in an optimal barrier and the necessary conditions for optimality proved in [5,11,23], while Section 6 describes a recent result concerning sufficient conditions for optimality. A numerical algorithm for computing the optimal barrier, introduced in [9], is here presented in Section 7. Finally, in Section 8 we discuss several remaining open problems.…”