Equivalent formulation and numerical analysis of a fire confinement problem

Department of Mathematics, Penn State University University Park, Pa. 16802, USA. bressan@math.psu.edu; wang_t@math.psu.edu

Received: 12 December 2008
Revised: 26 April 2009

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 ∈ 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) ⊂ . 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 ⊂ , rather than the multifunction t γ(t) ⊂ . Relying on this result, we then develop a numerical algorithm for the computation of optimal strategies, minimizing the total area burned by the fire.