Equivalent formulation and numerical analysis of a fire confinement problem

Bressan, Alberto  ; Wang, Tao

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 974-1001

Résumé

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 $\dot{x}$ ∈ 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 $\mapsto$ 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.

Détail

DOI : 10.1051/cocv/2009033

Classification : 49Q20, 34A60, 49J24, 93B03
Keywords: dynamic blocking problem, differential inclusion, constrained minimum time problem