Existence of optimal strategies for a fire confinement problem

Abstract

We consider a class of variational problems for differential inclusions related to the control of forest fires. The area burned by the fire at time t > 0 is modeled as the reachable set for a differential inclusion P x 2 F .x/ starting from an initial set R 0 . To block the fire, a barrier can be constructed progressively in time at a given speed. In this paper we prove the existence of an optimal strategy, which minimizes the value of the area destroyed by the fire plus the cost of constructing the barrier.

Dynamic Blocking Problems for a Model of Fire Propagation

“…Here B(x, r) denotes the open ball centered at x with radius r. As proved in [7], for every admissible strategy t → γ(t) one can construct a second admissible strategy t →γ(t) ⊇ γ(t), which is complete.…”

“…The following result on the existence of optimal blocking strategies was proved in [7]. The proof given in [7] relies on the direct method of the Calculus of Variations.…”

“…The proof given in [7] relies on the direct method of the Calculus of Variations. Consider a minimizing sequence γ n (·) of admissible strategies, such that…”

“…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.…”

“…The remainder of the paper is concerned with optimal strategies. The basic existence theorem [7] is presented in Section 4. 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.…”