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On the Optimal Strategy for an Isotropic Blocking Problem
, 2011
"... The paper is concerned with a dynamic blocking problem, originally motivated by the control of wild fires. It is assumed that the region R(t) ⊂ R 2 burned by the fire is initially a disc, and expands with unit speed in all directions. To block the fire, a barrier Γ can be constructed in real time, ..."
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The paper is concerned with a dynamic blocking problem, originally motivated by the control of wild fires. It is assumed that the region R(t) ⊂ R 2 burned by the fire is initially a disc, and expands with unit speed in all directions. To block the fire, a barrier Γ can be constructed in real time, so that the portion of the barrier constructed within time t has length ≤ σt, for some constant σ> 2. We prove that, among all barriers consisting of a single closed curve, the one which minimizes the total burned area is axisymmetric, and consists of an arc of circumference and two arcs of logarithmic spirals. 1
Dynamic blocking problems for a model of fire propagation
 In Advances in Applied Mathematics, Modeling, and Computational Science
, 2013
"... This paper contains a survey of recent work on a class of dynamic blocking problems. The basic model consists of a differential inclusion describing the growth of a set in the plane. To restrain its expansion, it is assumed that barriers can be constructed, in real time. Here the issues of major int ..."
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This paper contains a survey of recent work on a class of dynamic blocking problems. The basic model consists of a differential inclusion describing the growth of a set in the plane. To restrain its expansion, it is assumed that barriers can be constructed, in real time. Here the issues of major interests are: (i) whether the growth of the set can be eventually blocked, and (ii) what is the optimal location of the barriers, minimizing a cost criterion. After introducing the basic definitions and concepts, the paper reviews various results on the existence or nonexistence of blocking strategies. A theorem on the existence of an optimal strategy is then recalled, together with various necessary conditions for optimality. Sufficient conditions for optimality and a numerical algorithm for the computation of optimal barriers are also discussed, together with several open problems. 1