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.
Mots clés : dynamic blocking problem, differential inclusion, constrained minimum time problem
@article{COCV_2010__16_4_974_0,
author = {Bressan, Alberto and Wang, Tao},
title = {Equivalent formulation and numerical analysis of a fire confinement problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {974--1001},
publisher = {EDP-Sciences},
volume = {16},
number = {4},
year = {2010},
doi = {10.1051/cocv/2009033},
mrnumber = {2744158},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2009033/}
}
TY - JOUR AU - Bressan, Alberto AU - Wang, Tao TI - Equivalent formulation and numerical analysis of a fire confinement problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 974 EP - 1001 VL - 16 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2009033/ DO - 10.1051/cocv/2009033 LA - en ID - COCV_2010__16_4_974_0 ER -
%0 Journal Article %A Bressan, Alberto %A Wang, Tao %T Equivalent formulation and numerical analysis of a fire confinement problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 974-1001 %V 16 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2009033/ %R 10.1051/cocv/2009033 %G en %F COCV_2010__16_4_974_0
Bressan, Alberto; Wang, Tao. Equivalent formulation and numerical analysis of a fire confinement problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 974-1001. doi : 10.1051/cocv/2009033. http://archive.numdam.org/articles/10.1051/cocv/2009033/
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