# Equivalent formulation and numerical analysis of a fire confinement problem

• Volume: 16, Issue: 4, page 974-1001
• ISSN: 1292-8119

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## Abstract

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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 $\stackrel{˙}{x}$∈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$↦$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.

## How to cite

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Bressan, Alberto, and Wang, Tao. "Equivalent formulation and numerical analysis of a fire confinement problem." ESAIM: Control, Optimisation and Calculus of Variations 16.4 (2010): 974-1001. <http://eudml.org/doc/250702>.

@article{Bressan2010,
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 $\dot x$∈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) ⊂$\mathbb\{R\}^2$. 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⊂$\mathbb\{R\}^2$, rather than the multifunction t$\mapsto$γ(t) ⊂$\mathbb\{R\}^2$. Relying on this result, we then develop a numerical algorithm for the computation of optimal strategies, minimizing the total area burned by the fire. },
author = {Bressan, Alberto, Wang, Tao},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Dynamic blocking problem; differential inclusion; constrained minimum time problem; dynamic blocking problem},
language = {eng},
month = {10},
number = {4},
pages = {974-1001},
publisher = {EDP Sciences},
title = {Equivalent formulation and numerical analysis of a fire confinement problem},
url = {http://eudml.org/doc/250702},
volume = {16},
year = {2010},
}

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
DA - 2010/10//
PB - EDP Sciences
VL - 16
IS - 4
SP - 974
EP - 1001
AB - 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) ⊂$\mathbb{R}^2$. 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⊂$\mathbb{R}^2$, rather than the multifunction t$\mapsto$γ(t) ⊂$\mathbb{R}^2$. Relying on this result, we then develop a numerical algorithm for the computation of optimal strategies, minimizing the total area burned by the fire.
LA - eng
KW - Dynamic blocking problem; differential inclusion; constrained minimum time problem; dynamic blocking problem
UR - http://eudml.org/doc/250702
ER -

## References

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12. J. Nocedal and S.J. Wright. Numerical Optimization. Springer, New York (2006).

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