# Equivalent formulation and numerical analysis of a fire confinement problem

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

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topBressan, 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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- A. Bressan and C. De Lellis, Existence of optimal strategies for a fire confinement problem. Comm. Pure Appl. Math.62 (2009) 789–830.
- A. Bressan and T. Wang, The minimum speed for a blocking problem on the half plane. J. Math. Anal. Appl.356 (2009) 133–144.
- A. Bressan, M. Burago, A. Friend and J. Jou, Blocking strategies for a fire control problem. Anal. Appl.6 (2008) 229–246.
- C. De Lellis, Rectifiable Sets, Densities and Tangent Measures, Zürich Lectures in Advanced Mathematics. EMS Publishing House (2008).
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- K. Kuratovski. Topology, Vol. II. Academic Press, New York (1968).
- W.S. Massey, A Basic Course in Algebraic Topology. Springer-Verlag, New York (1991).
- J. Nocedal and S.J. Wright. Numerical Optimization. Springer, New York (2006).

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