Equivalent formulation and numerical analysis of a fire confinement problem

Alberto Bressan; Tao Wang

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

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

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 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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  1. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).  
  2. J.P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, Berlin (1984).  
  3. A. Bressan, Differential inclusions and the control of forest fires. J. Differ. Equ.243 (2007) 179–207 (special volume in honor of A. Cellina and J. Yorke).  
  4. A. Bressan and C. De Lellis, Existence of optimal strategies for a fire confinement problem. Comm. Pure Appl. Math.62 (2009) 789–830.  
  5. A. Bressan and T. Wang, The minimum speed for a blocking problem on the half plane. J. Math. Anal. Appl.356 (2009) 133–144.  
  6. A. Bressan, M. Burago, A. Friend and J. Jou, Blocking strategies for a fire control problem. Anal. Appl.6 (2008) 229–246.  
  7. C. De Lellis, Rectifiable Sets, Densities and Tangent Measures, Zürich Lectures in Advanced Mathematics. EMS Publishing House (2008).  
  8. H. Federer, Geometric Measure Theory. Springer-Verlag, New York (1969).  
  9. M. Henle, A Combinatorial Introduction to Topology. W.H. Freeman, San Francisco (1979).  
  10. K. Kuratovski. Topology, Vol. II. Academic Press, New York (1968).  
  11. W.S. Massey, A Basic Course in Algebraic Topology. Springer-Verlag, New York (1991).  
  12. J. Nocedal and S.J. Wright. Numerical Optimization. Springer, New York (2006).  

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