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

top
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) ⊂ 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⊂ 2 , rather than the multifunction t γ(t) ⊂ 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

top

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

top
  1. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).  Zbl0957.49001
  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).  Zbl1138.34002
  4. A. Bressan and C. De Lellis, Existence of optimal strategies for a fire confinement problem. Comm. Pure Appl. Math.62 (2009) 789–830.  Zbl1198.92045
  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.  Zbl1162.92041
  6. A. Bressan, M. Burago, A. Friend and J. Jou, Blocking strategies for a fire control problem. Anal. Appl.6 (2008) 229–246.  Zbl1160.49043
  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).  Zbl0176.00801
  9. M. Henle, A Combinatorial Introduction to Topology. W.H. Freeman, San Francisco (1979).  Zbl0527.55001
  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).  Zbl0725.55001
  12. J. Nocedal and S.J. Wright. Numerical Optimization. Springer, New York (2006).  Zbl1104.65059

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.