Equivalent formulation and numerical analysis of a fire confinement problem
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 974-1001.

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.

DOI: 10.1051/cocv/2009033
Classification: 49Q20, 34A60, 49J24, 93B03
Keywords: 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, Volume 16 (2010) no. 4, pp. 974-1001. doi : 10.1051/cocv/2009033. http://archive.numdam.org/articles/10.1051/cocv/2009033/

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | Zbl

[2] J.P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, Berlin (1984). | Zbl

[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). | Zbl

[4] A. Bressan and C. De Lellis, Existence of optimal strategies for a fire confinement problem. Comm. Pure Appl. Math. 62 (2009) 789-830. | Zbl

[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. | Zbl

[6] A. Bressan, M. Burago, A. Friend and J. Jou, Blocking strategies for a fire control problem. Anal. Appl. 6 (2008) 229-246. | Zbl

[7] C. De Lellis, Rectifiable Sets, Densities and Tangent Measures, Zürich Lectures in Advanced Mathematics. EMS Publishing House (2008). | Zbl

[8] H. Federer, Geometric Measure Theory. Springer-Verlag, New York (1969). | Zbl

[9] M. Henle, A Combinatorial Introduction to Topology. W.H. Freeman, San Francisco (1979). | Zbl

[10] K. Kuratovski. Topology, Vol. II. Academic Press, New York (1968). | Zbl

[11] W.S. Massey, A Basic Course in Algebraic Topology. Springer-Verlag, New York (1991). | Zbl

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

Cited by Sources: