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Equivalent formulation and numerical analysis of a fire confinement problem

Published online by Cambridge University Press:  11 August 2009

Alberto Bressan  and
Tao Wang
Alberto Bressan
Affiliation:
Department of Mathematics, Penn State University University Park, Pa. 16802, USA. bressan@math.psu.edu; wang_t@math.psu.edu
Tao Wang
Affiliation:
Department of Mathematics, Penn State University University Park, Pa. 16802, USA. bressan@math.psu.edu; wang_t@math.psu.edu
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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.

Keywords

Dynamic blocking problemdifferential inclusionconstrained minimum time problem
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 974 - 1001
Copyright
© EDP Sciences, SMAI, 2009

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References

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).
J.P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, Berlin (1984).
Bressan, A., 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). CrossRef
Bressan, A. and De Lellis, C., Existence of optimal strategies for a fire confinement problem. Comm. Pure Appl. Math. 62 (2009) 789–830. CrossRef
Bressan, A. and Wang, T., The minimum speed for a blocking problem on the half plane. J. Math. Anal. Appl. 356 (2009) 133–144. CrossRef
Bressan, A., Burago, M., Friend, A. and Jou, J., Blocking strategies for a fire control problem. Anal. Appl. 6 (2008) 229–246. CrossRef
C. De Lellis, Rectifiable Sets, Densities and Tangent Measures, Zürich Lectures in Advanced Mathematics. EMS Publishing House (2008).
H. Federer, Geometric Measure Theory. Springer-Verlag, New York (1969).
M. Henle, A Combinatorial Introduction to Topology. W.H. Freeman, San Francisco (1979).
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).