Equivalent formulation and numerical analysis of a fire confinement problem
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, p. 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 : https://doi.org/10.1051/cocv/2009033
Classification:  49Q20,  34A60,  49J24,  93B03
Keywords: dynamic blocking problem, differential inclusion, constrained minimum time problem
     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},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {4},
     year = {2010},
     pages = {974-1001},
     doi = {10.1051/cocv/2009033},
     zbl = {pre05821905},
     mrnumber = {2744158},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/COCV_2010__16_4_974_0/

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | Zbl 0957.49001

[2] J.P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, Berlin (1984). | Zbl 0538.34007

[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 1138.34002

[4] A. Bressan and C. De Lellis, Existence of optimal strategies for a fire confinement problem. Comm. Pure Appl. Math. 62 (2009) 789-830. | Zbl 1198.92045

[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. | Zbl 1162.92041

[6] A. Bressan, M. Burago, A. Friend and J. Jou, Blocking strategies for a fire control problem. Anal. Appl. 6 (2008) 229-246. | Zbl 1160.49043

[7] C. De Lellis, Rectifiable Sets, Densities and Tangent Measures, Zürich Lectures in Advanced Mathematics. EMS Publishing House (2008). | Zbl 1183.28006

[8] H. Federer, Geometric Measure Theory. Springer-Verlag, New York (1969). | Zbl 0874.49001

[9] M. Henle, A Combinatorial Introduction to Topology. W.H. Freeman, San Francisco (1979). | Zbl 0527.55001

[10] K. Kuratovski. Topology, Vol. II. Academic Press, New York (1968). | Zbl 0158.40802

[11] W.S. Massey, A Basic Course in Algebraic Topology. Springer-Verlag, New York (1991). | Zbl 0725.55001

[12] J. Nocedal and S.J. Wright. Numerical Optimization. Springer, New York (2006). | Zbl 1104.65059