Dynamic Blocking Problems for Models of Fire Propagation

Transcription

1 Dynamic Blocking Problems for Models of Fire Propagation Alberto Bressan Department of Mathematics, Penn State University Alberto Bressan (Penn State) Dynamic Blocking Problems 1 / 40

2 Dynamic Blocking Problems A set R(t) expands as time increases To restrain its growth, a barrier Γ can be constructed, in real time Models: blocking an advancing wild fire, the spatial spreading of a chemical contamination... Alberto Bressan (Penn State) Dynamic Blocking Problems 2 / 40

3 Blocking an advancing wildfire barrier Alberto Bressan (Penn State) Dynamic Blocking Problems 3 / 40

4 Models of fire propagation V. Mallet, D. E. Keyes, and F. E. Fendell, Modeling wildland fire propagation with level set methods. Computers and Mathematics with Applications 57 (2009), G. D. Richards, An elliptical growth model of forest fire fronts and its numerical solution, Internat. J. Numer. Meth. Eng. 30 (1990), R. C. Rothermel, A mathematical model for predicting fire spread in wildland fuels, USDA Forest Service, Intermountain Forest and Range Experiment Station, Research Paper INT-115, Ogden, Utah, USA, A. L. Sullivan, Wildland surface fire spread modelling, Internat. J. Wildland Fire, 18 (2009), Alberto Bressan (Penn State) Dynamic Blocking Problems 4 / 40

5 A Differential Inclusion Model for Fire Propagation R(t) R 0 x F(x) R(t) R 2 = set reached by the fire at time t 0 determined as the reachable set by a differential inclusion ẋ F (x) x(0) R 0 R 2 Fire may spread in different directions with different velocities { R(t) = x(t) ; x( ) absolutely continuous, x(0) R 0, ẋ(τ) F ( x(τ) ) } for a.e. τ [0, t] Alberto Bressan (Penn State) Dynamic Blocking Problems 5 / 40

6 Propagation speed of the fire front n(x) x F(x) R 0 x R(t) R(t) advancing speed of the fire front, in the normal direction: max v F (x) n(x), v Alberto Bressan (Penn State) Dynamic Blocking Problems 6 / 40

7 The minimum time function T (x) = inf { t 0 ; } x R(t) = minimum time taken by the fire to reach the point x The minimum time function provides a solution of the Hamilton-Jacobi equation H ( x, T (x) ) = 0, H(x, p). = max v F (x) p, v 1 with boundary data T (x) = 0 for x R 0 (in a viscosity sense) The level set {T (x) = t} describes the position of the fire front at time t > 0 Alberto Bressan (Penn State) Dynamic Blocking Problems 7 / 40

8 Confinement Strategies (A.B., J.Differential Equations, 2007) Assume: a controller can construct a wall, i.e. a one-dimensional rectifiable curve γ, which blocks the spreading of the fire. γ(t) R 2 = portion of the wall constructed within time t σ = speed at which the wall is constructed Definition 1. A set valued map t γ(t) R 2 is an admissible strategy if : (H1) For every t 1 t 2 one has γ(t 1 ) γ(t 2 ) (H2) Each γ(t) is a rectifiable set (possibly not connected). Its length satisfies m 1 (γ(t)) σ t Alberto Bressan (Penn State) Dynamic Blocking Problems 8 / 40

9 Definition 2. The reachable set determined by the blocking strategy γ is { R γ (t) =. x(t) ; x( ) absolutely continuous, x(0) R 0 ẋ(τ) F ( x(τ) ) } for a.e. τ [0, t], x(τ) / γ(τ) for all τ [0, t] REMARK: Walls must be constructed in real time! Γ γ R (t) R 0 R 0 Γ γ(t) An admissible strategy is described by a set-valued function t γ(t) R 2 γ(t) = portion of the wall constructed within time t Alberto Bressan (Penn State) Dynamic Blocking Problems 9 / 40

10 Optimal Confinement Strategies A cost functional should take into account - The value of the region destroyed by the fire. - The cost of building the wall. α(x) = value of a unit area of land around the point x β(x) = cost of building a unit length of wall near the point x COST FUNCTIONAL J(γ) { }. = lim α dm 2 + β dm 1 t R γ (t) γ(t) Alberto Bressan (Penn State) Dynamic Blocking Problems 10 / 40

11 Mathematical Problems 1. Blocking Problem. Given an initial set R 0, a multifunction F and a wall construction speed σ, does there exist an admissible strategy t γ(t) such that the reachable sets R γ (t) remain uniformly bounded for all t > 0? 2. Optimization Problem. Find an admissible strategy γ( ) which minimizes the cost functional J(γ). Alberto Bressan (Penn State) Dynamic Blocking Problems 11 / 40

12 Existence of an optimal strategy γ( ) Necessary conditions for optimality Sufficient conditions for optimality Regularity of the curves γ(t) constructed by an optimal strategy Numerical computation of an optimal strategy Alberto Bressan (Penn State) Dynamic Blocking Problems 12 / 40

13 Equivalent Formulation (A.B. - T. Wang, Control Optim. Calc. Var. 2009) Blocking Problems and Optimization Problems can be reformulated in terms of one single rectifiable set Γ (strategy) t γ(t) Γ (single wall) Γ γ(t) Γ R (t) R 0 Γ γ(t). = Γ R Γ (t) (walls touched by the fire within time t) Alberto Bressan (Penn State) Dynamic Blocking Problems 13 / 40

14 Blocking the Fire Fire propagates in all directions with unit speed: F (x) = B 1 Wall is constructed at speed σ Theorem (A.B., J.Differential Equations, 2007) On the entire plane, the fire can be blocked if σ > 2, it cannot be blocked if σ < 1. Blocking Strategy: If σ > 2, construct two arcs of logarithmic spirals along the edge of the fire Γ Γ 1+t R (t) γ(t) { } γ(t) =. (r, θ) ; r = e λ θ, 1 r 1 + t, λ =. 1 σ Alberto Bressan (Penn State) Dynamic Blocking Problems 14 / 40

15 No strategy can block the fire if σ 1 γ 1 R0 x0 x 1 γ 0 γ 3 Γ x _ γ 2 x = position of last brick of the wall T Γ ( x) = sup x Γ T Γ (x) Γ σ otherwise the fire escapes γ 2 + γ 3 2 Γ T Γ ( x) = γ 0 < γ 1 min { γ 2, γ 3 } Γ Alberto Bressan (Penn State) Dynamic Blocking Problems 15 / 40

16 The isotropic case on the half plane Fire propagates in all directions with unit speed. F (x) = B 1 Wall is constructed at speed σ Theorem. (A.B. - T.Wang, J.Math Anal.Appl. 2009) Restricted to a half plane, the fire can be blocked if and only if σ > 1 γ R 0 Alberto Bressan (Penn State) Dynamic Blocking Problems 16 / 40

17 When can the fire be blocked? Conjecture: Assume the fire propagates with speed 1 in all directions. On the entire plane the fire can be blocked if and only if σ > 2 Γ Γ R (t) 1 σ θ γ 1 (t) Q 2 R 0 Q 0 Q 1 P θ Single spiral strategy: curve closes on itself if and only if σ > σ = (M. Burago, 2006) Alberto Bressan (Penn State) Dynamic Blocking Problems 17 / 40

18 Non-isotropic fire propagation Assume : F = { } (r cos θ, r sin θ) ; 0 r ρ(θ) ρ( θ) = ρ(θ), 0 ρ(θ ) ρ(θ) for all 0 θ θ π. F F 1 2 F 3 0 k 0 0 Theorem. (A.B., M. Burago, A. Friend, J. Jou, Analysis and Applications, 2008) If the wall construction speed satisfies σ > [vertical width of F ] = 2 max θ [0,π] ρ(θ) sin θ then, for every bounded initial set R 0, a blocking strategy exists Alberto Bressan (Penn State) Dynamic Blocking Problems 18 / 40

19 Existence of Optimal Strategies Fire propagation: ẋ F (x) x(0) R 0 Wall constraint: γ(t) ψ dm 1 t (1/ψ(x) = construction speed at x) { Minimize: J(γ) = R γ(t) α dm 2 + γ(t) 1} β dm Assumptions: (A1) The initial set R 0 is open and bounded. Its boundary satisfies m 2 ( R 0 ) = 0. (A2) The multifunction F is Lipschitz continuous w.r.t. the Hausdorff distance. For each x R 2 the set F (x) is compact, convex, and contains a ball of radius ρ 0 > 0 centered at the origin. (A3) For every x R 2 one has α(x) 0, β(x) 0, and ψ(x) ψ 0 > 0. α is locally integrable, while β and ψ are both lower semicontinuous. Alberto Bressan (Penn State) Dynamic Blocking Problems 19 / 40

20 Theorem (A.B. - C. De Lellis, Comm. Pure Appl. Math. 2008) Assume (A1)-(A3), and inf γ S J(γ) <. Then the minimization problem admits an optimal solution γ. Direct method: Consider a minimizing sequence of strategies γ n( ) Define the optimal strategy γ as a suitable limit. γ m γ n? γ m γ n? γ n γ Alberto Bressan (Penn State) Dynamic Blocking Problems 20 / 40

21 Key step in the proof: For each rational time τ, order the connected components of γ n (τ) according to decreasing length: l n,1 l n,2 l n,3 γ n (t) = γ n,1 γ n,2 γ n,3 Taking a subsequence, as n we can assume We then define γ(τ). = l n,i (τ) l i (τ) γ n,i (τ) γ i (τ) τ Q i 1, l i (τ)>0 γ i (τ) Alberto Bressan (Penn State) Dynamic Blocking Problems 21 / 40

22 New approach: the minimum time function with obstacle (C. DeLellis & R. Robyr, Archive Rational Mech. Anal., 2011) ẋ F (x), x(0) R 0 R 2 Γ R(t) x x Γ R (t) Minimum time function with obstacle: T Γ (x). = inf {t 0 ; x R Γ (t)} R Γ (t) = set of points reached by F -trajectories which start in R 0 and do not cross Γ Goal: characterize T Γ in terms of a H-J equation Alberto Bressan (Penn State) Dynamic Blocking Problems 22 / 40

23 A family S Γ of subsolutions Definition. A function u : R 2 [0, ] is in the set S Γ if u SBV (Du = u + D jump u + D Cantor u, with D Cantor u 0). m 1 (J u \ Γ) = 0 (J u = set of jump points of u) u = 0 on R 0 H(x, u(x)) 0 for a.e. x u. = absolutely continuous part of the distributional derivative Du. Alberto Bressan (Penn State) Dynamic Blocking Problems 23 / 40

24 Theorem. (C.DeLellis & R.Robyr) The minimum time function T Γ is the unique maximal element of S Γ. = alternative proof of existence of optimal blocking strategies Take a minimizing sequence of admissible barriers Γ k Consider the corresponding minimum time functions T k. = T Γ k Using the Ambrosio-De Giorgi compactness theorem for SBV functions, one obtains a convergent subsequence T k U, with U SBV The jump set J U yields the optimal barrier Alberto Bressan (Penn State) Dynamic Blocking Problems 24 / 40

25 Necessary conditions for optimality Problem: find an admissible barrier Γ which minimizes J(Γ). = α [total burned area] + β [length of the curve] GOAL: derive a set of ODE s describing the walls built by an optimal strategy A.B., J.Differential Equations, 2007 A.B. - T.Wang, ESAIM, Control Optim. Calc. Var T.Wang, J. Optim. Theory Appl., to appear. Alberto Bressan (Penn State) Dynamic Blocking Problems 25 / 40

26 Classification of arcs in an optimal strategy Minimum time function T Γ (x) Set of times where the constraint is saturated { ( S. = t 0 ; meas. = inf Γ R Γ (t) Boundary arcs: Γ S. = {x Γ ; T Γ (x) S} constructed along the advancing fire front { t 0 ; ) = σt } } x R Γ (t) Free arcs: Γ F. = {x Γ ; T Γ (x) / S} constructed away from the fire front Γ S R 0 Γ F Γ S Alberto Bressan (Penn State) Dynamic Blocking Problems 26 / 40

27 Optimality conditions, minimizing the value of burned area 1. A free arc Γ. The curvature must be proportional to the local value of the land r(s) = radius of curvature α = land value r(s) α ( Γ(s) ) = const. Γ R (t) Γ F(x) n β n R(t) γ(t) 2. A single boundary arc Γ. The wall is constructed at maximum speed σ, always remaining at the edge of the burned set σ sin β = max n y y F (x) Alberto Bressan (Penn State) Dynamic Blocking Problems 27 / 40

28 3. Two or more boundary arcs: Γ 1,..., Γ ν, constructed simultaneously for t [a, b] h Γ 2 R(t+1) Γ 1 fast slow e 2 (t) η (t) 2 R(t) θ Sum of construction speeds σ At which speed should each wall be constructed? recast the problem in the standard setting of optimal control apply the Pontryagin Maximum Principle Alberto Bressan (Penn State) Dynamic Blocking Problems 28 / 40

29 Junctions between different arcs γ 1 γ 2 γ 2 γ 1 R Γ 8 P R Γ 8 Q Two boundary arcs originating at the same point are not optimal Non-parallel junctions between a free arc and a boundary arc are not optimal Alberto Bressan (Penn State) Dynamic Blocking Problems 29 / 40

30 Q R 0 P circle + two spirals (is better than two spirals only) Alberto Bressan (Penn State) Dynamic Blocking Problems 30 / 40

31 Further classification blocking arcs Γ b. = Γ R Γ delaying arcs Γ d Γ Ω 0 Ω 1 Ω 2 R 0 Necessary conditions for the optimality of delaying arcs (Tao Wang, J. Optim. Theory Appl., to appear) Alberto Bressan (Penn State) Dynamic Blocking Problems 31 / 40

32 Sufficient conditions for optimality? Standard Isotropic Problem: Fire starts on the unit disc, propagating with unit speed in all directions. Barrier can be constructed at speed σ > 2. Minimize the total burned area. Γ S R 0 Γ F Γ S Theorem (A.B. - T.Wang, 2010) The barrier consisting of circle + two logarithmic spirals is optimal among all simple closed curves enclosing R 0 Alberto Bressan (Penn State) Dynamic Blocking Problems 32 / 40

33 Γ Ω Γ original curve star shaped * Ω Γ r( ) θ θ symmetric, nondecreasing rearrangement Alberto Bressan (Penn State) Dynamic Blocking Problems 33 / 40

34 Polar coordinate representation: θ r(θ) non-decreasing, for θ [0, π] Admissibility constraint: m 1 ({x Γ ; x 1 + t} ) σt < σt = circumferences = σt = logarithmic spirals joining tangentially?? Alberto Bressan (Penn State) Dynamic Blocking Problems 34 / 40

35 Numerical Simulations (A.B. - T. Wang, ESAIM; Control Optim. Calc. Var. 2009) Assume: only blocking arcs, no delaying arcs. minimize total burned area: m 2 (R Γ ) subject to m 1 (Γ R Γ (t) ) σt for all t 0 Alberto Bressan (Penn State) Dynamic Blocking Problems 35 / 40

36 Approximate the barrier with a polygonal: fix an angle θ = 2π/n assign radii r k = r(kθ), k = 1,..., n starting with an admissible polygonal, search for a local minimizer subject to admissibility constraints double number of nodes (replace n by 2n), repeat local minimization... Q k 1 Q k S k Q 1 R 0 θ Q 0 Alberto Bressan (Penn State) Dynamic Blocking Problems 36 / 40

37 1. The isotropic case F (x) = R 0 = B 1 (unit disc), σ = 4. Minimize: total burned area. Alberto Bressan (Penn State) Dynamic Blocking Problems 37 / 40

38 2. A non-isotropic case { } F = (λx, λy) ; (x 3) 2 + y 2 1, λ [0, 1] y r = 1 F 0 κ x Choose: σ = 4.1, R 0 = unit disc. Analytic solution: A.Friend (2007). Numerical solution: T.Wang (2008) Alberto Bressan (Penn State) Dynamic Blocking Problems 38 / 40

39 Some open problems 1 (Isotropic blocking problem) On the whole plane, assume: fire propagates with unit speed in all directions barrier is constructed at speed σ Conjecture 1: A blocking strategy exists if and only if the wall construction speed is σ > 2. 2 (Sufficient optimality conditions) Not one single example is known where a blocking strategy can be proved to be optimal. Conjecture 2: The circle + two spirals strategy is optimal for the isotropic problem. Basic difficulty: delaying arcs Alberto Bressan (Penn State) Dynamic Blocking Problems 39 / 40

40 3 (Existence of optimal strategies). Determine whether an optimal strategy exists, in the general case where the velocity sets satisfy 0 F (x), but without assuming B(0, ρ) F (x) so that fire propagation speed is not uniformly positive in all directions 4 (Regularity). Assume: the initial set R 0 has a smooth boundary and the cost functions α, β are smooth. What is the regularity of an optimal strategy? Does it produce a finite number of piecewise C 1 arcs? Is the optimal barrier connected? Is it ever useful to construct purely delaying arcs? Alberto Bressan (Penn State) Dynamic Blocking Problems 40 / 40