Dynamic Blocking Problems for Models of Fire Propagation Alberto - PowerPoint PPT Presentation

Dynamic Blocking Problems for Models of Fire Propagation Alberto Bressan Department of Mathematics, Penn State University bressan@math.psu.edu Alberto Bressan (Penn State) Dynamic Blocking Problems 1 / 40

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

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

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), 1089–1101. G. D. Richards, An elliptical growth model of forest fire fronts and its numerical solution, Internat. J. Numer. Meth. Eng. 30 (1990), 1163–1179. 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, 1972. A. L. Sullivan, Wildland surface fire spread modelling, 1990-2007. Internat. J. Wildland Fire , 18 (2009), 349–403. Alberto Bressan (Penn State) Dynamic Blocking Problems 4 / 40

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 x (0) ∈ R 0 ⊂ R 2 x ∈ F ( x ) ˙ Fire may spread in different directions with different velocities � R ( t ) = x ( t ) ; x ( · ) absolutely continuous , � � � x (0) ∈ R 0 , x ( τ ) ∈ F ˙ x ( τ ) for a.e. τ ∈ [0 , t ] Alberto Bressan (Penn State) Dynamic Blocking Problems 5 / 40

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

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 ) = v ∈ F ( x ) � p , v � − 1 max 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

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

Definition 2. The reachable set determined by the blocking strategy γ is � R γ ( t ) . = x ( t ) ; x ( · ) absolutely continuous , x (0) ∈ R 0 � � � x ( 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 (t) Γ γ R (t) R 0 R 0 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

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

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

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

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 ( t ) . Γ − → = Γ ∩ R Γ ( t ) (walls touched by the fire within time t ) Alberto Bressan (Penn State) Dynamic Blocking Problems 13 / 40

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 1+t γ (t) Γ R (t) � � 1 γ ( t ) . λ . r = e λ | θ | , = ( r , θ ) ; 1 ≤ r ≤ 1 + t , = � σ 2 4 − 1 Alberto Bressan (Penn State) Dynamic Blocking Problems 14 / 40

No strategy can block the fire if σ ≤ 1 Γ γ 3 x 1 γ 1 R _ 0 x x 0 γ 0 γ 2 x = position of “last brick of the wall” ¯ T Γ ( x ) ≥ | Γ | T Γ (¯ x ) = sup otherwise the fire escapes σ x ∈ Γ | 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

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

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 σ 1 R θ Q 0 Q γ 1 (t) 0 1 Γ R (t) Q 2 P θ Single spiral strategy: curve closes on itself if and only if σ > σ † = 2 . 614430844 . . . (M. Burago, 2006) Alberto Bressan (Penn State) Dynamic Blocking Problems 17 / 40

Non-isotropic fire propagation � � Assume : F = ( r cos θ , r sin θ ) ; 0 ≤ r ≤ ρ ( θ ) for all 0 ≤ θ ≤ θ ′ ≤ π . 0 ≤ ρ ( θ ′ ) ≤ ρ ( θ ) ρ ( − θ ) = ρ ( θ ) , F F 3 F 2 1 k 0 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

Existence of Optimal Strategies Fire propagation: x ∈ 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 ) β dm 1 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

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 n ? γ m γ n γ Alberto Bressan (Penn State) Dynamic Blocking Problems 20 / 40

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 ℓ n , i ( τ ) → ℓ i ( τ ) γ n , i ( τ ) → γ i ( τ ) τ ∈ Q We then define γ ( τ ) . � = γ i ( τ ) i ≥ 1 , ℓ i ( τ ) > 0 Alberto Bressan (Penn State) Dynamic Blocking Problems 21 / 40