Traffic models on a network of roads

Transcription

1 Traic models on a network o roads Alberto Bressan Department o Mathematics, Penn State University bressan@math.psu.edu Center or Interdisciplinary Mathematics Alberto Bressan (Penn State) Traic low on Networks 1 / 23

2 A PDE model or traic low ρ = traic density (= number o cars per unit length) v = velocity o cars (is a decreasing unction o the traic density ρ) v v v( ρ) = velocity ( ρ) = ρv( ρ ) = lux 0 ρ jam ρ 0 ρ ρ jam lux: (t, x) = [number o cars crossing the point x per unit time] = [density] [velocity] = ρ(t, x) v(ρ(t, x)) Alberto Bressan (Penn State) Traic low on Networks 2 / 23 x

3 A conservation law or the traic density ρ = density o cars a b x d dt b a ρ(t, x) dx = b a ρ t (t, x) dx = (ρ(t, a)) (ρ(t, b)) = b a b a [ ] (ρ(t, x)) dx x {ρ t + (ρ) x } dx = 0 or every a < b = ρ t + (ρ) x = 0 Alberto Bressan (Penn State) Traic low on Networks 3 / 23

4 The LWR model Lighthill & Witham (1955), Richards (1956) ρ t + (ρ) x = 0 conservation law ρ(0, x) = ρ(x) initial data ρ = density o cars (ρ) = ρ v(ρ) = lux Alberto Bressan (Penn State) Traic low on Networks 4 / 23

5 Modeling traic low at a junction incoming roads: i I outgoing roads: j O j i Boundary conditions should relate the boundary densities depending on ρ i (t, 0 ) i I, ρ j (t, 0+) j O c i = relative priority o drivers rom road i (raction o time drivers rom road i get green light, on average) θ ij = raction o drivers rom road i that turn into road j θ ij = 1 j Alberto Bressan (Penn State) Traic low on Networks 5 / 23

6 Construction o a Riemann Solver (Coclite, Garavello, Piccoli... ) Basic approach: give a rule on how each Riemann problem should be solved i.e.: describe the solution in case where the initial data is constant on each road. j i Motivation: Every solution can be obtained as limit o piecewise constant approximations, by the wave-ront tracking algorithm C. Daermos, Polygonal approximations o solutions o the initial value problem or a conservation law, J. Math. Anal. Appl. 38 (1972), A. Bressan, Global solutions o systems o conservation laws by wave-ront tracking, J. Math. Anal. Appl. 170 (1992), Alberto Bressan (Penn State) Traic low on Networks 6 / 23

7 Step 1: determining the imum possible luxes Given: initial densities, constant on each road { ρi i I, ρ j j O Determine: i = imum lux that can exit rom the incoming road i, i I j = imum lux that can enter into the outgoing road j, j O ρ 1 ρ ρ 3 ρ Alberto Bressan (Penn State) Traic low on Networks 7 / 23

8 Step 2: the easible region Ω Ω On incoming roads one has the constraints 0 i i Moreover, outgoing roads impose the constraints i θ ij j i I i I j O Ω = set o incoming luxes satisying all constraints Riemann solver rule or selecting a point in the easible region Ω Alberto Bressan (Penn State) Traic low on Networks 8 / 23

9 Example: the Riemann solver generated by a stop sign STOP 3 0 Ω Cars coming rom road 2 have a STOP sign Alberto Bressan (Penn State) Traic low on Networks 9 / 23

10 Continuity o the Riemann Solver Example o a selection rule: imize the total lux i I i I the turning preerences θ ij remain constant, the luxes i depend Lipschitz continuously on the Riemann data ρ 1,..., ρ N. 2 Ω 0 1 The Riemann solver is discontinuous w.r.t. changes in the θ ij 2 2 Ω Ω Alberto Bressan (Penn State) Traic low on Networks 10 / 23

11 Why should the θ ij vary in time? Drivers turning preerences θ ij must be determined as part o the solution 2 1 A 3 B 4 5 # o vehicles on road i that wish to turn into road j is conserved: (ρθ ij ) t + (ρv i (ρ)θ ij ) x = 0 Alberto Bressan (Penn State) Traic low on Networks 11 / 23

12 Continuous Riemann Solvers (A.B. - F.Yu, Discr. Cont. Dyn. Syst., 2015) 2 _ Ω 0 1 The selection rule: imize i I i yields a Riemann solver which is Hölder continuous w.r.t. all variables (ρ i, θ ij ) i I, j O ( i ) i I One can also construct a Riemann solver which is Lipschitz continuous w.r.t. all variables Unortunately all this is useless, because i the θ ij are allowed to vary the Cauchy problem is ill posed anyway For some initial data one can ind two distinct solutions! Alberto Bressan (Penn State) Traic low on Networks 12 / 23

13 An intersection model with buer (A.B. - K.Nguyen, 2015) q 4 q 5 5 the intersection contains a buer with imum capacity M (a traic circle) t q j (t) = queues in ront o outgoing roads j O, within the buer incoming drivers are admitted to the intersection at a rate proportional to the empty space remaining the buer: M j q j drivers already inside the intersection low out to the road o their choice at the astest possible rate Alberto Bressan (Penn State) Traic low on Networks 13 / 23

14 Well-posedness o the Cauchy problem with buers Theorem (A.B.- K.Nguyen, Netw. Heter. Media 10 (2015), ). Consider an intersection modeled by (SBJ). For any L initial data ρ(0, x) = ρ k (x), θ ij (0, x) = θ ij, q j (0) = q j, the Cauchy problem has a unique entropy admissible solution, deined or all t 0. Moreover, the travel times depend continuously on the initial data, in the topology o weak convergence. ρ n k(x) ρ k ρ n i θ n ij ρ i θij, q n j q j Proo relies on: variational ormulation + ixed point argument Alberto Bressan (Penn State) Traic low on Networks 14 / 23

15 ρ 1 ρ 2 q 4 q 5 ρ 3 ρ 4 ρ 5 Lax ormula (q, q ) 4 5 length o queues boundary values (V, V, V, V, V ) (q, q ) ρ = V k k,x traic densities I the queue sizes q j (t) within the buer are known, then the PDEs or the densities ρ k, can be independently solved along each incoming and outgoing road. Conversely, i these value unctions V k are known, then the queue sizes q j can be determined by balancing the boundary luxes o all incoming and outgoing roads The solution o the Cauchy problem is obtained as the unique ixed point o a contractive transormation The present model accounts or backward propagation o queues along roads leading to a crowded intersection, it achieves well-posedness or general L data, and continuity o travel time w.r.t. weak convergence Alberto Bressan (Penn State) Traic low on Networks 15 / 23

16 The limit Riemann Solver or buer o vanishing size Theorem (A.B., A.Nordli, Netw. Heter. Media, to appear). Letting the size o the buer M 0 one obtains a Riemann Solver which is Lipschitz continuous w.r.t. all variables ρ i, θ ij 2 c 2 γ 2 γ 2 Ω γ _ c s γ(s) = ( 1 (s),..., m (s)), Then the incoming luxes are i { where: s = = i ( s) s 0 ; i (s) θ ij i (s). = min{c i s, i } j or all j O i I Alberto Bressan (Penn State) Traic low on Networks 16 / 23 }

17 Traic jams Traic jams occur not because o subtle properties o the mathematical equations, but simply because too many people choose to drive on the same road at the same time One should study traic low rom the point o view o optimal decision problems Alberto Bressan (Penn State) Traic low on Networks 17 / 23

18 Optimization Problems or Traic Flow on a Network n groups o drivers with dierent origins and destinations, and dierent costs Drivers in the k-th group depart rom A d(k) and arrive to A a(k) can use dierent paths Γ 1, Γ 2,... to reach destination Departure cost: ϕ k (t) arrival cost: ψ k (t) A d(k) Aa(k) Alberto Bressan (Penn State) Traic low on Networks 18 / 23

19 Basic assumptions (A1) The lux unctions ρ i (ρ) = ρ v(ρ) are all strictly concave down. i (0) = i (ρ jam i ) = 0, i < 0. (A2) For each group o drivers k = 1,..., N, the cost unctions ϕ k, ψ k satisy ( ) ϕ k < 0, ψ k, ψ k < 0, lim ϕ k (t) + ψ k (t) = + t (ρ) ϕ(t) ψ(t) 0 ρ ρ jam ρ t Alberto Bressan (Penn State) Traic low on Networks 19 / 23

20 Optima and Equilibria Choose optimally: the departure times the routes to destinations u k,p (t) = rate o departures o drivers o the k-th group, choosing path p to reach destination An admissible amily {ū k,p } o departure rates is globally optimal i it minimizes the sum o the total costs o all drivers. J(ū) = ( ϕ k (t) + ψ k (τ p (t))) ū k,p (t) dt k,p An admissible amily {ū k,p } o departure rates is a Nash equilibrium i no driver o any group can lower his own total cost by changing departure time or choosing a dierent path to reach destination. Alberto Bressan (Penn State) Traic low on Networks 20 / 23

21 Existence results Theorem (A.B.- K.Nguyen, Netw. Heter. Media 10 (2015), ). Under the assumptions (A1)-(A2), on a general network o roads, there exists at least one globally optimal solution. In addition, i the travel time admits a uniorm upper bound, then a Nash equilibrium solution exists. Proo is achieved by inite dimensional approximations + a topological argument (relying on the continuity o the travel time w.r.t. weak convergence o the departure rates) Alberto Bressan (Penn State) Traic low on Networks 21 / 23

22 Can traic get completely stuck? A A B C C A B B C Assume: at each node, equal numbers o cars are allowed to enter rom the two incoming roads. Then: or every two cars entering, only one exits the triangle o roads ABC at any time t, [# o cars that has reached destination] [# o cars inside the triangle ABC] only initely many cars can reach destination. All the others are stuck orever. Alberto Bressan (Penn State) Traic low on Networks 22 / 23

23 Further directions Necessary conditions or global optimality Models with partial inormation Dynamic stability o Nash equilibria Optimal control o luxes at intersections (to prevent stuck traic, or improve the overall traic pattern) Alberto Bressan (Penn State) Traic low on Networks 23 / 23