Conservation laws and some applications to traffic flows

Transcription

1 Conservation laws and some applications to traffic flows Khai T. Nguyen Department of Mathematics, Penn State University 46th Annual John H. Barrett Memorial Lectures May 16 18, 2016 Khai T. Nguyen (Penn State) Conservation laws and traffic flows 1 / 24

2 Traffic flows on a network of roads Several groups of drivers, with different departure and arrival locations Drivers in the k-th group depart from A d(k) and arrive to A a(k) A d(1) Aa(1) GOAL: Describe the evolution of traffic density Study optimization problems (for a central planner) Study Nash equilibrium problems Khai T. Nguyen (Penn State) Conservation laws and traffic flows 2 / 24

3 Overview Review of scalar conservation laws, the Lax formula Global optima and Nash equilibria on a network of roads Khai T. Nguyen (Penn State) Conservation laws and traffic flows 3 / 24

4 Cars on a highway ρ = traffic density = number of cars per unit length v = velocity of cars (is a decreasing function of the traffic density ρ) v v max v( ρ) = velocity f( ρ) = ρ v( ρ ) = flux 0 ρ jam ρ 0 ρ ρ jam flux: f (ρ(t, x)) = [number of cars crossing the point x per unit time] = [density] [velocity] = ρ(t, x) v(ρ(t, x)) Khai T. Nguyen (Penn State) Conservation laws and traffic flows 4 / 24 x

5 A scalar conservation law ρ = density of cars a b x d dt b a ρ(t, x) dx = b a ρ t(t, x) dx = f (ρ(t, a)) f (ρ(t, b)) = PDE model for traffic flow ρ t + f (ρ) x = 0 conservation law ρ(0, x) = ρ(x) initial data b a [ ] f (ρ(t, x)) dx x Lighthill & Witham (1955), Richards (1956) Khai T. Nguyen (Penn State) Conservation laws and traffic flows 5 / 24

6 Solving a linear wave equation ρ t + λρ x = 0 ρ(0, x) = ρ(x) The directional derivative of ρ(t, x) along the vector v = (1, λ) is zero ρ(t, x 0 + λ t) = ρ(x 0 ) ρ(0) λ t ρ(t) λ = wave speed _ ρ(t,x) = ρ (x λ t) x t 1 v λ 0 x 0 x Khai T. Nguyen (Penn State) Conservation laws and traffic flows 6 / 24

7 Solving a quasilinear wave equation ρ t + f (ρ) x = 0 ρ(0, x) = ρ(x) ρ t + f (ρ)ρ x = 0 wave speed = f (ρ) ( ) ρ t, x 0 + f ( ρ(x 0 )) t = ρ(x 0 ) ρ ρ(0) ρ(t) wave speed = f ( ρ) x t 0 x 0 x x Khai T. Nguyen (Penn State) Conservation laws and traffic flows 7 / 24

8 Shock formation τ t ρ = constant 0 x ρ(0,x) ρ(τ, x) ρ( t,x ) Points on the graph of ρ(t, ) move horizontally, with speed f (ρ) At a finite time τ the tangent becomes vertical and a shock is formed Khai T. Nguyen (Penn State) Conservation laws and traffic flows 8 / 24

9 From conservation laws to Hamilton Jacobi equations The conservation law ρ t + f (ρ) x = 0. Set V (t, x) := The Hamilton Jacobi equation x ρ(t, y)dy. V t (t, x) + f (V x (t, x)) = 0. V (t, x) is a value function of an optimization problem. Khai T. Nguyen (Penn State) Conservation laws and traffic flows 9 / 24

10 Legendre transform Legendre transform: g(v). = inf u [0, ρ jam ] {vu f (u)} max f vu jam f 0 ( ρ ) f (0) _ max f v f(u) g(v) 0 u *(v) jam ρ u g(v) = u (v) v f (u (v)) where f (u (v)) = v { v f (ρ jam ) = g(v) = ρ jam v v f (0) = g(v) = 0 Khai T. Nguyen (Penn State) Conservation laws and traffic flows 10 / 24

11 An optimization problem ρ t + f (ρ) x = 0 ρ(0, x) = ρ(x) [0, ρ jam ] Given the initial data V (x). = x ρ(y) dy and a terminal point (t, x). (t,x) z(s) t Maximize: V (z(0)) + g(ż(s)) ds, 0 subject to z(t) = x. 0 z(0) y x Khai T. Nguyen (Penn State) Conservation laws and traffic flows 11 / 24

12 The Lax formula Legendre transform: g(v). = inf u [0, ρ jam ] {vu f (u)} t f ( ρ (t,x)) = x y t * (t,x) The Lax formula V (t, x) = max y ( ) x y V (y) + t g t 0 y * is the solution of y x V t + f (V x ) = 0 for a.e. (t, x) Moreover ρ = V x yields a solution of conservation law ρ t + f (ρ) x = 0. Khai T. Nguyen (Penn State) Conservation laws and traffic flows 12 / 24

13 Overview Review of scalar conservation laws, the Lax formula Global optima and Nash equilibria on a network of roads Khai T. Nguyen (Penn State) Conservation laws and traffic flows 13 / 24

14 Modeling Traffic Flow on a Network of Roads A conservation law on each road γ k : ρ t + f k (ρ) x = 0 Boundary conditions at intersections. These depend on fraction θ ij of drivers coming from road i who wish to turn into road j priority given to different incoming roads γ 2 γ 1 γ 3 γ i γ j Khai T. Nguyen (Penn State) Conservation laws and traffic flows 14 / 24

15 Intersection Models Intersection Models (with θ ij constant): M. Garavello & B. Piccoli (2006, 2009) (A.Bressan. & N-, 2015) New intersection models with θ ij variable in time. Number of vehicles on road i that wish to turn into road j is conserved: (ρθ ij ) t + (ρv i (ρ)θ ij ) x = 0 Khai T. Nguyen (Penn State) Conservation laws and traffic flows 15 / 24

16 Remark: The basic property of a mathematical model is well posedness. M. Garavello - B. Piccoli (2009) prove well posedness for θ ij constant based on Riemann Solvers. A. Bressan, F. Yu (2014): Provide several examples on the ill posedness for θ ij variable. (A. Bressan & N-, 2015) prove well posedness with θ ij variable, for a new intersection model including buffers with finite capacity, using a Lax-type representation formula. Khai T. Nguyen (Penn State) Conservation laws and traffic flows 16 / 24

17 Variational formulation ρ 1 ρ 2 q 4 ρ 3 q 5 ρ 4 ρ 5 Lax formula (q, q ) 4 5 length of queues boundary values (V, V, V, V, V ) (q, q ) ρ = V k k,x traffic densities The solution of the Cauchy problem as the unique fixed point of a contractive transformation. Khai T. Nguyen (Penn State) Conservation laws and traffic flows 17 / 24

18 Avoiding traffic jams From a practical point of view, the most relevant feature of traffic flow are traffic jams. Look at traffic flow from an engineering point of view, as an optimal decision problem. (deciding the departure time and the route to destination) Khai T. Nguyen (Penn State) Conservation laws and traffic flows 18 / 24

19 An optimal decision problem Departure cost: ϕ k (t) arrival cost: ψ k (t) ϕ (t) k ψ (t) k T t ϕ k (t) < 0, ψ k (t) > 0 and lim (ϕ k(t) + ψ k (t)) =. t Cost for a driver β in the k-th group ϕ k (τ d (β)) + ψ k (τ a (β)). Khai T. Nguyen (Penn State) Conservation laws and traffic flows 19 / 24

20 Optima and equilibria t ū k,p (t) = departure rate of k-drivers traveling along the path Γ p A d(1) Aa(1) An admissible family {ū k,p } of departure rates is globally optimal if it minimizes the sum of the total costs of all drivers. An admissible family {uk,p } of departure rates is a Nash equilibrium solution if no driver of any group can lower his own total cost by changing departure time or switching to a different path to reach destination. Khai T. Nguyen (Penn State) Conservation laws and traffic flows 20 / 24

21 Existence results Thanks to the continuity properties of our new model. Theorem (A. Bressan & N-, 2015). On a general network of roads, there exists at least one globally optimal solution {ū k,p }. If an upper bound on the travel time (for all drivers, under all traffic conditions) is available, then there exists at least one Nash equilibrium solution {u k,p }. Proof: By finite dimensional approximations + topological methods No uniqueness, in general Khai T. Nguyen (Penn State) Conservation laws and traffic flows 21 / 24

22 Constructing Nash equilibria Φ k,p u ν k,p t t Piecewise constant (Galerkin) approximations of the departure rates. An approximate equilibrium exists, by the theory of variational inequalities As the mesh size t 0 the weak limit exists, and is a Nash equilibrium, thanks to the continuity properties of our new model. Khai T. Nguyen (Penn State) Conservation laws and traffic flows 22 / 24

23 Some open questions Find conditions that guarantee that traffic will not get stuck, namely: every driver reaches destination within finite time. Find necessary conditions for global optima. Study uniqueness. Determine explicit solutions in some simple cases Uniqueness and stability of Nash equilibrium. Khai T. Nguyen (Penn State) Conservation laws and traffic flows 23 / 24

24 Thank you for your attention Khai T. Nguyen (Penn State) Conservation laws and traffic flows 24 / 24