Mean Field Games in the context of crowd motion Y. Achdou (LJLL, - PowerPoint PPT Presentation

Mean Field Games in the context of crowd motion Y. Achdou (LJLL, Universit´ e Paris-Diderot) June, 2019 — CIRM Marseille-Luminy Y. Achdou Mean field games

Models of congestion Outline 1 Models of congestion 2 Numerical simulations 3 Common noise 4 Several populations, segregation 5 Mean field games of control 6 MFG versus mean field type control Y. Achdou Mean field games

Models of congestion Mean field games MFG: Nash equilibria with a continuum of identical agents, interacting via some global information. The dynamics of a representative agent is √ dX t = 2 νdW t + γ t dt where ( W t ) is a d -dimensional Brownian motion (idiosynchratic noise) ( γ t ) is the control of the agent. Individual optimal control problem: the representative agent minimizes 1 �� T � L ( X s , γ s ; m s ) ds + G ( X T ; m T ) , E t,x t where m s is the distribution of states (a single agent is assumed to have no influence on m s ). Dynamic programming yields an optimal feedback γ ∗ t and an optimal trajectory X ∗ t . 2 MFG equilibrium: m t = law of X ∗ t . Y. Achdou Mean field games

Models of congestion Congestion The cost of motion at x depends on m ( x ) in an increasing manner. A typical example was introduced by P-L. Lions (lectures at Coll` ege de France): L ( x, γ ; m ) ∼ ( µ + m ( x )) σ | γ | q ′ + F ( x, m ( x )) where µ ≥ 0, σ > 0 and q ′ > 1. The corresponding Hamiltonian is of the form | p | q H ( x, p ; m ) = q ( µ + m ( x )) α − F ( x, m ( x )) , with α = σ ( q − 1). Remarks Degeneracy of the Hamiltonian H as m → + ∞ Difficulty in empty regions if µ = 0 This model is named “Soft Congestion” by Santambrogio and his coauthors. Their “Hard Congestion” models include inequality contraints on m : m ≤ ¯ m Y. Achdou Mean field games

Models of congestion The system of PDEs | Du | q  − ∂ t u − ν ∆ u + 1 ( m + µ ) α = F ( m ) , ( t, x ) ∈ (0 , T ) × Ω  q        m | Du | q − 2 Du � � (1) ∂ t m − ν ∆ m − div = 0 , ( t, x ) ∈ (0 , T ) × Ω ( m + µ ) α        m (0 , x ) = m 0 ( x ) , u ( T, x ) = G ( x, m ( T )) , x ∈ Ω  + boundary conditions on ∂ Ω... Main assumption: Either µ > 0 (non singular case) or µ = 0 (singular case) Natural growth w.r.t. | Du | : 1 < q ≤ 2 0 < α ≤ 4 q − 1 = 4 q ′ q Y. Achdou Mean field games

Models of congestion The condition α ≤ 4( q − 1) /q is related to uniqueness and stability General MFG with local coupling lead to systems of the form  − ∂ t u − ν ∆ u + H ( x, p, m ) = F ( m ) ( t, x ) ∈ (0 , T ) × Ω ,       ∂ t m − ν ∆ m − div( mH p ( x, p, m )) = 0 , ( t, x ) ∈ (0 , T ) × Ω ,       m (0 , x ) = m 0 ( x ) , u ( T, x ) = G ( m ( T, x )) , x ∈ Ω . P-L. Lions proved that a sufficient condition for the uniqueness of classical solutions is that G be non decreasing, F be strictly increasing and that � − H m ( x, p, m ) 2 mH T 1 � m,p ( x, p, m ) > 0 , 1 2 mH m,p ( x, p, m ) mH p,p ( x, p, m ) for all x ∈ Ω, m > 0 and p ∈ R d . In the present congestion model, this condition ⇔ α ≤ 4( q − 1) /q . Y. Achdou Mean field games

Models of congestion Proof of uniqueness (in the periodic case) (1/2) Consider two solutions of ( ⋆ ): ( u 1 , m 1 ) and ( u 2 , m 2 ): multiply HJB 1 − HJB 2 by m 1 − m 2 : � T � T d ( u 1 − u 2 )( ∂ t m 1 − ∂ t m 2 ) + ν ∇ ( u 1 − u 2 ) · ∇ ( m 1 − m 2 ) 0 � T � � � + H ( x, ∇ u 1 , m 1 ) − H ( x, ∇ u 2 , m 2 ) ( m 1 − m 2 ) 0 T d � T � = T d ( F ( · ; m 1 ) − F ( · ; m 2 ))( m 1 − m 2 ) 0 � � � + G ( · ; m 1 | t = T ) − G ( · ; m 2 | t = T )( m 1 | t = T − m 2 | t = T . T d multiply FP 1 − FP 2 by u 1 − u 2 : � T � 0 = T d ( u 1 − u 2 )( ∂ t m 1 − ∂ t m 2 ) + ν ∇ ( u 1 − u 2 ) · ∇ ( m 1 − m 2 ) 0 � T � � ∂H ∂H � + m 1 ∂p ( x, ∇ u 1 , m 1 ) − m 2 ∂p ( x, ∇ u 2 , m 2 ) · ∇ ( u 1 − u 2 ) . T d 0 Y. Achdou Mean field games

Models of congestion Proof of uniqueness (in the periodic case) (2/2) subtract: � T � T  � � T d E ( m 1 , ∇ u 1 , m 2 , ∇ u 2 ) + T d ( F ( · ; m 1 ) − F ( · ; m 2 ))( m 1 − m 2 )    0 = 0 0 � + T d G ( · ; m 1 | t = T ) − G ( · ; m 2 | t = T ))( m 1 | t = T − m 2 | t = T )    where E ( m 1 , p 1 , m 2 , p 2 ) H ( p 1 , m 1 ) − H ( p 2 , m 2 ) � m 2 − m 1 � � � = · m 2 ∂H ∂p ( p 2 , m 2 ) − m 1 ∂H p 2 − p 1 ∂p ( p 1 , m 1 ) � − H ( p, m ) � m 2 − m 1 � � � m 2 � � m 1 �� � m � � = · Θ − Θ with Θ = . m ∂H p 2 − p 1 p 2 p 1 p ∂p ( p, m ) � mH T � − H m ( p, m ) m,p ( p, m ) / 2 E positive ⇔ Θ monotone ⇔ > 0 , ∀ m > 0 . mH m,p ( p, m ) / 2 mH p,p ( p, m ) If Θ, F and G are monotone, then the 3 terms vanish. If for example F is strictly increasing and G is nondecreasing, we get that u 1 ( t = T ) = u 2 ( t = T ) and m 1 = m 2 . Then, u 1 = u 2 is obtained from uniqueness results for the Bellman equation. Y. Achdou Mean field games

Models of congestion Some references P-L. Lions [ ∼ 2011]: lectures at Coll` ege de France. In particular, the condition for uniqueness of classical solutions. Gomes-Mitake [2015]: existence of classical solutions in a specific stationary case: purely quadratic Hamiltonian, i.e. H ( x, p, m ) = | p | 2 m α , with a special trick Gomes-Voskanyan[2015] and Graber[2015]: short-time existence results of classical solutions for evolutive MFG with congestion More generally, for the existence of classical solutions, restrictive assumptions (e.g. on the growth of F and G ) are needed. Moreover, if H ( x, p, m ) = | p | q m α , one needs to prove that m does not vanish. It seems more feasible to work with weak solutions: (Y.A-Porretta [2018]) Y. Achdou Mean field games

Models of congestion References on weak solutions of the MFG systems Weak solutions = distributional sol. with suitable integrability properties First introduced by Lasry and Lions in 2007 For Hamiltonians with separate dependencies: H ( x, p, m ) = H ( x, p ) − F ( m ), Porretta, [ARMA 2015], showed that weak sol. allow to build a very general well-posed setting: stability results for weak sol. of Fokker-Planck equations uniqueness of weak sol. of the MFG system (Ok but much harder than for classical sol.) Allow to prove general convergence results for numerical schemes [Y.A-Porretta 2016] When the MFG system can be seen as the optimality conditions of an optimal control problem driven by a PDE, weak solutions are the minima of a relaxed primal-dual pb [Cardaliaguet-Graber-Porretta-Tonon 2015] Allow to handle degenerate diffusion [C-G-P-T 2015] Difficulty with the present congestion model: it is not possible to use a variational approach. Yet, the proof of existence relies on the monotonicity assumption. Y. Achdou Mean field games

Models of congestion Weak solutions in the non singular case: µ > 0 Consider the model problem: | Du | q  − ∂ t u − ν ∆ u + 1 ( t, x ) ∈ (0 , T ) × T d , ( m + µ ) α = F ( m ) ,  q   m | Du | q − 2 Du � � ( t, x ) ∈ (0 , T ) × T d , ∂ t m − ν ∆ m − div = 0 , ( m + µ ) α   x ∈ T d . m (0 , x ) = m 0 ( x ) , u ( T, x ) = G ( x, m ( T, x )) ,  m 0 ∈ C ( T d ), F , G bounded from below, and Additional assumptions: λ f ( m ) − κ ≤ F ( t, x, m ) ≤ 1 λ f ( m ) + κ, ∀ m ≥ 0 , for some nondecreasing function f such that s �→ sf ( s ) is convex. Similar assumption on G . We set | p | q H ( x, p, m ) = 1 ( m + µ ) α . q Y. Achdou Mean field games

Models of congestion Weak solutions in the non singular case: µ > 0 Definition A weak solution ( u, m ) is a distributional solution of the boundary value problem s.t. mF ( m ) ∈ L 1 , m T G ( m T ) ∈ L 1 ( T d ) , | Du | q | Du | q ( µ + m ) α ∈ L 1 , ( µ + m ) α ∈ L 1 , m and the energy identity holds � T � � � T d m 0 u (0) dx = T d G ( x, m ( T )) m ( T ) dx + T d F ( t, x, m ) m dxdt 0 � T � + T d m [ H p ( t, x, m, Du ) · Du − H ( t, x, m, Du )] dxdt. 0 Theorem [Y.A- A.Porretta 2018] Under the previous assumptions, (mainly α ≤ 4 /q ′ ), there exists a weak solution. Uniqueness under further monotonicity assumptions on F and G . Y. Achdou Mean field games

Models of congestion Important steps in the proof Use renormalized solutions for the F-P. equation (Porretta 2014) Theorem : Crossed energy inequalities (also used in the proof of existence) Consider ( u, m ) such that mF ( m ) ∈ L 1 , m | t = T G ( m | t = T ) ∈ L 1 ( T d ) , | Du | q | Du | q ( µ + m ) α ∈ L 1 , ( µ + m ) α ∈ L 1 m � � m is a weak sol. of F.P. equation + m | t =0 = m 0 � � u is a distrib. subsol. of the Bellman equation + u | t = T ≤ G ( m t = T ) For any pair (˜ u, ˜ m ) with the same properties as ( u, m ), we have the crossed-integrability: | Du | q u | q | D ˜ ( µ + m ) α ∈ L 1 , m ) α ∈ L 1 , . . . m ˜ m ( µ + ˜ and the energy inequality: � T � � � ˜ m 0 , u (0) � ≤ T d G ( x, m ( T )) ˜ m ( T ) dx + T d F ( t, x, m ) ˜ m dxdt 0 � T � + T d [ ˜ m H p ( t, x, ˜ m, D ˜ u ) · Du − ˜ m H ( t, x, m, Du )] dxdt 0 Y. Achdou Mean field games