Convergence of some deterministic Mean Field Games to aggregation - PowerPoint PPT Presentation

Convergence of some deterministic Mean Field Games to aggregation and flocking models Martino Bardi joint work with Pierre Cardaliaguet Dipartimento di Matematica "Tullio Levi-Civita" Università di Padova "Crowds: models and control" CIRM Marseille, June 3–7, 2019 Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 1 / 26

Plan Connecting MFGs to kinetic models ? 1 ◮ Mean Field Games and their system of PDEs ◮ Agent-based models ◮ Large interest rate limit for stochastic MFG: Bertucci-Lasry-Lions ◮ The setting of Degond-Herty-Liu Convergence of MFGs to nonlocal continuity equations 2 ◮ a) MFG with controlled velocity → aggregation equation ◮ b) MFG with controlled acceleration → kinetic equations of flocking type ◮ Outline of the proof for controlled velocity ◮ Outline of the proof for controlled acceleration Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 2 / 26

1. Mean Field Games PDEs − ∂ u in ( 0 , T ) × R d  ∂ t − ν ∆ u + H ( ∇ u ) = F ( x , m )      ∂ m in ( 0 , T ) × R d (MFE) ∂ t − ν ∆ m − div ( m ∇ H ( ∇ u )) = 0      u ( T , x ) = g ( x ) , m ( 0 , x ) = m o ( x ) , m ( x , t ) = equilibrium distribution of the agents at time t ; u ( x , t ) = value function of the representative agent Data: ν ≥ 0 , H = L ∗ , e.g., H ( p ) = | p | 2 2 , F : R d × P 1 ( R d ) → R = running cost , g = terminal cost, � m o ≥ 0 = initial distribution of the agents, R d m o ( x ) dx = 1. 1st equation is backward H-J-B, 2nd equation is forward K-F-P eq. Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 3 / 26

Control interpretation of the MFE � T u ( x , t ) = inf E [ L ( α ( s )) + F ( y ( s ) , m ( s )) ds + g ( y ( T ))] t over controls α and trajectories of √ dy ( s ) = α ( s ) ds + 2 ν dW ( s ) , y ( t ) = x √ dy ( s ) = −∇ H ( ∇ u ( y ( s ) , s )) ds + 2 ν dW ( s ) = optimal trajectory of the representative agent m ( x , t ) = distribution of particles moving along optimal trajectories In particular, for ν = 0 and H ( p ) = | p | 2 / 2 the dynamics with ˙ optimal feedback is y ( s ) = −∇ u ( y ( s ) , s ) and the KFP equation becomes ∂ m ∂ t − div ( m ∇ u ) = 0 Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 4 / 26

Control interpretation of the MFE � T u ( x , t ) = inf E [ L ( α ( s )) + F ( y ( s ) , m ( s )) ds + g ( y ( T ))] t over controls α and trajectories of √ dy ( s ) = α ( s ) ds + 2 ν dW ( s ) , y ( t ) = x √ dy ( s ) = −∇ H ( ∇ u ( y ( s ) , s )) ds + 2 ν dW ( s ) = optimal trajectory of the representative agent m ( x , t ) = distribution of particles moving along optimal trajectories In particular, for ν = 0 and H ( p ) = | p | 2 / 2 the dynamics with ˙ optimal feedback is y ( s ) = −∇ u ( y ( s ) , s ) and the KFP equation becomes ∂ m ∂ t − div ( m ∇ u ) = 0 Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 4 / 26

Control interpretation of the MFE � T u ( x , t ) = inf E [ L ( α ( s )) + F ( y ( s ) , m ( s )) ds + g ( y ( T ))] t over controls α and trajectories of √ dy ( s ) = α ( s ) ds + 2 ν dW ( s ) , y ( t ) = x √ dy ( s ) = −∇ H ( ∇ u ( y ( s ) , s )) ds + 2 ν dW ( s ) = optimal trajectory of the representative agent m ( x , t ) = distribution of particles moving along optimal trajectories In particular, for ν = 0 and H ( p ) = | p | 2 / 2 the dynamics with ˙ optimal feedback is y ( s ) = −∇ u ( y ( s ) , s ) and the KFP equation becomes ∂ m ∂ t − div ( m ∇ u ) = 0 Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 4 / 26

Agent-based models They typically are nonlocal continuity equations of the form Q : P p ( R d ) → C 1 ( R d , R d ) ∂ t m − div ( m Q [ m ]) = 0 The aggregation equation (Bertozzi, Carrillo, Laurent and many others): � Q [ m ]( x , t ) = ∇ R d k ( x − y ) dm ( y ) k ( x ) = −| x | e − a | x | , a > 0 , ◮ k ( x ) = e −| x | − Fe −| x | / L , 0 < F < 1 , L > 1 ◮ Nonlinear friction equation of granular flows (Toscani et al.): same form with k ( x ) = | x | α /α, α > 0 Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 5 / 26

Agent-based models They typically are nonlocal continuity equations of the form Q : P p ( R d ) → C 1 ( R d , R d ) ∂ t m − div ( m Q [ m ]) = 0 The aggregation equation (Bertozzi, Carrillo, Laurent and many others): � Q [ m ]( x , t ) = ∇ R d k ( x − y ) dm ( y ) k ( x ) = −| x | e − a | x | , a > 0 , ◮ k ( x ) = e −| x | − Fe −| x | / L , 0 < F < 1 , L > 1 ◮ Nonlinear friction equation of granular flows (Toscani et al.): same form with k ( x ) = | x | α /α, α > 0 Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 5 / 26

Models of crowd dynamics (Cristiani-Piccoli-Tosin) � ◮ ∂ t m − div ( m ( v + Q [ m ])) = 0 , v = v ( x ) , Q [ m ] = ∇ R d k ( x − y ) dm ( y ) k = φ ( | x | ) with compact support, φ decreasing for small | x | , then increasing ◮ models with "social forces", or mesoscopic, or kinetic: state variables: position and velocity ( x , v ) ∈ R 2 d in ( 0 , T ) × R 2 d ∂ t m + v · D x m − div v ( mQ [ m ]) = 0 � Q [ m ]( x , v ) = ∇ v R 2 d k ( x − y , v − v ∗ ) m ( y , v ∗ , t ) dydv ∗ Flocking models: as the last one with different k , e.g. | v | 2 Cucker-Smale : k ( x , v ) = ( α + | x | 2 ) β , α > 0 , β ≥ 0 Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 6 / 26

Models of crowd dynamics (Cristiani-Piccoli-Tosin) � ◮ ∂ t m − div ( m ( v + Q [ m ])) = 0 , v = v ( x ) , Q [ m ] = ∇ R d k ( x − y ) dm ( y ) k = φ ( | x | ) with compact support, φ decreasing for small | x | , then increasing ◮ models with "social forces", or mesoscopic, or kinetic: state variables: position and velocity ( x , v ) ∈ R 2 d in ( 0 , T ) × R 2 d ∂ t m + v · D x m − div v ( mQ [ m ]) = 0 � Q [ m ]( x , v ) = ∇ v R 2 d k ( x − y , v − v ∗ ) m ( y , v ∗ , t ) dydv ∗ Flocking models: as the last one with different k , e.g. | v | 2 Cucker-Smale : k ( x , v ) = ( α + | x | 2 ) β , α > 0 , β ≥ 0 Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 6 / 26

Question: connection among MFGs and ABMs? For MFG with dynamics ˙ y = α the equation for the density m is ∂ m ∂ t − div ( m ∇ H ( ∇ u )) = 0 which is a continuity equation with Q [ m ] = ∇ H ( ∇ u ) and u depends on m in a non-local way via the HJB equation, so the dependence is not explicit. For MFG with dynamics ÿ = α the density m solves ∂ t m + v · D x m − div v ( m ∇ v H ( ∇ u )) = 0 which is a kinetic equation with Q [ m ] = ∇ v H ( ∇ u ) and u depends on m via the HJB equation. Q.: can one connect in a rigorous way the classical ABMs to some MFGs ? Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 7 / 26

Stochastic case: Bertucci-Lasry-Lions 2018 − ∂ t u λ + λ u λ − ∆ u λ + Q [ m λ ] · Du λ + | Du λ | 2  = F ( x ) ,    2     (MF0) in R d × R + ∂ t m λ − ∆ m λ − div ( m λ ( Du λ + Q [ m λ ])) = 0       in R d m λ ( 0 ) = m 0 ,  λ = the discount factor in the cost functional associated to the HJB equation = "inter temporal preference parameter that measures the weight of anticipation for a given agent", the dynamics of an agent in the MFG is √ dy ( s ) = ( α ( s ) − Q [ m λ ]) ds + 2 ν dW ( s ) , y ( t ) = x Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 8 / 26

The limit λ → ∞ Theorem (Bertucci-Lasry-Lions) Q : P 1 ( R d ) → Lip ( R d ) , � Q [ m ] � ∞ ≤ C , ∀ m = ⇒ any solution ( u λ , m λ ) of (MF0) is bounded uniformly in λ and for any λ n → ∞ such that m λ n → m the limit m is a solution of the continuity equation ∂ t m − ∆ m − div ( m Q [ m ]) = 0 . So "any" ABM model (with diffusion), defined by Q , has at least one solution that is the limit of the solution of a MFG. Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 9 / 26

The setting of Degond-Herty-Liu 2017 Here MPC = Model Predictive Control We address the horizontal ? = ⇒ ? with a different approach. Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 10 / 26

The control problem for a single agent is � T [ | α | 2 ˙ y = v ( y ) + α, y ( t ) = x , inf + F ( y ( s ) , m ( s ))] ds 2 α ( · ) t MPC approximation: ∆ t | α | 2 � � y ( t + ∆ t ) = x + ∆ t ( v ( x ) + α ) , min + F ( y ( t + ∆ t ) , m ( t )) 2 α Note that the scaling with ∆ t means that the control is cheap. Taking the derivative w.r.t. α we get the optimal control ¯ α if ∆ t [¯ α + DF ( x , m ( t ))] = 0 . This suggests that, for short horizon T and cheap control, the optimal feedback should be approximated by the α ≈ − DF ( x , m ( t )) . ¯ steepest decent of the running cost Martino Bardi (Università di Padova) MFGs and aggregation models Marseille, June 4, 2019 11 / 26