Mean Field Games problems for linear control system and ergodic behavior of Mean Field Games problems depending on acceleration Cristian Mendico Gran Sasso Science Institute (GSSI), L’Aquila, Italy & Universit´ e Paris Dauphine cristian.mendico@gssi.it ”Two Days Online Workshop on Mean Field Games” Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 1 / 21
Outline of the talk 1. MFG with linear control. 2. Example of MFG with control on the acceleration. 3. Asymptotic behavior of MFG with acceleration. Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 2 / 21
MFG with linear control Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 3 / 21
MFG system Given an Hamiltonian function H defined as H ( x , p , m ) = −〈 p , Ax 〉 + | B ∗ p | 2 − L ( x , − B ∗ p , m ) for any ( x , p , m ) ∈ R d × R d × P α ( R d ), where L is a Tonelli Lagrangian for any fixed m . The MFG system we want to study in the following ( t , x ) ∈ [0 , T ] × R d − ∂ t v ( t , x ) + H ( x , D x v ( t , x )) = F ( x , m t ) , ( t , x ) ∈ [0 , T ] × R d ∂ t m t − div( m t D p H ( x , D x v ( t , x ))) = 0 , x ∈ R d . m 0 = ¯ m , v ( T , x ) = G ( x , m T ) Main issues : H is not strictly convex and not coercive in p . Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 4 / 21
Presentation of the model Control problem This PDE system is associated with the following control problem. Fix T > 0 and let A , B be d × d and d × k real matrices, respectively. Consider the control system defined by γ ( t ) = A γ ( t ) + B γ ( t ) , ˙ t ∈ [0 , T ] where u : [0 , T ] → R k is a summable function. For x ∈ R d , u ∈ L 1 (0 , T ; R k ) and m ∈ C ([0 , T ]; P 1 ( R d )) set �� T � inf L ( γ ( s , x , u ) , u ( s ) , m s ) ds + G ( γ ( T ) , m T ) . u ∈ L 1 (0 , T ; R k ) 0 Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 5 / 21
Notation and hypothesis Let the Lagrangian L : R d × R k × P ( R d ) → R and the terminal costs G : R d × P ( R d ) → R be: 1. ( x , v ) → L ( x , v , m ) Tonelli; 2. m → L ( x , v , m ) continuous w.r.t. the d 1 distance; 3. G ∈ C b ( R d × P ( R d )) Set Γ T = { γ ( · , x , u ) : x ∈ R d , u ∈ L 1 ([0 , T ]; R k ) } Let m 0 ∈ P α ( R d ), let R ≥ [ m 0 ] α and set � � � γ � α P m 0 ( Γ T , R ) = η ∈ P ( Γ T ) : � ˙ 2 η ( d γ ) ≤ R , e 0 �η = m 0 . Γ T We take a particular form of m in the above functional. i.e. we consider m t = e t �η for η ∈ P m 0 ( Γ T , R ). Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 6 / 21
Equilibrium and mild solutions Definition We follow the Lagrangian approach to MFG system (see, for instance, Cannarsa–Capuani (2018) and Mazanti-Santabrogio (2018)). Definition Given m 0 ∈ P α ( R d ), we say that η ∈ P m 0 ( Γ T , R ) is a MFG equilibrium for m 0 if � Γ ∗ supp( η ) ⊂ η ( x ) . x ∈ R d Notation: Γ ∗ η ( x ) denotes the set of minimizing trajectories of the control problem associated with the measure e t �η . Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 7 / 21
Equilibrium and mild solutions Existence of an equilibrium We prove that there exists at least one equilibrium Define the set-valued map � � � � ⇒ E : P m 0 ( Γ T , R ) , d 1 P m 0 ( Γ T , R ) , d 1 such that E ( η ) = { µ ∈ P m 0 ( Γ T , R ) : supp( µ x ) ⊂ Γ ∗ η ( x ) , m 0 − a.e. } . 1. For R ≥ [ m 0 ] α the set-valued map has closed graph; 2. there exists a constant R ( α , [ m 0 ] α ) ≥ 0 such that E ( η ) is non-empty, convex and compact and moreover, E has closed graph. Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 8 / 21
Equilibrium and mild solutions Milds solutions Theorem (P. Cannarsa, M.C.) There exists at least one MFG equilibrium. Definition We say that ( V , m ) ∈ C ([0 , T ] × R d ) × C ([0 , T ]; P α ( R d )) is a mild solution if there exists a MFG equilibrium η such that 1. m t = m η t for any t ∈ [0 , T ] 2. V is the value function associated with the underlying control problem. Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 9 / 21
Weak solution Equivalence between weak solutions and mild solutions Definition ( v , m ) ∈ W 1 , + ∞ ([0 , T ] × R d ) × C ([0 , T ]; P α ( R d )) is a weak solution if: v is a continuous viscosity solution of the HJ-eq., D x v exists m -a.e. and m is a solution in the sense of distributions of the continuity equation. Theorem (P. Cannarsa, M.C.) Let m 0 be absolutely continuous w.r.t. the Lebesgue measure with compactly supported density. Then, ( V , m ) is a mild solutions if and only if it is a weak solution. Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 10 / 21
Example: MFG with acceleration Consider the control dynamics � x ( t ) ˙ = v ( t ) , v ( t ) ˙ = u ( t ) . and consider the Lagrangian of the form 2 | w | 2 + F ( x , v , m ). Then, we obtain the following MFG L ( x , v , w , m ) = 1 system 2 | D v u T | 2 − 〈 D x u T , v 〉 = F ( x , v , m T − ∂ t u T + 1 t ) ∂ t m T t − 〈 v , D x m T t 〉 − div( m T t D v u T )) = 0 u T ( T , x , v ) = g ( x , v , m T T ) , m T 0 = m 0 for ( x , v , t ) ∈ R d × R d × [0 , T ]. Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 11 / 21
Example: MFG with acceleration As a corollary of the above existence result we know that there exists at least one mild solution ( u , m η ) where η is a MFG equilibrium. Moreover, as proved in Y. Achdou, P. Mannucci, C. Marchi and N. Tchou (2020) the unique solution m is absolutely continuous w.r.t. the Lebesgue measure with bounded density and it is the image of m 0 by the flow � x ( t ) ˙ = v ( t ) v ( t ) ˙ = − D v u ( t , x ( t ) , v ( t )) . Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 12 / 21
Asymptotic behavior of MFG with acceleration Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 13 / 21
The problem We recall that the time-dependent MFG reads as − ∂ t u T + 1 2 | D v u T | 2 − 〈 D x u T , v 〉 = F ( x , v , m T t ) ∂ t m T t − 〈 v , D x m T t 〉 − div( m T t D v u T )) = 0 u T ( T , x , v ) = g ( x , v , m T T ) , m T 0 = m 0 where m T : [0 , T ] → P 1 ( T d × R d ) and the terminal costs g belongs to b ( T d × R d ) for any m ∈ P 1 ( T d × R d ) and m �→ g ( x , v , m ) is Lipschitz C 1 continuous w.r.t. the d 1 -distance, uniformly in ( x , v ). Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 14 / 21
It should be... From the recent literature on MFG on this subject (Cardaliaguet (2013) and Cannarsa-Cheng-M.-Wang (2019-2020)), we would expect that the limit of u T / T , as T → + ∞ , is described by the following ergodic system � 2 | D v u | 2 − 〈 D x u , v 〉 = F ( x , v , m ) 1 −〈 v , D x m 〉 − div( mD v u )) = 0 . Even for problems without mean field interaction, we cannot expect to have a solution to the corresponding ergodic Hamilton-Jacobi equation due to the lack of coercivity and due to the lack of small time controllability . Moreover, as the drift of the continuity equation is given in terms of solution to the ergodic Hamilton-Jacobi equation , there is no hope to formulate the problem in this way. Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 15 / 21
References on the context Even for first-order Hamilton-Jacobi equations without mean field interaction, i.e. ∂ t u ( t , x ) + H ( x , D x u ( t , x )) = 0 where H is not coercive in the gradient term, the long time behavior of u has been an open issue since several years. For the analysis of special cases we refer to: G. Barles (2007); P. Cardaliaguet (2010); Y. Giga, Q. Liu, H. Mitake (2012); M. Arisawa, P.L. Lions (1998); O. Alvarez, M. Bardi (2010);Z. Artstein, V. Gaitsgory (2000). Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 16 / 21
Presentation of the model We consider the Lagrangian L : T d × R d × R d × P ( T d × R d ) → R of the form L ( x , v , w , m ) = 1 2 | w | 2 + F ( x , v , m ) where F is such that 1. F is globally continuous; 2. there exists α > 1 and there exists c F ≥ 0 such that 1 | v | α − c F ≤ F ( x , v , m ) ≤ c F (1 + | v | α ); c F 3. there exists C F ≥ 0 such that | D x F ( x , v , m ) | + | D v F ( x , v , m ) | ≤ C F (1 + | v | α ) . Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 17 / 21
New problem and new method Ergodic MFG problem Let P α , 2 ( T d × R d × R d ) be the set of Borel probability measures satisfying � � | w | 2 + | v | α � µ ( dx , dv , dw ) < + ∞ . T d × R d × R d Definition Let η ∈ P α , 2 ( T d × R d × R d ). We say that η is a closed measure if for c ( T d × R d ) the following holds any test function ϕ ∈ C ∞ � � � 〈 D x ϕ ( x , v ) , v 〉 + 〈 D v ϕ ( x , v ) , w 〉 d η ( x , v , w ) = 0 . T d × R d × R d We denote by C the set of closed measures. Cristian Mendico Linear control MFG and ergodic behavior of MFG with acceleration 18 / 21
Recommend
More recommend
