Existence and uniqueness results for the MFG system...Existence/uniqueness of a solution for the MFG...

Existence and uniqueness results for the MFGsystem

P. Cardaliaguet

Paris-Dauphine

Based on Lasry-Lions (2006) and Lions’ lectures at Collège de France

IMA Special Short Course“Mean Field Games",

November 12-13, 2012

P. Cardaliaguet (Paris-Dauphine) MFG 1 / 86

The Mean Field Game systemThe general MFG system takes the form (with unknown(u,m) : [0,T ]× Rd → R2) :

(MFG)

(i) −∂tu − σ2∆u + H(x ,Du,m) = 0

in [0,T ]× Rd

(ii) ∂tm − σ2∆m − div(m DpH(x ,Du,m)) = 0in [0,T ]× Rd

(iii) m(0) = m0, u(x ,T ) = G(x ,m(T )) in Rd

where

σ ∈ R,

H = H(x ,p,m) is a convex Hamiltonian (in p) depending on the densitym,

G = G(x ,m(T )) is a function depending on the position x and thedensity m(T ) at time T .

m0 is a probability density on Rd .


System introduced in

Lasry-Lions ’06, as a finite dimensional reduction of a “master equation",

Huang-Caines-Malhame, ’06

to model large population differential games.


Outline

1 First order MFG equations with nonlocal couplingSemi-concavity properties of HJ equationsThe Kolmogorov equationExistence/uniqueness of a solution for the MFG system

2 First order MFG systems with local couplingThe quasilinear elliptic equationMFG as optimality conditions


First order MFG equations with nonlocal coupling

Outline





Discuss existence and uniqueness for the first order, non local system

(MFG)

(i) −∂tu +

12|Du|2 = F (x ,m(t , ·))

in Rd × (0,T )(ii) ∂tm − div (mDu) = 0

in Rd × (0,T )(iii) m(0) = m0 , u(x ,T ) = uf (x)

where

the first equation is backward in time

the second one is forward in time

the map F is nonlocal and regularizing.



Approach by fixed point

We fix a family (m(t))t∈[0,T ].Let u be the solution of the (backward) HJ equation

(HJ)

(i) −∂tu +

12|Du|2 = F (x ,m(t , ·))

in Rd × (0,T )(iii) u(x ,T ) = uf (x)

and m be the solution of the (forward) Kolmogorov equation

(K )

(ii) ∂tm − div (mDu) = 0

in Rd × (0,T )(iii) m(0) = m0

ProblemProve that the map m→ m has a fixed point.



Difficulties

(HJ) has a Lipschitz continuous solution, but not C1 solutions in general(shocks).

(K) is ill-posed for vector fields Du which are only L∞ (multiple solutions).

A system arising in geometric optics : The (forward-forward) system(i) ∂tu +

12|Du|2 = F (x ,m(t , ·))


in Rd × (0,T )(iii) m(0) = m0 , u(0, x) = u0(x)

was studied by Gosse and James (2002), Ben Moussa and Kossioris (2003))and Strömberg (2007).−→Well-posed of measured-valued solutions.



Existence and stability properties of the (MFG) system rely on :

Semi-concavity estimate for HJ equations

Well-posedness of the Kolmogorov equation for gradient vector fields ofsemi-concave functions

Further simplifying assumption :

m0 is bounded, with bounded support.


First order MFG equations with nonlocal coupling Semi-concavity properties of HJ equations

Outline





Semi-concave functions

DefinitionA map w : Rd → R is semi-concave if there is some C > 0 such that

w(λx + (1− λ)y) ≥ λw(x) + (1− λ)w(y)− Cλ(1− λ)|x − y |2

for any x , y ∈ Rd , λ ∈ [0,1]



Propositionw : Rd → R is semi-concave iff one of the following conditions is satisfied :

1 the map x → w(x)− C2 |x |

2 is concave in Rd ,

2 D2w ≤ C Id in the sense of distributions,

3 〈p − q, x − y〉 ≤ C|x − y |2 for any x , y ∈ Rd , t ∈ [0,T ], p ∈ D+x w(x) and

q ∈ D+x w(y), where D+

x w denotes the super-differential of w withrespect to the x variable, namely

D+x w(x) =

{p ∈ Rd ; lim sup

y→x

w(y)− w(x)− 〈p, y − x〉|y − x |

≤ 0}

In particular, a semi-concave function is locally Lipschitz continuous in Rd .



Lemma [Stability]Let (wn) be a sequence of uniformly semi-concave maps on Rd whichpoint-wisely converge to a map w : Rd → R.

Then the convergence is locally uniform and w is semi-concave.

Moreover, Dwn(x) converge to Dw(x) for a.e. x ∈ Rd .



Semi-concavity and HJ equations

Semi-concavity is the natural framework for HJ equations of the form

(HJ)

{−∂tu + 1

2 |Du|2 = f (t , x) in [0,T ]× Rd

u(T , ·) = g(x)

for data such that supt∈[0,T ]

‖f (·, t)‖C2 ≤ C, ‖g‖C2 ≤ C .

PropositionThe unique viscosity solution u of (HJ) is semi-concave in space withmodulus C = C(C) and given by the representation formula :

u(x , t) = infα∈L2([t,T ],Rd )

∫ T

t

12|α(s)|2 + f (s, x(s))ds + g(x(T )) ,

where x(s) = x +∫ s

t α(τ)dτ .



Lemma (Optimal synthesis)Let (x , t) ∈ Rd × [0,T ) and x(·) be an a.c. solution of

(grad − eq)

{x ′(s) = −Dxu(s, x(s)) a.e. in [t ,T ]x(t) = x

Then the control α := x ′ is optimal for the problem

infα∈L2([t,T ],Rd )

∫ T

t

12|α(s)|2 + f (s, x(s))ds + g(x(T ))

If u(·, t) is differentiable at x, then equation (grad − eq) has a unique solution,corresponding to the optimal trajectory.

In particular, for a.e. x ∈ Rd , equation (grad − eq) has a unique solution.



We denote by Φ = Φ(x , t , s) the flow of (grad − eq) :

(grad − eq)

{∂∂s Φ(x , t , s) = −Dxu(s,Φ(x , t , s)) a.e. in [t ,T ]Φ(x , t , t) = x

Remark : Φ(x , t , s) is defined for a.e. x ∈ Rd .

Lemma

The flow Φ has the semi-group property

Φ(x , t , s′) = Φ(Φ(x , t , s), s, s′) ∀t ≤ s ≤ s′ ≤ T



Lemma (contraction property of the flow)There is a constant C = C(C) such that

|x − y | ≤ C|Φ(x , t , s)− Φ(y , t , s)| ∀0 ≤ t < s ≤ T , ∀x , y ∈ Rd .

In particular the map x → Φ(x , s, t) has a Lipschitz continuous inverse on theset Φ(Rd , t , s).



Proof : Let x(τ) = Φ(x , t , s − τ) and y(τ) = Φ(y , t , s − τ). Then x(·) and y(·)solve

x ′(τ) = Dxu(x(τ), s − τ), y ′(τ) = Dxu(y(τ), s − τ) on [0, s − t)

with initial condition x(0) = Φ(x , t , s) and y(0) = Φ(y , t , s). Hence

ddτ

(12|(x − y)(τ)|2

)= 〈(x ′ − y ′)(τ), (x − y)(τ)〉 ≤ C|(x − y)(τ)|2

since u(τ, ·) is C−semiconcave. Therefore

|x(0)− y(0)| ≥ e−C(s−τ)|(x − y)(τ)| ∀τ ∈ [0, s − t ] ,

which proves the claim. �


First order MFG equations with nonlocal coupling The Kolmogorov equation

Outline





Fix u = u(t , x) the solution of (HJ). Recall that D2u ≤ C and Du ∈ L∞ whereC = C(C).

We study the Kolmogorov equation

(K )

{∂tµ− div (µDu) = 0 in Rd × (0,T )

µ(x ,0) = m0(x) in Rd

Goal : show that (K ) has a unique solution given by the flow Φ.



DefinitionLet µ ∈ L1

loc([0,T ]× Rd ). Then µ is a weak solution to

(K )

{∂tµ− div (µDu) = 0 in Rd × (0,T )

µ(x ,0) = m0(x) in Rd

if, for any ϕ ∈ C∞c ([0,T )× Rd ),∫ T

0

∫Rd

(−∂tϕ+ 〈Du,Dϕ〉)µ dxdt =

∫Rdϕ(0, x)m0(x)dx



Pull-back of a measure : Let µ be a Borel probability measure on Rd andΨ : Rd → Rd be a Borel map. The measure ν := Ψ]µ is defined by on of theequivalent statements :∫

Rdf (x)dν(x) =

∫Rd

f (Ψ(x))dµ(x) for any f ∈ C0b(Rd ).

ν(E) = µ(Ψ−1(E)) for any Borel set E ⊂ Rd .

The Monge-Kantorovitch distance : Let P1(Rd ) be the set of Borelprobability measures m on Rd such that

∫Rd |x |dm(x) < +∞.

The Monge-Kantorovitch distance between measures m,m′ ∈ P1(Rd ) isgiven by

d1(m,m′) = sup{∫

Qh d(m −m′)

}.

where the supremum is taken over the maps h : Rd → R which are1−Lipschitz continuous.



LemmaAny weak solution µ of

(K )

{∂tµ− div (µDu) = 0 in Rd × (0,T )

µ(x ,0) = m0(x) in Rd

such that µ(t) ∈ P1(Rd ) for any t, satisfies

d1(µ(t1), µ(t2)) ≤ ‖Du‖∞|t2 − t1|



Proof : Fix 0 < t1 < t2 ≤ T and let h ∈ C∞c (Rd ) be 1−Lipschitz continuous.Let ε > 0 and

ϕε(t , x) =

(t − t1)h(x)/ε if t ∈ [t1, t1 + ε]h(x) if t ∈ [t1 + ε, t2 − ε](t2 − t)h(x)/ε if t ∈ [t2 − ε, t2]0 otherwise

As ∫ T

0

∫Rd

(−∂tϕε + 〈Du,Dϕε〉)µ =

∫Rdϕε(0, x)m0(x)dx

we have ∫ t1+ε

t1

∫Rd

hµε

+

∫ t2

t1

∫Rd〈Du,Dh〉µ+

∫ t2

t2−ε

∫Rd−hµε≈ 0



Letting ε→ 0 gives for a.e. 0 < t1 < t2 < T :∫Rd

h(µ(t1)− µ(t2)) +

∫ t2

t1

∫Rd〈Du,Dh〉µ = 0 .

So ∫Rd

h(µ(t1)− µ(t2)) ≤ ‖Du‖∞‖Dh‖∞|t2 − t1| .

Take the sup over h :

d1(µ(t1), µ(t2)) ≤ ‖Du‖∞|t2 − t1| .

�



Recall that Φ = Φ(t , x , s) is the flow of (grad − eq) :

(grad − eq)

{∂∂s Φ(t , x , s) = −Dxu(s,Φ(t , x , s)) a.e. in [t ,T ]Φ(t , x , t) = x

Assume that m0 has compact support contained in B(0, C).

LemmaThere is C = C(C) such that, for any s ∈ [0,T ], the measureµ(s) := Φ(·,0, s)]m0

has a support contained in the ball B(0,C),

is absolutely continuous,

and its density µ(s, ·) satisfies ‖µ(s, ·)‖L∞ ≤ C.



Proof

Since ‖Dxu‖∞ ≤ C and Spt(m0) ⊂ B(0, C), the support of (µ(s)) iscontained in B(0,R) where R = C + TC.

Fix s ∈ [0,T ]. From the contraction Lemma, the map x → Φ(x ,0, s) hasa C−Lipschitz continuous inverse Ψ(s, ·) on the set Φ(Rd ,0, s).

If E is a Borel subset of Rd ,

µ(s,E) = m0(Φ−1(·,0, s)(E)) = m0(Ψ(s,E))

≤ ‖m0‖∞Ld (Ψ(s,E)) ≤ ‖m0‖∞CLd (E) .

Therefore µ(s) is a.c. with a density µ(t , ·) satisfying

‖µ(t , ·)‖∞ ≤ ‖m0‖∞C ∀t ∈ [0,T ] .

�



LemmaThe map s → µ(s) := Φ(·,0, s)]m0 is a weak solution of

(K )

{∂tµ− div (µDu) = 0 in Rd × (0,T )

µ(0, x) = m0(x) in Rd



Proof : Let ϕ ∈ C∞c (RN × [0,T )). Then

dds

∫Rdϕ(s, x)µ(s, x)

=dds

∫ N

Rϕ(s,Φ(x ,0, s))m0(x)dx

=

∫Rd

(∂sϕ(s,Φ(x ,0, s)) + 〈Dxϕ(s,Φ(x ,0, s)), ∂sΦ(x ,0, s)〉) m0(x)dx

=

∫Rd

(∂sϕ(s,Φ(x ,0, s))− 〈Dxϕ(s,Φ(x ,0, s)),Dxu(s,Φ(x ,0, s))〉) m0(x)dx

=

∫Rd

(∂sϕ(s, y)− 〈Dxϕ(s, y),Dxu(s, y)〉)µ(s, y)

We conclude by integrating between 0 and T . �



TheoremGiven a solution u to (HJ), the map s → µ(s) := Φ(·,0, s)]m0 is the uniqueweak solution of (K ).

Special case of Di Perna-Lions theory (1989).


First order MFG equations with nonlocal coupling Existence/uniqueness of a solution for the MFG system

Outline





We go back to the MFG system :

(MFG)

(i) −∂tu +

12|Du|2 = F (x ,m(t , ·))


in Rd × (0,T )(iii) m(0) = m0 , u(0, x) = G(x ,m(T ))



Standing assumptions :

1 F and G are continuous over Rd × P1.

2 There is a constant C such that, for any m ∈ P1,

‖F (·,m)‖C2 ≤ C, ‖G(·,m)‖C2 ≤ C ∀m ∈ P1 ,

where C2 is the space of function with continuous second orderderivatives endowed with the norm

‖f‖C2 = supx∈Rd

[|f (x)|+ |Df (x)|+ |D2f (x)|

].

3 Finally we suppose that m0 is absolutely continuous, with a density stilldenoted m0 which is bounded and has a compact support.



DefinitionA pair (u,m) is a solution to the (MFG) system

(MFG)

(i) −∂tu +

12|Du|2 = F (x ,m(t , ·))


in Rd × (0,T )(iii) m(0) = m0 , u(0, x) = G(x ,m(T ))

if u ∈W 1,∞loc ([0,T ]× Rd ) and m ∈ L1

loc([0,T ]× Rd ),

u is a (backward) viscosity solution to (MFG − (i)) with terminalcondition u(0, x) = G(x ,m(T )).

m is a weak solution to (MFG − (ii)) with initial condition m(0) = m0.



A stability property

Let (mn) be a sequence of C([0,T ],P1) which uniformly converges tom ∈ C([0,T ],P1(Rd )).

Let un be the solution to{−∂tun +

12|Dun|2 = F (x ,mn(t)) in Rd × (0,T )

un(x ,T ) = g(x ,mn(T )) in Rd

and u be the solution to{−∂tu +

12|Du|2 = F (x ,m(t)) in Rd × (0,T )

u(x ,T ) = g(x ,m(T )) in Rd

Let Φn (resp. Φ) the flow associated to un (resp. to u) as above and let usset µn(s) = Φn(·,0, s)]m0 and µ(s) = Φ(·,0, s)]m0.



Lemma (Stability)

The solution (un) locally uniformly converges u in Rd × [0,T ] while (µn)converges to µ in C([0,T ],P1(Rd )).



Proof :

As the F (x ,mn(t)) and the g(x ,mn(T )) locally uniformly converge toF (x ,m(t)) and g(x ,m(T )), local uniform convergence of (un) to u is aconsequence of the standard stability of viscosity solutions.

Since the un are uniformly semi-concave, Dun converges almosteverywhere in Rd × (0,T ) to Du.

The (µn(t)) have a uniformly bounded support (say in some compactK ⊂ Rd ) and are uniformly Lipschitz continuous :

d1(µn(t1), µn(t2)) ≤ ‖Dun‖∞|t2 − t1| ≤ C|t2 − t1| .

and uniformly bounded in L∞.

By Ascoli and Banach-Alaoglu, a subsequence (still denoted (µn)) of the(µn) converges in C([0,T ],P1(K )) and in L∞−weak-* to some m whichhas a support in K × [0,T ], belongs to L∞(Rd × [0,T ]) and toC([0,T ],P1(K )).

One can pass to the limit in the Kolmogorov equation to get that µ is theunique solution associated to u : µ(s) = Φ(·,0, s)]m0. �



Theorem (Lasry and Lions, 2006)There is a least one solution to the MFG system

(MFG)

(i) −∂tu +

12|Du|2 = F (x ,m(t , ·))


in Rd × (0,T )(iii) m(0) = m0 , u(T , x) = G(x ,m(T ))



Proof : Let C be the convex subset of maps m ∈ C([0,T ],P1) such thatm(0) = m0. To any m ∈ C one associates the unique solution u to{

−∂tu +12|Du|2 = F (x ,m(t)) in Rd × (0,T )

u(x ,T ) = G(x ,m(T )) in Rd

and to this solution one associates the unique solution to the continuityequation {

∂tm − div (m)Du) = 0 in Rd × (0,T )m(0) = m0

Then m ∈ C and, from the stability Lemma, the mapping m→ m iscontinuous. It is also compact because there is a constant C = C(C) suchthat, for any s ∈ [0,T ], m(s) has a support in B(0,C) and satisfies

d1(m(t1), m(t ′2)) ≤ C|t1 − t2| ∀t1, t ′2 ∈ [0,T ] .

One completes the proof by Schauder fix point Theorem. �



Comments

Semi-concavity properties of value function in optimal control areborrowed from the monograph by Cannarsa and Sinestrari 2004.

For the analysis of transport equations with discontinuous vector fields,see, in particular,

Di Perna-Lions, 1989, for vector fields in Sobolev spaces,Ambrosio, 2004, for vector fields in BV spaces.



Uniqueness

Assume furthermore that F and G satisfy the monotonicity condition :∫Rd

(F (x ,m1)− F (x ,m2))d(m1 −m2)(x) ≥ 0 ∀m1,m2 ∈ P1 ,

and ∫Rd

(G(x ,m1)−G(x ,m2))d(m1 −m2)(x) ≥ 0 ∀m1,m2 ∈ P1 .

TheoremUnder the above conditions, the (MFG) system has a unique solution.



Proof : Let (u1,m1) and (u2,m2) two solutions of (MFG). Set u = u1 − u2 andm = m1 −m2. Then u solves

(1) − ∂t u +12(|Du1|2 − |Du2|2

)− (F (x ,m1)− F (x ,m2)) = 0

while m solves

(2) ∂tm − div (m1Du1 −m2Du2) = 0 .

We compute

ddt

∫Rd

u(t)m(t) =

∫Rd

(∂t u(t))m(t) + u(t)(∂tm(t))

=

∫Rd

(12(|Du1|2 − |Du2|2

)− (F (x ,m1)− F (x ,m2))

)m

−∫Rd〈Du, (m1Du1 −m2Du2)〉

Note that

m2(|Du1|2 − |Du2|2

)− 〈Du,m1Du1 −m2Du2〉 = − (m1 + m2)

2|Du1 − Du2|2



As ∫Rd

(F (x ,m1)− F (x ,m2)) m ≥ 0 ,

we getddt

∫Rd

u(t)m(t) ≤ −∫Rd

(m1 + m2)

2|Du1 − Du2|2

Integrate over [0,T ] :∫Rd

m(T )u(T )−∫Rd

m(0)u(0) ≤ −∫ T

0

∫Rd

(m1 + m2)

2|Du1 − Du2|2

As ∫Rd

m(T )u(T ) =

∫Rd

(G(x ,m1)−G(x ,m2))(m1 −m2) ≥ 0

while m(0) = 0, we get

0 ≤ −∫ T

0

∫Rd

(m1 + m2)

2|Du1 − Du2|2



SoDu1 = Du2 in {m1 > 0} ∪ {m2 > 0}

Hence m1 is a solution of{∂tµ− div (µDu2) = 0 in Rd × (0,T )

µ(x ,0) = m0(x) in Rd

By uniqueness, m1 = m2.

Then u1 = u2 because u1 and u2 solve the same (HJ) equation. �



Conclusion for first order, nonlocal (MFG) systems

The existence/uniqueness theory can be easily generalized to generalHamiltonians,

Stability of solutions can be proved along the same lines,

Other existence proof : by vanishing viscosity,

Numerical approximation of the solution : Camilli and Silva (preprint,2012)


First order MFG systems with local coupling

Outline





The Mean Field Game system

We now study the MFG system

(MFG)

(i) −∂tu + H(x ,Du) = f (x ,m(x , t))(ii) ∂tm − div(mDpH(x ,Du)) = 0(iii) m(0) = m0, u(x ,T ) = uT (x)

where f : Rd × [0,+∞)→ [0,+∞) is a local coupling term.

Problem : Existence/uniqueness of a solutions - smoothness properties.



Similar systems

Adjoint methods for Hamilton-Jacobi PDE : analysis of the vanishingviscosity limit :

∂tuε + H(Duε) = ε∆uε,−∂tmε − div

(DH(Duε)mε

)= ε∆mε,

uε(0, x) = u0(x), mε(1, .) = m1.

−→ better understand the convergence of the vanishing viscositymethod when H is nonconvex.(Evans (2010))



A congestion model : An optimal transport problem related to congestionyields to the following system of PDEs : for α ∈ (0,1),

∂tu +α

4mα−1|Du|2 = 0,

∂tm + div(1

2mαDu

)= 0,

m(0, .) = m0, m(1, .) = m1.

(Benamou-Brenier formulation of Wasserstein distance : α = 1 -Dolbeault-Nazaret-Savaré, 2009)−→ analysis of the system : Existence and uniqueness of a weaksolution(C.-Carlier-Nazaret 2012.)



Back to the MFG system

(MFG)


under the assumptions that

f : Rd × [0,+∞)→ [0,+∞) is a local coupling term, strictly increasing inm with f (x ,0) = 0,

H is uniformly convex in p,

all the data are Zd−periodic in space : denoted by ].



Difficulties for the MFG with local coupling

For nonlocal coupling, existence/regularity of the MFG system comefrom

semi-concavity estimates for HJ equations with C2 RHS,preservation of the absolute continuity of m0 by Kolmogorovequation for gradient flows of semi-concave functions.

For local equations,

the RHS is a priori only measurable (or integrable),the HJ with discontinuous RHS is poorly understood,the Kolmogorov equation with Du ∈ L∞ is ill-posed as well.

−→ requires a completely different approach.



Ideas for the MFG system

Look at the MFG system

as a quasilinear elliptic equation for u in time-space.

or/and as necessary condition for optimal control problems. This is thecase

For an optimal control problem for an Hamilton-Jacobi equationFor an optimal control problem for a Kolmogorov equationThe second one being the dual of the first one.


First order MFG systems with local coupling The quasilinear elliptic equation

Outline





For simplicity of presentation, we assume that H = H(p), f = f (m). The MFGsystem becomes

(MFG)

(i) −∂tu + H(Du) = f (m(x , t))(ii) ∂tm − div(mH ′(Du)) = 0(iii) m(0) = m0, u(x ,T ) = uT (x)

Som(x , t) = f−1 (−∂tu + H(Du))



Therefore

0 = ∂tm − div(mH ′(Du))

= ∂t(f−1 (−∂tu + H(Du))

)− div

((f−1 (−∂tu + H(Du)) H ′(Du)

)= (f−1)′(. . . )

(−∂ttu + 2〈D∂tu,H ′(Du)〉 − 〈D2uH ′(Du),H ′(Du)〉

)−f−1(· · · )Tr

(H ′′(Du)D2u

)= −Tr

(A(Dt,xu)D2

t,xu)

where

A(Dt,xu) = (f−1)′(−∂tu + H(Du))

(1 H ′(Du)T

H ′(Du) H ′(Du)⊗ H ′(Du)

)+f−1(−∂tu + H(Du))

(0 00 H ′′(Du)

)≥ 0

(because (f−1)′ > 0 and f−1(−∂tu + H(Du)) = m ≥ 0)



The MFG systems reduces to the quasilinear elliptic equation−Tr

(A(Dt,xu)D2

t,xu)

= 0 in (0,1)× Td

−∂tu + H(Du) = f (m0) at t = 0

u(T , ·) = uT at t = T



A priori estimates

1 The map Φ(t , x) := −∂tu + H(Du) is bounded above.

Indeed :

Φ satisfies an equation of the form

−Tr(A(Dt,x Φ)− B.∆φ ≤ 0 in (0,1)× Td

So max Φ reached at the boundary,Φ(0, ·) = f (m0) is bounded,Φ(T , x) is bounded (barrier argument).

2 |Du| is bounded (Bernstein method)



Theorem (Lasry-Lions, 2009)

1 Existence/uniqueness of solution for the MFG system with u ∈W 1,∞

and m ∈ L∞,

2 If f (m) ∼ log(m) at 0, then the solution is smooth.


First order MFG systems with local coupling MFG as optimality conditions

Outline





Some assumptionsWe study the MFG system

(MFG)


under the following conditions :

f : Rd × [0,+∞)→ R is smooth and increasing w.r. to m with f (x ,0) = 0,with

−C +1C|m|q−1 ≤ f (x ,m) ≤ C(1 + |m|q−1) (where q > 1) .

There is r > d(q − 1) such that

1C|ξ|r − C ≤ H(x , ξ) ≤ C(|ξ|r + 1) ∀(x , ξ) ∈ Rd × Rd .

+ technical conditions of DxH...



The optimal control of HJ equation

inf

{∫ T

0

∫Q

F ∗(x , α(t , x)) dxdt −∫

Qu(0, x)dm0(x)

}where u is the solution to the HJ equation{

−∂tu + H(x ,Du) = α in (0,T )× Rd

u(T , ·) = uT in Rd

and F ∗(x ,a) = supm∈R

(am − F (x ,m)) where

F (x ,m) =

∫ m

0f (x ,m′)dm′ if m ≥ 0

+∞ otherwise

Note that F ∗(x ,a) = 0 for a ≤ 0.



The optimal control of Kolmogorov equation

inf

{∫ T

0

∫Q

mH∗(x ,−v) + F (x ,m) dxdt +

∫Q

uT (x)m(T , x)dx

}where the infimum is taken over the pairs (m, v) such that

∂tm + div(mv) = 0, m(0) = m0

in the sense of distributions.



Aim :

Provide a framework in which both problems are well-posed and induality,

Write the MFG system as optimality conditions.

Requires estimates on HJ

(HJ)

{−∂tu + H(x ,Du) = α in (0,T )× Rd

u(T , x) = uT (x) in Rd

where α is in some Lp.



Estimates for HJB equations

Assume that p > 1 + d/r and r > 1 such that

1C|ξ|r − C ≤ H(x , ξ) ≤ C(|ξ|r + 1) ∀(x , ξ) ∈ Rd × Rd .

and u be a subsolution to

−∂tu + H(x ,Du) ≤ α in (0,T )× Rd

Lemma (Upper bounds)There is a universal constant C such that

u(t1, x) ≤ u(t2, x) + C(t2 − t1)p−1−d/r

p+d/r ‖(α)+‖p

for any 0 ≤ t1 < t2 ≤ T (where p − 1− d/r > 0 by assumption).



Theorem (Regularity, C.-Silvestre, 2012)Let u be a bounded viscosity solution of{

−∂tu + H(x ,Du) = α in (0,T )× Rd

u(T , x) = uT (x) in Rd

where α ≥ 0, α ∈ Lp with p > 1 + d/r .

Then, for any δ > 0, u is Hölder continuous in [0,T − δ]× Rd :

|u(t , x)− u(s, y)| ≤ C|(t , x)− (s, y)|γ

whereγ = γ(‖u‖∞, ‖α‖p,d , r), C = C(‖u‖∞, ‖α‖p,d , r , δ).



Related results

Capuzzo Dolcetta-Leoni-Porretta (2010), Barles (2010) : stationaryequations, bounded RHS,

C. (2009), Cannarsa-C. (2010), C. Rainer (2011) : evolution equations,bounded RHS,

Dall’Aglio-Porretta (preprint) : stationary setting, unbounded RHS,

C.-Silvestre (2012) : evolution equations, unbounded RHS.



Summary

Let (u, α) solve {−∂tu + H(x ,Du) = α in (0,T )× Rd


with α = α(t , x) ≥ 0. Then

(Upper bound)

u(t1, ·) ≤ u(t2, ·) + C(t2 − t1)p−1−d/r

d+d/r ‖α‖p a.e.

for any 0 ≤ t1 < t2 ≤ T .

(Lower bound) u(t , x) ≥ uT (x)− C(T − t).

(Regularity) u is locally Hölder continuous in [0,T )× Rd with a modulusdepending only on ‖α‖p and ‖uT‖∞.



Back to the optimal control of HJBWe study the optimal control of the HJ equation :

(HJ− Pb) inf

{∫ T

0

∫Q

F ∗(x , α(t , x)) dxdt −∫

Qu(0, x)dm0(x)

}

where u is the solution to the HJ equation{−∂tu + H(x ,Du) = α in (0,T )× Rd


PropositionIf (un, αn) is a minimizing sequence, then

the (αn) are bounded in Lp.

the (un) are uniformly continuous in [0,T ]× Rd .

(Integral bounds) Dun is bounded in Lr] and (∂tun) is bounded in L1

] .



The relaxed problem

Let K be the set (u, α) ∈ BV]((0,T )× Rd )× Lp] ((0,T )× Rd ) such that

Du ∈ Lr]((0,T )× Rd )

α ≥ 0 a.e.

u(T , x) = uT (x) and

−∂tu + H(x ,Du) ≤ α in (0,T )× Rd .

holds in the sense of distribution.

Note that K is a convex set.

The relaxed problem is

(HJ− Rpb) inf(u,α)∈K

{∫ T

0

∫Q

F ∗(x , α(t , x)) dxdt −∫

Qu(0, x)dm0(x)

}



Theorem

The relaxed problem has (HJ-Rpb) at least one minimum (u, α), whereu is continuous and satisfies in the viscosity sense

−∂tu + H(x ,Du) ≥ 0 in (0,T )× Rd

The value of the relaxed problem (HJ-Rpb) is equal to the value of theoptimal control of the HJ equation (HJ-Pb) .



The optimal control problem of HJ equation (HJ-Pb) can be rewritten as

inf

{∫ T

0

∫Q

F ∗(x ,−∂tu + H(x ,Du)) dxdt −∫

Qu(0, x)dm0(x)

}

with constraint u(T , ·) = uT .

This is a convex problem.

−→ Suggests a duality approach.



The Fenchel-Rockafellar duality Theorem

Let

E and F be two normed spaces,

Λ ∈ Lc(E ,F ) and

F : E → R ∪ {+∞} and G : F → R ∪ {+∞} be two lsc proper convexmaps.

TheoremUnder the qualification condition : there is x0 such that F(x0) < +∞ and Gcontinuous at Λ(x0), we have

infx∈E{F(x) + G(Λ(x))} = max

y∗∈F ′{−F∗(Λ∗(y∗))− G∗(−y∗)}

as soon as the LHS is finite.



Remark : If x and y∗ are respectively optimal in

infx∈E{F(x) + G(Λ(x))} and sup

y∗∈F ′{−F∗(Λ∗(y∗))− G∗(−y∗)}

thenF(x) + F∗(Λ∗(y∗)) = 〈Λ∗(y∗), x〉

andG(Λ(x)) + G∗(−y∗) = 〈−y∗,Λ(x)〉

i.e.,Λ∗(y∗) ∈ ∂F(x)

andΛ(x) ∈ ∂G∗(−y∗)



The optimal control problem of HJ equation (HJ-Pb)

inf

{∫ T

0

∫Q

F ∗(x ,−∂tu + H(x ,Du)) dxdt −∫

Qu(0, x)dm0(x)

}

(with constraint u(T , ·) = uT ) can be rewritten as

infu∈E{F(u) + G(Λ(u))}

where

E = C1] ([0,T ]× Rd ) ,

F = C0] ([0,T ]× Rd ,R)× C0

] ([0,T ]× Rd ,Rd ) ,

F(u) = −∫

Qm0(x)u(0, x)dx if u(T , ·) = uT (+∞ otherwise).

G(a,b) =

∫ T

0

∫Q

F ∗(x ,−a(t , x) + H(x ,b(t , x))) dxdt .

Λ(u) = (∂tu,Du).



Then F is convex and lower semi-continuous on E while G is convex andcontinuous on F . Moreover Λ is bounded and linear.

The qualification condition is satisfied by u(t , x) = uT (x).

By Fenchel-Rockafellar duality theorem we have

infu∈E{F(u) + G(Λ(u))} = max

(m,w)∈F ′{−F∗(Λ∗(m,w))− G∗(−(m,w))}

where F ′ is the set of Radon measures (m,w) ∈ M]([0,T ]× Rd ,R× Rd ) andF∗ and G∗ are the convex conjugates of F and G.



Recall that

F(u) = −∫

Qm0(x)u(0, x)dx if u(T , ·) = uT (+∞ otherwise)

and

G(a,b) =

∫ T

0

∫Q

F ∗(x ,−a(t , x) + H(x ,b(t , x))) dxdt .

Lemma

F∗(Λ∗(m,w)) =

∫

QuT (x)dm(T , x) if ∂tm + div(w) = 0, m(0) = m0

+∞ otherwise

and

G∗(m,w) =

∫ T

0

∫Q−F (x ,m)−mH∗(x ,−w

m) dtdx if (m,w) ∈ L1

]

+∞ otherwise



Consequence of the Lemma :

max(m,w)∈F ′

{−F∗(Λ∗(m,w))− G(m,w)}

= max

{∫ T

0

∫Q−F (x ,m)−mH∗(x ,−w

m) dtdx −

∫Q

uT (x)m(T , x) dx

}

where the maximum is taken over the L1] maps (m,w) such that m ≥ 0 a.e.

and∂tm + div(w) = 0, m(0) = m0



Idea of proof :

F∗(Λ∗(m,w)) = supu(T )=uT

〈Λ∗(m,w),u〉 − F(u)

= supu(T )=uT

〈(m,w),Λ(u)〉+

∫Q

m0(x)u(0, x)dx

= supu(T )=uT

∫ T

0

∫Q

(m∂tu + 〈w ,Du〉) +

∫Q

m0(x)u(0, x)dx

= supu(T )=uT

∫ T

0

∫Q−u(−∂tm + div(w))

+

∫Q

m(T )uT +

∫Q

(m0 −m(0))u(0)dx”

=

∫

QuT (x)dm(T , x) if ∂tm + div(w) = 0, m(0) = m0

+∞ otherwise

�



The dual of the optimal control of HJ eqs

TheoremThe dual of the optimal control of HJ (HJ-Pb) equation is given by

(K− Pb) inf

{∫ T

0

∫Q

mH∗(x ,−wm

) + F (x ,m) dxdt +

∫Q

uT (x)m(T , x)dx

}

where the infimum is taken over the pairs(m,w) ∈ L1

]((0,T )× Rd )× L1]((0,T )× Rd ,Rd ) such that

∂tm + div(w) = 0, m(0) = m0

in the sense of distributions. Moreover this dual problem has a uniqueminimum.



(K-Pb) as an optimal control problem for Kolmogorov equation :

Set v = w/m. Then (K-Pb) becomes

inf

{∫ T

0

∫Q

mH∗(x ,−v) + F (x ,m) dxdt +

∫Q

uT (x)m(T , x)dx

}

where the infimum is taken over the pairs (m, v) such that

∂tm + div(mv) = 0, m(0) = m0

in the sense of distributions.



Back to the (MFG) system

We now define weak solutions of the (MFG) system

(MFG)


and explain the relation with the two optimal control problems

for the HJ equation (problem (HJ-Pb))

for the Kolmogorov equation (problem (K-Pb))



DefinitionA pair (m,u) ∈ L1((0,T )× Rd )× BV ((0,T )× Rd ) (with u continuous in[0,T ]× Rd , Du ∈ Lr ((0,T )× Rd ,m), mDpH(x ,Du) ∈ L1

] ) is a weak solution of(MFG) if

(i) −∂tu + H(x ,Du) ≤ f (x ,m) holds in the sense of distribution, withu(T , x) = uT (x) in the sense of trace,

(iii) ∂tm − div(mDpH(x ,Du)) = 0 holds in the sense of distribution in(0,T )× Rd and m(0) = m0,

(iv) Equality∫ T

0

∫Q

m (∂tuac − 〈Du,DpH(x ,Du)〉) =

∫Q

m(T )uT −m0u(0)

holds.

(where ∂tuac is the a.c. part of the measure ∂tu).



Remarks

If (i) holds with an equality and if u is in W 1,1, then (ii) implies (iii).

Conditions (i) and (iii) imply that−∂tuac(t , x) + H(x ,Du(t , x)) = f (x ,m(t , x)) holds a.e. in {m > 0}.



Existence/uniqueness of weak solutions

Theorem (Cannarsa, C., in preparation)There is a unique weak solution (m,u) of (MFG) such that u is locally Höldercontinuous in [0,T )× Rd and which satisfies in the viscosity sense

−∂tu + H(x ,Du) ≥ 0 in (0,T )× Rd .

Idea of proof :

Let (m,w) is a minimizer of (K-Pb) and (u, α) is a minimizer of (HJ-Rpb)such that u is continuous. Then one can show that (m,u) is a solution ofmean field game system (MFG) and w = −mDpH(·,Du) whileα = f (·,m) a.e..

Conversely, any solution of (MFG) such that u is continuous is such thatthe pair (m,−mDpH(·,Du)) is the minimizer of (K-Pb) while (u, f (·,m)) isa minimizer of (HJ-Rpb).



Link with the quasilinear equation

PropositionIf (u,m) is a weak solution of the MFG system, then u is a viscosity solution of

G(x , ∂tφ,Dφ, ∂ttφ,D∂tφ,D2φ

)= 0 in (0,T )× Rd

φ(T , ·) = φT in Rd

−∂tφ+ H(x ,Dφ) = f (m0) in Rd

where

G(x ,pt ,px ,a,b,C)

= −Tr(A(x ,pt ,px )

(a bT

b C

))− F ∗α,α〈Hp,Hx〉 − 〈F ∗x,α,Hp〉 − F ∗αTr(Hx,p)

with

A(x ,pt ,px ) = F ∗α,α

(1 −HT

p−Hp Hp ⊗ Hp

)+ F ∗α

(0 00 Hpp

)



ConclusionSummary

Existence/uniqueness is well understood for first order MFG systemswith nonlocal coupling,

For local coupling, existence of smooth solutions in some cases,otherwise weak solutions.

Open problems

Understand the regularity of solutions for 1rst order, local MFG systemsin full generality.

Existence/uniqueness for the MFG system of congestion type(α ∈ (0,2))

(i) −∂tu +|Du|2

2mα= f (x ,m(x , t))

(ii) ∂tm − div(m1−αDu)) = 0(iii) m(0) = m0, u(x ,T ) = uT (x)

Application to N−player games, periodic solutions...


Existence and uniqueness results for the MFG system...Existence/uniqueness of a solution for the MFG...

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Transcript of Existence and uniqueness results for the MFG system...Existence/uniqueness of a solution for the MFG...