Yves Achdou
email: achdou@ljll-univ-paris-diderot.fr

Paola Mannucci
email: mannucci@math.unipd.it

Claudio Marchi
email: claudio.marchi@unipd.it

Nicoletta Tchou
email: nicoletta.tchou@univ-rennes1.fr

Deterministic mean field games with control on the acceleration and state constraints: extended version

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The theory of mean field games (MFGs for short) is more and more investigated since the pioneering works [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen. II. Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Mean field games[END_REF] of Lasry and Lions: it aims at studying the asymptotic behaviour of differential games (Nash equilibria) as the number of agents tends to infinity. The dynamics of the agents can be either stochastic or deterministic. Concerning the latter case, we refer to [START_REF] Cardaliaguet | Notes on mean field games[END_REF] for a detailed study of deterministic MFGs in which the interactions between the agents are modeled by a nonlocal regularizing operator acting on the distribution of the states of the agents. They are described by a system of PDEs coupling a continuity equation for the density of the distribution of states (forward in time) and a Hamilton-Jacobi (HJ) equation for the optimal value of a representative agent (backward in time). If the interaction cost depends locally on the density of the distribution (hence is not regularizing), then, in the deterministic case, the available theory mostly deals with so-called variational MFGs, see [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF].

The major part of the literature on deterministic mean field games addresses situations when the dynamics of a given agent is strongly controllable: for example, in crowd motion models, this happens if the control of a given agent is its velocity. Under the strong controllability assumption, it is possible to study realistic models in which the agents are constrained to remain in a given region K of the space state, i.e. state constrained deterministic MFGs. An important difficulty in state constrained deterministic MFGs is that nothing prevents the agents from concentrating on the boundary BK of the state space; let us call mptq the distribution of states at time t. Even if mp0q is absolutely continuous, there may exist some t ą 0, such that mptq has a singular part supported on BK and the absolute continuous part of mptq with respect to Lebesgue measure blows up near BK. This was first observed in some applications of MFGs to macroeconomics, see [START_REF] Achdou | Partial differential equation models in macroeconomics[END_REF][START_REF] Achdou | Income and wealth distribution in macroeconomics: A continuous-time approach[END_REF]. From the theoretical viewpoint, the main issue is that, as we have already said, the distribution of states is generally not absolutely continuous with respect to Lebesgue measure; this makes it difficult to characterize the state distribution by means of partial differential equations. These theoretical difficulties have been addressed in [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints[END_REF]: following ideas contained in [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF][START_REF] Benamou | Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations[END_REF][START_REF] Cardaliaguet | First order mean field games with density constraints: pressure equals price[END_REF], the authors of [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints[END_REF] introduce a weak or relaxed notion of equilibrium, which is defined in a Lagrangian setting rather than with PDEs. Because there may be several optimal trajectories starting from a given point in the state space, the solutions of the relaxed MFG are probability measures defined on a set of admissible trajectories. Once the existence of a relaxed equilibrium is ensured, it is then possible to investigate the regularity of solutions and give a meaning to the system of PDEs and the related boundary conditions: this was done in [START_REF] Cannarsa | Mean field games with state constraints: from mild to pointwise solutions of the pde system[END_REF].

On the other hand, if the agents control their acceleration rather than their velocity, the strong controllability property is lost. In [START_REF] Achdou | Deterministic mean field games with control on the acceleration[END_REF], we have studied deterministic mean field games in the whole space R n with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration, see also [START_REF] Cannarsa | Mild and weak solutions of mean field game problems for linear control systems[END_REF]. The state variable is the pair px, vq P R n ˆRn where x and v respectively stand for the position and the velocity. The system of PDEs describing the MFG is then (1.1) $ & % piq ´Bt u ´v ¨Dx u `Hpx, v, D v uq ´F rmptqspx, vq " 0 in R 2n ˆp0, T q piiq B t m `v ¨Dx m ´div v pD pv Hpx, v, D v uqmq " 0 in R 2n ˆp0, T q piiiq mpx, v, 0q " m 0 px, vq, upx, v, T q " GrmpT qspx, vq , on R 2n

where u " upx, v, tq, m " mpx, v, tq, px, vq P R 2n , t P p0, T q and the Hamiltonian px, v, p x , p v q Þ Ñ ´v ¨px `Hpx, v, p v q is neither strictly convex nor coercive with respect to p " pp x , p v q. Hence the available results on the regularity of the value function u of the associated optimal control problem ( [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF][START_REF] Cardaliaguet | Notes on mean field games[END_REF]) and on the existence of a solution of the MFG system ( [START_REF] Cardaliaguet | Notes on mean field games[END_REF]) cannot be applied. In [START_REF] Achdou | Deterministic mean field games with control on the acceleration[END_REF], the existence of a weak solution of the MFG system is proved via a vanishing viscosity method; the distribution of states is characterized as the image of the initial distribution by the flow associated with the optimal control.

In traffic theory and also in economics, the models may require that the position of the agents belong to a given compact subset Ω of R n , and state constrained mean field games with control on the acceleration must be considered. In the present paper, we wish to investigate some examples of such mean field games and address the first step of the program followed by the authors of [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints[END_REF] in the strongly controllable case: we wish to prove the existence of a relaxed mean field equilibrium in the Lagrangian setting under suitable assumptions.

Our program

Most of the paper is devoted to the case when the running cost depends separately on the acceleration and on the other variables, and is quadratic with respect to the acceleration. We will assume that the acceleration can be chosen in the whole space. Thus, admissible trajectories are pairs of functions pξ, ηq, ξ P C 1 pr0, T s; Ωq, η P W 1,2 pr0, T s; R n q and ξ 1 " η.

An example of state constrained mean field games in which the acceleration takes its values in a compact of R n , (the optimal value may therefore take the value `8 in the interior of the x-domain), will be studied in a forthcoming work.

In view of the applications to traffic models, we will deal with the cases when 1. Ω is a bounded domain of R n with a smooth boundary 2. n " 1 and Ω is a bounded straight line segment [START_REF] Achdou | Deterministic mean field games with control on the acceleration[END_REF]. Ω is a bounded polygonal domain of R 2 (that we suppose convex for simplicity).

In the one-dimensional case, the simplicity of the geometry will allow us to obtain accurate information on the optimal trajectories, and in turn to get a more general existence result for the mean field game, yet under an additional assumption on the running cost. On the contrary, when dealing with the polygonal domain, we will make special assumptions in order to obtain an existence result.

Recall that the admissible states are the pairs px, vq P Ω ˆRn , where Ω is a bounded domain of R n . At first glance, we see that some restrictions will have to be imposed on the initial distribution of states: indeed, for x P BΩ and v pointing outward Ω at x, there is no admissible trajectory taking the value px, vq at t " 0; hence the optimal value upx, v, 0q is `8; the definition of the mean field equilibrium would then be unclear if the probability that the initial state takes such values px, vq was not zero.

As in [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints[END_REF], the aim is to prove the existence of relaxed MFG equilibria which are described by probability measures defined on a set of admissible trajectories. The proof involves Kakutani's fixed point theorem, see [START_REF] Glicksberg | A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points[END_REF], applied to a multivalued map defined on a convex and compact set of probability measures on a suitable set of admissible trajectories (itself endowed with the C 1 pr0, T s; R n q ˆC0 pr0, T s; R n q-topology). Difficulties in applying Kakutani's fixed point theorem will arise from the fact that all the optimal trajectories do not form a compact subset of C 1 pr0, T s; R n q ˆC0 pr0, T s; R n q (due to the lack of strong controllability). This explains why we shall need additional assumptions, either on the support of the initial distribution of states, or, in some cases, on the running cost.

Assumptions on the support of the initial distribution of states Note that if a set of trajectories is a compact metric space, then probability measures on this set form a compact set, as required by Kakutani's theorem. Therefore, a natural strategy is to identify a compact set of trajectories which contains the optimal trajectories whose initial value belongs to the support of the initial distribution of states. In such a strategy, we therefore need to identify a modulus of continuity common to all the velocity laws of the optimal trajectories; since the running cost is quadratic in the acceleration, the more natural idea is to look for a uniform bound on the W 1,2 norms of the velocity laws of the optimal trajectories. But, due the lack of strong controllability, if x and v respectively belong to BΩ and to the boundary of the tangent cone to Ω at x (the optimal value upx, v, 0q is finite), there exist sequences px i , v i q iPN tending to px, vq such that the optimal value upx i , v i , 0q blows up when i Ñ 8; in other words, the cost of preventing the trajectories with initial value px i , v i q from exiting the domain tends to `8 as i Ñ 8. Hence, to get uniform bounds on the W 1,2 norms of the velocity law, the support of the initial distribution of states must not contain such sequences px i , v i q. Sufficient conditions on the support of the initial distribution will be given.

Furthermore, Kakutani's fixed point theorem requires a closed graph property for the multivalued map which maps a given point px, vq to the set of optimal trajectories starting from px, vq. An important part of our work is therefore devoted to proving a closed graph property for the latter map. Note that this issue has its own interest in optimal control theory, independently from mean field games.

Assumptions on the running cost

We will see that if n " 1, we will able to get rid of the above-mentioned restrictions on the support of the initial distribution of states, if an additional assumption is made on the running cost, namely that it does not favor the trajectories that exit the domain. The existence of equilibria is then proved by approximating the initial distribution m 0 by a sequence m 0,k for which Kakutani's theorem can be applied, and by passing to the limit. To pass to the limit, accurate information on the optimal trajectories are needed. We managed to obtain them for n " 1 only.

Organization of the paper

The paper is organized as follows: Section 2 is devoted to state constrained optimal control problems in a bounded domain of R n with a smooth boundary, and in particular to the closed graph properties of the above mentioned multivalued map. Although this issue seems to be important in several applications, we were not able to find any relevant result in the available literature. Then, Section 3 deals with an existence result for a related mean field equilibrium in the Lagrangian setting, under sufficient conditions on the support of the initial distribution of states. A variant with a non quadratic cost will be investigated as well. In Section 4, we address the case when the dynamics take place in a bounded straight line segment, (n " 1): under a natural additional assumption on the running cost, we are able to prove the existence of mean field equilibria without any restriction on the initial distribution of states; the proof requires a quite careful study of the optimal trajectories. Finally, in Section 5, we discuss the case of bounded and convex polygonal domain of R 2 and put the stress on the closed graph result which requires a special care near the corners.

2 State constrained optimal control problems in a domain of R n

Setting and notation

Let Ω be a bounded domain of R n with a boundary BΩ of class C 2 . For x P BΩ, let npxq be the unitary vector normal to BΩ pointing outward Ω. We will use the signed distance to BΩ, d : R n Ñ R, dpxq "

" min yPBΩ |x ´y|, if x R Ω, ´min yPBΩ |x ´y|, if x P Ω.
Since BΩ is C 2 , the function d is C 2 near BΩ. In particular, for all x P BΩ, ∇dpxq " npxq. Given a time horizon T and a pair px, vq P Ω ˆRn , we are interested in optimal control problems for which the dynamics is of the form:

(2.1)

$ ' ' &

' ' % ξ 1 psq " ηpsq, s P p0, T q, η 1 psq " αpsq, s P p0, T q, ξp0q " x, ηp0q " v.

The state variable is the pair pξ, ηq and the state space is Ξ " Ω ˆRn . The optimal control problem consists of minimizing the cost (2. It is convenient to define the set of admissible trajectories as follows: (2.4) Γ " " pξ, ηq P C 1 pr0, T s; R n q ˆACpr0, T s; R n q : ˇˇˇξ 1 psq " ηpsq, @s P r0, T s pξpsq, ηpsqq P Ξ, @s P r0, T s * .

For any px, vq P Ξ, set (2.5) Γrx, vs " tpξ, ηq P Γ : ξp0q " x, ηp0q " vu.

Then, Γ opt rx, vs is the set of all pξ, ηq P Γrx, vs such that η P W 1,2 p0, T, R n q and pξ, η, η 1 q achieves the minimum of J in Γrx, vs. Note that Γrx, vs " H if x P BΩ and v points outward Ω. This is the reason why we introduce Ξ ad as follows:

(2.6) Ξ ad " tpx, vq : x P Ω, v ¨npxq ď 0 if x P BΩu Ă Ξ.

Lemma 2.1. For all px, vq P Ξ ad , the optimal value

(2.7) upx, vq " inf pξ,ηqPΓrx,vs
Jpξ, η, η 1 q is finite. The function u is lower semi-continuous on Ξ ad .

Proof. Let us consider px, vq P Ξ ad . We make out two cases:

1. x P Ω or x P BΩ and v ¨npxq ă 0: in this case, for t small enough, the trajectory pξ, ηq defined by # ηpsq " p1 ´s t qv and ξpsq " x `ps ´s2 2t qv, if 0 ď s ď t, ηpsq " 0 and ξpsq "

x `t 2 v, if t ď s ď T,
is admissible and Jpξ, η, η 1 q is finite.

2.

x P BΩ and v ¨npxq " 0. We make a simple observation that will also be used in the proof of Lemma 2.2 below: for all x P BΩ, there exists an open neighborhood V x of x in R n , a positive number R x and a C 2 -diffeomorphism Φ x from V x onto Bp0, R x q such that for all y P V x , the n th coordinate of Φ x pyq is dpyq, i.e. Φ x,n pyq " dpyq.

Hence, Φ x | VxXΩ is a C 2 -diffeomorphism from V x X Ω onto B ´p0, R x q " Bp0, R x q X tx n ă 0u, and Φ x | VxXBΩ is a C 2 -diffeomorphism from V x X BΩ onto Bp0, R x q X tx n " 0u. Let us also call Ψ x the inverse of Φ x , which is a C 2 -diffeomorphism from Bp0, R x q onto V x . Note that (2.8) ∇dpyq " DΦ T x pyqe n , for all y P V x ,
where e n is the n th vector of the canonical basis. In particular, npxq " DΦ T x pxqe n . In the present case, let us set x " Φ x pxq and v " DΦ x pxqv. It is easy to see that xn " 0 and vn " 0. Then, for t small enough, the trajectory pξ, ηq defined by ξpsq " Ψ x p ξpsqq, ηpsq " dξ ds psq " DΨ x p ξpsqqηpsq for all s P r0, T s, with # ηpsq " p1 ´s t qv and ξpsq " x `ps ´s2 2t qv, if 0 ď s ď t, ηpsq " 0 and ξpsq "

x `t 2 v, if t ď s ď T,
is admissible and Jpξ, η, η 1 q is finite.

The lower semi-continuity of u on Ξ ad stems from standard arguments in the calculus of variations. ✷

Closed graph properties

An important feature of the optimal control problem described above is the closed graph property:

Proposition 2.1. Consider a closed subset Θ of Ξ ad . Assume that for all sequences px i , v i q iPN such that for all i P N, px i , v i q P Θ and lim iÑ`8 px i , v i q " px, vq P Θ, the following holds: if x P BΩ, then (2.9)

`pv i ¨∇dpx i qq `˘3 " o `ˇd px i q ˇˇ˘;
then the graph of the multivalued map

Γ opt : Θ Ñ Γ, px, vq Þ Ñ Γ opt rx, vs
is closed, which means: for any sequence py i , w i q iPN such that for all i P N, py i , w i q P Θ with py i , w i q Ñ py, wq as i Ñ 8, consider a sequence pξ i , η i q iPN such that for all i P N, pξ i , η i q P Γ opt ry i , w i s; if pξ i , η i q tends to pξ, ηq uniformly, then pξ, ηq P Γ opt ry, ws.

Remark 2.1. In Proposition 2.1, the condition (2.9) is restrictive only for sequences px i , v i q P Θ which tend to px, vq P Θ such that x P BΩ and v is tangent to BΩ at x. We will see that this assumption makes it possible to control the cost associated to the optimal trajectories starting from px i , v i q.

Remark 2.2. In Section 4.1 below, we will see that in dimension one (Ω is then a bounded straight line), and under stronger assumptions on the running cost, the closed graph properties hold for Θ " Ξ ad .

Remark 2.3. In the context of mean field games, see Section 3 below, the assumptions in Proposition 2.1 will yield sufficient conditions on the support of the initial distribution for the existence of relaxed mean field equilibria.

The proof of Proposition 2.1 relies on Lemmas 2.2 and 2.3 below.

Lemma 2.2. Consider px, vq P Ξ ad , pξ, ηq P Γrx, vs such that η P W 1,2 p0, T ; R n q and a sequence px i , v i q iPN such that for all i P N, px i , v i q P Ξ ad and lim iÑ8 px i , v i q " px, vq.

Assume that one among the following conditions is true:

1. x P Ω 2.
x P BΩ and v ¨npxq ă 0 (hence v i ¨∇dpx i q ă 0 for i large enough)

3. x P BΩ, v ¨npxq " 0 and one among the following properties is true:

(a) for i large enough, v i ¨∇dpx i q ď 0 (b) for i large enough, v i ¨∇dpx i q ą 0 (hence dpx i q ă 0) and

(2.10) lim iÑ8 pv i ¨∇dpx i qq 3 |dpx i q| " 0.
Then there exists a sequence pξ i , η i q iPN such that pξ i , η i q P Γrx i , v i s, η i P W 1,2 p0, T ; R n q, and pξ i , η i q tends to pξ, ηq in W 2,2 p0, T ; R n q ˆW 1,2 p0, T ; R n q, hence uniformly in r0, T s.

Before proving Lemma 2.2, let us define a family of third order polynomials with values in R n : Definition 2.1. Given t ą 0 and x, v, y, w P R n , let Q t,x,v,y,w be the unique third order polynomial with value in R n such that

(2.11) Q t,x,v,y,w p0q " x, Q 1 t,x,v,y,w p0q " v, Q t,x,v,y,w ptq " y, Q 1 t,x,v,y,w ptq " w.
It is given by

(2.12) Q t,x,v,y,w psq " x `vs `ˆ3 y ´x ´vt t 2 ´w ´v t ˙s2 `ˆ´2 y ´x ´vt t 3 `w ´v t 2 ˙s3 .
The first and second derivatives of Q t,x,v,y,w are deduced immediately from (2.12).

Proof of Lemma 2.2. We are going to see that each of the three conditions mentioned in the statement makes it possible to explicitly construct families of admissible trajectories fulfilling all the desired properties (in particular with a finite energy or cost). The more tricky situations will arise when x P BΩ and v i ¨∇dpx i q ą 0 for i large enough, in which case the restrictive condition (2.10) will be needed. Since the construction is different in each of the three cases mentioned in Lemma 2.2, we discuss each case separately.

1. If x P Ω, then there exists t P p0, T s and c ą 0 such that dpξpsqq ă ´c for all s P r0, ts. We construct the sequence pξ i , η i q iPN as follows:

ξ i psq " " ξpsq `Qt,δx i ,δv i ,0,0 psq, if 0 ď s ď t, ξpsq, if t ď s ď T,
where δx i " x i ´x and δv i " v i ´v, see Definition 2.1 for the third order polynomial Q t,δx i ,δv i ,0,0 . It is clear that for i large enough, ξ i psq P Ω for all s P r0, T s, hence pξ i , η i q P Γrx i , v i s and η i P W 1,2 p0, T ; R n q. On the other hand, it can be easily checked that

(2.13) lim iÑ`8 ż T 0 ˇˇˇd η i dt psq ´dη dt psq ˇˇˇ2 ds " 0.
This achieves the proof in the first case.

2. x P BΩ and v ¨npxq ă 0, hence for i large enough, v i ¨∇dpx i q ă 0. We can always assume that the latter property holds for all i.

Notation We use the same geometric arguments as in the proof of Lemma 2.1: for the neighborhood V x mentioned there, there exists T , 0 ă T ď T such that ξpsq P V x X Ω for all s P r0, T s. Consider the local chart Φ x introduced in the proof of Lemma 2.1 and call Ψ x the inverse map, from Bp0, R x q onto V x . For t P r0, T s, let us set ξptq " Φ x pξptqq, ηptq " d ξ dt ptq " DΦ x pξptqq dξ dt ptq, x " Φ x pxq and v " ηp0q " DΦ x pxqv. It is clear that xn " 0 and that vn ă 0. We also set xi " Φ x px i q and vi " DΦ x px i qv i .

Since η P W 1,2 p0, T q, there exists t P p0, T q such that for all s P r0, ts,

3
# Q t i ,x i ,v i , ξpt i q,ηpt i q psq, if s P r0, t i s, ξpsq, if s P rt i , T s,
and ηi psq " d ξi dt psq for s P r0, T s. Then, we define ξ i as follows:

(2.17)

ξ i psq " # Ψ x ´ξ i psq ¯, if s P r0, T s, ξpsq, if s P r T , T s,
and η i " dξ i dt .
Let us first see why pξ i , η i q P Γrx i , v i s for t i small enough and i large enough. A straightforward calculation shows that for s ď t i ,

ξi psq ´x "px i ´xq ˆ1 ´3 s 2 t 2 i `2 s 3 t 3 i ˙`sv i ˆ1 ´2 s t i `s2 t 2 i ṗξ pt i q ´xq ˆ3 s 2 t 2 i ´2 s 3 t 3 i ˙`sηpt i q ˆ´s t i `s2 t 2 i ˙. (2.18)
Let us focus on p ξi psq ´xq ¨en " ξi psq ¨en : from the formula above, we see that ξi psq ¨en is the sum of four terms, the first three of them being nonpositive and the last one nonnegative for all s P r0, t i s. Let us consider the sum of the last three terms, namely:

Apsq " sv i ¨en ˆ1 ´2 s t i `s2 t 2 i ˙`ξ pt i q ¨en ˆ3 s 2 t 2 i ´2 s 3 t 3 i ˙`sηpt i q ¨en ˆ´s t i `s2 t 2 i ˙;
from (2.14) and (2.15), we see that

ξpt i q ¨en ˆ3 s 2 t 2 i ´2 s 3 t 3 i ˙`sηpt i q ¨en ˆ´s t i `s2 t 2 i ˙ďsv ¨en ˆ1 2 ˆ3 s 2 t 2 i ´2 s 3 t 3 i ˙`3 2 ˆ´s t i `s2 t 2 i ˙"sv ¨en ˆ´3 2 s t i `3 s 2 t 2 i ´s3 t 3 i ḟor
all s P r0, t i s. On the other hand, since lim iÑ8 vi " v, we see that for i large enough, vi ¨en ď 3v ¨en {4. Hence,

Apsq ď sv ¨en ˜3 4 ˆ1 ´s t i ˙2 ´3 2 s t i `3 s 2 t 2 i ´s3 t 3 i ¸" sv ¨en ˆ3 4 ´3 s t i `15 4
s 2 t 2 i ´s3 t 3 i ˙.
It is easy to check that the function θ Þ Ñ 3 4 ´3θ `15 4 θ 2 ´θ3 is positive for θ P r0, 1s, which implies that Apsq is negative for s P r0, t i s. Hence, for i large enough, ξi psq ¨en ď 0 for all 0 ď s ď t i ď t. On the other hand, since lim iÑ`8 `|x i ´x| `|v i ´v| ˘" 0, ξ and η are continuous, (2.18) implies that there exists I ą 0 and t P p0, t s such that, if i ě I and t i P p0, tq, then ξi psq P B ´p0, R x q for all s P r0, t i s. Hence, for i ě I and t i P p0, tq, pξ i , η i q P Γrx i , v i s.

Let us now turn to

› › › dη i dt › › › L 2 p0,t i q
: straightforward calculus shows that

dη i dt ptq " DΨ x p ξi ptqq dη i dt ptq `´D 2 Ψ x p ξi ptqqη i ptq ¯η i ptq.
This implies that

(2.19) › › › › dη i dt › › › › 2 L 2 p0,t i q ď C ˜› › › › dη i dt › › › › 2 L 2 p0,t i q `› › ηi › › 4 L 4 p0,t i q ¸,
for a constant C independent of x P BΩ. Hereafter, C may vary from line to line.

First, we focus on

› › › dη i dt › › › 2 L 2 p0,t i q : › › › › dη i dt › › › › 2 L 2 p0,t i q " ż t i 0 ˇˇˇˇ2 ˜3 ξpt i q ´v i t i ´x i t 2 i ´ηpt i q ´v i t i ¸`6s t i ˜´2 ξpt i q ´v i t i ´x i t 2 i `ηpt i q ´v i t i ¸ˇˇˇˇ2 ds ď2I 1 `2I 2 ,
where

I 1 " ż t i 0 ˇˇˇˇ2 ˜3 ξpt i q ´vt i ´x t 2 i ´ηpt i q ´v t i ¸`6s t i ˜´2 ξpt i q ´vt i ´x t 2 i `ηpt i q ´v t i ¸ˇˇˇˇ2 ds and I 2 " ż t i 0 ˇˇˇ2 ˆ3 x ´x i t 2 i `2 v ´v i t i ˙`6s t i ˆ´2 x ´x i t 2 i ´v ´v i t i ˙ˇˇˇ2 ds.
Standard arguments yield that

I 1 ď C › › › › dη dt › › › › 2 L 2 p0,t i q ,
for an absolute constant C ą 0. Therefore, given ǫ ą 0, there exists t : 0 ă t ă t such that 2I 1 ă ǫ{2 for all t i ă t.

On the other hand,

I 2 ď C 1 ˆ|x ´x i | 2 t 3 i `|v ´v i | 2 t i ˙ď C ˆ|x ´xi | 2 t 3 i `|v ´vi | 2 t i ˙.
It is possible to choose the sequence t i such that

• lim iÑ8 t i " 0, • lim iÑ8 |x´x i | 2 t 3 i `|v´v i | 2 t i " 0, • pξ i , η i q P Γrx i , v i s for i large enough. Such a choice of t i yields that lim iÑ8 › › › dη i dt › › › 2 L 2 p0,t i q
" 0. On the other hand, the choice made on t i also implies that lim iÑ8

|x´x i | t i
" 0, and in turn that › › ηi › › L 8 p0,t i q is uniformly bounded with respect to i; therefore, the quantity › › ηi › › 4 L 4 p0,t i q tends to 0 as i tends to 8; using (2.19), we have proved that lim iÑ8

› › › dη i dt › › › L 2 p0,t i q " 0.
Therefore, it is possible to choose a sequence t i ą 0 such that the trajectories pξ i , η i q are admissible for i large enough and lim

iÑ8 › › › › dη i dt ´dη dt › › › › L 2 p0,T q
" 0. This achieves the proof in case 2.

(a)

x P BΩ, v ¨npxq " 0 and v i ¨∇dpx i q ď 0 at least for i large enough. We may assume that v i ¨∇dpx i q ď 0 for all i. Using the same notation as in case 2, we see from (2.8) that vi ¨en " v i ¨DΦ T x px i qe n " v i ¨∇dpx i q ď 0. In the present case, the approximate trajectories will have three successive phases, see Remarks 2.4 and 2.5 for explanations on these different phases. Given t i,1 P p0, T q, we define pŷ i , ŵi q as follows:

(2.20) ŷi " xn e n `πe K n `x i `v i t i,1 ˘, ŵi " vn e n `πe K n `v i ˘" π e K n `v i ˘,
where π e K n stands for the orthogonal projector on e K n " R n´1 ˆt0u, and set ξi psq "

Q t i,1 ,x i ,v i ,ŷ i , ŵi psq and ηi psq " d ξi ds psq for 0 ď s ď t i,1 .
Remark 2.4. In this first phase of the approximate trajectory, i.e. for s P r0, t i,1 s,

π e K n ´Q2 t i,1 ,x i ,v i ,ŷ i , ŵi psq ¯" 0.
The effort only lies in driving the n th -components of ξi psq and ηi psq so that they match those of x and v at s " t i,1 .

As above, we first check that for t i,1 small enough and i large enough, ξi psq P B ´p0, R x q for all s P r0, t i,1 s: from the definition of

Q t i,1 ,x i ,v i ,ŷ i , ŵi , we see that (2.21) ξi psq ¨en " ˆx i ¨en ˆ1 `2 s t i,1 ˙`sv i ¨en ˙ˆ1 ´s t i,1
˙2 is nonpositive for s P r0, t i,1 s. On the other hand, we see that there exist I ą 0 and 0 ă t ď T such that if i ě I and 0 ă t i,1 ă t, then for all s P r0, t i,1 s, ξi psq P B ´p0, R x q.

As in case 2., we need to focus on

› › › dη i dt › › › L 2 p0,t i,1 q
: straightforward calculus shows that

› › › › dη i dt › › › › 2 L 2 p0,t i,1 q ď C ˜|x i ¨en | 2 t 3 i,1 `|pv ´v i q ¨en | 2 t i,1 ¸" C ˜d2 px i q t 3 i,1 `|v i ¨∇dpx i q| 2 t i,1
¸, and we see as above that there exists a sequence t i,1 such that

• lim iÑ8 t i,1 " 0, • lim iÑ8 d 2 px i q t 3 i,1 `|v i ¨∇dpx i q| 2 t i,1
" 0,

• ξi psq P B ´p0, R x q for all 0 ď s ď t i,1 .

Taking the derivative of ξi and arguing as in case 2., we also see that lim

iÑ8 }η i } L 4 p0,t i,1 q " 0, because lim iÑ8 dpx i q t i,1 " 0.
Next, for t i,1 ă t i,2 ă T , we set (2.22) ξi psq "

# Q t i,1 ,x i ,v i ,ŷ i , ŵi psq, if s ď t i,1 , Q t i,2 ´ti,1 ,ŷ i ´x, ŵi ´v,0,0 ps ´ti,1 q `ξps ´ti,1 q, if t i,1 ď s ď t i,2 ,

and

(2.23)

ξ i psq " # Ψ x ´ξ i psq ¯, if s ď t i,2 , ξps ´ti,1 q, if t i,2 ď s ď T.
As above ηi psq " d ξi dt psq for 0 ď s ď t i,2 and η i psq " dξ i dt psq for 0 ď s ď T . Remark 2.5. In the second phase of the approximate trajectory, i.e. for s P rt i,1 , t i,2 s, the components of ξi psq and ξps´t i,1 q parallel to e n coincide, i.e. Q t i,2 ´ti,1 ,ŷ i ´x, ŵi ´v,0,0 pst i,1 q ¨en " 0. The effort only consists of driving the projections of ξi psq and ηi psq on e K n such that they match those of ξps ´ti,1 q and ηps ´ti,1 q at s " t i,2 . We will see that is not necessary to have t i,2 tend to zero, because from the choice of t i,1 , the distance between p ξi pt i,1 q, ηi pt i,1 qq and px, vq tends to 0 as i Ñ `8.

It is possible to choose the sequence t i,2 bounded from below by a positive constant which depends on px, vq but not on i such that ξi psq stays in B ´p0, R x q for s P rt i,1 , t i,2 s. Hence, pξ i , η i q P Γrx i , v i s. Moreover, since t i,2 is bounded away from 0 and lim iÑ8 `|ŷ i ´x| `| ŵi ´v| ˘" 0, it

is not difficult to check that lim iÑ8 › › › › dη i dt ´dη dt › › › › L 2 p0,T q
" 0; this achieves the proof in subcase 3.(a).

(b) x P BΩ, v ¨npxq " 0, v i ¨∇dpx i q ą 0 for all i (or for i large enough), and (2. , for all 0 ď s ď t i,1 .

This happens if and only if

t i,1 ď 3|x i ¨en | vi ¨en " 3|dpx i q| v i ¨∇dpx i q ,
which should be supplemented with the other two conditions as in 3.(a):

lim iÑ8 t i,1 " 0, (2.24) lim iÑ8 |dpx i q| 2 t 3 i,1 `|v i ¨∇dpx i q| 2 t i,1 " 0. (2.25)
If (2.10) holds, then it is possible to choose such a sequence t i,1 . Then, as in 3.(a), it is possible to choose the sequence t i,2 bounded from below by a positive constant independent of i such that pξ i , η i q P Γrx i , v i s; the last part of the proof is identical as in subcase 3.(a). ✷ Lemma 2.3. Consider px, vq P Ξ ad and a sequence px i , v i q iPN such that for all i P N, px i , v i q P Ξ ad and px i , v i q Ñ px, vq as i Ñ 8. Suppose that Assumption 2.1 and one among the three conditions in Lemma 2.2 are satisfied. Let a sequence pξ i , η i q iPN be such that for all i P N, pξ i , η i q P Γ opt rx i , v i s. If pξ i , η i q tends to pξ, ηq uniformly in r0, T s, then η P W 1,2 p0, T ; R n q and pξ, ηq P Γ opt rx, vs.

Proof. We need to prove that for any p r ξ, r ηq P Γrx, vs such that r η P W 1,2 p0, T ; R n q,

(2.26) Jpξ, η, η 1 q ď Jp r ξ, r η, r η 1 q.

From Lemma 2.2 applied to p r ξ, r ηq, there exists a sequence p r ξ i , r η i q iPN , with p r ξ i , r η i q P Γrx i , v i s such that p r ξ i , r η i q Ñ p r ξ, r ηq uniformly on r0, T s as i Ñ 8, and

lim iÑ8 ż T 0 ˇˇˇd r η i dt psq ˇˇˇ2 ds " ż T 0 ˇˇˇd r η dt psq ˇˇˇ2 ds.
On the other hand, the optimality of pξ i , η i q yields that (2.27)

J ˆξi , η i , dη i dt ˙ď J ˆr ξ i , r η i , dr η i dt ˙.
From the properties of p r ξ i , r η i q, the right hand side of (2.27) converges to J ˆr ξ, r η, dr η dt

˙.
The left side (2.27) is thus bounded. Combining this fact with the uniform convergence of pξ i , η i q to pξ, ηq in r0, T s, we obtain that the sequence ş T 0 ˇˇd η i dt psq ˇˇ2 ds is bounded.

This implies that dη i dt á dη dt in L 2 p0, T ; R n q weakly and that lim inf

iÑ8 ż T 0 ˇˇˇd η i dt psq ˇˇˇ2 ds ě ż T 0 ˇˇˇd η dt psq ˇˇˇ2 ds. We deduce that J ˆξ, η, dη dt ˙ď lim inf iÑ8 J ˆξi , η i , dη i dt ˙.
Combining the information obtained above, we obtain (2.26), which achieves the proof. ✷

Proof of Proposition 2.1. Consider py, wq P Θ and a sequence py i , w i q iPN such that for all i P N, py i , w i q P Θ and py i , w i q Ñ py, wq as i Ñ 8. Consider a sequence pξ i , η i q iPN such that for all i P N, pξ i , η i q P Γ opt ry i , w i s and that pξ i , η i q tends to pξ, ηq uniformly. Thanks to the assumption made in the statement of Proposition 2.1, possibly after the extraction of a subsequence, we may suppose that one among the three conditions in Lemma 2.2 holds. Then the conclusion follows from Lemma 2.3. ✷ Proof. Take px, vq P Θ r and a sequence px i , v i q iPN , px i , v i q P Θ r such that lim iÑ8 px i , v i q " px, vq. From Lemma 2.1 we know that upx, vq is finite and from Assumption 2.1, that the infimum in (2.7) is achieved by a trajectory pξ, ηq P Γ opt rx, vs.

Bounds related to optimal trajectories

Possibly after the extraction of a subsequence, we may assume that px i , v i q satisfies one among the three points in Lemma 2.2. Then, there exists a sequence pξ i , η i q iPN such that for all i P N, pξ i , η i q P Γrx i , v i s, η i P W 1,2 p0, T ; R n q, and pξ i , η i q tends to pξ, ηq in W 2,2 p0, T ; R n q ˆW 1,2 p0, T ; R n q, hence uniformly in r0, T s. Hence,

lim iÑ8 J ˆξi , η i , dη i dt ˙" upx, vq.
On the other hand,

J ˆξi , η i , dη i dt ˙ě upx i , v i q.
The latter two observations yield that lim sup

iÑ8 upx i , v i q ď upx, vq.
This proves that u is upper-semi-continuous on Θ r . From Lemma 2.1,

u is continuous on Θ r . Since Θ r is a compact subset of R n ˆRn , u is bounded on Θ r .
Then from the definition of J and u and the boundedness of u on Θ r , it is clear that there exists a constant C " Cpr, M q such that Γ opt rx, vs Ă Γ C for any px, vq P Θ r . ✷

3 A mean field game with control on the acceleration and state constraints

Setting and notation

The bounded domain Ω of R n and the sets Ξ and Ξ ad have been introduced in Section 2.1. Let PpΞq be the set of probability measures on Ξ.

Let C 0 b pΞ; Rq denote the space of bounded and continuous real valued functions defined on Ξ and let F, G : PpΞq Ñ C 0 b pΞ; Rq be bounded and continuous maps (the continuity is with respect to the narrow convergence in PpΞq). Let ℓ be a real valued, continuous and bounded from below function defined on Ξ ˆr0, T s.

Let F rms and Grms denote the images by F and G of m P PpΞq. Set Let Γ be the set of admissible trajectories given by (2.4). It is a metric space with the distance dppξ, ηq, p ξ, ηqq " }ξ ´ξ} C 1 pr0,T s;R n q . Let PpΓq be the set of probability measures on Γ. For t P r0, T s, the evaluation map e t : Γ Ñ Ξ is defined by e t pξ, ηq " pξptq, ηptqq for all pξ, ηq P Γ. For any µ P PpΓq, let the Borel probability measure m µ ptq on Ξ be defined by m µ ptq " e t 7µ. It is possible to prove that if µ P PpΓq, then t Þ Ñ m µ ptq is continuous from r0, T s to PpΞq, for the narrow convergence in PpΞq. Hence, for all pξ, ηq P Γ, t Þ Ñ F rm µ ptqspξptq, ηptqq is continuous and bounded by the constant M in (3.1 Let C " Cpr, M q be the constant appearing in Proposition 2.2 (uniform w.r.t. µ), and Γ C be the compact subset of Γ defined by (2.29); clearly, Γ C is a Radon metric space. From Prokhorov theorem, see [5, Theorem 5.1.3], the set PpΓ C q is compact for the narrow convergence of measures.

Let P m 0 pΓq, (resp. P m 0 pΓ C q) denote the set of probability measures µ on Γ (resp. Γ C ) such that e 0 7µ " m 0 .

Hereafter, we identify PpΓ C q with a subset of PpΓq by extending µ P PpΓ C q by 0 outside Γ C . Similarly, we may consider P m 0 pΓ C q as a subset of P m 0 pΓq.

Note that for all µ P PpΓ C q and for all t P r0, T s, m µ ptq is supported in K C , where K C is defined in (2.28). Remark 3.2. Note that Γ C (endowed with the metric of the C 1 ˆC0 -convergence of pξ, ηq) is a Polish space (because it is compact). The multivalued map r Γ opt related for instance to F " 0 and G " 0 maps Θ r to non empty and closed subsets of Γ C (the closedness can be checked by usual arguments of the calculus of variations). Since the graph of r Γ opt is closed, r Γ opt is measurable. Therefore, there exists a measurable selection j : Θ r Ñ Γ C from Kuratowski and Ryll-Nardzewski theorem, [START_REF] Kuratowski | A general theorem on selectors[END_REF]. Then j7m 0 belongs to P m 0 pΓ C q. The set P m 0 pΓ C q is not empty.

Existence of a mean field game equilibrium

Lemma 3.1. Let a sequence of probability measures pµ i q iPN , µ i P PpΓq, be narrowly convergent to µ P PpΓq. For all t P r0, T s, pm µ i ptqq iPN is narrowly convergent to m µ ptq.

Proof. For all

f P C 0 b pΞ; Rq, ż Ξ f px, vqdm µ i ptqpx, vq " ż Γ f pξptq, ηptqqdµ i pξ, ηq Ñ ż Γ f pξptq, ηptqqdµpξ, ηq " ż Ξ f px, vqdm µ ptqpx, vq.

✷

An easy consequence of Lemma 3.1 is that for C " Cpr, M q as in Proposition 2.2, P m 0 pΓ C q is a closed subset of PpΓ C q, and is therefore compact. Lemma 3.2. If µ P PpΓ C q, the map t Þ Ñ m µ ptq is 1{2-Hölder continuous from r0, T s to PpK C q, (K C is defined in (2.28) and PpK C q is endowed with the Kantorovitch-Rubinstein distance).

Proof. Let φ be any Lipschitz function defined on K C with a Lipschitz constant not larger than 1. ż

K C φpx, vq pdm µ pt 2 qpx, vq ´dm µ pt 1 qpx, vqq " ż Γ C pφpξpt 2 q, ηpt 2 qq ´φpξpt 1 q, ηpt 1 qqq dµpξ, ηq ď ż Γ C p|ξpt 2 q ´ξpt 1 q| `|ηpt 2 q ´ηpt 1 q|q dµpξ, ηq ď C ż Γ C ´|t 2 ´t1 | `|t 2 ´t1 | 1 2 ¯dµpξ, ηq ď C |t 2 ´t1 | 1 2 ,
for a constant C which depends only on C and T . ✷

It is useful to recall the disintegration theorem:

Theorem 3.1. Let X and Y be Radon metric spaces, π : X Ñ Y be a Borel map, µ be a probability measure on X. Set ν " π7µ. There exists a ν-almost everywhere uniquely defined Borel measurable family of probability measures pµ y q yPY on X such that

(3.3) µ y pXzπ ´1pyqq " 0, for ν-almost all y P Y,
and for every Borel function f : X Ñ r0, `8s,

ż X f pxqdµpxq " ż Y ˆżX f pxqdµ y pxq ˙dνpyq " ż Y ˜żπ ´1pyq f pxqdµ y pxq ¸dνpyq. (3.4)
It is possible to apply Theorem 3.1 with X " Γ C , Y " Θ r , π " e 0 and ν " m 0 (identifying m 0 and its restriction to Θ r ): for any µ P P m 0 pΓ C q, there exists a m 0 -almost everywhere uniquely defined Borel measurable family of probability measures pµ px,vq q px,vqPΘr on Γ C such that (3.5) µ px,vq pΓ C ze ´1 0 px, vqq " 0, for m 0 -almost all px, vq P Θ r , and for every Borel function f : Γ C Ñ r0, `8s, ż

Γ C f pξ, ηqdµpξ, ηq " ż Θr ˆżΓ C f pξ, ηqdµ px,vq pξ, ηq ˙dm 0 px, vq " ż
Θr ˜że ´1 0 px,vq f pξ, ηqdµ px,vq pξ, ηq ¸dm 0 px, vq.

(3.6)

For px, vq P Θ r , m 0 supported in Θ r and µ P P m 0 pΓ C q (where C " Cpr, M q is the constant appearing in Proposition 2.2), let us set (3.7) Γ µ,opt rx, vs " # pξ, ηq P Γrx, vs : J µ pξ, ηq " min p r ξ,r ηqPΓrx,vs

J µ p r ξ, r ηq
+ .
Standard arguments from the calculus of variations yield that for each µ P P m 0 pΓ C q and px, vq P Ξ ad , Γ µ,opt rx, vs is not empty. Moreover, from Proposition 2.2, Γ µ,opt rx, vs Ă Γ C for all px, vq P Θ r .

Proposition 3.1. Under the assumptions made on ℓ, F and G in paragraph 3.1, and Assumption 3.1, let C " Cpr, M q be chosen as in Proposition 2.2. Let a sequence of probability measures pµ i q iPN , µ i P P m 0 pΓ C q, be narrowly convergent to µ P PpΓ C q. Let px i , v i q iPN , px i , v i q P Θ r , converge to px, vq. Consider a sequence pξ i , η i q iPN such that for all i P N, pξ i , η i q P Γ µ i ,opt rx i , v i s. If pξ i , η i q iPN tends to pξ, ηq uniformly, then pξ, ηq P Γ µ,opt rx, vs. In other words, the multivalued map px, v, µq Þ Ñ Γ µ,opt rx, vs has closed graph.

Proof. First, from Lemma 3.1, µ P P m 0 pΓ C q and for all t P r0, T s, pm µ i ptqq iPN is narrowly convergent to m µ ptq. From the continuity assumptions made on F and G and the dominated convergence theorem, we deduce that

ż T 0 F rm µ i ptqspξ i ptq, η i ptqqdt Ñ ż T 0 F rm µ ptqspξptq, ηptqqdt,
Grm µ i pT qspξ i pT q, η i pT qq Ñ Grm µ pT qspξpT q, ηpT qq.

The last part of the proof is completely similar to the proof of Proposition Proof. The proof follows that of Cannarsa and Capuani in [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints[END_REF]. Define the multivalued map E from P m 0 pΓ C q to P m 0 pΓ C q as follows: for any µ P P m 0 pΓ C q, (3.9) Epµq " μ P P m 0 pΓ C q : supppμ px,vq q Ă Γ µ,opt rx, vs for m 0 -almost all px, vq P Ξ ( , where pμ px,vq q px,vqPΞ is the m 0 -almost everywhere uniquely defined Borel measurable family of probability measures which disintegrates μ, see the lines after Theorem 3.1.

Then the measure µ P P m 0 pΓ C q is a constrained mean field game equilibrium if and only if µ P Epµq. This leads us to apply Kakutani fixed point theorem to the multivalued map E, see [START_REF] Aliprantis | Infinite dimensional analysis. A hitchhiker's guide[END_REF][START_REF] Glicksberg | A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points[END_REF]. Several steps are needed in order to check that the assumptions of Kakutani theorem are satisfied. First of all, we recall that P m 0 pΓ C q is compact.

Step 1: For any µ P P m 0 pΓ C q, Epµq is a non empty convex set. First, we have already seen that Γ µ,opt rx, vs " H and that the map px, vq Þ Ñ Γ µ,opt rx, vs has closed graph. Therefore, from [START_REF] Aubin | Set-valued analysis, volume 2 of Systems & Control: Foundations & Applications[END_REF], px, vq Þ Ñ Γ µ,opt rx, vs has a Borel measurable selection px, vq Þ Ñ pξ µ px,vq , η µ px,vq q. The measure μ defined by μpBq " ż Θr δ pξ µ px,vq ,η µ px,vq q pBqdm 0 px, vq, for all Borel subset B of Γ C , belongs to Epµq; indeed, the total mass of μ is one because m 0 is supported in Θ r and C " Cpr, M q as in Proposition 2.2 so Epµq is non empty. Second, take µ 1 , µ 2 in Epµq and λ P r0, 1s. We wish to prove that λµ 1 `p1 ´λqµ 2 P Epµq.

It is clear that λµ 1 `p1 ´λqµ 2 belongs to P m 0 pΓ C q. On the other hand, since µ 1 belongs to Epµq, there exist a m 0 -almost everywhere uniquely defined Borel measurable family pμ 1 px,vq q px,vqPΘr of probability measures which disintegrates µ 1 and a subset A 1 of Θ r such that m 0 pA 1 q " 0 and supppµ 1 px,vq q Ă Γ µ,opt rx, vs for all px, vq P Θ r zA 1 . Similarly, µ 2 can be disintegrated into a m 0 -almost everywhere uniquely defined Borel measurable family pμ 2 px,vq q px,vqPΘr of probability measures, and there exists a subset A 2 of Θ r such that m 0 pA 2 q " 0 and supppµ 2 px,vq q Ă Γ µ,opt rx, vs for all px, vq P Θ r zA 2 . Therefore, λµ 1 `p1 λqµ 2 can be disintegrated as follows: for each Borel function f defined on Γ C , ż and m 0 pA 1 Y A 2 q " 0. Hence, λµ 1 `p1 ´λqµ 2 P Epµq, so Epµq is convex.

Γ C f pξ, ηqd `λµ 1 `
Step 2: The multivalued map E has closed graph. Consider a sequence pµ i q iPN , µ i P P m 0 pΓ C q narrowly convergent to µ P P m 0 pΓ C q. Let a sequence pμ i q iPN , μi P Epµ i q be narrowly convergent to μ P P m 0 pΓ C q. We claim that μ P Epµq.

First, there exists a m 0 -almost everywhere uniquely defined Borel measurable family of probability measures pμ px,vq q px,vq on Γ C such that (3.5) and (3.6) hold for μ and μpx,vq .

In particular, there exists a subset A of Θ r with m 0 pAq " 0 such that for px, vq P Θ r zA, μpx,vq pΓ C ze ´1 0 px, vqq " 0. Take px, vq P Θ r zA and p ξ, ηq P supppμ px,vq q. The Kuratowski convergence theorem applied to pμ i q i , μ, see [START_REF] Billingsley | Convergence of probability measures[END_REF], implies that there exists a sequence p ξi , ηi q iPN , p ξi , ηi q P supppμ i q, which converges to p ξ, ηq uniformly in r0, T s. Set px i , v i q " p ξi p0q, ηi p0qq P Θ r . Since μi P Epµ i q, there holds that p ξi , ηi q P Γ µ i ,opt rx i , v i s.

From Proposition 3.1, we see that p ξ, ηq P Γ µ,opt rx, vs. Since px, vq is any point in Θ r zA, this implies that μ P Epµq.

All the assumptions of Kakutani theorem are satisfied: hence, there exists µ P P m 0 pΓ C q such that µ P Epµq. This achieves the proof. [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints[END_REF], the mild solution is unique.

✷

Non quadratic running costs

It is possible to generalize the results of Sections 2 and 3 to costs of the form (3.10) Jpξ, η, αq " ż T 0 ˆℓpξpsq, ηpsq, sq `1 p |α| p psq ˙ds `gpξpT q, ηpT qq, where 1 ă p, for dynamics given by (2.1) and staying in Ξ.

For brevity, we restrict ourselves to the closed graph result, whose proof is completely similar to that of Proposition 2.1. The generalization of Theorem 3.2 is then possible.

Proposition 3.2. Consider a closed subset Θ of Ξ ad . Assume that all sequence px i , v i q iPN such that for all i P N, px i , v i q P Θ and lim iÑ`8 px i , v i q " px, vq P Θ, the following holds: if x P BΩ, then

(3.11) ppv i ¨∇dpx i qq `q2p´1 " o ´ˇd px i q ˇˇp´1 ¯;
then the graph of the multivalued map Γ opt : Θ Ñ Γ, px, vq Þ Ñ Γ opt rx, vs is closed in the sense given in Proposition 2.1.

One dimensional problems: more accurate results

In dimension one and for a running cost quadratic in α, it is possible to obtain more accurate results under a slightly stronger assumption on the running cost, namely that it does not favor the trajectories which exit the domain. In particular, the closed graph property can be proved to hold on the whole set Ξ ad , and concerning mean field games, no assumptions are needed on the support of m 0 by contrast with Theorem 3.2.

Optimal control problem in an interval: a closed graph property

In this paragraph, we set Ω " p´1, 0q and Ξ " r´1, 0s ˆR. The optimal control problem consists of minimizing Jpξ, η, η 1 q given by (2.2) on the dynamics given by (2.1) and staying in Ξ. The definition of Ξ ad is then modified as follows:

(4.1) Ξ ad " Ξz ´t0u ˆp0, `8q Y t´1u ˆp´8, 0q ¯.

We make the following assumptions: An interpretation of Assumption 4.2 is that the running cost ℓ penalizes (or at least does not favor) the trajectories that exit Ξ ad . In that respect, Assumption 4.2 is rather natural.

For px, vq P Ξ, let Γ, Γrx, vs and Γ opt rx, vs be defined as follows:

Γ " " pξ, ηq P C 1 pr0, T s; Rq ˆACpr0, T s; Rq : ˇˇˇξ 1 psq " ηpsq, @s P r0, T s, pξpsq, ηpsqq P Ξ, @s P r0, T s * , Γrx, vs " tpξ, ηq P Γ : ξp0q " x, ηp0q " vu, Γ opt rx, vs " argmin pξ,ηqPΓrx,vs Jpξ, η, η 1 q.

Theorem 4.1. Under Assumptions 4.1 and 4.2, the graph of the multivalued map Γ opt : Ξ ad Ñ Γ, px, vq Þ Ñ Γ opt rx, vs, is closed, in the following sense: consider a sequence px i , v i q iPN , px i , v i q P Ξ ad , such that lim iÑ8 px i , v i q " px, vq P Ξ ad . Consider a sequence pξ i , η i q iPN such that for all i P N, pξ i , η i q P Γ opt rx i , v i s. If pξ i , η i q tends to pξ, ηq uniformly, then pξ, ηq P Γ opt rx, vs. Note also that the result stated in Theorem 4.1, namely the closed graph property of the multivalued map Γ opt , is obtained despite the fact that the value function of the optimal control problem is not continuous and not locally bounded on Ξ ad . This may seem surprising at first glance. Besides, the fact that the value function is singular at some points of Ξ ad will be an important difficulty in the proofs.

The proof of Theorem 4.1 relies on several lemmas. Lemma 4.1. Consider px, vq P Ξ ad , pξ, ηq P Γrx, vs such that η P W 1,2 p0, T ; Rq and a sequence px i , v i q iPN such that for all i P N, px i , v i q P Ξ ad and px i , v i q Ñ px, vq as i Ñ 8.

If one among the following assumptions is satisfied,

1. x P Ω 2.
x " 0, v ď 0 and for all integer i, v i ď 0 3. px, vq " p0, 0q, v i ą 0 for all integer i and lim

iÑ0 pv i q 3 |x i | " 0 4.
x " ´1, v ě 0 and for all integer i, v i ě 0 5. px, vq " p´1, 0q, v i ă 0 for all integer i and lim

iÑ0 |v i | 3 |x i `1| " 0
then there exists a sequence pξ i , η i q iPN such that for all i P N, pξ i , η i q P Γrx i , v i s, η i P W 1,2 p0, T ; Rq, and pξ i , η i q tends to pξ, ηq in W 2,2 p0, T ; Rq ˆW 1,2 p0, T ; Rq, hence uniformly in r0, T s.

Proof. Lemma 4.1 is the counterpart of Lemma 2.2 . The proof is quite similar, so we skip it for brevity. ✷ Corollary 4.1. Consider px, vq P Ξ ad and a sequence px i , v i q iPN such that for all i P N, px i , v i q P Ξ ad and px i , v i q Ñ px, vq as i Ñ 8. Suppose that Assumption 4.1 and one among the five conditions in Lemma 4.1 are satisfied. Let a sequence pξ i , η i q iPN be such that for all i P N, pξ i , η i q P Γ opt rx i , v i s. If pξ i , η i q tends to pξ, ηq uniformly in r0, T s, then η P W 1,2 p0, T ; Rq and pξ, ηq P Γ opt rx, vs.

Proof. Corollary 4.1 is the counterpart of Lemma 2.3. The proof is identical. ✷

Consider px, vq P Ξ ad and a sequence px i , v i q iPN such that for all i P N, px i , v i q P Ξ ad and px i , v i q Ñ px, vq as i Ñ 8. Because it is always possible to extract subsequences, we can say that the only cases that have not yet been addressed in Lemma 4.1 are the following:

(4.3)

#

px, vq " p0, 0q, v i ą 0, and there exists a constant C ą 0 s. t. for all i P N,

pv i q 3 |x i | ě C, or (4.4)
# px, vq " p´1, 0q, v i ă 0, and there exists a constant C ą 0 s. t. for all i P N,

|v i | 3 |x i `1| ě C.
Since the two cases are symmetrical, we may concentrate on (4.3). It is clear that (4.3) implies that |x i |{v i Ñ 0 as i Ñ `8, because v i Ñ 0. In the case when (4.3) is satisfied, we need two technical lemmas which provide a lower bound for the cost ş T 0 ˇˇd η i dt psq ˇˇ2 ds of the admissible trajectories starting at px i , v i q:

Lemma 4.2. Consider px, vq P Ξ ad such that x ă 0, v ą 0, 3|x|{v ă T , and θ P p0, T q. Given a real number w P r0, |x|{T s, set ; we have obtained the first line in (4.7). Note that if θ ă 2|x| v`w , then x `ż θ 0 ηpsqds ă 0.

(4.5) K θ,w " $ ' ' & ' ' % η P W 1,2
✷ Lemma 4.3. Consider a sequence px i , v i q iPN such that x i ă 0, v i ą 0 for all i P N, and v i Ñ 0, |x i |{v i Ñ 0 as i Ñ `8. Call I i pθ, wq the quantity given by (4.6) for v " v i , x " x i and w P " 0, x i {T ‰ . Then

(4.14) inf I i pθ, wq, θ P p0, T q ( " 2 9 pv i q 3 |x i | `op1q,
where op1q is a quantity that tends to 0 as i tends to infinity (which is in fact of the order of pv i q 2 or smaller).

Proof. Recall that I i pθ, wq is given by (4.7). It is easy to see that θ Þ Ñ I i pθ, wq is decreasing on `0, 2|x i |{pv i `wq ‰ and increasing on "

3|x i |{pv i `2wq, T ‰ . In " 2|x i |{pv i `wq, 3|x i |{pv i `2wq
‰ , I i pθ, wq " P p1{θq, where P is the third order polynomial:

P pzq " 6px i q 2 z 3 `6x i pv i `wqz 2 `2ppv i q 2 `vi w `w2 qz.

The roots of the second order polynomial P 1 pzq " 18px i q 2 z 2 `12x i pv i `wqz`2ppv i q 2 `vi wẁ

2 q are v i `w ˘?v i w 3|x i | . Hence, θ Þ Ñ I i pθ, wq is decreasing in " 2|x i | v i `w , 3|x i | v i `w `?v i w  and increasing in " 3|x i | v i `w `?v i w , 3|x i | v i `2w  .
Therefore, the minimizer of θ Þ Ñ I i pθ, wq on r0, T q is θ "

3|x i | v i `w`?v i w
and the minimal value is

P ´vi `w`?v i w 3|x i | " 2pv i q 3 9|x i | ˆ1 `c w v i `w v i ˙3 ´2pv i q 3 3|x i | ˆ1 `c w v i `w v i ˙2 `2pv i q 3 3|x i | ˆ1 `c w v i `w v i Ȯ ´pv i q 2 w |x i | " 2pv i q 3 9|x i | `O ˆpv i q 2 w |x i | ˙.

✷

The next lemma is the counterpart of Lemma 4.1 when (4.3) holds. By contrast with the situations considered so far, Assumption 4.2 is used. Lemma 4.4. Under Assumptions 4.1 and 4.2, consider a sequence px i , v i q iPN which tends to px, vq " p0, 0q as i Ñ 8, and which satisfies (4.3). Let a sequence pξ i , η i q iPN be such that for all i P N, pξ i , η i q P Γ opt rx i , v i s. If pξ i , η i q tends to pξ, ηq uniformly in r0, T s, then η P W 1,2 p0, T ; Rq and pξ, ηq P Γ opt rx, vs.

Proof. The proof is more difficult than that of Lemma 4.1 because we will see that in general, the sequence upx i , v i q does not converge to up0, 0q as i Ñ 8, and that ż T 0 ˆdη i dt psq ˙2 ds may tend to `8.

Step 1 : We start by building a particular competitor for the optimal control problem at px i , v i q. It will be used in Steps 2 and 3 below. Let us set r t i " 3|x i |{v i (observe that lim iÑ8 r t i " 0 since v i Ñ 0 and pv i q 3 {|x i | ě C ą 0). As in the proof of Lemma 4.2 with w " 0, we construct a pair of continuous functions p r ξ i , r η i q defined on r0, r t i s such that ´1 ď r ξ i ď 0 and d r ξ i dt " r η i , and

1. p r ξ i p0q, r η i p0qq " px i , v i q 2. r η i p r t i q " 0 3. 1 2 ż r t i 0 ˆdr η i dt psq ˙2 ds " 2 9 pv i q 3 |x i | ,
(we have also used Lemma 4.3 with w " 0 and Remark 4.2). Observe that x i ď r ξ i p r t i q ď 0, hence lim iÑ`8 r ξ i p r t i q " 0. Then, using the same arguments as in Lemma 2.2, it is possible to extend continuously p r ξ i , r η i q to r0, T s in such a way that

1. p r ξ i , r η i q P Γrx i , v i s 2. lim iÑ8 ż T r t i
ˇˇˇd r η i dt psq ´αps ´r t i q ˇˇˇ2 ds " 0, where α is an optimal control law for trajectories with initial values p0, 0q.

Combining all the information above, we obtain that (4.15)

J ˆr ξ i , r η i , dr η i dt ˙" 2 9 pv i q 3 |x i | `up0, 0q `op1q.
Step 2 : Since pξ i , η i q P Γ opt rx i , v i s, we know that for all t P r0, T s, ξ i ptq " x i ż t 0 η i psqds ď 0. We claim that there exists t i P p0, T s such that η i pt i q ď ´xi {T . Indeed, if it was not the case, then ξ i pT q would be larger than x i ´T px i {T q " 0, which is not true. Since η i is continuous, we may define θ i as the minimal time t such that η i ptq ď ´xi {T , and we see that η i pθ i q " ´xi {T .

Step 2 consists of proving that (4.16) lim

iÑ8 θ i " 0.
Suppose by contradiction that there exists δ ą 0 such that θ i ě δ. We may apply Lemma 4.2 with w " |x i |{T . Since v i Ñ 0 and pv i q 3 {|x i | ě C ą 0, we see that |x i |{v i Ñ 0, then that lim iÑ8

3|x i | v i `2|x i |{T " 0. Hence, for i large enough, θ i ě δ ą 3|x i | v i `2|x i |{T
, and the third line of (4.7) yields (4.17) 1 2

ż T 0 ˆdη i dt psq ˙2 ds ě 2T 9pT ´δq p x i T `vi q 3 |x i | " 2T 9pT ´δq pv i q 3 |x i | `op1q,
where op1q is a quantity that tends to zero as i Ñ 8, (in fact like pv i q 2 ). Note that η i ě ´xi {T ě 0 in r0, θ i s yields that ξ i ě x i in r0, θ i s. Therefore (4.18) lim iÑ8 }ξ i } L 8 p0,θ i q " 0.

Let us construct an admissible trajectory p p ξ i , p η i q starting from px, vq " p0, 0q as follows:

1. For s P r0, θ i s, p ξ i psq " Q θ i ,0,0,ξ i pθ i q,η i pθ i q psq and p η i psq " Q 1 θ i ,0,0,ξ i pθ i q,η i pθ i q psq, see Definition 2.1 2. p p ξ i psq, p η i psqq " pξ i psq, η i psqq for s P rθ i , T s.

It is easy to check that, if s ď θ i , then

p ξ i psq " `θi η i pθ i q ´2ξ i pθ i q ˘s3 θ 3 i ´`θ i η i pθ i q ´3ξ i pθ i q ˘s2 θ 2 i , (4.19) p η i psq " 3 `θi η i pθ i q ´2ξ i pθ i q ˘s2 θ 3 i ´2 `θi η i pθ i q ´3ξ i pθ i q ˘s θ 2 i , (4.20) dp η i dt psq " 6 `θi η i pθ i q ´2ξ i pθ i q ˘s θ 3 i ´2 `θi η i pθ i q ´3ξ i pθ i q ˘1 θ 2 i . (4.21)
Since η i pθ i q " ´xi {T ą 0 and ξ i pθ i q ď 0, we see that `θi η i pθ i q ´2ξ i pθ i q ˘ě 0 and that `θi η i pθ i q ´3ξ i pθ i q ˘ě 0. Hence for s P r0, θ i s, p ξ i psq " `θi η i pθ i q ´2ξ i pθ i q

˘ˆs 3 θ 3 i ´s2 θ 2 i ˙ξ i pθ i q s 2 θ 2 i
ď 0 as the sum of two nonpositive terms. Therefore p p ξ i , p η i q P Γr0, 0s. On the other hand, using (4.18) and the fact that θ i η i pθ i q " θ i |x i |{T , then (4. ´}p ξ i } L 8 p0,θ i q `}p η i q} L 8 p0,θ i q ¯" 0.

Moreover, since θ i ě δ ą 0, it is easy to check that (4.23) lim iÑ`8

ż θ i 0 ˆdp η i dt psq ˙2 ds " 0. Since p p ξ i , p η i q P Γr0, 0s, up0, 0q ď Jp p ξ i , p η i , 0q " ż T 0 ˜ℓp p ξ i psq, p η i psq, sq `1 2 ˆdp η i dt psq ˙2¸d s `gp p ξ i pT q, p η i pT qq " ż θ i 0 ˜ℓp p ξ i psq, p η i psq, sq `1 2 ˆdp η i dt psq ˙2¸d s `ż T θ i ˜ℓpξ i psq, η i psq, sq `1 2 ˆdη i dt psq ˙2¸d s `gpξ i pT q, η i pT qq " upx i , v i q `ż θ i 0 ˜ℓp p ξ i psq, p η i psq, sq `1 2 ˆdp η i dt psq ˙2¸d s ´ż θ i 0 ˜ℓpξ i psq, η i psq,
upx i , v i q ě 2T 9pT ´δq pv i q 3 |x i | `up0, 0q `op1q,
where op1q is quantity that tends to 0 as i Ñ 8. But for p r ξ i , r η i q constructed in Step 1, J ´r ξ i , r η i , dr η i dt ¯ě upx i , v i q. This fact and (4.15) lead to a contradiction with (4.26). We have proved (4.16).

Step 3 : Since lim iÑ8 θ i " 0 and pξ i , η i q converges uniformly to pξ, ηq, we see that ż θ i 0 ℓpξ i psq, η i psq, sqds " op1q. dt psq ˙2 ds `gpξ i pT q, η i pT qq ď up0, 0q `op1q. (4.30) Since pξ i , η i q converges uniformly to pξ, ηq, (4.30) implies that ´1pθ i ,T q dη i dt ¯iPN is a bounded sequence in L 2 p0, T q. Hence there exists φ P L 2 p0, T q such that, after the extraction of subsequence, 1 pθ i ,T q dη i dt á φ in L 2 p0, T q weak. By testing with compactly supported functions in p0, T q, it is clear that φ " dη dt . Hence, the whole sequence ´1pθ i ,T q dη i dt ¯iPN converges in L 2 p0, T q weak to dη dt P L 2 p0, T q. Moreover, the weak convergence in L 2 p0, T q implies that ż T

Hence upx i , v i q "J ˆξi , η i , dη i dt " 1 2
ż θ i 0 ˆdη i dt psq ˙2 ds `ż T θ i ℓpξ i psq, η i psq, sqds `1 2 ż T θ i ˆdη i dt psq ˙2 ds `gpξ i pT q, η i pT
0 ˆdη dt psq ˙2 ds ď lim inf iÑ`8 ż T 0 ˆ1pθ i ,T q dη i dt psq ˙2 ds.
This and (4.30) imply that Hence, pξ, ηq P Γ opt r0, 0s and the above inequality is in fact an identity. The proof is achieved. ✷ Proof of Theorem 4.1. Consider px, vq P Ξ ad and a sequence px i , v i q iPN such that for all i P N, px i , v i q P Ξ ad and px i , v i q Ñ px, vq as i Ñ 8. Consider a sequence pξ i , η i q iPN such that for all i P N, pξ i , η i q P Γ opt rx i , v i s and that pξ i , η i q tends to pξ, ηq uniformly. Under Assumption 4.1, for all r ą 0, there exists a positive number C " Cpr, M q (M is defined in Assumption 4.1) such that if px, vq P Θ r , then Γ opt rx, vs Ă Γ C . Moreover, as r Ñ `8, Cpr, M q " Op ? rq.

Proof. A possible proof consists of building a suitable map j from Θ r to Γ. We make out different cases: Then, using jpx, vq as a competitor for the optimal control problem leads to the desired result with a constant C that depends only on r and M and that can always be taken larger than r C. Note that j is piecewise continuous from Θ r to Γ. Note also that the construction of j is independent of ℓ and g. ✷ Remark 4.4. Note that the sets Θ r form an increasing family of compact subsets of Ξ ad and that (4.37) ď rě0 Θ r " Ξ ad .

Mean field games with state constraints

In the example considered here, we take Ξ " r´1, 0sˆR. Let PpΞq be the set of probability measures on Ξ. Let F, G : PpΞq Ñ C 0 b pΞ; Rq be bounded and continuous maps (the continuity is with respect to the narrow convergence in PpΞq) and ℓ be a continuous and bounded from below function defined on Ξ ˆr0, T s. Set Assumption 4.3. We assume that for all t P r0, T s, m P PpΞq and v ě 0, ℓp0, v, tq F rmsp0, vq ě ℓp0, 0, tq`F rmsp0, 0q and ℓp´1, ´v, tq`F rmsp´1, ´vq ě ℓp´1, 0, tq`F rmsp´1, 0q.

Using similar notations as in paragraph 3.1, we consider the cost given by (3.2). With M in (3.1), note that Proposition 4.1 can be applied to J µ defined in (3.2) with constants Cpr, M q uniform in µ. Lemma 4.5. Let r be a positive number. Under the assumptions made above on ℓ, F and G (including Assumption 4.3), let C " Cpr, M q be the constant appearing in Proposition 4.1. For any probability measure m 0 on Ξ supported in Θ r defined in (4.33), there exists a constrained mean field game equilibrium associated with the initial distribution m 0 , i.e. a probability measure µ P P m 0 pΓ C q such that (3.8) Proof. From (4.37) and (4.39), there exists n 0 ą 0 such that m 0 pΘ n q ą 0 for n ą n 0 .

For n ą n 0 , we set m 0,n " 1 m 0 pΘnq m 0 | Θn . With a slight abuse of notation, let m 0,n also denotes the probability on Ξ obtained by extending m 0,n by 0 outside Θ n , i.e. m 0,n pBq " 1 m 0 pΘnq m 0 pB X Θ n q, for any measurable subset B of Ξ. Let µ n P P m 0,n `ΓCpn,Mq ˘be a constrained mean field game equilibrium associated with the initial distribution m 0,n , the existence of which comes from Lemma 4.5. With a similar abuse of notations as above, let µ n also denote the probability on Γ obtained by extending µ by 0 outside Γ Cpn,M q .

We claim that tµ n , n ą n 0 u is tight in PpΓq, i.e. that for each ǫ ą 0, there exists a compact K ǫ Ă Γ such that (4.40)

µ n pΓzK ǫ q ă ǫ, for each n ą n 0 .

From the increasing character of the sequence Θ n , (4.37) and (4.39), we observe that for each ǫ ą 0, there exists n 1 ą 0 such that m 0 pΘ n 1 q ą 1 ´ǫ. Let us prove (4.40) with K ǫ " Γ Cpn 1 ,M q . Since for all n ą n 0 , µ n P P m 0,n pΓq is a MFG equilibrium, we see that for all measurable B Ă Ξ, m 0,n pBq " µ n tpξ, ηq P supppµ n q : pξp0q, ηp0qq P Bu ď µ n ¨ď px,vqPB Γ opt,µn rx, vs '.

Taking B " Θ n 1 and using Proposition 4.1, we see that m 0,n pΘ n 1 q ď µ n ¨ď px,vqPΘn 1 Γ opt,µn rx, vs 'ď µ n `ΓCpn 1 ,M q ˘, (note that the constant Cpn 1 , M q does not depend on µ n ).

On the other hand,

m 0,n pΘ n 1 q ě m 0 pΘ n 1 q ą 1 ´ǫ, if n ą n 1 , m 0,n pΘ n 1 q " 1, if n 0 ă n ď n 1 . Note that T Ω pxq " R 2 if x P Ω. A vector v P R 2 points outward Ω at x P BΩ if v R T Ω pxq.
Let pν i q 0ďiăN be the vertices of BΩ, labeled in such a way that BΩ " Ť N

´1

i"0 γ i , where γ i " rν i , ν i`1 s and ν N " ν 0 . We may assume that three successive vertices are not aligned. We are going to use the notation pν i , ν i`1 q for the open straight line segment between ν i and ν i`1 . For i P t0, . . . , N ´1u, let n i be the unitary normal vector to γ i pointing outward Ω. It is easy to see that T Ω pν i q " tx P R 2 : n i ¨x ď 0 and n i´1 ¨x ď 0u, setting n ´1 " n N ´1. Since Ω is convex, Ω coincides locally near ν i with ν i `TΩ pν i q.

The optimal control problem is set exactly as in Section 2: it consists of minimizing Jpξ, η, η 1 q given by (2.2) on the dynamics given by (2.1) and staying in Ξ " Ω ˆR2 . The set Ξ ad is defined by (2.6).

Closed graph properties

The closed graph result given in Proposition 5.1 below is similar to that contained in Proposition 2.1, but special conditions are needed near the vertices of BΩ: Proposition 5.1. Consider a closed subset Θ of Ξ ad . Assume that all sequence px i , v i q iPN such that for all i P N, px i , v i q P Θ and lim iÑ`8 px i , v i q " px, vq P Θ, the following holds:

1. If x P pν j , ν j`1 q for some j P t0, . . . , N ´1u (recall that ν N " ν 0 ), then (5.1) pv i ¨nj q 3 `" o ´px ´xi q ¨nj ¯;

2. if x " ν j for some j P t0, . . . , N ´1u and v " 0, then (5.2) pv i ¨nk q 3 `" o ´px ´xi q ¨nk ¯, for k " j ´1, j, recalling that n ´1 " n N ´1;

3. If x " ν j for some j P t0, . . . , N ´1u and v " 0, then

(5.3) pv i ¨nk q `´|x ´xi | 2 3 `|v i | 2 ¯" o ´px ´xi q ¨nk ¯, for k " j ´1, j;
then the graph of the multivalued map Γ opt : Θ Ñ Γ, px, vq Þ Ñ Γ opt rx, vs, is closed in the sense given in Proposition 2.1.

Remark 5.1. It is easy to find sets Θ which fulfill the assumption of Proposition 5.1, for example:

1. a compact subset Θ of R 2 ˆR2 contained in Ω ˆR2 ;
2. for j " 0, ¨¨¨, N ´1, let D j be the straight line containing ν j and ν j`1 ; for ρ ą 1, set

S j " ! px, vq P Ω ˆR2 : v ¨nj ď distpx, D j q ρ 3
) .

If Θ is a closed subset R 2 ˆR2 contained in ´Ξad X Ş N ´1 j"0 S j ¯z ´ŤN´1
j"0 tν j u ˆR2 ¯, then Θ fulfills the assumption of Proposition 5.1.

For brevity, we do not supply examples of sets Θ which intersect

Ť N ´1 j"0 tν j uˆR 2 , although it is possible to do so.

The proof of Proposition 5.1 relies on the counterpart of Lemma 2.3 whose proof is exactly the same, and on Lemma 5.1 below which is the counterpart of Lemma 2.2: Lemma 5.1. Consider px, vq P Ξ ad , pξ, ηq P Γrx, vs such that η P W 1,2 p0, T ; R 2 q and a sequence px i , v i q iPN such that for all i P N, px i , v i q P Ξ ad and lim iÑ8 px i , v i q " px, vq.

Assume that one among the following seven conditions is true:

1. x P Ω 2. x P pν j , ν j`1 q, v ¨nj ă 0 (hence v i ¨nj ă 0 for i large enough) 3. x P pν j , ν j`1 q, v ¨nj " 0 and one among the following properties is true:

(a) v i ¨nj ď 0 for i large enough (b) for i large enough, v i ¨nj ą 0 (hence px ´xi q ¨nj ą 0) and (5.4) lim iÑ8 pv i ¨nj q 3 px ´xi q ¨nj " 0 4. x " ν j , v ¨nj ă 0 and v ¨nj´1 ă 0 (setting n ´1 " n N ´1)

5.

x " ν j , v ¨nj " 0 and v ¨nj´1 ă 0 and one among the two following properties holds (a) v i ¨nj ď 0 for i large enough (b) v i ¨nj ą 0 and (5.4) holds for i large enough.

6.

x " ν j , v ¨nj´1 " 0 and v ¨nj ă 0 and one among the two following properties holds (a) v i ¨nj´1 ď 0 for i large enough (b) v i ¨nj´1 ą 0 and (5.4) holds (replacing j by j ´1) for i large enough 7.

x " ν j , v " 0 and

(5.5) lim iÑ8 max kPtj´1,ju pv i ¨nk q |px i ´xq ¨nk | ´|x ´xi | 2 3 `|v i | 2 ¯" 0, with the convention that pv i ¨nk q |px i ´xq¨n k | " 0 if pv i ¨nk q `" 0.
Then there exists a sequence pξ i , η i q iPN such that pξ i , η i q P Γrx i , v i s, η i P W 1,2 p0, T ; R 2 q, and pξ i , η i q tends to pξ, ηq in W 2,2 p0, T ; R 2 q ˆW 1,2 p0, T ; R 2 q, hence uniformly in r0, T s.

Proof of Lemma 5.1. Since many arguments are almost identical to those contained in the proof of Lemma 2.2, we just sketch the proof and put the stress on cases 4-7 which have no counterparts in Lemma 2.2. As for Lemma 2.2, each of the seven conditions mentioned in the statement makes it possible to explicitly construct families of admissible trajectories fulfilling all the desired properties. Since the construction is different in each of the seven cases, we discuss each case separately:

1. If x P Ω, then the construction is exactly the same as in the first case in the proof of Lemma 2.2. We do not repeat the argument.

2. x P pν j , ν j`1 q and v ¨nj ă 0, hence for i large enough, v i ¨nj ă 0. We can always assume that the latter property holds for all i. Since η P W 1,2 p0, T q, there exists t P p0, T q such that for all s P r0, ts, 3 2 v ¨nj ď ηpsq ¨nj ď 1 2 v ¨nj and 3s 2 v ¨nj ď pξpsq ´xq ¨nj ď s 2 v ¨nj . For t i P r0, ts, we set (5.6) ξ i psq " " Q t i ,x i ,v i ,ξpt i q,ηpt i q psq, if s P r0, t i s, ξpsq, if s P rt i , T s, and the remaining arguments are very close to those in case 2. in the proof of Lemma 2.2.

We skip the details.

3. (a) x P pν j , ν j`1 q, v ¨nj " 0 and v i ¨nj ď 0 at least for i large enough. We may assume that v i ¨nj ď 0 for all i. Given t i,1 P p0, T q, we define py i , w i q as follows:

(5.7) y i " px ¨nj qn j `ppx i `vi t i,1 q ¨nK j qn K j , w i " pv ¨nj qn j `pv i ¨nK j qn K j " pv i ¨nK j qn K j , and, for t i,1 ď t i,2 ă T , set (5.8) ξ i psq " [START_REF] Achdou | Income and wealth distribution in macroeconomics: A continuous-time approach[END_REF] ´ti,1 ,y i ´x,w i ´v,0,0 ps ´ti,1 q `ξps ´ti,1 q, if t i,1 ď s ď t i,2 , ξps ´ti,1 q, if t i,2 ď s ď T.

$ & % Q t i,1 ,x i ,v i ,y i ,w i psq, if s ď t i,1 , Q t i,
We argue as in case 3.(a) in the proof of Lemma 2.2. An important observation is that (5.9) pξ i psq ´xq ¨nj " ˆpx i ´xq ¨nj ˆ1 `2 s t i,1 ˙`sv i ¨nj ˙ˆ1 ´s t i,1

˙2

is non positive for s P r0, t i,1 s. We skip the other details.

(b) x P pν j , ν j`1 q, v ¨nj " 0, v i ¨nj ą 0 for all i (or for i large enough), and (5.4) holds. This case is the counterpart of case 3.(b) in the proof of Lemma 2.2. The trajectory ξ i is constructed as in (5.8), but a further restriction on t i,1 is needed in order to guarantee that the trajectory is admissible, namely that

t i,1 ď 3|px i ´xq ¨nj | v i ¨nj .
This condition should be supplemented with the other two conditions as in 3.(a), namely that lim iÑ8 t i,1 " 0, (5.10) lim iÑ8 |px ´xi q ¨nj | 2 t 3 i,1

`|pv ´vi q ¨nj | 2 t i,1 " 0. (5.11) If (5.4) holds, then it is possible to choose such a sequence t i,1 . The remaining part of the proof is as in case 3.(a).

4.

x " ν j , v ¨nj ă 0 and v ¨nj´1 ă 0. Since η P W 1,2 p0, T q, there exists t P p0, T q such that for all s P r0, ts, 3 2 v ¨nk ď ηpsq ¨nk ď 1 2 v ¨nk and 3s 2 v ¨nk ď pξpsq ´xq ¨nk ď s 2 v ¨nk , for k " j ´1, j. For t i P r0, ts, we choose ξ i as in (5.6) and the desired result is obtained as in case 2.

5.

x " ν j , v ¨nj " 0 and v ¨nj´1 ă 0. We make out two subcases: (a) v i ¨nj ď 0 at least for i large enough: the trajectory is constructed as in (5.8), with three different stages corresponding respectively to s P r0, t i,1 s, s P rt i,1 , t i,2 s and s ě t i,2 . As in point 3, it is always possible to choose the sequence t i,1 such that (5.10) and (5.11) hold in order to ensure the L 2 convergence of the accelerations. We need to prove that the trajectory is admissible for well chosen t i,1 and t i,2 .

Let us first check that ξ i psq remains in Ω for s P r0, t i,1 s. Since (5.9) holds, we see that pξ i psq ´xq ¨nj ď 0. On the other hand, after some algebra, we get that ˇˇpη i psq ´vq ¨nj´1 ˇˇ" 0.

η i psq
Hence, for i large enough, η i psq ¨nj´1 ă 0 for all s P t i,1 . This implies that pξ i psq ´xq ¨nj´1 ď 0 for all s P r0, t i,1 s. Combining the information above and arguing essentially as in case 3.(a), we see that it is possible to choose t i,1 satisfying (5.10) and (5.11), t i,2 bounded away from 0 uniformly w.r.t. i, such that pξ i , η i q P Γrx i , v i s for i large enough and

lim iÑ8 › › › › dη i ds ´dη ds › › › ›
L 2 p0,T q " 0.

(b) v i ¨nj ą 0 at least for i large enough and (5.4) holds: again, the trajectory is constructed as in (5.8) with t i,1 satisfying (5.10)- (5.11). As in 3.(b), a further restriction is needed on t i,1 such that the trajectory is admissible. For 0 ď s ď t i,1 , pξ i psq ´xq ¨nj is given by (5.9) and is non positive if (5.12) t i,1 ď 3|px i ´xq ¨nj | v i ¨nj .

On the other hand, the proof that pξ i psq ´xq ¨nj´1 ď 0 for i large enough and all s P r0, t i,1 s is the same as in subcase 5.(a). Hence if the sequence pt i,1 q satisfies (5.10)-(5.11) and (5.12), then for i large enough, ξ i psq P Ω for for all s P r0, t i,1 s. Constructing such a sequence pt i,1 q is possible thanks to (5.4). Then, using the fact that v ¨nj´1 ă 0 and arguing as in case 3., it is possible to choose the sequence t i,2 bounded from below by a positive constant independent of i such that pξ i , η i q P Γrx i , v i s and lim iÑ8

› › › dη i dt ´dη dt › › ›
L 2 p0,T q " 0.

6. Same arguments as for case 5., exchanging the roles of j and j ´1.

7.

The trajectory ξ i is constructed as follows:

(5.13) ξ i psq " Hence, a sufficient condition for ξ i psq to stay in Ω for all s P r0, t i s is that t i ď 3 min kPtj´1,ju

" Q t i ,x i ,v i ,
px´x i q¨n k pv i ¨nk q `,
with the convention that px´x i q¨n k pv i ¨nk q

`" `8 if pv i ¨nk q `" 0. Then, we also need that lim iÑ8 t i " 0 and that lim iÑ8 There exists a positive number C " Cpr, M q such that if px, vq P Θ r , then Γ opt rx, vs Ă Γ C , where Γ C is defined in (2.29).

|x´x i | 2 t 3 i `|v i | 2 t i "

Mean field games with state constraints

All the results obtained in Section 3 can be generalized to the case when Ω is a bounded and convex polygonal domain of R 2 , provided the initial distribution of states m 0 is supported in Θ r defined as in Proposition 5.2.

( 3 . 1 )

 31 M " max ˜sup px,v,sqPΞˆr0,T s ℓ ´px, v, sq `sup mPPpΞq }F rms} L 8 pΞq , sup mPPpΞq }Grms} L 8 pΞq ¸.

  ds `gpξpT q, ηpT qq ď up0, 0q.

( 4 .

 4 38) M " max ˜sup px,v,tqPΞˆr0,T s ℓ ´px, v, tq `sup mPPpΞq }F rms} L 8 pΞq , sup mPPpΞq }Grms} L 8 pΞq ¸.

  10) holds. The trajectory ξ i is constructed as in (2.22)-(2.23), but a further restriction on t i,1 is needed in order to guarantee that the trajectory is admissible. Using (2.21), we see that the trajectory is admissible if

	vi ¨en |x i ¨en |	ď	1 s `2 t i,1

  Hence, for all µ P PpΓq, Propositions 2.1 and 2.2 hold for the state constrained control problem related to J µ and the constants arising in these propositions can be chosen uniformly with respect to µ P PpΓq. There exists a positive number r such that the initial distribution of states is a probability measure m 0 on Ξ supported in Θ r , where Θ r is a closed subset of Ξ ad as in(2.30).

	Assumption 3.1.
	).
	With µ P PpΓq, we associate the cost (3.2) J µ pξ, ηq " ¨ż T 0 ˜F rm µ psqspξpsq, ηpsqq `ℓpξpsq, ηpsq, sq	`1 2	ˇˇˇd η dt	psq ˇˇˇ2 ¸ds

`Grm µ pT qspξpT q, ηpT qq ‹ '. Remark 3.1. It is clear from

(3.1)

that given µ P PpΓq, the running cost py, w, sq Þ Ñ F rm µ psqspy, wq `ℓpy, w, sq and the final cost py, wq Þ Ñ Grm µ pT qspy, wq satisfy Assumption 2.1, and that the constant arising in (2.3) can be chosen uniformly with respect to µ P PpΓq.

  2.1. It makes use of Assumption 2.1 and Lemma 3.2.

✷

Definition 3.1. The probability measure µ P P m 0 pΓq is a constrained mean field game equilibrium associated with the initial distribution m 0 if

(3.8) supppµq Ă ď px,vqPsupppm 0 q Γ µ,

opt rx, vs. Theorem 3.2. Under the assumptions made on F and G at the beginning of paragraph 3.1 and Assumption 3.1, let C " Cpr, M q be chosen as in Proposition 2.2. There exists a constrained mean field game equilibrium µ P P m 0 pΓ C q, see Definition 3.1. Moreover, t Þ Ñ e t 7µ P C 1{2 pr0, T s; PpK C qq, (K C is defined in (2.28) and PpK C q is endowed with the Kantorovitch-Rubinstein distance).

  Problem (4.6) is the minimization of a strictly convex and continuous functional under linear and continuous constraints, and the set K θ,w is non empty, as we shall see below, convex and closed. Hence there exists a unique minimizer, named η again. The Euler-Lagrange necessary conditions read as follows: there exists a real number µ ě 0 such that η is a weak solution of the linear complementarity problem (variational inequality)

	Remark 4.2. The partition of the interval r0, T s in (4.7) is justified by the assumptions of Lemma 4.2. Indeed • 3|x|{v ă T and w ě 0 imply that 3|x|{pv `2wq ă T (4.13) ηptq " v ´pv ´wq t θ .
	• 2|x|{pv `wq ă 3|x|{pv `2wq because 0 ď w ď |x|{T ă v{3. $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' % ´η2 ě ´µ, in p0, θq, η ě w, in p0, θq, p´η 2 `µqpη ´wq " 0 in p0, θq, x `ż θ 0 ηpsqds ď 0, µ ˆx ηpθq " w. ηp0q " v, 0 `ż θ ηpsqds ˙" 0, µ ě 0, Proof. (4.8) Note also that if |x|{v Ñ 0, then 3|x|{pv `2wq " 3|x|{v ! T . Then, Ipθ, wq " 1 2 pw´vq 2 θ
	The solution of (4.8) can be written explicitly. Skipping the details, it has the following
	form:
	1. If θ ě 3|x|{pv `2wq, then (4.9) # ηptq " v ´µτ t ηptq " w,	`µ 2	t 2 ,	0 ď t ď τ, τ ă t ď θ,
	with
	ηpsq ě w, @s P r0, θs, x `ż θ 0 ηpsqds ď 0 ż θ 0 ˇˇˇd η dt ´w and µ " Ipθ, wq " inf ηPK θ,w 1 2 τ " ´3 x `wθ v 2pv ´wq 3 9px `wθq 2 . v´w ď θ because θ ě 3|x| Note that ´3 x`wθ The quantity (4.6) (4.10) v`2w . Note also that x `ż θ 0 that Ipθ, wq " µ 2 2 ż τ 0 p´τ `tq 2 dt " µ 2 τ 3 6 " 2 9 pv ´wq 3 ; we have obtained the third line , / / . / / -. ηpsqds " 0. We see |x| ´wθ in (4.7). psq ˇˇˇ2 ds 2. If 2|x|{pv `wq ď θ ď 3|x|{pv `2wq, then for all t P r0, θs,
	(4.11) is achieved by a function η " η θ,w and is given by ηptq " v `kt `µ 2 (4.7) Ipθ, wq " $ ' ' ' ' ' & ' ' ' ' ' % 1 2 pw ´vq 2 t 2 , θ , with 6 x 2 θ 3 `6 xpv `wq θ 2 `2 v 2 `vw `w2 θ (4.12) k " ´6x `p4v `2wqθ θ 2 , and µ " 6 2 9 pv ´wq 3 |x| ´wθ , Note that x `ż θ	if θ P if θ P 2x `pv `wqθ θ 3 if θ P θ 3 `6 xpv`wq " 0, 2|x| v`w " 2|x| v`w , 3|x| ı , v`2w . " 3|x| v`2w , T ¯. θ 2 `2 v 2 `vw`w 2 ı ,

p0, θ; Rq : ˇˇˇˇˇˇˇη p0q " v, ηpθq " w, 0 ηpsqds " 0. Easy algebra leads to Ipθ, wq " 6 x 2 θ ; we have obtained the second line in (4.7).

3. If θ ď 2|x|{pv `wq, then for all t P r0, θs,

  Possibly after the extraction of a subsequence, we can always assume that either one among the five conditions in Lemma 4.1 or one among the two symmetrical conditions (4.3)-(4.4) holds.

	Then the conclusion follows from Corollary 4.1 in the former case or from Lemma 4.4 in
	the latter case.					✷
	Remark 4.3. For costs of the form
	(4.32)	Jpξ, η, αq "	ż T 0	ˆℓpξpsq, ηpsq, sq	`1 p	|α| p psq ˙ds `gpξpT q, ηpT qq,
	with 1 ă p " 2, it is not possible to reproduce the explicit calculations of Lemmas 4.2 and 4.3, which are crucial steps for Lemma 4.4 and finally for Theorem 4.1.
	4.2 Bounds related to optimal trajectories
	Proposition 4.1. For positive numbers r and C, let us set
	(4.33) (4.34) (4.35)	( Θ r " px, vq P Ξ : ´rpx `1q ď v 3 ď r|x| K C " tpx, vq P Ξ : |v| ď Cu, Γ C " # pξ, ηq P Γ : ˇˇˇˇp ξptq, ηptqq P K C , @t P r0, T s, , › › › dη dt › › › L 2 p0,T ;Rq ď C.	+	.

  ´3p1 `xq{T ď v ď 0: the situation is symmetric to Case 1, and jpx, vq is given by the same formula. the situation is symmetric to Case 2, and jpx, vq is constructed in the symmetric way as in Case 2.

	Case 1: 0 ď v ď ´3x{T : let jpx, vq " p r ξ, r ηq P Γrx, vs be defined by # r ηptq " v `1 ´3t 2T ˘and r ξptq " x `v ´t ´3t 2 4T ¯, r ηptq " 0 and r ξptq " x `vT 3 ,	3 , if 0 ď t ď 2T if 2T 3 ď t ď T.
	It is easy to check that there exists a constant r C " r Cpr, M q such that (4.36) }r η} L 8 p0,T ;Rq ď r C; › › › › dr η dt › › › › L 2 p0,T ;Rq ď r C.
	Case 2: ´3x{T ă v ď r|x| and r η is the solution of the linear complementarity problem (4.8) with θ " T . Here again 1 3 : in this case, we choose jpx, vq " p r ξ, r ηq P Γrx, vs where r η ξ 1 " r p r ξ, r ηq satisfies (4.36) for some constant r C " r Cpr, M q. From Lemma 4.2, we see that as r Ñ `8, r C " Op ? rq.
	Case 3: Case 4: ´rpx `1q	1 3 ď v ă ´3p1 `xq{T :

  holds.Proof. The proof is similar to that of Theorem 3.2. We skip it.✷ Compared to Theorem 3.2, the restrictions made on the support of m 0 are weaker in Lemma 4.5, but the latter requires the additional Assumption 4.3. In Theorem 4.2 below, we get rid of the assumptions on the support of m 0 .

	Remark 4.5. Theorem 4.2. Let m 0 be a probability measure on Ξ such that
	(4.39)	m 0 pΞzΞ ad q " 0.

Under the assumptions made above on ℓ, F and G (including Assumption 4.3), there exists a constrained mean field game equilibrium associated with the initial distribution m 0 , i.e. a probability measure µ P P m 0 pΓq such that

(3.8)

holds.

  ˆ1´s t i,1 ˙px i ´xq ¨nj n j ¨nj´1 .Since lim iÑ8 v i ¨nj " 0 and lim iÑ8 v i ¨nj´1 " v ¨nj´1 ă 0, we see that for i large enough,

	¨nj´1 "	dξ i ds	psq ¨nj´1 "v i ¨nj´1	´s t i,1	ˆ4	´3s t i,1	˙vi ¨nj n j ¨nj´1
						´6 s t 2 i,1
	lim iÑ8	ˆvi ¨nj´1	´s t i,1	ˆ4	´3s t i,1	˙vi ¨nj n

j ¨nj´1 ˙" v ¨nj´1 ă 0, uniformly with respect to s P r0, t i,1 s. On the other hand, the conditions (5.10) and

(5.11)

imply that lim iÑ8 px i ´xq¨n j t i " 0. Combining the latter two observations yields that lim iÑ8 max sPr0,t i,1 s

  x,0 psq, if s ď t i , ξps ´ti q, if t i ď t ď T,We see that, for k " j ´1, j and s P r0, t i s, pξ i psq ´xq ¨nk " ˆ1 ´s t i

	˙2	ˆˆ1	`2 s t i	˙px i ´xq ¨nk `sv i ¨nk	˙.

  0 in order to obtain that lim iÑ8 From (5.5), it is possible to construct a sequence pt i q i fulfilling all the desired properties.

	0.	› › › dη i dt	› › ›	L 2 p0,t i q	"

✷

Proposition 5.2 below is the counterpart of Proposition 2.2: Proposition 5.2. Given r ą 0, let us define Θ r by (2.30) where K r is defined by (2.28) and Θ is a closed subset of Ξ ad which satisfies the assumption in Proposition 5.1. Under Assumption 2.1, the value function u given by (2.7) is continuous on Θ r .

Acknowledgments. YA and NT were partially supported by the ANR (Agence Nationale de la Recherche) through project ANR-16-CE40-0015-01. YA was partially supported by the chair Finance and Sustainable Development and FiME Lab (Institut Europlace de Finance). Part of the research was completed while YA was on leave at INRIA-Paris in the project Matherials. PM and CM were partially supported by GNAMPA-INdAM and by the Fondazione CaRiPaRo Project "Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games". We would like to thank P. Cardaliaguet for an enlightening discussion concerning the argument in Paragraph 4.3.

In both cases, µ n `ΓCpn 1 ,M q ˘ě 1 ´ǫ and therefore µ n `ΓzΓ Cpn 1 ,M q ˘ě 1 ´ǫ, and the claim is proved.

Thanks to Prokhorov theorem, possibly after the extraction of subsequence that we still name µ n , we deduce that there exists µ P PpΓq such that µ n converges narrowly to µ.

We claim that µ is a MFG equilibrium related to m 0 . We already know that µ P PpΓq. There remains to prove that • µ P P m 0 pΓq, i.e. that e 0 7µ " m 0 • µ satisfies (3.8).

The fact that e 0 7µ " m 0 stems from Lemma 3.1 and from the fact that m 0,n narrowly converges to m 0 .

In order to prove (3.8), we recall that from Kuratowski's theorem, see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF],

which means that for all pξ, ηq P supppµq, there exists a sequence pξ n , η n q P supppµ n q such that pξ n , η n q Ñ pξ, ηq uniformly. As a consequence, setting px , v n q " pξ n p0q, η n p0qq and px, vq " pξp0q, ηp0qq, lim nÑ8 px n , v n q " px, vq and pξ n , η n q P Γ opt,µn rx n , v n s. Applying Proposition 4.2 below, which is a generalization of Theorem 4.1, we may pass to the limit and conclude that pξ, ηq P Γ opt,µ rx, vs, which achieves the proof. ✷ Proposition 4.2. Under the assumptions made above on ℓ, F and G (including Assumption 4.3), consider a sequence pµ i q iPN , µ i P PpΓq, such that µ i converges narrowly to µ P PpΓq. Consider a sequence pξ i , η i q iPN , pξ i , η i q P Γ, such that 1. pξ i , η i q P Γ opt,µ i rx i , v i s, where px i , v i q " pξ i p0q, η i p0qq 2. pξ i , η i q tends to pξ, ηq P Γrx, vs uniformly, where px, vq " lim iÑ8 px i , v i q.

Then pξ, ηq P Γ opt,µ rx, vs.

Proof. We skip the proof, because it follows the same lines as that of Theorem 4.1, (see paragraph 4.1). In particular, it includes an adaptation of Lemma 4.4. The necessary modifications are obvious. ✷ Remark 4.6. Finally, note that all the results contained in Section 4 can be generalized in a straightforward manner to the case when Ω is a half-space, for example Ω " tx P R n : x 1 ă 0u or a strip, for example Ω " tx P R n : ´1 ă x 1 ă 0u.

5 State constrained optimal control problems and mean field games in a convex polygonal domain of R 2

Let Ω be a bounded and convex domain of R 2 with a polygonal boundary BΩ. For x P Ω, the tangent cone to Ω at x is defined by T Ω pxq " v P R 2 : x `tv P Ω, for t ą 0 small enough ( .