Laurent Beaudoin
email: beaudoin@esiea.fr

Antoine Gademer

Ahmed Amir

Loïca Avanthey

Vincent Germain

Alexandre Pocheau

Near real time detection of hot spots on Meteosat Second Generation images : from forest fires to volcanic eruptions

Abstract-In the 2005 IGARSS' symposium, we presented the setting up of a data receiving station for METEOSAT SECOND GENERATION (MSG) satellites by undergraduates and we emphasized the pedagogical interest of the project. The experience has been successfully pursued by developing with the undergraduates involved the exploitation of the data received and its promotion. Among the different activities implemented in this context, we discuss in this article the work concerning the hot spots detection in near real time on MSG images. This approach, beyond the technical and pedagogical aspects, allows the future engineers to be involved in very important issues such as global warming and natural disasters.

In order to define the forest fire detection algorithms adapted to MSG data, we start by describing the physical properties that lead us to identify the more efficient spectral bands available in MSG. Then, we deal with the different algorithms developed and the accuracy and efficiency of the detection compared with other published results. Sometimes, we observed very localised and motionless fires during several days or even several weeks. Looking at more precise scales, we realized that these anomalies are mostly localised on active volcanoes. In order to validate the hypothesis that we really detect volcanic eruptions, we focused our attention on a very specific volcano called Karthala, Grand Comore Island, Indian Ocean. Comparing our results with ground data, we observe a very good detection of the eruption's activity from the start to the end with almost a quarter hour accuracy.

I. CONTEXT OF THE STUDY

A. About us

The Ecole Supérieure d'Informatique d' Électronique et d'Automatique (esiea) is a French engineering school and its main teaching topics are Electronics and Computer Sciences. esiea encourages its students to work on original and practical projects that efficiently complete their more classical and theoretical course. There is a wide range of possible subjects (from hardware to sofware ones) and some of them are connected with research laboratories of the School. This article presents one of these projects and deals with near real time detection of hot spots on Meteosat Second Generation images. The research department involved in this work is the one dedicated to image and signal processing (Acquisition et Traitement des Images et du Signal -ATIS dept.).

B. Hot spot detection

Life on Earth strongly depends on various parameters. One of them is the composition of the gases in the atmosphere. Atmospheric gases result from various and complex processes (photosynthesis, respiration decay ...). Biomass burning, i.e. burning of world's vegetation including forest, savannas and agricultural lands, impacts Earth's atmosphere and biosphere with short and long term effects [START_REF] Crutzen | Biomass burning as a source of atmospheric gases CO, H2, N2O, NO, CH3CL and COS[END_REF]. The main interactions concern [START_REF] Levine | Biomass Burning and Global Change[END_REF] :

• the modification by smoke aerosols of the transparency of the atmosphere and the reflectivity and the emissivity of the Earth's surface that therefore changes the global energy budget [START_REF] Kaufman | The effect of smoke particles on clouds and climate forcing[END_REF], [START_REF] Li | Influence of absorbing aerosols on the solar surface radiation budget[END_REF],

• the alteration of landcover that leads to changes in the hydrological cycle,

• changes the biogeochemical cycling of compounds from the biosphere to the atmosphere [START_REF] Houghton | Biomass burning from the perspective of the global carbon cycle[END_REF]. Biomass burning contributes in fact to about 40% of gross carbon dioxide and 38% of tropospheric ozone for example [START_REF] Levine | Biomass burning: A driver for global change[END_REF] A part of this burning is natural and essential to revitalize some ecosystems [START_REF] Bonn | Précis de télédétection[END_REF]. The rest, which represents the vast majority of the burning, results from human activities and has increased significantly over the last century [START_REF] Andrea | Biomass burning: its history, use and distribution and its impact on environmental quality and global change[END_REF]. It is supposed to have a key role in global warning. The main contribution to burning biomass comes from the savannas. Therefore, Africa becomes the "burn center" of the world because two-thirds of the Earth's savannas are located on this continent.

To understand long term effects of fires on the climate and the environment, it is fundamental to know the spatial and temporal distribution of fire on a global scale, and only spatial remote sensing technologies can be applied with efficiency at reasonable cost [START_REF] Justice | Remote sensing of fire: potential and limitations[END_REF]. In fact, there is no ideal sensor/platform. Each sensor/satellite has its own advantages and disavantages in fire monitoring, depending on spatial, spectral and temporal possibilities. The polar ones for example have a good spatial resolution but long revisiting time. The most used sensor for fire monitoring is AVHRR for Advances Very High Resolution Radiometer [START_REF] Li | A review of avhrr-based fire active fire detection algorithm: Principles, limitations, and recommendations[END_REF]. The geostationary ones have a degraded spatial resolution but a very high temporal resolution fully spectral channel (µm) 0.6, 0.8, 1.6, 3.9, 6.2, 7. compatible with near real time monitoring [START_REF] Prins | Global geostationary fire monitoring system[END_REF]. In this article, we will use geostionary data. Despite the fact that environmental issues are not the kernel of our teaching, it is essential for esiea to heighten its students' awareness of them, as they will later be in positions of high responsibilities. The work presented in this article uses satellite data received at esiea and detailled in section I-C. Because of their high temporal availability, of their well adapted spectral bands and because these data are centered on Africa, they fit particularly well with the detection and the monitoring of burning biomass which apears on these data like hot spots.

Hot spots (like forests fire seen from space for example) are very specific events that emit a lot of energy (i.e. that have a high temperature or high radiance emission). Specific events means for us events that have a spatial extent of one or only a few pixels. In other words, this kind of events have a great radiometric difference with their neighbours and specific spectral behaviours. So, the objective of the students was to capitalize on these specificities to develop a real time detection algorithm.

C. The data

The data used in this work are Meteosat Second Generation images that we received directly from the satellite. The lowcost receiving station has been developed by a previous generation of students and this work has already been presented in the 2005 IGARSS Symposium [START_REF] Beaudoin | A Meteosat Second Generation receiving, processing and storing images system developed by engineer students[END_REF]. So, the present work is in fact a next step of this pedagogical experience. In table I, you will find the main characteristics of the data [START_REF]MSG system[END_REF]. Obviously, as Meteosat Second Generation (MSG) is a geostationary satellite, it is particularly well adapted for near real time applications.

II. ALGORITHM

A. The physical phenomenon

A combustion process has short time and mid time effects [START_REF] Pereira | Potential fire applications from MSG/SEVIRI observations[END_REF]. Short time effects concern the emission of heat and smoke. The char residue left on the surface and the modified vegetation structure are mid time effects. In our case, we want to take advantage of the near real time capacity of MSG, that is why we focus on the major short time effect : hot spot emited by active fire.

The typical temperature of a flame is about 1000 ˚K and the one of smoldering combustion is 600 ˚K [START_REF] Lobert | Emission from the combustion process in vegetation[END_REF]. The peak emission of radiance corresponding to these effects would be located between 3µm and 5µm according to Wien's displacement law. To give a comparison, a surface with a temperature of 290 ˚K (17 ˚C) has a peak of radiance at about 10µm.

B. Four generations of algorithms

There are currently four major families of active fire detection algorithms but all are built upon this ability of the 3.9 µm channel to detect the rapid increase of temperature produced by the active fire [START_REF] Li | A review of avhrr-based fire active fire detection algorithm: Principles, limitations, and recommendations[END_REF].

Considering the strong correlation between the image high values and the observed fire, older algorithms used to be simple single-channel threshold methods with coarse but satisfying visual results. Nevertheless validation of the algorithms by fine observation of the results shows a lot of false alarms (pixel detected as fires but which are not) due to specular reflection on clouds, bare ground or open water. It is to be noticed that most of the sensors, which were not built for fire monitoring, present low saturation value (350 ˚K for 3.9 µm MSG channel) and thus do not allow fine discrimination.

To reduce the number of these false alarms, the second generation of algorithms takes advantage of the full multispectral potential of the satellites sensors. These multi-channel threshold algorithms try to sort the pixels between fire and non-fire classes by analysing the spectral signature of the observed surface. Generating or using clouds masks, land use, and ancillary data they progressively eliminate the false alarms without suppressing too many of the real fire signals.

The principal difficulty of these algorithms is the choice of the threshold values: indeed most approaches are datadriven and only work for regional data. Thence, you can find accurate algorithms for boreal forest, for tropical forest, etc. or global approaches with approximative results on every type of vegetation. Validation and cross-comparison of these algorithms is therefore a necessary but hard task [START_REF] Li | A review of avhrr-based fire active fire detection algorithm: Principles, limitations, and recommendations[END_REF]. The result of this validation shows obvious improvement since first generation but also the limitation of the model: masks cannot fit the ground data and the transition zones are a frequent source of misclassification: observations must take into account the spatial distribution of the pixels.

The third generation of algorithms emerges from the previous conclusions and makes the analysis of the spatial contextual information contained by the satellite images the heart of the fire detection algorithms. Fires are not only considered by their spectral signature but also by their neighbourhood. The thresholds are here deduced from statistical calculations and not from experimental tuning with a global detection goal. As for validation, these third generation algorithms show good results with very few false alarms but have two drawbacks: firstly, with their global approach they tend to miss the small fires and thus are less useful for early detection and operational response and secondly they still do not fully take into account the effect of sun reflections which depend in the majority of cases on the reflectivity of the objects.

Surface reflectivity is not easy to determine as it is dependent on the physical nature of all the components of the pixel. Futhermore, classic satellite resolution implies that each pixel contains a wide range of elements, but this composition may change in time. The next generation makes the bet of calculating the reflectivity by observing long time series. Knowing this mean reflectivity we can calculate the expected brilliance temperature and analyse the variation as potential fire. Another exploratory approach with time-series tries to use change detection to detect the onset of the fire. These approaches are still in progress and have not yet been fully validated.

C. Our algorithm

Our algorithm belongs to contextual fire detection algorithm generation. Started as student work the approach selected is quite original: our contextual approach is not based on statistical computation but on morphological mathematics. In this way, we hope to avoid some of the threshold determination problems.

Morphological filters are common image processing tools and therefore we will not explain the theory. The top-hat filter used here is a difference between the original and a filtered image with the opening morphological operator. The opening filter suppresses all objects of the original signal smaller than the morphological element size and therefore the top-hat filter focuses attention on all the small elements.

The filtered image is considered as a mask for the potential fire, but as in the other algorithms we have to set some tests to discriminate the non-fire hot spots. Our primary test is to avoid water pixels with a sea/land mask.

As a second test we use hot surface test analysing the difference between channel 3.9 µm and the 10.8 µm. As shown in [START_REF] Li | A review of avhrr-based fire active fire detection algorithm: Principles, limitations, and recommendations[END_REF] due to the non-linearity of the Planck function, this test is mostly correlated to the thermal homogeneity of the pixel: in the presence of fire the thermal homogeneity drops and the difference of channel jumps. The coarse spatial resolution of MSG validates this test. The last test yet implemented is the cloud mask. Imported from the IBGP cloud test on AVHRR data, it uses the radiance of 0.6 and 0.8 µm visible spectrum channels and the brilliance temperature of 12µm infrared channels. In the mid term we are planning to use full cloud information from SAFNWC [16] delivered by EUMETCAST along the MSG data.

III. RESULTS AND DISCUSSION

The main difficulty to validate our results lies in the comparason with ground data. A practical approach could be the use of available data produced by the other algorithms.

A. Available fire monitoring products

We can found fire map products for almost each algorithm published. Among them, some of the most popular are :

• for geostationary data, ABBA (GOES) and FIR (MSG) • for polar data, FIMMA (AVHHR) and MODIS Rapid Response System. The data are usually available in different formats : ascii, GIS products or image. The datasets vary from 15 minutes frequency for the last 24 hours to nine days cumulative results for the past 6 years.

These results tend to be very inhomogeneous and strongly dependant of the chosen set of parameters of the algorithms. It is the reason why direct comparison is difficult as no algorithm can be taken as absolute reference.

Nevertheless, the visual checking of the results with available fire monitoring products are coherent. Figure 1 shows for example the result obtained by the FIR algorithm (on the left) and our result (on the right) on Portugal during the summer 2005.

Then, a temporal checking has been done over Africa. The most unexpected observation was the duration of some fires (until several days). These exceptionnal situations always occur on erupting volcanoes. So, the work created for the monitoring of biomass burning could be extended to the monitoring of volcanic eruptions.

B. Monitoring volcanic eruptions: the Karthala's eruption example

Karthala volcano, an active basaltic shield, forms with the older La Grille Massif the island of Grande-Comore [START_REF] Lacroix | Le volcan actif de l'ile de la Réunion et celui de la Grande-Comore[END_REF], [START_REF] Bachèlery | Carte volcano-tectonique de la Grande-Comore ngazidja.au 1/50000 avec notice explicative[END_REF]. It has a summit caldera and two main rift zones [START_REF] Bachèlery | L'éruption phréatique du volcan karthala (grande-comore) en juillet 1991 : première crise sismo-volcanique suivie depuis l'implantation d'un réseau de surveillance[END_REF]. The center of the caldera has a pit crater 290m deep named "Choungou Chahale". The Karthala volcano forms the southern two-thirds of the Grande Comore Island in the Indian Ocean between latitude 11-13 ˚S and longitude 43-46 ˚E. This volcano was explored for the first time by Dr. Kresten in 1883 but there is little knowledge on the history, the structure, geology, the geophysics and the geodynamic context of the island. Grande-Comore is the youngest island of the Comoros archipelago which has four islands. A dating of the lava of low slopes of the top of the volcano by [START_REF] Bachèlery | Carte volcano-tectonique de la Grande-Comore ngazidja.au 1/50000 avec notice explicative[END_REF] indicated a date more recent than 130.000 years.

The eruptions of the volcano have been indexed since 1828. More than twenty eruptions have been recorded since the 19th century from both summit and flank vents [20]. The Karthala eruptive style is mostly effusive, however phreatic or phreato-magmatic explosions have occurred in 1918, 1948, 1952, 1991 and 2005 [21]. The one which took place on July 11, 1991 was purely phreatic and formed a new crater (250m Fig. 2. Observation of the Karthala eruption by MSG. In the low part of the chart the 3.9 and 10.8 mum channels. In the higher part the tests of the fire algorithm.

in diameter, 43m deep). In the bottom of Choungou-Chahale, which was partly filled by a lake. The eruption of November 2005 released ashes which covered the entire island. Most of the population was deprived of drinking water for a few days.

The eruption dealt with in this article is supposed to have begun on May 28, 2006 between 6:30pm and 8:00pm. It created a lake of lava in the crater Choungou Chahale. The Lake of lava apparently remained for five days. The flying over the crater on the fifth day of the eruption confirmed the cooling of the lake of lava.

This eruption was not preceded by seismic activities. It was thus a surprise for the inhabitants of the island. The observatory of Karthala which had two seismic stations working at the time of the eruption detected no abnormal activity before the eruption and the formation of the lake of lava within the crater. It was not the case of the last eruption of January 13, 2007 which had intense seismicity before and during the eruption. The latter reformed a lake of lava in the crater Choungou Chahalé.

Figure 2 shows the response of the algorithm for the pixels concerned by the eruption in orange. Above are represented the tests' responses of the components of the algorithm. Below the spectral response in 3.9 and 10.8mum channels. As you can see, there is a good correlation between the real start of the eruption and the hot spot detection. But we can see that the cloud test creates some lack of detection despite the fact that the eruption is still ongoing. Visual checking confirms the presence of the cloud, but the temperature of the eruption is so high that it nevertheless satures the pixel and should be detected. We conclued that classic fire detection algorithm has to be adapted to include the full detection of the volcanoes' eruptions.

IV. CONCLUSION

In this article, we have presented a student work dedicated to active fires' detection. This work follows through the previous pedagogical experience presented in 2005 [START_REF] Beaudoin | A Meteosat Second Generation receiving, processing and storing images system developed by engineer students[END_REF]. The first results are encouraging and multiple ways of improvement are planned. We have also presented another application for these algorithm with the volcano eruption detection and survey. Nevertheless, we have shown that full investigation will need some adaptations in the algorithms.

Fig. 1 .

 1 Fig. 1. Result of the FIR algorithm (on the left) and our result (on the right) on Portugal during the summer 2005.

ACKNOWLEDGMENT

The authors would like to thank Eumetsat and Météofrance organisations, Groupe De Recherche Meteosat Seconde Generation (GDR MSG) and Jean-Claude Bergès.


Yves Achdou

Mathieu Laurière

ON THE SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS ARISING IN MEAN FIELD TYPE CONTROL

Keywords: 2010 Mathematics Subject Classification, Primary: 49J20; Secondary: 35K55 mean field type control, existence and uniqueness, 1

We discuss the system of Fokker-Planck and Hamilton-Jacobi-Bellman equations arising from the finite horizon control of McKean-Vlasov dynamics. We give examples of existence and uniqueness results. Finally, we propose some simple models for the motion of pedestrians and report about numerical simulations in which we compare mean filed games and mean field type control.

(Communicated by the associate editor name) 1. Introduction. In the recent years, an important research activity has been devoted to the study of stochastic differential games with a large number of players. In their pioneering articles [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF]Jeux à champ moyen. II. Horizon fini et contrôle optimal[END_REF][START_REF]Mean field games[END_REF], J-M. Lasry and P-L. Lions have introduced the notion of mean field games, which describe the asymptotic behavior of stochastic differential games (Nash equilibria) as the number N of players tends to infinity. In these models, it is assumed that the agents are all identical and that an individual agent can hardly influence the outcome of the game. Moreover, each individual strategy is influenced by some averages of functions of the states of the other agents. In the limit when N → +∞, a given agent feels the presence of the other agents through the statistical distribution of the states of the other players. Since perturbations of a single agent's strategy does not influence the statistical distribution of the states, the latter acts as a parameter in the control problem to be solved by each agent. Another kind of asymptotic regime is obtained by assuming that all the agents use the same distributed feedback strategy and by passing to the limit as N → ∞ before optimizing the common feedback. Given a common feedback strategy, the asymptotics are given by the McKean-Vlasov theory, [START_REF] Mckean | A class of Markov processes associated with nonlinear parabolic equations[END_REF][START_REF] Sznitman | Topics in propagation of chaos[END_REF] : the dynamics of a given agent is found by solving a stochastic differential equation with coefficients depending on a mean field, namely the statistical distribution of the states, which may also affect the objective function. Since the feedback strategy is common to all agents, perturbations of the latter affect the mean field. Then, having each player optimize its objective function amounts to solving a control problem driven by the McKean-Vlasov dynamics. The latter is named control of McKean-Vlasov dynamics by R. Carmona and F. Delarue [START_REF] Carmona | Control of McKean-Vlasov dynamics versus mean field games[END_REF][START_REF] Carmona | Mean field forward-backward stochastic differential equations[END_REF] and mean field type control by A. Bensoussan et al, [START_REF] Bensoussan | Control and Nash games with mean field effect[END_REF][START_REF] Bensoussan | Mean field games and mean field type control theory[END_REF]. When the dynamics of the players are independent stochastic processes, both mean field games and control of McKean-Vlasov dynamics naturally lead to a coupled system of partial differential equations, a forward Fokker-Planck equation (which may be named FP equation in the sequel) and a backward Hamilton-Jacobi-Bellman equation (which may be named HJB equation). For mean field games, the coupled system of partial differential equations has been studied by Lasry and Lions in [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF]Jeux à champ moyen. II. Horizon fini et contrôle optimal[END_REF][START_REF]Mean field games[END_REF]. Besides, many important aspects of the mathematical theory developed by J-M. Lasry and P-L. Lions on MFG are not published in journals or books, but can be found in the videos of the lectures of P-L. Lions at Collège de France: see the web site of Collège de France, [START_REF] Lions | Cours du Collège de France[END_REF]. One can also see [START_REF] Gomes | Mean field games models-a brief survey[END_REF] for a brief survey. In the present paper, we aim at studying the system of partial differential equations arising in mean field type control, when the horizon of the control problem is finite: we will discuss the existence and the uniqueness of classical solutions. In the last paragraph of the paper, we briefly discuss some numerical simulations in the context of motion of pedestrians, and we compare the results obtained with mean field games and with mean field type control.

1.1. Model and assumptions. For simplicity, we assume that all the functions used below (except in § 4) are periodic with respect to the state variables x i , i = 1, . . . , d, of period 1 for example. This will save technical arguments on either problems in unbounded domains or boundary conditions. We denote by T d the d-dimensional unit torus: T d = R d /Z d . Let P be the set of probability measures on T d and P ∩L 1 (T d ) be the set of probability measures which are absolutely continuous with respect to the Lebesgue measure. For m ∈ P ∩L 1 (T d ), the density of m with respect to the Lebesgue measure will be still be noted m, i.e. dm(x) = m(x)dx. Let g be a map from P to a subset of C 1 (T d × R n ; R d ) ( the image of m ∈ P will be noted g

[m] ∈ C 1 (T d × R n ; R d ) ) such that
• there exists a constant M such that for all m ∈ P and

x ∈ T d , |g[m](x, 0)| ≤ M • there exists a constant L such that -for all m ∈ P, a ∈ R n and x, y ∈ T d , |g[m](x, a) -g[m](y, a)| ≤ Ld(x, y)
where d(x, y) is the distance between x and y in

T d . -for all m ∈ P, a, b ∈ R n and x ∈ T d , |g[m](x, a) -g[m](x, b)| ≤ L|a -b| -for all m, m ′ ∈ P, a ∈ R n and x ∈ T d , |g[m](x, a) -g[m ′ ](x, a)| ≤ Ld 2 (m, m ′
) where d 2 is the Wasserstein distance:

d 2 (m, m ′ ) ≡ inf γ∈Γ(m,m ′ ) T d ×T d d 2 (x, y)dγ(x, y) 1 2
, Γ(m, m ′ ) ≡ γ : transport plan between m and m ′ , and a transport plan γ between m and m ′ is a Borel probability measure on

T d × T d such that, for all Borel subset E of T d , γ(E × T d ) = m(E) and γ(T d × E) = m ′ (E).
• there exists a map g from

L 1 (T d ) to C 1 (T d × R n ; R d ) such that g| P ∩L 1 (T d ) = g| P ∩L 1 (T d ) and that for any x ∈ T d and a ∈ R n , m → g[m](x, a) is Fréchet differentiable in L 1 (T d ) and (x, a) → ∂g ∂m [m](x, a) belongs to C 1 (T d × R n ; L ∞ (T d ; R d ))
. Hereafter, we will not make the distinction between g and g.

Consider a probability space (Ω, A, P) and a filtration F t generated by a d-dimensional standard Wiener process (W t ) and the stochastic process (X t ) t∈[0,T ] in R d adapted to F t which solves the stochastic differential equation

dX t = g[m t ](X t , a t ) dt + √ 2ν dW t ∀t ∈ [0, T ], (1.1)
given the initial state X 0 which is a random variable F 0 -measurable whose probability density is noted m 0 . In (1.1), ν is a positive number, m t is the probability distribution of X t and a t is the control which we take to be

a t = v(t, X t ), (1.2)
where v(t, •) is a continuous function on T d . To the pair (v, m), we associate the objective

J(v, m) := E T 0 f [m t ](X t , a t )dt + h[m T ](X T ) (1.3) where f (resp. h) is a map from P to a subset of C 1 (T d × R n ), resp. to a subset of C 1 (T d ).
We assume that

• lim |a|→∞ inf m∈P,x∈T d f [m](x,a) |a| = +∞ • there exists a map f from L 1 (T d ) to C 1 (T d × R n ) such that f | P ∩L 1 (T d ) = f | P ∩L 1 (T d ) and that for any x ∈ T d and a ∈ R n , m → f [m](x, a) is Fréchet dif- ferentiable in L 1 (T d ) and (x, a) → ∂ f ∂m [m](x, a) belongs to C 1 (T d ×R n ; L ∞ (T d ))
. Hereafter, we will not make the distinction between f and f . We also assume that there exists a map h from L 1 (T d ) to C 1 (T d ) such that h| P ∩L 1 (T d ) = h| P ∩L 1 (T d ) and that for any

x ∈ T d , m → h[m](x) is Fréchet differentiable in L 1 (T d ) and x → ∂ h ∂m [m](x) belongs to C 1 (T d ; L ∞ (T d ))
. Hereafter, we will not make the distinction between h and h. It will be useful to define the Lagrangian and Hamiltonian as follows: for any

x ∈ T d , a ∈ R n and p ∈ R d , L[m](x, a, p) := f [m](x, a) + p • g[m](x, a) H[m](x, p) := min a∈R n L[m](x, a, p).
where p • q denotes the scalar product in R d . It is consistent with the previous assumptions to suppose that

• there exists a map H from L 1 (T d ) to C(T d × R d ) such that H| P ∩L 1 (T d ) = H| P ∩L 1 (T d ) and that for any x ∈ T d and p ∈ R d , m → H[m](x, p) is Fréchet dif- ferentiable in L 1 (T d ) and (x, p) → ∂ H ∂m [m](x, p) belongs to C 1 (T d ×R d ; L ∞ (T d )
). We will not make the distinction between H and H.

• if m ∈ P ∩L 1 (T d ) and a * = argmin a f [m](x, a) + p • g[m](x, a), then ∂H ∂m (x, p) = ∂f ∂m (x, a * ) + p • ∂g ∂m (x, a * ).
As explained in [START_REF] Bensoussan | Mean field games and mean field type control theory[END_REF], page 13, if the feedback function v is smooth enough and if m 0 ∈ P ∩L 1 (T d ), then the probability distribution m v (t, •) has a density with respect to the Lebesgue measure, m v (t, •) ∈ P ∩L 1 (T d ) for all t, and its density m v is solution of the Fokker-Planck equation

∂m v ∂t (t, x)-ν∆m v (t, x)+div m v (t, •)g[m v (t, •)](•, v(t, •)) (x) = 0, t ∈ (0, T ], x ∈ T d , (1.4) with the initial condition m v (0, x) = m 0 (x), x ∈ T d .
(1.5)

Therefore, the control problem consists of minimizing

J(v, m v ) = [0,T ]×T d f [m v (t, •)](x, v(t, x))m v (t, x)dxdt+ T d h[m v (T, •)](x)m v (T, x)dx,
subject to (1.4)-(1.5). In [START_REF] Bensoussan | Mean field games and mean field type control theory[END_REF], A. Bensoussan, J. Frehse and P. Yam have proved that a necessary condition for the existence of a smooth feedback function v * achieving

J(v * , m v * ) = min J(v, m v ) is that v * (t, x) = argmin v f [m(t, •)](x, v) + ∇u(t, x) • g[m(t, •)](x, v) ,
where (m, u) solve the following system of partial differential equations (1.8) It will be useful to write

0 = ∂u ∂t (t, x) + ν∆u(t, x) + H[m(t, •)](x, ∇u(t, x)) + T d ∂H ∂m [m(t, •)](ξ, ∇u(t, ξ))(x)m(t, ξ)dξ, (1.6)
G[m, q](x) := T d m(ξ) ∂ ∂m H[m](ξ, q(ξ))(x)dξ (1.9)
for functions m ∈ P ∩L 1 (T d ) and q ∈ C(T d ; R d ), so that (1.6) can be written

0 = ∂u ∂t (t, x) + ν∆u(t, x) + H[m(t, •)](x, ∇u(t, x)) + G[m(t, •), ∇u(t, •)](x
:= (0, T ) × T d .
We shall need to use spaces of Hölder functions in Q: For α ∈ (0, 1), the space of Hölder functions C α/2,α ( Q) is classically defined by

C α/2,α ( Q) := w ∈ C( Q) : ∃C > 0 s.t. ∀(t 1 , x 1 ), (t 2 , x 2 ) ∈ Q, |w(t 1 , x 1 ) -w(t 2 , x 2 )| ≤ C d(x 1 , x 2 ) 2 + |t 1 -t 2 | α/2
and we define

|w| C α/2,α ( Q) := sup (t1,x1) =(t2,x2)∈ Q |w(t 1 , x 1 ) -w(t 2 , x 2 )| (d(x 1 , x 2 ) 2 + |t 1 -t 2 |) α/2 and w C α/2,α ( Q) := w C( Q) +|w| C α/2,α ( Q) .
Then the space C (1+α)/2,1+α ( Q) is made of all the functions w ∈ C( Q) which have partial derivatives ∂w ∂xi ∈ C α/2,α ( Q) for all i = 1, . . . , d and such that for all (t 1 , x)

= (t 2 , x) ∈ Q, |w(t 1 , x) -w(t 2 , x)| ≤ C|t 1 -t 2 | (1+α)/2 for a positive constant C. The space C (1+α)/2,1+α ( Q), endowed with the semi-norm |w| C (1+α)/2,1+α ( Q) := d i=1 ∂w ∂x i C α/2,α ( Q) + sup (t1,x) =(t2,x)∈ Q |w(t 1 , x 1 ) -w(t 2 , x 2 )| |t 1 -t 2 | (1+α)/2 and norm w C (1+α)/2,1+α ( Q) := w C( Q) + |w| C (1+α)/2,1+α ( Q) is a Banach space.
Finally, the space C 1+α/2,2+α is made of all the functions w ∈ C 1 ( Q) which are twice continuously differentiable w.r.t. x, with partial derivatives ∂w ∂xi ∈ C (1+α)/2,1+α ( Q) for all i = 1, . . . , d, and ∂w ∂t ∈ C α/2,α ( Q). It is a Banach space with the norm

w C 1+α/2,2+α ( Q) := w C( Q) + d i=1 ∂w ∂x i C (1+α)/2,1+α ( Q) + ∂w ∂t C α/2,α ( Q) .
2.2. The case when ∂ p H is bounded. We make the following assumptions on h, m 0 , H and G, in addition to the regularity assumptions on H already made in § 1:

(H 0 ) For simplicity only, the map h is invariant w.r.t. m, i.e. h[m](x) = u T (x)
, where u T is a smooth function defined on T d . Moreover, m 0 is a smooth positive function.

(H 1 ) There exists a constant γ 0 > 0 such that

|H[m](x, 0)| ≤ γ 0 ∀(m, x) ∈ (P ∩L 1 (T d )) × T d (H 2 ) There exists a constant γ 1 > 0 such that ∂H ∂p [m] Lip(T d ×R d ) ≤ γ 1 ∀m ∈ P ∩L 1 (T d ) (H 3 ) For all (m, x, p) ∈ (P ∩L 1 (T d )) × T d × R d , ∂H ∂m [m](x, p) is a C 1 function on T d and there exists a constant γ 2 > 0 such that for all (m, x, p) ∈ (P ∩L 1 (T d )) × T d × R d , ∂H ∂m [m](x, p) C 1 (T d ) ≤ γ 2 (1 + |p|) (H 4 )
There exists a constant γ 3 > 0 such that:

∂H ∂p [m 1 ](•, 0) - ∂H ∂p H[m 2 ](•, 0) C(T d ) ≤ γ 3 m 1 -m 2 L 1 (T d ) ∀m 1 , m 2 ∈ L 1 (T d ). (H 5 ) There exists γ 4 > 0 such that for m 1 , m 2 ∈ P ∩L 1 (T d ), p 1 , p 2 ∈ L ∞ (T d ), G[m 1 , p 1 ] -G[m 2 , p 2 ] L ∞ (T d ) ≤ γ 4 p 1 -p 2 L ∞ (T d ) + m 1 -m 2 L 1 (T d ) .
Example. All the assumptions above are satisfied by the map H :

H[m](x, p) = - Φ(p) (c + (ρ 1 * m)(x)) α + F (x, (ρ 2 * m)(x)),
where Φ is a C 2 function from R d to R + such that D 2 Φ and DΦ are bounded, α and c are positive numbers, ρ 1 and ρ 2 are smoothing kernels in C ∞ (T d ), ρ 1 is nonnegative, and F is a C 2 function defined on

T d × R d . Here, ρ * m(x) = T d ρ(x -z)m(z)dz. It is easy to check that G[m, q](x) = αρ 1 * m Φ(q) (c + ρ 1 * m) α+1 (x) + ρ2 * (mF ′ (•, ρ 2 * m))(x)
where ρ1 (x) = ρ 1 (-x) and ρ2 (x) = ρ 2 (-x). Such a Hamiltonian models situations in which there are congestion effects, i.e. the cost of displacement increases in the regions where the density is large. The term F (x, (ρ 2 * m)(x)) may model aversion to crowded regions. The prototypical situation is g[m](x, a) = a and Φ(q) = min b∈K (q

• b + Φ * (b)), where K is a compact subset of R d . Setting Φ * (b) = +∞ if b / ∈ K, H corresponds to the cost f [m](x, a) = 1 (c+(ρ1 * m)(x)) α Φ * (a(c + (ρ 1 * m)(x)) α ) + F (x, (ρ 2 * m)(x)).

2.2.1.

A priori estimates. We first assume that (1.6)-(1.8) has a sufficiently smooth solution and we look for a priori estimates.

Step 1: uniform bounds on m L p (0,T ;

W 1,p (T d )) + m C α/2,α ( Q) , p ∈ [1, ∞), α ∈ [0, 1). First, standard arguments yield that m(t, •) ∈ P for all t ∈ [0, T ]. From Assumption (H 2 ), the function b : (t, x) → ∂ p H[m(t, •)](x, ∇u(t, x)) is such that b L ∞ (Q) ≤ γ 1 .
The Cauchy problem satisfied by m can be written

∂m ∂t (t, x) -ν∆m(t, x) + div(b(t, •)m(t, •))(x) = 0, m(0, x) = m 0 (x), (2.1)
and from the classical theory on weak solutions to parabolic equations, see e.g. Theorem 6.1 in [START_REF] Lieberman | Second order parabolic differential equations[END_REF], there exists a constant C 0 depending only on

m 0 L 2 (T d ) such that m L 2 0,T ;H 1 (T d ) + m C [0,T ];L 2 (T d ) ≤ C 0 .
Moreover, since the operator in (2.1) is in divergence form, we have maximum estimates on m, see Corollary 9.10 in [START_REF] Lieberman | Second order parabolic differential equations[END_REF]: there exists a constant C 1 depending only on m 0 ∞ and γ 1 such that

m(t, x) ≤ C 1 ∀(t, x) ∈ [0, T ] × T d . (2.2)
Therefore, the Fokker-Planck equation in (2.1) can be rewritten

∂m ∂t (t, x) -ν∆m(t, x) + div(B(t, •))(x) = 0, (2.3)
where

B ∞ ≤ γ 1 C 1 .
From from standard results on the heat equation, see [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF], this implies that for all p ∈ [1, ∞) there exists a constant C 2 (p) which depends on m 0 ∞ and γ 1 , such that

m L p 0,T ;W 1,p (T d ) + ∂m ∂t L p 0,T ;W -1,p (T d ) ≤ C 2 (p). (2.4)
Finally, Hölder estimates for the heat equation with a right hand side in divergence form, see for example Theorem 6.29 in [START_REF] Lieberman | Second order parabolic differential equations[END_REF], yield that for any α ∈ (0, 1), there exists a positive constant C 3 (α) ≥ C 1 which only depends on γ 1 and on

m 0 C α (T d ) such that m C α/2,α ( Q) ≤ C 3 (α). (2.5) Step 2: uniform bounds on u C (1+θ)/2,1+θ ( Q) , θ ∈ (0, 1). Defining a(t, x) := -H[m(t, •)](x, 0) and A(t, x) := 1 0 ∂H ∂p [m(t, •)](x, ζ∇u(t, x))dζ, the HJB equation (1.6) can be rewritten ∂u ∂t (t, x) + ν∆u(t, x) + A(t, x) • ∇u(t, x) = a(t, x) -G[m(t, •), ∇u(t, •)](x). (2.6)
For some smooth function û, let us consider ∂u ∂t

(t, x) + ν∆u(t, x) + A(t, x) • ∇u(t, x) = a(t, x) -G[m(t, •), ∇û(t, •)](x) (2.7)
instead of (2.6), with the same terminal condition as in (1.8). From Assumption (H 1 ) and

(H 2 ), a ∞ ≤ γ 0 and A ∞ ≤ γ 1 . From Assumption (H 3 ), G[m, ∇û] L 2 (T d ) ≤ c(1 + ∇û L 2 (T d ) ), (2.8)
where c > 0 depends on C 1 in (2.2) and γ 2 . Multiplying (2.7) by u(t, x)e -2Λt and integrating on T d , then using the bounds on a ∞ , A ∞ ≤ γ 1 and (2.8), a standard argument yields that there exist constants Λ and C4 which depend only on γ

0 , γ 1 , γ 2 , m 0 ∞ such that - d dt u(T -t, •) 2 L 2 (T d ) e -2Λ(T -t) + ν ∇u(T -t, •) 2 L 2 (T d ) e -2Λ(T -t) ≤ C4 + ν 2 ∇û(T -t, •) 2 L 2 (T d ) e -2Λ(T -t) .
(2.9)

Hence, if

ν T t=0 ∇û(T -t, •) 2 L 2 (T d ) e -2Λ(T -t) dt ≤ C 4 , (2.10)
with

C 4 = 2 C4 T + 2 T d u 2 T (x)dx, (2.11) then sup t e -2Λ(T -t) T d u 2 (T -t, x)dx + ν T t=0 ∇u(T -t, •) 2 L 2 (T d ) e -2Λ(T -t) dt ≤ C 4 .
(2.12) Similarly, a solution of (1.6)-(1.8) satisfies (2.12) with the same constants Λ and C 4 . Note that Λ can be chosen large enough such that the function (t, x) → u T (x) satisfies (2.12).

For a solution of (1.6)-(1.8), this implies that

∂ t u + ν∆u is bounded in L 2 (Q), hence that u is bounded in C 0 (0, T ; H 1 (T d )) by a constant C4 > u T H 1 (T d ) which depends on Λ, C 4 , γ 1 and u T H 1 (T d ) , i.e. ∇u L ∞ (0,T ;H 1 (T d )) ≤ C4 . (2.13)
As a consequence, the left-hand side of (2.6) is bounded in L ∞ (Q), and this yields Hölder estimates on u: by using Theorem 6.48 in [START_REF] Lieberman | Second order parabolic differential equations[END_REF], we see that for all θ ∈ (0, 1), there exists a constant

C 5 (θ) which depends on θ, m 0 ∞ , u T C 1+θ (T d ) , γ 0 , γ 1 , γ 2 such that u C (1+θ)/2,1+θ ( Q) ≤ C 5 (θ), (2.14)
which holds for a solution of (2.7) with the terminal condition (1.8), as soon as û satisfies (2.12) and (2.13).

Step 3: uniform bound on m C (1+θ)/2,1+θ ( Q) , θ ∈ (0, 1). Let us go back to (1.7). From Assumptions (H 1 ) -(H 4 ), and from the previous two steps, we see that for any θ ∈ (0, 1), m and ∂H ∂p [m](•, ∇u) are both bounded in C θ/2,θ ( Q) by constants which depend on m 0 and u T , and γ 0 , . . . , γ 3 . Thus, the function B in (2.3) is bounded in C θ/2,θ ( Q). Using Theorem 6.48 in [START_REF] Lieberman | Second order parabolic differential equations[END_REF] for the heat equation with a data in divergence form, we see that for all θ ∈ (0, 1), there exists a constant C 6 (θ) which depends on θ, m 0

C 1+θ (T d ) , u T C 1+θ (T d ) , γ 0 , . . . , γ 3 such that m C (1+θ)/2,1+θ ( Q) ≤ C 6 (θ).
Step 4: uniform bounds on u C 1+θ/2,2+θ ( Q) , θ ∈ (0, 1). From the previous steps and Assumptions (H 1 ) -(H 4 ), we see that there exists a constant c such that the functions in (2.6)

satisfy a C θ/2,θ ( Q) ≤ c and A C θ/2,θ ( Q) ≤ c. Similarly, from Assumptions (H 3 ) and (H 5 ), G[m, ∇u] C θ/2,θ ( Q) ≤ c.
Standard regularity results on parabolic equations, for instance Theorem 4.9 in [START_REF] Lieberman | Second order parabolic differential equations[END_REF] lead to the existence of

C 7 (θ) such that u C 1+θ/2,2+θ ( Q) ≤ C 7 (θ).
2.2.2. The existence theorem.

Theorem 1. Under the Assumptions (H 0 )-(H 5 ), for α ∈ (0, 1) there exist functions u ∈ C 1+α/2,2+α ( Q) and m ∈ C (1+α)/2,1+α ( Q) which satisfy (1.6)-(1.8), ( note that (1.7) is satisfied in a weak sense).

Proof. The argument is reminiscent of that used by J-M. Lasry and P-L. Lions for mean field games: it is done in two steps

Step A. For R > 0, let η R : R → R be a smooth, nondecreasing and odd function such that

1. η R (y) = y if |y| ≤ R, η R (y) = 2R if y ≥ 3R 2. η ′ R ∞ ≤ 1 We consider the modified set of equations 0 = ∂u ∂t (t, x) + ν∆u(t, x) + H[m(t, •)](x, ∇u(t, x)) + η R (G[m(t, •), ∇u(t, •)](x)), (2.15) 0 = ∂m ∂t (t, x) -ν∆m(t, x) + div m(t, •) ∂H ∂p [m(t, •)](•, ∇u(t, •)) (x). (2.16)
We are going to apply Leray-Shauder fixed point theorem to a map χ defined for example in

X = m ∈ C 0 ([0, T ]; L 2 (T d ) ∩ P) : consider first the map ψ : X → X × L 2 (0, T ; H 1 (T d )), m → (m, u)
where u is a weak solution of (2.15) and u| t=T = u T . Existence and uniqueness for this problem are well known. Moreover, from the estimates above, for every 0 < α < 1, u C 1/2+α/2,1+α ( Q) is bounded by a constant independent of m and m → u is continuous from X to C 1/2+α/2,1+α ( Q). Fix θ ∈ (0, 1), and consider the map ζ :

X × C 1/2+θ/2,1+θ ( Q) → L 2 (0, T ; H 1 (T d )), (m, u) → m where m is a weak solution of the Fokker-Planck equation 0 = ∂ m ∂t (t, x) -ν∆ m(t, x) + div m(t, •) ∂H ∂p [m(t, •)](•, ∇u(t, •)) (x).
and m| t=0 = m 0 . Existence and uniqueness are well known, and moreover, the estimates above tell us that for all 0 < α < 1, there exists R α > 0 such that m C α/2,α ( Q) ≤ R α uniformly with respect to m and u. Moreover from the assumptions, it can be seen that ζ maps continuously X × C 1/2+θ/2,1+θ ( Q) to X. Let K be the set { m C α/2,α ( Q) ≤ R α ; m| t=T = m T } ∩ X: this set is a compact and convex subset of X and the map χ = ζ • ψ: m → m is continuous in X and leaves K invariant. We can apply Leray-Shauder fixed point theorem the map χ, which yields the existence of a solution (u R , m R ) to (2.15)- (2.16). Moreover the a priori estimates above tell us that u R ∈ C 1+α/2,2+α ( Q) and m R ∈ C (1+α)/2,1+α ( Q).

Step B. Looking at all the a priori estimates above, it can be seen that m R , (resp u R ) belongs to a bounded subset of

C α/2,α ( Q) (resp. C 1/2+α/2,1+α ( Q)) independent of R. Hence, for R large enough, η R (G[m R , ∇u R ]) = G[m R , ∇u R ], and (m R , u R ) is a weak solution of (1.6)-(1.8), with u R ∈ C 1+α/2,2+α ( Q) and m R ∈ C (1+α)/2,1+α ( Q).
Remark 3. It is possible to weaken some of the assumptions in Theorem 1: for example, we can assume the following weaker version of (H 2 ), namely: (H ′

2 ) There exists a constant γ 1 > 0 and η ∈ (0, 1) such that

• ∀m ∈ P ∩L 1 (T d ), ∂H ∂p [m] C(T d ×R d ) ≤ γ 1 • ∀m ∈ P ∩L 1 (T d ), x, y ∈ T d , p, q ∈ R d , | ∂H ∂p [m](x, p) - ∂H ∂p [m](y, q)| ≤ γ 1 (d(x, y) + |p -q| η )
Indeed, the regularity of ∂H ∂p with respect to p is only used in Steps 3 and 4 above: with this weaker assumptions, the conclusions of steps 3 and 4 hold with 0 < θ < η, and this is enough for proving the existence of u ∈ C 1+α/2,2+α ( Q) and m ∈ C (1+α)/2,1+α ( Q) for some α, 0 < α < η which satisfy (1.6)-(1.8).

2.3.

Hamiltonian with a subquadratic growth in p: a specific case. For a smooth nonnegative periodic function ρ, two constants α > 0 and β, 1 < β ≤ 2, let us focus on the following Hamiltonian:

H[m](x, p) := - |p| β (1 + (ρ * m)(x)) α .
(2.17)

The map G defined in (1.9) is

G[m, q](x) = α ρ * m |q| β (1 + (ρ * m)) α+1 (x),
where ρ(x) := ρ(-x).

Assuming that m 0 is smooth, let us call m0 = m 0 ∞ : for all x ∈ T d , 0 < m 0 (x) ≤ m0 . We assume that

ρ L 1 (T d ) < β -1 α m0 . ( 2
[m](•, ∇u) L 1 (Q) , see (2.20)
.

2.3.1.

A priori estimates. We first assume that (1.6)-(1.8) has a sufficiently smooth weak solution and we look for a priori estimates.

Step 1: a lower bound on u. Since G is non negative, by comparison, we see that

u(t, x) ≥ u T := min ξ∈T d u(T, ξ) ∀(t, x) ∈ [0, T ] × T d .
Step 2: an energy estimate and its consequences. Let us multiply (1.6) by m -m0 and (1.7) by u and integrate the two resulting equations on T d . Summing the resulting identities, we obtain:

Q ∂ ∂t (u(t, x)(m(t, x) -m0 ))dxdt + Q H[m(t, •)](x, ∇u(t, x))(m(t, x) -m0 )dxdt + Q G[m(t, •), ∇u(t, •)](x)(m(t, x) -m0 )dxdt + Q div m(t, x) ∂ ∂p H[m(t, •)](x, ∇u(t, x)) u(t, x)dxdt = 0 Hence T d u(T, x)(m(T, x) -m0 )dx + T d u(0, x)( m0 -m(0, x))dx = Q H[m(t, •)](x, ∇u(t, x))( m0 -m(t, x))dxdt + Q G[m(t, •), ∇u(t, •)](x)( m0 -m(t, x))dxdt + Q m(t, x) ∂ ∂p H[m(t, •)](x, ∇u(t, x)) • ∇u(t, x)dxdt (2.19)
In (2.19), the first term in the left hand side is bounded from below by -u T ∞ (1 + m0 ), The second term is larger than u T T d ( m0 -m(0, x))dx = ( m0 -1)u T . Therefore, the left hand side of (2.19), is bounded from below by a constant c which only depends on m0 and u T ∞ ; we obtain that

c ≤(β -1) Q m(t, x)H[m(t, •)](x, ∇u(t, x))dxdt + Q m0 H[m(t, •)](x, ∇u(t, x))dxdt + α Q ( m0 -m(t, x))ρ * m(t, •) |∇u(t, •)| β (1 + ρ * m(t, •)) α+1 (x)dxdt.
We see that last term can be bounded as follows:

Q ( m0 -m(t, x))ρ * m(t, •) |∇u(t, •)| β (1 + ρ * m(t, •)) α+1 (x)dxdt ≤ m0 Q ρ * m(t, •) |∇u(t, •)| β (1 + ρ * m(t, •)) α (x)dxdt ≤ m0 ρ L 1 (T d ) Q m(t, x) |∇u(t, x)| β (1 + ρ * m(t, x)) α dxdt = -m0 ρ L 1 (T d ) Q m(t, x)H[m(t, •)](x, ∇u(t, x))dxdt. Therefore, c ≤ β -1 -α m0 ρ L 1 (T d ) Q m(t, x)H[m(t, •)](x, ∇u(t, x))dxdt + Q m0 H[m(t, •)](x, ∇u(t, x))dxdt.
From (2.17) and (2.18), we see that there exists a constant C 1 which depends on m0 and u T ∞ such that

mH[m](•, ∇u) L 1 (Q) + H[m](•, ∇u) L 1 (Q) ≤ C 1 .
(2.20)

Using (2.20), we deduce from a comparison argument applied to the HJB equation that there exists a constant C 2 which depends on m0 and u T ∞ such that

u L ∞ (Q) ≤ C 2 . (2.21) Since 1 < β ≤ 2, there exists a constant c such that | ∂H[m] ∂p (x, p)| 2 ≤ c(1 - H[m](x, p)).
We deduce from (2.20) and the latter observation that there exists a constant

C 3 > 0 such that Q (m(t, x) + 1) ∂H[m(t, •)] ∂p (x, ∇u(t, x)) 2 dxdt ≤ C 3 . ( 2

.22)

Step 3: uniform estimates from the Fokker-Planck equation. The following estimates can be proved exactly as in [START_REF]On the planning problem for the mean field games system[END_REF], Lemma 2.3 and Corollary 2.4, (see also [START_REF] Cardaliaguet | Long time average of mean field games[END_REF], Lemma 2.5 and Corollary 2):

Lemma 2. For γ = d+2 d if d > 2 and all γ < 2 if d = 2, there exists an constant c > 0, (independent from m 0 and u T ) such that

sup t∈[0,T ] m(t, •) log(m(t, •)) L 1 (T d ) + √ m 2 L 2 (0,T ;H 1 (T d )) + m γ L γ (Q) ≤c Q m(t, x) ∂H[m(t, •)] ∂p (x, ∇u(t, x)) 2 dxdt + T d m 0 (x) log(m 0 (x))dx .
(2.23)

Corollary 1. For q = d+2 d+1 if d > 2 and q < 4/3 if d = 2
, there exists a constant c > 0 such that

∇m q L q (Q) + ∂m ∂t q L q (0,T ;W -1,q (T d )) ≤c Q m(t, x) ∂H[m(t, •)] ∂p (x, ∇u(t, x)) 2 dxdt + T d m 0 (x) log(m 0 (x))dx .
(2.24)

From (2.24) and (2.22), we have a uniform bound on ∂m ∂t q L q (0,T ;W -1,q (T d )) by a constant depending only on u T and m 0 . We infer that (1.6) can be written

∂u ∂t (t, x) + ν∆u(t, x) + a(t, x)|∇u| β (t, x) = b(t, x), (2.25)
where a is a function which belongs to C([0, T ]; C p (T d )) for all p ∈ N, with corresponding norms bounded by constants depending only on u T and m 0 . From (2.20), we deduce that for all p ∈ N, b L 1 (0,T ;W p,∞ (T d )) is bounded by a constant depending only on u T and m 0 , because

b L 1 (0,T ;W p,∞ (T d )) = G[m, ∇u] L 1 (0,T ;W p,∞ (T d )) ≤ c m|∇u| β (1 + (ρ * m)) α+1 L 1 (Q) ≤ c mH[m](•, ∇u) L 1 (Q) .
Step 4: uniform estimates on |∇u|. Since a ∈ C([0, T ]; C p (T d )) and b ∈ L 1 (0, T ; W p,∞ (T d )), we can apply Bernstein method to (2.25) and estimate |∇u|. By a slight modification of the proof of Theorem 11.1 in [START_REF] Lieberman | Second order parabolic differential equations[END_REF], (the only difference is that in [START_REF] Lieberman | Second order parabolic differential equations[END_REF], b is supposed to belong to L ∞ (0, T ; W p,∞ (T d )), but it can be checked that this assumption can be weakened), we prove that there exists a constant C 4 which depends on u T and m 0 such that

∇u L ∞ (Q) ≤ C 4 .
(2.26)

The proof adapted from [START_REF] Lieberman | Second order parabolic differential equations[END_REF] is rather long, so we do not reproduce it here.

Step 5: stronger a priori estimates. Since |∇u| is bounded, we can recover all the a priori estimates in § 2.2.1, except that the estimates in Step 3 and 4 of § 2.2.1 only hold with 0 < θ < β -1, in view of Remark 3. We obtain that for all γ ∈ (0, 1), there exist two constants C 5 (γ) and C 6 (γ) such that m C γ/2,γ ( Q) ≤ C 5 (γ) and u C (1+γ)/2,1+γ ( Q) ≤ C 5 (γ), and that for all θ ∈ (0, β -1), there exist two constants

C 7 (θ) and C 8 (θ) such that m C (1+θ)/2,1+θ ( Q) ≤ C 7 (θ) and u C 1+θ/2,2+θ ( Q) ≤ C 8 (θ).

The existence theorem.

Theorem 3. We assume (H 0 ) and (2.18). For γ, 0 < γ < β -1, there exists a function u ∈ C 1+γ/2,2+γ ( Q) and m ∈ C (1+γ)/2,1+γ ( Q) which satisfy (1.6)-(1.8) with H given by (2.17).

Proof. We start by suitably truncating the Hamiltonian H and the map G: for R > 1, define

H R [m](x, p) =        - |p| β (1 + (ρ * m)(x)) α if |p| < R, - βR β-1 |p| + (1 -β)R β (1 + (ρ * m)(x)) α if |p| ≥ R, (2.27) and G R [m, q](x) = α ρ * m min(|q| β , R β ) (1 + (ρ * m)) α+1 (x). (2.28) Note that -H R [m](x, p) + ∂ ∂p H R [m](x, p) • p =        -(β -1) |p| β (1 + (ρ * m)(x)) α if |p| ≤ R, -(β -1) R β (1 + (ρ * m)(x)) α if |p| ≥ R.
(2.29) Thanks to Remark 3, we can use a slightly modified version of Theorem 1: for some γ, 0 < γ < β -1, there exists a solution (u R , m R ) of

0 = ∂u R ∂t (t, x) + ν∆u R (t, x) + H R [m R (t, •)](x, ∇u R (t, x)) + G R [m R (t, •), ∇u R (t, •)](x), 0 = ∂m R ∂t (t, x) -ν∆m R (t, x) + div m R (t, •) ∂H R ∂p [m R (t, •)](•, ∇u R (t, •) (x),
with the initial and terminal conditions (1.8), such that u R ∈ C 1+γ/2,2+γ ( Q) and m R ∈ C (1+γ)/2,1+γ ( Q).

Then it is possible to carry out the same program as in Step 1 and 2 in § 2.3.1: using (2.27)-(2.29), we obtain that there exists a constant c independent of R such that

c ≤ -β -1 -α m0 ρ L 1 (T d ) Q m R (t, x) |∇u R (t, x)| β (1 + ρ * m R (t, x)) α 1 {|∇uR(t,x)|<R} dxdt -β -1 -α m0 ρ L 1 (T d ) Q m R (t, x) R β (1 + ρ * m R (t, x)) α 1 {|∇uR(t,x)|≥R} dxdt + Q m0 H R [m R (t, •)](x, ∇u R (t, x))dxdt,
and this implies the counterpart of (2.20): there exists a constant C independent of R such that

(1 + m R ) min(|∇u R | β , R β ) (1 + (ρ * m R )) α L 1 (Q) ≤ C. (2.30)
From this, we obtain the counterpart of (2.22):

Q (m R (t, x) + 1) ∂H R [m R (t, •)] ∂p (x, ∇u R (t, x)) 2 dxdt ≤ C, (2.31)
where C is a constant independent of R. This estimate allows one for carrying out Steps 3 and 4 in § 2.3.1 and obtaining estimates independent of R: in particular, the same Bernstein argument can be used, and we obtain that there exists a constant independent of R such that ∇u R L ∞ (Q) ≤ C. In turn, step 5 in § 2.3.1 can be used and leads to estimates independent of R.

From this, taking R large enough yields the desired existence result.

3. Uniqueness.

+ T d ∂H ∂m [m(t, •)](ξ, ∇u(t, ξ))(x)m(t, ξ)dξ, (3.1) 0 = ∂m ∂t (t, x) -ν∆m(t, x) + div m(t, •) ∂H ∂p [m(t, •)](•, ∇u(t, •)) (x), (3.2)
and

0 = ∂ ũ ∂t (t, x) + ν∆ũ(t, x) + H[ m(t, •)](x, ∇ũ(t, x)) + T d ∂H ∂ m [ m(t, •)](ξ, ∇ũ(t, ξ))(x) m(t, ξ)dξ, (3.3) 0 = ∂ m ∂t (t, x) -ν∆ m(t, x) + div m(t, •) ∂H ∂p [ m(t, •)](•, ∇ũ(t, •)) (x). (3.4)
We subtract (3.3) from (3.1), multiply the resulting equation by (m(t, x) -m(t, x)), and integrate over Q. Similarly, we subtract (3.4) from (3.2), multiply the resulting equation by (u(t, x) -ũ(t, x)), and integrate over Q. We sum the two resulting identities: we obtain

0 = T d (u(T, x) -ũ(T, x))(m(T, x) -m(T, x))dx - T d (u(0, x) -ũ(0, x))(m(0, x) -m(0, x)) dx + T t=0 E[m(t, •), ∇u(t, •), m(t, •), ∇ũ(t, •)]dt. (3.5)
where

E[m 1 , p 1 , m 2 , p 2 ] = T d (H[m 1 ](x, p 1 (x)) -H[m 2 ](x, p 2 (x)))(m 1 (x) -m 2 (x))dx + T d (m 1 (x) -m 2 (x)) T d ∂H ∂m [m 1 ](ξ, p 1 (ξ))(x)m 1 (ξ) - ∂H ∂m [m 2 ](ξ, p 2 (ξ))(x)m 2 (ξ) dξdx - T d m 1 (x) ∂ ∂p H[m 1 ](x, p 1 (x)) -m 2 (x) ∂ ∂p H[m 2 ](x, p 2 (x)) (p 1 (x) -p 2 (x))dx.
Call δm = m 2 -m 1 and δp = p 2 -p 1 and consider the function e : [0, 1] → R defined by

e(θ) = 1 θ E[m 1 , p 1 , m 1 + θδm, p 1 + θδp], θ > 0, e(0) = 0. (3.6)
It can be checked that e is C 1 on [0, 1] and that its derivative is

e ′ (θ) = 2 T d T d ∂H ∂m [m 1 + θδm](ξ, p 1 + θδp(ξ))(x)δm(ξ)δm(x) + T d T d T d (m 1 (ξ) + θδm(ξ)) ∂ 2 H ∂m∂m [m 1 + θδm](ξ, p 1 + θδp(ξ))(x)(y)δm(x)δm(y) - T d (m 1 (x) + θδm(x))δp(x) • D 2 p,p H[m 1 + θδm](x, p 1 (x) + θδp(x))δp(x). (3.7)
Let us introduce the functional defined on

C(T d ) × C(T d ; R d ) by H[m, p] := T d m(x)H[m](x, p(x))dx. (3.8)
The second order Fréchet derivative of H with respect to m (respectively p) at (m, p) is a bilinear form on C(T d ), (resp.

C(T d ; R d )), noted D 2 m,m H[m, p], (resp. D 2 p,p H[m, p]) . For all m ∈ C(T d ) ∩ P and all p ∈ C(T d ; R d ), let us define the quadratic form Q[m, p] on C(T d ) × C(T d ; R d ) by Q[m, p](µ, π) = D 2 m,m H[m, p](µ, µ) -D 2 p,p H[m, p](π, π). (3.9)
We see that (3.7) can be written as follows:

e ′ (θ) = Q[m 1 + θδm, p 1 + θδp](δm, δp). (3.10)
Theorem 4. We assume (H 0 ) and that (m,

x, p) → H[m](x, p) is C 2 on C(T d ) × T d × R d . A sufficient condition for the uniqueness of a classical solution of (1.6)- (1.8) is that 1. for all m ∈ C(T d ) ∩ P and all p ∈ C(T d ; R d ), the quadratic form µ → D 2 m,m H[m, p](µ, µ) is positive definite 2. for all m ∈ C(T d )∩P and x ∈ T d , the real valued function p ∈ R d → H[m](x, p) is strictly concave. Proof. From the concavity of p → H[m](x, p), -D 2 p,p H[m, p] is positive semi-definite. Therefore, Q[m, p] is positive semi-definite, and Q[m, p](µ, π) = 0 implies that D 2
m,m H[m, p](µ, µ) = 0 and -D 2 p,p H[m, p](π, π) = 0, and therefore µ = 0. From (3.5), two solutions (u, m) and (ũ, m) of (1.6)-(1.8) satisfy

T t=0 E[m(t, •), ∇u(t, •), m(t, •), ∇ũ(t, •)]dt = 0, (3.11)
because m(0, •) = m(0, •) and ũ(T, •) = u(T, •). But, from (3.6) and (3.10), the properties of the quadratic form

Q[(1 -θ)m(t, •) + θ m(t, •), (1 -θ)∇u(t, •) + θ∇ũ(t, •)] imply that T t=0 E[m(t, •), ∇u(t, •), m(t, •), ∇ũ(t, •)]dt > 0 if m = m.
Therefore, (3.11) implies that m = m. Then,

0 = Q m(t, x) ∂H ∂p [m(t, •)](x, ∇u(t, x)) - ∂H ∂p [m(t, •)](x, ∇ũ(t, x)) • (∇u -∇ũ)(t, x). (3.12)
If ν > 0, then the maximum principle implies that m(t, x) > 0 for all t > 0, x ∈ T d . This observation, (3.12) and the strict concavity of H with respect to p imply that ∇u(t, x) = ∇ũ(t, x) > 0 for all t, x, which yields immediately that u = ũ by using (1.6).

Remark 5. Let us give an alternative argument which does not require the knowledge that m(t, x) > 0 for all t > 0, x ∈ T d . Such an argument may be useful in situations when ν = 0 or ν is replaced in (1.1) by a function of x which vanishes in some regions of T d . The strict concavity of H with respect to p and (3.12) yield the fact that u = ũ in the region where m > 0. This implies that

G[m(t, •)](x, ∇u(t, x)) = G[m(t, •)](x, ∇ũ(t, x)): hence, for all t and x, ∂u ∂t (t, x) + ν∆u(t, x) + H[m(t, •)](x, ∇u(t, x)) = ∂ ũ ∂t (t, x) + ν∆ũ(t, x) + H[m(t, •)](x, ∇ũ(t, x)).
We can then apply standard results on the uniqueness of the Cauchy problem with the HJB equation ∂u ∂t (t, x) + ν∆u(t, x) + H[m(t, •)](x, ∇u(t, x)) = g and obtain that u = ũ.

Corollary 2. In the case when H depends locally on m, i.e.

Example. Consider for example the Hamiltonian

H[m](x, p) = H(x, p, m(x)) = - |p| β (c + m(x)) α + F (m(x)), (3.13)
with c > 0, α > 0, β > 1, F a smooth function defined on R + . One can check that if α ≤ 1 and F is strictly convex, then uniqueness holds. Such a Hamiltonian arise in a local model for congestion, see [START_REF] Lions | Cours du Collège de France[END_REF].

Remark 6. The same analysis can be carried out for mean field games, see [START_REF] Lions | Cours du Collège de France[END_REF]: for example, under Assumption (H 0 ) and in the case when H depends locally on m, i.e. H[m](x, p) = H(x, p, m(x)), a sufficient condition for the uniqueness of a classical solution of (1.10)-(1.12) is that

2 ∂ H ∂m (x, p, m) -∂ ∂m ∇ T p H(x, p, m) -∂ ∂m ∇ p H(x, p, m) -2D 2
p,p H(x, p, m) be positive definite for all x ∈ T d , m > 0 and p ∈ R d . Here, we see that the sufficient condition involves the mixed partial derivatives of H with respect to m and p, which is not the case for mean field type control. If H depends separately on p and m as in [START_REF]Mean field games[END_REF], then ∂ ∂m ∇ p H(x, p, m) = 0 and the condition becomes: H is strictly concave with respect to p for m > 0 and non decreasing with respect to m, (or concave with respect to p and strictly increasing with respect to m).

Remark 7. The extension of the result on uniqueness to weak solutions is not trivial. In the context of mean filed games, one can find such results in [18] and [START_REF] Porretta | Weak solutions to Fokker-Planck equations and mean field games, to appear. 18. , On the planning problem for a class of mean field games[END_REF]: roughly speaking they rely on some new uniqueness results for weak solutions of the Fokker-Planck equation and on crossed regularity lemmas, see Lemma 5 in [18]. In the context of mean field type control, the same kind of analysis has not been done yet.

In the case when

n = d, g[m](x, v) = v and v → f [m](x, v) is strictly convex for all m ∈ P and x ∈ T d , it is well known that f [m](x, v) = sup q∈R d (H[m](x, q) -q • v). Furthermore if p → H[m](x, p
) is strictly concave for all m ∈ P and x ∈ T d , then

f [m](x, v) = max q∈R d (H[m](x, q) -q • v) (3.14)
and the maximum is achieved by a unique q. This observation leads to the following necessary condition for the assumption of Theorem 4 to be satisfied.

Proposition 1. Assume that n = d, g[m](x, v) = v, that v → f [m](x,
; R d ), m → H[m, p] is strictly convex in P ∩C(T d ), then for all v ∈ C(T d ; R d ), m → T d m(x)f [m](x, v(x))dx is strictly convex in P ∩C(T d ).
Proof. Take λ 1 > 0 and λ 2 > 0 such that λ 1 + λ 2 = 1 and m 1 = m 2 in P ∩C(T d ).

From (3.14),

T d (λ 1 m 1 (x) + λ 2 m 2 (x))f [λ 1 m 1 + λ 2 m 2 ](x, v(x))dx = T d max q∈R d (λ 1 m 1 (x) + λ 2 m 2 (x)) (H[λ 1 m 1 + λ 2 m 2 ](x, q) -qv(x)) dx.
If for all x ∈ T d , the maximum in the latter integrand is achieved by q * (x), then x → q * (x) is a continuous function (from the continuity of v) and we have

T d (λ 1 m 1 (x) + λ 2 m 2 (x))f [λ 1 m 1 + λ 2 m 2 ](x, v(x))dx = max q∈C(T d ;R d ) T d (λ 1 m 1 (x) + λ 2 m 2 (x)) (H[λ 1 m 1 + λ 2 m 2 ](x, q(x)) -q(x)v(x)) dx.
From this and the convexity of m → H[m, p], we deduce that

T d (λ 1 m 1 (x) + λ 2 m 2 (x))f [λ 1 m 1 + λ 2 m 2 ](x, v(x))dx < max q∈C(T d ;R d ) λ 1 T d (m 1 (x)H[m 1 ](x, q(x)) -q(x)v(x)) dx+ λ 2 T d (m 2 (x)H[m 2 ](x, q(x)) -q(x)v(x)) dx ≤λ 1 max q∈C(T d ;R d ) T d (m 1 (x)H[m 1 ](x, q(x)) -q(x)v(x)) dx + λ 2 max q∈C(T d ;R d ) T d (m 2 (x)H[m 2 ](x, q(x)) -q(x)v(x)) dx =λ 1 T d m 1 (x)f [m 1 ](x, v(x))dx + λ 2 T d m 2 (x)f [m 2 ](x, v(x))dx.
(z, m z ) = Q f [m z (t, •)] x, z(t, x) m z (t, x) m z (t, x)dxdt + T d u T (x)m z (T, x)dx, (3.15) subject to the linear constraints ∂m z ∂t (t, x) -ν∆m z (t, x) + divz(t, x) = 0 t ∈ (0, T ], x ∈ T d , (3.16)
with the initial condition

m z (0, x) = m 0 (x), x ∈ T d . (3.17)
For simplicity, we assume that f depends locally on m, i.e.

f [m] (x, v(x)) = f (x, v(x), m(x)) , ∀x ∈ T d .
We are going to look for sufficient conditions for (z, m) → m f (x, z m , m) be a convex function. This condition will thus yield the uniqueness for the above control problem. Assuming that all the following differentiations are allowed, we see that the Hessian of the latter function is

Θ(x, v, m) = 1 m 3 z • D 2 vv f z m , m z -1 m 2 z • D 2 vv f z m , m -1 m 2 D 2 vv f z m , m z 1 m D 2 vv f z m , m + 2 ∂ f ∂m z m , m + m ∂ 2 f ∂m 2 z m , m -2 z m • ∂∇v f ∂m z m , m ∂∇ T v f ∂m z m , m ∂∇v f ∂m z m , m 0
where we have omitted the dependency on x for brevity. This is better understood when expressed in terms of (v, m): Proof. We observe first that the positive definiteness of Θ implies that D 2 vv f (x, v, m) is positive definite for all x ∈ T d , m > 0 and v ∈ R d . Let us call v * ∈ R d the vector achieving H(x, p, m) = p • v * + f (x, v * , m). We know that ∇ p H(x, p, m) = v * . Differentiating the optimality condition for v * with respect to p, we find that

Θ(x, v, m) = ∂ 2 ∂m 2 m f (x, v, m) m ∂∇ T v f ∂m (x, v, m) m ∂∇v f ∂m (x, v, m) mD 2 vv f (x, v, m) . ( 3
D 2 p,p H(x, p, m) = -D 2 v,v f (x, v * , m) -1 . ( 3
∂ H ∂m (x, p, m) = ∂ f ∂m (x, v * , m)+∇ v f (x, v * , m)• ∂v * ∂m +p• ∂v * ∂m = ∂ f ∂m (x, v * , m), (3.20)
where the last identity comes from the definition of v * . Differentiating once more with respect to m, we find that

∂ 2 H ∂m 2 (x, p, m) = ∂ 2 f ∂m 2 (x, v * , m) + ∂∇ v f ∂m (x, v * , m) • ∂v * ∂m . ( 3

.21)

Then the implicit function theorem applied to the optimality condition for v * yields that

∂v * ∂m = -D 2 v,v f (x, v * , m) -1 ∂∇ v f ∂m (x, v * , m). ( 3
∂ 2 ∂m 2 m H(x, p, m) = ∂ 2 ∂m 2 m f (x, •, m) (v * ) -m ∂∇ v f ∂m (x, v * , m) • mD 2 v,v f (x, v * , m) -1 m ∂∇ v f ∂m (x, v * , m) . (3.23) Hence, ∂ 2 ∂m 2 m H(x, p, m) is a Schur complement of Θ(x, v * , m)
. Therefore, it is positive definite and we have proved the second condition on H. 4. Numerical Simulations. Here we model a situation in which a crowd of pedestrians is driven to leave a given square hall (whose side is 50 meters long) containing rectangular obstacles: one can imagine for example a situation of panic in a closed building, in which the population tries to reach the exit doors. The chosen geometry is represented on Figure 1. The aim is to compare the evolution of the density in (1 + m)

while (4.2) and the boundary condition are unchanged. The initial density m 0 is piecewise constant and takes two values 0 and 4 people/m 2 , see Figure 1. At t = 0, there are 3300 people in the hall. We use the finite difference method originally proposed in [START_REF] Achdou | Mean field games: numerical methods[END_REF], see [START_REF] Achdou | Finite difference methods for mean field games, Hamilton-Jacobi equations: approximations, numerical analysis and applications[END_REF] for some details on the implementation and [START_REF] Achdou | Mean field games: convergence of a finite difference method[END_REF] for convergence results. On Figure 2, we plot the density m obtained by the simulations for the two models, at t = 1, 2, 5 and 15 minutes. With both models, we see that the pedestrians rush towards the narrow corridors leading to the exits, at the left and right sides of the hall, and that the density reaches high values at the intersections of corridors; then congestion effects explain why the velocity is low (the gradient of u) in the regions where the density is high. On the figure, we see that the mean field type control leads to a slower exit of the hall, with lower peaks of density.

  x) -ν∆m(t, x) + div m(t, •) ∂H ∂p [m(t, •)](•, ∇u(t, •)) (x), (1.7) with the initial and terminal conditions m(0, x) = m 0 (x) and u(T, x) = h[m(T, •)](x) + T d ∂h ∂m [m(T, •)](ξ)(x)m(T, ξ)dξ.

H

  [m](x, p) = H(x, p, m(x)), the sufficient condition in Theorem 4 is implied by the strict concavity of p ∈ R d → H(x, p, m) for all m > 0 and x ∈ T d and the strict convexity of the real valued function m ∈ R + → m H(x, p, m), for all p ∈ R d .

3. 2 .

 2 Back to the control of McKean-Vlasov dynamics. As in the end of the previous paragraph, we assume that n = d and g[m](x, v) = v. The control of McKean-Vlasov dynamics can be written as a control problem with linear constraints by making the change of variables z = mv: it consists of minimizing J

Proposition 2 .Proposition 3 .

 23 We assume that n = d and g[m](x, v) = v, and that f[m] (x, v(x)) = f (x, v(x), m(x)), for all x ∈ T d ,where f is a smooth function. A sufficient condition for the uniqueness of a minimum (z * , m * ) such that m * > 0 is that Θ(x, v, m) be positive definite for all x ∈ T d , m > 0 and v ∈ R d . We make the same assumptions as in Proposition 2. The positive definiteness of Θ(x, v, m) for all x ∈ T d , m > 0 and v ∈ R d implies the sufficient conditions on H in Corollary 2.

Figure 1 .

 1 Figure 1. Left: the geometry. Right: the density at t = 0 two models: 1. Mean field games: we choose ν = 0.012 and the Hamiltonian to be of the form (3.13), i.e. which takes congestion effects into account and depends locally on m; more precisely: H(x, p, m) = -8|p| 2

2 ) 2 ( 1

 221 The horizon T is T = 50 minutes. There is no terminal cost. There are two exit doors, see Figure1. The part of the boundary corresponding to the doors is called Γ D . The boundary conditions at the exit doors are chosen as follows: there is a Dirichlet condition for u on Γ D , corresponding to an exit cost; in our simulations, we have chosen u = 0 on Γ D . For m, we may assume that m = 0 outside the domain, so we also get the Dirichlet condition m = 0 on Γ D . The boundary Γ N corresponds to the solid walls of the hall and of the obstacles. A natural boundary condition for u on Γ N is a homogeneous Neumann boundary condition, i.e. ∂u ∂n = 0 which says that the velocity of the pedestrians is tangential to the walls. The natural condition for the density m is thatν ∂m ∂n + m ∂ H ∂p (•, ∇u, m) • n = 0,therefore ∂m ∂n = 0 on Γ N . 2. Mean field type control: this is the situation where pedestrians or robots use the same feedback law (we may imagine that they follow the strategy decided by a leader); we keep the same Hamiltonian, and the HJB equation becomes ∂u ∂t + 0.012 ∆u -+ m)

Figure 2 .

 2 Figure 2. The density computed with the two models at different dates. Left: Mean field game. Right: Mean field type control. The scales vary from one date to the other

  ).

	Remark 2. At least formally, it is possible to consider situations when H and h depend locally on m, i.e. H[m](x, p) = H(x, p, m(x)) and h[m](x) = h(x, m(x)): in
	this case, (1.6)-(1.8) become
	0 =	∂u ∂t +m(t, x) (t, x) + ν∆u(t, x) + H(x, ∇u(t, x), m(t, x)) ∂ ∂m (x, ∇u(t, x), m(t, x)), H	(1.13)
	0 =	∂m ∂t	(t, x) -ν∆m(t, x) + div m(t, •)	∂ ∂p H	(•, ∇u(t, •), m(t, •)) (x), (1.14)
	with the initial and terminal conditions
	m(0, x) = m 0 (x) and u(T, x) = h(x, m(T, x)) + m(T, x)	∂ ∂m h	(x, m(T, x)). (1.15)
	2. Existence results. We focus on the system (1.6)-(1.8). We are going to state
	existence results in some typical situations.
	2.1. Notations. Let Q be the open set Q
	Remark 1. Note the difference with the system of partial differential equations
	arising in mean field games, namely
	0 =	∂u ∂t	(t, x) + ν∆u(t, x) + H[m(t, •)](x, ∇u(t, x)),	(1.10)
	0 =	∂m ∂t	(t, x) -ν∆m(t, x) + div m(t, •)	∂H ∂p	[m(t, •)](•, ∇u(t, •)) (x), (1.11)
	with the initial and terminal conditions
			m(0, x) = m 0 (x) and u(T, x) = h[m(T, •)](x).	(1.12)
	Both the HJB equation (1.6) and the terminal condition on u in (1.8) involve ad-
	ditional nonlocal terms, which account for the variations of m v caused by variations
	of the common feedback v.

  3.1. Uniqueness for (1.6)-(1.8): a sufficient condition.In what follows, we prove sufficient conditions leading to the uniqueness of a classical solution of (1.6)-(1.8). For simplicity, we still assume that the final cost does not depend on the density, i.e. that there exists a smooth function u T such that h[m](x) = u T (x). In order to simplify the discussion, we assume that the operator H depends smoothly enough on its argument to give sense to the calculations that follow.

We consider two classical solutions (u, m) and (ũ, m) of 0 = ∂u ∂t (t, x) + ν∆u(t, x) + H[m(t, •)](x, ∇u(t, x))

  v) is strictly convex for all m ∈ P and x ∈ T d , and that p → H[m](x, p) is strictly concave for all m ∈ P and x ∈ T d . If for all p ∈ C(T d

  .19) Note that(3.19) implies the strict concavity of p → H(x, p, m) which is the first desired condition on H. The second condition on H will be a consequence of the implicit function theorem: differentiating H with respect to m, we find that

Acknowledgements. We warmly thank A. Bensoussan for helpful discussions. The first author was partially funded by the ANR projects ANR-12-MONU-0013 and ANR-12-BS01-0008-01. The second author was partially funded by the Research Grants Council of HKSAR (CityU 500113).