Marco Caponigro
email: marco.caponigro@cnam.fr

Benedetto Piccoli
email: piccoli@camden.rutgers.edu

Francesco Rossi
email: francesco.rossi@lsis.org.

Emmanuel Trélat
email: emmanuel.trelat@upmc.fr

Mean-Field Sparse Jurdjevic-Quinn Control

Introduction and main result 1.The context

In recent years, the study of collective behavior of a crowd of autonomous agents has drawn a great interest from scientific communities, e.g., in civil engineering (for evacuation problems [START_REF] Cristiani | Multiscale Modeling of Pedestrian Dynamics, Springer MS & A: Modeling, Simulation and Applications[END_REF][START_REF] Helbing | Simulating dynamical features of escape panic[END_REF]), robotics (coordination of robots [START_REF] Bullo | Distributed control of robotic networks: a mathematical approach to motion coordination algorithms[END_REF][START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules, Automatic Control[END_REF][START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF][START_REF] Rahmani | Controllability of multi-agent systems from a graph-theoretic perpective[END_REF]), computer science and sociology (social networks [START_REF] Hegselmann | Opinion dynamics and bounded confidence: models, analysis and simulation[END_REF]), and biology (animals groups [START_REF] Bellomo | On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms[END_REF][START_REF] Couzin | Collective memory and spatial sorting in animal groups[END_REF][START_REF] Giardina | Collective behavior in animal groups: theoretical models and empirical studies[END_REF]). In particular, it is well known that some simple rules of interaction between agents can promote formation of special patterns, like lines in ants formations and migrating lobsters, or V-shaped formation in migrating birds. This phenomenon is often referred to as self-organization.

Beside the problem of analyzing the collective behavior of a "closed" system [START_REF] Cristiani | Effects of anisotropic interactions on the structure of animal groups[END_REF], it is interesting to understand what changes of behavior can be induced by an external agent (e.g., a policy maker) to the crowd. For example, one can try to enforce creation of patterns when they are not formed naturally, or break the formation of such patterns [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF][START_REF] Caponigro | Sparse Stabilization and Control of Alignment Models[END_REF][START_REF] Fornasier | Mean-field sparse optimal control[END_REF][START_REF] Piccoli | Control to flocking of the kinetic Cucker-Smale model[END_REF][START_REF] Herty | Mean-field control and Riccati equations[END_REF]. This is the problem of control of crowds, that we address in this article in a specific case.

From the mathematical point of view, problems related to models of crowds are of great interest. From the analysis point of view, one needs to pass from a big set of simple rules for each individual to a model capable of capturing the dynamics of the whole crowd. This can be solved via the so-called mean-field process, that permits to consider the limit of a set of ordinary differential equations (one for each agent) to a partial differential equation (PDE in the following) for the whole crowd [START_REF] Piccoli | Control to flocking of the kinetic Cucker-Smale model[END_REF]. The resulting equation is a transport equation with non-local velocity, of the form

∂ t µ + ∇ • (f [µ] µ) = 0, ( 1
)
where µ is the measure representing the density of agents, ∇• is the divergence operator and f [µ] is a vector field depending on the measure, taking into account interactions between agents. Many kinetic equations are of this form. We recall fundamental properties of such classes of equations in Section 2. We just highlight here that we assume in the following that f [µ] is a bounded Lipschitz vector field for any µ, Lipschitz with respect to the Wasserstein distance W p , p ∈ [1, +∞), as a function of µ. This ensures existence and uniqueness of the solution of the associated Cauchy problem [START_REF] Ambrosio | Hamiltonian ODEs in the Wasserstein Space of Probability Measures[END_REF][START_REF] Piccoli | Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes[END_REF]. Since such an equation generates a semigroup, we will use the notation e tf µ 0 to denote the unique solution of (1) at time t with initial data µ 0 . We recall existence results for (1) in Section 2.

Control of transport equations with non-local velocity

To control systems described by [START_REF] Agrachev | Controllability on the group of diffeomorphisms[END_REF], we assume to act only on a small part of the crowd. Since agents are indistinguishable when one only knows µ(t) at time t, controls can only be state-dependent and cannot focus on specific agents. For this reason, we model the control action by means of a vector field g, and a control gain u(t, x) localized in a small control set ω(t) (itself depending on time), modeling our choice of the gain on the vector field. We choose the control gain u and the control set ω, both of them varying in time while the vector field g is fixed, and it depends on the density µ. The resulting control system is given by

∂ t µ + ∇ • ((f [µ] + χ ω ug[µ])µ) = 0. (2)
Here, the function χ ω is the indicator function of ω, defined almost verywhere by χ ω (x) = 1 if x ∈ ω and χ ω (x) = 0 otherwise. We focus on the following continuous sparse space constraint: we assume to act only on a small portion of the configuration space, with finite strength. Accordingly, we assume the following constraints. Here, given a measurable subset ω of R d , we denote by |ω| its Lebesgue measure.

Control Constraints (U)

Fix c > 0. For each time t ≥ 0, we have:

Sparsity space constraint: |ω(t)| ≤ c, (3)
Finite strength:

u(t, •) L ∞ ≤ 1. (4)
Control sparsity constraints have been first introduced in [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF][START_REF] Caponigro | Sparse Stabilization and Control of Alignment Models[END_REF], for a population with a finite number of agents. The sparsity space constraint was considered in [START_REF] Piccoli | Control to flocking of the kinetic Cucker-Smale model[END_REF]. In the mean-field approach, this is actually the most natural sparsity constraint when one wants to use space-dependent vector fields and to act on "small sets". A sparsity population constraint was also considered in [START_REF] Piccoli | Control to flocking of the kinetic Cucker-Smale model[END_REF].

For the sparsity space constraint, one can easily deal both with measures that are absolutely continuous with respect to the Lebesgue measure, and with measures containing singular (Dirac) parts. We denote by P c (R d ) the space of probability measures on R d with compact support and by P ac c (R d ) the subspace of probability measures on R d that are absolutely continuous with respect to the Lebesgue measure.

In the following, we will define a control strategy that satisfies the following property: if the initial data, at time 0, is absolutely continuous with respect to the Lebesgue measure, then it remains absolutely continuous for any positive time. This does not prevent µ(t) of converging to some Dirac mass as t → +∞, as this is the case for consensus problems for crowd models.

In this paper, our objective is to generalize the Jurdjevic-Quinn stabilization method [START_REF] Jurdjevic | Controllability and stability[END_REF] to mean-field controlled equations, under the sparsity constraint (U) described above. Following the Jurdjevic-Quinn approach, we assume to have available a Lyapunov function V for which:

• the uncontrolled dynamics f [µ] gives no increase of V ;

• the control ug[µ] allows one to increase-decrease V , except for some specific configurations of the population µ in the subset Z of the set of measures with compact support P c (R d ) defined as the set on which the Lie derivatives of V vanish (see the precise definition in ( 8)).

We will then define a sparse control strategy, steering the population exactly to the set Z, in complete analogy with the standard finite-dimensional Jurdjevic-Quinn method.

The vector fields f and ug defined on the space R d play the role of derivatives for the function V [µ], in the following sense: the vector field f induces an infinitesimal change in V that can be estimated as the derivative lim t→0

V [e tf µ]-V [µ] t
. This limit needs to be well-defined, and this is why we will assume from now on the following regularity assumptions on f : for any

µ ∈ P c (R d ), the function t → V [e tf µ] is of class C 2 .
Then, in analogy with the definition of a Lie derivative in finite dimension, we give the following definition. Definition 1.1. We define the Lie derivative of V along f as the limit

L f V [µ] = lim t→0 V [e tf µ] -V [µ] t . (5)
Requiring the non-increase of V along the flow of f is equivalent to require dissipativity for the system.

Definition 1.2. We say that the system (2) is dissipative if there exists a Lyapunov function

V : P c (R d ) → R such that t → V [e tf µ] is of class C 2 and L f V [µ] ≤ 0 ∀µ ∈ P c (R d ). (6)
Regularity conditions need to be satisfied as well for the controlled vector field ug. For this reason, we define the space U of admissible control functions u, and from now on, we assume that U = Lip(R d , R). In what follows, we impose the following regularity assumption: for all µ ∈ P c (R d ) and u ∈ U, the function t → V [e tug µ] is of class C 2 . As a consequence, the limit

L ug V [µ] = lim t→0 V [e tug µ] -V [µ] t (7)
is well defined.

Remark 1. Accordingly, the notion of Lie derivative can be extended to piecewise constant functions t → u(t, •) ∈ U.

Definition 1.3. Assume that V : P c (R d ) → R is such that t → V [e tf µ] and t → V [e tug µ] are of class C 2
for all µ ∈ P c (R d ) and u ∈ U. We define

Z = µ ∈ P c (R d ) | L f V [µ] = L ug V [µ] = 0 ∀u ∈ Lip(R d , R) . (8)
Note that the definition of Lie derivative implies the multiplicative property

L kf V [µ] = kL f V [µ] and L kug V [µ] = kL ug V [µ]. (9)
This continuity condition also implies additivity of Lie derivatives. Indeed, one can easily see that e t(u+u )g = e tug+o(t) e tu g , which in turn implies that

L (u+u )g V [µ] = L ug V [µ] + L u g V [µ].
While conditions ( 5)-( 7) are equivalent to differentiability of V along f and ug, we also need a kind of differentiability for V along directions of the dynamics. By the additivity property, we can state it as follows: there exists K > 0 such that, for all µ ∈ P c (R d ) and u ∈ U, we have

|L ug V [µ]| ≤ K u L 1 (µ) . (10)
This yields a metric for the space of controls u similar to the zero-order metric in a more general sub-Riemannian structure for metrics on the space of diffeomorphisms on a manifold [START_REF] Arguillere | Sub-Riemannian structures on groups of diffeomorphisms[END_REF] (see also [START_REF] Agrachev | Controllability on the group of diffeomorphisms[END_REF]). The main difference here is that we choose the L 1 norm weighted with respect to the measure µ(t), and not with respect to the Lebesgue measure. While conditions ( 5)-( 7)-( 10) hold for a fixed µ ∈ P c (R d ), we also require the continuity of the Lie derivatives of first and second order. We require that

lim i→+∞ L f V [µ i ] = L f V [µ], lim i→+∞ L 2 f V [µ i ] = L 2 f V [µ], lim i→+∞ L ug V [µ i ] = L ug V [µ], lim i→+∞ L 2 ug V [µ i ] = L 2 ug V [µ], lim i→+∞ L f L ug V [µ i ] = L f L ug V [µ], (11)
for all µ i and µ ∈ P c (R d ), with µ i µ (weak convergence of measures), i.e., lim i→+∞ φ dµ i = φ dµ for every φ ∈ C 0 c (R d , R). The conditions [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF] imply in particular that L f +ug V [µ] exists and satisfies

L f +ug V [µ] = L f V [µ] + L ug V [µ],
i.e., that additivity holds also for the vector field f + ug.

Remark 2. Clearly, the choice of the set of admissible controls U has an impact on the set of admissible functionals V for which ( 7)-( 10)-( 11) are satisfied. We choose here the set of Lipschitz functions because existence is then ensured for (2) (see [START_REF] Ambrosio | Hamiltonian ODEs in the Wasserstein Space of Probability Measures[END_REF][START_REF] Piccoli | Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes[END_REF]).

Note that reducing the space of admissible controls to some proper subset of Lip(R d , R) (such as

C ∞ c (R d , R
)) may enlarge the set of functionals V for which the regularity conditions ( 5)-( 7)-( 10)-( 11) are satisfied. In Section 3, we will enforce the decrease of the functional V by a steepest descent method on the space U, by (approximately) solving an optimization problem in the space of Lipschitz functions.

Following the classical Lyapunov theory for finite-dimensional systems, we need to impose some conditions ensuring compactness of trajectories. In finite dimension, this is often imposed by requiring V to be proper, i.e., lim |x|→+∞ V (x) = +∞, hence the fact that d dt V (x(t)) ≤ 0 implies compactness. In the present meanfield setting, instead, such a condition cannot be imposed by a simple evaluation of the function V : in the case where V is the variance of the measure, one can have measures µ with arbitrarily small variance and arbitrarily large support. For this reason, we impose compactness of trajectories by assuming that the dynamics of the system, i.e., the vector fields f and g, have a compact support. More precisely, we assume the existence of a ball B(0, R) such that for all µ ∈ P c (B(0, R)),

f [µ](x) = g[µ](x) = 0 ∀x ∈ B(0, R). (12)
Note that this condition implies that µ(t) ∈ P c (B(0, R)) for any t ≥ 0. Then, all assumptions described above will be assumed to hold for measures in P c (B(0, R)) only.

Assumptions (H)

The vector fields f, g :

P c (R d ) → Lip(R d , R d ) and the Lyapunov function V : P c (R d ) → R satisfy Assump- tions (H) if:
• there exists R > 0 such that f, g satisfy the compact support property (12);

• there exist L > 0, Q > 0 and p ≥ 1 such that, for all µ, ν ∈ P c (B(0, R)) and for all x, y ∈ R d ,

|f [µ](x) -f [µ](y)| ≤ L|x -y|, |g[µ](x) -g[µ](y)| ≤ L|x -y|, |f [µ](x) -f [ν](x)| ≤ QW p (µ, ν), |g[µ](x) -g[ν](x)| ≤ QW p (µ, ν); (13)
• the functions t → V [e tf µ] and t → V [e tug µ] are of class C 1 for all µ ∈ P c (B(0, R)) and u ∈ U; in particular, the Lie derivatives ( 5) and ( 7) exist;

• the uncontrolled system is dissipative, i.e., L f V [µ] ≤ 0 for any µ ∈ P c (B(0, R));

• the Lie derivative [START_REF] Bertozzi | Blow-up in multidimensional aggregation equations with mildly singular interaction kernels[END_REF] satisfies the Lipschitz condition [START_REF] Canuto | An Eulerian approach to the analysis of Krause's consensus models[END_REF] and the continuity condition [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF] for any µ ∈ P c (B(0, R)).

Remark 4. The uniform Lipschitz property of f and g in ( 13) and the uniform compactness of their support in [START_REF] Caponigro | Sparse Stabilization and Control of Alignment Models[END_REF] imply that there exists

M > 0 such that f [µ] L ∞ ≤ M and g[µ] L ∞ ≤ M for any µ ∈ P c (B(0, R)).
These facts imply existence and uniqueness of the solution of the Cauchy problem for (2) (see, e.g., [START_REF] Ambrosio | Hamiltonian ODEs in the Wasserstein Space of Probability Measures[END_REF][START_REF] Piccoli | Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes[END_REF]). We recall existence results in Section 2.

The main result

The main idea of our control strategy is to choose the controller to make V decrease along trajectories. We will do this choice with a steepest descent method, similarly to the finite-dimensional approach described in [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF][START_REF] Caponigro | Sparse Stabilization and Control of Alignment Models[END_REF][START_REF] Caponigro | Sparse Jurdjevic-Quinn stabilization of dissipative systems[END_REF].

Since the space of admissible controls χ ω ug[µ] is infinite dimensional, we restrict ourselves to a finitedimensional set by imposing the following structure. Consider the class of Lipschitz mollified indicator functions χ η [a,b] : R → R, defined by

χ η [a,b] (x) =          1 for x ∈ [a, b], 0 for x ∈ [a -η, b + η],
x-a+η η for x ∈ [a -η, a];

-x+b+η η for x ∈ [b, b + η],
and then, consider the d-dimensional version of such functions. Given a = (a 1 , . . . , a d ), b = (b 1 , . . . , b d ) and

x = (x 1 , . . . , x d ) in R d , we define χ η [a,b] (x) = min i=1,...,d χ η [a i ,b i ] (x i ). ( 14
)
Now, for any choice of the three parameters (a, b, η), we take ω = ω(a, b, η)

as the multi-interval [a 1 - η, b 1 + η] × [a 2 -η, b 2 + η] × . . . × [a d -η, b d + η].
Then we reduce the choice of the sparse control in an infinite-dimensional space of controls to the choice of three parameters (a, b, η). In what follows, we set

U (a, b, η) = χ [a 1 -η,b 1 +η]×[a 2 -η,b 2 +η]ו••×[a d -η,b d +η] χ η [a,b] . (15)
We then define the "slope function" by

s t (a, b, η) = |L U (a,b,η)g[µ(t)] V [µ(t)]|,
which describes the instantaneous variation of V in µ(t) as a consequence of the action of the control U (a, b, η). Note that [START_REF] Canuto | An Eulerian approach to the analysis of Krause's consensus models[END_REF] and the fact that the function t → V [e tug µ] is of class C1 imply the continuity of the slope function with respect to its arguments (t, a, b, η).

We then apply a steepest descent method by choosing the control corresponding to one of the maximizers 1 (a * , b * , η * ) of s t in the space

Ω t = (a, b, η) | |ω(a, b, η)| ≤ c and η ≥ t -1 . ( 16
)
The condition |ω(a, b, η)| ≤ c in [START_REF] Cristiani | Multiscale Modeling of Pedestrian Dynamics, Springer MS & A: Modeling, Simulation and Applications[END_REF] ensures that the space constraint (3) is satisfied. We will see in Lemma 4 that the condition η ≥ t -1 implies that the control function is uniformly Lipschitz for any bounded time interval [0, θ], thus ensuring that µ(t) remains absolutely continuous with respect to the Lebesgue measure. At the same time, when t → +∞, this constraint allows to consider controls with an arbitrarily large Lipschitz constant, since Lip(χ η [a,b] ) = 1 η . This Lipschitz constraint is somehow unavoidable if one wants to ensure regularity of the measure µ(t) within finite time; otherwise, the steepest descent method might either generate a non-Lipschitz vector field (for which existence for (2) holds for small times only) or a timevarying Lipschitz vector field converging to a non-Lipschitz vector field within finite time (see an example for a problem of crowd dynamics in Section 3).

Choosing the control as the instantaneous maximizer of s t may cause chattering (in time) phenomena, as it has been already noticed in finite dimension (see [START_REF] Caponigro | Sparse Jurdjevic-Quinn stabilization of dissipative systems[END_REF]). For this reason, we regularize the control by means of an hysteresis: we introduce a parameter h ∈ (0, 1) and, given the control U (a * , b * , η * ), maximizer of s t at time t n , we keep it constant over an interval

[t n , t n + δ] along which (s t (a * , b * , η * ) ≥ (1 -h)s t (a, b, η).
Summing up, the combination of a steepest descent method with an hysteresis provides a control making V decrease and steering the density µ(t) to Z. Our main result is the following.

Theorem 1 (Main Theorem). Let f, g : P c (R d ) → Lip(R d , R d ) and V : P c (R d ) → R satisfy Assumptions (H) for some R > 0.
Consider the controlled transport equation with non-local velocity

∂ t µ + ∇ • (f [µ] + χ ω(t,y) u(t, y)g[µ])µ = 0, µ(0) = µ 0 , (17)
where µ 0 ∈ P ac c (R d ) is such that supp(µ 0 ) ⊂ B(0, R). Fix the hysteresis parameter h ∈ (0, 1). Fix the following initial parameters n = 0 and t 0 = 0. Define the following algorithm step.

Step n At time t n , choose one of the maximizers (a * , b * , η * ) of s tn (a, b, η) in the set Ω tn defined in [START_REF] Cristiani | Multiscale Modeling of Pedestrian Dynamics, Springer MS & A: Modeling, Simulation and Applications[END_REF]. Then, we have two cases:

• If either s tn (a * , b * , η * ) < t -1
n or Ω tn is empty, then choose the zero control

χ ω(t) u(t, x) ≡ 0, (18)
(thus, ω(t) needs not be defined) and let the measure µ(t), starting at µ(t n ), evolve according to [START_REF] De Gennaro | Decentralized control of connectivity for multi-agent systems[END_REF] over the time interval [t n , t n+1 ], where t n+1 is the smallest time greater than t n for which there exists

(ā, b, η) ∈ Ω t such that s tn+1 (ā, b, η) ≥ 2t -1 n+1 ,
where

Ω t = (a, b, η) ∈ Ω t | η ≥ 2t -1 . ( 19
)
• If s tn (a * , b * , η * ) ≥ t -1
n , then choose the control defined by

ω(t) = [a * 1 -η * , b * 1 + η * ] × [a * 2 -η * , b * 2 + η * ] × • • • × [a * d -η * , b * d + η * ], u(t, •) = -χ η * [a * ,b * ] sign(L U (a,b,η)g[µ(t)] V [µ(t)]), ( 20
)
where U is given in [START_REF] Cristiani | Effects of anisotropic interactions on the structure of animal groups[END_REF], and let the measure µ(t), starting at µ(t n ), evolve according to (17) over the time interval [t n , t n+1 ], where t n+1 is the smallest time greater than t n satisfying at least one of the following conditions:

-either s tn+1 (a * , b * , η * ) ≤ t -1 n+1 2 ; -or there exists (ā, b, η) ∈ Ω tn+1 such that s tn+1 (a * , b * , η * ) ≤ (1 -h)s tn+1 (ā, b, η). ( 21
)
If t n+1 is finite, then go to Step (n + 1).

If t n+1 = +∞, then keep the control [START_REF] Di Costanzo | A hybrid mathematical model for self-organizing cell migration in the zebrafish lateral line[END_REF] or (20) over the time interval [t n , +∞).

For this control strategy, the control χ ω u satisfies the control constraint (U), the unique solution µ(t) of ( 17) is such that µ(t) ∈ P ac c (R d ) for any t ∈ [0, +∞), and µ(t) converges to Z ∩ P c (B(0, R)), i.e.

• lim t→+∞ inf ν∈Z∩Pc(B(0,R)) W p (µ(t), ν) = 0,

• or equivalently, there exists a choice ν(t)

∈ Z ∩ P c (B(0, R)) for each t ≥ 0 such that µ(t) ν(t), i.e. for all φ ∈ C ∞ c (R d ) it holds lim t→∞ φd(µ(t) -ν(t)) = 0.
Remark 5. The three threshold time-dependent functions used in the definition of control algorithm in Theorem 1 satisfy t -1 /2 < t -1 < 2t -1 . One can easily see that they can be replaced with three positive functions satisfying φ 1 (t) < φ 2 (t) < φ 3 (t) converging to 0 as t → +∞. In particular, the functions can take finite values for t = 0, by maybe allowing one control to be active on the starting interval [0, t 1 ].

The rest of the paper is structured as follows. In Section 2, we recall the main definitions and results for transport PDEs with non-local velocities as (1) and (2). In Section 3, we discuss some examples of dynamics of the form (2), and we show explain some differences with respect to the finite dimensional setting. Theorem 1 is proved in Section 4. In Section 5, we study a generalization of the Theorem 1 to a system of the form (2) with several control potentials. Finally, in Section 6, we present an application of Theorem 1 to the control of kinetic multi-agent systems.

Transport equations with non-local velocities

In this section, we recall existence and uniqueness results for (1) and [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF]. In (1), the variable µ ∈ P c (R d ) is a probability measure on R d . The term f [µ] is called the velocity field and it is a non-local term. Since the value of a measure at a single point is not well defined, it is important to observe that f [µ] is not a function depending on the value of µ in a given point, as it is often the case in the setting of hyperbolic equations in which f [µ](x) = f (µ(x)). Instead, one has to consider f as an operator taking an as input the whole measure µ and giving as an output a global vector field f [µ] on the whole space R d . These operators are often called "non-local", as they consider the density not only at a given point, but in a whole neighbourhood.

We first recall two useful definitions to deal with measures and solutions of (1), namely the Wasserstein distance and the push-forward of measures (for more details see, e.g., [START_REF] Villani | Topics in Optimal Transportation[END_REF]). Definition 2.1. Given two probability measures µ and ν on R d and p ∈ [1, +∞), the p-Wasserstein distance between µ and ν is

W p (µ, ν) = inf R 2d |x -y| p dπ(x, y) | π ∈ Π(µ, ν) 1/p
, where Π(µ, ν) is the set of transference plans from µ to ν, i.e., of the probability measures π on R d × R d such that Proj x #π = µ and Proj y #π = ν with Proj x : (x, y) → x and Proj y : (x, y) → y.

The topology induced by W p on the space of probability measures P(X) on a compact space X coincides with the weak- * topology of measures (see [START_REF] Villani | Topics in Optimal Transportation[END_REF]Theorem 7.12]). As a consequence of condition [START_REF] Caponigro | Sparse Stabilization and Control of Alignment Models[END_REF], each trajectory µ(t) of the controlled system (2) is contained in the compact space P c (B(0, R)) (compact if endowed with the Wasserstein topology). Thus, from now on, we will state equivalently convergence with respect to the weak- * topology of measures and with respect to the Wasserstein distance. We now define the push-forward of measures.

Definition 2.2. Given a Borel map γ : R d → R d , the pushforward of a measure µ ∈ P c (R d ) is defined by γ#µ(A) = µ(γ -1 (A)) for every measurable subset A of R d .
We now recall an existence and uniqueness result for (1) (see a complete proof in [START_REF] Piccoli | Control to flocking of the kinetic Cucker-Smale model[END_REF]).

Theorem 2. We assume that, for every µ ∈ P c (R d ), the velocity field f [µ] is a function of (t, x) with the regularity f [•] :

P(R d ) -→ Lip(R d ) ∩ L ∞ (R d ) µ -→ f [µ]
satisfying the following assumptions:

• there exist functions L(•) and

M (•) in L ∞ loc (R) such that f [µ](t, x) -f [µ](t, y) ≤ L(t) x -y , f [µ](t, x) ≤ M (t)(1 + x ), for every µ ∈ P c (R d ), every t ∈ R and all (x, y) ∈ R d × R d ; • for a given p ∈ [1, +∞), there exists a function K(•) in L ∞ loc (R) such that f [µ] -f [ν] L ∞ (R;C 0 (R d )) ≤ K(t)W p (µ, ν), for all (µ, ν) ∈ (P c (R d )) 2 .
Then, for every µ 0 ∈ P c (R d ), the Cauchy problem

∂ t µ + ∇ • (f [µ]µ) = 0, µ |t=0 = µ 0 , (22)
has a unique solution µ(•) ∈ C 0 (R; P c (R d )), where P c (R d ) is endowed with the weak- * topology of measures. Moreover, t → µ(t) is Lipschitz in the sense of the Wasserstein distance W p . Moreover, if µ 0 ∈ P ac c (R d ), then µ(t) ∈ P ac c (R d ) for every t ∈ R.

Furthermore, for every T > 0, there exists C T > 0 such that

W p (µ(t), ν(t)) ≤ e C T t W p (µ(0), ν(0)), (23)
for all solutions µ and ν of (22

) in C 0 ([0, T ]; P c (R d )).
Finally, the solution µ of the Cauchy problem (22) can be made explicit as follows. Let Φ(t) be the flow of diffeomorphims of R d generated by the time-dependent vector field f [µ], defined as the unique solution of the Cauchy problem

Φ(t) = f [µ(t)] • Φ(t), Φ(0) = Id R d , or in other words, ∂ t Φ(t, x) = f [µ(t)](t, Φ(t, x)), Φ(0, x) = x.
Then, we have

µ(t) = Φ(t)#µ 0 , that is, µ(t) is the push-forward of µ 0 under Φ(t).
Theorem 2 can be generalized to mass-varying transport PDEs, that is, in presence of sources (see [START_REF] Piccoli | Generalized Wasserstein distance and its application to transport equations with source[END_REF]). We now observe that Theorem 2 can be applied to (2) as well, under Assumptions (H) and provided that the control u be a Lipschitz function of the space variable for all times.

Corollary 1. Under Assumptions (H), if u is a uniformly Lipschitz function of x on the time interval [0, θ], satisfying the constraint (U), then, given any initial data µ(0

) = µ 0 ∈ P c (B(0, R)), the equation (2) has a unique solution µ(•) ∈ C 0 ([0, θ], P c (B(0, R))). Moreover, if µ 0 ∈ P ac c (B(0, R)), then µ(t) ∈ P ac c (B(0, R)) for every t ∈ [0, θ].
Denoting by Ψ the flow of diffeomorphims of R d generated by the time-dependent vector field

f [µ] + u(t, x)g[µ], we have µ(t) = Ψ(t)#µ 0 .
Proof. It suffices to check that Theorem 2 can be applied to the vector field f [µ] + u(t, x)g [µ]. As already stated, the existence of a uniform bound M for f +ug L ∞ is a consequence of the uniform Lipschitz property and of the uniform boundedness of the support of both f [µ] and g[µ], together with the bound u L ∞ ≤ 1 imposed by (U) in (4). Similarly, we have a uniform bound on the Lipschitz constant Lip(f + ug). Indeed, by [START_REF] Ambrosio | Hamiltonian ODEs in the Wasserstein Space of Probability Measures[END_REF],

Lip(f [µ(t)] + ug[µ(t)]) ≤ L + Lip x (u) g[µ(t)] L ∞ + u L ∞ Lip(g[µ(t)]) ≤ 2L + Lip x (u)M. ( 24
)
Finally, we have

f [µ] + ug[µ] -f [ν] + ug[ν] L ∞ ≤ f [µ] -f [ν] L ∞ + u L ∞ g[µ] -g[ν] L ∞ ≤ W p (µ, ν) + 1 • QW p (µ, ν) = 2QW p (µ, ν).
This proves the corollary.

We end this section with an estimate of the L ∞ norm of the solution µ(t) to (1), when it is absolutely continuous with respect to the Lebesgue measure.

Proposition 1. Let µ(•) be the unique solution of (1) for a given Lipschitz vector field f with

µ 0 ∈ P ac c (R d ). Then d dt µ(t) L ∞ ≤ µ(t) L ∞ ∇ • f L ∞ . ( 25
)
Proof. The proof follows [22, Proposition 3.1]. Let ρ(t) be the density of µ(t) with respect to the Lebesgue measure. For each p ∈ [1, +∞), by dropping the dependence with respect to time, we write

d dt ρ p dx = -p ρ p-1 ∇ • (f ρ) dx = -p ρ p ∇ • f + ρ p-1 f, ∇ρ dx. Since ∇ • (f ρ p ) = ρ p ∇ • f + f, ∇(ρ p ) = ρ p ∇ • f + pρ p-1 f, ∇ρ , we infer that d dt ρ p dx = -(p -1)ρ p ∇ • f dx -∇ • (f ρ p ) dx.
The last term is zero as a consequence of the divergence theorem. Then [START_REF] Hegselmann | Opinion dynamics and bounded confidence: models, analysis and simulation[END_REF] as p → +∞.

d dt ρ p L p ≤ (p -1) ρ p L p ∇ • f L ∞ , which in turn implies

Steepest descent under population constraint induces mass concentration

In this section, we discuss a remarkable phenomenon for controlled equations of the form (2): starting from a measure µ 0 ∈ P ac c (R d ), i.e., a measure that is absolutely continuous with respect to the Lebesgue measure in R d , a time-dependent choice of the control might drive the measure outside P ac c (R d ) in finite time, in particular with emergence of Dirac deltas. In fact, we will show that such a phenomenon arises when trying to minimize a Lyapunov function V , in particular when one chooses the control u(t) as the instantaneous minimizer of the Lie derivative of V as time evolves. This example also shows that some key ideas coming from control of finite-dimensional systems cannot be extended straightforwardly to infinite dimension.

In this section, we discuss the interest and the drawbacks of a control constraint different than (U), namely the following:

Alternative control Constraints (U')

Fix c > 0. For each time t ≥ 0 it holds: Sparsity population constraint:

ω(t) dµ(t) ≤ c, (26)
Finite strength:

u(t, .) L ∞ ≤ 1. ( 27
)
The population constraint represents the idea of acting on a small part of the crowd itself, and not on a small part of the configuration space, as we require in the space constraint in (U). Even though the sparse population constraint is interesting from the theoretical point of view, it has a surprising drawback on the modeling point of view: when a crowd is extremely concentrated, the constraint (U') implies that the control cannot act on the crowd anymore. This is somehow unnatural, since a crowd that is already concentrated is the best configuration to steer. On the other hand, the space constraint (U) permits to act on the whole crowd, when it is concentrated in a set of size c, i.e., exactly when it is concentrated.

We now show that the population constraint also induces some formal mathematical problems when using a sparse Jurdjevic-Quinn approach. Consider the following system on the real line:

∂ t µ + ∂ x (uµ) = 0. ( 28
)
This is a particular case of (2) with f = 0 and g = 1. We consider the initial data µ 0 = χ [0,1] , i.e., a uniform probability density on the interval [0, 1]. We consider the Lyapunov function

V [µ] = x 2 dµ(x),
i.e., the second moment with respect to zero. We have L f V = 0, and we have L ug V [µ] = 0 for µ = δ 0 only, i.e., Z = {δ 0 }. Then, minimizing V is equivalent to steer µ(t) to the Dirac mass δ 0 . We now apply a rough form of the steepest descent method to the problem of minimizing V : given the initial measure µ 0 , we look for a control function u that maximizes the descent L ug V [µ], while taking into account the control constraints (U'). An easy computation shows that no optimal choice for u exists. Indeed, for every ε > 0, consider the C ∞ function

u ε (x) =      -1 for x ∈ [1 -c + ε, 1], 0 for x ∈ (-∞, 1 -c] ∪ [1 + ε, +∞), C ∞ -spline with values in [-1, 0] for x ∈ [1 -c, 1 -c + ε] ∪ [1, 1 + ε].
Then, for a sufficiently small time t > 0, each particle x ∈ (1 -c + ε, 1] is displaced to x -t while each particle x ∈ [0, 1 -c] undergoes no displacement. The particles in the small interval [1 -c, 1 -c + ε] are displaced toward 1 -c, then giving a reduction of the value of the functional V . Then, we have

L uεg V [µ] = d dt |t=0 1-c+ε 1-c (x + tu(x) + o(t)) 2 dµ(x) + d dt |t=0 1 1-c+ε (x -t) 2 dµ(x) ≤ -2 1 1-c+ε x dµ(x).
As a consequence, by decreasing the parameter ε > 0, one can reach a larger decrease of V . Nevertheless, the limit for ε → 0 would result in the discontinuous control function

u 0 = χ [1-c,1]
, for which the solution to the corresponding dynamics [START_REF] Jurdjevic | Controllability and stability[END_REF] does not satisfy Assumptions2 (H).

The fact that a maximizer of the steepest descent does not exist in the space of Lipschitz functions can be overcome by fixing a value ε 0 > 0 and applying the control u ε0 over a small interval of time [0, t 0 ] with t 0 < ε 0 . As a result, the component of the measure µ with x ∈ [1 -c, 1] concentrates in the interval [1 -c, 1 -t 0 ], while its density keeps being constantly equal to 1 for x ∈ [0, 1 -c]. At time t, one can observe that the largest contribution to V comes anyway from the mass in the interval [1 -c, 1 -t 0 ], on which the control already acted. Moreover, the mass in such interval keeps being c.

For the reasons described above, any strategy maximizing the descent of V acts on the mass in the interval [1 -c, 1] for all times. In particular, we can choose a sequence ε i acting on the time interval [t i-1 , t i ], with the condition ε i < c -t i for all t ∈ [0, c). Applying the time-dependent control u(t, x) = u εi (x) for t ∈ [t i-1 , t i ), the corresponding solution µ(t) of ( 28) has support in [0, 1 -t]. More precisely, it has the following structure: the measure keeps having density 1 in the interval [0, 1 -c], while the rest of the mass

c is contained in the interval [1 -c, 1 -t].
Then, the solution µ(t) converges as t c to the singular measure

µ(c) = χ [0,1-c] + cδ 1-c . (29)
This is not in contradiction with the fact that any solution of (1) with initial data µ 0 ∈ P ac c (R d ) and Lipschitz vector field f satisfies µ(t) ∈ P ac c (R d ). Indeed, the condition ε(t) < c -t implies lim t→c ε(t) = 0, hence the control u(t, x) converges to a non-Lipschitz function.

Starting from the singular measure (29) at time t = c, one finds several problems to steer it towards the minimizer δ 0 of V . First, the main contribution to V is given by the Dirac delta c δ 1-c : then, the control set ω chosen to maximize the descent for V would certainly contain such mass. But the condition 1 -c ∈ ω together with u(1 -c) = 0 would directly impose to choose ω containing a whole neighbourhood of 1 -c. This would imply ω µ(c) > c, hence the population constraint would be automatically violated.

This would in turn enforce us to focus our control on the absolutely continuous part, possibly leading to the formation of a new Dirac delta c δ 1-2c , and so on. The final result would be a set of Dirac deltas, not concentrated at 0, on which control with population constraint cannot be applied.

Summing up, the steepest descent method with population constraint in (U') might not steer the measure to a configuration in Z, but rather to a configuration in which the population constraint itself may not be satisfied.

Proof of Theorem 1

For the moment, we assume that the solution µ(•) of ( 17), with the control strategy defined by Theorem 1, is well defined on [0, θ] with θ ∈ (0, +∞], and we establish some lemmas describing its evolution. Recall that supp(µ 0 ) ⊂ B(0, R).

Lemma 3. We have supp(µ(t)) ⊂ B(0, R) for every t ∈ [0, θ].

Proof. Since the vector field f + ug is zero outside B(0, R), the corresponding flow φ t (•) coincides with the identity in R d \B(0, R). Since we have µ(t) = φ t # µ 0 by Theorem 2, we get that, for any Borel set E satisfying E ∩ B(0, R) = ∅, we have µ(t)(E) = µ 0 (φ -t (E)) = µ 0 (E) = 0. The lemma follows.

Recall that L is the Lipschitz constant for f [µ] and g[µ] given in [START_REF] Caponigro | Sparse Jurdjevic-Quinn stabilization of dissipative systems[END_REF], and recall that f [µ] L ∞ ≤ M and g[µ] L ∞ ≤ M for every µ ∈ P ac c (B(0, R)) (as in Remark 4).

Lemma 4. We have µ(t) L ∞ ≤ e dθ(2L+M θ) µ(0) L ∞ , for every t ∈ [0, θ].

Proof. Since the vector field f + ug satisfies [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules, Automatic Control[END_REF], and since Lip x (u(t, •)) ≤ 1 η ≤ t ≤ θ, the lemma follows from Proposition 1.

Lemma 5. The function (t, a, b, η) → s t (a, b, η) is continuous with respect to t, and uniformly Lipschitz with respect to (a, b, η) on ∪{Ω t , t ∈ [0, θ]}.

Proof. Let us first establish the Lipschitz property for (a, b, η)

∈ ∪{Ω t , t ∈ [t n , t n+1 ] ∩ [0, θ]}. Note that the condition |ω(a, b, η)| ≤ c implies that η ≤ c 2 . Besides, we have η ≥ t -1 ≥ θ -1
. By definition of U (a, b, η) in ( 15), with simple geometric arguments, it is clear that in the 1D case we have

U (a, b, η) -U (a , b , η ) L 1 (dx) ≤ U (a, b, η) -U (a, b, η ) L 1 (dx) + U (a, b, η ) -U (a , b , η ) L 1 (dx) ≤ |η -η | + |a -a | + |b -b |,
where dx is the standard Lebesgue measure on R. By applying the estimate componentwise, the same result follows in dimension d. By Lemma 4, there exists P > 0 such that µ(t)

L ∞ ≤ M for every t ∈ [0, θ], hence U (a, b, η) -U (a , b , η ) L 1 (µ(t)) ≤ P (|a -a | + |b -b | + |η -η |),
and thus [START_REF] Canuto | An Eulerian approach to the analysis of Krause's consensus models[END_REF] implies that s t (a, b, η) is Lipschitz with respect to (a, b, η), with Lipschitz constant KP .

The function s t (a, b, η) is continuous with respect to t, as a consequence of the continuity of L ug V given by [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF].

Lemma 6. Assume that µ(t n ) ∈ P ac c (R d ).
If Ω tn is nonempty then s tn (a, b, η) has a maximizer in Ω tn .

Proof. It suffices to observe that Ω tn can be considered as being compact: indeed, each choice (a, b, η) ∈ Ω tn can be replaced by an equivalent choice (a , b , η) with a, b ∈ B 0 (R + 2c) since U (a, b, η) -U (a , b , η) L 1 (µ) = 0, since µ has zero mass outside of B(0, R). In other terms, one can restrict the choice of the parameters a, b to a compact set. Similarly, we have η

∈ [t -1 n , c 2 ]
. Since, by Lemma 5, s tn is a continuous function of its arguments, then it admits a maximizer.

We are now in a position to prove Theorem 1. We split the proof into three steps:

Step 1. For each time t n , the n-th step of the algorithm univocally determines a control satisfying the constraint (U), the corresponding solution of (2) and a time t n+1 > t n .

Step 2. We have t n → +∞.

Step 3. We have lim t→+∞ L f +ug V [µ(t)] = 0. This fact, together with the choice of maximizing controls and of the hysteresis, provides convergence to the sets in which the maximizers of s t (a, b, η) give zero control. Since the constraint η ≥ t -1 is negligible for t → +∞, the strategy provides convergence of µ(t) to Z.

Proof of Step 1. Let us prove that the algorithm of Theorem 1 univocally defines a control strategy, by induction. We have t 0 = 0 and µ(0) = µ 0 ∈ P ac c (R d ). Let us prove that, for a given time t n , the time t n+1 is well defined and satisfies t n+1 > t n .

We first observe that the control χ ω u is a well-defined function, Lipschitz with respect to x.

Setting t = 1 2 c |B0(1)| 1/d
, we note that Ω t = ∅ for every t ∈ [0, t) since any function of the form χ η [a,b] has a support of size larger than (2η) d |B 0 (1)|. For t ≥ t, the set Ω t is nonempty and Lemma 6 yields the existence of a maximizer (a * , b * , η * ) in Ω t . We thus have two cases:

• If s tn (a * , b * , η * ) < t -1
n or Ω t empty, then the control χ ω u = 0 is well defined and is Lipschitz.

• If s tn (a * , b * , η * ) ≥ t -1 n , then the control χ ω u = U (a * , b * , η *
) is well defined and is Lipschitz, as a consequence of the Lipschitz property in [START_REF] Couzin | Collective memory and spatial sorting in animal groups[END_REF].

Let us now prove that there exists a unique minimum t n+1 defined by the algorithm, and that it satisfies t n+1 > t n . For t ∈ [0, t), there is nothing to prove, since t 1 ≥ t. For t ≥ t, we have two cases:

• If s tn (a * , b * , η * ) < t -1
n , then the set

A = t ≥ t n | s t (a, b, η) ≥ t -
A = t ≥ t n | s t (a * , b * , η * ) ≤ t -1 n 2
is closed or empty, and it does not contain t n . We now consider the set

B = t ≥ t n such that there exists (ā, b, η) ∈ Ω t for which s t (ā, b, η) ≥ (1 -h) -1 s t (a * , b * , η * ) ≥ (1 -h) -1 t -1 2 .
.

Let us prove that it is closed and that t n ∈ B . Take a sequence (t i , a i , b i , η i ) such that t i ∈ B is a sequence converging to some t, and (a i , b i , η i ) satisfy

s t i (a i , b i , η i ) ≥ (1 -h) -1 s t i (a * , b * , η * ) ≥ τ (t i ) 2 .
Observing that the compact set Ω t varies smoothly with respect to time, we can restrict ourselves to a sequence (a i , b i , η i ) converging to some (ā, b, η) ∈ Ω t. Then, by continuity of s t , we have st(ā, b, η) ≥ (1 -h) -1 st(a * , b * , η * ), hence t ∈ B . Moreover, t = t n , otherwise (a * , b * , η * ) would not be a maximizer of s tn .

Since both A and B are closed or empty, not containing t n , then A ∪ B is closed or empty and does not contain t n . If it is closed, then it admits a minimal element t n+1 > t n ; if it is empty, then we have +∞ = t n+1 > t n .

Proof of Step 2. We now prove that the sequence t n of times given by the algorithm converges to +∞. Since t n is increasing, it has a limit T . By contradiction, if T < +∞, then µ(t) is defined for every t ∈ [0, T ]. Indeed, since f + ug L ∞ ≤ 2M , the curve t → µ(t) is Lipschitz, and thus µ(T ) is well defined.

If we have s tn (a * , b * , η * ) < t -1 n at time t n , then at the next time t n+1 we must have s tn+1 (ā, b, η) ≥ t -1 n+1 for some (ā, b, η) ∈ Ω tn+1 , by definition of the algorithm itself. As a consequence, the sequence t n converging to T contains an infinite number of times t ni such that

s tn i (a i , b i , η i ) ≥ t -1 ni , where (a i , b i , η i ) is a maximizer of s tn i in Ω tn i .
The sequence (a i , b i , η i ) is bounded, and its converging subsequences have their limit in Ω T . Indeed, we can restrict ourselves to (a i , b i ) ∈ B 0 (R + 2c), and we have η i ≥ T -1 and η i ≤ c 2 . Hence, taking a subsequence if necessary, we have the existence of a limit (â, b, η) ∈ Ω T .

Observe now that, at time t ni+1 , one of the two conditions leading to switching of the control holds. Since the sequence t ni+1 has an infinite number of terms, at least one of the conditions holds for an infinite subsequence (that we do not relabel). We show now that this is in contradiction with the fact that t n converges to a finite time T . We have two cases:

• If s tn i (a i , b i , η i ) ≥ t -1 ni and s tn i +1 (a i , b i , η i ) ≤ t -1 n i
2 , then, taking a subsequence converging to (â, b, η) ∈ Ω T , we have a contradiction with the continuity of s T in (â, b, η). Indeed, we have

s T (â, b, η) = lim i→+∞ s tn i (a i , b i , η i ) ≥ t -1 ni > t -1 ni 2 ≥ lim i→+∞ s tn i +1 (a i , b i , η i ) = s T (â, b, η).
• If there exists (ā i , bi , ηi ) ∈ Ω tn i +1 such that

s tn i +1 (a i , b i , η i ) ≤ (1 -h)s tn i +1 (ā i , bi , ηi ), (30)
then, for n i → +∞, we have Ω tn i +1 ⊂ Ω tn i , since 2t -1 ni+1 ≥ t -1 ni , as a consequence of the fact that lim i→+∞ (t ni+1 -t ni ) = 0. Since (a i , b i , η i ) is a maximizer of s tn i in Ω tn i +1 , we have

s tn i (ā i , bi , ηi ) ≤ s tn i (a i , b i , η i ). (31)
One can take a converging subsequence of (ā i , bi , ηi ), for the same reasons given above for the sequence (a i , b i , η i ). Denoting by (ā * , b * , η * ) and (a * , b * , η * ) the two limits, and using continuity of s t (a, b, η) with respect to all its arguments, we get from ( 30)-( 31) that

s T (a * , b * , η * ) ≤ (1 -h)s T (ā * , b * , η * ) ≤ (1 -h)s T (a * , b * , η * ), which is in contradiction with s T (a * , b * , η * ) ≥ T -1 > 0.
Then t n cannot converge to a finite value T . Therefore either t n → +∞ or there exists a t n such that t n+1 = +∞. In both cases, the control strategy is defined for every t ∈ [0, +∞).

Proof of Step 3. It remains to prove that µ(t) converges to Z. This is the hardest part of the proof, in which the choice of the admissible controls in Ω t plays a crucial role. Thanks to Step 2, we have, for every time, µ(t) ∈ P c (B(0, R)), that is compact with respect to the weak topology, which coincides with the topology of the Wasserstein distance. Then Assumptions (H) imply that V is a continuous function and thus is bounded below.

We now prove that the function

V (t) = V [µ(t)
] is differentiable for almost every t, and that it satisfies

V (t) = lim t→0 V [e t(f +ug) µ]-V [µ] t ≤ 0.
Differentiability on the open time interval (t n , t n+1 ) follows from the fact that V (t) = L f +ug V [µ(t)] is continuous, as a consequence of Assumptions (H). Clearly, the set of times t n on which differentiability is not ensured is countable, hence V (t) is differentiable for almost every t.

For t ∈ (t n , t n+1 ), we have

V = L f +ug V [µ] = L f V [µ] + L ug V [µ].
If at time t n the algorithm defines the control χ ω u ≡ 0, then clearly

V = L f V [µ] ≤ 0, (32)
for every t ∈ (t n , t n+1 ). If instead the control given by the algorithm is χ ω u in (20), we have

V = L f V [µ] + L χωug V [µ] ≤ -sign(L U (a * ,b * ,η * )g[µ(tn)] V [µ tn ])(L U (a * ,b * ,η * )g[µ(t)] V [µ t ]). (33)
It is clear that, at the beginning of the interval, we have

lim t→t + n sign(L U (a * ,b * ,η * )g[µ(tn)] V [µ tn ])(L U (a * ,b * ,η * )g[µ(tn)] V [µ tn ]) ≥ s tn (a * , b * , η * ) ≥ τ (t n ) > 0,
and hence V (t + n ) < 0. Since V (t) is a continuous function, we either have V (t) < 0 for every t ∈ (t n , t n+1 ), or there exists t

∈ (t n , t n+1 ) such that L U (a * ,b * ,η * )g[µ(t)] V [µ(t)] = 0. This is equivalent to state that s t (a * , b * , η * ) = 0, which is in contradiction with s t (a * , b * , η * ) > τ (t) 2
> 0 for every t ∈ (t n , t n+1 ), by definition of the time t n+1 .

We now prove that lim t→∞ µ(t) ∈ Z. Since P c (B(0, R)) is compact, all sequences have limits. Consider a sequence t j → ∞ such that lim j→∞ µ(t j ) = µ * . We are going to prove that µ * ∈ Z.

Since V is continuous, bounded below and V (t) ≤ 0 for almost every t, we have lim t→+∞ V (t) = V * for some V * . The existence and continuity of the second-order derivatives L f +ug L f +ug V [µ] on the compact space P c (B(0, R)) implies the existence of a uniform bound on V . As a consequence, we have lim t→∞ V (t) = 0. Since V ≤ L f V ≤ 0 by either [START_REF] Piccoli | Control to flocking of the kinetic Cucker-Smale model[END_REF] or [START_REF] Rahmani | Controllability of multi-agent systems from a graph-theoretic perpective[END_REF], this in turn implies lim j→∞

L f V [µ(t j )] = 0, hence L f V [µ * ] = 0 by continuity of L f V .
We now prove that L ug V [µ * ] = 0 for all u ∈ U. By contradiction, assume that there exists u * ∈ U such that |L u * g V [µ * ]| = 0. Without loss of generality, by using (9), we assume that u * L ∞ = 1. Similarly, by decomposing u * = u + -u -with u + , u -non-negative Lipschitz functions, and using additivity of the Lie derivative, we can replace u * with either u + or u -and assume that it is nonnegative and that |L u * g V [µ * ]| = C * = 0. Finally, by observing that µ * has compact support, we can replace u * with a nonnegative Lipschitz function with compact support. We now approximate u * in L ∞ by a family of functions of the form

I i=1 k i χ η [a i ,b i ] ,
where the number I of terms depends on the approximation error, but not on the (sufficiently small) parameter η.

For simplicity, we only give the construction in the 1D case. Since u * is Lipschitz with bounded support, it is Riemann integrable. In particular, by using an approximation of u * from below, we have the following: take a grid step ∆x and define the rectangles k i χ [ã i , bi ] with bi -ãi = ∆x, for which

I i=1 k i χ [ã i , bi ] ≤ u * and u * - I i=1 k i χ [ã i , bi ] L ∞ ≤ ε. ( 34
)
for some ε. The Riemann integrability of u * implies that, for any ε > 0 there exists ∆x such that (34) is satisfied. Note that u * L ∞ = 1 also implies k i ≤ 1. We now prove that we can replace k i χ [ã i , bi ] with their mollified version k i χ η [a i ,b i ] for any sufficiently small η, while keeping (34) satisfied. We provide here the explicit construction. First denote with L the Lipschitz constant of u * . To replace

k i χ [ã i , bi ] with k i χ η [a i ,b i ] ,
we have two cases:

• If bi-1 < ãi , then keep both b i-1 = bi-1 and a i = ãi , and choose η ≤ min ãi -bi-1 2

, k i-1 L , k i L . • If bi-1 = ãi , then choose η ≤ min k i-1 L , k i L . If k i-1 > k i , then define b i-1 = bi and a i = ãi + η.
Otherwise, take b i-1 = bi -η and a i = ãi . Note that the constraints imposed on η are higher bounds, and they are in finite number. Then, η can be chosen in a whole interval (0, η ], where η depends on ε only. By construction, we have [START_REF] Tanner | Flocking in fixed and switching networks[END_REF] are satisfied. Note that this construction depends only on u * and not on µ * . We denote this new function by

I i=1 k i χ [ã i , bi ] ≤ I i=1 k i χ η [a i ,b i ] ≤ u * , thus both conditions in
u η = I i=1 k i χ η [a i ,b i ]
. Consider now the sequence of measures µ(t j ) converging to µ * . If t j is a switching time t n for the algorithm, replace it with a slightly larger time. Then we can assume that µ(t j ) keeps converging to µ * , with no switching times. By continuity of L u * g V [µ] given by [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF], we have |L u * g V [µ(t j )]| ≥ C * 2 for sufficiently large indices. Note that

|L (u * -u η )g V [µ(t j )]| ≤ K u -u η L 1 (µ(t j )) ≤ K u -u η L ∞ ≤ Kε.
Then, choose a sufficiently small ε ≤ C * 2K and a corresponding ∆x > 0 such that (34) is satisfied. If ∆x ≥ c 2 , then replace it with c 2 : by integrability of u * , ( 34) is still satisfied when refining the grid. Then, the previous construction shows that there exists η such that u η satisfies (34) for every η ∈ (0, η ]. Choose then η * ∈ (0, η ]. For a sufficiently large j, we have

|L u η * g V [µ(t j )]| ≥ C * 2 . Note now that u η * is a sum of I terms of the form k i χ η * [a i ,b i ] .
Then, for each j there exists at least one term such that

|L χ η * [a i ,b i ] g V [µ(t j )]| ≥ |L k i χ η * [a i ,b i ] g V [µ(t j )]| ≥ C * 2I ,
where we have used that k i ≤ 1 and that the number I does not depend on the parameter η * . Observe now that, for a sufficiently large T , we have (a i , b i , η * ) ∈ Ω t for every t ≥ T . Similarly, taking a larger T if necessary, we have s t j (a i , b i , η * ) ≥ C * 2I ≥ t -1 j . As a consequence, the control algorithm provides a maximizer (a j , b j , η j ) ∈ Ω t j of s t j , for which

s t j (a j , b j , η j ) ≥ (1 -h)s t j (a i , b i , η * ) ≥ (1 -h) C * 2I .
The corresponding derivative satisfies

V (t j ) = L f +U (a j ,b j ,η j )g V [µ(t j )] ≤ 0 -(1 -h) C * 2I .
In particular, this is in contradiction with the fact that lim t→∞ V (t) = 0. The theorem is proved.

Generalization to several controls

In this section, we show how to extend our result to a transport equation with a finite number of controlled vector fields,

∂ t µ + ∇ • f [µ] + m i=1 χ ωi u i g i [µ] µ = 0.
Now, Assumptions (H) are done for all vector fields f, g 1 , . . . , g m . In this setting, one can require the control constraint (C), together with the following additional constraint.

Componentwise sparsity constraint

For every t ∈ [0, +∞), there exists at most one index i such that u i (t, •) is not identically zero.

This sparsity constraint was first considered in the finite-dimensional setting for crowd models in [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF][START_REF] Caponigro | Sparse Stabilization and Control of Alignment Models[END_REF]. We recently generalized the Jurdjevic-Quinn stabilization method with this additional constraint in [START_REF] Caponigro | Sparse Jurdjevic-Quinn stabilization of dissipative systems[END_REF]. Under such additional assumptions, we can adapt the control algorithm of Theorem 1 as follows.

for some Lipschitz vector field (g 1 , . . . , g N ) on (R d ) N and controls (u 1 , . . . , u N ) in some subset of (R d ) N . In the case in which the control vector field is defined, for every agent i, only via the interaction between the other agents and the action of the control is the same on any agent, namely if there exists an interaction kernel

G ∈ Lip(R d × R d , R d ) such that g i (x 1 , . . . , x N ) = 1 N j =i G(x i , x j ),
for every i = 1, . . . , N , and if u i = u j for all i, j ∈ {1, . . . , N }, then we can consider the limit of (37) as N → +∞, which gives the mean-field equation ( 2) with [START_REF] Wongkaew | On the control through leadership of the Hegselmann-Krause opinion formation model[END_REF] and

g[µ](x) = G(x, y) dµ(y). (38)
The controllability problem is then the following: given an initial measure µ 0 and a final measure µ 1 , find a suitable control function (t, x) → u(t, x) steering the system (2) from µ 0 to µ 1 . We refer to [START_REF] Piccoli | Control to flocking of the kinetic Cucker-Smale model[END_REF] for a first result on the control of a mean-field equation of the form (2) with constraint (U) and (U'). In particular the paper focuses on the controlled version of the kinetic Cucker-Smale system introduced in [START_REF] Ha | From particle to kinetic and hydrodynamic description of flocking[END_REF] with constant g, and the existence of a control steering the system to a neighborhood of a Dirac measure is proved. Existence and uniqueness for the mean-field equation (2) when the vector fields are given by ( 36) and (38) are ensured by Theorem 2 provided that F (x, y) and G(x, y) have compact support. Indeed if F (x, y) and G(x, y) have compact support, then the vector fields f [µ] and g[µ] satisfy Assumptions (H).

Multi-agent models with a compactly supported interaction potential are sometimes called "bounded confidence" or homophilous models. The idea is that the agents interact only with the ones having closer states. This kind of interaction is used, for instance, to model opinion formation in first-order systems. One of the most influential models in opinion formation is, indeed, the Bounded Confidence Model by Hegselmann and Krause [START_REF] Hegselmann | Opinion dynamics and bounded confidence models, analysis, and simulation[END_REF] (see also [START_REF] Blondel | Continuous-time average-preserving opinion dynamics with opinion-dependent communications[END_REF]). The main feature of this model is that the interaction is zero when the distance between two opinions is larger than a certain threshold:

F (x i , x j ) = (x j -x i ) if |x i -x j | ≤ 1, 0 otherwise.
It has been proved in [START_REF] Blondel | Continuous-time average-preserving opinion dynamics with opinion-dependent communications[END_REF] that, for almost every initial configuration, the opinions converge asymptotically to clusters. In particular, the system does not reach global consensus in general. Since the right-hand side is discontinuous with respect to the state variable, for some configurations, the system has no unique solution, hence we consider the more general first-order consensus model ẋi

= 1 N j =i φ(x j -x i )(x j -x i ) i = 1, . . . , N, (39)
where the function φ is defined by

φ(x) =      1 if |x| < 1, -|x| ε + 1 + 1 ε if |x| ∈ [1, 1 + ε], 0 if |x| > 1 + ε, (40)
for some small ε > 0. This is a variant of the Hegselmann-Krause model, in which the Lipschitz property of φ ensures existence and uniqueness of solutions of (39). Therefore the associated vector field for the mean-field equation ( 2) is

f [µ](x) = φ(y -x)(y -x) dµ(y). (41)
The kinetic version of the Hegselmann-Krause model has been first studied in [START_REF] Canuto | An Eulerian approach to the analysis of Krause's consensus models[END_REF] for discrete-time dynamics.

Existence of solutions has been first proved in [START_REF] Bertozzi | Blow-up in multidimensional aggregation equations with mildly singular interaction kernels[END_REF] for a general bounded decreasing φ(x) such that |xφ (x)| ≤ φ(x). Moreover, if φ(x) is everywhere nonzero, then the system converges unconditionally to consensus, meaning that for every µ 0 the solution µ(t) converges asymptotically to a Dirac mass. If φ(x) is compactly supported, as in our case, however, then the large time behavior of the dynamics is not yet completely understood and a precise description of the asymptotic dynamics is, in general, a hard task. As in the finitedimensional analogue, generically the solution µ(t) converges to a finite sum of Dirac deltas, representing the clusters of opinion, but sufficient conditions for global consensus are still unknown. Theorem 1 provides then a useful tool to establish convergence to global consensus.

Here we consider the controlled kinetic Hegselmann-Krause model in dimension d = 1 with drift vector field given by (41), control vector field g = 1, and Lyapunov function

V [µ] = x 2 dµ(x). Then L ug V [µ] = 0 for every u ∈ U ⇔ µ = δ 0 . Since L f [δ 0 ] = 0, it follows that Z = {δ 0 }. We claim that if µ = 1 2 (δ x + δ y ) for some x, y ∈ R, then L f V [µ] ≤ 0.
The same statement is valid for the Lie derivative of V along any combination of Dirac 1 N N i=1 δ xi for some x i ∈ R, but for the sake of readability let us prove it for the sum of two Dirac masses. Noting that

f δ x + δ y 2 (z) = 1 2 φ(x -z)(x -z) + 1 2 φ(y -z)(y -z),
we have

L f V [µ] = 1 2 d dt |t=0 V (e tf [δ x ]) + 1 2 d dt |t=0 V (e tf [δ y ]) = xf δ x + δ y 2 (x) + yf δ x + δ y 2 (y) = 1 2 φ(y -x)(y -x)x + 1 2 φ(x -y)(x -y)y = - φ(x -y) 2
x -y 2 ≤ 0.

Then, using the continuity conditions [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF] and the density of the sum of Dirac deltas in P c (R), one can extend the estimate on P c (R). In particular this system fits into the framework of Theorem 1, which thus provides the existence of a control strategy concentrating the mass at 0, in other words, steering the system to global consensus.

Theorem 1 also gives an explicit construction of a control achieving consensus. Assume that at a certain switching time, say t, the solution is µ [START_REF] Agrachev | Controllability on the group of diffeomorphisms[END_REF] . In this case, we can write explicitly the slope function

(t) = 1 2 χ [-1,
s t (a, b, η) = |L U (a,b,η)g[µ] V [µ]| = | d dt |t=0 V (e tU (a,b,η) µ)| = 2| R xU (a, b, η)(x)dµ(x)| = 2 1 η a a-η x(x -a + η)dµ(x) + b a xdµ(x) + 1 η b+η b x(-x + b + η)dµ(x) ,
where b -a + 2η ≤ c. For η large, the biggest contribution is given by the second integral term If c < 2, then the control set ω cannot cover the whole support of µ and it will be close to -1 or 1. The action of the control steers the mass in the region ω toward 0 breaking the symmetry of the measure µ. It may happen therefore that the region ω will lose the optimality of the slope function and the control will switch to another region on the opposite side. In general, if the measure µ is symmetric with respect to the origin, then the control may chatter. This is the rationale for the introduction of an hysteresis parameter h: the control acts on a set and holds it also sometimes after losing optimality in order to prevent high-oscillating controls.

Here we present numerical simulations for this system. We consider an initial data µ 0 randomly distributed on the interval [0, 10] and we apply the control given in Theorem 1 with three different choice of the hysteresis parameter h. In Figure 2 the free evolution of the system, i.e. with u = 0. Notice that the solution tends to a finite combination of Dirac deltas representing clusters. The action of Theorem 1, with the variance as Lyapunov function, is represented in Figure 3. In this case the whole mass tends to a single Dirac delta, representing consensus. Chattering may usually happen when dealing with sparse controls designed with optimality criteria. Sparsity of the control in finite dimension is usually coded in terms of control acting on the smallest number of components/agents and the term sparsity comes from the fact that the control operator g i in (37) is a sparse vector. This notion has been introduced in [START_REF] Caponigro | Sparse stabilization and optimal control of the Cucker-Smale model[END_REF][START_REF] Caponigro | Sparse Stabilization and Control of Alignment Models[END_REF] for second-order alignment systems (see also [START_REF] Wongkaew | On the control through leadership of the Hegselmann-Krause opinion formation model[END_REF] for the controllability via leader of the Hegselmann-Krause finite-dimensional model).

In the infinite-dimensional framework, the assumption that agents are indistinguishable is crucial for defining mean-field limits; therefore, the notion of componentwise sparsity loses its sense. The infinitedimensional analogue of componentwise sparsity is the sparsity population constraint (U'). In Section 3, we have described the issues arising from this definition.

The sparsity space constraint given in (U) is, on the other hand, the natural definition of sparsity for mean-field equation of the form (2). The constraint is in some sense geographical, since the control can act only on a region of the space. The finite-dimensional analogue of such a control is the so-called decentralized control. A decentralized control acts based on partial information on the agents inside a certain neighborhood of the controlled ones. The decentralized control for multi-agent systems is a well-established topic, we refer for instance to the works [START_REF] De Gennaro | Decentralized control of connectivity for multi-agent systems[END_REF][START_REF] Tanner | Flocking in fixed and switching networks[END_REF][START_REF] Zavlanos | Flocking while preserving network connectivity[END_REF] for decentralized consensus algorithms, see also [START_REF] Bongini | Un)conditional consensus emergence under perturbed and decentralized feedback controls[END_REF] for a recent result with L ∞ constraints on the control of the form (4).

Conclusion and open questions

In this paper, we have generalized the classical Jurdjevic-Quinn stabilization method to infinite-dimensional control systems described by transport partial differential equations with non-local terms. Such equations arise in crowd models that are mean-field limits of particle systems for a finite number of agents: for this reason, it is natural to require some sparsity constraint to the control.

We established a mean-field Jurdjevic-Quinn stabilization method under the sparsity constraint (U): the control acts on a small set of the configuration space, with a bounded strength.

Improving Theorem 1 in the original spirit of Jurdjevic and Quinn may be done in several ways: on the one side, by reducing the target goal to the largest subset of Z that is invariant under the uncontrolled dynamics f [µ]; on the other side, by reducing the target by imposing zero higher-order derivatives, i.e., when Z is defined by

Z = µ ∈ P c (B(0, R)) | L f V [µ] = L k f L ugi V [µ] = 0 ∀u ∈ Lip(R d , R), i = 1, . . . , m, k ∈ N .
Dealing with iterated Lie derivatives, that is, with Lie brackets, in the kinetic setting is an open perspective. Addressing more general systems than those presented in Section 5 is also of great interest. In particular, it would be interesting to develop similar approaches to enforce stabilization of a transport equation to a specific set Z of configurations, such as steady-states or periodic trajectories. In this spirit, a remarkable result for describing cell migrations is given in [START_REF] Di Costanzo | A hybrid mathematical model for self-organizing cell migration in the zebrafish lateral line[END_REF], where steady-states are "rosettes", that are symmetric configurations of cells leading to emergence of specific macroscopic structures.

Figure 1 :

 1 Figure 1: Construction of the approximation k i χ η [a i ,b i ] .

2 -

 2 max(a, -1) 2 .

Figure 2 :

 2 Figure 2: Uncontrolled kinetic Hegselmann-Krause model. Blue lines represent the evolution µ(t). The evolution of max and min of the support of µ(t) are represented with red lines.

  (a) h = 0.9 (b) h = 0.5 (c) h = 0.2

Figure 3 :

 3 Figure 3: Application of the control strategy of Theorem 1 for different values of the hysteresis parameter h. The yellow region is the controlled area.

  1 for some (a, b, η) ∈ Ω t is closed or empty. If it is nonempty, there exists a minimal element t n+1 ≥ t n . Moreover,t n+1 = t n , since s tn (a, b, η) ≤ s tn (a * , b * , η * ) < t -1n for all (a, b, η) ∈ Ω tn . If A is empty then +∞ = t n+1 > t n .• If s tn (a * , b * , η * ) ≥ t -1 n , then, similarly to the previous case, since the function s t (a * , b * , η * ) is continuous with respect to the time t, the set

Remark 3. The results of this paper can be stated for transport equations with non-local terms (2) defined on bounded manifolds without boundary, or on bounded manifold with no-flux boundary condition.Summing up, we make the following assumptions on the system (2) and on the Lyapunov functional V :

The method to select a maximizer plays no role in the convergence of the method. One may consider for instance the lexicographic order in R

2n+1 , and choose the smallest maximizer.

Existence and uniqueness for the solution of (2) in small times, with possibly discontinuous controls, can be derived from results of[START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF][START_REF] Goatin | A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit[END_REF]. Nevertheless, it is shown in[START_REF] Goatin | A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit[END_REF] that, under such assumptions, one can have formation of singularities such as Dirac deltas in finite time.

Acknowledgement

The second author acknowledges the support of the NSF grant #1107444 (KI-Net). The third author acknowledges the support of the ANR project CroCo ANR-16-CE33-0008. The last author acknowledges the support of the ANR project Finite4SoS ANR-15-CE23-0007-01 and of the project FA9550-14-1-0214 of the EOARD-AFOSR.

Step n At time t n , choose the maximizer (a * , b * , η * , i * ) of |L U (a,b,η)gi V [µ(t n )]| in the space Ω tn × {1, . . . , m}. Observe that we only added the index i in the maximization process. Then, we have two cases:

n or if Ω tn is empty, then choose the zero control χ ω u(t, x) ≡ 0 and let the measure µ(t), starting at µ(t n ), evolve according to [START_REF] De Gennaro | Decentralized control of connectivity for multi-agent systems[END_REF] over the time interval [t n , t n+1 ], where t n+1 is the smallest time greater than t n such that there exists (ā, b, η, ī) ∈ Ω tn+1 × {1, . . . , m} for which

and let the measure µ(t), starting at µ(t n ), evolve according to [START_REF] De Gennaro | Decentralized control of connectivity for multi-agent systems[END_REF] over the time interval [t n , t n+1 ], where t n+1 is the smallest time greater than t n satisfying at least one of the following conditions:

The proof of convergence of µ(t) to Z is obtained by combining the proof of Theorem 1 in Section 4 with the proof of the finite-dimensional sparse Jurdjevic-Quinn stabilization method with hysteresis given in [START_REF] Caponigro | Sparse Jurdjevic-Quinn stabilization of dissipative systems[END_REF]. We do not provide details.

Application to crowd models

In this section, we give some relevant models to which Theorem 1 can be applied. Control problems for equations of the form [START_REF] De Gennaro | Decentralized control of connectivity for multi-agent systems[END_REF] arise naturally when studying large crowds of interaction agents. Consider a system of N interacting agents in which the dynamics of the state x i ∈ R d of the i-th agent are influenced by the state of the other N -1 agents, according to the time evolution

where

) represents interaction rules, that are the same for any pair of agents. When the number N of agents is large, it is often convenient to describe the evolution of the system as a mean-field equation. In the mean-field limit, when N → +∞, the evolution of the mass of the agents µ ∈ P c (R d ) is described by [START_REF] Agrachev | Controllability on the group of diffeomorphisms[END_REF] with

Indeed, to derive the mean-field model (1) from the finite-dimensional multi-agent models [START_REF] Villani | Topics in Optimal Transportation[END_REF], it suffices to consider the empirical measure µ(t) = 1 N N i=1 δ xi(t) . We consider then the controlled version of the multi-agent system [START_REF] Villani | Topics in Optimal Transportation[END_REF], given by ẋi = 1 N j =i F (x i , x j ) + u i g i (x 1 , . . . , x N ), i = 1, . . . , N,


Jocelyn Prigent

Jean-Marc Joubert

Michel Latroche
email: michel.latroche@icmpe.cnrs.fr

Study of the ternary system Al-H-RE (RE = Er, La and Y) in liquid state

Keywords: rare earthhydrides, aluminum, hydrogensolubility, ternary phase diagram, Calphad, Thermodynamic modelling

Hydrogen is the only gas able to dissolve in aluminum. The solubility Sof hydrogen in Al obeys the Sieverts' law andSis much larger in the liquid phase (above 660.4°C) than in the solid one. This might lead to the formation of porosity during aluminum casting. In the present work, the ternary system Al-H-RE (RE=Er, La, Y) is investigated. The equilibria between the different phases are determined in the presence of liquid Al, RE and H 2 gas by both experimental measurements and phase diagram calculations. The possible formation of metallic hydrides and the change in the hydrogen solubility S are discussed regarding pressure and temperature determining the thermodynamics of the system.

Introduction

Aluminum is a metal widely used in metallurgy for various applications in transportation, packaging or building. Aluminum alone exhibits rather poor mechanical properties but they can be significantly improved by alloying Al with other elements like Li, Cu, Mn, Mg or Si.

Hydrogen is the only gas soluble in aluminum and its alloys. The quantity of hydrogen absorbed by solubilization in the liquid phase in an aluminum alloy remains low, of the order of 10 -4 at.% H. It can dissolve in both the solid and the liquid phase following the Sieverts' law. The solubility in the solid phase is systematically lower than in the liquid phase. Therefore, upon solidification, a large fraction of the atomic hydrogen dissolvedin the liquid phase recombines in gaseous form in the solid, generating bubbles in the alloys. The presence of alloying elements affects the solubility and favors the presence of structural defects, which increases the solubility. In the liquid phase, it is noted that elements which have a strongaffinity with hydrogen (Mg, Ti, Li, etc.) tend to increase the solubility [START_REF] Anyalebechi | Analysis of the effects of alloying elements on hydrogen solubility in liquid aluminum alloys[END_REF]. On the contrary alloying elements such as Si, Cu or Fewhich form unstable hydrides decrease the solubility of hydrogen in the alloy [START_REF] Anyalebechi | Analysis of the effects of alloying elements on hydrogen solubility in liquid aluminum alloys[END_REF].

Rare Earths (RE) are known to form very stable hydrides in the range 2 to 3 H per metal [START_REF] Manchester | H-Mg (Hydrogen-Magnesium)[END_REF][START_REF] Fagerstroem | Phase Diagrams of Binary Hydrogen Alloys[END_REF].

A study of the binary diagrams and enthalpies of formation shows that severalrare earths like Er, La or Y form solid hydrides at temperatures above 660 °C, i.e. the temperature of aluminum solidification. The addition of a 4f element in aluminummighttherefore have two effects: a change in the hydrogen solubility Sin the aluminum-hydrogen solution and/or the precipitation of rare earth hydride in the liquid phase. Such phenomenon has been already reported by ChiottiandWoerner [START_REF] Chiotti | Metal hydride reactions[END_REF] for magnesium and Mg-Zn alloys for which they observed precipitation of hydrides in the temperature range 500-900°C from the following added solutes Ca, Y and Th. Similarly, Smithells [START_REF] Smithells | Metals Reference Book[END_REF] reported precipitation of ZrH 2 in liquid magnesium.

In the present work, aluminum alloyed with small quantities of Rare Earths(RE=Er, La, Y) is investigated in liquid state in the presence of hydrogen gas. Solubility measurements are made and compared to ternary phase diagram calculations. Effects of the RE addition on the Al sorption properties are discussed.

Experimental

The alloys were obtained by induction melting of the pure elements (Al, La 5N; Y 3N; Er 4N purities) under vacuum in a water-cooled copper crucible under secondary vacuum (10 -6 mbar). The ingots wereused as-cast without any further thermal treatment. Composition were chosen according to the known binary Al-RE phase diagrams for RE=Er, La and Y [START_REF] Massalski | Binary Alloy Phase Diagrams[END_REF] in the range 1 to 2% mol.,i.e. in the aluminum-rich part of the diagram.

Structural analysiswas made at room temperature by X-ray powder diffraction using a Bruker D8 Advance diffractometer with Cu-K  radiation, flat plate, Bragg-Brentano geometry and backscattered rear graphite monochromator. Experimental data were refined withthe Rietveld method using the Fullprof program [START_REF] Rodríguez-Carvajal | Fullprof: a program for Rietveld refinement and pattern matching analysis[END_REF].Chemical analysis was performed by electron probe microanalysis (EPMA) using a Cameca SX-100 to check the composition of the alloys. Sample observation were made with a Zeiss LEO Scanning Electron Microscope (SEM) equipped with a field emission gun and an energy dispersive X-ray detector (EDX).

Hydrogen solubilityS was determined with a high temperature hydrogen rig by manometric measurements using the Sieverts' method. The sample holder was heated at two temperatures 675 and 725°C (i.e. above T liq =660.4°C for pure Al) and the working hydrogen pressure was set to 3 50 kPa(0.5 bar). The sample holder was made ofa boron nitride bed filled with Al and introduced in a quartz tube to limit hydrogen permeation out of the rig. The metal to dead-volume ratio was optimized to accurately detect small fraction of sorbed hydrogen. Strong temperature gradient between the thermostated bath of the rig and the heated sample holder leads to cautious measurements. Therefore,the mass of the samples (7~8 g) and the amount of introduced dihydrogen were adjusted so that there was much less gas than the quantity necessary to form the RE hydride in order to improve the detection level.

The Al-Er-H ternary phase diagrams have been assessed with the Calphadmethod using the Thermo-Calc software [START_REF] Sundman | The Thermo-Calc databank system[END_REF]. This technique consists in describing the Gibbs energies of all the phases present in the system as a function of temperature and composition.The thermodynamic description of the constitutive binary systems have been taken from published assessments: Er-H [START_REF] Mascaro | Experimental study and thermodynamic assessment of the erbium-hydrogen binary system[END_REF],Al-Er [START_REF] Saccone | The Al-Er-Mg ternary system Part I: Experimental investigation[END_REF],

Al-H [START_REF] Palumbo | Thermodynamic and ab initio investigation of the Al-H-Mg system[END_REF].The phases taken into account in the present work are: gas, liquid, bcc, fcc, hcp, Al 2 Er 3 , Al 2 Er, Al 3 Er, AlEr, AlEr 2 , -AlH 3 , -AlH 3 , -AlH 3 , ErH 3 , and ErH 2 . The gas phase contains Al, H and H 2 species. The solid and liquid solutions contain the three elements. No hydrogen solubility is considered in the binary Al-Er compounds and no Al solubility in considered in the erbium hydrides. Calculations are made by combining the three databases and performing ternary extrapolations considering that no ternary phase is formed and that the ternary interaction parameters are negligible.

Results

Sample characterizations

All samples have been characterized by X-ray diffraction. Results are given inTable 1. All patterns can be indexed with cubic aluminum and variousbinary phases when rare earths are added. Al 3 Er respectively [START_REF] Massalski | Binary Alloy Phase Diagrams[END_REF]) are observed. A typical powder diffraction pattern is shown in Figure 1 for the composition Al 98 La 2 that exhibits the two-phase coexistence. However, this is not the case for Y for which some phases are observed but cannot be indexed correctly with the known binary phases of the Al-Y binary phase diagram. Further investigations, as described in the next section, have shown that those phases are metastable occurring in the as cast sample but not present any more after thermal treatment. All crystallographic data for the identified phases are in agreement with literature data.

Hydrogen absorption in liquid state

Sorption measurements were made for the different samples at two temperatures (675 and 725 °C), i.e. above the melting of aluminum. Results are given for all samples in Table 2. Pure aluminum was also measured to assess the accuracy of the equipment. The observed value for pure Al is 0.02 mmol of H dissolved in the tested sample (67 ppm mol), one order of magnitude larger than the value calculated with the Sieverts' law with such a sample of 8g. For binary systems, the quantity of hydrogen absorbed after 1h is in the range 0.02-0.07 mmol H,slightly higher that pure Al indicating some influence of the rare earth addition. However, these values are two orders of magnitude lower than that expected in case of the formation of the dihydridesREH 2 .

6 After solidification at room temperature, the samples were analyzed by XRD to check any hydride formation. The observed phases are all metallic and the same asin the as-cast samples for La and Er. For Y, it is worth to note that the unidentified phases reported for the as-cast samples are no longer present whereas the intermetallic Al 3 Y is observed.

Calphad calculations

For a better understanding of the equilibrium existing in the ternary system Al-H-REin the Al liquid state (667 °C), Calphad calculations have been carried out at this temperature and atvarious pressures in the range 1 to 10 8 Pa for the system Al-Er-H. First calculation at 1 Pa shows that ErH 2

does not form at this low pressure (Figure 2a). The binary hydride appears for 10 Pa in equilibrium with the intermetallic phase AlEr 2 (Figure 2b). This is in agreement with the equilibrium pressure of ErH 2 at this temperature which is calculated at 3.87 Pa. At 100 Pa, a three-phase equilibrium between ErH 2 and Al-rich intermetallic occurs (Figure 2c). Increasing the pressure up to 10 MPa (Figure 2d) does not allow to obtain an equilibrium between ErH 2 and liquid aluminum. Such an equilibrium is only observedunder 100 MPa (Figure 2e). This agrees with the calculated pressure (33.2 MPa) at which equilibrium between liquid Al, ErH 2 and gaseous H 2 is expected to occur.For a better interpretation of our experimental data, the ternary phase diagram has been also calculated at 50 kPa and 667°C (Figure 2f), i.e. using the experimentalthermodynamic conditions of section 3.2.The equilibrium is foundbetween liquid Al, Al 3 Er and H 2 gas only, in fairly good agreement with our data obtained from Sieverts' measurements and XRD analysis.

From the calculation of the phase diagram, it is also possible to estimate the solubility of H in the liquid phase depending on the rare earth composition. The variation of S is shown in Figure 3 0,0 0,5

Er (% mol)

Discussion

Aluminum can be alloyed with small quantities of rare earths in the range 12%mol. After solidification, equilibrium between pure Al and either Al 11 La 3 or Al 3 Er is observed. For Y, some unidentified phases are observed but after the thermal treatment done during the sorption experiment, they transform into the expected Al 3 Y phase showing that they are not related to contamination but rather to some out-of-equilibrium state. It can then be concluded that for the three rare earths, the obtained phases are in agreement with the Al-rich part of the published phase diagrams [START_REF] Massalski | Binary Alloy Phase Diagrams[END_REF].

H sorption experiments have been performed in liquid state at two temperatures (675 and 725°C). Though slightly overestimated, the H-sorbed quantities are close to what is expected according to the Sieverts' law for pure Al. When adding small quantities of rare earths, a small increase of the solubility is observed, except for Er. However, the absorbed hydrogen values are well below the quantities expected in the case hydrides would be formed(7 th column in Table 2).

There are two orders of magnitude between the measured values (6 th column in Table 2) as compared to the calculated ones. Moreover, XRD analysis of the samples made after the experiments confirmed the presence of the Al-rich phases only without any traces of REH 2 . This rules out the possible formation of rare earth hydrides, at least in our thermodynamic conditions, as reported by ChiottiandWoerner for Mg(Al).

At this stage, to determine whether or not precipitation of a solid hydride is possible in these systems, the Al-Er-H phase diagram has been calculated using the Calphad method at a temperature close to the aluminum melting temperature (667 °C) with increasing hydrogen pressure. From these data, it is possible to determine the phase equilibria. We show that at this temperature,ErH 2 does form at low pressure (3.87 Pa) but not in equilibrium with liquid aluminum. Increasing the pressure leads to phase equilibrium between ErH 2 and Al-rich intermetallics but again not liquid Al. Such an equilibrium between liquid Al, ErH 2 and H 2 gasoccurs only above 33.2 MPa. This well explained why the hydride precipitation was not observed in our experimental conditions made at lower pressure (50 kPa).

Additionally, Calphad calculations allows to determine the solubility S as a function of the erbium composition. We observed a slight decrease of the solubility from 0.0253 down to 0.0236 Ncm 3 /100g(Al) for 2% mol. of added erbium at 940 K (667°C) and 100 Pa. This resultdoes not contradict our experimental measurements for which the solubility does not increase for 2% mol. of added Er (Table 2) showing a positive effect of this rare earth compared to La or Y. At this stage, more calculated data are needed to predict the possible formation of solid hydrides within liquid Al at low pressure but also to determine whether or not element addition can induce significant changes in the H solubility in liquid aluminum. On this basis, other metal forming hydrides might be investigated to propose aluminum alloy composition with lower porosity related to hydrogen solubility.

Conclusion

The system Al-H-REhas been investigated regarding H solubility in the metallic liquid phase using both experimental and calculated data.Our study shows that hydride precipitation in solid state in the liquid phase is possible for these systems only at high pressure (33.2 MPa). However, Calphad calculations show that adding 2% mol. of erbium could slightly decrease the H solubility in liquid aluminum (by approximately 7%).

Table caption

Table 1 -XRD characterizations of as cast samples for different amounts of added rare earths.

Table 2 -Weight, temperature and hydrogen quantities observed for the different sorption measurements. The phases observed by XRD after exposure in liquid state to H 2 gas.

Figure caption

Figure 1 -

 1 Figure 1 -Refined X-ray powder diffraction pattern for the composition Al 98 La 2 showing the coexistence of two phases: pure Al (Fmm) and Al 11 La 3 (Immm). Measured (blue dots), calculated (red solid line), and difference curves (black line below); vertical bars correspond to line positions of the two observed phases from top to bottom; radiation Cu-K  .

Figure 2 -

 2 Figure 2 -Ternary phase diagrams calculated at 940K (667°C) for the system Al-Er-H with increasing hydrogen pressures: (a) 1 Pa; (b) 10 Pa;(c) 100 Pa;(d) 10 MPa;(e) 100 MPa;(f) 50 kPa.

Figure 3 -

 3 Figure 3 -Variation of the H concentration (% mol) as a function of the Er content (% mol)in the ternary system Al-Er-H at 940K (667°C) and 100 Pa.

Figure 1 -

 1 Figure 1 -Refined X-ray powder diffraction pattern for the composition Al 98 La 2 showing the

Figure 2 -

 2 Figure 2 -Ternary phase diagrams calculated at 940K (667°C) for the system Al-Er-H with

Figure 3 -

 3 Figure 3 -Variation of the H concentration (% mol) as a function of the Er content (% mol) in

Table 1 -XRD characterizations of as cast samples for different amounts of added rare earths.

 1

	Rare earth	Nominal composition (%. mol)	As cast observed phases (XRD)	Expected mass fraction	Obs. mass fraction (XRD)	Space group	Cell parameters (Å)
	None	0	Al	100%	100%	Fmm	a = 4.0486(1)
	Y	1.00	1 unidentified phase Al	-	--	Fmm	a=4.0492(3)
	Y	2.00	2 ununidentified phases Al	--	--	Fmm	a= 4.0489(9)
			Al	95.6	92.3	Fmm	a = 4.0484(1)
							a = 4.4341(5);
	La	1.00	Al 11 La 3	4.4	7.7	Immm	b=10.125(1);
							c = 13.149(2)
			Al	90.9	85.4	Fmm	a = 4.0487(1)
							a = 4.4349(4);
	La	2.00	Al 11 La 3	9.1	14.6	Immm	b = 10.1261(9);
							c = 13.149(1))
			Al	96.2	92.8	Fmm	a = 4.0495(1)
	Er	1.98
			ErAl 3	3.8	7.2	Pmm	a = 4.2116(2)

Table 2 -Weight, temperature and hydrogen quantities observed for the different sorption measurements. The phases observed by XRD after exposure in liquid state to H 2 gas.

 2

	Samples	Weight	T	Total H in the rig	H absorbed after 1h	Global solubility	Expected H absorbed for REH 2	Phases observed (XRD) after exposure in liquid state to H 2 gas
		(g)	(°C)	(mmol H)	(mmol H)	(ppmmol)	(mmol H)
	Pure Al	8.0254	675	0.444	0.02	67	-	Al
	Al 99 Y 1	8.1500	675	0.443	0.05	169	5.906	Al. Al 3 Y
	Al 98 Y 2	7.3654	675	0.526	0.05	192	5.220	Al. Al 3 Y
	Al 98 Y 2	6.6938	725	0.528	0.04	169	4.744	Al. Al 3 Y
	Al 99 La 1	8.5655	675	0.522	0.04	131	6.097	Al. Al 11 La 3
	Al 98 La 2	6.8248	725	0.532	0.07	300	4.672	Al. Al 11 La 3
	Al 99 Er 1	6.5092	725	0.527	0.02	87	4.717	Al. Al 3 Er

Acknowledgements

The authors are thankful to Dr. E. Leroy for technical assistance in the EMPA analysis. This work was financially supported by ANR under the contract Project ANR-2010-RMNP-007-01 PRINCIPIA in partnership withConstellium, Saint-Gobain, University of Lorraine, Ecole Centrale de Paris and ICMPE.