Martino Bardi
email: bardi@math.unipd.it

Juan Pablo

Maldonado Lopez
email: pablo.maldonado@imj-prg.fr

Juan Pablo Maldonado

Maldonado López

Juan Pablo Maldonado López

A Dijkstra-type algorithm for dynamic games

Keywords: Zero-sum dynamic games, Dijkstra algorithm, Pursuit-evasion games, Discrete time games Mathematics Subject Classification (2000) 91A25, 65Kxx, 49N70, 91A23, 91A24, 49N75

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

Introduction

In this paper we study two-person zero-sum dynamic games where the players move a state variable in a finite state space X . Each action has a (possibly discounted) positive cost for player 1, that he pays to player 2, which depends on the position and actions. Player 1 aims at reaching a given terminal set X f and once this is done the game is finished and a final cost is incurred. We adopt a rule of alternating moves that gives an informational advantage to one of the players.

Our purpose is to provide an efficient algorithm to compute the value. We follow an approach inspired by the classical Dijkstra algorithm [START_REF] Dijkstra | A note on two problems in connection with graphs[END_REF]) for finding shortest paths in finite graphs, which has running time O(e + v log v) if a suitable data structure is used, where v, e denote respectively the number of vertices and edges, see [START_REF] Fredman | Fibonacci heaps and their uses in improved network optimization algorithms[END_REF]. The algorithm we propose updates the approximate value function only in the immediate neighbours of those nodes where the value is already computed, thus reducing the computation time, and converges in a finite number of steps. In particular, if the running and terminal costs are constant the algorithm is single-pass as Dijkstra's, namely, the value function is computed only once in each node.

Our assumptions are designed to cover discrete approximations of generalised pursuit-evasion differential games [START_REF] Bardi | Approximation of differential games of pursuit-evasion by discrete-time games. Differential games-developments in modelling and computation[END_REF][START_REF] Bardi | Fully discrete schemes for the value function of pursuit-evasion games[END_REF][START_REF] Bardi | Convergence of discrete schemes for discontinuous value functions of pursuitevasion games[END_REF]; see also the surveys by [START_REF] Bardi | Numerical methods for pursuit-evasion games via viscosity solutions[END_REF] and Cardaliaguet, Quincampoix, and Saint-Pierre 1999 and the references therein) if the grid of the numerical scheme is suitably adapted to the controlled dynamical system. We discuss this connection in detail and prove a convergence result of the value of the discrete game to the value of the differential game as the step of approximation tends to zero. Different from the known theory (e.g., Bardi, Falcone andSoravia 1994 and[START_REF] Sethian | Level set methods and fast marching methods[END_REF] we do not require the usual condition ∆x/∆t → 0, where ∆x is the mesh size and ∆t is the time step.

Our motivation comes from the so-called Fast Marching Methods (briefly, FMM) for Hamilton-Jacobi equations with convex Hamiltonian arising in deterministic control and front propagation problems, introduced in Tsitsikilis (1995) and [START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF] and developed by [START_REF] Sethian | Level set methods and fast marching methods[END_REF], [START_REF] Sethian | Ordered upwind methods for static Hamilton-Jacobi equations: theory and algorithms[END_REF], [START_REF] Cristiani | A fast marching method for Hamilton-Jacobi equations modeling monotone front propagations[END_REF], see also the references therein and Cacace, Cristiani, and Falcone (2014) for some recent improvements. These numerical methods approximate time-optimal control in continuous time and space with a fully discrete Bellman equation on a grid, and then rely on the classical Dijkstra algorithm for an efficient solution of the discrete approximation. We recall that the methods based on Dynamic Programming have several good properties, especially robustness, but they face the well-known "curse of dimensionality". A large amount of research in the last twenty years was devoted to overcoming this difficulty in some cases and FMM played an important role for problems with positive costs. We refer to [START_REF] Mceneaney | Max-plus methods for nonlinear control and estimation[END_REF], Cacace, Cristiani, Falcone, and Picarelli (2012), and the references therein for other approaches.

Recently various forms of FMM were also proposed for solving some Hamilton-Jacobi-Isaacs equations arising from differential games [START_REF] Cristiani | A Fast Marching Method for Pursuit-Evasion Games[END_REF][START_REF] Von Lossow | A min-max version of Dijkstra's algorithm with application to perturbed optimal control problems[END_REF][START_REF] Grüne | Global optimal control of perturbed systems[END_REF][START_REF] Cristiani | A fast marching method for Hamilton-Jacobi equations modeling monotone front propagations[END_REF]), with possible applications to the stabilization of perturbed systems and to front propagation problems with non-convex Hamiltonian. They approximate the continuous problem with a discrete Isaacs equation on a grid. However, so far there is no theoretical justification for using Dijkstra-type algorithms for discrete dynamic games associated to such equations. One of the goals of this paper is contributing to a rigorous foundation of these methods.

Let us mention that some discrete games related to ours are the reachability games studied in [START_REF] Alfaro | Concurrent reachability games[END_REF], where also some numerical algorithms are provided. Moreover, if the players move simultaneously and use random strategies, the game becomes a special case of the zero-sum stochastic games introduced in the seminal paper of [START_REF] Shapley | Stochastic games[END_REF]. Several algorithms have also been proposed to compute the value function of stochastic games, starting with the value iteration algorithm in [START_REF] Shapley | Stochastic games[END_REF]. Some variants designed to accelerate the convergence can be found in the survey by Filar and Vrieze (1991), the more recent paper by [START_REF] Raghavan | A policy-improvement type algorithm for solving zero-sum two-person stochastic games of perfect information[END_REF] and the references therein, and in [START_REF] Kushner | The Gauss-Seidel numerical procedure for Markov stochastic games[END_REF], where a Gauss-Seidel procedure for value iteration is studied. The extension of Dijkstra-type algorithms to some stochastic games is an interesting open problem.

The discrete dynamic game

The model

Let X be a finite set belonging to an Euclidean space R d , representing the state space of a system and whose elements we call nodes. Let A, B be finite sets where the players choose their controls. For a function S :

X × A × B → X define the trajectory x • = x • (x, a • , b • ) recursively by x n+1 = S(x n , a n , b n ), x 0 = x. (1)
Let X f ⊂ X , denote a terminal set of nodes (which player 1 wishes to attain) and let γ ∈ (0, 1] be a discount factor. We introduce the running and terminal cost :

X × A × B → R, 0 < 0 ≤ (x, a, b) ≤ L, ∀(x, a, b) ∈ X × A × B (2) g : X f → R, g 0 ≤ g(x) ≤ g 1 , ∀x ∈ X f (3)
and additionally define the arrival time n :

X × A N × B N → R by n(x, a • , b • ) = min{n ∈ N : x n ∈ X f }, if {n ∈ N : x n ∈ X f } = ∅ +∞ else,
where x n is the trajectory of (1) corresponding to the control sequences a • , b • . To alleviate the notation, we will often write n instead of the more explicit n(x, a • , b • ) when no confusion arises. We have then the following total cost functional J :

X × A N × B N → R J(x, a • , b • ) := n-1 n=0 (x n , a n , b n )γ n + γ ng(x n),
where the discount factor γ satisfies 0 < γ ≤ 1. Observe that if n = +∞ the cost is finite for γ < 1 and +∞ for γ = 1 (i.e., no discount). Player 1 chooses a • ∈ A N and player 2 chooses b • ∈ B N . The aim of player 1 is to minimize the cost functional, whereas player 2 has the opposite goal. We assume that both players observe each other's actions and the state x • .

We refer to G = G X , X f , S, A, B, , g, γ as the game.

The lower value function

We consider an information pattern where player 1 is informed in advance of the action that player 2 will play at each time. Although not realistic in many situations, this is relevant in the discretization of the lower value of a differential game, as we will see in Section 3, and also in the case of discrete robust control problems, where player 2 represents a disturbance.

Definition 1 A map α : B N → A N is a non anticipating strategy for player 1 if b n = bn , ∀n ≤ m =⇒ α[b • ] n = α[ b• ] n , ∀n ≤ m.
Denote with A the set of non anticipating strategies for player 1. The definition of the set B of non anticipating strategies for player 2 is completely analogous. This allows us to introduce the lower value function

V -(x) := inf α∈A sup b•∈B N J(x, α[b • ], b • ).
The following result follows from familiar arguments, see for instance Chapter 8, Theorem 3.18 in Bardi and Capuzzo-Dolcetta 1997.

Proposition 1

The lower value function satisfies

V -(x) = inf α∈A sup b•∈B N k∧n-1 n=0 (x n , α[b • ] n , b n )γ n + γ k∧n V -(x k∧n ) , ∀k ∈ N. (4)
V

-(x) = max b∈B min a∈A (x, a, b) + γV -(S(x, a, b)) , ∀x / ∈ X f (5) V -(x) = g(x), ∀x ∈ X f . (6)
The first equality ( 4) is the well known dynamic programming property. By taking k = 1 in (4) one can easily prove [START_REF] Bardi | Numerical methods for pursuit-evasion games via viscosity solutions[END_REF]. The last equality (6) follows directly from the definition.

The following form of the dynamic programming property will be useful later.

Proposition 2 Let X f ⊂ X ⊂ X and let ñ denote the arrival time to X , i.e. ñ = ñ(x, a • , b • ) = inf{n ∈ N : x n ∈ X }. Then V -(x) = inf α∈A sup b•∈B N ñ-1 n=0 (x n , α[b • ] n , b n )γ n + γ ñV -(x ñ) .
Proof This is a direct consequence of the dynamic programming property (4) since ñ ≤ n.

The algorithm

The following algorithm computes the lower value function:

Require: n = 0, Acc 0 := X f , W 0 (x) := +∞, ∀x ∈ X , V - 0 (x) = g(x), ∀x ∈ X f while Acc n = X do for x ∈ X \ Acc n , b ∈ B do A n (x, b) := {a ∈ A : S(x, a, b) ∈ Acc n } end for end while Cons n := {x ∈ X \ Acc n : A n (x, b) = ∅ ∀b ∈ B} while Cons n = ∅ do W n+1 (x) := max b∈B min a∈An(x,b) { (x, a, b) + γV - n (S(x, a, b))}, ∀x ∈ Cons n Acc n+1 := Acc n ∪ argminW n+1 V - n+1 (x) := W n+1 (x), ∀x ∈ argminW n+1 V - n+1 (x) := V - n (x), ∀x ∈ Acc n n ← n + 1 end while
The notations introduced in the algorithm have the following meaning -Acc n is the set of nodes accepted at the step n, at such nodes the approximate value is not re-computed in the next steps; -A n (x, b) ⊆ A is the set of controls that take the state x to Acc n if player 2 uses the control b; -Cons n is the set of nodes considered at the step n, they are the nodes from which player 1 can reach Acc n no matter what player 2 does;

-x ∈ argminW n+1 if W n+1 (x) = min X W n+1 , such nodes become accepted at step n + 1.
Note that Acc n is strictly increasing as long as Cons n = ∅, so the algorithm terminates in a finite number N of steps, at most the cardinality of X \ X f , which we denote with |X \ X f |. Denote also with R the set of nodes from which player 1 can reach the terminal set for any behavior of player 2, i.e., R := {x ∈ X : inf

α∈A sup b•∈B N n(x, α[b • ], b • ) < +∞}.
It is easy to see that if N is the terminal step of the algorithm, i.e., Cons N = ∅ or Acc N = X , then Acc N = R, i.e., the algorithm identifies the set R reachable by the the first player.

The main result of this section states that the algorithm indeed computes the value function. It requires the following additional assumption in the discounted case γ < 1 (see Remark 3 below for a discussion about this condition).

Condition 1 If γ < 1 L + γg 1 ≤ 0 1 -γ .
Proposition 3 Assume either γ = 1, or γ < 1 and Condition 1. Then, for any n ≤ N ,

V - n (x) = V -(x)
, for all x ∈ Acc n , and the algorithm converges in N ≤ |X \ X f | steps to the value function V -on the reachability set R.

Proof Observe that for n = 0 the conclusion holds by definition. It suffices to prove that V - 1 (x) = V -(x) for x ∈ Acc 1 since by Proposition 2, if we know V -on X = Acc 1 , then we can obtain V -as the value of the new problem with X f replaced by X and g by V -| X and thus conclude by induction.

Observe first that V - 1 (x) ≥ V -(x) follows easily from the definitions. Now for x ∈ argmin x∈Cons1 W 1 (x) consider an optimal pair (α * , b * • ) ∈ A × B N and the corresponding optimal trajectory x n starting from x, that is, 4) and > 0 we have that

x n+1 = S(x n , α * [b * • ] n , b * n ), x 0 = x, V -(x) = J(x, α * [b * • ], b * • ). If n(x, α * [b * • ], b * • ) = 1 then V -(x) = W 1 (x) = V - 1 (x), which is the desired conclusion. If, instead, n := n(x, α * [b * • ], b * • ) > 1 we will distinguish two cases. -Case γ = 1. From (
V -(x) = n-2 n=0 (x n , α * [b • ] n , b * n ) + V -(x n-1 ) > V -(x n-1 ).
On the other hand, we have an optimal pair strategy-control and corresponding optimal trajectory starting from x n-1 that reaches X f in one step. Then V -(x n-1 ) = W 1 (x n-1 ) and so

V -(x n-1 ) = W 1 (x n-1 ) ≥ W 1 (x) = V - 1 (x) ≥ V -(x)
which is a contradiction.

-Case γ < 1.
Here (4) gives

V -(x) = n-2 n=0 (x n , α * [b • ] n , b * n )γ n + γ n-1 V -(x n-1 ) = n-2 n=0 (x n , α * [b • ] n , b * n )γ n + γ n-1 W 1 (x n-1 ) ≥ n-2 n=0 (x n , α * [b • ] n , b * n )γ n + γ n-1 W 1 (x) ≥ n-2 n=0 (x n , α * [b • ] n , b * n )γ n + γ n-1 V -(x).
Then,

V -(x)(1 -γ n-1 ) ≥ 0 1 -γ n-1 1 -γ =⇒ V -(x) ≥ 0 1 -γ .
On the other hand, since

V - 1 (x) ≤ L + γg 1 we get the inequality V - 1 (x) ≤ V -(x) by Condition 1.

Remarks and some open problems

Remark 1 The main advantage of our algorithm is that at each step the approximate value function is updated only on some nodes, namely on Cons n . Moreover, at least one of these nodes becomes accepted and will not be considered in the next steps. In particular, if the costs and g are constant (generalized pursuit-evasion games), then W n+1 is constant on Cons n and all considered nodes are accepted. In other words, the value function is computed only once on each node, which considerably speeds up the algorithm in comparison to iterative methods. Algorithms with this property are often called single-pass.

Remark 2 If we stop the algorithm before it terminates, say at ñ < N , Proposition 3 says that we have anyway computed the value function in the set Acc ñ, which may be enough for some practical problems with very large grids.

Remark 3 Condition 1 requires that the oscillation of the running cost and the size of the terminal cost g are not too high compared with 1/(1 -γ). It essentially says that, if player 1 is on a node where he can reach X f , it is convenient for him to do so even if the cost of this step is high, rather than following forever a trajectory with cheap running costs. For any and g verifying ( 2) and (3) the condition is satisfied for γ sufficiently close to 1. It is satisfied also for all γ ∈ (0, 1) in discounted pursuit-evasion games, where ≡ 1 and g ≡ 0.

Remark 4 If γ = 1, we can add a final step to the algorithm by setting V - N +1 (x) := W 0 (x) = +∞ for all x ∈ X \ Acc N , so V - N +1 (x) = V -(x) for x ∈ X \ R and we have convergence on the whole state space X . On the other hand, if γ < 1 the algorithm gives no information on the value V -outside R.

Remark 5 If γ = 1, ≡ 1, and g ≡ 0, the problem for player 1 is finding the path of shortest length that reaches X f , whereas player 2 seeks to make such length the longest possible (generalized pursuit-evasion). If, in addition, there is no player 2, i.e., B is a singleton, the problem reduces to the shortest path and the algorithm of this section is the classical Dijkstra algorithm.

In reachability games there is no running cost and all the states in X f have the same cost. Then the algorithm is essentially the same as Algorithm 2 in Alfaro et al. (2007), where the set Acc n+1 is updated by Acc n+1 := Acc n ∪ Cons n .

Then the reachability set R is computed exactly as in the above generalized pursuit-evasion game, although here the length of the path is not of interest. When the moves are alternating, this algorithm runs in a linear time with respect to the size of the game, defined as1

G := |X| + |A| + |B|.
Remark 6 All the results of this section can be adapted to the upper value function

V + (x) := sup β∈B inf a•∈A N J(x, a • , β[a • ]),
where B is the set of non-anticipating strategies for the second player, defined in the obvious symmetric way. Although in general V -(x) < V + (x), we will see in the next Section that the informational advantage disappears in the continuum limit of discrete approximations of differential games, under suitable assumptions, see Remark 12.

Remark 7 It would be interesting to study the model of Section 2.1 in the case of simultaneous moves, where players select their actions randomly. This falls within the theory of stochastic games initiated by [START_REF] Shapley | Stochastic games[END_REF] (for more recent accounts, see the book by [START_REF] Filar | Competitive Markov Decision Processes[END_REF] or Chapter 5 in Sorin (2002) , although the transition map S is deterministic here. Even in the case of deterministic transitions, there are algebraic obstructions that make the extension of our approach difficult, see for instance the first example provided by [START_REF] Parthasarathy | An orderfield property for stochastic games when one player controls transition probabilities[END_REF]. In their example, the data of the problem consist of rational numbers but the value is irrational. The authors suggest that any algorithm that terminates in a finite number of steps, such as ours, would require that the data and the value of the game remain in the same ordered field. See, for instance, [START_REF] Filar | Competitive Markov Decision Processes[END_REF] for several classes of games with this so-called ordered field property.

Remark 8

The adaptation of the algorithm to the case of stochastic transitions presents several additional difficulties. For a single player the Dijkstra algorithm was extended to the stochastic case in Bertsekas (2001) if there exists a consistently improving optimal policy. A deeper study of the causality properties needed in stochastic shortest path problems with one controller is in [START_REF] Vladimirsky | Label-setting methods for multimode stochastic shortest path problems on graphs[END_REF]. As hinted in the preceding remark, the case of zero-sum two-person stochastic games appears completely open.

Discretisation of differential games

Consider a continuous-time dynamical system controlled by two players

y (t) = f (y(t), a(t), b(t)), y(0) = x, (7)
where It is well known that under suitable conditions v -coincides with the upper value function and it is therefore the value of the game, see Remark 12 for more details. It is also known (see, e.g., Chapter VIII in Bardi and Capuzzo-Dolcetta 1997) that it is a viscosity solution of the (lower) Hamilton-Jacobi-Isaacs equation

f : R d × A × B → R is
λv --max b∈B min a∈A f (x, a, b) • Dv -+ l(x, a, b) = 0 in Ω := R d \ T . ( 8
)
Now take a finite grid X and set X f := X ∩ T , i.e., the final nodes are those in the target. Next fix h > 0 and set

S(x, a, b) = x + hf (x, a, b), (x, a, b) = hl(x, a, b), γ = e -λh .
Then the dynamic game of Section 2.1 is a natural discretisation of the differential game, with the control system replaced by its Euler scheme with step h and the integral cost functional replaced by a Riemann sum. This discretisation is the basis of some semi-Lagrangian approximation schemes for the Hamilton-Jacobi-Isaacs equation, see, e.g., [START_REF] Bardi | Numerical methods for pursuit-evasion games via viscosity solutions[END_REF] and [START_REF] Falcone | Semi-Lagrangian approximations schemes for linear and Hamilton-Jacobi equations[END_REF]. In such schemes one solves the finite difference equation

W (x) = max b∈B min a∈A { (x, a, b) + γW (S(x, a, b))} , ∀ x ∈ X \ X f , (9)
with the boundary condition

W (x) = g(x), ∀ x ∈ X f . (10)
Note, however, that in general S(x, a, b) might not be in X , so in the right hand side W is extended by interpolation among the neighbouring nodes. The main assumption of this section is that the grid is adapted to the dynamics (with time step h), that is,

S(x, a, b) ∈ X ∀ x ∈ X \ X f , a ∈ A, b ∈ B. (11)
Then W can be computed only on the nodes, without any interpolation procedure.

Proposition 4 Under the assumption (11) the solution W of the discrete Isaacs equation (9) coincides with the lower value function V -defined in Section 2.2. Thus it can be computed by the algorithm of Section 2.3.

Proof Under the assumption (11) the discrete Isaacs equation ( 9) coincides with the one-step Dynamic Programming Principle (5) satisfied by V -. On the other hand, from (5) one gets the general Dynamic Programming Principle (4) by induction, and therefore the equality W = V -.

Note that, given a finite set X f , the dynamics f , and h, a grid adapted to the dynamics can be easily constructed, at least in principle, by setting

X 0 := X f , X n+1 := {x ∈ R d : S(x, a, b) ∈ X n , for some a ∈ A, b ∈ B}
and iterating a finite number of times, where the set X n+1 is truncated to a finite set in case it is not finite.

Example 1 Assume that f = f (a, b) does not depend on x. Then the previous construction of a grid adapted to the dynamics becomes explicit

X 0 := X f , X n+1 := {x -hf (a, b) for some x ∈ X n , a ∈ A, b ∈ B},
and after a finite number of iterations it produces a finite grid X , since X f , A and B are finite.

Example 2 Consider the convex-concave eikonal equation in R 2 with Dirichlet boundary conditions

|u x | -δ|u y | = l(x, y) in Ω, u(x, y) = g(x, y) on ∂Ω,
with δ, l > 0 and Ω a rectangle, say Ω = (0, c) × (0, d) (see Bardi and Terrone 2013 for other results on such equation). An associated differential game can be chosen with dynamics

x = a, y = b, a ∈ {-1, 0, 1}, b ∈ {-δ, 0, δ}.
A rectangular grid X = {(jh, kδh) : j = 1, . . . , c h , k = 1, . . . , d δh } is adapted to the dynamics if c h , d δh ∈ N, and this occurs for a suitable sequence of steps h going to 0 if cδ/d = m/n for some m, n ∈ N. In fact we can take h = d δnK for any K ∈ N, because c h = mK and d δh = nK.

The locally uniform convergence of the solution W of the discrete Isaacs equation ( 9) to the lower value function v -as h and the mesh size k of the grid X tend to zero was proved in [START_REF] Bardi | Fully discrete schemes for the value function of pursuit-evasion games[END_REF] in the case v -is continuous, T has a Lipschitz boundary, and k/h → 0, see also [START_REF] Cristiani | Fully-discrete schemes for the value function of pursuit-evasion games with state constraints. Advances in dynamic games and their applications[END_REF], the survey by [START_REF] Bardi | Numerical methods for pursuit-evasion games via viscosity solutions[END_REF], and the recent book by [START_REF] Falcone | Semi-Lagrangian approximations schemes for linear and Hamilton-Jacobi equations[END_REF]. In the case of discontinuous v -the convergence is weaker and the precise statements are more technical, see [START_REF] Bardi | Convergence of discrete schemes for discontinuous value functions of pursuitevasion games[END_REF]. For grids adapted to the dynamics it is unlikely that k/h → 0 because the space and time step are of the same order, as it can be seen in the preceding examples. However, by taking advantage of the property [START_REF] Cacace | A patchy dynamic programming scheme for a class of Hamilton-Jacobi-Bellman equations[END_REF] we can remove such restriction, as we show now.

We consider a sequence h n → 0 and sequences X n , X n f = X n ∩ T , such that X n is a grid adapted to the dynamics with time step h n and

∀ x ∈ R d ∃ x (n) ∈ X n such that lim n x (n) = x. (12)
We call this an admissible sequence of grids. For each pair h n , X n we solve the discrete Isaacs equation [START_REF] Bertsekas | Dynamic Programming and Optimal Control[END_REF] with boundary conditions [START_REF] Cacace | Can local single-pass methods solve any stationary Hamilton-Jacobi equation?[END_REF] and call W n its solution. Then we define for x ∈ Ω the viscosity semi-limits

W (x) := sup lim sup n W n (x (n) ) : X n x (n) → x , W (x) := inf lim inf n W n (x (n) ) : X n x (n) → x .
(13) Note that they are finite at x if W n are equibounded in a neighborhood of x.

The following lemma about the semi-limits is a straightforward extension of a standard result in viscosity theory, see Lemma V.1.6 in [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF].

Lemma 1 Assume W n : X n → R are locally equibounded, X n satisfies (12), φ ∈ C(Ω), and W -φ attains at y a strict maximum relative to the ball B centered at y with radius r > 0. Choose x n ∈ argmax X N ∩B (W n -φ). Then there is a subsequence x n k → y and such that

W n k (x n k ) → W (y) as k → ∞.
The next proposition is the main result of this section.

Proposition 5 Assume h n → 0, X n is an associated admissible sequence of grids, W n solves (9) on X = X n , and the sequence W n is locally equibounded. Then W and W are, respectively, a viscosity suband supersolution of the Isaacs equation [START_REF] Bardi | On the Homogenization of some Non-coercive Hamilton-Jacobi-Isaacs Equations, Commun[END_REF].

Proof We prove that W is a subsolution, the statement about W is obtained in a similar way. Take φ ∈ C 1 (Ω) such that W -φ attains at y ∈ Ω a strict maximum relative to the ball B centered at y with radius r > 0. We must prove that λW (y) -max

Consider the subsequence of maximum points of W n -φ produced in Lemma 1 and relabel it so that x n → y and W n (x n ) → W (y). Since x n ∈ X n \ X n f , the discrete Isaacs equation ( 9) gives

W n (x n ) = max b∈B min a∈A {h n l(x n , a, b) + γ n W n (x n + h n f (x n , a, b))} , γ n := e -λhn .
Then for all n there exists b n ∈ B such that

γ n [W n (x n ) -W n (x n + h n f (x n , a, b n ))] ≤ h n l(x n , a, b n ) + (γ n -1)W n (x n ) ∀a ∈ A.
Since x n is a local maximum point of W n -φ we have, for h n small enough,

φ(x n ) -φ(x n + h n f (x n , a, b n )) ≤ W n (x n ) -W n (x n + h n f (x n , a, b n )).
Combining the last two inequalities gives

min b∈B max a∈A γ n φ(x n ) -φ(x n + h n f (x n , a, b)) h n -l(x n , a, b) ≤ γ n -1 h n W n (x n ) and letting n → ∞ we get min b∈B max a∈A {-Dφ(y) • f (y, a, b)) -l(y, a, b)} ≤ -λW (y),
which implies the desired inequality ( 14).

Now we can get the convergence of the scheme under suitable assumptions.

Corollary 1 Assume λ > 0, ∂T is a Lipschitz hypersurface, f is bounded on ∂T × A × B, h n → 0, X n
is an associated admissible sequence of grids, and the lower value v -∈ C(Ω). Then the semilimits (13) of the solutions W n of ( 9) and (10) satisfy

W (x) = W (x) = v -(x) ∀ x ∈ Ω. ( 15
)
Proof It is easy to see that for λ > 0 the sequence W n is uniformly bounded by sup R d ×A×B |l|/λ+sup R d g.

The rest of the proof is similar to that of Theorem 1 in Bardi, Falcone and Soravia (1994) or Theorem 2.3 in Bardi, Falcone and Soravia (1999). In addition to Proposition 5 we use the following boundary conditions in the viscosity sense:

W ≤ g or λW -max b∈B min a∈A f (x, a, b) • W + l(x, a, b) ≤ 0 on ∂Ω, (16)
W ≥ g or λW -max b∈B min a∈A {f (x, a, b) • W + l(x, a, b)} ≥ 0 on ∂Ω.
To prove, for instance, ( 16), we proceed as in Proposition 5. We take φ ∈ C 1 (Ω) such that W -φ attains a strict local maximum at y ∈ ∂Ω. By extracting a subsequence from the sequence of maximum points of W n -φ produced in Lemma 1 we can assume that either x n ∈ X n \ X n f for all n or x n ∈ X n f or all n. In the former case the argument of Proposition 5 gives the inequality [START_REF] Cristiani | A Fast Marching Method for Pursuit-Evasion Games[END_REF]. In the latter case g(x n ) = W n (x n ) → W (y), and the continuity of g implies W (y) = g(y).

Next we use a comparison principle between a continuous semi-solution of (8) attaining the boundary data g and a semicontinuous semi-solution of (8) with boundary condition in viscosity sense. It can be found in [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF] for Ω bounded and in [START_REF] Bardi | Fully discrete schemes for the value function of pursuit-evasion games[END_REF] for Ω unbounded. Here we need the Lipschitz regularity of ∂Ω, the boundedness of f on ∂Ω, and the fact that, if v -∈ C(Ω), it solves (8) and v -= g on ∂Ω. Then we get the inequalities

W ≤ v -≤ W in Ω.
Then W = v -= W .

Remark 9

The equality (15) implies the following form of uniform convergence of W n to v -: for all > 0 and compact set K there exists n and δ > 0 such that

|W n (x (n) ) -v -(x)| < ∀ x ∈ K, x (n) ∈ X n , n ≥ n, |x (n) -x| < δ,
as it is easy to prove by contradiction.

Remark 10 Proposition 5 and Corollary 1 appear to be new even in the case of a single player, although the convergence of FMM schemes for Hamilton-Jacobi equations with convex Hamiltonian has been studied, e.g., in (Sethian and[START_REF] Sethian | Ordered upwind methods for static Hamilton-Jacobi equations: theory and algorithms[END_REF][START_REF] Cristiani | A fast marching method for Hamilton-Jacobi equations modeling monotone front propagations[END_REF].

Remark 11

The case of discontinuous v -, which may occur in pursuit-evasion games, could also be treated in the framework of [START_REF] Bardi | Convergence of discrete schemes for discontinuous value functions of pursuitevasion games[END_REF].

Remark 12 All the results of this section can be adapted to the upper value of the differential game the upper H-J-I equation coincides with [START_REF] Bardi | On the Homogenization of some Non-coercive Hamilton-Jacobi-Isaacs Equations, Commun[END_REF]. By the uniqueness of viscosity solutions to the Dirichlet problem, if either v -or v + is known to be continuous, we get that v -= v + and the differential game has a value. In this case the informational advantage of player 1 in the discrete game of Section 2 vanishes in the continuum limit.

  Lipschitz with respect y, and A, B are given finite sets, as in the previous section. The admissible controls for the players are à := {a : [0, +∞) → A, measurable} , B := {b : [0, +∞) → B, measurable} . We are given a closed target T ⊆ R n , and define the first time the trajectory hits T as t x (a, b) := inf{t : y x (t; a, b) ∈ T }, where y x (•; a, b) is the solution of (7) corresponding to a ∈ à and b ∈ B, and t x (a.b) := +∞ if the target is never reached. Consider also the cost functional J(x, a, b) := tx 0 l(y(t), a(t), b(t))e -λt dt + e -λtx g(y x (t x ; a, b)), t x = t x (a, b), a ∈ Ã, b ∈ B; where λ ≥ 0 is the discount rate, and the running and terminal costs l : R d × A × B → R and g : T → R are continuous and bounded, with l > 0. In differential games non-anticipating strategies for the first player are maps α : B → à such that, for all t > 0, b(s) = b(s) for all s ≤ t implies α[b](s) = α[ b](s) for all s ≤ t. We denote with Γ the set of such strategies. The lower value function of the game, in the sense of Varaiya, Roxin, Elliott and Kalton, is defined as v -(x) := inf α∈Γ sup b∈ B J(x, α[b], b).

  b∈B min a∈A {Dφ(y) • f (y, a, b)) + l(y, a, b)} ≤ 0.

v

  + (x) := sup β∈∆ inf a∈ Ã J(x, a, β[a]), where ∆ is the set of non-anticipating strategies for the second player. It is known that v + satisfies the upper Hamilton-Jacobi-Isaacs equation λv + -min a∈A max b∈B f (x, a, b) • Dv + + l(x, a, b) = 0 in Ω := R d \ T . If the Isaacs condition holds, namely, min a∈A max b∈B {f (x, a, b) • p + l(x, a, b)} = max b∈B min a∈A {f (x, a, b) • p + l(x, a, b)} ∀ p ∈ R d , x ∈ Ω,

This definition is slightly different than the one in[START_REF] Alfaro | Concurrent reachability games[END_REF], since there only the actions available at each state are considered. Without loss of generality, we assume that all actions are available in all states.

Partially supported by the Fondazione CaRiPaRo Project "Nonlinear


N Morellini

S Grehl

J Mariani

R M Sherrard
email: rachel.sherrard@upmc.fr

What does low intensity rTMS do to the cerebellum?

Keywords: Purkinje cell, climbing fibres, rTMS, magnetic stimulation, reinnervation, spatial learning

Abstract:

Non-invasive stimulation of the human cerebellum, such as by transcranial magnetic stimulation (TMS), is increasingly used to investigate cerebellar function and identify potential treatment for cerebellar dysfunction. However, the effects of TMS on cerebellar neurons remain poorly defined. We applied low intensity repetitive TMS (LI-rTMS) to the mouse cerebellum in vivo and in vitro and examined the cellular and molecular sequelae. In normal C57/Bl6 mice, 4 weeks of LI-rTMS using a complex biomimetic high-frequency stimulation (BHFS) alters Purkinje cell (PC) dendritic and spine morphology; the effects persist 4 weeks after the end of stimulation. We then evaluated whether LI-rTMS could induce climbing fibre (CF) reinnervation to denervated PCs. After unilateral pedunculotomy in adult mice and 2 weeks sham or BHFS stimulation, VGLUT2 immunohistochemistry was used to quantify CF reinnervation. In contrast to sham, LI-rTMS induced CF reinnervation to the denervated hemicerebellum. To examine potential mechanisms underlying the LI-rTMS effect, we verified that BHFS could induce CF reinnervation using our in vitro olivocerebellar explants in which denervated cerebellar tissue is co-cultured adjacent to intact cerebella and treated with BDNF (as a positive control), sham or LI-rTMS for 2 weeks. Compared with sham, BDNF and BHFS LI-rTMS significantly increased CF reinnervation, without additive effect. To identify potential underlying mechanisms, we examined intracellular calcium flux during the 10 min stimulation. Complex high frequency stimulation increased intracellular calcium by release from intracellular stores. Thus even at low intensity, rTMS modifies PC structure and induces CF reinnervation.

Introduction:

Non-invasive stimulation to the cerebellum is increasingly being investigated not only to clarify cerebellar function and its influence on other regions of the brain, but also as a potential treatment for neurological dysfunction. Such stimulation can be induced through repetitive transcranial magnetic stimulation (rTMS) or transcranial direct current stimulation (tDCS) and their effects vary according to the method of stimulation delivery (rTMS frequency, tDCS anode/cathode) [START_REF] Pell | Modulation of cortical excitability induced by repetitive transcranial magnetic stimulation: Influence of timing and geometrical parameters and underlying mechanisms[END_REF][START_REF] Grimaldi | Noninvasive cerebellar stimulation--a consensus paper[END_REF]. While both rTMS and tDCS have been shown to modulate function of cerebello-thalamo-cortical loops [see review 2], the long-term consequences of cerebellar stimulation remain unknown including the effects of such stimulation on the cerebellum itself and its intrinsic circuits. In order to better understand the behavioural or psychiatric outcomes of cerebellar stimulation, animal studies are necessary to identify underlying cellular or network mechanisms of non-invasive cerebellar stimulation.

Cerebellar Purkinje cells (PCs) are large macro-neurons at the centre of the cerebellar cortical circuit, which modulate activity in the deep cerebellar nuclear outflow. Synchronous activity in groups of PCs selects appropriate muscle groups to perform skilled movements [START_REF] Welsh | Dynamic organization of motor control within the olivocerebellar system[END_REF][START_REF] Lang | Olivocerebellar modulation of motor cortex ability to generate vibrissal movements in rat[END_REF]. Purkinje cell synchronous firing is strongly regulated by their climbing fibre afferents from the inferior olive (ION) [START_REF] Welsh | Dynamic organization of motor control within the olivocerebellar system[END_REF]. Olivocerebellar axons cross the midline in the medulla and ascend the contralateral inferior cerebellar peduncle to terminate on PCs organised in narrow parasagittal zones. This precise parasagittal organisation is refined during the developmental process of transient PC multiple innervation by CFs arriving from both contralateral and ipsilateral inferior olives [START_REF] Fournier | Developmental modifications of olivocerebellar topography: the granuloprival cerebellum reveals multiple routes from the inferior olive[END_REF] and the subsequent selective synapse stabilisation and elimination of supernumerary synapses [START_REF] Sugihara | Organization and remodeling of the olivocerebellar climbing fiber projection[END_REF] and ipsilateral projections [START_REF] Fournier | Developmental modifications of olivocerebellar topography: the granuloprival cerebellum reveals multiple routes from the inferior olive[END_REF].

Thus any modulation of Purkinje neurons and/or their CF afferents by magnetic stimulation is likely to affect cerebellar-mediated behaviours.

Since it has been proposed that homeostatic mechanisms render the brains of healthy volunteers less responsive to rTMS than those of patients with neurological conditions [START_REF] Siebner | Preconditioning of low frequency repetitive transcranial magnetic stimulation with transcranial direct current stimulation: evidence for homeostatic plasticity in the human motor cortex[END_REF] we also lesioned the olivocerebellar path to induce a condition in which we could more easily identify rTMS-induced cerebellar plasticity. Plasticity within the olivocerebellar pathway can be induced by cutting it unilaterally (pedunculotomy: Px) in rodents and injecting neurotrophic factors, such as brain-derived neurotrophic factor (BDNF) [START_REF] Dixon | Brain-derived neurotrophic factor induces post-lesion transcommissural growth of olivary axons that develop normal climbing fibers on mature Purkinje cells[END_REF], into the denervated hemi-cerebellum. This stimulates the remaining (ipsilateral) ION to grow a new pathway, which reinnervates the denervated hemicerebellum [START_REF] Dixon | Brain-derived neurotrophic factor induces post-lesion transcommissural growth of olivary axons that develop normal climbing fibers on mature Purkinje cells[END_REF][START_REF] Willson | BDNF increases homotypic olivocerebellar reinnervation and associated fine motor and cognitive skill[END_REF]. The induced reinnervation is incomplete, but respects the parasagittal organisation [START_REF] Dixon | Brain-derived neurotrophic factor induces post-lesion transcommissural growth of olivary axons that develop normal climbing fibers on mature Purkinje cells[END_REF] and supports recovery of motor and spatial function [START_REF] Willson | BDNF increases homotypic olivocerebellar reinnervation and associated fine motor and cognitive skill[END_REF]. Given that intracranial injection of BDNF is not readily transferrable to the clinic and that rTMS can upregulate BDNF synthesis [START_REF] Wang | Repetitive Transcranial Magnetic Stimulation Enhances BDNF-TrkB Signaling in Both Brain and Lymphocyte[END_REF], we wanted to investigate whether non-invasive cerebellar stimulation could induce similar olivocerebellar reinnervation.

Low intensity rTMS alters Purkinje cell morphology

To investigate the effect of magnetic stimulation on cerebellar PCs, we applied a low intensity (10mT) high frequency complex pattern of stimulation (LI-rTMS), which we have previously shown modulates axonal projections in the visual system [START_REF] Rodger | Transcranial pulsed magnetic field stimulation facilitates reorganization of abnormal neural circuits and corrects behavioral deficits without disrupting normal connectivity[END_REF], to the cerebellum of healthy mice aged 3 or 7 months, 10 min per day for 4 weeks. Sham treated animals underwent the same handling but the stimulator was not turned on. We then evaluated gait on a CatWalk XT (Noldus Information Technology, Wageningen, The Netherlands), motor coordination on an accelerating rotarod (TSE Systems, Bad Homburg Germany) and spatial learning in a Morris Water maze according to our established protocols [START_REF] Willson | BDNF increases homotypic olivocerebellar reinnervation and associated fine motor and cognitive skill[END_REF][START_REF] Chen | Mature Purkinje cells require the retinoic acid-related orphan receptor alpha (RORα) to maintain climbing fiber monoinnervation and other adult characteristics[END_REF] General motor activity and gait were normal in all groups with no differences between LI-rTMS treated mice and sham controls. However when we evaluated spatial learning and memory in the Morris water maze, we found that while LI-rTMS did not alter parameters of spatial learning, in the probe test there was a positive effect of LI-rTMS on spatial memory in comparison to sham-treated controls (p<0.05; Fig 1).

We then wanted to see if these modifications in spatial behaviour were accompanied by morphological changes. After behavioural testing, mice were euthanized and the cerebellum prepared for electrophysiology [START_REF] Willson | BDNF increases homotypic olivocerebellar reinnervation and associated fine motor and cognitive skill[END_REF]. We carried out whole cell patch clamp recordings of Purkinje cells in cerebellar slices, with biocytin in the internal pipette solution to allow visualisation of the cell morphology after fixation and streptavidin revelation.

Confocal imaging of fluorescent PCs was followed by simple measurements of dendritic arbor dimensions. In the LI-rTMS treated mice, PC dendritic arbors were taller and had a greater surface area (p< 0.05; Fig 1). Higher power images were used to quantify spine density on these same PCs; we found that spine density was increased in LI-rTMS treated mice (p<0.01). Whether these spine changes are directly due to the LI-rTMS, or indirectly through the improved spatial learning and memory, cannot be identified from this study. However, since these morphological studies were undertaken after the 3 weeks of behavioural testing, these data reveal that one month of daily LI-rTMS can induce stable changes to PC morphology in conjunction with modified spatial learning and memory.

Low intensity rTMS induces CF reinnervation

In a second series of experiments we also tested whether magnetic stimulation parameters can promote PC reinnervation post-lesion. These experiments were undertaken using our explant model of the olivocerebellar system in vitro [START_REF] Chédotal | The embryonic cerebellum contains topographic cues that guide developing inferior olivary axons[END_REF][START_REF] Letellier | Synapse elimination in olivo-cerebellar explants occurs during a critical period and leaves an indelible trace in Purkinje cells[END_REF], in which the cerebellum and brainstem are cultured "en bloc" to retain all cell populations and circuits. These explants develop as in vivo and can be manipulated to replicate the pedunculotomy lesion and PC reinnervation [START_REF] Letellier | Synapse elimination in olivo-cerebellar explants occurs during a critical period and leaves an indelible trace in Purkinje cells[END_REF]. They are highly reproducible and thus form an easily controlled experimental paradigm for examining the effects of magnetic stimulation on olivocerebellar neosynaptogenesis. Following deafferentation of the hemicerebellum and co-culture adjacent to an intact explant, we treated them with our complex pattern low intensity repetitive magnetic stimulation (LI-rMS) [START_REF] Rodger | Transcranial pulsed magnetic field stimulation facilitates reorganization of abnormal neural circuits and corrects behavioral deficits without disrupting normal connectivity[END_REF] for two weeks.

VGLUT2-positive CF reinnervation was induced by our LI-rMS. To evaluate potential mechanisms underlying this process, we used qPCR to identify that BDNF was upregulated in the explant. Moreover, the density of reinnervation induced by LI-rMS is not additive with BDNF-induced repair, suggesting that the same pathways mediate these two responses. We are currently investigating whether this effect of LI-rTMS is reproducible in vivo. Our preliminary data indicate that our complex frequency pattern LI-rTMS can induce post pedunculotomy olivocerebellar reinnervation.

Finally, to try to understand the cellular signalling processes likely to underlie these LI-rTMS effects we have undertaken experiments in cell culture [START_REF] Grehl | Cellular and molecular changes to cortical neurons following low intensity repetitive magnetic stimulation at different frequencies[END_REF]. In primary cultures of cortical neurons, we performed calcium imaging using Fura2AM ratiometric fluorescence changes from 5 minutes pre-to 5 minutes post-repetitive magnetic stimulation. We found rMS increased cytoplasmic calcium and pharmacological analysis showed that this was due to release from intracellular stores rather than influx from the extracellular milieu (Fig 1). This increase in calcium was associated with altered expression of genes of interest for neuronal survival (JunD, hexokinase) and cytoskeleton modification (Cyr61) [START_REF] Grehl | Cellular and molecular changes to cortical neurons following low intensity repetitive magnetic stimulation at different frequencies[END_REF].

Conclusion

In conclusion, our data show that repeated low-intensity magnetic stimulation to the cerebellum modifies Purkinje cell dendrites and can facilitate reorganisation of the olivocerebellar circuit; effects which need to be considered during the clinical application of cerebellar stimulation for ameliorating cerebellar dysfunction or modulating extra-cerebellar motor circuits.

  . All our experiments followed guidelines established by le Comité National d'Ethique pour les Sciences de la Vie et de la Santé, in accordance with the European Communities Council Directive 2010/63/EU.

Fig 1 A

 1 Fig 1

Acknowledgments

We wish to thank Tom Dufor, Anaëlle Dumas, Jennifer Fransson and Alix Noel de la Font Reaulx for technical assistance. The study was supported in part by the Institut de la Recherche sur la Moelle épinière et l'Encéphale in France and the Neurotrauma Research Program Western Australia.

Conflict of Interest

There are no current or potential conflicts of interest for any of the authors.


Marie-Nour Kaydouh
email: marie.kaydouh@std.balamand.edu

Nissrine El Hassan
email: nissrine.hassan@balamand.edu

Anne Davidson
email: anne.davidson@upmc.fr

Sandra Casale
email: sandra.casale@upmc.fr

Pascale Massiani
email: pascale.massiani@upmc.fr

Effect of the order of Ni and Ce addition in SBA-15 on the activity in dry reforming of methane Effet de l'ordre d'ajout du Ni et du Ce dans SBA-15 sur l'activité en reformage à sec du methane

Keywords: methane dry reforming, mesoporous SBA-15 silica, nickel, cerium, impregnation Résumé

Dry reforming of methane has been carried out on SBA-15 catalysts containing 5 wt% Ni and 6 wt% Ce. The effect of the order of Ni and Ce impregnation on the catalytic activity has been studied. Both metals were added using the "two solvents" method that favors metal dispersion inside the pores. Characterizations by XRD (low and high angles), N 2 sorption, SEM and TEM of the materials after metal addition and calcination indicate good preservation of the porosities and high NiO and CeO 2 dispersion inside the porous channels.

Reduction was carried out before catalytic tests and followed by TPR measurements. The most active reduced catalyst was the Ni-Ce/SBA-15 sample prepared by impregnating cerium first, then nickel. All catalysts were highly active and selective towards H 2 and CO at atmospheric pressure. Full CH 4 conversion was obtained below 650 o C. The higher performances compared to those reported in literature for mesoporous silica with supported Ni and Ce catalysts are discussed.

Le reformage à sec du méthane a été étudié sur des catalyseurs SBA-15 contenant 5% en poids de Ni et 6% en poids de Ce. L'effet de l'ordre d'imprégnation de Ni et Ce sur l'activité catalytique a été étudié. Ces deux métaux ont été ajoutés en utilisant la méthode à "deux solvants" qui favorise la dispersion du métal à l'intérieur des pores. La caractérisation par DRX (petits et grands angles), adsorption de N 2 , MEB et MET des matériaux après ajout du métal et calcination montre une bonne préservation de la porosité et une grande dispersion Mots clés: reformage à sec du méthane, silice mésopoureuse SBA-15, nickel, cérium, imprégnation

Introduction

Dry reforming of methane (CH 4 + CO 2 → 2H 2 + 2CO) is an attractive process that allows the production of recoverable compounds from the utilization of two major greenhouse gases, namely methane and carbon dioxide. This reaction has an important positive environmental impact since behind the widely known deleterious effect of CO 2 , methane has a global warming potential roughly 25 times greater than that of carbon dioxide, even with lower amounts in the atmosphere [1]. The second important benefit of the dry reforming reaction is both economic and energetic. Methane is indeed the major component of natural gas that is mainly found in remote areas. Thus, to prevent cost ineffective CH 4 gas transportation, onsite gas-to-liquid transformation via the DRM would be interesting. The dry reforming reaction (DRM) could therefore play a critical role by converting CH 4 and CO 2 into a syngas with hydrogen and carbon monoxide in equimolar ratio that can be directly used in Fischer Tropsch Synthesis for the production of liquid hydrocarbons with high selectivity. However, a drawback of DRM is its endothermic character and the requirement of large amounts of energy.

Recent studies on DRM catalysts have mainly focused on Ni-based systems that are comparably as active as noble metals but more available and less expensive [2,3].

Nevertheless, the high sensitivity of the dispersed Ni active phase to both carbon deposition and to sintering constitutes essential drawbacks that still hinder the use of these catalysts in industrial applications. A suggested solution to inhibit carbon deposition is to promote the catalyst with a metal oxide, especially CeO 2 . The latter is characterized by redox properties (Ce 4+ ↔ Ce 3+ ) that promote oxygen vacancies formation, thus enhancing the mobility of surface oxygen whose presence contributes in lowering carbon deposition on the metallic particles [4][5][6][7]. It was also reported that both metal dispersion and metal-support interaction are improved in presence of CeO 2 [5,7]. Moreover, the recent use of organized mesoporous silica [8] and carbon [9,10] supports to disperse the Ni active phase appears promising.

Mesoporous SBA-15 materials in particular have distinct properties (high specific surface area, uniform narrow pore size distribution, confinement effect…) known to favor metal dispersion and limit particles sintering, with consequently interesting performances in DRM [5,6].

Few papers, listed in Table 1, have already discussed the effect of the addition of CeO 2 to nickel catalysts on organized mesopourous silica supports. In a Ce-free Ni/SBA-15 catalyst, Wang et al. [5] reported the formation of large Ni particles resulting in heavy coke deposition whereas in presence of cerium high nickel dispersion inside the SBA-15 channels was obtained, the Ce-enriched catalyst was moreover rather stable with minor effect of the formed carbon nanotubes on the main catalytic reaction. Nevertheless, the Ce/Si atomic ratio had to be kept below 0.04 to avoid pore blocking. In another work focused on the promotion of Ni-based SBA-15 catalysts with Ceria-Zirconia mixed oxides (CZ) [6], formation of clustered layers of CZ and NiO phases occurred, which also led to partial or complete blockage of the pores.

Consequently, low metal dispersion and very slight enhancement of the catalytic stability were obtained in spite of the presence of ceria. In SBA-16 supported catalysts [7], the Ce addition also improved both nickel dispersion and catalytic stability but comparable coke deposition (mainly of C β type) was observed with and without Ce. However, migration of Ni particles to the external surface of the support and structural collapse of SBA-16 were inhibited in presence of Ce.

From these previous works, the effect of CeO 2 on mesoporous Ni/oxide catalysts seems to be important in increasing the dispersion of the Ni particle size and stabilizing it during reaction.

Nevertheless, in a recent study on preparation of Ni/SBA-15 catalysts for DRM [11], we obtained high and stable nickel dispersions by applying a "two solvent" method of impregnation [13][14][15] that allowed full deposition of the metal nanoparticles inside the pores. Moreover, at the medium reaction temperatures applied (500-650°C), the addition of cerium had only a minor effect on both Ni dispersion and catalytic activity. In the continuation of this work, our purpose here is to study in more details the influence of the addition of Ce on the DRM performances of Ni/SBA-15 catalysts prepared by the "two solvents" impregnation method. More precisely, the focal point will be to identify the possible effect of the order of Ni and Ce addition. The dispersion of the active phase after thermal activation as well as the catalytic activity and stability of the catalysts after reduction will be presented.

(Table 1)

Experimental part 2.1. Catalysts preparation

The SBA-15 silica support was synthesized according to the method established by Zhao et al. [12]. After filtration, the powders were calcined at 500 o C (heating rate 2 o C.min -1 ) for about 9 hours in a muffle furnace (Nabertherm, LE6/11) in order to remove the organic template and the Cl ions. The nickel nitrate (Ni(NO 3 ) 2 .6H 2 O) or cerium nitrate (Ce(NO 3 ) 3 .6H 2 O) salts (from Sigma-Aldrich) were then impregnated using the "two solvents" technique, described elsewhere [13][14][15], that favors the deposition of the metal precursor species inside the pores of the SBA-15 supports. The cerium content was fixed to 6 wt% and that of nickel to 5 wt%. After each impregnation, the sample was calcined in air at 450°C for 5h previous to further treatment. Even if calcination in air has been reported to be detrimental to the dispersion of the active phase compared to treatment in a neutral or reducing atmosphere reduction [4,16,17], this calcination atmosphere was chosen in order to obtain catalysts that are similar to those usually reported in literature for this type of materials. The three prepared samples were designated Ni-Ce/SBA-15 when the Ce addition preceded the Ni addition, Ce-Ni/SBA-15 when the nickel precursor was added first and co-NiCe/SBA-15 when Ni and Ce were co-impregnated together. Two additional reference catalysts were prepared following the same procedure but with only Ni or Ce addition (samples Ni/SBA-15 and Ce/SBA-15, respectively).

Catalytic test

The catalytic tests were conducted at atmospheric pressure in a continuous U-shaped fixedbed flow reactor using 20 mg of catalysts loaded on a quartz wool plug. Prior to the catalytic reaction, each sample was reduced in situ at 650°C for about 2 hours in 5 volume% H 2 /Ar (30 mL.min -1 ). After that, the temperature was cooled down to 200°C then increased from about 200°C up to 800°C at a rate of 5°C.min -1 . The molar ratio of CH 4 to CO 2 was 1:1 and the gas hourly space velocity (GHSV) was 264 L.g -1 .h -1 . For stability measurements, the temperature was cooled down to 500°C and maintained at this temperature for 12 hours. The effluent gases were analyzed using a Micro-GC Inficon equipped with a Thermal Conductivity Detector (TCD). The conversions during activity and stability measurements were calculated as follows (in and out relate to the measurements at the reactor entrance and exit, respectively):

CH 4 Conversion (%) = 100*(CH 4in -CH 4out )/CH 4in CO 2 Conversion (%) = 100*(CO 2in -CO 2out )/CO 2in

Characterization techniques

The pore volumes and surface areas were determined from N 2 adsorption-desorption isotherms recorded on an ASAP 2020 (Micromeritics) apparatus. Prior to experiments, the samples were degassed under vacuum for 2 hours at 250 o C. XRD patterns were registered on a Brüker D8 diffractometer (Bragg-Brentano, Copper λ= 1.5418 Å). The diffractograms were recorded both at low angles (2 theta between 0.5° and 5°) and wide angles (2 theta between 5° and 95°) ranges to respectively (i) ascertain the pore structure arrangement of the supports and (ii) evaluate the nature and mean size (calculated using the Scherrer equation) of the supported metal nanoparticles. The morphology and homogeneity of the silica grains were identified by SEM (Scanning Electron Microscopy) on a Hitachi SU-70 SEM-FEG. TEM (Transmission Electron Microscopy) images were registered on a 2010 JEOL JEM-200 electron microscope operating at 200 keV (LaB6 gun) and equipped with an energy dispersive X-ray spectroscopy detector (EDS PGT, spatial resolution below 10 nm) for chemical analyses. TPR experiments were carried out on a Micromeritics Autochem 2910 Instrument equipped with a trap for water removal before TCD detection. The H 2 consumption was followed during heating in 5 volume% H 2 /Ar (20 mL.min -1 ) at a rate of 10°C.min -1 from 200°C to 650°C.

Results and discussion

Catalytic activity

The comparison of activity between the different samples is presented first in order to identify if the order of addition of the metals played a role towards catalytic performances of the samples. After reduction of the catalysts at 650°C under 5% H 2 /Ar for 2h, the reactor was cooled down to about 200°C and catalytic measurements were carried out while raising again the temperature up to around 750°C. The variations of the conversions of reactants (CH 4 and CO 2 ) as a function of temperature are shown in Figure 1. Both conversions increased with temperature, reflecting the endothermic character of the dry reforming reaction. However, the CH 4 conversion was always slightly higher than that of CO 2 , over the entire temperature range, suggesting some (although limited) decomposition of methane into hydrogen and carbon (CH 4 → 2H 2 + C) as side reaction [6,18].

In the absence of nickel (Ce/SBA-15 reference catalyst, data not shown), no catalytic activity was observed, thus validating that the active sites for the dry reforming reaction are the metallic nickel nanoparticles. The most active catalyst was Ni-Ce/SBA-15 (Ce impregnated first) on which 100% CH 4 conversion and 90% CO 2 conversion were reached at around 600°C (Fig. 1a,a'). On the contrary, the lowest activity was observed for Ce-Ni/SBA-15 (Ni impregnated first) on which the same conversions as above were attained at 700 (Fig. 1c,c').

The co-impregnation led to better results than Ce-Ni/SBA-15, yet lower than Ni-Ce/SBA-15, but similar to those of the reference catalyst Ni/SBA-15. Therefore, it appears that Ce has to be added previous to Ni in order to ensure a positive effect towards activity, whereas its addition after Ni leads on the contrary to an activity loss. This could be due to a blocking by Ce of some channels where Ni species were previously deposited, thus leading to a hindering of the access to the metal active phase.

All catalysts were relatively stable during 12 hours of catalytic running (data not shown) not only in terms of activity but also with respect to their selectivity. However, the selectivity (reflected in the H 2 :CO molar ratio near unity) varied between the different catalysts, as for their activity, in the order Ni-Ce/SBA-15>co-NiCe/SBA-15>Ni/SBA-15>Ce-Ni/SBA-15.

Thus, the H 2 :CO molar ratio was 0.96 on Ni-Ce/SBA-15 whereas it was only 0.77 on Ce-Ni/SBA-15. Such ratio below 1 suggests the occurrence of side reactions such as RWGS reverse water gas shift reaction [4,6] that therefore occur at higher extent on the Ce-Ni/SBA-15 catalyst than on the Ni-Ce/SBA-15 one. The more limited access to active sites assumed above from conversion data could play a role, enhancing the possibility of side reactions.

Therefore, the order of addition of nickel and cerium is important and the best catalyst in terms of both activity and selectivity is obtained when Ni is deposited on the mesoporous support that already contains CeO 2 .

(Figure 1)

Structural properties and morphology

The low angles X-ray diffractograms of the SBA-15 support (uncalcined and calcined) and of the calcined Ni-Ce/SBA-15 and Ce-Ni/SBA-15 samples are shown in Figure 2. After calcination of the fresh SBA-15 silica at 500°C for 9h, the structuring organic agent in the pores was fully eliminated and the silica walls were condensed. This resulted in a decrease in the unit cell parameter from 94.5 Å for the parent uncalcined SBA-15 (Fig. 2 a) to 81.6 Å for the calcined SBA-15 support (Fig. 2b) as calculated from the (100) main diffraction peak. For all samples, the well-resolved peak indexed as (100) and the tiny (110) and (200) peaks are characteristic of the highly ordered p6mm hexagonal mesoporous structure. This structure was preserved after impregnation and calcination treatments (Fig. 2c,d) with however some collapse as indicated by the lower intensity of the peaks.

(Figure 2)

In spite of this collapse attributable to successive impregnation in solution and calcination, high pore volumes and surface areas were still detected after Ni and Ce deposition. All the Ce-Ni/SBA-15, Ni-Ce/SBA-15 and co-NiCe/SBA-15 samples showed a type IV N 2 adsorptiondesorption isotherm (Fig. 3) typical of mesoporous materials. The H2-type hysteresis loop observed with a smooth adsorption step and a sharper desorption step revealed non-uniform pore shapes and/or sizes. For all three samples, the BET surface area was in the range 550-650 m 2 .g -1 , and the total pore volume was between 0.3 and 0.4 cm 3 .g -1 . As for the conversion data, the values were the highest for Ni-Ce/SBA-15 in which Ce was impregnated first. Nevertheless, the pore diameter was around 3.4 nm whatever the order of Ni and Ce addition suggesting that, if present, the effect of accessibility inhibition discussed above would rather consist in partial channel blocking than in channel diameter restrictions.

(Figure 3)

The structure and morphology of the samples are further depicted in Figure 4. As seen on the MEB pictures (Fig. 4a,b), the SBA-15 grains of the support were small, almost spherical and homogeneous in size with a length of about 0.3-0.4 µm and diameters in the range 0.4-0.5 µm. This morphology was maintained after catalytic study (spent catalysts). Moreover, TEM images taken both in parallel (Fig. 5a) and perpendicular (Fig. 5b) grain orientations clearly showed the highly ordered linear mesoporous system and hexagonal pore structure characteristic of the SBA-15 structure for all samples.

(Figure 4) (Figure 5)

Nanoparticles dispersion and consequences on activity

The darker regions seen on the grey silica grains in the TEM images correspond to metal nanoparticles with higher electronic density than the silica support (Fig. 5). These nanoparticles often appeared regularly aligned along the channels (Fig. 5c) and they were well dispersed inside the pores, even after catalytic tests (Fig. 5d), whatever the sample.

Such preservation of the dispersion upon reaction was also illustrated by EDS measurements.

Indeed, in spite of some heterogeneity in the Ni and Ce enrichment between grains, the local compositions were on average not different before and after catalytic tests, with for instance atomic Ni/Si ratios between 0.06 and 0.18 and Ce/Si ratios in the range 0.02 to 0.07 for the co-NiCe/SBA-15 catalyst. This shows the efficiency of the two solvents impregnation method for dispersing rather homogeneously the nanoparticles inside the grains of the SBA-15 support. Nevertheless, some differences in mean particle sizes were found depending on the order of metal impregnation. This is even better illustrated by the XRD data of calcined samples (Fig. 6 and Table 3). The broad signal observed on all the diffractograms at around 22° is due to the presence of silica with amorphous silica walls. The narrower peaks at 2Θ = 37°, 44°, and 63° visible above this signal correspond to nickel oxide nanoparticles (JCPDS 89-5881) with a face-centered cubic lattice (Fm-3m) and a unit cell parameter equal to 8.35 Å. The peaks at 2Θ = 29°, 34°, 48°, 57°, and 70° are assignable to cerium oxide (JCPDS 81-0792) with a face-centered cubic lattice and a cell parameter of 5.41 Å. All these peaks are slightly asymmetric indicating that the nanoparticles are not spherical but rather slightly elongated as also seen on TEM images.

(Figure 6) (Table 3)

The average nanoparticles sizes were calculated by applying the Scherrer equation to the (400) peak for NiO and to the (220) peak for CeO 2 (Table 3), assuming spherical shapes. It thus appeared that NiO and CeO 2 nanoparticles as small as 5-6 nm and 3-4 nm, respectively, were formed over the Ni-Ce/SBA-15 and co-NiCe/SBA-15 catalysts. With respect to NiO, this size is the same as for the Ce-free Ni/SBA-15 sample (NiO nanoparticles with diameter of 6 nm). It is worth noting that the crystalline domains thus calculated for NiO are greater than the diameter of the pores (around 3.5 nm, Table 1). However, this can be explained by the elongated shape of the particles (zoom in Fig. 5b) with diameter around 3-4 nm, smaller than their length (between 4 and 8 nm), forming worm-like aggregates along the channels (Fig. 5c). This also agrees with the slightly asymmetric XRD diffraction peaks for NiO (Fig. 6).

Therefore, contrarily to previous reports (references in Table 1), it is seen that the addition of Ce does not necessarily reduce the nickel particle size. In the present samples, this size was indeed the same in the three following cases: sample with only Ni, sample co-impregnated with Ni and Ce at the same time, and sample submitted to Ce impregnation first followed by Ni addition. In contrast, the addition of Ni first followed by Ce (Ce-Ni/SBA-15 sample) induced an enlargement of the NiO particle size up to 14 nm (Table 3). At this point, it has to be recalled that the deposited Ni-based nanospecies in this last sample were submitted, during preparation, to a second impregnation step in solution (for Ce deposition) followed by a new calcination step, which may have provoked some instability of the nanoparticles and additional sintering. In fact, this could be the main explanation of the lowest catalytic activity of the Ce-Ni/SBA-15 catalyst, in addition to the accessibility effects discussed above. Indeed, the Ni nanoparticles were on the contrary kept very small on the most active, stable and selective Ni-Ce/SBA-15 catalyst. Therefore, the impregnation of cerium first followed by the addition of nickel is more favorable and this order of impregnation was interestingly also validated in a study on Ni/La 2 O 3 /γ-Al 2 O 3 catalyst [19] in which lower Ni dispersions and catalytic performances were obtained when impregnating nickel first and then lanthanum.

Interestingly, further information on the spent catalysts was provided by the electron microscopy study. Firstly, the nickel dispersion after stability tests, carried out in reaction conditions for 12 h at 500°C (after heating at 800°C), was found as good as before reaction, as for instance visualized in Figure 5d. Secondly, at this temperature, only very rare carbon nanotubes were observed, attesting of the absence of significant coke formation, in line with the above-mentioned close CH 4 and CO 2 conversions found on all catalysts.

Finally, additional insights on the influence of the order of Ni and Ce addition on the Ni active phase were obtained by TPR (Figure 7 and Table 4). It is known that Ni 2+ is directly reduced to Ni 0 without passing by intermediate oxides [20]. Two main reduction processes can be distinguished in the ranges 200-400 and 400-650°C. The first peak (200-400°C) is relatively the less intense for the Ce-Ni/SBA-15 sample, while it is the highest for Ni-Ce/SBA-15 catalyst (Table 4). This suggests that easily accessible and more reducible NiO nanoparticles were present on the Ni-Ce/SBA-15 catalyst, in line with porosity data, thus contributing to its highest performances compared to the other samples. On the contrary, the addition of Ce after Ni reduces the accessibility to the NiO nanoparticles as well as their reducibility (Ni Ce-Ni/SBA-15 catalyst). (Figure 7)

Comparative catalytic performances

As indicated in the introduction, the preservation of high nickel dispersion combined to an easy access to the active sites appears as a prerequisite to ensure good catalytic performances in dry reforming of methane. In this respect, the smaller Ni-based particles obtained with or without pre-addition of Ce are more active than the larger ones formed when the Nicontaining sample is submitted to new impregnation/calcination for Ce addition.

However, regardless of the order of impregnation, the catalysts tested at medium temperatures in this study showed a better catalytic behavior than the mesoporous catalysts from literature listed in Table 1. The activity performances at 600°C of our catalysts and of Ni/SBA-15 catalysts from bibliography containing or not Ce are compared in Figure 8. At this temperature, more than 80% CH 4 and CO 2 conversions are achieved on our samples while they are around 50% [5] and 60% [6] in bibliographic reports.

A first reason of the good performances of our catalysts lies in the formation of very small NiO nanoparticles; this is not only favored by the mesoporosity of the SBA-15 support, but also strongly ameliorated by the two-solvents method used for metal deposition that helped dispersing and stabilizing the active phase inside the channels, with almost absence of metal deposited outside. Thus, excellent nickel dispersions were obtained, with very small particles sizes below 6 nm, compared to higher sizes reported for comparable materials in literature (Table 1). The high dispersion in the Ni-Ce/SBA-15 sample is also the result of the enhanced reducibility of the metal species in this sample (Table 4).

(Figure 8)

Besides, thermodynamic features have to be also considered since the dry reforming reaction is known to be thermodynamically limited, thus usually requiring high temperatures to operate in pure reactant mixtures. Nevertheless, thermodynamic data calculated with the HSC chemistry software prove that 90% CH 4 conversion can be achieved at a temperature of 575°C, calculated with respect to the dilution conditions used in our study. Similarly, another study [21] reported an equilibrium temperature of 710°C for 90% CH 4 conversion at the dilution conditions used in their study. Therefore, dilution effects could play a role in this reaction in which the number of formed molecules is doubled in comparison to reactants. Also, we may assume that the SBA-15 silica support can act as a membrane where the diffusion of H 2 outside the pores is higher than that of the remaining products and reactants, shifting the equilibrium to the side of products and increasing the conversions slightly above thermodynamics. Work is in progress to clarify these points.

Conclusion

Dry reforming of methane was investigated on Ni and Ce based catalysts supported on SBA-15. Metal sequential impregnations and co-impregnations were employed during the preparation. The characterizations by XRD, SEM and TEM show preserved structure and morphology after impregnation, calcination and catalytic test as well as high dispersion of the active phases inside the porosity due to the "two solvents" impregnation method employed. This method minimized the crystallization of NiO on the external surface of the silica grains.

The most active, stable and selective catalyst was Ni-Ce/SBA-15 sample, prepared by impregnating Ce first then Ni. This was attributed to the smaller particles size obtained on this sample and to its higher reducibility. The activity of the co-NiCe/SBA-15 catalyst was comparable to the Ce-free Ni/SBA-15 catalyst. Contrarily, the impregnation of Ce after Ni causes additional plugging of the pores and formation of larger NiO particles, hindering thus the accessibility of the reactants to the active sites and lowering the activity in dry reforming of methane. Table 1: Summary of the silica supported Ni-Ce based catalysts used for DRM.

List of Figures

List of Tables

  des nanoparticules de NiO et CeO 2 à l'intérieur des pores. La réduction des catalyseurs suivie par RTP a été effectuée avant tests catalytiques. Le catalyseur le plus actif est le Ni-Ce/SBA-15 réduit préparé par imprégnation du cérium tout d'abord, puis du nickel. Les catalyseurs étaient très actifs et sélectifs en H 2 et CO sous pression atmosphérique, avec une conversion complète de CH 4 atteinte avant 650°C. Les performances supérieures à celle décrites dans la littérature pour des catalyseurs à base de silice mésoporeuse contenant du Ni et du Ce sont discutées.

Figure 1 :

 1 Figure 1: variation of CH 4 (left) and CO 2 (right) conversions as a function of temperature on (a,a') Ni-Ce/SBA-15, (b,b') co-NiCe/SBA-15, (c,c') Ce-Ni/SBA-15 and (d,d') Ni/SBA-15.

Figure 2 :

 2 Figure 2: low angles XRD patterns of (a) uncalcined SBA-15, (b) calcined SBA-15, (c) calcined Ni-Ce/SBA-15 and (d) calcined Ce-Ni/SBA-15.

Figure 3 : 15 Figure 4 :

 3154 Figure 3: N 2 adsorption-desorption isotherms for samples (a) Ce-Ni/SBA-15, (b) Ni-Ce/SBA-15 and (c) co-NiCe/SBA-15

Figure 5 :

 5 Figure 5: representative TEM micrographs of the (a-c) calcined and (d) spent co-NiCe/SBA-15 catalyst; images taken (a,c,d) parallel or (b) perpendicular to the silica grains.

Figure 6 :

 6 Figure 6: wide angles XRD patterns of calcined (a) co-NiCe/SBA-15, (b) Ni-Ce/SBA-15 and (c) Ce-Ni/SBA-15.

Figure 7 :

 7 Figure 7: H 2 -TPR profiles for (a) Ni-Ce/SBA-15, (b) co-NiCe/SBA-15 and (a) Ce-Ni/SBA-15.

Figure 8 :

 8 Figure 8: comparisons of the CH 4 (left) and CO 2 (right) conversions obtained at 600°C in this study (GHSV=264 L.g -1 .h -1 , CO 2 /CH 4 =1, 20 mg loading) and in bibliography (Ref. 5: GHSV=36L.g -1 .h -1 ,CO 2 /CH 4 =1, 100 mg loading; Ref. 6: GHSV=20 L.g - 1 .h -1 ,CO 2 /CH 4 =1, 300 mg loading).

Figure 1 :

 1 Figure 1: variation of CH 4 (left) and CO 2 (right) conversions as a function of temperature on (a,a') Ni-Ce/SBA-15, (b,b') co-NiCe/SBA-15, (c,c') Ce-Ni/SBA-15 and (d,d') Ni/SBA-15.

Figure 2 :

 2 Figure 2: low angles XRD patterns of (a) uncalcined SBA-15, (b) calcined SBA-15, (c) calcined Ni-Ce/SBA-15 and (d) calcined Ce-Ni/SBA-15.

Figure 3 : 15 Figure 4 :

 3154 Figure 3: N 2 adsorption-desorption isotherms for samples (a) Ce-Ni/SBA-15, (b) Ni-

Figure 5 :

 5 Figure 5: representative TEM micrographs of the (a-c) calcined and (d) spent co-NiCe/SBA-

Figure 6 :

 6 Figure 6: wide angles XRD patterns of calcined (a) co-NiCe/SBA-15, (b) Ni-Ce/SBA-15 and (c) Ce-Ni/SBA-15.

Figure 7 : 15 Figure 8 :

 7158 Figure 7: H 2 -TPR profiles for (a) Ni-Ce/SBA-15, (b) co-NiCe/SBA-15 and (c) Ce-Ni/SBA-15

Table 2 :

 2 Textural properties of the calcined Ni and Ce containing samples

Table 3 :

 3 average sizes (nm) of the NiO and CeO 2 nanoparticles in the prepared catalysts.

Table 1 :

 1 Summary of available bibliographic data on DRM using Ni,Ce/mesoporous silica catalysts prepared by wetness impregnation.

	Catalysts a	Ni content (wt%)	Co-metal	Mean NiO (*Ni 0 ) particles sizes (nm)	T (°C)	GHSV (L.g -1 .h -1 )	Time (h) on stream	Initial conversions (%)	Final conversion s (%)	Ref
								CH 4 CO 2 CH 4 CO 2
	Ni/SBA-15	10	5wt% Ce	7.6*				70	70	65	63
	Ni/Ce-SBA-15		Ce/Si=0. 04	5.3*	700	36	40	75	76	73	75	5
		12
	Ni/Ce-SBA-15		Ce/Si=0. 06	6.3*				72	76	62	67
	Ni/SBA-15	10	-	12				53	83	39	70
	Ni/CZ/SBA-15	5		7				42	39	49	45
	Ni/CZ/SBA-15 Ni/CZ/SBA-15	10 15	10wt% ceria-zirconia	11 11	600	20	24	50 52	66 43	40 57	58 48	6
			10wt%
	Ni/CZ/SBA-15m.l.	10	mulit-ceria-layered	11				48	51	38	43
			zirconia
	Ni/SBA-16		-	9.6*				73	78	52	63
	NiCe /SBA-16	5	14.6 wt% CeO 2	7.5*	700	45	100	72	77	68	74	7

Table 2 :

 2 Textural properties of the calcined Ni and Ce containing samples

		BET surface area	Total pore volume	Pore diameter
	Sample
		(m 2 .g -1 )	(cm 3 .g -1 )	(nm)
	Ce-Ni/SBA-15	650	0.40	3.4
	Ni-Ce/SBA-15	607	0.38	3.5
	co-NiCe/SBA-15	542	0.32	3.4

Table 4 :

 4 relative quantification of H 2 -TPR peak area (in % of total area) for the different catalysts.

	Temperature	co-NiCe/SBA-15 Ce-Ni/SBA-15 Ni-Ce/SBA-15
	range (°C)
	200-400	24	34	40
	400-650	76	66	60

Table 4: relative quantification of H 2 -TPR peak area (in % of total area) for the different catalysts.

Acknowledgements

Authors are grateful to the UOB research council (BIRG 14/2012), to the "Agence Universitaire de la Francophonie" (AUF) and to the CEDRE France-Lebanon Hubert Curien (PHC) Program for their financial supports.

Conflicts of Interest

Please note that no conflict of interests affect the corresponding author and the co-authors of this manuscript.