Camille Bourgaux

Ana Ozaki

Rafael Peñaloza

Livia Predoiu

Provenance for the Description Logic ELH r (Extended Abstract)

Important reasoning tasks performed over description logics (DL) ontologies are axiom entailment, i.e. deciding whether a given DL axiom follows from the ontology; and query answering. In many settings it is crucial to know how a consequence-e.g. an axiom or a query-has been derived from the ontology. In the database community, provenance has been studied for nearly 30 years [START_REF] Cheney | Provenance in databases: Why, how, and where[END_REF][START_REF] Buneman | The providence of provenance[END_REF] and gained traction when the connection to semirings, so called provenance semirings [START_REF] Green | Provenance semirings[END_REF][START_REF] Green | The semiring framework for database provenance[END_REF] was discovered. Provenance semirings serve as an abstract algebraic tool to record and track provenance information; that is, to keep track of the specific database tuples used for deriving the query, and of the way they have been processed in the derivation. Besides explaining a query answer, provenance has many applications like: computing the probability or the degree of confidence of an answer, counting the different ways of producing an answer, handling authorship, data clearance, or user preferences. Semiring provenance has drawn interest beyond relational databases, and in particular, it has recently been considered for ontology-mediated query answering [START_REF] Bourgaux | Querying attributed DL-Lite ontologies using provenance semirings[END_REF] and for ontology-based data access [START_REF] Calvanese | Enriching ontology-based data access with provenance[END_REF], a setting where a database is enriched with an ontology and mappings between them. In the latter, the ontology axioms are annotated with provenance variables. Queries are then annotated with provenance polynomials that express their provenance information.

Example 1. Consider the facts mayor(Venice, Brugnaro) and mayor(Venice, Orsoni), stating that Venice has mayors Brugnaro and Orsoni, annotated respectively with provenance information v 1 and v 2 , and the DL axiom ran(mayor) Mayor, expressing that the range of the role mayor is the concept Mayor, annotated with v 3 . The query ∃x.Mayor(x) asks if there is someone who is a mayor. The answer is yes and it can be derived using ran(mayor) Mayor together with any of the two facts, interpreting x by Brugnaro or Orsoni. This is expressed by the provenance polynomial v 1 × v 3 + v 2 × v 3 . Intuitively, × expresses the joint use of axioms in a derivation path of the query, and + the alternative derivations.

An important contribution of [START_REF] Calvanese | Enriching ontology-based data access with provenance[END_REF] was a semantic definition of provenance for annotated DL-Lite R ontologies-which contains axioms of the form (α, v) where α is a DL axiom and v a provenance variable-based on annotated models. An annotated interpretation I interprets concepts (resp. roles) by sets of pairs of a domain element and a provenance monomial (resp. triples of two domain elements and a provenance monomial). It satisfies e.g. an annotated concept assertion (A(a), v) if (a I , v I ) ∈ A I , and an annotated general concept inclusion (GCI) (C D, v) if (d, m I ) ∈ C I implies that (d, (v × m) I ) ∈ D I . Annotated models and entailment of DL axioms annotated with provenance monomials are then defined as expected. Finally, entailment of conjunctive queries annotated with sums of monomials is defined by considering the matches of the query (extended with binary and ternary predicates to handle provenance information) in the annotated models of the ontology. However, [START_REF] Calvanese | Enriching ontology-based data access with provenance[END_REF] only considered DL-Lite R , which has the particularity of not allowing for conjunctions or qualified role restrictions in the TBox axioms. Our first contribution is to adapt the provenance semantics of [START_REF] Calvanese | Enriching ontology-based data access with provenance[END_REF] for the ELH r variant of the lightweight DL EL, extending the semantics to those ELH r axioms that do not occur in DL-Lite R . It turns out that handling the conjunction allowed in ELH r axioms is not trivial. To obtain models from which we can derive meaningful provenance-annotated consequences, we adopt ×-idempotent semirings and a syntactic restriction on ELH r that forbids conjunctions and qualified restriction of a role to appear in the right-hand side of GCIs (preserving the expressivity of full ELH r when annotations are not considered). We then devise methods to handle annotated ontologies in this context.

1. We present a completion algorithm for annotated ELH r ontologies that computes all axioms annotated with provenance monomials that follow from the ontology in exponential time. We show that it allows us to solve annotated axiom entailment and instance queries in ELH r in polynomial time in the size of the ontology and polynomial space in the size of the provenance polynomial.

2. We also consider the problem of computing the set of relevant provenance variables for the entailment of an axiom (or instance query) α from an annotated ontology O, i.e., the set of variables v such that O |= (α, v × m) for some monomial m. We show that this can be done in polynomial time, using an adaptation of the completion algorithm.

3. Finally, we investigate conjunctive query answering. The query answering methods developed in [START_REF] Calvanese | Enriching ontology-based data access with provenance[END_REF] cannot be extended to ELH r since they rely on the FO-rewritability of conjunctive queries in DL-Lite R , a property that does not hold for ELH r . Therefore, we adapt the combined approach for query answering in EL [START_REF] Lutz | Conjunctive query answering in the description logic EL using a relational database system[END_REF] to provenance-annotated ELH r ontologies. We define a finite canonical model and a rewriting of the query such that the ontology entails the original annotated query if and only if the canonical model satisfies the rewritten query.


Alexis Galland
email: <alexis.galland@gmail.com>

Marc Lelarge

Invariant embedding for graph classification

Learning on graphs requires a graph feature representation able to discriminate among different graphs while being amenable to fast computation. The graph isomorphism problem tells us that no fast representation of graphs is known if we require the representation to be both invariant to nodes permutation and able to discriminate two non-isomorphic graphs. Most graph representations explored so far require to be invariant. We explore new graph representations by relaxing this constraint. We present a generic embedding of graphs relying on spectral graph theory called Invariant Graph Embedding (IGE). We show that for a large family of graphs, our embedding is still invariant. To evaluate the quality and utility of our IGE, we apply them to the graph classification problem and show that IGE reaches the state-of-the-art on benchmark datasets.

Introduction

Many scientific fields study data with an underlying graph or manifold structure such as social networks, sensor networks, biomedical knowledge graphs. The need for new algorithms and neural network architectures that can accommodate these non-Euclidean structures is becoming increasingly clear. In parallel, there is a growing interest in how we can leverage insights from these domains to incorporate new kinds of relational and non-Euclidean inductive biases into deep learning. Recent years have seen a surge in research on these problems, see the recent surveys [START_REF] Goyal | Graph embedding techniques, applications, and performance: A survey[END_REF] and [START_REF] Battaglia | Relational inductive biases, deep learning, and graph networks[END_REF].

Graph classification is an important problem with practical applications in a diverse set of fields: bioinformatics, chemoinformatics or social networks. To solve this problem, one needs to compute graph features that help discriminate between graphs of different classes.

Presented at the ICML 2019 Workshop on Learning and Reasoning with Graph-Structured Data Copyright 2019 by the author(s).

Methods to learn representations that encode structure information about the graph can be categorized to be graphkernel based or graph neural network (GNN) based. Interestingly, in both cases, best perfoming methods are closely related to the Weisfeiler-Lehman (WL) graph isomorphism test [START_REF] Weisfeiler | A reduction of a graph to a canonical form and an algebra arising during this reduction[END_REF], see [START_REF] Shervashidze | Weisfeiler-lehman graph kernels[END_REF] for its connection with graph kernel and [START_REF] Xu | How powerful are graph neural networks[END_REF] for its connection with GNN. The WL test iteratively updates a given node's feature vector by aggregating feature vectors of its neighbors. [START_REF] Babai | Isomorphism of graphs with bounded eigenvalue multiplicity[END_REF] proposes an alternative algorithm for the graph isomorphism problem based on spectral graph theory. Our aim in this paper is to demonstrate the relevance of ideas from spectral graph theory [START_REF] Spielman | Spectral graph theory and its applications[END_REF] to the graph representation learning problem. We introduce a novel and powerful graph feature representation called Invariant Graph Embedding (IGE). We show that IGE is invariant for a large family of graphs and also give intuition on why IGE has a great discrimminative power by explicitly showing the steps in its construction where some graph information is lost. We demonstrate the excellent performances of our IGE on graph classification tasks.

Related Work

Graph classification techniques can be grouped into different categories. Graph kernel algorithms define a distance between graphs in order to compute the similarity of two graphs and classify them using this metric. Among these techniques we list the graphlets kernel [START_REF] Shervashidze | Efficient graphlet kernels for large graph comparison[END_REF] that computes a list of subgraphs and counts the apparitions of these subgraphs in each network of our dataset. There exist algorithms that define different distances between graphs such as Random walk kernel [START_REF] Gärtner | On graph kernels: Hardness results and efficient alternatives[END_REF], shortest path kernels [START_REF] Borgwardt | Shortest-path kernels on graphs[END_REF], Weisfeiler Lehman subtree kernel [START_REF] Shervashidze | Efficient graphlet kernels for large graph comparison[END_REF] or deep graph kernels [START_REF] Yanardag | Deep graph kernels[END_REF]. Other articles are based on recent works in the field of signal processing and define convolutional neural networks on graphs. Convolutions can be defined in many ways, with Graph Fourier Transform and spectral methods [START_REF] Defferrard | Convolutional neural networks on graphs with fast localized spectral filtering[END_REF][START_REF] Henaff | Deep convolutional networks on graph-structured data[END_REF][START_REF] Hammond | Wavelets on graphs via spectral graph theory[END_REF][START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF], or by defining aggregation processes on the features of the neighboors of each nodes [START_REF] Scarselli | The graph neural network model[END_REF][START_REF] Atwood | Diffusion-convolutional neural networks[END_REF][START_REF] Niepert | Learning convolutional neural networks for graphs[END_REF] and other compute graph features without training using a generalization of scattering transform to graphs, initially defined on images [START_REF] Gama | Diffusion scattering transforms on graphs[END_REF]. A fundamental issue when using neural networks defined on graphs is that they are dependant on the number of nodes in the graph and thus their use is not straigh-forward for a graph classification task. A way to cope with this issue would be to add nodes that are not linked to the graph to have a corpus of graphs that are all of the same size. Another way is to define a parametrization of the neural network that is independant with the number of nodes of the graph [START_REF] Bruna | Community detection with graph neural networks[END_REF].

Invariant graph embedding

Notations

We consider connected undirected graphs without self-loops, G = (V, E), where V is the vertex set and E the edge set. The size of the graph is the number of nodes and is denoted by n = |V |. We denote by A its adjacency matrix. In the absence of edge weights, this is a binary, symmetric matrix, with A ij = 1 if and only if there is an edge between nodes i and j. In the presence of edge weights, A ij is the weight of the edge between nodes i and j, if any, and is equal to 0 otherwise. We will mostly deal with unweighted graphs but our analysis carries over provided that all edge weights are positive.

Let d = A1, where 1 is the n-dimensional vector of ones. The components d(1), . . . , d(n) of the vector d are equal to the actual degrees in the absence of edge weights and to the total weights of incident edges otherwise. Let D = diag(d) be the diagonal matrix of these generalized degrees. The Laplacian matrix is defined by L = D -A. We will also use the matrix P = D -1 A which is the transistion matrix for the random walk on the graph G.

Recall that two graphs G = (V, E) and H = (V, F ) are isomorphic if there exists a permutation π of V such that (a, b) ∈ E ⇔ (π(a), π(b)) ∈ F . We can express this relation in terms of matrices associated with the graphs.

Every permutation may be realized by a permutation matrix. For the permutation π, this is the matrix Π with entries given by

Π(a, b) = 1 if π(a) = b, 0 otherwise.
For a vector x, we see that (Πx)(a) = x(π(a)). Let A be the adjacency matrix of G and let B be the adjacency matrix of H. We see that G and H are isomorphic if and only if there exists a permutation matrix Π such that ΠAΠ T = B.

Note that the same relation can be written for the respective Laplacians. In this case, we will write π(G) = H.

A graph embedding (or graph feature) is a function F map-ping graphs to vectors in R d , where d is called the dimension of the embedding. A graph embedding is invariant if for any two isomorphic graphs G and H, we have F(G) = F(H).

From an embedding of nodes, it is straightforward to create a graph embedding. We need only to deal with the fact that graphs can have different sizes. A very simple choice is just to sum the nodes' embeddings. Now, if the embedding of nodes E is equivariant, this will produce an invariant graph embedding F, but the invariance property would be preserved if instead of the sum, we apply any symmetric function as defined below. Definition 1. A function f : S n → R m is symmetric if for any permutation σ and any x ∈ S n , we have

f (x(1), . . . , x(n)) = f (x(σ(1)), . . . , x(σ(n))).
This generic construction of graph embedding from nodes'embeddings will be used repeatedly in the sequel.

Invariant embedding from eigenvalues

W define our first invariant embedding. Recall that the Laplacian of a graph of size n is positive semi-definite with eigenvalues

λ 1 = 0 < λ 2 ≤ λ 3 ≤ • • • ≤ λ n . Note that λ 2 > 0 because we assumed that the graph is connected. For a fixed k 1 , we define F 1 (G) = (λ 2 , . . . , λ k1+1 ) if n ≥ k 1 + 1 and if n ≤ k 1 , then F 1 (G) = (0, . . . , 0, λ 2 , . . . , λ n )
where this last vector is completed by zeros to reach size k 1 .

Note that adding zeros 'on the left preserves the ordering of the λ i 's. It is clear that this embedding is invariant and as noted in (de Lara & Pineau, 2018) it already has a good discriminative power on real graphs. However, there are still many pairs of graphs that are non-isomorphic but which have the same eigenvalues and we will now improve our embedding to deal with those.

Space embedding

We now construct an embedding capturing the property of a random walk on the graph. Remember that P k ij is the probability for a random walk started in node i to be in node j after k steps. If we have features F available on nodes like atoms encoded in a one-hot vector for biochemical datasets, we consider P k F = (x k 1 , ..., x k n ) which is, for each node, the aggregation of the features of its k-hope neighbors. Clearly P k F is an equivariant embedding of nodes so that for any symmetric fucntion f : R n → R m , f (x k (1), . . . , x k (n)) is an invariant feature of the graph. If we don't have features available on nodes, we use F = (d 1 , ..., d n ) the vector of node degrees.

We now specify the symmetric function f that we take: for a graph of size n, we define

f (x 1 , . . . , x n ) = hist(x 1 , . . . , x n ; t), ( 1
)
where hist is the histogram function and t is the number of bins of the histogram. Note that hist(.; t) is a symmetric function for any t and it characrerizes the multiset {x 1 , . . . , x n }.

In summary, our second graph embedding F 2 (G) ∈ R k2t2 is obtained by concatenating the features defined in (1) with number of bins t 2 and applied to the vector P k F for values of k ∈ {1, . . . , k 2 } for a fixed parameter k 2 .

Spectral theory for graph embedding

So far, we constructed invariant graph embeddings, so that two isomorphic graphs will produce exactly the same embeddings. We will now relax this constraint. In this section, we define our spectral embedding and show that it is still invariant for graphs in which all eigenvalues have multiplicity 1.

Invariant graph theory

Recall that L = D -A is the Laplacian of the graph and the spectral theorem yields L = U ΛU T , where

Λ = diag(λ 1 , . . . , λ n ) is the diagonal matrix of eigenvalues of L, with 0 = λ 1 < λ 2 ≤ • • • ≤ λ n , and U = (u 1 , . . . , u n )
is the matrix of corresponding eigenvectors, with U T U = I and u 1 = 1/ √ n.

We now define a classical spectral embedding for nodes. Let X = √ Λ + U T , where Λ + = diag(0, 1/λ 2 , . . . , 1/λ n ) denotes the pseudo-inverse of Λ. The columns x(1), . . . , x(n) of the matrix X define an embedding of the nodes in R n , each dimension corresponding to an eigenvector of the Laplacian. Note that the first component of each vector x(1), . . . , x(n) is equal to 0, reflecting the fact that the first eigenvector u 1 of L is not informative. Moreover since Xu 1 = 0, the centroid of the n vectors is the origin:

n i=1 x(i) = 0.
(2)

The Gram matix of X is the pseudo-inverse of the Laplacian:

X T X = U Λ + U T = L + .
In particular, we can recover the Laplacian matrix by taking the pseudo-inverse of X T X and hence the adjacency matrix and the graph itself. In words, the graph is completly encoded in the nodes' embedding X, i.e. without any loss of information.

We now give a random walk interpretation of this embedding. We consider the random walk with transition rate A ij from node i to node j: the walker stays at node i an exponential time with parameter d i , then moves from node i to node j with probability P ij = A ij /d i . This defines a continuous-time Markov chain with generator matrix -L and uniform stationary distribution. The sequence of nodes visited forms a discrete-time Markov chain with transition matrix P . Let H ij be the mean hitting time of node j from node i. Observe that H ii = 0. The following results are standard, see [START_REF] Lovász | Random walks on graphs: A survey[END_REF], we follow the notations in [START_REF] Bonald | Weighted spectral embedding of graphs[END_REF]: H ij = n(x(j) -x(i)) T x(j), hence the mean commute time between nodes i and j is:

C ij = H ij + H ji = n x(i) -x(j) 2 .
Hence, the geometry of the nodes'embedding x(i) is related to the geometry of the graph. Indeed, the matrix of commute times characterizes the graph: Proposition 1. We can reconstruct the adjacency matrix from the matrix of commute times.

Proof. Since C ij /n = x(i) 2 -2x(i) T x(j) + x(j) 2 , we can recover the matrix X T X = L + from C if we know the x(i) 2 's. But thanks to (2), we have j C ij /n = n x(i) 2 + j x(j) 2 and ij C ij /n 2 = 2 j x(j) 2 , hence the claim follows.
We should stress that also all the information about the graph is contained in the relative geometry of the x(i)'s, their intrinsic values also carries relevant information. Here, we state Fiedler's nodal domain theorem [START_REF] Fiedler | A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory[END_REF]:

for any k ≥ 2, let W k = {i ∈ V, x k (i) ≥ 0}.
Then the graph induced by G on W k has at most k -1 connected components. Together with (2), this implies that for low values of k, nodes i with non-negative entries x k (i) tends to be well-connected.

From commute times to graph embedding

As shown by Proposition 1, the commute times contain all the information about the graph. In order to incorporate it to our embedding, we proceed in a very straightforward manner. Instead of the euclidean distances x(i) -x(j) 2 , we compute the dot products x(i) T x(j) for 1 ≤ i, j ≤ n. Then, we 'flatten' this matrix to obtain a vector of size n 2 and pass this vector through an histogram.

Formally, our embedding is defined by:

F 3 (G) = hist((x(i) T x(k)) 1≤i,k≤n ; t 3 ) ∈ R t3 , (3)
which is clearly an invariant embedding of the graph. Note that even if we can reconstruct the original graph from the matrix (x(i) T x(k)) 1≤i,k≤n , our embedding F 3 defined in (3) will not allow to reconstruct the graph.

Invariant graph embedding (IGE)

Handcrafted embeddings

So far, we defined three different graph embeddings:

F 1 (G) ∈ R k1 defined in Section 3.2; F 2 (G) ∈ R k2t2
defined in Section 3.3 and F 3 (G) ∈ R t3 defined in Section 4.2. Our final graph embedding is the concatenation of these embeddings for fixed parameters k 1 , k 2 and t 2 , t 3 :

F IGE (G) = (F 1 (G), F 2 (G), F 3 (G))
which is a vector of size k 1 + t 2 k 2 + t 3 , independent of the size of the graph.

All these embeddings are invariant graph embeddings. But two graphs that are not identical can produce an identical embedding F IGE . Finding the non-isomorphic graphs for which our embedding is not discriminative is out of the scope of this paper. In Section 6, we will assess the discriminative power of our embedding on benchmark datasets for graph classification.

Experiments and results

Datasets: We choose a wide variety of benchmark datasets for graph classification to evaluate our model. The datasets can be separated in two classes. 6 bioinformatics datasets: NCI1, NCI109, PROTEINS, MUTAG, PTC, D&D. And 5 social network datasets: REDDIT-BINARY and REDDIT-MULTI, IMDB-BINARY and IMDB-MULTI and COLLAB. In the 6 bioinformatics datasets, graphs represent chemical compounds. Nodes are atoms and edges represent connections between two atoms. The task is to discriminate molecules active against a cancer from those that are inactive against that same cancer. The social network datasets are composed of ego-networks the labels of the graphs are the nature of the entity from which we have generated the ego-network. More details can be found in [START_REF] Yanardag | Deep graph kernels[END_REF].

Experimental setup: After having computed the features for each graph, we use a Random Forest for the classification. We compared the results with an SVM and we got slightly better results with a random foretst. We perform cross-validation by splitting each datasets in 10 folds and we use a grid search to fixe the parameters k 1 , k 2 , t 2 and t 3 respectively chosen from the sets {5, 10, 20, 50}, {1, 2, 3, 4, 5}, {5, 11, 21, 31} and {5, 11, 21, 31}. We report the mean accuracy and the standard deviation over the 10 folds. We compare our method with 3 state-of-art algorithms: Family of Graph Spectral Distances [START_REF] Verma | Hunt for the unique, stable, sparse and fast feature learning on graphs[END_REF], Diffusion CNNs (DCNN) [START_REF] Atwood | Diffusion-convolutional neural networks[END_REF], Deep Graph Kernel (DGK) [START_REF] Yanardag | Deep graph kernels[END_REF]. For those algorithms we use the same setup as in [START_REF] Verma | Hunt for the unique, stable, sparse and fast feature learning on graphs[END_REF].

Results: From the results of Table 1 we can observe that IGE outperforms the other algorithms on all biochemichal datasets except for MUTAG dataset. We can note that MUTAG is a very small dataset. Thus the standard deviation is high because the size of the validation set is very small (19 graphs) when we split the dataset in 10 folds. From

Conclusion

We presented a novel and simple method to handcraft invariant features for a task of graph classification based on graph spectral decomposition. We showed that these features were relevant for a graph classification task because they are easy to compute, invariant by node permutation and contain sufficient information of the structure of the graph.

In our future work we plan to develop deeper architectures based on these features and use the Invariant Graph Embedding to initialize the neural network in order to achieve better classification.

Table 2 ,

 2 IGE is challenging with all other algorithms on social network datasets and outperforms FSGD on both REDDIT datasets.

	Dataset MUTAG	PTC	PROTEINS NCI1	NCI109	D&D
	Max	28	109	620	111	111	5748
	Avg	17.93	25.56	39.06	29.87	29.68	284.32
	#Graphs	188	344	1113	4110	4127	1178
	DCNN	66.51	55.79	65.22	63.10	60.67	-
	DGK	86.17	59.88	71.69	64.40	67.14	72.95
	FSGD	92.12	62.80	73.42	79.80	78.84	77.10
		90.01 ±	64.54 ±	74.21 ±	81.46 ±	79.36 ±	78.86 ±
	IGE	6.6	5.8	3.1	0.9	1.0	3.2

Table 1 .

 1 Classification accuracy on bioinformatics datasets

	Dataset	IMDB-B	IMDB-M	REDDIT-B	REDDIT-M	COLLAB
	Avg	19.77	13.00	429.63	508.52	74.49
	#Graphs	1000	1500	2000	4999	5000
	DGK	66.96	44.55	78.04	41.27	73.09
	FSGD	73.62	52.41	86.50	47.76	80.02
	IGE	74.10 ± 3.2	48.87 ± 2.1	89.12 ± 2.2	50.92 ± 2.4	78.95 ± 2.0

Table 2 .

 2 Classification accuracy on social datasets


Ouahib Guenounou
email: wguenounou@yahoo.fr

Boutaieb Dahhou
email: dahhou@laas.fr

Billal Athmani
email: billal1983@hotmail.fr

Optimal design of PID controller by Multi-objective genetic algorithms

Keywords: NSGA-II, PID controller, Laplace crossover, Pareto front

A novel design method for PID controller with optimal parameters is proposed based on the Improved Nondominated Sorting Genetic Algorithm II(NSGA-II). The design of PID controller is formulated as multi-objective optimization problem where the integral of time multiplied by absolute error and integral of the square of the error (ISE), are optimized simultaneously. By testing two control systems, the proposed method has been able to produce a good performance control.

I. INTRODUCTION

Most real control problems are characterized by several objectives, often contradicting, that must be satisfied simultaneously. For example, the engineers are often faced with design problems, where a controller is needed that provide a small overshoot, fast response and economical control action. Since the 1980's several Multi-objective Genetic (Algorithms MOGAs) have been proposed and applied in Multi-objective optimization problems [START_REF] Schaffer | Multiple objective optimization with vector evaluated genetic algorithms[END_REF][START_REF] Horn | Multiobjective optimization using the niched pareto genetic algorithm[END_REF][START_REF] Zitzler | Comparison of multiobjective evolutionary algorithms: Empirical results[END_REF][START_REF] Deb | A fast and elitist multiobjective genetic algorithm: NSGA-II[END_REF][START_REF] Ferentinos | Optimal design of plant lighting system by genetic algorithms[END_REF][START_REF] Guenounou | Multi-objective optimization of TSK fuzzy models[END_REF]. These algorithms are mainly classified in two approaches: [START_REF] Schaffer | Multiple objective optimization with vector evaluated genetic algorithms[END_REF] aggregative approach which consists to transform the Multi-objective optimization problem into a single objective optimization problem. [START_REF] Horn | Multiobjective optimization using the niched pareto genetic algorithm[END_REF] non aggregative approach which solves the Multi-objective optimization problem, based on Pareto's principle. The aggregative approach is relatively simple but on other hand many design iterations are required to obtain the set of trade-off (Pareto front), leading to a prohibitive number of evaluations. For this reason, non aggregative approach is often used as solving method of Multi-objective optimization problems and generates a Pareto set in a single run. In this work we use the improved non-dominated sorting genetic algorithm (NSGA-II), which is one of the most powerful non aggregative multi-objective techniques, and able to locate the Pareto front in complex reach space. The design of PID controller is considered here and it formulated as multi-objective optimization problem where the integral of time multiplied by absolute error and integral of the square of the error (ISE), are optimized simultaneously. This controller type is still the one most widely use in industrial control. However the optimal tuning of its three parameters is not an easy task, especially if several performance indices are simultaneously considered. Genetic algorithms (GA) are well suited for searching in large objective space. The remainder of the paper is organized as follows: Section 2 describes the PID design problem. Section 3 gives a brief overviews of multi-objective optimization problem and NSGA-II algorithm. Section 4 presents a design method based on NSGA-II. Section 5 gives simulation results and Section 6 presents some conclusions.

II. PROBLEM FORMULATION

PID (proportional integral derivative) control is one of the earlier control strategies. The control-loop system is illustrated in Fig. 1, where it can be seen that in a PID controller, the error signal e(t) is used to generate the proportional, integral, and derivative actions, with the resulting signals weighted and summed to form the control signal u(t) applied to the plan model. A mathematical description of the PID controller is

u(t) = K p e(t) + K i t 0 e(τ )dτ + K d de(t) dt (1)
where K p , K i and K d are the parameters of PID controller. Formally, a "performance index" is defined as a quantitative measure to depict the system performance. To a PID control system, there are often four indices to depict the system performance: Integral of the Square of the Error (ISE), Integral of the Absolute magnitude of the Error (IAE), Integral of Time multiplied by Absolute Error (ITAE) and Integral of Time multiplied by the Square Error (ITSE). ITAE provides the best selectivity of the performance indices [START_REF] Dorf | Modern control systems[END_REF] and it is commonly referred to as a good criterion in designing PID controllers [START_REF] Martins | Tuning PID Controllers using the ITAE Criterion[END_REF][START_REF] Seborg | Process Dynamics and Control[END_REF]. In this work, two different objectives are considered, the first objective is the ITAE performance index and the second objective is the ISE. Whereas minimizing the first objective will provide good reference tracking and better disturbance rejection, minimizing the second reduces the rise time. The mathematical descriptions of these objectives are given by:

Ob 1 = IT AE = Ts 0 t|e(t)|dt
(2)

Ob 2 = ISE = Ts 0 e(t) 2 dt (3)
where the t is the time, the error signal e(t) is the difference between reference input signal r(t) and controlled variable y(t) and T s is the simulation time.

III. DESCRIPTION OF MULTI-OBJECTIVE OPTIMIZATION PROBLEM AND NSGA-II IV. MULTI-OBJECTIVE OPTIMIZATION PROBLEM

Without loss of generality, we consider here a multiobjective minimization problem which can be formulated as:

   M inimize (F (X) = (f 1 (X), f 2 (X), ..., f n (X))) s.t g i (X) ≤ 0 (i = 1, 2, . . . , k) h i (X) = 0 (i = 1, 2, . . . , l) (4
) where (f 1 (X),f 2 (X), ..., f n (X)) are the n different objective functions to be minimized, X = (x 1 , x 2 , ..., x r ) is the rdimensional decision space, g i (X) are the k inequality constraints, h i (X) are the l equality constraints. The objectives in the multi-objective optimization problems are often conflict each other, so it does not exist an absolutely optimal solution to optimize all objectives in same time, but can just get a set of satisfactory solutions i.e. Pareto solutions, which describe the trade-off among contradicted objectives. For a multi-objective minimization problem mentioned above, any two solutions X 1 and X 2 can have one of two possibilitiesone covers or dominates the other or none dominates the other. We say that the solution X 1 dominates X 2 , if the following two conditions are satisfied:

∀i ∈ {1, 2, ..., n} : f i (X 1 ) ≤ f i (X 2 ) ∃j ∈ {1, 2, ..., n} : f j (X 1 ) < f j (X 2 ) (5)
If X 1 dominates solution X 2 , X 1 is called the nondominated solution. The solutions that are nondominated within the entire search space are denoted as Pareto-optimal and constitute the Pareto-optimal front. The goal of a multi-objective optimization algorithm is not only to guide the search towards the Pareto-optimal front, but, also to maintain population diversity in the set of the nondominated solutions.

A. GENERAL PRINCIPLE OF NSGA-II

The genetic algorithm, used in this paper, is an adaptation of a general structure of multi-objective genetic algorithms, called NSGA-II (non-dominated sorting genetic algorithm II) proposed by Deb [4].This algorithm provides excellent results compared to others proposed multi-objective genetic algorithms such as its first version [START_REF] Srinivas | Multiobjective function optimisation using nondominated sorting function genetic algorithms[END_REF]. In this section, we present the general principle of NSGA-II, using the same notations adopted in [START_REF] Deb | A fast and elitist multiobjective genetic algorithm: NSGA-II[END_REF]. Then we specify its components, for our problem, in Section 4. NSGA-II algorithm uses an elitist approach which can significantly speed up the performance of GA [START_REF] Meunier | A multiobjective genetic algorithm for radio network optimization[END_REF], a sorting procedure, based on a fast algorithm and a comparison operator based on calculus of a crowding distance instead a sharing function [START_REF] Horn | A niched pareto genetic algorithm for multiobjective optimization[END_REF] which has been proved to be problematic (depends largely on the chosen sigma value). The general principle of NSGA-II is presented Fig. 2. NSGA-II procedure in Fig. 1: at each generation t, a parent population P t of size N and an offspring population Q t of the same size are merged for forming a population R t (P t ∪ Q t ) , of size 2N . Then, the population R t is partitioned into a number of sets called fronts F , which are constructed iteratively. Front F 1 consists of the non-dominated solutions from R t . F 2 consists of non-dominated solution from the set (Rt -F1) and so on. In general, F i consists of the non-dominated solutions from the set ( R t -(

F 1 ∪ F 2 ∪ ... ∪ F i-1 ) ). Deb et al.
have proposed a fast partitioned algorithm called fast nondominated sorting algorithm. Once all fronts are identified, a new parent population P t+1 of size N is formed by adding the fronts to P t+1 in order ( front one F 1 followed by front two and so on) as long as the size of P t+1 do not exceed N individuals. If the number of individuals present in P t+1 is lower than N , a crowding procedure is applied to the first front F i not included in P t+1 . The aim of this procedure is to insert the N -P t+1 best individuals which miss to fill all population Rt . For each solution i in F i a crowding distance d i is calculated based on each objective Ob k . The front F i is sorted according to objective function value Ob k in ascending order of magnitude. The first and the last individuals of the front are assigned infinity as their distance with respect to Ob k . For all other intermediate individuals are assigned a distance value equal to the absolute normalized difference in the objective function values of two adjacent solutions. This calculation is repeated for other objective functions. The overall crowding distance for each solution i is calculated as the sum of individual distance values corresponding to each objective function. The individuals with the highest crowding distance values are added to P t+1 until P t+1 =N . Once the individuals appertaining to the population P t+1 are identified, a new population Q t+1 of size N is created by selection, crossover and mutation using individuals of the population P t+1 . It is important to note that with NSGA-II, the selection procedure is based on a crowded-comparison operator defined as follows: to select an individual, two solutions are chosen randomly and uniformly from P t+1 and that of smaller number of front i rank is preserved. If both solutions belong to the same front, one separates them by calculating the crowding distance for each solution and the solution with the higher distance is preferred. The NSGA-II operation will repeat the procedure until the stopping criteria is satisfied.

V. DESIGN OF PID CONTROLLER BY NSGA-II

A detailed method of the NSGA-II learning algorithm for the PID is introduced in this section. The proposed NSGA-II has been implemented using realcoded genetic algorithm (RCGA). Each coefficient of the PID is encoded by two real variables {p 1 , p 2 } using the exponential form K = p 1 10 p2 . This allows the NSGA-II to search a wide range of values, with a small change of the exponential variable. So the chromosome uses strings of length 6. In this study the Laplace crossover operator, recently proposed in [START_REF] Deep | A new crossover operator for real coded genetic algorithms[END_REF] for RCGA, has been employed. This operator provides a good results and outperforms other crossover operators such as heuristic crossover [START_REF] Wright | Genetic algorithms for real parameter optimization[END_REF]. Using LX, two offsprings Y 1 = (y First, a uniformly distributed random number u ∈ [0, 1] is generated. Then, a random number β is generated which follows the Laplace distribution by simply inverting the distribution function of Laplace distribution as follows:

β = a -b log e (u) if u ≤ 1 2 a + b log e (u) if u > 1 2 (6)
The offsprings are given by the equations

y (1) i = x (1) i + β|x (1) i -x (2)
i |

y (2) i = x (2) i + β|x (1) i -x (2) i | (7)
A real value mutation is designed [START_REF] Michalewicz | Genetic Algorithms + Data Structures = Evolution Programs[END_REF][START_REF] Makinen | Multidisciplinary shape optimization in aerodynamics and electromagnetic using genetic algorithms[END_REF][START_REF] Deep | A new mutation operator for real coded genetic algorithms[END_REF] for the realstring. Here we use the non-uniform mutation [START_REF] Michalewicz | Genetic Algorithms + Data Structures = Evolution Programs[END_REF] which is one of the most widely used mutation operators in RCGA. So, at the t th generation, a parameter x i of the chromosome X will be transformed to other parameter x i with a probability P m as follows:

x i = x i + ∆(t, x u i -x i ) if r ≤ 1 2 x i -∆(t, x i -x l i ) otherwise (9)
where r is a uniformly distributed random number between 0 and 1. x l i and x u i are lower and upper bounds of x i , respectively. The function ∆(t, y) given below takes value in the interval [0, y].

∆(t, y) = y(1 -u (1-t T ) b ) ( 10
)
where u is a uniformly distributed random number in the interval [0, 1], T is the maximum number of generations and b is a parameter, determining the strength of the mutation operator. In the initial generations nonuniform mutation tends to search the space uniformly and in the later generations it tends to search the space locally, i.e. closer to its descendants [START_REF] Michalewicz | Genetic Algorithms + Data Structures = Evolution Programs[END_REF].

VI. SIMULATION EXPERIMENTS AND RESULTS

In order to examine the effectiveness of our approach, two typical control systems as used in [START_REF] Cheong | Constraining the Optimization of a Fuzzy Logic Controller Using an Enhanced Genetic Algorithm[END_REF][START_REF] Chin | Genetic algorithm based optimal fuzzy controller[END_REF] were tested. The transfer functions of the plants (Plant A and Plant B) in the two control systems are given as follows: G A (s) = (2/s(s + 1.4) + 2), G B (s) = 2/((s + 1)(s + 2)). Fig. 3 shows the closed-loop step responses of the two plants without a PID controller. The performance of the PID is evaluated based on the unit step responses and we used the ITAE (eq. 2) and ISE (eq. 3) criterions as objective functions. In order to acquire better performance, several parameters of NSGA-II should be set appropriately. In this work, the used parameters are summarized as follows: Fig. 4 and Fig. 5 show the distribution of the two objectives corresponding to chromosomes of initial population for both plants A and B respectively. The values of two objectives are relatively important and we have only two nondominated solutions in the case of Plant A and three nondominated solutions in the case of plant B. This is due to initialization step that creates chromosomes in randomly way. The PID's parameters and two objective functions (IT AE and ISE) corresponding to the nondominated solutions of each study case (Plant A and Plant B) were summarized in table 1. Using these PID controller parameters, the unit step responses of each study case were obtained as shown in Fig. 6 and Fig. 7. 2 gives PID's parameters and two objective functions corresponding to some solutions (four solutions) in the Pareto Front of each study case. Using these PID controller parameters, the unit step responses of each study case were obtained as shown in Fig. 10 and Fig. 11.

Observing the values of the two objectives in Table 2 and the unit responses in figures 10 and 11, we can find that the PID controller has good unit step responses in each study case. Therefore, we can conclude that the NSGA-II algorithm is able to find the optimal parameters of PID controller that provide a good performance control.

VII. CONCLUSION

A method to design a PID controller has been proposed. All the parameters of PID (K p , K i and K d ) are optimally adjusted using a NSGA-II. Two systems are used as test problems to analyze the performance of the proposed method. The effectiveness of the approach is well justified through simulation results. We have a good performance control for each system.

Fig. 1 .

 1 Fig. 1. A typical PID control structure

6 )

 6 and are generated from a pair of parentsX 1 = (x

Fig. 3 .

 3 Fig. 3. Step reponses

Fig. 4 .Fig. 5 .

 45 Fig. 4. Initial population (Plant A)

Fig. 6 .Fig. 7 .

 67 Fig. 6. Unit step response of Plant A

Fig. 8 .Fig. 9 .

 89 Fig. 8. Final population (Plant A)

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Unit step response of Plant A

TABLE II SUMMARY

 II OF SIMULATION RESULTS RELATED TO SOME SOLUTIONS IN PARETO FRONT

	Plant	Ke	Ki	K d	IT AE	ISE
		19.9759	3.9966	4.4507	0.0578	0.1758
	A	19.9269	3.9950	4.0031	0.1074	0.1698
		19.9934	3.9737	2.5517	0.3514	0.1575
		19.9931	3.9845	2.4766	0.3623	0.1574
		19.9066	3.9833	3.5269	0.0598	0.1744
	B	19.9453	3.9871	3.1740	0.1141	0.1693
		19.9759	3.9908	1.7540	0.3474	0.1576
		19.9759	3.9896	1.9285	0.3178	0.1580

• crossover probability P c = 0.85; • mutation probability P m = 0.01; • population size pop -size = 40;• number of generations T = 100;


Laurene Casal

Hélène Tribouillois

Renaud Misslin

Julie J Constantin

Magali M Willaume

Olivier Therond

Integrated assessment and modelling of agro-ecological practices on water flow at catchment level: what potential of the Maelia platform?

come   L'archive ouverte pluridisciplinaire

Scenario : Irrigation optimisation

34%

Potential gain on total withdrawals through maximizing irrigation efficiency 13

  a , Tribouillois H. a , Misslin R. b , Constantin J. a , Willaume M. a , Therond O. b a UMR AGIR, INRA, Université de Toulouse, 31326 Castanet-Tolosan, France b UMR LAE, INRA, Université de Lorraine, Colmar Territorial issues for the sustainability of agriculture What are the effects of agro-ecological practices such as diversification and cover crop? farm practices • The Maelia platform is able to reproduce the behaviour of the stakeholders Murgue

MAELIA


S Guené-Girard
email: simon.guene-girard@icmcb.cnrs.fr

V Jubera

L Viers

R Boulesteix

A Maître

J.-M Heintz

Liquid phase synthesis and sintering of Y 3 NbO 7

Keywords: Y3NbO7, Ceramics, Sintering, Niobate, Densification, Co-precipitation, liquid route synthesis

A new liquid phase synthesis of Y3NbO7 is presented. The co-precipitation method was chosen and both nitrate or chloride precursors and as well as aqueous ammonium or ammonia hydrogen carbonate as precipitant were used to obtain the cubic fluorite phase. Purity of the powders was controlled by XRD and through the luminescence of the Eu 3+ ions introduced as doping element during the synthesis process. Effect of the synthesis route on particle morphology was also investigated. Then, heat treatments in air were conducted to study the sintering behavior of the material obtained from chloride precursors precipitated in ammonia. The purity of the sintered pellets was controlled. Results demonstrate the Y3NbO7 phase is stable even for the highest temperatures and that a relative densification of 99.6 % is obtained.

1) Introduction

Rare earth niobates (RE3NbO7) are materials that present interesting magnetic [START_REF] Wakeshima | Magnetic properties and structural transitions of orthorhombic fluoriterelated compounds Ln3MO7 (Ln=rare earths, M=transition metals)[END_REF][START_REF] Vente | The Structure and Magnetic Properties of Pr3MO7 with M = Nb, Ta, and Sb[END_REF][START_REF] Doi | Crystal structures and magnetic properties of fluorite-related oxides Ln3NbO7 (Ln=lanthanides)[END_REF], dielectric [START_REF] Cai | Phase formation and dielectric properties of Ln2(Ln′0.5Nb0.5)2O7 (Ln=rare earth element)[END_REF][START_REF] Cai | Structure and dielectric properties of Ln3NbO7 (Ln=Nd, Gd, Dy, Er, Yb and Y)[END_REF], photocatalytic [START_REF] Abe | Photocatalytic Activity of R3MO7 and R 2 Ti 2 O 7 (R = Y, Gd, La; M = Nb, Ta) for Water Splitting into H 2 and O 2[END_REF][START_REF] Zhao | Hydrothermal synthesis of Y3NbO7 nanowires for the photocatalytic degradation of omeprazole sodium[END_REF], electric or ionic conductivity [START_REF] Yamamura | Electrical conductivity of the systems (YM)3NbO7 (M=Ca, Mg) and Y3NbMO7 (M=Zr and Ce)[END_REF][START_REF] Marrocchelli | Cation composition effects on oxide conductivity in the Zr2Y2O7-Y3NbO7 system[END_REF][START_REF] Norberg | Local structure and ionic conductivity in the Zr2Y2O7-Y3NbO7 system[END_REF][START_REF] Chesnaud | Hightemperature anion and proton conduction in RE3NbO7 (RE = La, Gd, Y, Yb, Lu) compounds[END_REF][START_REF] López-Conesa | Local Structure of Rare Earth Niobates (RE3NbO7[END_REF] and optical properties [START_REF] Kim | Spectral evolution of Eu3+ doped Y3NbO7 niobate induced by temperature[END_REF][START_REF] Kim | Tailoring the Composition of Eu3+ -Doped Y3NbO7 Niobate: Structural Features and Luminescent Properties Induced by Spark Plasma Sintering[END_REF].

Such properties could be related to the crystal structures of RE3NbO7 that exists in three different crystalline forms depending on the ionic radii of the rare earth atoms [START_REF] Doi | Crystal structures and magnetic properties of fluorite-related oxides Ln3NbO7 (Ln=lanthanides)[END_REF][START_REF] Allpress | Fluorite-related phases Ln3MO7, Ln= rare earth, Y, or Sc, M= Nb, Sb, or Ta. I. Crystal chemistry[END_REF][START_REF] Rossell | Fluorite-related phases Ln3MO7, Ln = rare earth, Y or Sc, M = Nb, Sb, or Ta[END_REF][START_REF] Barker | The systems Y2O3-Nb2O5 and Ho2O3-Nb2O5: the cubic solid-solution region[END_REF][START_REF] Siqueira | Synchrotron X-ray diffraction and Raman spectroscopy of Ln3NbO7 (Ln=La, Pr, Nd, Sm-Lu) ceramics obtained by molten-salt synthesis[END_REF][START_REF] Subramanian | Oxide pyrochlores -A review[END_REF]. Orthorombic (Pnma) and cubic pyrochlore (Fd3 ̅ m) structures are found for the largest cations, while the smallest radii RE (Dy, Ho, Er, Tm, Eu, Yb…) crystallize into a defect cubic fluorite structure (Fm3 ̅ m) [START_REF] Allpress | Fluorite-related phases Ln3MO7, Ln= rare earth, Y, or Sc, M= Nb, Sb, or Ta. I. Crystal chemistry[END_REF][START_REF] Barker | The systems Y2O3-Nb2O5 and Ho2O3-Nb2O5: the cubic solid-solution region[END_REF]. Rare earth and niobium atoms are statistically distributed on cationic sites, as well as oxygen vacancies on anionic sites. This relative disordered structure leads to peculiar optical properties with a significant broadening of the 4f-4f transitions observed in an Eu-doped Y3NbO7 compound [START_REF] Kim | Spectral evolution of Eu3+ doped Y3NbO7 niobate induced by temperature[END_REF].

Because many optical applications require massive component, the implementation of these properties requires the elaboration of full densified RE3NbO7-based ceramics. Sintering is one of the classical methods to obtain such materials. Thus, it is possible to obtain rare earth doped YAG transparent ceramic with laser properties similar to YAG single crystal [START_REF] Ikesue | Fabrication and Optical Properties of High-Performance Polycrystalline Nd:YAG Ceramics for Solid-State Lasers[END_REF]. Thus, concerning the RE3NbO7 composition, Spark Plasma Sintering (SPS) technics was successfully used by An et al. to obtain the cubic Lu3NbO7 transparent ceramics [START_REF] An | Fabrication of transparent Lu3NbO7 by spark plasma sintering[END_REF][START_REF] An | Transparent Lu3NbO7 bodies prepared by reactive spark plasma sintering and their optical and mechanical properties[END_REF]. More recently, we studied the effect of SPS process on the sintering of isostructural Eu 3+ -doped Y3NbO7 [START_REF] Kim | Tailoring the Composition of Eu3+ -Doped Y3NbO7 Niobate: Structural Features and Luminescent Properties Induced by Spark Plasma Sintering[END_REF]. However, SPS remains quite a constraining method of densification in terms of apparatus, pressure and atmosphere, especially for optical properties. In addition, the strong reducing condition favors the reduction of Nb 5+ into Nb 4+ , which is detrimental to a satisfying transparency in the visible range. Therefore, pressure-less sintering of rare earth niobates appears as a potential alternative method to obtain massive component. Consequently, powder preparation optimization is necessary to ensure high purity of the phase and to optimize particle morphology and grain size distribution. The synthesis of Y3NbO7 matrix has been reported through different experimental routes : hydrothermal [START_REF] Zhao | Hydrothermal synthesis of Y3NbO7 nanowires for the photocatalytic degradation of omeprazole sodium[END_REF], glycothermal [START_REF] Kominami | Formation of niobium double oxides by the glycothermal method[END_REF] and Pechini method [START_REF] Kim | Spectral evolution of Eu3+ doped Y3NbO7 niobate induced by temperature[END_REF][START_REF] Kim | Tailoring the Composition of Eu3+ -Doped Y3NbO7 Niobate: Structural Features and Luminescent Properties Induced by Spark Plasma Sintering[END_REF][START_REF] Okubo | Low temperature synthesis of Y3NbO7 by polymerizable complex method: Utilization of a methanol-citric acid solution of NbCl5 as a novel niobium[END_REF]. In this work, we propose an alternative synthesis process based on a co-precipitation method, leading to pure Y3NbO7 phase with a wellcontrolled particle morphology. The sintering behavior of the powder was then studied as a function of the temperature of heat treatment and dwell time.

2) Materials and methods

2.1) Powder synthesis

Different chemical routes have been investigated in this work to synthesize fine yttrium niobate powders (Y3NbO7), showing a high sinterability. Since europium cations can substitute yttrium ions with no influence on the fluorite structure due to close radii and equal oxidation number [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF],

Eu 3+ doping (1 mol%) has been performed in order to use the lanthanide luminescence as structural probe. For all tested synthesis routes, the precursor for Nb was a niobium chloride powder (NbCl5, 99.9 %, Alfa Aesar). Concerning rare earth elements, two different precursors were tested: yttrium/europium nitrate hydrate (RE(NO3),6H2O, 99.9 %, Alfa Aesar) and yttrium/europium chloride hydrate (RECl3,6H2O, Alfa Aesar, 99.99 % purity).

Two different aqueous precursor solutions were considered: (i) the first one consisted in a solution of yttrium/europium nitrates while (ii) the second one was a solution of yttrium/europium chlorides, both dissolved in distilled water at 1.2 mol.L -1 . Each solution was then mixed with a solution of niobium chloride (0.4 mol.L -1 ), dissolved in hydrochloric acid (HCl, 37 wt%, Scharlau), according to the Y3NbO7 stoichiometry. Chemical precipitation was conducted using a reverse-strike precipitation in order to obtain a better cations and particles size homogeneity [START_REF] Marlot | Synthesis of YAG nanopowder by the coprecipitation method: Influence of pH and study of the reaction mechanisms[END_REF]. The different cation solutions were added dropwise to a precipitant solution under continuous stirring. Two different basic aqueous precipitating solutions were also considered: Ammonia, (A, Fisher Chemicals) and Ammonia Hydrogen Carbonate (AHC, Sigma Aldrich) giving rise to four different synthesis routes. In the case of aqueous ammonia, the pH value was maintained to 9 during all the process by addition of supplementary ammonia. In the case of ammonia hydrogen carbonate, a solution in 20 % molar excess of AHC was used in order to insure the full precipitation of the precursors as well as the neutralization of the excess HCl. After precipitation of all the precursors, the suspension obtained was aged during 3 days at room temperature under continuous stirring. The obtained amorphous product was recovered and washed by centrifugation several times, using successively distilled water and absolute ethanol.

Once dried, the precipitate was grinded in a mortar and calcined in air at different temperatures.

Four different powders were finally prepared. Table 1 provides the synthesis parameters and designations of these powders. Also, Eu:YNbO4 and Eu:Y2O3 powders were synthetized by similar chemical route. Such phases are expected to be found as impurities in the desired Y3NbO7 phase. Thus Eu 3+ photoluminescence will be measured on these compounds and will be used as standards to verify the purity of Eu:Y3NbO7 (1 %mol) material. Eu:YNbO4 (1 %mol) was prepared with the Cl-A route, by changing the cationic ratio. Eu:Y2O3 (1 %mol) was prepared by solid mixing of yttrium and europium nitrate and calcination at 1100 °C.

2.2) Sintering process

The sintering study was carried out only for the purest powder, obtained through the chloride/ammonia process (Cl-A powder). This Cl-A crystallized powder was planetary ball milled, sieved through a 100 µm mesh and pressed into green pellets by uniaxial pressing at room temperature (applied stress equal to 125 MPa). Sintering experiments were performed in air, in a muffle furnace, up to 1700 °C with a heating rate of 10 °C/min. The sintering temperature varied from 1400 °C to 1650 °C and the sintering time from 1 h to 24 h.

2.3) Characterizations

Powders and pellets were analyzed at room temperature using X-ray diffraction technique (XRD).

Diffraction patterns were collected on a PANalytical X'Pert MDP-PRO X-ray diffractometer with a Bragg-Brentano θ-θ geometry using Cu Kα1,2 radiation (8 °< 2θ < 80 °), with a step size of 0.02 ° and a counting time of 30 s. The photoluminescence properties of the Eu-doped powders and pellets were analyzed using a Horiba Fluorolog 3 spectrofluorimeter. The emission spectra were corrected for the transmission of the monochromator and the response of the photomultiplier. The equipment used consisted of a 450 W xenon lamp, an excitation double monochromator, an emission double monochromator and a thermo-electrically cooled photomultiplier tube.

The morphology of the powders was observed by scanning electron microscopy (TESCAN Vega II SBH SEM). Granulometry measurements were realized on a Pratica LA-950V2 apparatus from Horiba and the specific surface area was obtained by BET approximation (Brunauer-Emmett-Teller theory, 3Flex Micrometrics under N2).

Shrinkage evolution of the powder was determined by dilatometric measurements, using a thermomechanical analyzer SETSYS Evolution TMA (SETARAM instrumentation). The pellet was firstly pressed under an uniaxial applied at 125 MPa at room temperature and then heated up to 1700 °C with a heating rate of 10 °C/min under flowing air. Green density of the pellets was determined from geometric measurements while sintered density was measured by Archimede's method in diethyl phthalate. These values are expressed as relative densities (drel) corresponding to the density of the pellets divided by the theoretical density of Eu-Y3NbO7, taken as dth = 5.436 g.cm -3 (JCPDS N° 01-074-6421). Grain size measurements were performed on SEM micrographs of fractured samples for sintering temperature below 1525 °C and using the intercept method (on 3 SEM micrographs) on a polished section after thermal etching for temperatures higher than 1525 °C.

3) Results and discussion

3.1) Role of the precipitant and the precursors

The diffraction patterns of the washed and dried precipitates are shown in Fig. 1. The nature of the precipitant has a strong impact on the structure of the powder. The dried powders synthetized with ammonia solution are mainly amorphous, whereas synthetized powders with AHC show diffraction peaks that can be attributed to different phases of yttrium oxychloride, carbonate and hydroxide and niobium oxychloride. Numerous peaks were not attributed and can be related to nitrate, ammonium, carbonate and chloride sub-products from the different reactants. As the washing processes were the same for all routes, we concluded the observed species result from reaction between reactants.

Fig.1. XRD patterns of the as washed and dried powders.

For example, the TGA curve of Cl-A powder reveals the elimination of several species, all along the temperature increase (Fig. 2). As expected water and OH groups are decomposed up to 200 °C, followed by the decomposition of nitrate compounds up to 700 °C and finally elimination of chlorides group at higher temperatures. A minimum calcination temperature of 900 °C was therefore used, as it allowed elimination of the majority of the synthesis' residues. than the fluorite one can be explained by a poor cation homogeneity due to the presence of the carbonate groups. YAG synthesis mechanism' by co-precipitation has been previously described in the literature [START_REF] Palmero | Co-Precipitation of YAG Powders for Transparent Materials: Effect of the Synthesis Parameters on Processing and Microstructure[END_REF], we propose a similar process. We suggest that the formation of the niobate phase may occur by precipitation of Y(OH)3 particles followed by the diffusion of niobium ions into those particles. The carbonate groups might be an obstacle for the diffusion of niobium ions, explaining the poor homogeneity of the AHC co-precipitated powders. Furthermore, an increase of the temperature results in the disappearance of the Y2O3 sesquioxide. This could be explained by the reaction between Y2O3 and amorphous niobium compounds, leading to the increase of Y3NbO7 phase proportion via homogenization of cations by diffusion at high temperature.

However, small diffractions peaks relative to YNbO4 are still present at that temperature. Another hypothesis may be based on the fluorite domain of existence. The Y3NbO7 composition belongs to a solid solution domain in which the cationic ratio Nb/(Y+Nb) varies from about 21 % to 28 % [START_REF] Miida | Locally Modulated Structures of Fluorite-Related Y2O3-Nb2O5 Solid Solutions[END_REF][START_REF] Lee | Phase Diagram and Oxygen-Ion Conductivity in the Y2O3-Nb2O5 System[END_REF]. The decrease of the Y2O3 amount in the solution may also reflect an enrichment of the cubic solid solution in yttrium leading to a cationic ratio inferior to the targeted 25 %. The modification of the fluorite composition may occur with precipitation of the YNbO4 monoclinic phase.

The precipitates obtained with aqueous ammonia (N-A and Cl-A) show diffraction peaks consistent with the Y3NbO7 fluorine type structure, up to 1100 °C for both cation precursors (Fig. 3 (a) and (b)). However, at 1400 °C, the N-A powder XRD pattern shows the presence of YNbO4 peaks. This could be attributed to a partial decomposition of the fluorite phase into the monoclinic one, as it has already been observed for heat treatment of Y3NbO7 powders obtained from Pechini chemical routes [START_REF] Kim | Spark plasma sintering and decomposition of the Y3NbO7:Eu phase[END_REF]. The Cl-A powder synthesis route seems to be the most appropriate process to stabilize a pure and stable Y3NbO7 powder according to these first XRD analyses. Because of its high sensitivity, photoluminescence of Eu 3+ was then used to check the absence of secondary phases. Emission spectra of the Cl-A powder calcined at 1100 °C and of two main secondary phases that can be expected in Y-Nb-O ternary system are presented in Fig. 4. As already shown by Kim et al. [START_REF] Kim | Spectral evolution of Eu3+ doped Y3NbO7 niobate induced by temperature[END_REF],

Y2O3:Eu and YNbO4:Eu present sharp maximum emission lines for the hypersensitive 5 D0  7 F2 electric dipole transition, whereas Eu 3+ ions present in the fluorite phase show large response due to the intrinsic disorder of this structure. By consequence, the spectral responses of the three phases can be easily discriminated and the photoluminescence can be applied as an efficient probe to validate the purity of the powder after synthesis. Since no sharp luminescence peaks have been detected in the 600-620 nm range corresponding to the energy of the hypersensitive transition mentioned above, we can conclude that the Cl-A powder is definitely pure without any secondary phase. The morphology of the powders calcined at 1100 °C is presented in Fig. 5. Significant variations in morphology can be observed depending on the chosen synthesis route. Firstly, nitrate precursor (N-A and N-AHC) leads to a weakly aggregated powder, with almost individual particles.

Whereas chloride precursor (Cl-A and Cl-AHC) provides a more aggregated powder with microsized aggregates. Secondly, particles which precipitated by aqueous ammonia (N-A and Cl-A) present a spherical morphology, similar in size for both precursors. Conversely, powders synthetized using AHC as precipitant display various morphologies and sizes. Fine nonaggregated particles are present as well as flakes. This combination of morphologies can be related to the XRD phase analysis that shows for this precipitant, the presence of three different phases, such as Y3NbO7, Y2O3 and YNbO4. Similar morphologies have been observed in the literature relative to the preparation of YAG powders using AHC as a precipitant and for high AHC concentration [START_REF] Wang | The effect of precipitant concentration on the formation procedure of yttrium aluminum garnet (YAG) phase[END_REF]. In this case, yttrium hydroxide precipitates as micrometer flake-shaped particles. Such large particles lead then to an increase of the diffusion distance needed to homogenize yttrium and niobium atoms, explaining the presence of several phases in the powder.

Fig.5. SEM micrographs of a) N-A, b) Cl-A, c) N-AHC, d) Cl-AHC powders, calcined at 1100 °C.

From a morphological point of view, the most interesting powder for sintering is the one resulting from the nitrate precursor and precipitated with ammonia. Indeed, in this latter case, particles appear non agglomerated and spherical. However, XRD showed that this N-A powder was not pure at high temperature. Therefore, only the monophasic Cl-A powder that is stable at high temperature was studied in the following.

3.2) Study of the sintering behavior of the Cl-A powder

Fig. 6. SEM micrograph of the calcined Cl-A powder after ball milling.

As the morphology was not fully satisfying for sintering, the Cl-A powder was planetary ball milled after calcination at 1100 °C. SEM micrograph (Fig. 6) shows the effective disaggregation of many agglomerates initially present in this powder. BET measurements gave a specific surface area of 17.1 m 2 .g -1 . A BET equivalent particle size can be deduced from the surface area according to the following formula:

dBET = k 𝜌 * 𝑆𝑆𝐴
where k, is a shape factor taken here as 6 assuming that particles are spherical,  the theoretical density (5.436 g.cm -3 ) and SSA the Specific Surface Area.

Under these conditions, a particle size of dBET = 68 nm can be obtained. This result is quite consistent with the values of the average elementary particle size estimated from both image analysis of particles on SEM micrographs (60 ± 20 nm) and granulometry measurements (70 nm). This means that the particles of this powder can be considered almost spherical and weakly agglomerated, resulting in a high green relative density value (51 %), considering the small grain size (less than 100 nm) and the compression method (dry and uniaxial).

The sintering curve of Cl-A sample is reported up to 1700 °C in Fig. 7. The shrinkage starts at around 1200 °C. It also indicates that the initial niobate powder must not be calcined at temperatures higher than 1100 °C in order to avoid formation of hard aggregates that could not be released by ball milling. The shrinkage rate increases rapidly above 1200 °C and two steps can be observed. The first one, around 1500 °C, could be likely related to the densification within some more packed zones of the granular compact, those zones can results of remaining agglomerates that can be seen Fig. 6. The second step consist of the final densification of the pellet with a maximum of the shrinkage rate observed at 1650 °C. The final relative density of that sample sintered at 1700 °C was measured to be equal to 94.5 % which shows the pretty good densification ability of this powder, considering that Y3NbO7 phase is a refractory material that needs high temperatures to be fully densified. The study of the isothermal sintering of the Cl-A powder at 1600 °C is presented in Fig. 8. High relative densities are obtained from 1 h of treatment (Fig. 8(a), drel > 97 %), confirming the good sinterability of this powder. The highest density is observed for a 6 h heat treatment: drel = 99.6 %.

Grain growth is noticeable in these conditions, with an average grain size higher than 1 µm after a 1 h sintering treatment. Naming G the grain size at a given temperature and G0 the initial particle size, a classical G m -Go m law with m = 3 is observed for the last stage of sintering, up to 12 h of sintering (Fig. 8(b)). It can be related to a pore control mechanism, trough lattice diffusion, considering that the niobate phase is pure [START_REF] Rahaman | Ceramic processing and sintering[END_REF][START_REF] Brook | Treatise on materials science and technology[END_REF]. To complete this study, the influence of the sintering temperature was studied for a 6 h dwell time (Fig. 9). The maximum of densification is reached at 1600 °C with drel = 99.6 % and the relative density remains stable up to 24 h (Fig. 9(a)). The evolution of the microstructure of the sintered Cl-A pellets as a function of the temperature is presented in Fig. 10. From 1400 °C, a very homogeneous microstructure can be observed with a regular distribution of porosity and a significant growth compared to the initial powder. At 1550 °C (Fig. 10(c)), the third stage of sintering is reached with only closed porosity and microsized grains. At 1600 °C (Fig. 10(d)), very few pores remain in the ceramic and the grains are homogeneous in size, around 2.6 µm. Grain growth evolution as a function of temperature is presented in Fig. 9(b). A clear increase in size is noticed above 1550 °C, which is consistent with SEM micrographs (Fig. 10) and with achieving a final density close to 100 %. The representation of all these experiments can be summarized on a sintering map (Fig. 11). A unique trajectory is obtained, showing a clear predominance of densification over grain growth up to a relative density of 99 %, meaning that full density can be achieved with our powder. X-ray diffraction pattern of a pellet sintered at 1600 °C for 6 h and the corresponding photoluminescence spectrum are presented in Fig. 12. No diffraction peaks of secondary phases can be observed on the XRD analysis. In addition, the photoluminescence spectrum shows a unique spectral repartition which is characteristic of the Eu 3+ -doped Y3NbO7. The Y3NbO7 ceramic, synthesized by the ammonia precipitation method (Cl-A) and sintered at 1600 °C is not only dense but is also stable at high temperature, showing no phase decomposition. The best density achieved by natural sintering of this Cl-A powder is drel = 99,6 %, for a 1600 °C heat treatment during 6 h. It shows that the composition as well as the particles morphology of the initial powder were well controlled. Considering that this powder has a high sintering potential (see the sintering map), some improvements could likely be made, in order to reach full density (drel = 100 %). For example, the initial packing of the powder before sintering can be improved using slip casting technics or the densification itself with the use of alternative sintering technics like vacuum sintering or high isostatic pressure sintering.

4) Conclusions

The synthesis of a pure Y3NbO7 powder, presenting a high sinterability was investigated through a liquid phase synthesis. A co-precipitation route was developed using chloride precursors and aqueous ammonia as precipitant. The synthetized product was calcined at 1100 °C. Crystallization of a pure cubic fluorite phase was validated by XRD and the Eu 3+ spectral distribution. Preparation of the powder by ball-milling leads to high green densities and an excellent sinterability: a 99.6 % relative density was measured on a pellet sintered in air at 1600 °C during 6 h. The sintering map indicates that a full densification could be reached. Effort on green preparation and alternative sintering technics will have to be performed in order to reach full densification and transparency.

Fig. 2 .

 2 Fig.2. TGA curve of the Cl-A powder.

Fig. 3 .

 3 Fig.3. XRD patterns of a) N-A, b) Cl-A, c) N-AHC, d) Cl-AHC powders, heat-treated at different

Fig. 4 .

 4 Fig.4. Photoluminescence spectra of a) Y2O3:Eu, b) YNbO4:Eu, and c) Cl-A Y3NbO7:Eu powder

Fig. 7 .

 7 Fig.7. Dilatometric curve of the Cl-A powder calcined at 1100 °C, ball milled and uniaxially

Fig. 8 .

 8 Fig.8. Influence of the sintering time at 1600 °C on a) relative density and b) grain size of sample Cl-A (insert shows the linear evaluation of G 3 -Go 3 = f(t)).

Fig. 9 .

 9 Fig.9. Evolution of a) the relative density and b) grain growth evolution of sample Cl-A as a

Fig. 10 .

 10 Fig.10. SEM micrographs of pellets sintered 6 h at a) 1400 °C (fracture), b) 1500 °C (fracture),

Fig. 11 .

 11 Fig.11. Sintering map of sample Cl-A.

Fig. 12 .

 12 Fig.12. a) XRD pattern and b) photoluminescence spectrum of a pellet sintered 6 h at 1600 °C.

Table 1 .

 1 Sample designations according to cation precursors and precipitant

	Designation	Precursors		Precipitant
	N-A	Y,Eu(NO3).6H2O NbCl5	NH4OH
	N-AHC	Y,Eu(NO3).6H2O NbCl5	NH4HCO3
	Cl-A	Y,EuCl3.6H2O	NbCl5	NH4OH
	Cl-AHC	Y,EuCl3.6H2O	NbCl5	NH4HCO3

Acknowledgments

The authors thank the CNRS, the Region Nouvelle-Aquitaine. This study was carried out with financial support of the Region Nouvelle-Aquitaine (MISTRAL program n° 2017-1R50311) in the frame of "the Investments for the future" Program IdEx Bordeaux -LAPHIA (ANR-10-IDEX-03-02)