Léoplod Carron

Lorenzo Concia

Stefan Grob

Fredy Barneche

Alessandra Carbone

Julien Mozziconacci

A layer cake model for plant and metazoan chromatin

Keywords: Hi-C, genome 3D organisation, chromatin, DNA repeats 1

Early structural studies of chromosome organisation in eukaryotic nuclei led to the identification of euchromatin and heterochromatin, two main archetypes of chromatin visible through microscopy. Subsequently, the diversity of chromatin composition along the linear genome sequence was resolved using high throughput sequencing techniques to map a multitude of chromatin marks and unravel their functional organisation. Recent analyses of human chromosome conformation capture experiments tend to point to the existence of several chromosome sub-compartments that may correspond to epigenomic variations in chromatin composition. Here, we compare genome 3D organisations in representative eukaryotic species to explore the links between chromosomal sub-compartments and chromatin marks, genome replication timing, and genomic repeats in six model organisms, including vertebrates, plants and insects. We report that the 3D organisation of chromatin in organisms with different genome content and size can be described as layers characterised by distinct chromatin marks and activities. We propose a "layer cake" model for the genome 3D organisation as a more refined view than the prevalent "two compartments" model of chromatin organisation in multi-cellular organisms.

Introduction

Chromatin is formed by the association of DNA and histones in the nucleus. The name chromatin was originally coined by Walther Flemming in the 19th century and it constitutes today the subject of a very active field of research due to the role of chromatin in genome packaging, gene expression and cell epigenomic inheritance. Electron microscopy observations of the interphase nucleus led to the first distinction between heterochromatin, which comprises the dense regions of chromatin, and euchromatin, often located in the centre of the nucleus and less dense. These two types of chromatin were identified as respectively repressing and permissive to transcription and are since then recognised as key players to sustain cell-type-specific genome 3D conformation and transcription. With the rise of high throughput sequencing, the mapping of the epigenome revealed a mechanistic link between chromatin functional status and its composition and organisation. Among characteristic histones and histone posttranslational modification, histone H3 di-or trimethylation (H3K9me2/H3K9me3) are enriched at condensed and silent constitutive heterochromatin whereas histone acetylation is typically found at transcribing genes in euchromatin. The case of H3K27me3, which co-localize with the silencing factor Polycomb, is somewhat subtle since it is found in smaller islands within euchromatic active marks. These regions have been termed "facultative heterochromatin" because they can be either expressed or repressed in different cell types compared to "constitutive heterochromatin" regions which are usually considered as "constitutively inactive".

Over the past decade, Chromosome Conformation Capture (3C) experiments, which map the contact frequencies between different loci along chromosomes, have linked the classical euchromatin/heterochromatin view obtained from microscopy experiments to the epigenomic maps. The main result of the first, and low resolution, genome-wide 3C analysis (Hi-C) of intact human nuclei was that euchromatin and heterochromatin were forming two spatially distinct compartments, referred to as A and B, which strongly correlate with gene expression [START_REF] Lieberman-Aiden | Comprehensive mapping of long-range interactions reveals folding principles of the human genome[END_REF]. Interestingly, facultative heterochromatin, which contains repressed genes, was at the time found together with euchromatin in the A compartment. This organisation in two compartments is conserved in multi-cellular eukaryotes, including plants [START_REF] Dong | 3D Chromatin Architecture of Large Plant Genomes Determined by Local A/B Compartments[END_REF] and species with holocentric chromosomes [START_REF] Das | Condensin i folds the c. elegans genome[END_REF]. Principal features of chromatin compartmentalisation were later on shown to correlate with replication timing [START_REF] Suhas | A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping[END_REF], TE content [START_REF] Cournac | The 3D folding of metazoan genomes correlates with the association of similar repetitive elements[END_REF] and many other epigenetic marks [START_REF] Rowley | Evolutionarily conserved principles predict 3d chromatin organization[END_REF].

Improvements in Hi-C resolution have recently allowed to shed new light on nucleus 3D organisation and its link to chromatin marks, notably suggesting existence of intermediate sub-compartments within A and B compartments (that we will call here I for "intermediate") [START_REF] Suhas | A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping[END_REF][START_REF] Spracklin | Heterochromatin diversity modulates genome compartmentalization and loop extrusion barriers[END_REF][START_REF] Liu | Systematic inference and comparison of multi-scale chromatin sub-compartments connects spatial organization to cell phenotypes[END_REF][START_REF] Michael | Principles of 3d compartmentalization of the human genome[END_REF][START_REF] Carron | The 3d organization of chromatin colors in mammalian nuclei[END_REF]. Interestingly, these studies point towards a strong enrichment of facultative heterochromatin in the I compartments. The number of sub-compartments considered in these studies ranges from 3 to 6. Here we compare and contrast chromatin compartments in five distantly-related multi-cellular species based on Hi-C data. We correlate compartments with distinctive histone marks and DNA repeat families and report a common organising principle of 3D chromosome architecture as a multi-layered compartment structure. These layers, corresponding to the formerly described compartments ranging from A to B through I, come with different chromatin flavours and are conserved in all species studied, with a main difference between relatively short, more compact genomes, and longer genomes. In the plant Arabidopsis thaliana and the fly Drosophila melanogaster, for which estimated genome sizes are 135 and 180 Mbp long, respectively, TEs are enriched in peri-centromeric 2 regions and are more dispersed in the chromosomal arms which exhibit a compartment layer-like organisation. In human, mouse and chicken, for which the estimated genome sizes are in between 1 and 3 Gbp, TEs are found scattered throughout the whole genome, in the A, B and I compartments, with specific families being enriched in A and B compartment types [START_REF] Cournac | The 3D folding of metazoan genomes correlates with the association of similar repetitive elements[END_REF].

Results and discussion

Layered compartment organisation of human chromatin

In this study, we performed a detailed analysis of chromatin compartments in five representative species of metazoans and plants and highlight the existence of A/B intermediary states that subtly partitions epigenomic features. We hereafter start by illustrating this partition in human chromatin and later extend it to Mouse, Drosophilia and Arabidopsis Thaliana. Two emblematic features can be observed on the correlation matrix obtained from one (here the 16th) of the intra-chromosomal Hi-C contact maps from human nuclei (Fig. 1A). The first one is the white cross in the middle, which corresponds to repetitive, un-mappable (peri-) centromeric regions which are filtered out from further analysis. The second one, which was already reported in the founding Hi-C study [START_REF] Lieberman-Aiden | Comprehensive mapping of long-range interactions reveals folding principles of the human genome[END_REF], is the existence of a checker-board pattern resulting from alternating regions of two different types exhibiting a strong affinity for the same type. This pattern is interpreted as an indication of the presence of two spatially disjointed compartments, which correspond to eu-and heterochromatin hallmarks. In line with this initial study, and with the many to come, we computed the first eigenvector of the correlation matrix (Fig. 1B). When properly oriented, a positive value of the eigenvector corresponds to the A, euchromatic compartment and a negative value to the B, hetero-chromatic compartments (respectively in red and blue on Fig . 1B). When reordering the matrix according to this eigenvector values, the resulting map shows that B and A regions, respectively at the top left and bottom right, have a highly correlated contact profiles, while the contact profile between regions from the B type and the A type are negatively correlated. Since the transition between A and B is progressive, we identify additional compartments within A, B and the A/B transition zone. To this end, we used a six states Hidden Markov Model (HMM) trained on the original correlation map. The resulting states were found to alternate along the chromosomal arms (Fig 1D). When ordered with respect to the first eigenvector, we can see that these states correspond to gradual transitions in the transition region from A to B (Fig 1E). The number six that we chose is here arbitrary and could be higher or lower, the only requirement being that this number should be high enough to describe the gradual change of the first eigenvector values when going from A to B compartments.

In order to illustrate how the states are organised in 3D, we coloured a 3D representation of the contact map obtained with ShRec3D [START_REF] Lesne | 3D genome reconstruction from chromosomal contacts[END_REF][START_REF] Morlot | Network concepts for analyzing 3d genome structure from chromosomal contact maps[END_REF]. Since Hi-C data come from a population of cells, this 3D structure represents an average consensus representation of the data. The six HMM states form layers in 3D, stacked from the most B-like to the most A-like (Fig 1F). We use here the term layer that differs from the term compartment since the genomic regions in these layers are not completely isolated from each others, but rather gradually organised in the 3D space. Reflecting the observed 3D arrangement of the regions in layers corresponding to the different states, regions that are found in the same state also exhibit similar values of the first eigenvector.

The distribution of these values for each state illustrates the gradual changes from B to A (Fig1G). The classical distinction between region from the A and B compartment is done by looking at the sign of the value of the first eigenvector, the distribution in Fig1G is nevertheless far from bimodal, as would be expected in presence of two compartments, but rather in agreement with a gradual layered organisation as described in this paper, with layers 1,2,3 and 4,5,6 respectively corresponding to B and A [START_REF] Liu | Systematic inference and comparison of multi-scale chromatin sub-compartments connects spatial organization to cell phenotypes[END_REF]. In order to tackle the question of the spatial organisation of these layers in the nuclear space we used the results from the SPIN methodology [START_REF] Wang | SPIN reveals genome-wide landscape of nuclear compartmentalization[END_REF], an integrative computational method to reveal genome-wide intra-nuclear positioning of genomic loci relative to nuclear speckles, lamina, and putative associations with nucleoli. We recovered that the six layers strongly correlates with SPIN states, although these were obtained on a different human cell type (K562,

Layered compartment organisation in plants and metazoans

In order to know whether the organisation of chromosomes is similar in other metazoans and plants we applied the same methodology on Hi-C data from mouse, chicken, drosophila and A. thaliana In all species studied here, the genome is alternating between two ground states 1 and 6, hence the checker-board pattern, with a smooth transition between those two states, hence the layered organisation. However, we observed a difference between the organisation of these compartments in longer genomes (human, mouse and chicken) compared to shorter ones (Drosophila and A. thaliana). While the checker-board pattern could be seen at large scale (i.e. on several chromosomes or on a full chromosome) for long genomes, it could be retrieved when zooming on shorter regions of the chromosomal arms for shorter ones (Fig S2). This specific feature of shorter genomes is not specifically due to the genome size but 5 to the higher mappability of (peri-)centromeric sequences in the smaller genomes [START_REF] Nurk | The complete sequence of a human genome[END_REF]. When they are included in the Hi-C maps, mappable centromeres form squares of high contact frequency values that are spatially dissociated from chromosomal arms [START_REF] Muller | The Impact of Centromeres on Spatial Genome Architecture[END_REF]. The correlation matrix obtained from the Hi-C contact map thus reflects this specific organisation, as highlighted by the first eigenvector which delineates the position of the centromeres (Fig. S2,left). When the centromeres are excluded from the contact maps by zooming in on the chromosomal arms, the checker-board pattern characteristic of the presence of A/B compartments appears and correlates with gene activity as previously reported by Dong & al. [8] in Arabidopsis Thaliana (Fig. S2,right). In human, mouse and chicken, centromeres consist of highly repetitive and therefore masked sequences and it is not necessary to zoom on the chromosome arms to extract the checker-board pattern. Layered compartments correspond to different epigenomic features, DNA Repeat contents and replication timings

Now that we have split the genomes of these different species into seven regions, the 6 layers given by the HMM together with the regions that have been filtered out, corresponding mainly to the (peri-) centromeres of the chromosomes, we used publicly available epigenome profiling datasets to test whether these regions correspond 6 to distinct chromatin states. For this purpose, we first used the chromatin states that have been previously described for Human [START_REF] Ernst | ChromHMM: automating chromatin-state discovery and characterization[END_REF], Drosophila [START_REF] Guillaume J Filion | Systematic Protein Location Mapping Reveals Five Principal Chromatin Types in Drosophila Cells[END_REF] and A. thaliana [START_REF] Roudier | Integrative epigenomic mapping defines four main chromatin states in Arabidopsis[END_REF] chromatin. These states were obtained using 1D annotations of the genomes, corresponding mostly to histone modifications and association with the lamina. For human (Fig 3A ), we find that active chromatin marks are enriched in the A-like layer 6 while inactive chromatin is enriched in the B-like layer 1. We also see that the three states corresponding to polycomb repressed chromatin are enriched in intermediate layers 3 and 4. This finding is in agreement with recently published studies [START_REF] Carron | Boost-HiC: computational enhancement of long-range contacts in chromosomal contact maps[END_REF][START_REF] Spracklin | Heterochromatin diversity modulates genome compartmentalization and loop extrusion barriers[END_REF][START_REF] Liu | Systematic inference and comparison of multi-scale chromatin sub-compartments connects spatial organization to cell phenotypes[END_REF][START_REF] Michael | Principles of 3d compartmentalization of the human genome[END_REF] and suggests that polycomb and bivalent domains, that can be activated or inhibited depending on the cellular context, are typically located in between constitutively active and inactive regions. Interestingly, we found that this particular positioning of polycomb is conserved in Drosophila (Fig 3B ) andA. thaliana (see also [START_REF] Grob | Hi-C Analysis in Arabidopsis Identifies the KNOT, a Structure with Similarities to the flamenco Locus of Drosophila[END_REF], Fig 3C ), suggesting that this property could be a hallmark of all species that have polycomb. Of note, the tendencies observed in Fig 3 do not depend on the number of layers that was chosen but are rather due to the distribution of chromatin marks along the chromosomes.

We finally checked for the enrichment of DNA repeats families within the different layers described here. In mammalian chromosomes, the enrichment of SINEs in the A compartment and LINES in the B compartments [START_REF] Cournac | The 3D folding of metazoan genomes correlates with the association of similar repetitive elements[END_REF] is reflected by a correlation between these DNA repeats and the layer organisation (Fig 3D). On the other hand, for shorter genomes such as the Drosophila and Arabidopsis, the majority of repeats (tandem repeats as well as dispersed repeats) are found in the (peri-)centromeric regions and we did not find any repeat family significantly over-represented in any of the layers (Fig 3FE). The few repeats families that were not found to be enriched in the (peri-)centromeric regions were preferentially found in the intermediate layers (2 to 5) in both Drosophila (e.g. low complexity and simple repeats) and Arabidopsis (e.g. Tc1 and Mariners).

Taken together, our results highlight the existence of chromosome architecture as 3D layers corresponding to different "flavours" of chromatin and DNA sequence. This partition goes beyond the mere distinction between eu-and hetero-chromatin, with the noteworthy observation that Polycomb repressed domains were on average found in intermediate layers between active and inactive chromatin. We also described here a difference between relatively shorter, more compact, genomes and longer genomes. In Arabidopsis and Drosophila, for which the genome is respectively 135 and 180 Mbp long, DNA repeats are preferentially found in peri-centromeric regions while in mammals, for which the genomes are 3 Gbp long, these elements are found scattered in the whole genome with specific families being enriched in the extreme layers 1 and 6. The origin of this difference remains an important question and the relationship uncovered here between the repeat content, the chromatin composition and the 3D genome folding adds a new perspective to this matter.

Material and Methods

Datasets used in the present study

All the datasets used here are summarised in figure Supplementary Table S1. We downloaded Hi-C from Human, Mouse, Chicken, Arabidopsis thaliana and Maize genome [START_REF] Hu | Plant lamin-like proteins mediate chromatin tethering at the nuclear periphery[END_REF][START_REF] Suhas | A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping[END_REF][START_REF] Foissac | Multi-species annotation of transcriptome and chromatin structure in domesticated animals[END_REF][START_REF] Dong | 3D Chromatin Architecture of Large Plant Genomes Determined by Local A/B Compartments[END_REF][START_REF] Ogiyama | Polycomb-Dependent Chromatin Looping Contributes to Gene Silencing during Drosophila Development[END_REF][START_REF] Teano | Histone h1 protects telomeric repeats from h3k27me3 invasion in arabidopsis[END_REF] from SRA archive and reanalyse them with Hi-C pro [START_REF] Servant | HiC-Pro: an optimized and flexible pipeline for Hi-C data processing[END_REF]. For chromatin colors, we used the datasets presented in [START_REF] Ernst | ChromHMM: automating chromatin-state discovery and characterization[END_REF][START_REF] Guillaume J Filion | Systematic Protein Location Mapping Reveals Five Principal Chromatin Types in Drosophila Cells[END_REF][START_REF] Roudier | Integrative epigenomic mapping defines four main chromatin states in Arabidopsis[END_REF]. For the replication 7 timing, we used the datasets presented in [START_REF] Scott Hansen | Sequencing newly replicated dna reveals widespread plasticity in human replication timing[END_REF][START_REF] Wear | Genomic analysis of the dna replication timing program during mitotic s phase in maize (zea mays) root tips[END_REF][START_REF] Concia | Genome-wide analysis of the arabidopsis replication timing program[END_REF].

Hi-C analysis

HiC data where downloaded from previously published study described in Supplementary Table S1. Hi-C reads were processed using HiC-Pro [START_REF] Servant | HiC-Pro: an optimized and flexible pipeline for Hi-C data processing[END_REF]. All generated map were produce at a resolution 10kb and 100kb.

To extract A and B compartments, we first normalise the intra-chromosomal contact count matrix by Sequential Component Normalization (SCN) [START_REF] Cournac | Normalization of a chromosomal contact map[END_REF] in order to have the same sum of contact in each row and column. Then each contact is divide by the mean number of contact of his diagonal [START_REF] Nynke L Van Berkum | Hi-C: a method to study the three-dimensional architecture of genomes[END_REF] in order to obtain the matrix ratio of observed contact under expected. The first eigenvector of the correlation matrix C is then oriented with respect to the gene density to get the A and B compartments [START_REF] Lieberman-Aiden | Comprehensive mapping of long-range interactions reveals folding principles of the human genome[END_REF].

3D structures were produced with the Matlab (https://www.mathworks.com/products/matlab.html) implementation of ShRec3D [START_REF] Lesne | 3D genome reconstruction from chromosomal contacts[END_REF] with an alpha parameter of 0.2 and using the "strain" dimension reduction method.

Compartment extraction using the Hidden Markov Model

We extracted the genomic compartments with a hidden markov model (HMM) as described in Rao et al. [START_REF] Suhas | A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping[END_REF] using the HMMlearn library in python direclty on the correlation matrix C. We used the gaussianHMM function with parameters covari-ance='diag', niter=100 using 6 component in the model. We then used the function predict in order to have the most probable compartment sequence. Since compartment states from HMM are not ordered we used the first eigenvector of the correlation map as computed in the previous section to order them from the most "A-like" to the most "B-like". Fro comparison, we also generated compartments from inter-chromosomal contacts, using the same pipeline.

Partition of DNA repeat families, epigenomic states and replication timing in the different regions of the genome

For each dataset (Table S1), we downloaded bed files in order to have the genome coverage of each tracks. We summed the signal over 10kb non-overlapping bins for each track and computed the overall signal in each compartment, as determined using HMM. This sum was then normalised to one in order to get the proportion of the feature of interest in each of the six compartments Table S1: Table of Hi-C data-sets used this study with the reference genome used in the mapping step.

Figure 1 :

 1 Figure 1: 3D chromosome structure exhibits a layer cake pattern A Pearson correlation map C of human chr 16 at a resolution of 100kb. B First eigenvector of C. C Pearson correlation map C sorted by the first eigenvector. D The six states obtained with the HMM. They are coloured blue, light blue, green, yellow, orange and red. E The six states found in D sorted by the first eigenvector, as in C. F Average 3D structure of chromosome 16 coloured with the six states obtained with the HMM, as in D. G Distribution of the values of the first eigenvector (grey). The distribution for each of the six states is shown with the colours corresponding to each compartments. Color-scale as in D, centromeres have been filtered out.

  Fig S1). The most B-like 3D layer (state 1 of our HMM model) corresponds to the lamina-associated SPIN state, 4 whereas most A-like 3D layer (state 6 of our HMM model) is associated with the SPIN comprising nuclear speckles and active chromatin regions at the interior of the nucleus. Intermediate layers 2-3 are associated with lamina and near lamina states while layers 4-5 are associated with euchromatic interior states (Fig S1). This comparison shows that while they are only derived from Hi-C contacts, HMM-defined states closely mirror SPIN states, which are derived from a larger set of experimental data. This similarity prompted us to investigate the organisation of chromatin in other species and assess the conservation of the layered organisation described here in human cell lines.

Figure 2 :

 2 Figure 2: Universality of the layer cake organisation in metazoan and plant chromosomes Overview of averaged consensus 3D structures from different species at 10kb resolution with the overlap of the six HMM-derived states. For each chromosome or segment of chromosome, the figure includes the 3D representation labelled with 6 layers (top) and the correlation map (bottom). A Human chromosome 15. B Mouse chromosome 18. C Chicken chromosome 7. D Drosophila chromosome arm 2L. E A. thaliana chromosome 1 (region between 16 and 23 Mbp).

(Supp Fig 1 ,

 1 Fig 2). Our results showed the presence of a checker-board pattern in the correlation maps (Fig 2, bottom) as well as a layered organisation of HMM-derived compartments on the average 3D structures (Fig 2, top).

Figure 3 :

 3 Figure 3: Chromatin and sequence composition of the layers. Epigenomic annotations (ChrommHMM[9]) in Human. B The chromatin colors [10] in Drosophila. C Epigenomic annotations [25] marks in Arabidopsis Thaliana. D,E,F Enrichment in repetive sequences in the layers in Human, DrosophilaArabidopsis[START_REF] Bao | Repbase update, a database of repetitive elements in eukaryotic genomes[END_REF]

Figure S1 : 13 Figure

 S113 Figure S1: Heatmap of contingency between the SPIN compartments and the 6 layers inferred from the intra-chromosomal Hi-C contacts in human cells at a resolution of 25kb.

Acknowledgements

The authors are very grateful to Vincent Colot teams for precious advise during common lab meetings ; Irc chat of bioinfo-fr canal for technical guidance with repeat element. Work by L Carron was funded by a collaborative research grant from Agence Nationale de la Recherche ANR-18-CE13-0004-01 to AC and FB. Work in FB laboratory was also supported by a research grant from the Velux Foundation (Switzerland), the CNRS EPIPLANT Action (France) and by the COST Action CA16212 INDEPTH (EU).

Competing interests

The authors declare that they have no competing interests.

Author's contributions

LeC developed the Hi-C analysis tools, analysed Hi-C datasets, prepared annotation and made the figures. LeC and LoC analysed the Chip-seq experiments in Arabidopsis thaliana. FB, AC and JM supervised research. LeC, and JM wrote the manuscript with input from FB, LoC, AC and SG.


Elodie Pillon
email: epillon@cesi.fr

Exploring the Activities and Outcomes of Living Labs: Insights for meeting stakeholder expectations

Keywords: Living Lab, stakeholders, activities, outcomes, collaboration 1. Introduction

Open innovation is a key concept in innovation management that encourages collaboration, knowledge sharing, and co-creation. Living Labs, as collaborative innovation structures, are well-suited for implementing open innovation. They serve as user-centered ecosystems for testing and validating user-oriented solutions in real-life conditions. While existing literature explores various dimensions of Living Labs, empirical research on practical implementation activities and outcomes is lacking. This study aims to address this gap by investigating the activities deployed within a Living Lab to support innovation projects. By examining a specific Living Lab in a territorial context, the research aims to provide insights into the mechanisms and processes that foster synergies among Living Lab actors and territorial stakeholders. The findings highlight the importance of collaboration, co-creation, and animating diverse ecosystems to maximize the impact of Living Lab initiatives on territorial ecosystems. It also underscores the role of the Living Lab as an ecosystem facilitator. For Living Lab managers and policymakers, the study provides practical insights on key activities that foster collaboration and meet stakeholder expectations.

characteristics [START_REF] Hossain | A systematic review of living lab literature[END_REF][START_REF] Westerlund | Key Constructs and a Definition of Living Labs as Innovation Platforms[END_REF], the role of users [START_REF] Leminen | A typology of creative consumers in living labs[END_REF], user motivation [START_REF] Bergvall Kareborn | Living Lab: an open and citizen-centric approach for innovation[END_REF], types of services they provide [START_REF] Schuurman | Innovation management in living lab projects: The innovatrix framework[END_REF], and their main contribution to the innovation process [START_REF] Dekker | The Living Lab as a Methodology for Public Administration Research: a Systematic Literature Review of its Applications in the Social Sciences[END_REF]. One of the most commonly used approaches to understanding Living Labs seems to be the study of innovation and collaboration activities [START_REF] Hossain | A systematic review of living lab literature[END_REF], including trials, validation, experimentation, and co-creation. The specialized literature explores these different dimensions in detail, providing valuable insights into the characteristics of Living Labs and the benefits they offer. However, while it is essential to understand how Living Labs work in specific situations [START_REF] Hossain | A systematic review of living lab literature[END_REF], empirical research on the practical implementation of activities and the resulting outcomes is lacking [START_REF] Paskaleva | Are living labs effective? Exploring the evidence[END_REF][START_REF] Veeckman | Linking Living Lab Characteristics and Their Outcomes: Towards a Conceptual Framework[END_REF]. Our study aims to fill this gap by addressing the following research question: What activities are deployed within a Living Lab to support innovation projects? By deeply exploring a specific Living Lab situated within a particular territorial context, we strive to provide a better understanding of the specific activities that can be carried out within a Living Lab. From a theoretical standpoint, by identifying key activities and their resulting outcomes, we will shed light on the mechanisms and processes that foster the development of synergies between Living Lab actors and territorial stakeholders. On a managerial level, our research will offer practical insights to Living Lab managers and policymakers by highlighting activities that can be undertaken to maximize the impact of these initiatives on the territorial ecosystem while addressing the needs of stakeholders. In doing so, we hope to contribute to a better understanding of management practices and the enhancement of open innovation policies.

2. Theoretical background.

Definition of Living Lab.

Living Labs have emerged from the growing awareness of the challenges (and frequent failures) of commercializing science, which often struggles to move beyond the stages of fundamental research and development (Leminen & Westerlund, 2019). The value of innovation no longer resides solely in superior technical qualities but rather in its alignment with the needs, capabilities, and aspirations of users [START_REF] Veeckman | Linking Living Lab Characteristics and Their Outcomes: Towards a Conceptual Framework[END_REF]. As such, Living Labs have emerged as collaborative and open innovation environments, providing a third place where businesses, users, researchers, and other stakeholders can collaborate to develop and support innovation projects that address the challenges of a specific territory. To anchor themselves in local realities and address the specificities of territories, Living Labs are often situated in urban or regional environments [START_REF] Bergvall Kareborn | Living Lab: an open and citizen-centric approach for innovation[END_REF]. These cocreation spaces encourage interaction among different actors, facilitate the sharing of knowledge and resources, and enable real-life experimentation [START_REF] Eriksson | State-of-the-art in utilizing Living Labs approach to user-centric ICT innovation -a European approach[END_REF]. They facilitate interaction among various stakeholders, encouraging the sharing of expertise, codesign, and co-implementation of innovation projects [START_REF] Baccarne | Facilitating Quintuple Helix innovation with Urban Living Labs Basti[END_REF]. The active involvement of users in the innovation process promotes the adoption and diffusion of innovations [START_REF] Bergvall Kareborn | Living Lab: an open and citizen-centric approach for innovation[END_REF][START_REF] Leminen | A typology of creative consumers in living labs[END_REF]. By providing a realworld context for testing prototypes, evaluating their functionality, and gathering feedback from users [START_REF] Hossain | A systematic review of living lab literature[END_REF], and with an iterative approach, Living Labs allow for adjustments and improvements to the proposed solutions, thus enhancing their market acceptance. Additionally, [START_REF] Ballon | The effectiveness of involving users in digital innovation: Measuring the impact of living labs[END_REF] identify several objectives of a Living Lab, such as aligning innovation processes and outcomes with user preferences and practices, discovering unexpected uses, identifying solid business models and revenue sources, promoting cooperation among stakeholders, allowing specific stakeholder groups to influence design characteristics, fostering acceptability, and reducing failures.

Living Labs are characterized by their participatory approach, involving users from the early stages of the innovation process. Their active contribution helps in better understanding their needs and expectations, thus promoting the development of more suitable solutions [START_REF] Almirall | Innovation: A question of fit -The Living Labs approach[END_REF]. User-centered innovation can significantly reduce costs compared to producer-centered innovation [START_REF] Hippel | Horizontal innovation networks-by and for users[END_REF]. The involvement of users can range from passive participation, where their role is limited to providing feedback on specific technologies, products, or services, to active participation where they co-create new solutions [START_REF] Leminen | A typology of creative consumers in living labs[END_REF]. However, although Living Labs are associated with active user participation, in practice, users often remain passive during the innovation process and are less involved [START_REF] Nyström | Actor roles and role patterns influencing innovation in living labs[END_REF]. [START_REF] Leminen | On becoming creative consumers -user roles in living labs networks[END_REF] identify four roles that users can adopt within a Living Lab: informants, testers, contributors, and co-creators. Beyond users, the existing literature on Living Labs emphasizes the presence of multiple stakeholders [START_REF] Bergvall Kareborn | Living Lab: an open and citizen-centric approach for innovation[END_REF] in innovation activities, resulting in complex partnerships and fostering links between universities and industry, as well as between large companies, SMEs, and startups. This often leads to the formation of what is called public-private-person partnerships [START_REF] Hyysalo | What difference does a living lab make? Comparing two health technology innovation projects[END_REF]. These stakeholders, represented by actors from different spheres of society, can play various roles. [START_REF] Westerlund | Managing the Challenges of Becoming an Open Innovation Company: Experiences from Living Labs[END_REF] identify four roles: enablers, providers, utilizers, and users. Enablers support Living Lab activities by promoting them or providing financial support or space. They are typically public actors, funders, or non-governmental organizations such as cities or municipalities. Providers bring essential knowledge and expertise to innovation support activities. They mainly consist of universities, training institutes, or consultants. Utilizers are the organizations (public or private) that benefit from the innovations resulting from Living Lab activities. Finally, as mentioned earlier, users are citizens who participate in Living Lab activities. By integrating a Living Lab, stakeholders generally have specific expectations. For example, communities may seek to initiate citizen-government dialogue, integrate innovation into an environment, or stimulate a value chain with structural effects on the territory. Investors may aim to reduce time to market or research and development time. Users may want to actively contribute to the improvement/creation of innovations or enhance their skills. Businesses may seek to develop, validate, and integrate new ideas or establish partnerships with other companies or users. Finally, researchers may benefit from privileged access to communities in the field and new ways to valorize their research. Thus, by integrating a Living Lab, each stakeholder can contribute to the territory while addressing their own needs.

In summary, Living Labs play a crucial role in promoting open innovation by providing a space for collaboration, co-creation, and experimentation. Their participatory approach and proximity to users foster the development of innovative solutions that are tailored to the needs of stakeholders and the specificities of territories. They enable a range of actors to identify mutual interests, address emerging challenges related to society and businesses, and pursue common projects with practical applications and real-world impact.

Activities and outcomes of Living Labs.

Living Labs are collaborative innovation environments that enable the detection, prototyping, validation, and refinement of complex solutions in multiple real-life contexts [START_REF] Ballon | The effectiveness of involving users in digital innovation: Measuring the impact of living labs[END_REF].

These spaces provide a platform for co-creation, exploration, experimentation, and evaluation, deploying a set of iterative activities throughout the different stages of an innovation process. [START_REF] Huang | A Review of Living Lab Research and Methods for User Involvement[END_REF] highlight that studies on co-creation methods and activities are relatively scarce due to a lack of conceptual and methodological knowledge in early studies [START_REF] Bergvall Kareborn | Living Lab: an open and citizen-centric approach for innovation[END_REF][START_REF] Leminen | Categorization of Innovation Tools in Living Labs[END_REF].

Through participatory research methods such as discussions, surveys, or observations [START_REF] Hossain | A systematic review of living lab literature[END_REF], Living Labs work closely with stakeholders to identify emerging needs and challenges. Co-creation is facilitated by recruiting user panels with specific characteristics to develop and test products and services [START_REF] Schuurman | Technology Innovation Management Review Structuring User Involvement in Panel-Based Living Labs[END_REF], as well as by actively involving users in the innovation process through workshops, roundtables, or hackathons [START_REF] Huang | A Review of Living Lab Research and Methods for User Involvement[END_REF]. Although specific co-creation methods have not yet been established according to [START_REF] Følstad | Living Labs for innovation and development of information and communication technology: a literature review[END_REF], various methods such as analysis of system logs or automatically collected behavioral data, ethnographic methods, questionnaires, focus groups, and observations are listed, allowing for data collection to stimulate co-creation potential. With these tools, Living Labs provide a framework for conducting real-world experiments in real environments. This enables the testing and evaluation of prototypes, gathering user feedback, and making iterative improvements to ongoing innovations under development [START_REF] Hossain | A systematic review of living lab literature[END_REF]. Lastly, by organizing various types of activities or workshops, Living Labs raise awareness among users and stakeholders about innovation, new technologies, and collaborative working methods. This contributes to strengthening the skills of the actors involved in the innovation process. [START_REF] Veeckman | Linking Living Lab Characteristics and Their Outcomes: Towards a Conceptual Framework[END_REF] emphasize that in a Living Lab, innovation projects are shaped through the interaction and engagement of users, researchers, businesses, and other stakeholders. By understanding the specific contributions of each stakeholder and identifying the desired outcomes, innovation activities can be tailored to the needs and goals of each party. For example, if the goal is to develop a new product that meets user needs, co-creation and experimentation activities can be strengthened to foster active user involvement and obtain valuable feedback. Similarly, if the expected outcome is the successful adoption of an innovation in the market, it is essential to consider user expectations and preferences from the early stages of the development process. Validation and evaluation activities can be implemented iteratively to gather user feedback and adjust innovations accordingly. Therefore, the environment of a Living Lab influences the nature and direction of innovation projects. Thus, innovation activities are connected to the contributions and expected outcomes. To assess the effectiveness of a Living Lab's activities, particular attention should be given to the resulting outcomes. [START_REF] Schaffers | Living Labs for Cross-Border Systemic Innovation[END_REF] explain that product and service innovations are typical outcomes of Living Labs. However, [START_REF] Hossain | A systematic review of living lab literature[END_REF] reveal in their literature review that outcomes are not only tangible (designs, products, prototypes, and solutions) but can also be intangible (concepts, ideas, intellectual property rights, knowledge, and services). [START_REF] Veeckman | Linking Living Lab Characteristics and Their Outcomes: Towards a Conceptual Framework[END_REF] specify that outcomes are closely linked to the environment and chosen innovation approach. [START_REF] Leminen | A recipe for innovation through living lab networks[END_REF] show that outcomes depend on:

Strategic intention, which refers to individual or common collaboration motives.

 Passion or the willingness of users or ecosystem partners to participate and collaborate within the community.

 Knowledge and skills of the Living Lab.

 The quantity and availability of resources.

 The number of different partners in the Living Lab network.

Research methodology.

Given the exploratory approach in this study and the research question, we conducted a qualitative study, and more specifically a participatory case study. This research approach involves active participation of researchers in all phases of the case study. It aims to integrate the perspectives and knowledge of the actors involved in the studied situation.

The Living Lab Mix.

Our case study is conducted within the framework of the project « Rouen Mobilité Intelligente pour Tous" (Rouen Smart Mobility for All) which seeks to completely rethink the mobility system to design an offering that encompasses all modes, services, and uses. One of the actions is to  Networking with other national, European, and international Living Labs or third places.

 Attractiveness and promotion of the territory by organizing and participating in exhibitions and seminars focused on citizen inclusion in innovation projects.

 Communication and promotion to enhance visibility of the territory and highlight ongoing collaborative projects.

 Contribution to third-party initiatives such as Rouen European Capital of Culture 2028 and digital, environmental, and energy transitions.

To date, 36 partners divided into 6 colleges have joined the Mix. The 6 colleges are:

 The main funders.

 Local authorities and government departments.

 Users.

 Professional associations and sectors.

 Companies and technology.

 Research and higher education.

Data collection.

The data were collected as part of a research-intervention approach. This involved In order to gather as much data as possible on both the context of the Living Lab Mix and its activities, we also attended the general assemblies of Mix, participated in partner-focused events, and collected the documents produced during various co-creation workshops. In addition, we had an meeting with the director of the Living Lab to have a better understanding of the aim of the Living Lab Mix and its missions. Additionally, we collected information on events designed to introduce the Living Lab to potential partners or provide progress updates to funders.

Results and discussion.

Results.

As innovation activities are connected to stakeholders, their needs, and the targeted objectives [START_REF] Veeckman | Linking Living Lab Characteristics and Their Outcomes: Towards a Conceptual Framework[END_REF], it is essential to identify these stakeholders and their needs. The collection of individual motivations (by structure) regarding the Living Lab project (Table 1) has highlighted four main axes of motivations:  Initiating multi-partner projects and collaborations with the ecosystem.

 Experimenting and testing solutions that meet identified territorial needs.

 Imagining and co-constructing mobility services with citizens and prefiguring new uses.

 Identifying and understanding user expectations.

Examples of verbatim comments Motivation

"Collaborating to explore new avenues. These four main axes of motivations encompass two different scales. On the macro scale, which refers to the scale of service provision: Initiating multi-partner projects and collaborations with the ecosystem. This dynamic initiated with the ecosystem should align with its needs. This can include projects, as well as events, roundtable discussions. On the micro scale, for the other three levers, referring to methodologies and approaches related to user-centered innovation that can be applied to projects. Regarding participants' expectations of the living lab, the main purpose of the Living Lab is identified unanimously by the majority of groups: to promote the cross-pollination of actors (professionals, general public) and their participation in the development of future mobility through user-centered innovation methodologies, known as Living Lab approaches. This notion of user-centered innovation is reflected in a number of verbatim: "expression and gathering of needs", "co-creation", "testing", and "experimentation". However, we highlight that companies refer more to the concept of experimentation than other participants do: "testing services and benefiting from user panels."

We also note that the partners expect the Living Lab to provide an "access to a unique database for the region". Overall, the Living Lab Mix should thus address three ambitions: i) Supporting and equipping Living Lab projects and partners, ii) initiating and conducting innovative collaborative projects, and iii) fostering and animating a diverse ecosystem and an open space. Accompaniment activities address the need to experiment and test solutions that meet identified territorial needs. Activities related to the initiation of innovative collaborative projects will enable the envisioning and co-construction of mobility services with citizens, prefiguring new uses, as well as identifying and understanding user expectations. Meanwhile, fostering and animating a diverse ecosystem and an open space allow for initiating multipartner projects and collaborations with the ecosystem.

About the expectations of stakeholders towards the Living Lab, a quarter of the participants were local authorities, a quarter from mobility-related players, a quarter from ecosystem structures (incubators and clusters), followed by a smaller proportion of 16% from academic players and 6% energy-related players. Stakeholders in mobility, energy, and academic research expect to benefit from a forward-looking perspective on trends and innovations, integrate into a structured ecosystem, receive support in project calls, receive individual support in project implementation, enhance project maturity, and improve project quality through public exposure and feedback. Incubators and clusters have the specific need to identify innovative startups. All actors want to be able to identify the right interlocutor to address an identified need, have an assessment of project profitability, support innovation diffusion, and provide public/user access. The stakeholders seek to identify mutual interests, address emerging societal challenges, and undertake joint projects with practical applications and real-world impact. Finally, the following activities have been selected:  Innovation and usage monitoring in coordination with the ecosystem: Keeping track of emerging trends and user behaviors to inform the development of innovative solutions.

 Organization of networking events (B2B, general public): Facilitating opportunities for stakeholders to connect, collaborate, and share knowledge.

 Organization of working groups, pooling of resources, and structuring projects: Bringing together diverse stakeholders to collaborate on projects, share resources, and ensure project success.

 Support for local councilors to encourage and train them in the challenges of innovative mobility: Assisting and empowering local councilors to understand and address the challenges of implementing innovative mobility solutions.

 Nugget detection: Identifying promising ideas, concepts, or technologies with potential for further exploration and development.

 Monitoring of RFPs and funding: Keeping track of relevant project funding opportunities and providing guidance on the application process.

 Connection with relevant stakeholders: Facilitating connections and collaborations between the Living Lab and stakeholders who can contribute to and benefit from its activities.

 Project evaluation: Assessing the performance, impact, and outcomes of projects to ensure their effectiveness and identify areas for improvement.

 Support for strategic project definition: Assisting in the strategic planning and definition of projects to align with the Living Lab's objectives and maximize their impact.

 RFPs identification: Identifying and disseminating information about request for proposals (RFPs) that align with the Living Lab's focus areas.

 Targeted partner search: Identifying potential partners who can contribute expertise, resources, or collaboration opportunities to projects.

 Support in project implementation: Providing guidance, resources, and assistance in implementing projects effectively.

 Support for small-scale implementation: Assisting in the implementation of pilot projects or small-scale initiatives to test and validate innovative ideas.

 Final project evaluation: Conducting a comprehensive assessment of completed projects to evaluate their outcomes, impact, and success.

 Innovation promotion: Promoting and showcasing innovative solutions and ideas to inspire others and foster a culture of innovation.

 Public readiness for innovation: Engaging and preparing the public for embracing and adopting innovative mobility solutions through awareness campaigns and education initiatives.

 Consultation on challenges: Seeking input and feedback from stakeholders on challenges and opportunities in the field of innovative mobility.

 Conducting satisfaction surveys: Gathering feedback from stakeholders and users to assess their satisfaction, gather insights, and identify areas for improvement.

In termed of deliverables, stakeholders expected that the Mix propose:  Events, conferences, and publications to disseminate information and knowledge.

 Newsletter to keep stakeholders updated on the latest developments.

 Webinars/conferences for remote participation and knowledge sharing.

 Address book to maintain a network of relevant contacts.

 Collaborative digital platform for effective collaboration and knowledge exchange.

 Collaborative projects that foster cooperation among stakeholders.

 Conferences specifically designed for elected representatives to educate and engage them in the challenges of innovative mobility.

 Training courses/workshops to enhance skills and knowledge.

 Challenges to encourage innovative solutions and ideas.

 Coordinated events that bring together various stakeholders for networking and collaboration.

 Web page as a dedicated online presence for the Living Lab.

 Targeted networking activities to connect with specific stakeholders.

 Evaluation files/project notes to assess and measure the success of projects.

 Advice on gaining maturity in project development and implementation.

 Workshops and working groups for in-depth discussions and collaboration.

 Recommendations for improvements and future actions.

 Proposal/presentation to relevant RFPs for funding opportunities.

 Meetings with thematic players to explore synergies and collaborations.

 Editing workshops to refine project documentation and materials.

 Iterations to fine-tune and improve project outcomes.

 (Co)production of project components to achieve desired results.

 Experimentation strategy, implementation, and results to test and validate innovative solutions.

 Creation of support, implementation, and administration of evaluation to ensure effective project management.

 Marketing campaign to raise awareness and promote innovation.

 Solution referencing to highlight successful solutions and encourage their adoption.

 Communication with the public through social networks and events to engage and inform them.

 Public access and feedback during project implementation to incorporate user perspectives.

 Demonstrator to showcase and demonstrate innovative solutions.

 Social acceptance measures to assess the impact and acceptance of projects.

 Survey results to gather feedback and insights from stakeholders and users.

These outcomes will enable us to assess whether the Living Lab is meeting stakeholders' expectations, and to monitor the activities implemented.

Stakeholder need Activity Outcome

A forward-looking vision of trends and innovations.

Discussion

Through the activities and outcomes identified by stakeholders, we can define the expected impacts of the Living Lab Mix on the territory:

 Foster the creation of an inclusive and sustainable mobility innovation community.

 Develop and test innovative mobility solutions to meet specific territorial needs.

 Enhance existing mobility services through the integration of technologies and new approaches.

 Actively involve citizens in co-creating and making decisions regarding mobility projects.

 Strengthen the skills of stakeholders in the field of sustainable and innovative mobility.

 Facilitate the diffusion and transfer of knowledge among the actors of the Living Lab Mix.

 Establish strong partnerships among different actors in the mobility sector.

 Position Rouen Normandy Metropolis as an innovation hub in mobility.

These impacts contribute to achieving a more inclusive, sustainable, and efficient mobility in the region of Rouen Normandy Metropolis, while fostering collaboration among stakeholders and promoting the adoption of innovative mobility solutions. The expected impacts of the Living Lab Mix on the territory need to be measured using a dashboard, which is often identified as a gap in Living Lab practices. Through this study, we can identify several insights:

 The main categories of actors involved in the Living Lab Mix are communities, mobilityrelated entities, academic research institutions, and energy-related organizations.

 The motivations of the actors include initiating multi-partner projects, experimenting with solutions to meet territorial needs, co-creating mobility services with citizens, and understanding user expectations.

 The expectations of the actors towards the Living Lab Mix include networking opportunities, information dissemination, collaborative projects, skill enhancement through training, and support for project implementation.

 The activities of the Living Lab Mix are focused on monitoring the ecosystem, initiating multi-partner projects, evaluating and supporting proposed projects, experimenting and testing solutions to address identified territorial needs, co-creating mobility services with citizens, identifying user expectations, and preparing the public for innovation.

 A key outcome identified is the creation of a diverse and open ecosystem, which is an activity expected by all actors.

These insights provide a deep understanding of Living Lab management practices and the improvement of open innovation policies. They highlight the importance of collaboration, cocreation, and animating diverse ecosystems to maximize the impact of Living Lab initiatives on territorial ecosystems. The presence of multiple stakeholders, as mentioned in the literature [START_REF] Bergvall Kareborn | Living Lab: an open and citizen-centric approach for innovation[END_REF], is well reflected in the Living Lab Mix framework through the 6 partner colleges and the participants in the various co-creation workshops. However, we notice that citizens were not consulted during the co-creation workshops and are not part of the partners. The Mix explains that their involvement in the project will be the subject of related workshops. It is also worth noting that in the defined activities, citizens (the primary target users of soft mobility systems) seem to be consulted only for feedback on specific technologies, products, or services. In line with the work of [START_REF] Nyström | Actor roles and role patterns influencing innovation in living labs[END_REF], users remain less involved, and therefore, they appear to have a passive role in the innovation process, as defined by [START_REF] Leminen | A typology of creative consumers in living labs[END_REF]. However, a significant number of workshop participants expressed support for "the creation and provision of panels." This underscores the importance of fully involving citizens throughout the innovation process, from identifying needs to experimenting with future services.

The literature indicated that the activities of a Living Lab can be summarized as detecting, prototyping, validating, and refining complex solutions in real-life settings through a loop of activities (co-creation, exploration, experimentation, evaluation). While we do find these steps within the Living Lab Mix framework, the federation and animation of the diverse ecosystem emerge as a key activity expected by all stakeholders. Just as the environment of a Living Lab shapes the undertaken projects [START_REF] Veeckman | Linking Living Lab Characteristics and Their Outcomes: Towards a Conceptual Framework[END_REF], it also appears to shape the activities conducted within it. In the case of the Living Lab Mix, although the identified objective is to make Rouen Normandy Metropolis a reference territory for inclusive innovations and experiments in the field of mobility, it seems that the stakeholders expect the Living Lab Mix to help them strengthen collaborations and expand their network, particularly by facilitating their interaction with other actors in the territory or by providing access to a unique database consolidating regional data. This highlights the importance of facilitating and developing access to local data. Thus, the Living Lab Mix must not only facilitate the innovation process but also address the relational dynamics gaps within the ecosystem. Lastly, although [START_REF] Schaffers | Living Labs for Cross-Border Systemic Innovation[END_REF] explain that product and service innovations are typical outcomes of Living Labs, most of the defined outcomes for the Living Lab Mix, such as the address book, a collaborative digital platform, or collaborative projects, are neither services nor products and appear to be more intangible. This can be explained by the fact that the activities were defined based on the articulated needs of stakeholders, and these same stakeholders participated in defining the outcomes that will enable a relatively reliable measurement of whether the Living Lab Mix meets their expectations or not.

Conclusion.

Open innovation emphasizes collaboration, knowledge sharing, and co-creation to tap into diverse sources of ideas and resources. Living Labs have emerged as collaborative innovation structures that are particularly well-suited for implementing open innovation strategies [START_REF] Westerlund | Managing the Challenges of Becoming an Open Innovation Company: Experiences from Living Labs[END_REF]. These user-centered innovation ecosystems serve as physical or virtual spaces for co-creation, testing, and validation of user-oriented solutions in real-life conditions, addressing a wide range of social, technological, and economic challenges in a given territory [START_REF] Almirall | Mapping Living Labs in the Landscape of Innovation Methodologies[END_REF][START_REF] Della Santa | How does the space influence Living Labs? Evidence from two automotive experiences[END_REF][START_REF] Huang | A Review of Living Lab Research and Methods for User Involvement[END_REF][START_REF] Westerlund | Managing the Challenges of Becoming an Open Innovation Company: Experiences from Living Labs[END_REF]. While extensive research has been conducted on various aspects of Living Labs [START_REF] Hossain | A systematic review of living lab literature[END_REF], there is a lack of empirical studies on the practical implementation of activities and their outcomes [START_REF] Paskaleva | Are living labs effective? Exploring the evidence[END_REF][START_REF] Veeckman | Linking Living Lab Characteristics and Their Outcomes: Towards a Conceptual Framework[END_REF]. This study aims to bridge this gap by examining the activities deployed within a specific Living Lab, providing insights into the mechanisms and processes that facilitate collaboration and synergies among Living Lab actors and territorial stakeholders.

We employed a participatory case study, to explore the Living Lab Mix, a physical space in Normandy, that aims to foster inclusive innovation and experimentation in the field of mobility.

The study involved active participation from researchers and various stakeholders through co- Different outcomes have been also identified by stakeholders according to their expectations and allow to identifiy the expected impacts of the Living Lab Mix on the territory. In this way, the Living Lab will be able to set up a dashboard to measure its impact on the region (a point often identified as a lack for Living Labs).

More broadly, this study contributes to a better understanding of Living Lab management practices and the enhancement of open innovation policies. The results underscore the importance of collaboration, co-creation, and animating diverse ecosystems to maximize the impact of Living Lab initiatives on territorial ecosystems. It also highlights the role of ecosystem facilitator expected of the Living Lab by stakeholders. From a managerial perspective, this research provides practical insights for Living Lab managers and policymakers by identifying key activities that can be undertaken to foster collaboration among stakeholders and address their specific expectations.

The remainder of this study will focus on the implementation and monitoring of these actions, in order to better understand how they can impact the region and whether they really meet the expectations of stakeholders.

  launch a Living Lab, with the objective of making the Rouen Normandy Metropolis a reference territory for inclusive innovations and experiments in the field of mobility. The Living Lab "Mix" is a physical space based in Rouen (France) launched in January 2021. It focuses on 12 general interest missions:  Territory observation: Study current and future users of mobility.  Creation and valorization of common goods: Promote citizen inclusion and social acceptability of innovation projects.  Public and local stakeholders' awareness of environmental and societal transition related to mobility.  Constitution and development of a citizen innovation panel to generate innovative ideas and support projects led by third parties.  Establishment of citizen-user specifications as a basis for future collaborative projects in the territory.  Identification and selection of innovation projects in the mobility of goods and people.  Facilitation of access to open data from the territory (Open Data) and utilization of nonopen data to derive added value in innovation projects. Connecting and facilitating collaboration processes among partners to implement mobility experiments in the territory.

  participation in four co-creation workshops organized by Mix (between December 2020 and May 2023) with the different stakeholders. For the first co-creation workshop, before the official launch of the Living Lab, 25 participants representing 20 different structures (belonging to the overall mobility ecosystem of the region) were present. This session was dedicated to gather individual motivational factors for joining the Living Lab project and identify the actors in the territory and their positioning within the project. For the second co-creation workshop, 49 participants representing 39 different structures were present. This session was dedicated to present a first draft of the service offering based on the needs identified during the first session and to gather participants' expectations towards the Living Lab project. To define the service offer, each participant had to vote for the most interesting proposed services. It aimed to coconstruct the MIX collaborative innovation process, which will enable all partners to bring forward innovative mobility issues that will give structure to the region. The third and fourth cocreation workshop were only for partner membership in the aim to identify the key success factors for a collaborative project conducted within the Living Lab framework. So 14 participants representing 13 different structures were present to:  Build a project compass to select projects  Share visions of the needs of such a collaborative project  Identify for each partner: how are they involved? What resources do they bring ? And what are your constraints?

  The 20 participants' structures at the first session for identify individual motivational factors for joining the Living Lab project could be grouped into 4 categories:  Communities (3 structures)  Mobility (5 structures)  Academic research (6 structures)  Energy (6 structures)

  creation workshops and data collection from different sources. The findings revealed four main categories of stakeholders: communities, mobility-related entities, academic research institutions, and energy-related organizations. These stakeholders expressed motivations such as initiating multi-partner projects, experimenting with solutions to meet territorial needs, coconstructing mobility services with citizens, and understanding user expectations. The stakeholders' expectations from the Living Lab Mix included networking opportunities, dissemination of information, collaborative projects, skill enhancement through training, and support in project implementation. The activities of the Living Lab Mix were found to align with these expectations. They are focused on watching the ecosystem, initiating multi-partner projects, assessing and supporting proposed projects, experimenting and testing solutions to address identified territorial needs, co-creating mobility services with citizens, identifying user expectations and preparing the public for innovation. Additionally, federating and animating a diverse and open ecosystem was identified as a key activity expected by all stakeholders.

Table 1 :

 1 Individual motivation of participants regarding the Living Lab project

	"

Table 2 :

 2 Summary of activities and outcomes in relation to identified needs

		Innovation and usage	 Events
		monitoring coordination with	 Conferences
		the ecosystem	 Publications
	Integrating a structured	Organization of networking	 Newsletter
	ecosystem	events (B2B, general public)	 Webinaires / conferences
		Organization of working	 Address book
		groups, pooling of resources,	 Collaborative digital
		structuring projects.	platform
			 Collaborative projects
		Support for local councilors	 Conferences for elected
		to encourage and train	representatives
		them in the challenges of	 Training courses /
		innovative mobility	workshops
	Identify innovative startups	Nugget detection	 Challenges
	Description of RFPs	Monitoring of RFPs and	 Coordinated events
		funding	 Web page
	Identify the right player to	Connection with relevant	 Targeted networking
	meet an identified need	stakeholders
	Reinforcement of project	Project evaluation	 Evaluation file / project
	maturity.		note
			 Advice on gaining maturity
		Support for strategic project	 Workshops
		definition	 Working groups
			 Recommendations
	Individual support in setting	RFPs identification	 Proposal / presentation to
	up projects		relevant RFPs
		Target partner search	 Meet thematic players
		Support in projects	 Editing workshops
		implementation	 Iterations
			 (co)production of parts
	Enhance project quality	Support for small-scale	 Experimentation strategy,
	through public exposure	implementation	implementation and results
	and feedback
	Evaluation of project returns	Final project evaluation	 Creation of the support,
			implementation and
			administration of the
			evaluation.


Ion Pauline Kergus
email: pauline.kergus@cnrs.fr

Mihaly Victor Gosea

Mihaly Petreczky

Loewner functions for bilinear systems

whether they are published or not. The documents may come

Introduction

Accurate modeling of physical phenomena often leads to large-scale dynamical systems that require long simulation times and large data storage. In this context, model order reduction (MOR) aims at obtaining much smaller and simpler models that are still capable of accurately representing the behavior of the original process. Many different MOR techniques have been proposed: SVD-based (e.g., balanced truncation) methods, Krylov-based or moment-matching methods, proper orthogonal decomposition (POD) methods, and reduced basis methods. Most of these methods are included in the broad family of projection-based methods: the internal state x is approximated by a vector of smaller dimension x obtained through a projection into a particular subspace. For more insights, the reader can refer to [START_REF] Antoulas | A survey of model reduction methods for large-scale systems[END_REF], [START_REF] Athanasios | Approximation of large-scale dynamical systems[END_REF], [START_REF] Athanasios C Antoulas | Interpolatory model reduction of large-scale dynamical systems[END_REF], [START_REF] Baur | Model order reduction for linear and nonlinear systems: a system-theoretic perspective[END_REF], and to [START_REF] Benner | A survey of projection-based model reduction methods for parametric dynamical systems[END_REF].

Among them, the Loewner framework (LF) [START_REF] A C Antoulas | A tutorial introduction to the Loewner framework for model reduction[END_REF] is very appealing due to its data-driven nature, which makes it nonintrusive as it does not use the full/exact description of the model. It is based on the Loewner pencil that allows solving the generalized realization problem for linear time-invariant (LTI) systems [START_REF] Mayo | A framework for the solution of the generalized realization problem[END_REF], and obtaining reduced-order models through rational interpolation [START_REF] Anderson | On the scalar rational interpolation problem[END_REF]. In particular, the Loewner matrix is the cornerstone of the LF as it can be factored into the tangential generalized controllability matrix and the tangential generalized observability matrix, which can then be used to construct LTI models.

To broaden its applicability, extensions of the LF have been proposed for specific classes of nonlinear systems: bilinear [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF], quadratic-bilinear [START_REF] Gosea | Data-driven model order reduction of quadratic-bilinear systems[END_REF], and linear switched systems [START_REF] I V Gosea | Data-driven model order reduction of linear switched systems in the loewner framework[END_REF]. These extensions rely on higher-order transfer functions based on Volterra series, and the definition of associated observability and controllability matrices.

In parallel, an interconnection approach is proposed in [START_REF] Simard | An interconnection-based interpretation of the Loewner matrices[END_REF], based on the definition of left and right Loewner matrices. This approach allowed to develop a model order reduction procedure for linear time-varying systems [START_REF] Simard | Loewner functions for linear time-varying systems with applications to model reduction[END_REF]. This idea is taken further in [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF] to extend the LF to general nonlinear input-affine systems by defining Loewner functions, which can be seen as time-varying nonlinear counterparts of the matrices classically used in the LF. This branch remains mostly theoretical since the explicit computation Loewner functions requires solving Partial Differential Equations (PDEs).

This work aims at bringing these two interpretations of the Loewner framework together, in order to benefit from the practical implementations of [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF] and the theoretical guarantees regarding steady-state behaviour coming with Loewner equivalence as in [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF]. To start with, this paper focuses on bilinear systems. New Loewner functions, inspired by [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF], are proposed for bilinear systems, allowing to derive a moment matching procedure for this category of systems. These Loewner functions are built based on the generalized reachability and observability matrices from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF]. In particular, they consist in infinite power series, which are not suitable for practical implementation. In this work, it is proposed to truncate these Loewner functions and a concept of approximate Loewner equivalence, namely κ-Loewner equivalence is introduced. Finally, it is shown that the approach from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF] preserves bilinearity at the expense of a higher order than the moment matching approach resulting from the proposed Loewner function, though the latter results in a more complex model structure. Both models are κ-Loewner equivalent to the underlying bilinear full-order model, i.e., they both match the same first κ-th order moments.

The outline of this paper is as follows: in Section 2, the interconnectionbased approach of the LF for input-affine nonlinear systems as in [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF] and the LF for bilinear systems from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF] are recalled. Section 3 introduces Loewner functions for the case of bilinear systems, allowing to derive a reduced-order Loewner equivalent model. As the resulting model cannot be implemented in practice, it is approximated by truncating the Lowner functions, and the concept of κ-Loewner equivalence is introduced. Connections between the Figure 1: Interconnected system according to [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF].

Bilinear Loewner Framwork (BLF) from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF] and [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF] are then established. Conclusions are provided in Section 4.

Preliminaries

Nonlinear model order reduction based on Loewner functions

In [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF], nonlinear input-affine systems of the following form are considered:

ẋ=f (x) + g(x)u y=h(x) , (1)
with state x ∈ C n , input u ∈ C m and output y ∈ C p . This system is interconnected with two linear generators (2) and (3) as visible on Figure 1:

ζr =Λζ r + ∆ v=Rζ r , (2)
ζl =Mζ l + Lχ η=ζ l , (3)
with states ζ r ∈ C ρ and ζ l ∈ C ν , inputs ∆ ∈ C ρ and χ ∈ C p , and outputs v ∈ C m and v ∈ C ν , and with matrices Λ ∈ C ρ×ρ , R ∈ C m×ρ , M ∈ C ν×ν , and L ∈ C ν ×p. The matrices Λ and M have all eigenvalues on the imaginary axis, and these eigenvalues have geometric multiplicity one. For more details on the interconnection interpretation, see [START_REF] Simard | An interconnection-based interpretation of the Loewner matrices[END_REF]. Loewner functions are introduced as generalizations of the Loewner matrices to extend the Loewner method for model reduction to nonlinear input-affine systems. The tangential generalized controllability function X : C ρ → C n is defined as the solution, provided it exists, to the PDE with boundary condition:

∂X ∂ζr Λζ r = f (X(ζ r )) + g(X(ζ r ))Rζ r X(0) = 0 (4)
while the tangential generalized observability function Y : C n → C ν is defined as the solution, provided it exists, to the PDE with boundary condition

∂Y ∂x f (x) = MY (x) -Lh(x) Y (0) = 0 . (5)
In [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF], the existence of X and Y is proven under the following assumptions: the unforced system ẋ = f (x) is locally exponentially stable at the origin and the matrices Λ and M have all eigenvalues on the imaginary axis and of geometric multiplicity one. The Loewner function L is then defined as

L(ζ r ) = -Y (X(ζ r )). (6)
The nonlinear counterparts V and W of the data matrices V and W are defined as follows:

V(ζ r ) = ∂Y ∂x (X(ζ r ))g(X(ζ r )), W(ζ r ) = h(X(ζ r )). ( 7
)
The left-Loewner function

L l : C ρ → C ν is the solution (provided it exists) of ∂L l ∂ζ r Λζ r = ML l (ζ r ) -V(ζ r )Rζ r , L l (0) = 0. ( 8
)
The right-Loewner function L r : C ρ → C ν is defined as:

L r (ζ r ) = L(ζ r ) -L l (ζ r ). (9)
The shifted Loewner function L s : C ρ → C ν is defined as: The property of Loewner equivalence between two locally exponentially stable systems implies that, for initial conditions on the manifold x = X(ζ), the two systems interconnected with the generators have the same steadystate behavior, provided it exists.

L s (ζ r ) = ML(ζ r ) + LW (ζ r ). (10)
Theorem 1 (From [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF]) Consider the interconnected system with ρ = ν. Let L l , L r , L, L s , V and W be the associated Loewner function. Assume that ∂L ∂ζr is non-singular. The system

∂L ∂ζr (x r ) ẋr =L s (x r ) -V(x r )u y r =W(x r ) ( 11
)
with state x r ∈ C ρ , input u ∈ C m and output y r ∈ C p , is Loewner equivalent at (Λ, R, M, L) to the system (1).

Locally, the original model and the interpolating model produce the same steady-state response, provided that it exists, when interconnected with generators corresponding to the Loewner functions.

The bilinear Loewner framework

The bilinear Loewner framework (BLF) presented here first appears in [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF]. In this paper, single input -single output (SISO, m = p = 1) bilinear systems of the form:

E ẋ=Ax + Nxu + Bu y=Cx (12)
are considered, with state x ∈ R n , input u ∈ R and output y ∈ R, with matrices E, A, N ∈ R n×n , and B, C T ∈ R n . The cornerstone of the BLF is that the behavior of bilinear systems is characterized by the following serie of generalized transfer function, defined for l ≥ 1:

H l (s 1 , s 2 , . . . , s l ) = CΦ(s 1 )NΦ(s 2 )N . . . NΦ(s l )B, (13)
where

Φ(s) = (sE -A) -1 (14)
is the resolvent of the matrix pencil (A, E). These generalized transfer functions are obtained by taking the Laplace transform of the kernels of the Volterra series expansion of the system, see [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF] for more details.

The BLF interpolates the generalized transfer functions at interpolation points which are grouped into left and right multi-tuples, denoted µ (j) and λ (i) respectively, with j = 1 . . . q and i = 1 . . . k. The multi-tuples are defined as follows:

µ (j) =                µ (j) 1 µ (j) 1 , µ (j) 2 . . . µ (j) 1 , µ (j) 2 , . . . , µ (j) p λ (i) =                λ (i) 1 λ (i) 2 , λ (i) 1 . . . λ (i) m , . . . , λ (i) 2 , λ (i) 1 . ( 15
)
The generalized reachability matrix R ∈ C n×k associated with the right multi-tuples λ (1) , λ (2)

, . . . , λ ( k) is R = R (1) , R (2) , • • • , R ( k) , (16)
where the matrices R (j) ∈ C n×m , j = 1, . . . , k, are associated with the j-th multi-tuple λ (j) in (15) are given by

R (j) = Φ(λ (j) 1 )B, Φ(λ (j)
2 )NΦ(λ

(j) 1 )B, . . . . . . , Φ(λ (j) m i )N . . . NΦ(λ (j)
2 )NΦ(λ

(j) 1 )B . (17)
Similarly, the generalized observability matrix O ∈ C k×n associated with the left multi-tuples µ (1) , µ (2) , . . . , µ ( k) is given by

O = O (1) T , O (2) T , . . . O ( k) T T ∈ C k×n , (18)
where O (i) ∈ C p×n , i = 1, . . . , k, correspond to the j-th multi-tuple µ (j) in (15) and

O (j) =       C T Φ(µ (j) 1 ) C T Φ(µ (j) 1 )NΦ(µ (j) 2 ) . . . C T Φ(µ (j) 1 )NΦ(µ (j) 2 )N . . . NΦ(µ (j) p j )       . ( 19
)
Next, similarly to the linear case, the Loewner matrix L and the shifted Loewner matrix L s are defined using the generalized reachability (16) and observability (18) matrices as

L = -O E R, L s = -O A R . ( 20
)
The fact that the Loewner matrices are factorized in terms of the pairs of matrices (E, A) and (O, R) is an inherent property of the Loewner framework, which holds true for both the bilinear and quadratic-bilinear case, and even for more general extensions involving higher-order polynomial structures. Next, introduce the following matrices:

V = OB, W = CR, T = ONR, (21)
Lemma 1 (from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF]) Assuming that the pencil (L, L s ) is regular, then the quintuple (-L, -L s , T, V, W)

defining the following model

-L ẋr =-L s x r + Tx r u + Vu y r =Wx r (22)
is a minimal realization of an interpolant of the generalized transfer functions H l for l = 1 . . . k + q at the points defined by the multi-tuples (15).

In case of redundant data, i.e. the pencil (L, L s ) is not regular, then a reduced-order model can be constructed by performing and truncating a singular value decomposition of the pencil, see [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF].

Moment matching for bilinear systems

In this section, Loewner functions are proposed in Section 3.1 for bilinear systems as described in [START_REF] Simard | An interconnection-based interpretation of the Loewner matrices[END_REF] with E = I. To start with, the left generator is given by Λ = λ and R = 1, and the right generator is given by M = µ and L = 1, i.e. ρ = ν = 1. Connections are then established with the BLF from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF] in Section 3.3. The general case ρ ≥ 1 is presented in Section 3.4.

Loewner functions for bilinear systems

Theorem 2 The tangential generalized controllability function X

X(ζ r ) = ∞ k=1 Φ k ζ k r , (23)
with coefficients {Φ k } defined by

Φ 1 =(λI -A) -1 B Φ k =(kλI -A) -1 NΦ k-1 for k > 1 (24)
solves (4).

Proof: First, X(0) = 0. Note that, from (24), we have kλΦ k = AΦ k + NΦ k-1 for k > 1, and λΦ 1 = AΦ 1 + B. Therefore, we have:

∂X ∂ζr λζ r = ∞ k=1 kλΦ k ζ k r =(AΦ 1 + B)ζ r + ∞ k=2 (AΦ k + NΦ k-1 )ζ k r =A ∞ k=1 Φ k ζ k r + N ∞ k=1 Φ k ζ k r + B ζ r =f (X(ζ r )) + g(X(ζ r ))Rζ r .
The tangential observability function

Y (x) = C(µI -A) -1 x, (25)
solves [START_REF] Benner | A survey of projection-based model reduction methods for parametric dynamical systems[END_REF] as Y (0) = 0 and ∂Y ∂x (µI -A)x = Cx. Then, the corresponding Loewner functions can be expressed as in ( 6), ( 7) and ( 10) respectively:

L(ζ r ) = -C(µI -A) -1 ∞ k=1 Φ k ζ k r , (26)
V(ζ r ) = C(µI -A) -1 B + N ∞ k=1 Φ k ζ k r , (27)
W(ζ r ) = C ∞ k=1 Φ k ζ k r , (28)
L s (ζ r ) = -C(µI -A) -1 A ∞ k=1 Φ k ζ k r . ( 29
)
In addition, the left and right Loewner functions are defined as:

L l (ζ r ) = C(µI -A) -1 B λ -µ ζ r + ∞ k=2 NΦ k-1 (kλ -µ) ζ k r , ( 30
) L r (ζ r ) = -C(µI -A) -1 Φ 1 -B λ-µ ζ r -C(µI -A) -1 ∞ k=2 (Φ k - NΦ k-1 (kλ-µ) )ζ k r . (31)
Then the Loewner equivalent model ( 11) at (λ, 1, µ, 1) to the system ( 12) is given by:

∞ k=1 kC(µI -A) -1 Φ k r k-1 ẋr = ∞ k=1 C(µI -A) -1 AΦ k r k + + ∞ k=1 C(µI -A) -1 NΦ k r k u r +C(µI -A) -1 Bu r y r = ∞ k=1 CΦ k r k (32)
Remark 1 If R ̸ = 1 and/or L ̸ = 1, the basis functions Φ k are multiplied by R in (24) and the generalized observability function becomes Y (x) = LC(µI -A) -1 x.

Truncation of the Loewner functions and κ-Loewner equivalence

In practice, it is difficult to implement the Loewner equivalent model (32) as the Loewner functions consist in infinite series. To overcome this issue, let the truncated (generalized) controllability function be defined for a given κ as:

X(ζ r ) = κ k=1 Φ k ζ k r . ( 33
)
As a result, the truncated left and right Loewner functions are given by

Ll (ζ r ) = C(µI -A) -1 B λ -µ ζ r + κ+1 k=2 NΦ k-1 (kλ -µ) ζ k r , (34)
and

Lr (ζ r ) = -C(µI -A) -1 Φ 1 -B λ-µ ζ r -C(µI -A) -1 κ k=2 (Φ k - NΦ k-1 (kλ-µ) )ζ k r -C(µI -A) -1 NΦκ ((κ+1)λ-µ) )ζ κ+1 r . (35)
Hence the truncated model ( 11) is given by:

κ k=1 kC(µI -A) -1 Φ k r k-1 ẋr = κ k=1 C(µI -A) -1 AΦ k r k + + κ k=1 C(µI -A) -1 NΦ k r k u r +C(µI -A) -1 Bu r y r = κ k=1 CΦ k r k (36)
The truncation introduced previously for practical implementation is supported by the following concept of κ-Loewner equivalence. Intuitively, κ-Loewner equivalence means that the Loewner functions are close for sufficiently small signals ζ r .

Definition 2 (κ-Loewner equivalence) Two systems Σ and Σ are called κ-Loewner equivalent at (Λ, L, M, R), if the left-and right-Loewner functions of Σ and Σ, denoted by L r (ζ r ), L l (ζ r ) and Lr (ζ r ), Ll (ζ r ) respectively, are smooth and have the same k-order derivatives at zero for any k ≤ κ:

d k dζ k r L r (ζ r )| ζr=0 = d k dζ k r Lr (ζ r )| ζr=0 d k dζ k r L l (ζ r )| ζr=0 = d k dζ k r Ll (ζ r )| ζr=0 (37)
Note that Loewner equivalence corresponds to the case κ = ∞.

Theorem 3 Given λ, µ, the truncated model (36) is κ-Loewner equivalent to the original system at (λ, 1, µ, 1).

Proof: Considering the left and right Loewner functions in (30) and (31), their truncated counterparts in (34) and ( 35) satisfy (37).

Connection to the bilinear Loewner framework

In order to establish a first connection between the moment matching method by [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF] for bilinear systems with the BLF from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF], the following interpolation multi-tuples are considered:

µ (1) = {µ} and λ (1) =          {λ} {2λ, λ} . . . {κλ, . . . , 2λ, λ} . (38)
The reachability matrix R from the BLF can be written as follows:

R = Φ 1 Φ 2 • • • Φ κ ∈ C n×κ , (39)
and the observability matrix is

O = C(µI -A) -1 ∈ C 1×n .
Then, the generalized controllability function can be written as

X(ζ r ) = Rv(ζ r ), (40)
with v(ζ r ) ∈ R κ defined as:

v(ζ r ) = ζ r , ζ 2 r , . . . , ζ κ r . (41)
The generalized observability function can be written as

Y (x) = Ox. (42)
The Loewner functions can then be expressed according to the Loewner matrices as follows:

L(ζ r ) = -ORv(ζ r ) = -Lv(ζ r ), V(ζ r ) = O (B + NRv(ζ r )) = V + Tv(ζ r ), W(ζ r ) = CRv(ζ r ) = Wv(ζ r ), L s (ζ r ) = -MORv(ζ r ) + LCRv(ζ r ) = L s v(ζ r ). ( 43
)
The BLF then results in the following bilinear model of order κ:

-L ẋr =-L s x r -Tx r u -Vu y r =Wr ( 44
)
while the Loewner equivalent model obtained based on the Loewner functions is of order 1 and is characterized by:

-L ∂v ∂ζr (x r ) ẋr =L s v(x r ) -(Tv(x r ) + V)u y r =Wv(x r ) (45)
Note that when κ = 1, the Loewner equivalent model ( 45) is the same than the one resulting from the BLF (44), and is written in descriptor form.

For κ ≥ 2, the BLF results in a bilinear model of order κ while the moment matching approach results in a first-order model but that does not preserve bilinearity.

Remark 2 Note that for κ ≥ 2, the model (45) from the BLF is not a descriptor model, but an ordinary differential equation of order κ. This is due to the choice of the interpolation multi-tuples. In the classical BLF, the data is usually split equally between left and right multi-tuples, and the left and right multi-tuples have the same number of components, which leads to descriptor models (with square Loewner matrices).

Theorem 4 The BLF model (44) is κ-Loewner equivalent to the original one [START_REF] Simard | An interconnection-based interpretation of the Loewner matrices[END_REF].

Proof: The Loewner functions corresponding to the BLF models are derived as in the moment matching procedure, based on the generalized controllability and observability functions which can be written as follows:

Y BLF (x) = W(-µL + L s )x, X BLF (ζ r ) = κ k=1 Φ BLF k ζ k r , (46) with
Φ BLF 1 =(-λL + L s ) -1 V Φ BLF k =(-kλ i L + L s ) -1 TΦ BLF k-1 for k > 1 .
The rest of the Loewner functions can be derived accordingly, in a similar manner as to (26)-(31).

which implies that X from (48) solves (4). The generalized observability function can be defined in a similar way.

Theorem 6

The tangential observability function is defined as:

Y (x) =    Y (1) (x) . . . Y (ρ) (x)    , (51)
with, for i = 1 . . . ρ, the function Y (i) defined as in the first-order generator case as

Y (i) (x) = L i C(µ i I -A) -1
x. The tangential generalized controllability function X from (51) solves (5).

Proof: First, as Y (i) (0) = 0 for all i = 1 . . . ρ, we have Y (0) = 0. In addition, from Section 3.1, we have for all i = 1 . . . ρ:

∂Y (i) ∂x f (x) = µ i Y (i) x -L i h(x)
which implies that Y from (51) solves [START_REF] Benner | A survey of projection-based model reduction methods for parametric dynamical systems[END_REF].

Similarly to the first-order generators case, a truncated version of the tangential generalized controllability function can be introduced so that the resulting model can be implemented in practice:

∀i = 1 • • • ρ, X(i) (ζ (i) r ) = κ k=1 Φ (i) k ζ k r i . (52)
For any value of κ (finite or infinite), the connection with the BLF from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF] can be established as in Section 3.3 by choosing ρ interpolation multi-tuples:

µ (i) = {µ i } and λ (i) =          {λ i } {2λ i , λ i } . . . {κλ i , . . . , 2λ i , λ i } . (53)
Then, equation (40) becomes:

L(ζ r ) = -Lw(ζ r ), V(ζ r ) = V + Tw(ζ r ), W(ζ r ) = Ww(ζ r ), L s (ζ r ) = -L s w(ζ r ), (54)
with w(ζ r ) ∈ R ρκ×ρ defined as:

w(ζ r ) =       v(ζ r,1 ) 0 • • • 0 0 v(ζ r,2 ) . . . . . . . . . 0 0 • • • 0 v(ζ r,ρ )      
with v defined in (41).

In the end, the BLF results in a bilinear model of order ρκ while the moment matching approach results in a model of order ρ that does not preserve bilinearity. As previously shown, the two models are exactly the same for κ = 1. Finally, the two models are κ-Loewner equivalent to the original one at (Λ, L, M, R).

Conclusion

In this work, new Loewner functions have been proposed for bilinear systems, using the generalized reachability and observability matrices used in the bilinear Loewner framework from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF]. A reduced-order Loewner equivalent model is then derived as in [START_REF] Simard | Nonlinear model reduction in the Loewner framework[END_REF]. Inpractice, it is quite challenging to implement. That is why, in this work, the Loewner functions are truncated in order to obtain approximate Loewner equivalence.This is defined in this work as κ-Loewner equivalence. It appears that the approach from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF] preserves bilinearity at the expense of a higher order than the moment matching approach resulting from the proposed Loewner function, although the latter results in a more complex model structure. Regarding the choice of κ, practical implementations of the BLF from [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF] are performed with κ = 2 but based on different multi-tuples definition. Future work will investigate the impact of κ and could provide approximation error bounds for the truncated κ-Loewner equivalent model. In addition, the authors aim at expanding this concept and proposing Loewner functions in a similar manner for more general polynomial extensions of the Loewner framework.

Definition 1

 1 Two systems Σ and Σ of the form (1) are called Loewner equivalent at (Λ, R, M, L) if their left-and right-Loewner functions satisfy L l (ζ r ) = L l (ζ r ) and L r (ζ r ) = L r (ζ r ) in a neighborhood of the origin.

Due to the interpolatory properties of the BLF, see [START_REF] Antoulas | Model reduction of bilinear systems in the Loewner Framework[END_REF], we have that, for all l = 1 . . . κ: H l (λ, 2λ, . . . , lλ) = H BLF l (λ, 2λ, . . . , lλ) H l+1 (µ, λ, 2λ, . . . , lλ) = H BLF l+1 (µ, λ, 2λ, . . . , lλ)

where H BLF l are the generalized transfer functions of the BLF model, as defined in [START_REF] Simard | Loewner functions for linear time-varying systems with applications to model reduction[END_REF]. According to the definition of the generalized transfer functions in [START_REF] Simard | Loewner functions for linear time-varying systems with applications to model reduction[END_REF] and to the expressions of the Loewner functions in the bilinear case, it implies that the truncated Loewner functions corresponding to the BLF model ( 44) are equal to the truncated ones from the moment matching procedure proposed in Section 3.1. Consequently, according to Theorem 3, the BLF model is κ-Loewner equivalent to the original system.

Extension to higher-order generators

These Loewner functions can be extended to the case ρ ≥ 1 (recall that ρ = ν). Consider the general case where the left generator is given by Λ and R, see [START_REF] Athanasios | Approximation of large-scale dynamical systems[END_REF], and the right generator is given by M and L, see [START_REF] Athanasios C Antoulas | Interpolatory model reduction of large-scale dynamical systems[END_REF]. Up to a similarity transformation, the left generator can be defined by a diagonal matrix Λ, with distinct coefficients λ 1 , . . . , λ ρ on the imaginary axis and R = [R 1 . . . R ρ ]. The same goes for the right generator, and the coefficients of M are denoted µ 1 , . . . , µ ρ and

Theorem 5 For Λ ∈ C ρ×ρ , R T ∈ C ρ , with ρ ≥ 1, the tangential generalized controllability function X is defined by

where, for i = 1 . . . ρ, ζ r i denotes the i-th component of ζ r , and the functions X (i) are defined as in the first-order generator case as follows:

with the coefficients Φ (i) k defined by

The tangential generalized controllability function X from (48) solves (4).

Proof: First, as X (i) (0) = 0 for all i = 1 . . . ρ, we have X(0) = 0. In addition, from Theorem 2, we have, for all i = 1 . . . ρ, that: