Laura Odasso
email: laura.odasso@college-de-france.fr

Family Rights-Claiming as Act of Citizenship: An Intersectional Perspective on the Performance of Intimate Citizenship

Keywords: intimate citizenship, intermarriages, performativity, rights claims, intersectionality, non-governmental organisations

Ethnographic and biographical research conducted with mixed-status couples and nongovernmental organisations in France and Belgium provides insights into how the citizen partners of mixed-status relationships define and assert their family rights. In response to injustices suffered, from the state or from the migrant (non-citizen) partner, these citizens turned to organisations with contrasting discourses on marriage migration. These organisations encouraged them to participate in collective actions, and to give voice to their intimate experiences. Drawing on accounts of 'intimate citizenship', this article explores the citizenship-belonging nexus through lenses of performativity and intersectionality. Gender and ethnicity interact to influence interactions between citizen partners and the state, the tension between their virtual and actual social identities, andultimately - their assertions of citizenship, with personal status underpinning public claims. By speaking and acting in the name of their private lives and choices, these citizen partners affirm their (intimate) citizenship through its public performance.

Introduction

Since the mid-1970s, family and private ties have constituted the principal route for migration to France and Belgium. Due to the increasing number of residence permits issued to migrants from the Global South on this basis, since the 1990s both states have tightened the relevant legislation and procedures, to discourage this form of migration and the entrance of foreigners considered 'undesirable'. During the 2000s, this trend evolved in a socio-political environment that distinguished between 'desirable' (i.e., highly qualified) and 'undesirable' (i.e., dependent family members, asylum seekers) migrants. Restrictions on family migration had a marked impact on mixed-status relationships, i.e. intimate relationships between citizens and migrants. Marriage to a citizen in such circumstances was turned into 'an object of governmental intervention and a site of discursive proliferation' (Martin 2012, 867). For its part, the state framed such relationships as a shortcut, via the rights of the citizen partner, to legal residency. Beyond this, it has also been claimed that marriages between citizens with migration backgrounds and non-citizen partners from their country of origin reinforced 'ethnic separatism'.

Whilst technically binational, these unions are generally perceived as homogamous in their essence. Both states used these constructs as justification for shaping legal instruments intended to prevent fraudulent and forced marital unions - and to preserve a distinct conception of 'proper' intermarriage.

In their management of mass migration, France and Belgium had already institutionalised differential treatment for migrants [START_REF] Sayad | Immigration et 'pensée d'État[END_REF]; but these new measures, directed at mixed-status couples, went further by imposing the duty to satisfy migration law requirements on their own citizensan ostensibly neutral requirement, but in practice creating a 'politics of belonging [START_REF] Davis | Belonging and the Policies of Belonging[END_REF]. These measures mirror particular conceptualisations of social cohesion and membership that have specific implications for nation building, and the belonging of citizen partners in such relationships [START_REF] Turner | Citizenship, Reproduction and the State: International Marriage and Human Rights[END_REF]. By governing mixed-status family-making practices, both states adjusted the contours of the national community, and stratified their citizenry.

Relatively little is known of the experience of de facto citizen partners vis-à-vis such restrictions, imposed due to their intimate choices. This article addresses this gap by assessing this citizenship-belonging nexus from an intersectional perspective. It presents the narratives of the citizen partners in mixed-status relationships, as participants in the activities of non-governmental organisations (NGO) advocating for the rights of binational couples. By providing legal advice and social support, such organisations support the citizen partner in asserting their rights. Thus, they respond to immigration regimes by 'perform[ing] their right to have rights [and] by asking questions about justice and injustice' (Isin 2013, 22). I argue that the characteristics of the 'performer' matters: gender, ethnicity and class all shape personal and institutional experiences, as well as individual and collective claims. These attributes and their intersections are not relevant per se, but do inform the stereotypes that underpin interactions with the immigration apparatus. The credit and the discredit experienced via immigration procedures, together with gendered and racialised political and media discourses on family migration, impact their agency (Odasso 2016a) and mould their 'performative acts of citizenship' [START_REF] Isin | Theorizing Acts of Citizenship[END_REF]. The recognition of the tension between the virtual and actual social identities of these citizens - imposed by another or proven to be possessed, respectively [START_REF] Goffman | Stigma[END_REF]) -nuances the presupposition that citizens enjoy the benefits of their civic rights more successfully than non-citizens.

In what follows, I first establish the theoretical framework underpinning my conceptualisation of citizenship as a performative practice. I then outline the contextual background for the four NGOs discussed in the study. I then explore archetypical citizen partner narratives of immigration-related experiences, and the participation of these citizen partners in the collective activities promoted by these NGOs. An intersectional analysis informed this process, highlighting the place of gender and ethnicity in the interactions between these citizens with the state - and consequently, in their acts of citizenship and belonging.

Rights claims and performative acts of citizenship

The concept of citizenship covers status, identity and practices.

Citizenship, as a status, is defined as a set of reciprocal civic, political, and social rights and responsibilities existing between a nation state and its citizens [START_REF] Marshall | Citizenship and Social Class and Other Essays[END_REF].

Even if conceived as universal, this conception of citizenship still legitimises inequalities.

In its earliest iterations, the status of the citizen was limited to adult malesreinforcing gender inequalities in intermarriages [START_REF] Guerry | Married Women's Nationality in the International Context (1918-1935)[END_REF][START_REF] De Hart | Regulating Mixed Marriages through Acquisition and Loss of Citizenship[END_REF]. More recently, neoliberal policies have underscored inequities anchored by social class, mass migration forcing a re-evaluation of the rights owed to non-national residents. Nevertheless, for some, the way of thinking about 'what citizens have in common […] and that law and rules are blind to individual and group difference' (Young 1989, 250), cannot lead to genuine inclusion. Thus, Young conceptualised 'differentiated citizenship' as a pathway for protecting the rights of full citizenship for all social categories, specifically marginalised and oppressed groups.

Citizenship as identity consists of the behaviours and feelings nurtured by membership of a collective. From this perspective, [START_REF] Joppke | Transformation of Citizenship: Status, Rights, Identity[END_REF] theorised citizenship as a combination of status, rights and identity. This far, citizenship remains a state-related concept, pertaining to the individual in the public sphere. However, recent literature has explored multilayered conceptualisations of citizenship, disrupting the public-private divide that characterised the classic paradigm of citizenship.

The private sphere - namely the 'intimate' and the 'sexual' - also underpins citizenship. The notion of 'intimate citizenship' [START_REF] Lister | Sexual Citizenship[END_REF][START_REF] Plummer | Intimate Citizenship. Private decisions and public dialogues[END_REF]) positions issues of gender, family choices, sexual autonomy and reproduction at the heart of citizenship as an embodied and inclusive practice (Kabeer 2005). It should be noted that 'intimate citizenship is not to be confused with intimacy itself: it concerns public talks and action about the intimate' (Lister 2002, 199).

Citizenship, as I will show in the following, is thus also a practice: it is more what individuals do than what individuals have. This understanding feeds into the capacity of individuals to act, either to change power relations or to maintain the status quo.

Three arguments can be adduced in support of this statement.

First, the intrusion of migration law into the family life of mixed-status couples reveals new perspectives on the notion of the 'alien', highlighting the fragility of the insider-outsider dichotomy in citizenship [START_REF] Bosniak | The Citizen and the Alien. Dilemmas of Contemporary Membership[END_REF]. In fact, the path towards legitimising the status of the migrant partner can have a profound effect on the citizen partner (Odasso 2016a). For the latter, having previously enjoyed the de jure rights of citizenship, intimate choices reconfigure expectations via the context of encounters with migration law. Family migration policies weaken presumptions of citizenship status and force the reconsideration of one's belonging. This combination of factors cannot be interpreted in terms of citizenship as 'status'. It is one's intimate choices that inform deeper appreciation of citizenship through the nexus between personal experiences, legal consciousness -'the ways people understand and use the law' along their experiences (Merry 1990, 5) - and identity changes, namely how a person's sense of is shaped by law and bureaucracy.

Second, even within the framework of the European Union, member states retain control over granting the rights of residence and citizenship. Studies analysing the enforcement of marriage migration policies have shown that decision-making processes are situated and arbitrary. The encounters with the 'state' - whether the human face of migration processes (i.e., state officers, policemen) or the non-human one (i.e., laws, administrative documents) - are steeped in the myth of an 'imagined sameness' [START_REF] Lavanchy | L'amour à l'état civil. Des régulations institutionnelles de l'intimité à la fabrique de la ressemblance nationale en Suisse[END_REF], coupled with paternalistic and gendered attitudes [START_REF] Maskens | Screening for Romance and Compatibility in Brussels's Civil Registrar Office: Practical Norms of Bureaucratic Feminism[END_REF][START_REF] Odasso | Le genre de l'engagement pour la défense du droit à la vie familiale en France[END_REF]. While these administrative hurdles affect all citizen partners, it seems that the processes specifically drag women back to the primordial role of reproducing the nation and, preferably, maintaining its purity (Yuval [START_REF] Davis | Gender & Nation[END_REF]. Claims of couplehood are tested by interviews, home visits, the minute examination of personal documentation; ultimately, decisions are based on personal appraisal and emotion, such as love (D'Aoust 2013). A subjective conceptualisation of 'true' romantic love intersects with other representations of the partners based on migration background and presumed identities.

In this manner, moral and civic ideologies become entangled within daily considerations of access to the nation [START_REF] Eggebø | With a Heavy Heart': Ethics, Emotions and Rationality in Norwegian Immigration Administration[END_REF][START_REF] Longo | Keeping it in 'the Family': How Gender Norms Shape U.S. Marriage Migration Politics[END_REF]. In identifying fake relationships, state agents define 'informal hierarchies of desirability that result in the differential treatment of applicants across intersecting lines of class, race, gender and so forth' (Scheel and Gutekunst 2019, 859). Through boundary maintenance work between the citizen and the alien, migration politics frame gendered and racialised values that define individuals' social position, and their acceptability to the national communities [START_REF] Davis | Belonging and the Policies of Belonging[END_REF]. This practice does not weigh on the migrant partner alone; it places a burden on citizen partners, reifying suppositions of the 'good' family and 'good' citizens [START_REF] Bonjour | A Proper Wife, a Proper Marriage: Constructions of "Us" and "Them" in Dutch Family Migration Policy[END_REF]. In addition, the socio-political context (i.e., the rise of the political Right) contributes to the negative experience of mixed-status couples. These differ according to the race, gender, age and class [START_REF] Crenshaw | Mapping the Margins: Intersectionality, Identity Politics, and Violence against Women of Color[END_REF] of the partners, revealing the limits of one's status of citizen. Third, actions informed by the understanding of one's entitlement to fundamental rights by virtue of citizenship shape the assertion of such rights (Kabeer 2005, 31); a power results from recognition of the private and intimate as a public and political issue.

The NGOs discussed in this paper have become the 'town squares' for couples to present their bureaucratic-legal experiences to the wider world. They acquire awareness of their rights and develop a legal consciousness and a 'repertoire of actions' -'a set of means […] for making claims of different types on different individuals' (Tilly 1986, 2). Thus, they learn to articulate in the public sphere statements of grievances on the basis of their 'intimacy' - which thus becomes a matter of citizenship [START_REF] Lister | Sexual Citizenship[END_REF]. When individuals struggle to assert their rights or the rights of other, they 'constitute themselves as citizens […], transform themselves and others, the rights under which they make claims, and the rights to which they make claims' (Isin 2017, 507-508). This transformative potential of asserting one's rights is observed here from the perspective of acts of citizenship conceived by the citizen partner within the framework of an NGO, or with its support, as a performance of intimate citizenship. This performative approach to rights focuses on 'what rights do' (Zivi 2012, 9) - going beyond the formal definition of rights as things owned, and engaging with them as social and political practices - echoing the definition of citizenship embraced here (supra 'what individuals do rather than what individuals have'). Such acts of citizenship are 'those acts through which citizens, strangers, outsiders and aliens emerge not as being already defined but as beings acting and reacting with others' (Isin 2008, 39).

Doing ethnography and biographies

The empirical material that informs this article is drawn from three research studies (2010-2018) which explored socio-institutional representations of mixed-status couples, their experiences of migration procedures, and their interactions with the nongovernmental actors from whom they sought support. The studies were conducted in urban areas in Belgium (2014Belgium ( -2017) ) and France (2010France ( -2018)). Within the framework of a broader, multi-sited ethnography, I conducted observant participations with four NGOs, collecting life histories of one hundred couples recruited by snowballing sampling. Some of the respondents were participants in the activities of these NGOs, which had helped strengthen their advocacy skills. The NGOswith, it should be noted, contrasting approaches to marriage migrationare Amoureux au Ban Public (ABP) and Association Nationale des Victimes de l'Insécurité, Mariages Gris (ANVI) in France; and Amoureux, vos Papiers ! (AVP) and Coeurs Piégés (CP) in Belgium. Data on the aims, structures, and advocacy-building capacities of the organisations were drawn from their websites, interviews with coordinators 1 and volunteers (in mixed relationships and not), and participant observations. While I was able to establish in-depth engagement with ABP and AVP, with ANVI and CP I was only able to observe limited meetings and public activities (i.e., parliamentary hearings), as these NGOs functioned principally through online and private contacts. Nevertheless, members of both associations were happy to share their respective histories. Observant participation in their activities made it possible to build relationships of trust with members. I recorded in-depth life histories of ten members of CP and ANVI, and twenty members of ABP and AVPtogether with shorter ad hoc interviews during observations. The interviews were conducted in private locations (i.e., respondents' homes, researcher's office), more rarely in public ones (i.e., café, associations' offices). An initial question ('How did you join this organisation?') allowed the respondents to narrate their personal history (i.e., affective choices, family background and social information), detail encounters with the bureaucracy of migration, and expand on their motives for their activism. So as to trace their histories over time, I met with some of the respondents more than once. Narratives were analysed using the 'biographical policy evaluation' method [START_REF] Apitzsch | The Method of Biographical Policies Evaluation[END_REF]; I sought to identify action schemes, institutional expectations, manifestations of loss of control over life plans (i.e.: related to relationship, administrative encounters), and the pivotal events that prompted their activism and the identity changes related. Furthermore, a historical and temporal analysis helped overlay personal trajectories with the contextual local and national socio-political dynamics and power relations. My profile - a female foreign national from a southern European country of migration, and a fluent French speaker, with a good contextual knowledge of the countries of origin of the migrant partners - afforded me a privileged position for developing empathy with respondents.

All of the organisations had more online followers (several hundred per organisation) than regular activists (a few dozen per organisation, many only sporadically active). A majority were citizens; nearly all of these autochthonous women [START_REF] Odasso | Le genre de l'engagement pour la défense du droit à la vie familiale en France[END_REF]. Thus, two-thirds of my respondents were women in their thirties and forties, mainly of middle-socioeconomic profiles, and in relationships with men from North Africa, the Middle East and sub-Saharan Africa. This sample has ramifications, in that it reflected societal and political discourses, racialised and gendered preconceptions, and colonial and ancient migration legacies (Odasso 2016b) - a reminder that migration policies are not neutral.

Migration policies as contexts for contrasting ideologies on mixed-status families

The evolution and enforcement of marriage migration policies informed the emergence of the NGOs discussed here. In France and Belgium, marriage no longer confers automatic residency rights on the migrant partner. Marriage procedures are subjected to a range of controls. Municipality and embassy officers - the initial points of contact for processing marriage and civil partnership applications - may commission a police investigation to confirm the veracity of the union. Specific additional requirements, assessed over a period of time, must be fulfilled before the right to residency is granted. This time span is presently five years in Belgium, and four in France. During this period, the claimant is dependent on the citizen spouse; marital breakdown will terminate any right to residency. In France, the initial residence permit must be renewed periodically, the applicant required to prove (with fresh documentation in each instance) continued shared family life; in Belgium, the initial permit is valid for five years, but may be revoked on the basis of checks carried out collaboratively by immigration authorities and the social security system 2 . At the end of this period, eligible spouses may apply for nationality. This procedure actualises verification of the mixed-union relationship, confirming the precarious position of foreign nationals even when married to a citizen.

Besides these basic commonalities, the countries differ with regard to the status of the citizen partner in family migration law. In France, citizens retain some degreealbeit shrinking - of privileged access to family rights [START_REF] Slama | Le long Dimanche de fiançailles des couples mixtes[END_REF]. In Belgium, such privileges were withdrawn in 2011. Citizens must comply with strict income 3 and housing requirements, hitherto relevant only to the family reunification of (legal) migrant residents and their spouses. This reform codified the downgraded status of Belgian citizen partners of mixed unions [START_REF] Mascia | Le contrôle du mariage binational en Belgique: acteurs et rationalités en jeu[END_REF].

Nevertheless, the law-in-theory only partially corresponds with the law-in-action.

Beyond the abstraction of the laws themselves, the procedure itself operates as a deterrent.

Partners are at the mercy of documentation deemed unsatisfactory and low-level agents.

The street-level clerks of local prefectures (France) and of the municipal foreign office (Belgium) who process applications employ some personal appraisal of couples and their documents; but the decision itselfonce the documentation is acceptedis taken on the basis of documents submitted by other local prefectural officers (France) and the officers of the Office des Étrangers, the federal Immigration Office (Belgium). In France, the multiple prefecture arrangement allows some room for discretion. But, ultimately, both national mechanisms, commonly experienced by applicants as obscure and intangible, underpin legal-bureaucratic deadlocks.

In both countries, the legal restrictionsand their somewhat haphazard implementationare accompanied by a political discourse averring 'correct' family circumstances (i.e., future family plans, long-lasting relationships) and moral arguments against unacceptable forms of marital union (i.e., forced and arranged marriages). By implying that fraudulent intent is often concealed behind social and affective relationships, these arguments have had a deleterious effect on representations of mixedstatus marriages. The migrant spouse is portrayed as a danger to the national community, its ideals and its values. In 2009, France's Ministry of Immigration adopted the term 'grey marriage' (marriage gris) 4 to warn against migrants taking advantage of unsuspecting citizens in order to secure residency and the other social benefits that accrue through marriage. This concept has similarly gained traction in Belgium, and the migration paths of foreign partners subjected to greater state scrutiny. This set of circumstances can also explain why some citizen partners seek legal advice and human support from ABP and AVP to resolve problems emerging from migration policiesand why others turn to ANVI and CP to unpick marriage subterfuges.

Autochthonous women and state paternalism

In France, following a 2006 legislative reform conceived to block fraudulent unions institutionalising systematic checks on marriages, a Montpellier-based lawyer linked to an established association for the defence of migrants' rights identified a growing population in need of advice and support. Some couples had an 'irregular' migrant partner; some experienced increased scrutiny due to the significant age gap between the parties; others experienced difficulties because the citizen spouse's migration origins were the same as that of the future partner. Some migrant partners had already been ordered to leave French territory, and others had had residency applications rejected.

After assessing the needs of this 'first' wave of couples, he founded the ABP in 2007, to mobilise French-migrant couples into defending their rights as a group. With time, parallel groups were established across France, volunteers and members of mixed-status couples offering guidance to couples in need. ABP also invited couples impacted by the legislative regime to participate in collective action lobbying activities, to press for policy changes and to influence public opinion.

Alice, a 34-year-old saleswoman, embraced the ideologies of ABP from the start. Over the course of several interviews, first at the French consulate then at the police headquarters of her home city in France, Alice realised that French citizenship was not enough to protect her from the stigma created by her intimate choice to marry a Moroccan. Furthermore, whiteness did not count: the French apparatus was racist and sexist, and not only towards ethnic minorities, as she had believed previously. From an upper middleclass family, her social, cultural and economic capital did help negotiate the situation, but did not spare her the cumbersome interactions with the state. One bureaucrat remarked 'that I was very thin for Moroccans, who like shapely women!'; another, a police officer, telephoned her father to check if he had consented to the marriage - even though Alice was above the age of majority. She explained that the migration administration: try to make people crack with all means at hand […] for French-foreigner couples, they think that the French [citizen] is an idiot who was duped by the foreigner [laugh]; when the French [citizen] is of foreign origins, instead, they suspect an arranged fake marriage… I spent hours explaining to them that I am legitimate, and I [want to] marry a Moroccan, who, moreover, does not need to immigrate.

Alice was obliged to defend her choices proving repeatedly that she was a 'good' citizen.

They were too young; their relationship had not endured long enough; she was too thin and too delicate to fit 'archetypal' Moroccan tastes: these prejudices marked her interactions with the state. But, in order to contract the marriage and secure Mourad's visa, she had to confront them. Obliged to narrate her history to migration agents repeatedly, Alice was confronted on account of her affective choices, honesty towards the state and personal autonomy. The police 'advised' her that by marrying a Moroccan, she risked losing her independence; as far as the mechanism of migration enforcement was concerned, Alice was incapable of making valid decisions concerning her own family life and thus required the paternalistic protection of the state. Administrative incongruences, repeated over time, coalesce into the stigmatised representations of citizens. The rights and autonomy that accrue to citizen women are, in the eyes of the bureaucracy, placed at risk by marriage to an Arab man; because they are responsible for producing the next generation of citizens, women's choices must be controlled. In parallel, citizenship and whiteness are not guaranteed enough that they are not accomplices to (marriage) fraud. The mechanism of migration enforcement impacts their citizenship status, by framing them as 'suspicious insiders'. Considered alternatively a 'criminal for love' [START_REF] Ferran | La politique d'immigration contre les couples mixtes[END_REF]) and a 'second-class citizen' by the state (Odasso 2016a), she was obliged to defend, publicly, her intimate choice. It took almost ten months of bureaucratic inquiry before Mourad was allowed to travel to France. 'I came for her, not thanks to her,' he emphasised. In common with many of my encounters with migrant spouses during my fieldwork, he preferred not to dwell too much on that unpleasant period, or his continuing precarious status. Instead, by making her history visible and sharing the legal consciousness she had acquired, Alice accomplished an act of citizenship, emerging from her loss of trust in French state values. As for other women who I encountered through ABP in France (and in AVP in Belgium), the unexpected bureaucratic treatments triggered a common sentiment of outrage and injustice and inspired activist activity to assert their rights as legal citizens. This also affected their relationship to their state, and reshaped their belonging. Given their first-hand experiences, they could no longer subscribe to the promise of equality for all citizens, given their first-hand experiences, and now understood that their own state could be sexist and racist. This ideological frame, elaborated through collective action, created a path for overcoming the peculiarity of their situation by generalising their rights claim with those of all - citizens and foreign nationals - with similar experiences. Arab nationals experience significant stigmatisation in both France and Belgium, due to the stereotypical representations of their culture and religion, underlined by presumed conception of gender, family relations, and loyalty (Odasso 2016a). But, as already filtered through Alice's archetypical story, it is not just gender and nationality that shape the state's approach to mixed-status unions and changes to conceptualisations of citizenship.

Ethnicity is not diluted by citizenship

Operating along similar lines to ABP, AVP was founded in Belgium in July 2015, by a group of women with migrant partners - all held in Belgian detention centres following the rejection of their marriage applications. The women formed an alliance with volunteers protesting the detention and deportation of migrants, working to strengthen the legal consciousness of mixed-status couples and to lobby for changes in marriage migration policies and ground-level practices impacting such couples. Collective actions directed at local authorities and intended to raise public awareness about the mistreatment of mixed-status couples are convened for prominent dates such as Valentine's Day and Loving Day (12 th June - a day celebrating interracial relationships). 5 AVP's repertoire of collective action is similar to ABP's; the narratives of couples in love, fighting against bureaucratic discretion, are similarly pivotal for AVP's cause.

Mounia was one of the first members of AVP. A 43-year-old Belgian hairdresser, born to Moroccan parents and divorced with three children, Mounia submitted an application to marry Youssef, an undocumented Moroccan national, to her municipality in June 2015. They had met in 2014 and lived together subsequently. Mounia's kids considered Youssef as their father. But family life was interrupted when, without advance notification of the rejection of their application, police raided the family home and arrested Youssef. The municipality, after determining doubts concerning the application due to an order of leave the territory still pending on Youssef, had flagged up their place of residence to immigration enforcement officials. In short, their steps to regularise their union and Youssef's administrative status facilitated this ad hoc intervention.

During this period, Belgium was contending with the arrival of large numbers of refugees from the Middle East seeking asylum and protection. Describing the situation as an 'emergency', Belgium reinforced its border controls; and irregular migrants across the territory became the principal target of the migration apparatus. This intensification of deportations of such population presented indicative signals of the intent to maintain national security and public order, in a country where right-wing anti-immigrant parties were - and are - gaining increased prominence. In line with this political orientation, municipal authorities used ad hoc instruments like police raids and street controls to enforce their agenda. Mounia reported that From the municipality, our application was referred to the police, who interrogated us to verify that we know each other, and we have a common life, and even about our sex life! Can you imagine? A few days later, they arrived, armed, at our home at around half past five in the morning. We were together, pretty good evidence of common family life [laughs], but not for them! My daughter is still in shock from that event! Shocked by the lack of respect demonstrated by the Belgian authorities who invaded her intimacy, Mounia came to understand through her engagement with the first activists of AVP that such feelings could present a good point for strengthening private personal claims by turning them into public claims. She agreed to narrate her history during the very first public meetings of AVP, together with a jurist who contextualised the rights of the individual with regards to threats of police action. In articulating what she described as unwarranted intrusion by the state in her intimate life, Mounia sought to encourage others faced with similar circumstances to similarly speak out. Accordingly, many other people in the audiences presented similar experiences. Mounia, by doing this, performed an act of re-evaluation upon her Belgian citizenship that had been mistreated by the administration. AVP, like ABP, campaigns for respect for the rule of law by the servants of the state; Mounia embraced this fight against differentiated treatment in encounters with the state.

Thanks to a lawyer recommended by AVP's network, Youssef was released and the couple were finally able to celebrate their marriage. But a few months later, the Belgian Immigration Office rejected Youssef's residence permit application. As a single mother working part-time, Mounia could not satisfy the income threshold required to sponsor Youssef's application. Youssef himself was employed, but the letter of the law deemed his income ineligible for consideration until after three years of marriage. The harsh words of the municipality clerk who notified them of the rejection reinforced the sense of differentiated treatment. 'Madam,' he snapped, 'you should have found a Belgian husband rather than a Moroccan.' Mounia understood that her citizenship had been doubly devalued by her mixedstatus marriage, and further impacted by a political discourse of supposed ethnic separatism fuelled by the intra-community marriage practices of Moroccans and Turks.

Her choice to marry a Moroccan revealed the tribal stigma hitherto obscured by her legal citizenship, namely her origin. Her new awareness of the tensions between Belgian belonging, immigrant origins, and class-and-gendered economic precariousness prompted her to redefine her citizenship through activism. To achieve her objectives, she tacitly consented to the state's requirements, and took a full-time employment; the couple reapplied, and in due course secured Youssef's residency rights. Nevertheless, her experience of migration law in action reshaped her conceptualisation of citizenship and belonging.

The 'victim' or 'repentant' seeking state protection

For members of ANVI and CP, it was the mixed-status union itself that constituted the negative experience. Immigration fraud, for migratory and/or financial purposes, exemplifies the 'social insecurity' underlying one strand of ANVI's activism. Established in France in 1999 to fight social insecurity at large, the NGO claims that foreign nationals target potential citizen partners in order to benefit from the rights bestowed by such unions. ANVI's objective is to challenge the 'scourge of fraudulent marriages', thus protecting citizens and 'good' mixed-status marriages. Volunteers and two employees (both with personal experience of the association's raison d'être) support mainly French nationals in gaining redress for the moral and material damage suffered through such unions. ANVI describes its target audience during public events as 'victims', underlining the serious consequences of marriage subterfuge on the affected partner. In 2011, 'grey' marriages were declared illegal under French law, creating a legal remedy for wronged partners seeking to overcome the unexpected consequences of their unions - and, as an ancillary benefit, to ensure that the errant partner was sanctioned by law.

Annette, a 42-year-old white French teacher, was in economic and emotional distress caused by the breakdown of her marriage to Mamadou - a 38-year-old black Senegalese national - when she encountered ANVI. She said, I felt abused when he disappeared after four years and a baby together. Before meeting ANVI, I was not aware that I have the right to be protected by French law in this situation. It has been a long legal process to annul the marriage, but now I feel better. Even if it is still emotionally hard, I take part in public debates, to narrate my story and help people in similar situations to find the courage to claim their rights.

In practice, the law extends penalties intended for unions of convenience to situations where the migrant who contracted the marriage concealed his intentions from the spouse.

Annette identified with this specific situation. She described how Mamadou had been nice and their relationship perfect until their baby was born. The couple had completed standard bureaucratic procedures before their marriage - Mamadou's status was irregular at that time - without major incident. He obtained an initial residence permit, which could be renewed twice. The couple then discovered that they were going to have a baby.

Excited, Mamadou suggested that he formally acknowledge paternity of the baby at the earliest possible legal opportunity - the seventh month of pregnancy. 'I was not aware of the legal opportunities that this act opened for him,' Annette remarked. After this but before the birth, Mamadou disappeared from Annette life. She fell into a deep depression, which had a profound impact on her pregnancy and subsequent motherhood. Thanks to ANVI, she was able to find a means of articulating the intimate deception inflicted upon her and, in line with ANVI's ideology, to assert her rights as a victim. She sought material and emotional compensation for what she had suffered: the experience had convinced her that the 'relationship' had been nothing but a pre-conceived administrative swindle. She joined ANVI in order to warn other citizens of the stratagem deployed by non-citizens to defeat the migration regime, and to clarify the legal implications of her situation. Annette believed that this topic is uncomfortable! It is still easier for a swindler to enter the nation than for us to denounce frauds. People think that I am a weak woman; situations such as mine are not about love, but about public and national security, for all citizens.

During the lengthy process that ensued, the authorities alternated between treating her with the compassion due an unfortunate lover, and - on occasions - the disdain reserved for a weak woman fooled by a migrant. As with Alice and Mounia, the state implicitly asserted its paternalistic role here, albeit in a different form: Annette, in fact, is a 'repentant'. Marrying out of the national collective ended badly, and so she needed forgiveness from the state for her 'error', stressing its role of paternal protector. Annette's experience, among others, is used by ANVI to demonstrate the softness of French migration policies towards non-citizen partners and to call for better state intervention.

Between these virtual social identities, she experienced changes to her understanding of what it meant to be a citizen. Consequently, she emphasised her membership by presenting as an autochthonous French national interesting in protecting other nationals while seeking intimate justice.

Not all 'victims' are equal

Following the example of the ANVI, CP was founded in Belgium in 2012 to support the victims of fraudulent unions. The founder - a former 'victim' herselflobbied for increased awareness of the impact of the phenomenon on citizens. CP collaborated with immigration enforcement authorities, mainly the police, to identify marriage fraud patterns and to support victims of such fraud. Aligned with political challenges to so-called 'marriages of convenience', the organisation succeeded in garnering media and political visibility (i.e. parliamentary interventions, television reports), and mobilised 'victims' (at the beginning, slightly more men than women) and lawyers into addressing contentious cases of marriage annulment. Both organisations campaigned for the simplification of legal procedures, and for stiffer sanctions for the fraudulent 'spouse'.

Mounir, a 35-year-old Belgian taxi driver born to Moroccan parents, came across CP while in a desperate frame of mind. He succeeded in annulling his marriage with Samia, a 24-year-old Moroccan woman, with the support of the organisation. He admitted My romantic expectations were not met. Samia lied and used me. I wanted to go the whole hog and to secure justice. But it was hard. Belgian institutions do not easily believe a citizen of Arab origin denouncing a scam marriage, and they accused me of violence and… I spent months with police, lawyers and judges.

He met Samia during a vacation in Morocco. Attracted to one another, after some back and forth they decided to marryfacilitating Samia's path to residency in Belgium. The formalities went smoothly; Mounir easily satisfied family reunification requirements. But after a few months of cohabitation, marital tensions began to emerge. 'She was never happy nor satisfied,' Mounir stated, saying that he began to notice major changes in Samia's behaviour after she obtained her residence permit. She was no longer interested in him, trivial disagreements provoking fierce arguments. Matters went from bad to worse after Samia became pregnant. After accusing him of domestic violence, she left the family home. 6 Later, after the birth of the baby, Mounir discovered that she was working in a nightclub and abusing drugs, endangering the baby's safety. Even though he had implicitly been tainted by the stereotype of the violent Arab male, the allegations of domestic violence were never objectively proved. With the benefit to hindsight, Mounir now believes that Samia was merely seeking any possible route to securing a residency permit. He realised this when he started to participate in CP's collective actions. 'To exchange stories with people in my situation was very good for me; CP helped me articulate my genuine claims during this personal and legal nightmare,' he explained.

Mounir agreed to present his intimate history at public events. His case presented as a contrast to the engrained presumption that 'grey' marriages only affected white women, and thus justified CP's universal discourse on abuses - occurring regardless of gender, nationality, ethnicity or social class. Although Mounir ultimately obtained the sought-for protection of the Belgian state, he admitted that during the lengthy process he felt discredited, due to his Arab origins. When I first met Mounir, he had just presented his story to the French-Belgian Parliament in Brussels. I was struck by a question asked by a politician in the audience: 'But why did you marry a Moroccan woman?' - with the implied codicil, 'and not a Belgian one?' Male foreign partners, Arabs and Africans especially, are the principal target of political and institutional suspicion (Charsley and Wray 2015); but here Mounir found himself at the crossroads linking many stigma and attributed identities. On one hand, in common with citizens of migrant backgrounds in homogamous marriages, he was regarded with suspicion by Belgian authorities worried about ethnic separatism. But on the other hand, associative actors like CP were able to use his ethnicity and sex to portray an (a)typical victim, for lobbying purposes. His marital and legal experiences, and the engagement with CP, led Mounir to generate a specific consciousness of his positioning among Belgian citizens, and from this to reshape his belonging.

Conclusion

The emblematic narratives of citizen partners involved with two divergent types of NGOs, in two national contexts, presents a solid perspective for exploring the citizenship-belonging nexus as a performative practice, through the lens of intersectionality. Even though France's family migration policy appears more liberal than Belgium's, the policing of the worthiness of family has diminished the rights of citizens in both countries. Social and bureaucratic practices and discourses regarding mixed-status unions converge to similar effect, informing the emergence of similar NGOs. Motivated by different grievances - whether mistreatment by state agents or by their partners - citizen partners in mixed-status relationships approach the NGO whose ideology and repertoire of actions matches their outlook. The ideological framework they find informs their participation. But, their significant ideological differences aside, both sets of NGOs actively employ the narratives of mixed-status couples to strengthen their lobbying, recognising that personal stories are generally more effective in generating empathy. Consequently, with all the NGOs discussed here, the citizens who participated in their activities introduced intimate life into their activism. By agreeing to speak publicly about their intimate experiences in collective actions, they performed their intimate citizenship. In the case of ABP and AVP, citizens presented themselves as lovers prevented by administrative obstacles from establishing the genuine family life they desired. But with ANVI and CP, they presented as duped lovers: victims of legal fraud and emotional deceit, but nevertheless 'suitably repentant' citizens. Although diametrically opposed, the reformulation of personal and administrative injustices into assertions of rights provided relief after their experiences of private frustration and institutional disqualification. This process impacted the actual selves of the citizens, who produced 'new subject of law and action, new subjectivities and identities, new site of struggle and new scale of identification' (Isin 2008, 16).

The narratives all emphasise how the citizenship-belonging experience and the attribution of identity framed the citizen partners' ideal vision of their state. Discovering the weak boundary between intimacy and state intervention informed a process of awareness, which guided them in discovering the limits of legal citizenship as a status. In this context, their affective choices as citizens became a powerful tool with which to defend their rights. Notwithstanding the origin of the injustice suffered or the ideology embraced by the specific NGO, the intersectional analysis reveals unequal experiences of practical intimate citizenship, and a common transversal transformation among partners.

In relation to family rights and the nation, gender and ethnicity define specific dimensions of (mis)trust in both the embodied and symbolic visages of the state. Autochthonous white women and citizens with migration backgrounds, whether men and women, experience similar problems: stigmatisation by the apparatus of the state apparatus, which had eroded the boundaries between their intimacy and their supposed universal citizenship. Treated as second-class citizens due to their partner choices, whether seeking the annulment of a marriage or actually marrying to secure the right of residence, they similarly must prove that they deserve the rights of citizenship that should accrue to them despite their intimate choices.

Following

  their script for action, she established a local group to support other couples, campaigned at the local level, and established a local media presence. Her story is an archetypical example of the experiences of ABP couples' experiences. She married Mourad in Morocco in 2006, against a backdrop of increasing legal controls over marriages contracted abroad. At this point in time, such marriages would only be considered as valid in France after issuance of a certificate 'of legal capacity to marry' (certificate de capacité à marriage) - to be produced before the marriage. Consular authorities enforced this procedure by verifying a genuine intention to start family life together. These reforms were put in place a few months after the 'banlieues crisis' of October 2005, citizens of migrant background - mainly from Africa - protesting police violence and segregation across the inner cities of France. Conflating several issues, politicians asserted the need to sanction the perpetuators of the violence - thus justifying the new immigration regime. The implementation, at much the same time, of an aggressive regime of expulsions, linked to strict controls on the entrance of non-selected migration, had a marked impacted on mixed-status couples too. Alice and Mourad had been together for a few months when they sought permission to contract their marriage; the migration apparatus identified their union as a sham, intended to facilitate Mourad's legal status in France. Alice stated: I'm French, I've never been particularly proud of it, but now I am really ashamed! I grew up with the Déclaration des Droits de l'Homme [the 1789 Declaration of the Rights of Man and of the Citizen]; but gradually, I amended my idyllic vision in the light of colonialism, Vichy regime, and witnessing racism against migrant background friends around me. But, by marrying Mourad, it was against me, a French white citizen!'

Acknowledgements

I would like to thank Saskia Bonjour and Betty de Hart for their guidance, the two anonymous reviewers and the editors who generously commented on this article, and Akin Ajayi for his proofreading. Special thanks are due to the many interviewees whose narratives and experiences informed this article.

Funding:

This work was supported by the FP7-PEOPLE-2013-IEF Marie Curie Action, project AMORE under grant 625364; and by the French Collaborative Institute on Migration, coordinated by the CNRS under the reference ANR-17-CONV-0001.

Oxford University Press.

Notes

1 Five for APB (across the fieldwork period), two for AVP, one for ANVI, and one for CP. The coordinators were white citizens, except CP's, a Black Belgian. All except two (ANVI and the first ABP coordinator) were women.

2 Following the 2016 France reform, a plurennial residence permit is issued, at the discretion of the immigration services, after the second renewal. Immigration authorities are also permitted to perform random checks with utility providers to evidence of ongoing family life. [START_REF] Odasso | Des 'mariages noirs'. Les violences conjugales et le contrôle de la migration familiale en Belgique[END_REF].


M Louédec

L Jaulin
email: lucjaulin@gmail.com

C Viel

Outer enclosures of nonlinear mapping with degenerate ellipsoids ⋆

Keywords: N-dimensional systems, Stability of nonlinear systems, p-invariance, degenerate ellipsoids, guaranteed enclosures, set propagation, hybrid system

The propagation of ellipsoids is a tool that can be used to study the stability and the convergence of non-linear systems with time. Recent works have presented a method to enclose the propagation of ellipsoids via nonlinear functions. While this method is effective in high dimensions, it requires the Jacobian matrix of the function to be invertible. In this paper, we propose to generalize this method to include non-invertible Jacobians, using degenerate ellipsoids. The generalized method is then applied to a nonlinear and/or synchronous hybrid system.

INTRODUCTION

Non-degenerate ellipsoids have been used in a variety of approaches to study nonlinear systems in terms of reachability analysis [START_REF] Kousik | Ellipsotopes: Uniting Ellipsoids and Zonotopes for Reachability Analysis and Fault Detection[END_REF], stability analysis [START_REF] Lian | Stabilization of Switched Linear Systems Subject to Actuator Saturation via Invariant Semiellipsoids[END_REF]; [START_REF] Rauh | Verifying Provable Stability Domains for Discrete-Time Systems Using Ellipsoidal State Enclosures[END_REF] ,and state estimation [START_REF] Althoff | Comparison of guaranteed state estimators for linear time-invariant systems[END_REF]; [START_REF] Valiauga | Set-membership Estimation using Ellipsoidal Ensembles[END_REF]. These approaches often involve the propagation of ellipsoids via nonlinear mappings. Since the result of this propagation is often no longer an ellipsoid, the result is often approximated by inner and outer ellipsoidal enclosures.

These approaches follow previous works such as [START_REF] Polyak | Rejection of Bounded Disturbances via Invariant Ellipsoids Technique[END_REF]; [START_REF] Messerer | An Efficient Algorithm for Tube-based Robust Nonlinear Optimal Control with Optimal Linear Feedback[END_REF] which studied linear or linearized systems where the propagation of ellipsoids follow an affine arithmetic and where stability problems are often equivalent to LMI constraints.

Zonotope set type is an alternative to ellipsoids and is used in some papers such as [START_REF] Kochdumper | Sparse Polynomial Zonotopes: A Novel Set Representation for Reachability Analysis[END_REF]; [START_REF] Rego | Guaranteed methods based on constrained zonotopes for set-valued state estimation of nonlinear discrete-time systems[END_REF]. According to [START_REF] Althoff | Comparison of guaranteed state estimators for linear time-invariant systems[END_REF], for linear time-invariant systems, the zonotopic method has better tightness than ellipsoidal methods but higher computational costs when using set propagation. Thus, ellipsoidal methods are often more relevant to study highdimensional systems.

Several works such as [START_REF] Dompierre | On ellipse intersection and union with application to anisotropic mesh adaptation[END_REF]; [START_REF] Blanchini | Set-Theoretic Methods in Control[END_REF]; [START_REF] Poznyak | Attractive Ellipsoids in Robust Control[END_REF]; [START_REF] Calbert | An Efficient Method to Verify the Inclusion of Ellipsoids[END_REF] have proposed various mathematical tools for operations on ellipsoids. In the recent paper [START_REF] Rauh | A computationally inexpensive algorithm for determining outer and inner enclosures of nonlinear mappings of ellipsoidal domains[END_REF], a numerical guaranteed method has been proposed to compute the ellipsoidal enclosures of a nonlinear mapping. The polynomial complexity of this ⋆ This work has been supported by the French Defence Innovation Agency (AID) and the Brittany Region method allows the study of high-dimensional mappings. This method can also study mapping whose analytical expression is unknown, provided that the Jacobian matrix of the mapping is enclosed in a known box. The main assumption of this method is that the Jacobian matrix is invertible. In [START_REF] Louedec | Computational tractable guaranteed numerical method to study the stability of n-dimensional time-independent nonlinear systems with bounded perturbation[END_REF], this method has been used to prove the existence of positive invariant ellipsoids with respect to n-dimensional time-independent nonlinear systems.

However, some mappings do not correspond to this assumption. The Jacobian matrix can be non-invertible when there are equality constraints. This singularity is common in systems with discrete measurement updates or shock effects. Some papers, such as [START_REF] Balandin | Ellipsoidal reachable sets of linear time-varying continuous and discrete systems in control and estimation problems[END_REF]; Becis-Aubry (2021), have proposed the use of degenerate ellipsoids in the singular case when the system is linear. Degenerate ellipsoids are harder to use in practice because they don't have the quadratic form of non-degenerate matrix because they don't have an invertible shape matrix. Linear methods work well in practice with degenerate ellipsoids because they don't need matrix inversion. However, for non-linear systems, the method of [START_REF] Rauh | A computationally inexpensive algorithm for determining outer and inner enclosures of nonlinear mappings of ellipsoidal domains[END_REF] relies on matrix inversion.

In this paper, we propose to generalize the method of Rauh and Jaulin (2021) by using degenerate ellipsoids to solve the singular case where the Jacobian matrix is not invertible. This generalization is made using the singular value decomposition of the shape matrix and the addition of new singular values when necessary. The performances of the generalized method in terms of computational complexity and pessimism are similar to those of the initial method.

The paper is structured in the following way. The key concepts and the original method are presented in Section 2.

The generalized method is presented in Section 3. Section 4 presents an algorithmic implementation of the method using interval analysis. Finally, Section 5 discusses the performance of the method in applications with degenerate ellipsoids.

NOTATION AND HYPOTHESIS

The theory of the propagation method presented in this paper is a modified version of the method presented in Section 2.1. It relies on degenerate ellipsoids presented in Section 2.2, used to solve the problem presented in Section 2.3.

Initial method

The method provided in [START_REF] Rauh | A computationally inexpensive algorithm for determining outer and inner enclosures of nonlinear mappings of ellipsoidal domains[END_REF] considers the multi-dimensional state space of state vector x ∈ R n , the nonlinear mapping

y = f (x) ,f : R n → R n ,
(1) and the non-degenerate ellipsoids E (µ, Γ ), given by the form

E (µ, Γ ) = x ∈ R n | (x -µ) T Γ -T Γ -1 (x -µ) ≤ 1 ,
(2) with the midpoint µ ∈ R n and the positive definite shape matrix Γ Γ T ≻ 0.

Given the initial domain x ∈ E (µ x , Γ x ), the method can compute an enclosing ellipsoid E µ y , Γ y such that y ∈ E µ y , Γ y using Theorem 1.

Theorem 1. [START_REF] Rauh | A computationally inexpensive algorithm for determining outer and inner enclosures of nonlinear mappings of ellipsoidal domains[END_REF], Theorem 1) Consider the nondegenerate ellipsoid E (µ x , Γ x ). For a differential function (1) with

A = ∂f ∂x (µ x ) (3) invertible, E µ y , Γ y is an outer enclosure of the solution set f (E (µ x , Γ x )) with µ y = f (µ x ) , (4)
Γ y = (1 + ρ) • A • Γ x ,
(5) where

ρ = max ∥ x∥≤1 b ( x) , (6)
and b :R n → R n x → Γ -1 x • A -1 • (f (Γ x x + µ x ) -f (µ x )) -x. (7)

Degenerate ellipsoids

While the ellipsoids are often described by the quadratic form (2), they can also be presented as an affine transformation of the unit sphere, given by

E (µ, Γ ) = {x ∈ R n |∃ x ∈ R n , x = µ + Γ • x, ∥ x∥ ≤ 1} , = µ + Γ • E (0, I n ) .
(8) A degenerate ellipsoid is an ellipsoid whose shape matrix Γ is not invertible. While degenerate ellipsoids can't be described by the quadratic form (2), they can be defined as a singular affine transformation with (8). In the problem presented in Section 2.1, degenerate ellipsoid can appear if the matrix A or Γ x are not invertible.

The general method proposed in this paper uses the singular value decomposition (SVD) of the matrix AΓ x , given by

AΓ x = U • Σ • V T , (9)
with U ∈ R n×n and V ∈ R n×n orthonormal and Σ ∈ R n×n diagonal. The diagonal elements of Σ are the singular values σ i ≥ 0.

For a matrix M , let us define pinv (M ) the pseudo-inverse of M .

Problem formulation

In this paper, we propose to generalize Theorem 1 by considering the case of degenerate ellipsoids, i.e. singular case where the Jacobian matrix is not invertible. Consider the mapping (1), the initial domain x ∈ E (µ x , Γ x ) given by ( 8), and the matrix A given by (3). Assuming Γ x A is noninvertible, the problem is to find an outer enclosure E µ y , Γ y such that y ∈ E µ y , Γ y .

GENERAL METHOD

The modifications of Theorem 1 to solve the problem of Section 2.3 are described in the Theorem 2 and it illustrated in Figure 1. Theorem 2. Consider the non-degenerate ellipsoid

E (µ x , Γ x ).
For a differential function (1) with

A = ∂f ∂x (µ x ) , (10)
E µ y , Γ y is an outer enclosure of the solution set f (E (µ x , Γ x )) with

µ y = f (µ x ) , (11)
Γ y = (1 + ρ) • M , (12)
where

ρ = max ∥ x∥≤1 b ( x) , (13)
and

b :R n → R n x → W • f (Γ x x + µ x ) -µ y -Z x, (14)
and where the matrices M ∈ R n×n , W ∈ R n×n and Z ∈ R n×n are given by

• case 1 (General): if AΓ is invertible, then M = AΓ x , (15)
W = M -1 , (16) Z = I n . ( 17
)
• case 2 (Singularity): if AΓ is non invertible, then

M = U S, (18)
W = pinv (S) U T , ( 19
) Z = W AΓ x , (20)
S = diag (s 1 , s 2 , . . . , s n ) , (21)
where

s i =    σ i if σ i > 0, q i • max x∈E(µ,Γ) e T i U T f (x) -µ y else, (22)
where U , V and σ i are defined from the SVD decomposition (9), the unitary vector e i from the Cartesian base of R n written {e 1 , . . . , e n } and the design parameters q i > 0.

Proof. Case 1 has been studied in [START_REF] Rauh | A computationally inexpensive algorithm for determining outer and inner enclosures of nonlinear mappings of ellipsoidal domains[END_REF]. Let us study case 2.

Let x ∈ E (µ x , Γ x ). From (8), there is x ∈ R n such that ∥ x∥ ≤ 1 and

x = µ x + Γ x • x.
(23) Then, in order to have a vector y which plays the role of x for the point y ∈ E µ y , Γ y , we propose

y = Z x + b ( x) .
(24) Let us show ( 24) is a relevant solution. From ( 14), one has

y = W • f (Γ x x + µ x ) -µ y , ( 25
) (23) = W • f (x) -µ y , (26)
(1)

= W • y -µ y , (27) (19)
= pinv (S) U T • yµ y .

(28) Then, from (18), it follows that

M • y = U Spinv (S) U T y -µ y .
(29) From Appendix (7.1), it can be shown that

Spinv (S) U T y -µ y = U T y -µ y .
(30) Using (30) and since U is orthonormal, (29) can be rewritten as

M • y = U U T y -µ y , (31)
M • y = y -µ y , (32) y = µ y + M • y, (33) y
(12) = µ y + Γ y • 1 1 + ρ • y (34)
Then, from Appendix (7.2), one has ∥Z∥ ≤ 1. (35) Therefore, because ∥ x∥ ≤ 1, one gets ∥Z x∥ ≤ 1. Then, from ( 24), ( 13) and the triangle inequality

1 1 + ρ • y ≤ 1 1 + ρ • ∥Z x∥ + b ( x) , (36)
≤ 1 + ρ 1 + ρ , ( 37
) ≤ 1. ( 38
)
Finally, from (8), ( 34) and ( 38), one has y ∈ E µ y , Γ y which implies f (E (µ x , Γ x )) ⊆ E µ y , Γ y .

(39) Remark 3. The parameter q i can be tuned to reduce the value of ρ to reduce pessimism. One should consider q i = 1 by default.

IMPLEMENTATION

Theorem 2 can be implemented in an algorithm which computes E µ y , Γ y , as in [START_REF] Rauh | A computationally inexpensive algorithm for determining outer and inner enclosures of nonlinear mappings of ellipsoidal domains[END_REF]. In practice, the values of ρ and S are computed using tools from interval analysis such as those in [START_REF] Jaulin | Applied Interval Analysis[END_REF].

To solve (13), the parameter ρ is overestimated by the upper interval bound of the box b as

ρ ≤ sup b , ( 40
) with b ( x) ∈ b := (W • [J f ] • Γ x -Z) • [ x] , (41)
x

∈ [ x] := [-1; 1] n , (42)
where [J f ] is the interval extension of ∂f ∂x over the tightest axis-aligned box [x] which contains the ellipsoid E (µ x , Γ x ). ( 41) is deduced from the center form representation of b as presented in (Rauh and Jaulin, 2021, Equation (35)).

Then, to solve ( 22) when σ i = 0, the parameter s i is overestimated by the upper interval bound of the box

[s i ] as 0 < s i ≤ sup {[s i ]} , (43) with [s i ] := q i • e T i U T • [J f ] • Γ x • [ x] .
(44) In practice, if s i = 0 is known, selecting q i = 0 limit pessimism and is recommended. Note that overestimation of ρ and s i don't invalidate the proof of Theorem 2.

These new parameter estimations can then be implemented in Algorithm 1, a modified version of algorithm [START_REF] Rauh | A computationally inexpensive algorithm for determining outer and inner enclosures of nonlinear mappings of ellipsoidal domains[END_REF], Algorithm 1). The parameter γ > 0 is used to detect singularity and should have a small value.

Algorithm 1. General Outer ellipsoidal enclosure

Require: {µ x , Γ x , A, [J f ] , γ} 1: if |det (AΓ x )| > γ then 2:
do as in [START_REF] Rauh | A computationally inexpensive algorithm for determining outer and inner enclosures of nonlinear mappings of ellipsoidal domains[END_REF], Algorithm 1) 3: else 4:

U ,Σ,V = full svd decomposition(AΓ x ) 5:

for i ∈ [1 : n] do 6: if σ i < γ then 7: [s i ] = q i • e T i U T • [J f ] • Γ x • [ x]
8:

s i = sup {[s i ]} 9:
else 10:

s i = σ i 11: end if 12:
end for 13:

M = U S 14: W = S + U T 15: Z = S + ΣV T 16: b = (W • [J f ] • Γ x -Z) • u 17: ρ = sup b 18: µ y = f (µ x )
19:

Γ y = (1 + ρ) • M 20: end if return µ y , Γ y

APPLICATION ON NONLINEAR SYSTEM

A common case of singularity appears when one considers hybrid systems like robots, whose dynamics are continuous physical variables but their memory store discrete estimations.

Robots are usually controlled by their state estimation which is often corrected when the robots make measurements, and predicted between measurements.

When there is no measurement noise, the estimation correction can be perfect, making the estimation vector equal to the state vector. This correction is a common cause of singularity as it can create degenerate ellipsoids. Thus, to illustrate our method, we propose to apply the general ellipsoidal propagation to this type of system. The system is defined in Sections 5.1. Section 5.2 gives an example of ellipsoidal propagation. Section 5.3 presents another example where the ellipsoidal propagation is used to prove the p-invariance of the initial ellipsoid. Section 5.4 comments on the obtained results.

System description

Consider the dynamical system described by ẋ

= f x (x, x, α) , ẋ = f x (x, x, 0) , for t ∈ ]t k ; t k+1 [ , x t + k = x (t k ) (45
) where x ∈ R n is the state vector, x ∈ R n is the state estimation, α ∈ R n is a constant disturbance, and where f x is differentiable and the time t + k represents the time t k after the measurement. At every measurement time t k = k • T with k ∈ N and the period T > 0, the piece-wise continuous estimation x is corrected on the value of the state x (t k ).

The evolution of the system can be described by the augmented state vector z = x T xT α T T ∈ R 3n whose evolution is described by ż

= f z (z) , for t ∈ ]t k ; t k+1 [ , ( 46
) z t + k = P z • z (t k ) , (47)
where

f z (z) = f x (x, x, α) f x (x, x, 0) 0 , (48)
and

P z = 1 0 0 1 0 0 0 0 1 ⊗ I n . (49)
Thus, the evolution of the system from one measurement time t k to the next one t k+1 can be described by the nonlinear mapping

z k+1 = f (z k ) = z k + ϕ (T, z k ) , (50)
where ϕ is the flow function of (46).

In order to illustrate the system in low dimensions, let us consider a system with n = 1 and

f x (x, x, α) = 0.5 • sin (x) -x + α. (51)
Note that our method can easily treat high-dimension problems with polynomial calculation time, see, [START_REF] Louedec | Computational tractable guaranteed numerical method to study the stability of n-dimensional time-independent nonlinear systems with bounded perturbation[END_REF].

Ellipsoidal propagation

The first test consider the initial ellipsoid k with k ∈ {1, 2, 3} using the Algorithm 1. The parameters q = [ q 1 q 2 q 3 ] T and γ are empirically set to q = [ 0 0 10 ] T and γ = 10 -6 . The result is illustrated in Figure 2, where the ellipsoids are projected on the two first dimensions.

E µ + 0 , Γ + 0 a time t + 0 with µ + 0 = [ 1 1 0 ] T , (52)
Γ + 0 = 0.05 • 1 0 0 1 0 0 0 0 1 . ( 53

p-invariant neighborhood

The second test consider the initial ellipsoid E µ + 0 , Γ + 0

with µ + 0 = [ 0 0 0 ] T , (54)
Γ + 0 = 0.5 • 1 0 0 1 0.5 0 0 0 0.1 . ( 55
)
The ellipsoids E (µ 1 , Γ 1 ) and E µ + 1 , Γ + 1 are computed. The parameters q and γ are unchanged.

Then, the inclusion

E µ + 1 , Γ + 1 ⊆ E µ + 0 , Γ + 0 is tested to verify that E µ + 0 , Γ + 0 is p-invariant for the time t + k
according to (Soyer et al., 2020, Theorem IV.5). The ellipsoids are illustrated in Figure 3.

Results and discussion

As expected, the ellipsoids are degenerate at the times t + k with x (t k ) = x (t k ) but become non-degenerate at time t k+1 because of the bias α. Moreover, the nonlinearity of the system creates some pessimism in the ellipsoidal propagation. A smaller initial set leads to smaller pessimism. Pessimism has also been reduced by selecting the value of q. For now, one needs to find it empirically, but a method to find it automatically will be the subject of future works. Note that pessimism is incremental and thus increase at every iteration.

CONCLUSION

In this paper, the ellipsoid propagation method from Rauh and Jaulin (2021) has been generalized for non-degenerate ellipsoids with similar computational time and precision.

The method propagates degenerate ellipsoids through an n-dimensional nonlinear mapping. This method makes it possible to analyze high-dimensional synchronous nonlinear hybrid systems, which is a challenging topic in the field of mobile robotics.

Proof of equation (30)

Consider S from ( 21), e i from ( 22), U from (20), y from (2) and µ y from (11). the matrix S is diagonal, so its pseudo inverse pinv (S) is also diagonal and its diagonal elements are

p i =    1 s i if s i > 0, 0 else. ( 56
)
Let c = SS + U T yµ y and d = U T yµ y . One has e T i c = e T i SS + U T yµ y , = s i s + i e T i U T yµ y , = s i s + i e T i d.

(57)

From ( 22), if e T i d = 0, then s i s + i = 0, so e T i c = e T i d. Moreover, if e T i d ̸ = 0, then s i s + i = 1, so e T i c = e T i d. Therefore c = d and (30) is verified.

≤ max i (p i σ i ) . (56)
Moreover, from (21), if σ i ̸ = 0, then p i = 1 σi . Thus ∥Z∥ ≤ 1.

Fig. 1 .

 1 Fig. 1. Illustration of Theorem 2 in the singular case.

)Fig. 2 .

 2 Fig. 2. Ellipsoidal propagation for the system (46) withz (t k ) ∈ E (µ k , Γ k ) and z t + k ∈ E µ + k , Γ + k for k ∈ {0, 1, 2, 3}. Ellipsoids are projected on {x, x}.The ellipsoids E (µ k , Γ k ) and E µ + k , Γ + k are computed for respectively t k and t + k with k ∈ {1, 2, 3} using the Algorithm 1. The parameters q = [ q 1 q 2 q 3 ] T and γ are

Fig. 3 .

 3 Fig. 3. Propagation the p-invariant ellipsoid E µ + 0 , Γ + 0 until the next measurement of the system 46. Ellipsoids are projected on {x, x}.

=

  ∥W AΓ x ∥ ,

  ) U T U ΣV T , = pinv (S) ΣV T . (58)Then, since V is orthonormal, ∥Z∥ = ∥pinv (S) Σ∥ ,

Future work will study the stability of such a system using the proposed method, to prove the existence of p-invariant ellipsoids.