G De Sousa
email: gil.de-sousa@cemagref.fr

J.-P Chanet

A Jacquot

D Boffety

G André

K.-M Hou

Data Collection and Management Solution for Wireless Sensor Networks

Keywords:

Wireless sensors networks (WSN) use can be very interesting in agricultural and environmental data collection. The first WSN generations operated in a continuous data stream mode which generates high energy consumption. This article presents a new WSN platform that limits data exchanges and has an increased lifetime. All of its components are designed in a resource aware mode in relation with energy, memory and processing. This platform is built on wireless sensors, implementing a hardware component-based concept, that allow them to be combined to form a more evolved wireless device. To manage this wireless sensor, a hybrid operating system, both multithreading and based on events, has been developed and is associated to a micro-file system. A WSN management tool is in charge of monitoring the wireless sensors and of the data collection. A first evaluation of this WSN platform has been realised in an agricultural context data collection.

Introduction

The current emergence of Wireless Sensor Networks (WSN) research topic is based on the recent advances in embedded technologies and on the various possible applications. The development of miniaturized wireless access medium has, for example, enabled the implementation of more integrated wireless sensor node. Moreover, with the availability of this kind of devices, new applications have been defined even simply imagined. WSN has then been used in military, industrial, transport and "smart care, smart home" applications [START_REF] Stojmenovic | Handbook of Sensor Networks: Algorithms and Architectures[END_REF].

Naturally, agricultural and environmental WSN applications are also numerous and varied. With regard to this last point, rather than develop a WSN solution for a specific application, our approach aims at the design of a complete versatile and adaptive solution dedicated to agricultural and environmental data collection. Thus, this article presents a new WSN platform that integrates different components designed, based on resource-aware concept, and being able to adapt to diverse applications constraints. The following sections describe, in this order, the LiveNode (LIMOS Versatile Embedded Node) wireless sensor node, the LIMOS (LIghtweight Multithreading Operating System) kernel, the LiveFile (LIMOS Versatile Embedded File system) system and the LiveNCM (LiveNode Non invasive Context-aware and modular Management tool) tool that form the WSN platform proposed. The penultimate section shows some examples of application of this WSN platform. The last section concludes this article.

The LiveNode wireless sensor node

Classical wireless sensor node has the following components:

• A microcontroller.

• A wireless access medium.

• Different types of sensor(s).

• A memory or storage unit.

• A power or energy supply. The LiveNode has a modular architecture which enables to implement a hardware component-based concept. Indeed, several LiveNodes can be combined to form a more evolved wireless sensor node. For example, two LiveNodes using different wireless access medium (e.g. Wi-Fi and ZigBee) can be grouped together to realise either a wireless gateway or a multi-support module (Fig. 1). This latter is part of existing Quality of Service (QoS) methods in communication. If one of the technologies is unavailable, the other can be used in order to transmit data successfully. Like other existing wireless sensor nodes, the LiveNode has limited resources in terms of energy, memory and processing power. Moreover, these resources are closely linked. For example, energy can be saved by reducing the size of sending message which may be achieved either by storing or by compressing some part of the collected data. Then, all the software embedded in a wireless sensor node have to be designed by taking into account the constraints generated by the lack of available resources.

The LIMOS operating system

In wireless sensor network (WSN), there are three families of operating systems:

• Multi-threading (multitask) operating system.

• Event-driven operating system.

• Hybrid operating system.

The first one included operating system using classical multi-threading functioning mode. MANTIS (MultimodAl system for Networks of In-situ wireless Sensors) OS is part of this family [START_REF] Bhatti | MANTIS OS: An Embedded Multithreaded Operating System for Wireless Micro Sensor Platforms[END_REF]).

In the second group, the operating mode is based on the event notion. The main difference between thread and event is that this latter can be pre-empted. In other words, event can be considered as a single task which runs to completion. However, both thread and event can be interrupted. TinyOS is the most representative of the existing event-driven operating system [START_REF] Hill | System Architecture Directions for Networked Sensors[END_REF]. Hybrid operating systems try to combine multi-threading and event-driven properties.

Nevertheless, most of these kinds of operating systems are not natively hybrid. For example, Contiki is natively event-driven and can propose a multi-threading behaviour thanks to an optional library [START_REF] Dunkels | Contiki -a Lightweight and Flexible Operating System for Tiny Networked Sensors[END_REF]).

In our purpose to build a versatile WSN platform, a natively hybrid operating system called LIMOS has been designed. In this kernel, each thread is associated to an event which allows three operating modes: multi-threading, event-driven and a combination of the two others (Fig. 2). LIMOS kernel used also a two-level scheduling policy based on an event manager and a thread scheduler. In the multi-threading mode, all the threads are associated to the same event. In the event-driven mode, each event has only one thread.

The LiveFile micro-file system

The main purpose of the LiveFile system is an intelligent use of the non volatile memory embedded in a wireless node. It has been elaborated from the study of different memory flash management systems, dedicated to WSN, such as Capsule [START_REF] Mathur | Capsule: An Energy-Optimized Object Storage System for Memory-Constrained Sensor Devices[END_REF]. It provides different preservation mechanisms and data formats for the storage.

Flash memory is divided into pages. Each page allows a restricted and limited number of writings. This limit varies from 10,000 to 1,000,000 depending of the flash memory technology. For example, the microcontroller AT91SAM7S256 has an internal flash memory divided into 1024 pages of 256 bytes which support 10,000 writings.

Then, preservation mechanisms are essential in order to increase memory lifetime. The first mechanism found on LiveFile and also in other systems is the use of RAM Buffers in order to make page-size writings (Fig. 3). Then, effective writing only occurs when the size of the data to store is equal to a given threshold. Memory is more preserved than this threshold is closed to the page size. The second mechanism is based on the principle of balancing the distribution of the writing load between the different pages. It aims at the preservation of the most used pages and a wear evenly distributed of the memory. To implement such functionality, the number of writings of each page is necessary. Then, a header is integrated into the beginning of each memory page and contains different kind of information about it such as:

• The state.

• The writing number.

• The free space.

• The type of stored data.

About this latter attribute, to reduce data management complexity, each page only stores data of the same format. Three formats are available in the LiveFile system: "RECORD", "FILE" and "CHECKPOINT". The first one is highly adapted to the supported application with a definition of its size and its type according to the kind of collected data. The "File" format is used to store data stream with variable size. Finally, information useful to the recovery process after an eventual breakdown are stored thanks to the "CHECKPOINT" format.

The LiveNCM management tool

In order to manage an important number of wireless sensor node deployed during an application, a Wireless sensor network Management Tools (WMT) is required. Thus, a new WMT called LiveNCM has been designed to achieve this task in the context of the WSN platform presented in this article. The LiveNCM management tool integrates an extension of the SNMP (Simple Network Management Protocol) protocol by taking into account WSN constraints. More precisely, the LiveNCM tool consists of four main elements (Fig. 4):

• A SNMP Server with a sub-agent extension.

• A wireless gateway.

• A distributed component embedded on each LiveNode wireless sensor node.

• A supervisor.

Classical functionalities of a SNMP, though the implementation of an agent, are the storage of management data and to answer user requests. In the case of the LiveNCM tool, a subagent, with its own data structure called MIB (Management Information Base), has been developed to complete the SNMP agent. The SNMP Server uses a dedicated wireless gateway to communicate with a given WSN. Various WSN can be managed by the SNMP server. To answer to the requests sent by the SNMP server, LiveNodes are equipped with a specific software component. Collected data and information about the functioning state of the nodes can be consulted using a supervisor that integrates an end user interface. This supervisor is based on a database which takes its information from the SNMP MIB. The purpose of this project is the development and the validation of methods or techniques allowing the fast and accurate deployment of different wireless technologies to ensure communication between isolated areas. Then, our WSN platform was integrated in a wireless communication infrastructure installed in an isolated farm.

Conclusion

This article introduces a new WSN platform and its components. These platform is based on a wireless sensor node LiveNode which resources are managed by an operating system LIMOS and a micro-file system LiveFile. This latter operates essentially on the non volatile Flash memory. Administration of the sensor nodes and collected data storage is obtained thanks to the LiveNCM management tool. This platform has been designed and implemented gradually until now. It reaches a level of performance that allows us to evaluate it in real experimentations associated to different research projects.

Fig 1 :

 1 Fig 1: The combination of two LiveNode wireless sensor nodes

Fig. 2 :

 2 Fig. 2: LIMOS operating mode

Fig 3 :

 3 Fig 3: Memory management in the LiveFile system

Fig

  Fig 4: The LiveNCM tool

Fig 5 :

 5 Fig 5: Deployment of the WSN platform during the NeT-ADDeD project

  Then, in the same way, the LiveNode wireless sensor node is based on an ARM7TDMI ATMEL microcontroller AT91SAM7S256. The main characteristics of this microcontroller are a variable processor frequency from 50 Hz to 50 MHz, 256 KB of Flash and 64 KB of SRAM memories, different interfaces such as RS232, USB and I²C. This sensor node can be equipped with an IEEE 802.11 Wi-Fi or an IEEE 802.15.4 ZigBee wireless communication module, connected to the microcontroller through a RS232 serial port. The energy used to power this wireless sensor node can be provided by standard AA battery or by a photovoltaic.


Génia Babykina Cemagref

Vincent Couallier
email: couallier@sm.u-bordeaux2.fr

Modeling failures of repairable systems under "worse than old" assumption

The objective of the work is to model the failure process of a repairable system under "worse than old", or harmful repairs, assumption. The proposed model is founded on the counting process probabilistic approach and interprets harmful repairs as the accumulation of failures on the same system. The explicit form of likelihood function is given. Fixed and time-varying covariates are taken into account. Monte-Carlo simulations are carried out to asses the quality of the MLE. The method of data generation is detailed.

Introduction

Recurrent failures of a repairable system are commonly modeled by the non-homogeneous Poisson process (NHPP) or the renewal process (RP), implying "as bad as old" and "as good as new" repair effects respectively. Other after-repair states of a system ("better than old but worse than new", "worse than old", "better than new") can be modeled within the framework of the generalized renewal process (GRP), using for example the concept of virtual age. Refer to [START_REF] Krivtsov | Recent advances in theory and applications of stochastic point process models in reliability engeneering[END_REF], [START_REF] Lindqvist | On the statistical modeling and analysis of repairable systems[END_REF] for the review of models and references and, for example, to [START_REF] Lawless | Regression methods for Poisson process data[END_REF], [START_REF] Yãnez | Generalized renewal process for analysis of repairable systems with limited failure experience[END_REF], [START_REF] Doyen | Classes of imperfect repair models based on reduction of failure intensity or virtual age[END_REF] for statistical inference details.

The counting process theory as defined in [START_REF] Andersen | Statistical models based on counting process[END_REF]) and as reviewed below, is widely used to construct the probability models for the mentioned processes. Let us denote (T i ) i≥1 the failure times (assumed to be confounded with repair times) and N (t) = i≥1 1 Ti≤t the corresponding counting process (N (0) = 0). Modeling is then based on the conditional failure intensity [START_REF] Andersen | Statistical models based on counting process[END_REF])

λ(t) = lim dt→0 1 dt P(N (t + dt) -N (t) = 1 | H t-)
with H t-the history of the process up to time t.

The corresponding likelihood function for m failures of a system, used for statistical inference, is defined by [START_REF] Andersen | Statistical models based on counting process[END_REF])

L(θ) =   m j=1 λ(t j )   × exp   - m j=0 t (j+1) tj λ(u) du  
(1) (t j , j = 1, . . . , m) being the times of failures and θ the parameter vector.

The "worse than old" after-repair state can be captured, for instance, by a particular value of the maintenance effect parameter in intensity-reduction models [START_REF] Doyen | Repair efficiency estimation in the ARI1 imperfect repair model. Modern statistical and mathematical methods in reliability[END_REF], or by stratifying the NHPP model according to the failure rank [START_REF] Røstum | Statistical Modelling of Pipe Failures in Water Networks[END_REF].

In this paper we propose an alternative approach to model the failure process under the "worse than old" assumption. The general framework is presented in Section 2, Section 3 deals with simulation tools. Section 4 presents the results of Monte simulations.

Harmful repairs, i.e. the after-repair state of a system is worse than that just before failure, can result in accumulation of failures on a system. Starting from this assumption Le [START_REF] Gat | Une extension du processus de Yule pour la modélisation stochastique des événements récurrents[END_REF] proposes to link the conditional failure intensity function to the number of previous failures of a system. The resulting model is derived from the birth process theory (see [START_REF] Ross | Stochastic Processes[END_REF]) and is defined by

λ(t) = (1 + αj) δt δ-1 e Z(t) β , (α > 0)
with j the number of failures occurred up to time t, Z(t) a 1×p covariate vector (eventually time-varying) and θ = {α, δ, β 0 , . . . , β p } the vector of parameters to estimate. We note λ 0 (t) = δt δ-1 e Z(t) β (baseline intensity) and Λ 0 (s) = s 0 δu δ-1 e Z(u) β du (cumulative baseline intensity). The proposed model implies the following:

α > 0: instantaneous failure intensity augments with the number of previous failures (note that for α = 0 the model is reduced to an NHPP);

δ > 1: system wears-out;

-∃ k = 0 β k = 0: external risk factors and/or system's individual characteristics influence the failure intensity.

Le [START_REF] Gat | Une extension du processus de Yule pour la modélisation stochastique des événements récurrents[END_REF] shows that for a system with a known number of observed failures during a time-period [a, b) (a not necessarily being the system installation time), the number of failures to occur in the future period [b, c]

follows the Negative Binomial distribution (NBD) [N (c) -N (b) | N (b-) -N (a) = k] ∼ N B α -1 + k, e αΛ0(b) -e αΛ0(a) + 1 e αΛ0(c) -e αΛ0(a) + 1 (2)
This result enables us to give an explicit expression for the likelihood function (1), and to estimate the parameters of the model. For exemple, the log-likelihood for m failures of a system during the period [a, b] is as follows ln L(θ) = m ln α + ln Γ(α -1 + m) -ln Γ(α -1 )

-(α -1 + m) ln(e αΛ0(b) -e αΛ0(a) + 1) + m j=1 (ln λ 0 (t j ) + αΛ 0 (t j )) ( 3
)
where Γ is the gamma function resulting from the Negative Binomial distribution.

Data generation technique

Numerical study of the MLE properties requires generation of times of events and of covariate vector. Whilst the latter is quite easy to obtain using standard probability distributions (exponential, normal, uniform, binomial, etc.), simulation of event times is less straightforward. The employed method is detailed in the present section.

The times of events {t j , j ∈ N} are simulated using the inverse transform algorithm (see for example [START_REF] Ross | Simulation[END_REF].

Consider random variable T j , the time of j-th event, X j+1 the holding time between the j-th and the (j + 1)-th events, and u the uniform (0, 1) random variable. The conditional distribution function F of X j+1 can be used for inversion. Specifically,

F (X j+1 ) = P [X j+1 > x | T j = t j ] = P [N (t j+1 ) -N (t j ) = 0 | N (t j ) = j] since {T j = t j } ≡ {N (t j ) = j} = exp (-(1 + αj) [Λ(t j + x) -Λ(t j )])
using Eq. 2 and NBD properties Successively solving F (X j+1 ) = u j+1 for t j+1 with t 0 = 0 we obtain the event times.

For a time-varying covariate vector Z(t), Λ(.) is a piecewise function (see Fig. 1 for schematic representation) and its inversion becomes more involved. The solution for piecewise constant covariate vector is to divide the time into sub-intervals [τ h , τ h+1 ), where Z(t) remains constant at value Z h+1 , and evaluate Λ k = Λ(τ k ) at each predefined point τ k . Generated random number u j at each iteration j gives the interval for Λ(t j ) and, thus, the interval for the constant covariate vector, allowing the inversion of Λ(t j ). Specifically, the time of (j + 1)-th event is calculated by

t j+1 = Λ(t j ) - ln(uj+1) 1+αj -Λ(τ h ) e Z h+1 β + τ δ h 1/δ (4)
The data-generation algorithm for time-varying covariate vector is detailed below (refer to Fig. 1 for graphical support)

τ0 = 0 τ1 τ2 τ3 Λ1 Λ2 Λ3 Λ(t1) Λ(t2) Λ(t3) t1 t2 t3 t4 t Λ(t) Λ0 = 0 Λ(t4) E1 E3 E4 E2
Figure 1: Data simulation schema.

1. Calculate k at predefined points {τ k }.

2. Set iteration counter: j = 0 (t 0 = 0, Λ 0 = 0).

3. Generate uniform (0, 1) random variable u j+1 .

Calculate

E j+1 = - ln(uj+1) (1+αj) , Λ(t j+1 ) = Λ(t j ) + E j+1 . 5. Identify h for which Λ(t j+1 ) ∈ [Λ h , Λ h+1 ) ⇒ t j+1 ∈ [τ h , τ h+1
). Use Z h+1 in Eq. 4 to inverse Λ(.)

6. Reset counter: j = j + 1.

7. Repeat steps 3-6 until the end of observational period.

Monte Carlo study of the MLE

The simulated data consists of 10 000 simultaneously observed systems over the time period The times of events are simulated according to (4). The likelihood defined by ( 3) is maximized to estimate the parameter vector θ = {α, δ, β 0 , β 1 , β 2 }, β 0 corresponding to the intercept, β 1 and β 2 being the coefficients of the fixed and time-dependent covariates respectively. The initial parameter vector is θ = {1.5, 1.2, -6, 0.2, 0.5}.

The 100-runs Monte-Carlo simulations are carried out. For comparison purposes the 10-, 20-, 30and 40-year observational periods are considered.

The quality of the MLE for θ is assessed by means of the empirical mean square error, MSE, and the relative square error, RSE. The MSE is calculated as the sum of the empirical variance and the squared empirical biais of θ. The RSE j for each parameter θ j over K Monte-Carlo runs is:

RSE j = 1 K K j=1 θj -θ j θ j 2 .
In general the estimators are better for longer observational periods, and thus, longer failure history (refer to Fig. 2 and Fig. 3 for the MSE and the RSE respectively).

The Shapiro-Wilks test is used to empirically verify the asymptotic normality of the estimator. The p-values are presented in Table 1. 0.10 0.12 q q q q q q q q q q q q q q q q α α δ δ β β 0 β β 1 β β 2 q q q q q q q q q q q q q q q q α α δ δ β β 0 β β 1 β β 2

  [a, b]. Time unit is year. Installation moments are uniformly distributed over [a -10, b]. One fixed exponential covariate is considered. The time-dependent covariate is a dummy for the period [b -6, b -5].

Figure 2 :

 2 Figure 2: Empirical mean square error.

Figure 3 :

 3 Figure 3: Relative square error.

Table 1 :

 1 P-values for Shapiro-Wilks normality test.

	Length of [a, b]	α	δ	β0	β1	β2
	10 20 30 40	0.8887 0.5286 0.0797 0.0222 0.6668 0.2251 0.5095 0.4498 0.7890 0.2354 0.0965 0.3514 0.7361 0.9396 0.5519 0.7057 0.8197 0.9691 0.1629 0.7938


Génia Babykina
email: genia.babykina@cemagref.fr

Vincent Couallier

Yves Le Gat

Prise en compte des covariables temporelles dans la mod élisation des év énements r écurrents

Keywords: événements récurrents, NHPP, processus de naissance, variables dépendantes du temps, défaillances de conduites d'eau recurrent events, NHPP, birth process, time-dependent covariates, water pipe failures

L'étude porte sur la modélisation des événements récurrents d'un système réparable avec maintenances imparfaites, fondée sur la généralisation du modèle NHPP. L'effet des maintenances sur l'intensité est pris en compte par le nombre de défaillances précédentes. L'introduction des covariables dépendantes du temps dans le modèle a été détaillée ainsi que l'algorithme de simulation des données. L'application à la modélisation des défaillances d'un réseau d'eau est présentée.

Introduction

Il est naturel de modéliser les événements récurrents d'un système par le processus de comptage de type NHPP, Processus de Poisson Non-Homogène [START_REF] Andersen | Statistical Models Based on Counting Processes[END_REF]), dont l'intensité s'écrit

E [dN (t) | H] = E [dN (t)] = λ(t) dt λ(t) = λ 0 (t)e Z β (Z = Z t )
1 avec dN (t) : le saut du processus à l'instant t, H : l'historique, λ 0 (t) : l'intensité de base, Z : vecteur des covariables, β : vecteur des paramètres à estimer.

En fiabilité le NHPP peut être utilisé dans la modélisation des défaillances d'un système réparable avec maintenances neutres (ABAO : As Bad As Old ), i.e. après la réparation le système revient à l'état dans lequel il était juste avant la défaillance. En situation de maintenances imparfaites, chaque défaillance et/ou action de réparation déteriore l'état du système. Le [START_REF] Gat | Une extension du processus de Yule pour la modélisation stochastique des événements récurrents[END_REF] propose dans ce cas d'élargir le modèle de NHPP en y intégrant le facteur du nombre des défaillances précédentes. Le modèle ainsi construit est basé sur le processus de naissance (voir Ross (1983)) et son intensité s'écrit

E [dN (t) | N (t-)] = (1 + αj)λ(t), (α > 0) λ(t) = δt δ-1 e Z β (1
Cela nous permet d'écrire la vraisemblance du processus défini par le système (1) et d'estimer ses paramètres.

Notre étude s'applique à la modélisation du processus des casses de conduites dans un réseau d'eau, dont l'intensité est sujette à des fluctuations temporelles qui ne sont pas directement prises en compte par le modèle (1), ce qui introduit un biais dans les estimations. Nous proposons ici de modéliser les fluctuations d'intensité du processus en intégrant les variables dépendantes du temps dans la partie Z β du modèle. Les détails des calculs adaptés à ce cas sont présentés dans la Section 2. Dans la Section 3 nous décrivons la méthode de simulation des données suivant (1) en présence des variables temporelles.

Intégration des variables dépendantes du temps

Deux situations sont envisageables :

(a) Variable temporelle constante par morceaux, ce qui se traduit par une ou plusieurs période(s) marquée(s) dans le processus de défaillances (ex.: une année froide, campagne de recherche de fuites, augmentation de pression dans le réseau, etc.) (b) Variable temporelle dont l'effet sur le processus est continu (ex.: la température moyenne journalière, l'humidité, etc.)

Variable temporelle constante par morceaux

Nous supposons dans ce cas que la période d'observation est découpée en k sous-intervalles, sur chacun desquels l'effet de la variable temporelle, noté βk , est constant (cf. Figure 1 pour illustration).

a=τ 0 τ 2 τ k-2 τ k-1 τ 1 s t τ k = b 1 β 2 β 1 ~-k β k β
Zl (t) = 1 si t ∈ [τ l-1 , τ l [ , l = 1, . . . , k 0 sinon
Ainsi, le vecteur de covariables sera Zl = {Z, 0, . . . , 0 (p+1,...,p+(l-1))

, 1, 0, . . . , 0

(p+(l+1),...,p+k)

} et le vecteur des coefficients des covariables : β = {β, β1 , . . . , βk }.

Selon le modèle (1), l'écriture de la vraisemblance des événements survenus aux instants t j dans l'intervalle [a, b] nécessite le calcul des intégrales Λ(s), s ∈ {a, b, t j }. La modification de ces calculs en présence des covariables temporelles est detaillée ci-dessous.

Les covariables étant fixes sur chaque intervalle τ l-1 , τ l , nous pouvons calculer l'intégrale :

Λ l = τ l τ l-1 λ(u) du = τ l τ l-1 δt δ-1 e Z l β = τ δ l -τ δ l-1 e Z l β .
En fixant s ∈ [τ h , τ h+1 ], les Λ(s) dans la vraisemblance sont calculés selon :

Λ(s) = h-1 l=0 τ δ l+1 -τ δ l e Z l+1 β + s δ -τ δ h e Z h+1 β (3)
Les inconvénients de la méthode présentée sont les suivants :

-le grand nombre de paramètres supplémentaires à estimer β1 , . . . , βk ; -la nécessité de connaître ou supposer connus le début et la fin de la période marquée, [τ l-1 , τ l [ ; -la transformation des données pour l'estimation est compliquée. En revanche, l'écriture explicite des intégrales Λ(s) facilite les calculs et accelère la procédure d'optimisation de la vraisemblance.

Variable temporelle continue

Notons X(t) la variable continue qui influence l'intensité du processus tout au long de la période d'observation. L'intensité instantanée s'écrit

E [dN (t) | N (t-)] = (1 + αj)λ(t) λ(t) = δt δ-1 e Z β+γX(t) (4)
En supposant que la variable X(t) est observée avec une certaine périodicité aux instants {τ 1 , . . . , τ i , . . . , τ m }, nous pouvons calculer λ(t) à chacun de ces instants. Sur les intervalles [τ i , τ i+1 [ entre les observations, λ(t) peut être approchée par une droite y(t).

L'intégrale Λ(s) pour s ∈ [τ h , τ h+1 ], s'écrit Λ(s) = h-1 i=0 Λ(τ i , τ i+1 ) + Λ(τ h , s) avec Λ(τ i , τ i+1 ) = τ i+1 τ i y(u) du.
Un exemple d'intégration approchée pour une covariable temporelle périodique est présenté sur la Figure 2.

Les avantages de la méthode sont les suivants :

-supposer continu l'effet de la variable sur l'intensité est plus réaliste ; -l'approche ne nécessite l'estimation que d'un paramètre supplémentaire, γ. Les limites de l'approche sont :

-il est nécessaire de supposer que la forme de X(t) est connue ; -l'intégration de λ(t) approchée pour chaque individu ralentit le calcul de la vraisemblance à l'étape d'estimation.

∫ ∫ ∑ ∫ - = +         = = Λ + s s h i h i i du u du u du u s τ τ τ λ λ λ ) ( ) ( ) ( ) (

Simulation des données

Les temps des événements, t j (j ∈ N), sont simulés successivement par la méthode de la transformée inverse. La fonction utilisée est celle de la survie conditionnelle (notée S(.)) du temps (noté X j+1 ) entre les défaillances j et j + 1. Cette fonction est la probabilité conditionnelle de ne pas avoir d'événement dans l'intervalle ]t j , t j + x] sachant que j événements se sont produits sur [0, t j [ (N (t j ) = j). En utilisant (2) Le Gat (2009) montre que :

S (X j+1 ) = P [X j+1 > x | T j = t j ] = e -(1+αj)[Λ(t j +x)-Λ(t j )]
Avec la méthode de la transformée inverse (voir par exemple Shih et Leems (1993)), nous avons : e -(1+αj)[Λ(t j+1 )-Λ(t j )] = u j+1 ⇒ Λ (t j+1 ) = Λ(t j ) -ln(u j+1 ) 1+αj et t j+1 = Λ -1 (t j+1 ) avec u la réalisation de la variable aléatoire uniforme sur [0, 1] (u ∼ U (0,1) ) et t j+1 le temps de (j + 1) ème événement.

Le problème qui se pose dans le cadre des simulations est la nécessité d'identifier l'intervalle de temps, [τ h , τ h+1 [, dans lequel se trouve la prochaine défaillance avant de calculer son temps exact t j+1 . En effet, si t j+1 ∈ [τ h , τ h+1 [, alors Λ (t j+1 ) = Λ(τ h ) + t δ j+1 -τ δ h e Z h+1 β et d'après (3) nous avons :

t j+1 =   Λ(t j ) -ln(u j+1 ) 1+αj -Λ(τ h ) e Z h+1 β + τ δ h   1/δ
Il est donc nécessaire de connaître le vecteur des futures covariables, Z h+1 , afin de trouver le temps du futur événement, t j+1 .

L'algorithme itératif de simulation utilisé est le suivant (cf. Figure 3 pour l'illustration et un exemple) :

1. Calculer Λ l = Λ(τ l ), l = 0, . . . , k du début et de la fin de chaque sous-intervalle où la variable temporelle est constante.

2. j = 0 3. Tirer u j+1 ∼ U (0,1) . Calculer

E j+1 = -ln(u j+1 ) (1+αj) et Λ(t j+1 ) = Λ(t j ) + E j+1 (Λ(0) = 0). 4. Identifier la position de Λ(t j+1 ). Si Λ(t j+1 ) ∈ [Λ(τ h ), Λ(τ h+1 )[ alors t j+1 ∈ [τ h , τ h+1 [
et Z h+1 est utilisé dans l'inversion de Λ j+1 . Ainsi, le temps t j+1 peut être calculé.

5. j = j + 1 6. Répéter les étapes de 3 à 5 jusqu'à ce que l'on sorte de la période d'observation. Les résultats de l'étude par simulations appliquée à la modelisation des défaillances de conduites d'eau seront présentés.

Bibliographie

  ) avec j : le nombre d'événements précédents, t : l'âge du système, Z : vecteur p -dimensionnel des covariables fixes dans le temps, θ = {α, δ, β 1 , . . . , β p } : paramètres à estimer. Le modèle (1) est un NHPP pour α = 0. On note Λ(t) = t 0 λ(s) ds = t δ e Z β . Le Gat (2009) montre que le nombre de défaillances dans une période de temps donnée [a, b], conditionnellement au nombre de défaillances précédentes suit la distribution Binomiale Négative : [N (b) -N (a) | N (a-) = m] ∼ N B α -1 + m, e α[Λ(b)-Λ(a)] .

Figure 1 :

 1 Figure

Figure 2 :

 2 Figure 2: Calcul de Λ(s) avec une variable temporelle continue observée ponctuellement

Figure 3 :

 3 Figure 3: Simulation des données : schéma de l'algorithme et exemple de simulation


Mehdi Badra
email: mehdi.badra@univ-pau.fr

Fabien Caubet
email: fabien.caubet@univ-pau.fr

Marc Dambrine
email: marc.dambrine@univ-pau.fr

F Badra

M Caubet

Dambrine

DETECTING AN OBSTACLE IMMERSED IN A FLUID BY SHAPE OPTIMIZATION METHODS

Keywords: stationary Stokes problem, shape sensitivity, geometric inverse problem AMS Subject Classification: 35R30,49Q10, 35Q30, (76D07)

The paper presents a theoretical study of an identification problem by shape optimization methods. The question is to detect an object immersed in a fluid. Here, the problem is modeled by the Stokes equations and treated as a nonlinear least-squares problem. We consider both the Dirichlet and Neumann boundary conditions. Firstly, we prove an identifiability result. Secondly, we prove the existence of the first order shape derivatives of the state, we characterize them and deduce the gradient of the least-squares functional. Moreover, we study the stability of this setting. We prove the existence of the second order shape derivatives and we give the expression of the shape Hessian. Finally, the compactness of the Riesz operator corresponding to this shape Hessian is shown and the ill-posedness of the identification problem follows. This explains the need of regularization to numerically solve this problem.

Introduction, notations and setting of the problem

The detection and the reconstruction of an object immersed in a fluid is a source of many investigations. Recently, in 2005, Alvarez et al. studied in Ref. [START_REF] Alvarez | Identification of immersed obstacles via boundary measurements[END_REF] the following inverse problem: an inaccessible rigid body ω is immersed in a viscous fluid, in such a way that ω plays the role of an obstacle around which the fluid is flowing in a greater bounded domain Ω. They wish to determine ω (i.e. its shape and location) via boundary measurement on the boundary ∂Ω. Under reasonable smoothness assumptions on Ω and ω, they prove that one can identify ω via the measurement of both the velocity of the fluid and the Cauchy forces on some part of the boundary ∂Ω. They also give a directional stability result for the inverse problem (see Theorem 1.3 in Ref. 4). After this first result, more investigations were conducted. In 2007, Heck et al. estimate in Ref. [START_REF] Heck | Reconstruction of obstacles immersed in an incompressible fluid[END_REF] the distance of a chosen point from the obstacle when the fluid motion is governed by the Stokes equations. In Ref. [START_REF] Alves | Integral equations for an inverse boundary value problem for the two-dimensional Stokes equations[END_REF], Alves et al. use a method based on the analysis of a system of nonlinear integral equations to determine the shape and the location of a rigid body immersed in a viscous and incompressible fluid. In 2008, Conca et al. investigate in Ref. 11 the problem of the detection of a moving obstacle in a perfect fluid by a boundary measurement. They show that, when the obstacle is a ball, one may identify the position and the velocity of its center of mass from a single boundary measurement. In 2010, using complex analysis, Conca et al. prove in Ref. 12 that this result can not be generalized to any solid. However, they extend this one to moving ellipses and they proved that when the solid enjoys some symmetry properties, it can be "partially detected". Numerical experiments were also conducted: the numerical reconstruction is difficult as expected for an inverse problem.

This paper aims to explain these numerical difficulties. We focus on the question of stability of the shape optimization problem in the case of Stokes equations with Dirichlet and Neumann boundary conditions. Following previous works on electrical impedance tomography (by Eppler et al. in Ref. [START_REF] Eppler | A regularized Newton method in electrical impedance tomography using shape Hessian information[END_REF] 3), our strategy to solve this inverse problem is to minimize a least-squares functional. Then, we present the basis of the numerical analysis of this problem by characterizing the gradient of this functional. To study the stability, we compute the associated shape Hessian. We prove that this shape Hessian has a pathological behavior at possible solutions of the inverse problem: the Riesz operator corresponding to this shape Hessian is compact. This property explains why only simple shapes can be recovered. A result converging in the same direction was pointed out to us by one of the referee: this year, Ballerini proved in Ref. 8 a log-log stability estimate for a slightly different problem.

This paper is organized as follows. Firstly, we introduce the notations and the two problems we consider: the Dirichlet case, which is physically relevant, and the Neumann case much less physically relevant but more challenging from the mathematical point of view. Secondly, we state the main results of this work concerning these two cases in section 2. In particular, we provide an identification result, we prove the existence of the first order shape derivative of the state, we characterize this derivative and we give the expression of the gradient of the least-squares functional. Furthermore, we discuss higher order shape derivatives and we characterize the shape Hessian at a critical point. Finally, we justify the instability of the problem with the compactness of the Riesz operator corresponding to the shape Introduction of the general notations. Let us introduce the notations that we adopt in this paper. We denote by L p , W m,p and H s the usual Lebesgue and Sobolev spaces. For k ∈ N and an open set Ω ⊂ R N (N = 2 or 3), we denote by C k c (Ω) the space of functions with continuous k first derivatives compactly supported in Ω and by • k,∞ its natural norm. We also note by D(Ω) the space of C ∞ functions compactly supported in Ω. We note in bold the vectorial functions and spaces: L p , W m,p , H s , . . . Moreover we will note respectively •, • Ω and •, • ∂Ω the dual product between H 1 (Ω)

′ and H 1 (Ω) and the dual product between H -1/2 (∂Ω) and H 1/2 (∂Ω). We denote by |Ω| the measure of Ω and by 1 O the characteristic function of a measurable subset O. Moreover, n represents the external unit normal to ∂Ω, and for a smooth enough function u, we note respectively ∂ n u and ∂ 2 nn u the normal derivative and the second normal derivative of u. We also denote by M N,N the space of the matrix of size N × N . The tangential differential operators will be noted by the subscript Γ. In particular, for w ∈ W 1,1 (∂Ω) and W ∈ W 1,1 (∂Ω, M N,N ), the following operators are defined on ∂Ω:

• ∇ Γ w := ∇w -(∇w n) ⊗ n, where ⊗ denotes the tensorial product,(1.1)

• div Γ W := div W -(∇W n) n, (1.2)
• ∆ Γ w := div Γ (∇ Γ w), which is the Laplace-Beltrami operator.

(

For more details on tangential differential operators, we refer to Section 5.4.3 in Ref. [START_REF] Henrot | Variation et optimisation de formes[END_REF].

The problem setting. Let Ω a bounded, connected and Lipschitz open subset of R N (with N = 2 or N = 3). Let δ > 0 fixed (small). We define O δ the set of all open subsets ω of Ω with a C 2,1 boundary such that d(x, ∂Ω) > δ for all x ∈ ω and such that Ω \ ω is connected. We also define Ω δ an open set with a C ∞ boundary and such that

{x ∈ Ω ; d(x, ∂Ω) > δ/2} ⊂ Ω δ ⊂ {x ∈ Ω ; d(x, ∂Ω) > δ/3} . Let U := {θ ∈ W 2,∞ (R N ); Supp θ ⊂ Ω δ } and U := {θ ∈ U ; θ 2,∞ < 1}
(I + θ)(ω) which is so that Ω \ ω θ ∈ O δ .
Let f b be an admissible boundary measurement and O a non-empty open subset of ∂Ω. We will study separately both the Dirichlet case and the Neumann case.

The Dirichlet case. Let g ∈ H 5/2 (∂Ω) such that g = 0 and satisfying the following condition:

∂Ω g • n = 0, (1.4)
and let us consider, for ω ∈ O δ , the following overdetermined Stokes boundary values problem:

           -div (σ(u, p)) = 0 in Ω \ ω, div u = 0 in Ω \ ω, u = g on ∂Ω, u = 0 on ∂ω, σ(u, p)n = f b on O, (1.5)
where σ(u, p) := ν(∇u + t ∇u)p I is the stress tensor, with ν > 0 a given constant representing the kinematic viscosity of the liquid. Since div u = 0 in Ω \ ω, one has

-div (σ(u, p)) = -ν∆u + ∇p,
in Ω \ ω and we will use indifferently both expressions.

We suppose here that there exists ω such that (1.5) has a solution. This means that the measurement f b is perfect, that is to say without error. Thus, we consider the following geometric inverse problem: find ω ∈ O δ and a pair (u, p) which satisfies the overdetermined system (1.5).

(1.6) To solve this inverse problem, we consider, for ω ∈ O δ , the least-squares functional

J D (ω) := O |σ(u(ω), p(ω)) n -f b | 2 , where (u(ω), p(ω)) ∈ H 1 (Ω \ ω) × L 2 (Ω \ ω) is a solution of the Stokes problem        -div (σ(u, p)) = 0 in Ω \ ω, div u = 0 in Ω \ ω, u = g on ∂Ω, u = 0 on ∂ω.
(1.7)

Since we imposed the compatibility condition (1.4), problem (1.7) has a unique solution once a normalization condition on the pressure p is imposed (see for example Chapter 1 in Ref. [START_REF] Temam | Navier-Stokes equations, Theory and numerical analysis[END_REF]. This solution will be called the state. Here, we choose the normalization

O (σ(u, p)n) • n = O f b • n.
Then, we try to minimize the least-squares criterion J D : The Neumann case. We mimic the study of the Dirichlet case. Let f ∈ H 3 (R N ) such that f = 0, f ≡ 0 in Ω δ and satisfying the condition:

ω * = argmin ω∈O δ J D (ω). ( 1
Ω f = 0.
(1.9)

For ω ∈ O δ , we consider the overdetermined Stokes boundary values problem:

           -ν∆u + ∇p = f in Ω\ω, div u = 0 in Ω\ω, -ν∂ n u + p n = 0 on ∂Ω, -ν∂ n u + p n = 0 on ∂ω, u = f b on O, (1.10)
where ν > 0 is a given constant representing the kinematic viscosity of the liquid. We suppose here that there exists ω such that (1.10) has a solution. This means that the measurement f b is perfect. Thus, we consider the geometric inverse problem:

find ω ∈ O δ and a pair (u, p) which satisfies the overdetermined system (1.10).

(1.11) To study this inverse problem, we introduce, for ω ∈ O δ , the least-squares functional

J(ω) := O |u(ω) -f b | 2 , where (u(ω), p(ω)) ∈ H 1 (Ω \ ω) × L 2 (Ω \ ω) satisfies the Stokes boundary values problem        -ν∆u + ∇p = f in Ω\ω, div u = 0 in Ω\ω, -ν∂ n u + p n = 0 on ∂Ω, -ν∂ n u + p n = 0 on ∂ω.
(1.12) Since f satisfies the condition (1.9) and since f ≡ 0 in Ω δ , the compatibility condition of problem (1.12) is automatically satisfied. According to Appendix A.1, it has a unique solution once a normalization condition on u is imposed. Here, we take

O u = O f b .
Then, we try to minimize the least-squares criterion J:

ω * = argmin ω∈O δ J(ω).
(1.13) Indeed, if ω * is solution of the inverse problem (1.11), then J(ω * ) = 0 and (1.13) holds. Conversely, if ω * solves (1.13) with J(ω * ) = 0, then this domain ω * is a solution of the inverse problem.

Remark 1.1. In this Neumann case, we can also consider a non-homogeneous Neumann data of the kind -ν∂ n u + p n = g on ∂Ω instead of / or in addition to

Introduction of the needed functional tools. Let T > 0, that we will have to fix small. We will use the shape calculus introduced by Murat and Simon in Ref. [START_REF] Murat | Sur le contrôle par un domaine géométrique[END_REF]. Thus, we consider the function

φ : t ∈ [0, T ) → I + t V ∈ W 2,∞ (R N ). (1.14)
where V ∈ U . Note that for small t, φ(t) is a diffeomorphism of R N and that φ ′ (0) = V vanishes on ∂Ω and even on the tubular neighborhood Ω \ Ω δ of ∂Ω. For t ∈ [0, T ), we define ω t := φ(t)(ω) where φ is defined by (1.14). For the rest of the paper, we use a subscript " t " to indicate that the quantity is defined on the time t dependent domain. For instance, n t is the external unit normal of Ω \ ω t .

Statement of the main results

We will state the main results in the two cases we study: the Dirichlet and the Neumann cases exposed in the previous section. We obtain similar results.

* = 1, let (u j , p j ) for j = 0, 1, be a solution of        -div (σ(u j , p j )) + ǫ * div (u j ⊗ u j ) = 0 in Ω \ ω j , div u j = 0 in Ω \ ω j , u j = g on ∂Ω, u j = 0 on ∂ω j .
Assume that (u j , p j ) are such that σ(u j , p j ) n = σ(u j , p j ) n on O.

Then ω 0 ≡ ω 1 . Sensitivity with respect to the domain Secondly, we aim to make a sensitivity (with respect to the shape) analysis. The Stokes problem on

Ω \ ω t        -div (σ(u t , p t )) = 0 in Ω \ ω t , div u t = 0 in Ω \ ω t , u t = g on ∂Ω, u t = 0 on ∂ω t , admits a unique solution (u t , p t ) ∈ H 1 (Ω \ ω t ) × L 2 (Ω \ ω t ) satisfying the norma- lization condition O (σ(u t , p t )n) • n = O f b • n.
Proposition 2.1 (First order shape derivatives of the state). The solution (u, p) is differentiable with respect to the domain. Moreover, the derivatives u ′ and

p ′ belong to H 2 (Ω δ \ ω) and to H 1 (Ω δ \ ω). The pair (u ′ , p ′ ) ∈ H 1 (Ω \ ω) × L 2 (Ω \ ω)
is the only solution of the following boundary values problem

       -div (σ(u ′ , p ′ )) = 0 in Ω \ ω, div u ′ = 0 in Ω \ ω, u ′ = 0 on ∂Ω, u ′ = -∇u V on ∂ω, with the normalization condition O (σ(u ′ , p ′ )n) • n = 0.
Proposition 2.2 (First order shape derivatives of the functional). For V in U , the least-squares functional J D is differentiable at ω in the direction V with

D J D (ω) • V = - ∂ω [(σ(w, q)n) • ∂ n u] (V • n),
where (w, q) ∈ H 1 (Ω\ω)×L 2 (Ω\ω) is the solution of the following Stokes boundary values problem:

       -div (σ(w, q)) = 0 in Ω \ ω, div w = 0 in Ω \ ω, w = 2(σ(u, p)n -f b )1 O on ∂Ω, w = 0 on ∂ω, with the normalization condition O (σ(w, q)n) • n = 0.
Remark 2.1. The differentiability with respect to the domain of u and p as well as Proposition 2.2 remain true under weaker assumptions. Indeed, the proof is still valid with g ∈ H 1/2 (∂Ω) and ω with a C 1,1 boundary (but in this case, the expression of D J D (ω) • V has to be seen as a duality product H -1/2 × H 1/2 ).

Second order analysis: justification of the instability Finally, we want to study the stability of the optimization problem (1.8) at ω * .

Proposition 2.3 (Characterization of the shape Hessian at a critical shape). The solution (u, p) is twice differentiable with respect to the domain. Moreover, for V ∈ U , we have

D 2 J D (ω * ) • V • V = ∂ω * -(σ(w ′ , q ′ )n) • (∇u V ),
where

(w ′ , q ′ ) ∈ H 1 (Ω \ ω * ) × L 2 (Ω \ ω * ) is the solution of the following problem:        -div (σ(w ′ , q ′ )) = 0 in Ω \ ω * , div w ′ = 0 in Ω \ ω * , w ′ = 2 σ(u ′ , p ′ )n 1 O on ∂Ω, w ′ = 0 on ∂ω * , with the normalization condition O (σ(w ′ , q ′ )n) • n = 0.
Proposition 2.4 (Compactness at a critical point). The Riesz operator corresponding to D 2 J D (ω * ) defined from H 1/2 (∂ω * ) to H -1/2 (∂ω * ) is compact.

Remark 2.2. We refer to Theorem IV-5-1 in Ref. 17 for the local regularity result for the solutions in the Dirichlet case. This point is crucial for proving Proposition 2.4 with the method we adopt here.

The Neumann case

Identifiability result

We first present a new identifiability result for the Neumann case.

Theorem 2.2 (Identifiability result).

Let Ω ⊆ R N , N = 2 or N = 3, be a bounded Lipschitz domain and O a non-empty open subset of ∂Ω. Let ω 0 , ω 1 ∈ O δ and f ∈ L 2 (Ω) with f = 0 and f ≡ 0 in Ω δ satisfying the condition (1.9). Let (u j , p j ), for j = 0, 1, be a solution of

       -ν∆u j + ∇p j = f in Ω\ω j , div u j = 0 in Ω\ω j , -ν∂ n u j + p j n = 0 on ∂Ω, -ν∂ nj u j + p j n j = 0 on ∂ω j .
Assume that (u j , p j ) are such that u 0 = u 1 on O. Then ω 0 ≡ ω 1 .

Remark 2.3. This proof of identifiability will be done under the weaker assumption: ω 0 and ω 1 have a Lipschitz boundary. Sensitivity with respect to the domain. Secondly, we aim to make a sensitivity (with respect to the shape) analysis, as in the Dirichlet case. The Stokes problem on

Ω \ ω t        -ν∆u t + ∇p t = f in Ω \ ω t , div u t = 0 in Ω \ ω t , -ν∂ n u t + p t n = 0 on ∂Ω, -ν∂ nt u t + p t n t = 0 on ∂ω t , (2.1) admits a unique solution (u t , p t ) ∈ H 1 (Ω \ ω t ) × L 2 (Ω \ ω t ) satisfying the norma- lization condition O u t = O f b .
Notice that the compatibility condition of problem (2.1) is automatically satisfied since f satisfies the condition (1.9) and since f ≡ 0 in Ω δ . Proposition 2.5 (First order shape derivatives of the state). The solution (u, p) is differentiable with respect to the domain. Moreover, the shape derivatives u ′ and p ′ belong to H 2 (Ω δ \ω) and to H 1 (Ω δ \ω). The pair

(u ′ , p ′ ) ∈ H 1 (Ω\ω)×L 2 (Ω\ω)
is the only solution of the following boundary values problem

       -ν∆u ′ + ∇p ′ = 0 in Ω \ ω, div u ′ = 0 in Ω \ ω, -ν∂ n u ′ + p ′ n = 0 on ∂Ω, -ν∂ n u ′ + p ′ n = ν∂ 2 nn u -∂ n p n (V •n) + p ∇ Γ (V •n) -ν∇u∇ Γ (V •n) on ∂ω, (2.2) with the normalization condition O u ′ = 0. Remark 2.

(Meaning of high order derivatives on ∂ω). By the local regularity result Appendix

A.2, u ∈ H 3 (Ω δ \ ω), p ∈ H 2 (Ω δ \ ω), then the quantity ν∂ 2 nn u -∂ n p n belongs to H 1/2 (∂ω).
Proposition 2.6 (First order shape derivatives of the functional). For V in U , the least-squares functional J is differentiable at ω in the direction V with Remark 2.5 (Meaning of high order derivatives on ∂ω). The expression of shape derivative involves high order derivatives of the couple (u, p) that have to be well defined. By the local regularity result Appendix A.2, w ∈ H 3 (Ω δ \ ω). It follows that w ∈ H 5/2 (∂ω) and p Iν t ∇u ∈ H 3/2 (∂ω). Therefore, the function (p Iν t ∇u)w ∈ W 1,1 (∂ω) has a tangential divergence on ∂ω.

D J(ω) • V = ∂ω ν∂ 2 nn u -∂ n p n • w -div Γ (p I -ν t ∇u)w (V • n), where (w, q) ∈ H 1 (Ω \ ω) × L 2 (Ω \ ω) is the solution of the following problem:        -ν∆w + ∇q = 0 in Ω \ ω, div w = 0 in Ω \ ω, -ν∂ n w + q n = 2(u -f b )1 O on ∂Ω, -ν∂ n w + q n = 0 on ∂ω, (2.3)
Moreover, using the Sobolev embeddings in dimension N = 2 or N = 3,

w ∈ H 3 (Ω δ \ ω) ֒→ L ∞ (Ω δ \ ω) and ∇w ∈ H 2 (Ω δ \ ω) ֒→ L ∞ (Ω δ \ ω).
Thus, w ∈ W 2,∞ (Ω δ \ ω), then (p Iν t ∇u)w ∈ H 1 (Ω δ \ ω) and finally we get that div Γ (p Iν t ∇u)w ∈ H -1/2 (∂ω).

Remark 2.6. The differentiability with respect to the domain of u and p as well as Proposition 2.6 remain true under weaker assumptions. Indeed, the proof is still valid with f ∈ H 1 (R N ) and ω with a C 1,1 boundary (but in this case, the expression of D J(ω) • V has to be seen as a duality product H -1/2 × H 1/2 ).

Second order analysis: justification of the instability. Finally, we study the stability of the optimization problem (1.13) at ω * .

Proposition 2.7 (Characterization of the shape Hessian at a critical shape). The solution (u, p) is twice differentiable with respect to the domain. Moreover, for V ∈ U , we have Statements 2.4 and 2.8 explain the difficulties encountered to solve numerically this problem. Indeed, the gradient has not a uniform sensitivity with respect to the deformation directions. Since J is twice differentiable, it behaves about ω * as its second order approximation. the compactness result means, roughly speaking, that in a neighborhood of ω * (i.e. for t small) one cannot expect an estimate of the kind C t ≤ J(ω t ) with a constant C uniform in V .

D 2 J(ω * )•V •V = ∂ω * ν∂ 2 nn u -∂ n p n (V • n) + p∇ Γ (V • n) -ν∇u∇ Γ (V • n) •w ′ , where (w ′ , q ′ ) ∈ H 1 (Ω \ ω * ) × L 2 (Ω \ ω * ) satisfies the following Stokes boundary values problem:        -ν∆w ′ + ∇q ′ = 0 in Ω \ ω * , div w ′ = 0 in Ω \ ω * , -ν∂ n w ′ + q ′ n = 2 u ′ 1 O on ∂Ω, -ν∂ n w ′ + q ′ n = 0 on ∂ω * , (2.4
However we prove that, for a finite dimensional space of deformation fields, the discrete shape Hessian is coercive: our identifiability result proves that the domain to be recovered is a local strict minimum of the least-squares functional. Let us take the example of a starshaped domain ω in dimension two. Assume that ∂ω is parametrized by

∂ω = g 0 g 1 + g 2 + ∞ k=1 (g 2k+1 cos(k t) + g 2k+2 sin(k t)) cos(t) sin(t) = ∞ k=0 g k V k (t); t ∈ (0, 2π) ,
where g k ∈ R. Then, for all n ∈ N, we have an estimate of the kind

∀V ∈ Span(V k ) 0≤k≤2n+2 , D 2 J(ω * ) • (V , V ) ≥ C n |V | 2 ,
where C n is a positive constant. But this constant C n tends to 0 when n tends to +∞. We even expect that this decreasing is exponential as observed in Ref. 2 for the Laplacian case. Therefore, our functional is degenerate for the highly oscillating deformations, i.e. for the deformation directions V k with k >> 1. Thus, for a numerical resolution, one has to exclude these highly oscillating deformations. We therefore explain the result of Conca et al. in Ref. 12. This instability can be avoided by regularization, for example, by adding to the least-squares functional a penalization in terms of the perimeter. This term leads to well posed problems (see Ref. 10 and Ref. 13). This lack of real stability is also pointed out in the recent paper of Ballerini (Ref. 8).

Differentiability results

Existence and characterization of the shape derivatives of the state and of the gradient of the least-squares functional are proved by classical arguments in shape calculus. Differentiability of the solution of the Stokes problem and of the shape functional is obtained through a generalized implicit function theorem proved by • an open set U in a Banach space U , u 0 ∈ U , two reflexive Banach spaces

E 1 and E 2 , • a map F : U × E 1 → E 2 , such that F (u, •) ∈ L(E 1 , E 2 ) for all u ∈ U, • a function m : U → E 1 and a function f : U → E 2 such that F (u, m(u)) = f (u) ∀u ∈ U. (i) Assume that • u → F (u, •) is differentiable at u 0 into L(E 1 , E 2 ), • f is differentiable at u 0 , • F (u 0 , x) E2 ≥ α x E1 ∀x ∈ E 1 , for some α > 0.
Then, the map u → m(u) is differentiable at u 0 . Its derivative m ′ (u 0 , •) is the unique solution of

F (u 0 , m ′ (u 0 , v)) = f ′ (u 0 , v) -∂ u F (u 0 , m(u 0 ), v) ∀v ∈ U.
(ii) In addition, assume that for some integer k ≥ 1, u → F (u, •) and f are k times differentiable at u 0 . Then, the map u → m(u) is k times differentiable at u 0 .

We recall the main assumptions made in the introduction: f ∈ H 3 (R N ), Ω is Lipschitz and ω has a C 2,1 boundary. Let θ ∈ U . We set (u θ , p θ ) the unique solution in

H 1 (Ω \ ω θ ) × L 2 (Ω \ ω θ ) of        -ν∆u θ + ∇p θ = f in Ω \ ω θ , div u θ = 0 in Ω \ ω θ , -ν∂ n u θ + p θ n = 0 on ∂Ω, -ν∂ nθ u θ + p θ n θ = 0 on ∂ω θ , (3.1) with O u θ = O f b .
This boundary values problem has the following variational formulation:

               Ω\ω θ {ν ∇u θ : ∇ϕ θ -p θ div ϕ θ } = Ω\ω θ f • ϕ θ ∀ϕ θ ∈ H 1 (Ω \ ω θ ), Ω\ω θ ξ θ div u θ = 0 ∀ξ θ ∈ L 2 (Ω \ ω θ ), O u θ = O f b . (3.2)
Let us define the key objects of our differentiability proof: For k, m ∈ N, k < m, we note X k,m (Ω \ ω, Ω δ \ ω) the space of functions in H k (Ω \ ω) such that their restriction to Ω δ \ ω belongs to H m (Ω δ \ ω). This space endowed with the norm

v θ := u θ • (I + θ) ∈ H 1 (Ω \ ω) and q θ := p θ • (I + θ) ∈ L 2 (Ω \ ω).
u X k,m (Ω\ω,Ω δ \ω) := u 2 H k (Ω\ω) + u 2 H m (Ω δ \ω) 1/2
, is hilbertian. We define similarly the space X * ,m (Ω\ω, Ω δ \ω) the space of functions in H 1 (Ω \ ω) ′ such that their restriction to Ω δ \ ω belongs to H m (Ω δ \ ω).

Before establishing the first order differentiability, we prove the following three lemmas.

Lemma 3.1 (Characterization of (v θ , q θ )). For θ ∈ U , the pair

(v θ , q θ ) satisfies for all ϕ ∈ H 1 (Ω \ ω) and all ξ ∈ L 2 (Ω \ ω)                Ω\ω {[∇v θ A(θ)] : ∇ϕ -q θ B(θ) : ∇ϕ} = Ω\ω [f • (I + θ)] • ϕJ θ , Ω\ω (∇v θ : B(θ)) ξ = 0, O v θ = O f b , with
J θ := det (I + ∇θ) ∈ W 1,∞ Ω δ , A(θ) := νJ θ (I + ∇θ) -1 (I + t ∇θ) -1 ∈ W 1,∞ Ω δ , M N,N , B(θ) := J θ (I + t ∇θ) -1 ∈ W 1,∞ Ω δ , M N,N . Lemma 3.2 (Differentiability of θ → (v θ , q θ )). The function θ ∈ U → (v θ , q θ ) ∈ X 1,2 (Ω \ ω, Ω δ \ ω) × X 0,1 (Ω \ ω, Ω δ \ ω)
is differentiable in a neighborhood of 0. Lemma 3.3 (Differentiability of θ → (u θ , p θ )). There exists u θ , p θ some respective extension of

u θ ∈ H 1 (Ω \ ω), p θ ∈ L 2 (Ω \ ω) such that the functions θ ∈ U → u θ ∈ H 1 (Ω) and θ ∈ U → p θ ∈ L 2 (Ω)
are differentiable at 0. Remark 3.1. We will prove this three lemmas under weaker geometrical assumptions: f ∈ H 1 (R N ) and ω with a C 1,1 boundary.

Proof. (of Lemma 3.1: characterization of

(v θ , q θ )). Let ϕ ∈ H 1 (Ω \ ω), ξ ∈ L 2 (Ω \ ω) and θ ∈ U . Using ϕ θ := ϕ • (I + θ) -1 ∈ H 1 (Ω \ ω θ ) and ξ θ := ξ • (I + θ) -1 ∈ L 2 (Ω \ ω θ )
                       Ω\ω θ ν ∇v θ (I + ∇θ) -1 • (I + θ) -1 : ∇ϕ (I + ∇θ) -1 • (I + θ) -1 -q θ • (I + θ) -1 div (ϕ • (I + θ) -1 ) = Ω\ω θ f • ϕ • (I + θ) -1 , Ω\ω θ ξ • (I + θ) -1 div (v θ • (I + θ) -1 ) = 0, O v θ • (I + θ) -1 = O f b .
However, div ϕ

• (I + θ) -1 = ∇ϕ • (I + θ) -1 : (I + t ∇θ) -1 • (I + θ) -1
, and a similar equality holds with v θ instead of ϕ. Thus, the conclusion follows after the change of variables x = (I + θ)y and by noticing that

O v θ J θ = O [f b • (I + θ)] J θ ⇐⇒ O v θ = O f b , since θ ≡ 0 on O ⊂ ∂Ω.
Proof. (of Lemma 3.2: differentiability of θ → (v θ , q θ )). Let us check the assumptions of Simon's Theorem. First step: notations. We need some additional tools: a third domain Ω δ which is an open set with a C ∞ boundary and such that Ω δ ⊂⊂ Ω δ ⊂⊂ Ω and a truncation function Φ ∈ C ∞ c ( Ω δ ) such that Φ ≡ 1 in Ω δ . Using the notations introduced in Lemma 3.1, we define the spaces:

E 1 := (v, q) ∈ H 1 (Ω \ ω) × L 2 (Ω \ ω) ; (Φv, Φq) ∈ H 2 (Ω \ ω) × H 1 (Ω \ ω) , E 2 := (f , g) ∈ H 1 (Ω \ ω) ′ ×L 2 (Ω \ ω) ; (Φf , Φg) ∈ L 2 (Ω \ ω)×H 1 (Ω \ ω) × R N .
Note that E 1 and E 2 are Hilbert spaces with respective norms

(v, q) 2 E1 := v 2 H 1 (Ω\ω) + q 2 L 2 (Ω\ω) + Φv 2 H 2 (Ω\ω) + Φq 2 H 1 (Ω\ω) , ((f , g), r) 2 E2 := f 2 [H 1 (Ω\ω)] ′ + g 2 L 2 (Ω\ω) + Φf 2 L 2 (Ω\ω) + Φg 2 H 1 (Ω\ω) + |r| 2 . Moreover, for (v, q) ∈ E 1 , (v, q) ∈ X 1,2 (Ω \ ω, Ω δ \ ω) × X 0,1 (Ω \ ω, Ω δ \ ω), and for ((f , g), r) ∈ E 2 , (f , g) ∈ X * ,0 (Ω \ ω, Ω δ \ ω) × X 0,1 (Ω \ ω, Ω δ \ ω),
and there is a constant C E such that the following estimates hold:

(v, q) X 1,2 (Ω\ω,Ω δ \ω)×X 0,1 (Ω\ω,Ω δ \ω) ≤ C E (v, q) E1 (f , g) X * ,0 (Ω\ω,Ω δ \ω)×X 0,1 (Ω\ω,Ω δ \ω) ≤ C E ((f , g), r) E2 . (3.3)
We also define the functions: Detecting an obstacle immersed in a fluid by shape optimization methods 15

• f 1 (θ, (v, q)) ∈ H 1 (Ω \ ω) ′ by ∀ϕ ∈ H 1 (Ω \ ω), f 1 (θ, (v, q)) , ϕ Ω\ω := Ω\ω [f • (I + θ)] • J θ ϕ ,
• F 1 (θ, (v, q)) ∈ H 1 (Ω \ ω) ′ by ∀ϕ ∈ H 1 (Ω \ ω), F 1 (θ, (v, q)) , ϕ Ω\ω := Ω\ω {[∇v A(θ)] : ∇ϕ -q B(θ) : ∇ϕ} , • m(θ) := (v θ , q θ ) and f (θ) := f 1 (θ, (v, q)) , 0 , O f b , • F (θ, (v, q)) := F 1 (θ, (v, q)) , ∇v : B(θ) , O v .
By the characterization of (v θ , q θ ) obtained in Lemma 3.1,

F (θ, m(θ)) = f (θ) ∀θ ∈ U .
Second step: differentiability of F and f at 0. We use the chain rule. Since the functions

θ ∈ U → J θ ∈ L ∞ (Ω) and θ ∈ U → (I + ∇θ) -1 ∈ L ∞ (Ω, M N ×N ) are C ∞ , the functions θ ∈ U → A(θ) ∈ L ∞ (Ω, M N ×N ) and θ ∈ U → B(θ) ∈ L ∞ (Ω, M N ×N ) are C ∞ . Moreover, the following functions are C ∞ : L ∞ (Ω, M N×N )× v ∈ H 1 (Ω\ω); Φv ∈ H 2 (Ω\ω) → g ∈ L 2 (Ω\ω); Φg ∈ H 1 (Ω\ω) (B, v) →∇v : B, v ∈ X 1,2 (Ω \ ω, Ω δ \ ω) → O v ∈ R N , F 1 : (θ, (v, q)) ∈ U × E 1 → F 1 (θ, (v, q)) ∈ f ∈ H 1 (Ω \ ω) ′ ; Φf ∈ L 2 (Ω \ ω) .
Finally, since f ∈ H 1 (R N ) by hypothesis, the function

θ ∈ W 1,∞ (Ω) → f • (I + θ) ∈ L 2 (R N ) is C 1 in a neighborhood of 0 by Lemma 5.3.3 in Ref. 20.
Then, the function

f 1 : (θ, (v, q)) ∈ U × E 1 → f 1 (θ, (v, q)) ∈ f ∈ H 1 (Ω \ ω) ′ ; Φf ∈ L 2 (Ω \ ω) is C 1 . By composition, F is C ∞ and f is C 1 in a neighborhood of 0.
Third step: existence of α > 0 such that F (0, (v, q)) E2 ≥ α (v, q) E1 . We consider a pair (v, q) ∈ E 1 and we define (ξ, η, r) ∈ E 2 by F (0, (v, q)) = (ξ, η, r). Then, we have, for all ϕ ∈ H Setting ϕ i := t (0, . . . , 0, 1, 0, . . . , 0), (i = 1, . . . , N ), as test functions in the first line of the previous system, we check the compatibility condition of Appendix A.1.

1 (Ω \ ω),            Ω\ω {ν∇v : ∇ϕ -q div ϕ} = ξ , ϕ Ω\ω , div v = η in Ω \ ω, O v = r.
Thus, according to Appendix A.1, there exists w ∈ H 1 (Ω \ ω) such that Ω\ω w = 0 and such that, for all ϕ ∈ H

1 (Ω \ ω),    Ω\ω {ν∇w : ∇ϕ -q div ϕ} = ξ , ϕ Ω\ω , div w = η , in Ω \ ω,
and there exits a constant α 1 > 0 such that

w H 1 (Ω\ω) + q L 2 (Ω\ω) ≤ α 1 ξ [H 1 (Ω\ω)] ′ + η L 2 (Ω\ω) . (3.4)
Such a problem has a unique solution, we then get that

v = w - 1 |O| O w + r.
Thus, by computation, there exists a constant α 2 > 0 such that

v H 1 (Ω\ω) -|Ω \ ω| -1/2 |r| ≤ α 2 w H 1 (Ω\ω) .
Hence, using the above inequality and (3.4), there exists a constant α 3 > 0, such that

v H 1 (Ω\ω) + q L 2 (Ω\ω) ≤ α 3 ξ [H 1 (Ω\ω)] ′ + η L 2 (Ω\ω) + |r| . (3.5)
Mimicking the proof of the local regularity result Appendix A.2 (see inequality (A.9)), we use the argument stated above to prove the existence of a constant α 4 such that

Φv H 2 ( Ω δ \ω) + Φq H 1 ( Ω δ \ω) ≤ α 4 Φξ L 2 ( Ω δ \ω) + Φη H 1 ( Ω δ \ω) + v H 1 (Ω\ω) + q L 2 (Ω\ω) + |r| . Thus, since Φ is compactly supported in Ω δ , Φv H 2 (Ω\ω) + Φq H 1 (Ω\ω) ≤ α 4 Φξ L 2 (Ω\ω) + Φη H 1 (Ω\ω) + v H 1 (Ω\ω) + q L 2 (Ω\ω) + |r| . (3.6)
Gathering (3.5) and (3.6), there exists a constant α > 0 such that F (0, (v, q)) E2 ≥ α (v, q) E1 .

Fourth step: conclusion. By Simon's Theorem,

θ ∈ U → (v θ , q θ ) ∈ E 1
is differentiable in a neighborhood of 0. We conclude using the fact that

E 1 is continuously embedded in X 1,2 (Ω \ ω, Ω δ \ ω) × X 0,1 (Ω \ ω, Ω δ \ ω) (see (3.3)).
• ν ∇u 0 : ∇ϕ-p 0 div ϕ = ν ∇u : ∇ϕ-p div ϕ ∈ W 1,1 (Ω δ \ω) since ϕ belongs to H 2 (R N ), (u, p) belongs to H 2 (Ω δ \ω)×H 1 (Ω δ \ω) and Ω δ \ω is bounded.

Thus, using Hadamard's formula (Corollary 5.2.5 in Ref. [START_REF] Henrot | Variation et optimisation de formes[END_REF], we decompose the derivative I into I = J 1 + J 2 with

J 1 := Ω\Ω δ {ν ∇u ′ : ∇ϕ -p ′ div ϕ} , J 2 := Ω δ \ω {ν ∇u ′ : ∇ϕ -p ′ div ϕ} + ∂(Ω δ \ω) (ν ∇u : ∇ϕ -p div ϕ)(V • n).
Since V = 0 on ∂Ω δ , we have:

J 2 = Ω δ \ω {ν ∇u ′ : ∇ϕ -p ′ div ϕ} + ∂ω (ν ∇u : ∇ϕ -p div ϕ)(V • n). Therefore I = Ω\ω {ν ∇u ′ : ∇ϕ -p ′ div ϕ} + ∂ω (ν ∇u : ∇ϕ -p div ϕ)(V • n). (4.6)
By the same technics, we differentiate the right hand side of (4.5):

d dt Ω\ωt f • ϕ t=0 = Ω\ω div (f • ϕ) V = ∂ω (f • ϕ)(V • n). (4.7)
Gathering (4.5), (4.6) and (4.7), we get

Ω\ω {ν ∇u ′ : ∇ϕ -p ′ div ϕ} + ∂ω (ν ∇u : ∇ϕ -p div ϕ)(V • n) = ∂ω (f • ϕ)(V • n).
∇u : ∇ϕ = ∇ Γ u : ∇ Γ ϕ + ∂ n u • ∂ n ϕ.
Plugging this relation in (4.8), we obtain:

Ω\ω {ν ∇u ′ : ∇ϕ -p ′ div ϕ} + ∂ω (ν ∇ Γ u : ∇ Γ ϕ -p div Γ ϕ)(V • n) = ∂ω {(f • ϕ)(V • n) -ν∂ n u • ∂ n ϕ(V • n) + ∂ n ϕ • (p n)(V • n)} .
We use the boundary condition -ν∂ n u + p n = 0 on ∂ω to simplify this relation in:

Ω\ω {ν ∇u ′ : ∇ϕ -p ′ div ϕ}+ ∂ω (ν ∇ Γ u : ∇ Γ ϕ-p div Γ ϕ)(V •n) = ∂ω (f •ϕ)(V •n).
(4.9) Fifth step: derivative of the boundary values condition. We differentiate with respect to t the relation -ν∂ n u t + p t n = 0 on ∂Ω. At t = 0, we obtain on ∂Ω that -ν∂ n u ′ + p ′ n = 0. Let us set v t := u t • φ(t). By differentiation with respect to t (see Corollary 5.2.5 in Ref. [START_REF] Henrot | Variation et optimisation de formes[END_REF], we obtain u ′ = v ′ -∇u V on Ω δ \ ω, where v ′ is the material derivative of v t at t = 0. Thus, u ′ belongs to H 2 (Ω δ \ ω) as a sum of two terms in this space. Indeed, the derivative v ′ is the limit of the differential quotient (v t -v)/t that belongs to H 2 (Ω δ \ ω) since v t ∈ H 2 (Ω δ \ ω) for all t ∈ [0, T ). The second term ∇u V is also in H 2 (Ω δ \ ω) by the local regularity result Appendix A.2. The crucial point here is the C 2,1 regularity of ∂ω. Therefore, ∆u ′ ∈ L 2 (Ω δ \ ω) and, using (4.4),

∇p ′ ∈ L 2 (Ω δ \ ω). Now, consider ϕ ∈ H 2 (Ω δ \ ω) with compact support in Ω δ and compute Ω\ω {ν ∇u ′ : ∇ϕ -p ′ div ϕ}= Ω δ \ω {ν ∇u ′ : ∇ϕ -p ′ div ϕ} = Ω δ \ω (-ν ∆u ′ + ∇p ′ ) • ϕ + ∂(Ω δ \ω) (ν ∂ n u ′ -p ′ n) • ϕ = ∂(Ω δ \ω) (ν ∂ n u ′ -p ′ n) • ϕ by (4.4) .
Since ϕ = 0 on ∂Ω δ , the boundary term on ∂Ω δ cancels. Inserting (4.9), we obtain

∂ω (ν∂ n u ′ -p ′ n) • ϕ + ∂ω (ν∇ Γ u : ∇ Γ ϕ -p div Γ ϕ) (V • n) = ∂ω f • ϕ (V • n) . ( 4
p div Γ ϕ (V • n) = ∂ω {Hp (V • n) ϕ • n -ϕ • ∇ Γ (p (V • n))} , ∂ω -ν∇ Γ u : ∇ Γ ϕ (V • n) = ∂ω {-ϕ • div Γ (-ν∇ Γ u (V • n)) + Hϕ • (-ν∇ Γ u(V • n)n)} ,
where H is the mean curvature of ∂ω. Inserting these results in (4.10), we obtain

-ν∂ n u ′ + p ′ n = -f (V • n) -div Γ (ν∇ Γ u(V • n)) + νH∇ Γ u(V • n)n -Hp(V • n)n + (V • n)∇ Γ p + p∇ Γ (V • n). (4.11)
We used the fact that ∇

Γ (p (V • n)) = (V • n)∇ Γ p + p∇ Γ (V • n)) and
(see (1.1)), ∇ Γ u∇ Γ (V • n) = ∇u∇ Γ (V • n). We expand the tangential divergence div Γ (ν∇ Γ u(V • n)) = ν div Γ (∇ Γ u) (V • n) + ν∇ Γ u∇ Γ (V • n)
Γ (ν∇ Γ u(V • n)) = ν∆ Γ u(V • n) + ν∇u∇ Γ (V • n). ( 4
-ν∂ n u ′ + p ′ n = ν∂ 2 nn u(V • n) -∇p(V • n) -ν∇u∇ Γ (V • n) + νH∇ Γ u(V • n)n + (V • n)∇ Γ p + p∇ Γ (V • n) on ∂ω,
since (u, p) satisfies the boundary conditions -ν∂ n u + pn = 0 on ∂ω. We conclude the proof by noting that ∇ Γ u n = 0.

First order shape derivatives of the functional. Define, for all t ∈ [0, T ),

J(ω t ) := j(t) := O (u t -f b ) 2 .
(of Proposition 2.6). First step: derivative of j and adjoint problem. By Proposition 2.5, (u, p) is differentiable with respect to the shape. We denote their respective derivative by u ′ and p ′ . Differentiating j with respect to t at 0, we obtain

j ′ (0) = O 2u ′ (u -f b ).
Then, we consider the adjoint problem (2.3). According to Appendix A.1, it admits a unique solution (w, q) ∈ H 1 (Ω \ ω) × L 2 (Ω \ ω) with O w = 0. Indeed, the compatibility condition for the adjoint is true as consequence of our choice of compatibility condition for the state.

Second step: writing of j ′ (0) as an integral on ∂ω. We proceed by successive integrations by parts. We multiply the first equation of the adjoint problem (2.3) by u ′ to get: since div w = 0 in Ω \ ω. Therefore, gathering (4.14) and (4.15) and using the boundary conditions of (u ′ , p ′ ) and (w, q) (see problems (2.2) and (2.3)), we obtain Characterization of the shape Hessian at a critical point. We consider ω * a critical shape of the functional J. We assume that ω * has a C 2,1 boundary. Thanks to the local regularity result Appendix A.2,

O 2(u -f b )u ′ = ∂ω ν∂ 2 nn u -∂ n p n (V •n) + p∇ Γ (V •n) -ν∇u ∇ Γ (V • n) • w = ∂ω ν∂ 2 nn u -∂ n p n • w (V •n) + t (p I -ν∇u) w • ∇ Γ (V •n) .
(u, p) ∈ H 3 (Ω δ \ ω * ) × H 2 (Ω δ \ ω * ) and (u ′ , p ′ ) ∈ H 2 (Ω δ \ ω * ) × H 1 (Ω δ \ ω * ).
(of Proposition 2.7). First step : second order shape differentiability. By Lemma 3.5, (u, p) is twice differentiable with respect to the domain. We denote by (u ′′ , p ′′ ) the second order shape derivatives.

Second step: second derivative of j and derivative of the adjoint problem. Consider V ∈ U . We differentiate the function j twice with respect to t. At t = 0, it holds

j ′′ (0) = D 2 J(ω) • V • V = O 2(u ′ ) 2 + 2 u ′′ (u -f b ) .
Since ω * solves the inverse problem, u = f b on O. Therefore

D 2 J(ω * ) • V • V = O 2(u ′ ) 2 .
We introduce (w, q) ∈ H 1 (Ω\ω)×L 2 (Ω\ω) with O w = 0 the solution of the adjoint system (2.3). As we characterized u ′ and p ′ (see Proposition 2.5), we characterize w ′ and q ′ , the shape derivatives of w and q to get

-ν∂ n w ′ + q ′ n = ν∂ 2 nn w -∂ n q n (V • n) + q ∇ Γ (V • n) -ν∇w∇ Γ (V • n) on ∂ω.
In particular, for ω = ω * , u = f b on O. Uniqueness of the solution of Stokes problem then enforces that (w, q) = (c, 0) where c is constant. Using the fact that w = c and q = 0, we obtain system (2.4).

Third step: writing of j ′′ (0) as an integral on ∂ω. We multiply the first equation of problem (2.4) by u ′ to get Ω\ω * ν∇w ′ : ∇u ′ = --ν∂ n w ′ + q ′ n , u ′ ∂Ω . (4.16)

  be the space of admissible deformations. If θ ∈ U , then (I + θ) is a diffeomorphism by Banach's fixed point Theorem. For such a θ with Supp θ ⊂ Ω and ω ∈ O δ , we check Ω = (I + θ)(Ω) and we define the perturbed domain ω θ :=

2. 1 .

 1 The Dirichlet case Identifiability result We first quote an identifiability result in the Dirichlet case (see Theorem 1.2 in Ref. 4) proved by Alvarez et al.: it states that given a fixed g, two different geometries ω 0 and ω 1 in O δ yield two different measures f b1 and f b2 . Hence problem (1.6) admits a unique solution. Theorem 2.1 (C. Alvarez, C. Conca, L. Friz, O. Kavian, J.H. Ortega, Ref. 4). Let Ω ⊆ R N , N = 2 or N = 3, be a bounded Lipschitz domain, and O a non-empty open subset of ∂Ω. Let ω 0 , ω 1 ∈ D ad := {ω ⊂⊂ Ω; ω is open, Lipschitz and Ω \ ω is connected} and g ∈ H 3/2 (∂Ω) with g = 0, satisfying the flux condition (1.4). For ǫ * = 0 or ǫ

  Detecting an obstacle immersed in a fluid by shape optimization methods 7

  Detecting an obstacle immersed in a fluid by shape optimization methods 9

8 ( 11 2. 3 .

 8113 Compactness at a critical point). The Riesz operator corresponding to D 2 J(ω * ) defined from H 1/2 (∂ω * ) to H -1/2 (∂ω * ) is compact.Detecting an obstacle immersed in a fluid by shape optimization methods About the lack of stability.

Theorem 3 . 1 (

 31 J.Simon, Ref. 23). We give us

  Detecting an obstacle immersed in a fluid by shape optimization methods 13

  Badra, F. Caubet, M. Dambrine as test functions in the variational formulation (3.2), we have

  Badra, F. Caubet, M. Dambrine

(4. 8 )

 8 We use the definition of tangential gradient and divergence and the formulae(∂ n u⊗n) : (∂ n ϕ⊗n) = ∂ n u•∂ n ϕ and ∇u : (∂ n ϕ⊗n) = ∂ n u•∂ n ϕ = ∇ϕ : (∂ n u⊗n) to get

  Detecting an obstacle immersed in a fluid by shape optimization methods 23

  Badra, F. Caubet, M. Dambrine and obtain the decomposition div

  Ω\ων∇w : ∇u ′ = --ν∂ n w + q n , u ′ ∂(Ω\ω) ,(4.14)since div u ′ = 0 in Ω \ ω (see Proposition 2.5). Then, we multiply the first equation of the problem (2.2) by w to obtainΩ\ω ν∇u ′ : ∇w = -∂(Ω\ω) (-ν∂ n u ′ + p ′ n) • w ,Detecting an obstacle immersed in a fluid by shape optimization methods 25

Finally, we integrate

  by parts on the boundary ∂ω to obtainO 2(uf b )u ′ = ∂ω ν∂ 2 nn u -∂ n p n • wdiv Γ (p Iν t ∇u)w (V • n) .

  and Afraites et al. in Ref. 2, Ref.

  Simon (see Theorem 6 in Ref. 23) that we recall the statement for the reader's convenience. All the proofs are done for the Neumann case. Adaption to the Dirichlet case is straightforward. Auxiliary results on Stokes equations with Neumann conditions (an existence and uniqueness of the solution theorem and a local regularity result) are recalled in Appendix A.

	January BadraCaubetDambrine-m3as 7,	2011 9:22 WSPC/INSTRUCTION	FILE
	12 M. Badra, F. Caubet, M. Dambrine

  the density in L 2 (∂ω) of the traces on ∂ω of functions H 2 (Ω δ \ ω) with compact support in Ω δ .Let us simplify this expression. According to the definition of the tangential gradient

  .12) Using Stokes equations -ν∆u + ∇p = f in Ω \ ω and the decomposition of the Laplace-Beltrami operator (see Proposition 5.4.12 in Ref. 20), we compute f = -ν ∆ Γ u + H∂ n u + ∂ 2 nn u + ∇p on ∂ω.

	(4.13)
	Inserting (4.12) and (4.13) in (4.11), we obtain

Detecting an obstacle immersed in a fluid by shape optimization methods 17

Proof. (of Lemma 3.3: differentiability of θ → (u θ , p θ )). We use the chain rule. We only prove the result on u θ because the ideas are exactly the same for the result on p θ .

For θ ∈ U , u θ = v θ • (I + θ) -1 ∈ H 2 (Ω δ \ ω θ ). According to Lemma 3.2 and Stein's extension Theorem (Theorem 5.24 in Ref. [START_REF] Adams | Pure and Applied Mathematics[END_REF], there exists v θ , an extension of v θ , such that θ ∈ U → v θ ∈ H 2 (Ω δ ) is differentiable at 0. Moreover,

is differentiable at 0. Thus,

is differentiable at 0. We apply Lemma 5.3.9 in Ref. [START_REF] Henrot | Variation et optimisation de formes[END_REF], to get that

is C 1 in a neighborhood of 0. By composition, ϕ 2 • ϕ 1 is differentiable at 0. Then, we define an extension of u θ as

Using Lemma 3.2, the function

is differentiable at 0. Moreover, on a neighborhood V(∂Ω δ ) of ∂Ω δ , u θ = u θ = v θ . Thus, using Lemma 3.2, θ ∈ U → u θ V(∂Ω δ ) ∈ H 1 (V(∂Ω δ )) is differentiable at 0. Therefore θ ∈ U → u θ ∈ H 1 (Ω) is differentiable at 0.

Higher order differentiability. To prove that (u, p) is twice differentiable with respect to the shape, we use the second order differentiability of θ → (v θ , q θ ). Therefore, we first prove the following lemma:

Lemma 3.4. The function

is twice differentiable in a neighborhood of 0.

Proof. We use Simon's Theorem 3.1. Let the open set Ω δ defined in the proof of Lemma 3.2 and Φ ∈ C ∞ c ( Ω δ ) such that Φ ≡ 1 in Ω δ . We consider the functions F and f defined in the proof of Lemma 3.2 with modified sets of departure and destination: We adapt the proof of Lemma 3.2 to obtain that F and f are C 2 in a neighborhood of 0. Indeed, since f ∈ H 3 (R N ), then the function Lemma 5.3.9 in Ref. 20). Moreover, we check last assumption of Theorem 3.1 (the estimation) like in the proof of Lemma 3.2. We use in particular the local regularity result Appendix A.2. Therefore, the function

is twice differentiable in a neighborhood of 0. Lemma 3.5 (Second order shape differentiability). The solution (u, p) is twice differentiable with respect to the domain.

Proof. Let us set v t := u t • φ(t) and q t := p t • φ(t) where φ(t) = I + tV . Using the Fréchet differentiability Lemma 3.4, we obtain the Gâteaux differentiability in the direction V , the functions

are two times differentiable at 0. Using Stein's extension Theorem, there exists

two extension operators linear continuous. Thus, using Lemma 5.3.9 in Ref. [START_REF] Henrot | Variation et optimisation de formes[END_REF], the functions

are twice differentiable in a neighborhood of 0. Moreover, the functions

are twice differentiable in a neighborhood of 0 by composition. Thus, defining

we conclude as in the proof of Lemma 3.3.

with u = 0 in D 0 , then u = 0 in Ω 0 and p is constant in Ω 0 .

We deduce from the above theorem the following corollary:

where O ⊂ ∂Ω 0 is a relatively open non-empty subset, then u = 0 and p = 0 in Ω 0 .

Proof. (of Corollary 4.1). We enlarge the domain by a ball: we fix a point y in O, we set ρ := d(y, ∂O)/2 and set Ω ρ := Ω 0 ∪ B(y, ρ). We define

Since u ∈ H 1 (Ω 0 ) and u = 0 on O, we check that p ∈ L 2 (Ω ρ ) and u ∈ H 1 (Ω ρ ). By summing on Ω 0 and on Ω ext ρ := {Ω ρ ∩ B(y, ρ)} \ O, we get that for all ϕ in C ∞ c (B(y, ρ)):

Proceeding as above, we also get div u = 0 in D ′ (Ω ρ ). Moreover, u = 0 in Ω ext ρ ⊂ Ω ρ . Thus, according to the unique continuation Theorem 4.1, u = 0 and p is constant in Ω ρ . Since p = 0 in Ω ext ρ , p = 0 in Ω ρ then on Ω 0 by restriction. We proceed by contradiction. Assume that ω 0 \ ω 1 is non-empty. We know that

Proof. (of Theorem 2.2). We define u

Multiplying this equation by u 1 and integrating by parts in ω 0 \ ω 1 with f ≡ 0 in Ω δ and -ν∂ n u 1 + p 1 n = 0 on ∂ω 1 , we obtain

Since u 0 = u 1 and p 0 = p 1 in Ω \ ω, the boundary condition satisfied by u 0 on ∂ω 0 provides

Hence, equality (4.2) is simply

Hence, u 1 is constant in ω 0 \ ω 1 . By Theorem 4.1, u 1 and p 1 are then constant in Ω\ω 1 and -ν∆u 1 +∇p 1 = 0 in Ω\ω 1 . This contradicts f = 0 and then

First order shape derivatives of the state. We recall the variational formulation of the problem (2.1) in Ω \ ω t :

Proof. (of Proposition 2.5).

First step: first order shape differentiability. Using the Fréchet differentiability Lemma 3.3, we obtain the Gâteaux differentiability in the direction V : there exist u t and p t , respective extensions in Ω of u t and p t such that the functions, defined on [0, T ), t → u t ∈ H 1 (Ω) and t → p t ∈ L 2 (Ω) are differentiable at 0 by composition. We denote by u ′ and p ′ their respective derivative at 0.

Second step: derivative of the normalization condition. The function

is constant on [0, T ). Thus, its derivative is 0. By differentiation under the sum sign, we get Third step: derivative of the equalities on Ω \ ω t . Let ϕ ∈ D(Ω \ ω). As Ω \ ω is open, we have, for t small enough, ϕ ∈ D(Ω \ ω t ). Therefore, using the first line of the variational formulation (4.3), we have for all t ∈ [0, T )

We differentiate with respect to t at t = 0

As it is true for all ϕ ∈ D(Ω \ ω), we get

Proceeding with the second line of the variational formulation (4.3) as previously, we check that div u ′ = 0 in D ′ (Ω \ ω).

Fourth step: variational characterization of u ′ and p ′ . Let ϕ ∈ H 2 (Ω \ ω). Using Stein's extension Theorem, ϕ admits an extension in H 2 (R N ) still noted ϕ. This extension is in particular in H 2 (Ω \ ω t ). Using ϕ as a test function in the first line of the variational formulation (4.

so that for t in a neighborhood of 0,

). Then, we have

We multiply the first equation of problem (2.2) by w ′ to get

Therefore, gathering (4.16) and (4.17), we obtain

Justifying the ill-posedness of the problem. The compactness of the Riesz operator corresponding to the shape Hessian at possible solutions of the inverse problem is here proved using a local regularity argument. We do not use the hydrodynamical potential layers as it was done in Ref. 2 for the Laplacian case. This should provide an alternative proof.

To prove Proposition 2.8, we first decompose the shape Hessian at a critical point: for

Here, T u,p :

and where

where ψ is the trace on ∂ω * of the solution v of the following Stokes problem

We study each operator: Lemma 4.1 states that T u,p and M 1 are linear continuous and Lemma 4.2 claims that M 2 is compact. Hence, compactness is obtained by composition of linear continuous operator with a compact one.

Lemma 4.1. The operators T u,p and M 1 are linear continuous.

Proof. Since u and p don't depend on V and using the continuity of the tangential gradient operator ∇ Γ , the operator T u,p is linear continuous as multiplier by a smooth function (see Ref. [START_REF] Maz | Theory of multipliers in spaces of differentiable functions[END_REF]). Let V ∈ H 1/2 (∂ω * ). We define M 1 (V ) =: u ′ and

Since ∇ Γ is linear continuous from H 1/2 (∂ω * ) into H -1/2 (∂ω * ), Detecting an obstacle immersed in a fluid by shape optimization methods 27 is linear continuous. Moreover the following operator is linear continuous:

where ψ is the trace on ∂Ω of the solution v of the problem

Proof. According to the local regularity result Appendix A.2, the operator

is linear continuous. Moreover, the domain Ω δ \ ω * is bounded and the embedding of H 2 (Ω δ \ ω * ) in H 1 (Ω δ \ ω * ) is then compact. Thus, since the trace operator of ∂ω * from H 1 (Ω δ \ ω * ) to H 1/2 (∂ω * ) is linear continuous, we get, by composition, that M 2 is compact.

Appendix A. Some results on the Stokes problem with Neumann conditions

In order to be complete, we choose to recall classical results with short proofs about the Stokes problem with Neumann conditions: a theorem of existence and uniqueness of the solution and a local regularity result. We refer to Ref. 9 and Ref. [START_REF] Grubb | Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods[END_REF] for a detailed study, in particular for the regularity of the solutions. In all this appendix, we note C a generic positive constant only depending on the geometry of the domain and which may change from line to line.

For beginning, let us introduce two notations: for Ω an open set of R N , we define

Theorem Appendix A.1 (Existence and uniqueness of the solution).

Let

admits a unique solution and the following estimate holds:

Moreover, for k ≥ 0, if the open set Ω has a C k+1,1 boundary and if

then the pair (u, p) belongs to

and the following estimate holds:

.

Proof. Existence and uniqueness. For the proof of the first part of Appendix A.1, let us begin by studying the case of null divergence. According to Lax-Milgram's Theorem, there exists a unique u ∈

and we have

(Ω) as a test function in (A.2). According to the compatibility condition, we check that (A.2) is also true for all v ∈ V 1 (Ω) and in particular for all v ∈ V 1 (Ω) ∩ H 1 0 (Ω). Then, using De Rham's Theorem (see for example Lemma 2.7 in Ref. 7), there exists p ∈ L 2 (Ω), up to an additive constant, such that We define ϕ N (x) := t (x 1 , ..., x N )/N , where N is the dimension and (x 1 , ..., x N ) are the coordinates of x: ϕ N is such that div ϕ N = 1. Let v ∈ H 1 (Ω) and

Using Theorem 3.2 in Ref. 17 or Lemma 3.

According to (A.2) (which remains valid for v ∈ V 1 (Ω)) and (A.4) and choosing the additive constant for p such that

Therefore, there is a unique pair (u, p)

(Ω) (see Theorem 3.2 in Ref. 17 or Lemma 3.3 in Ref. 7). Using v in (A.5), and according to (A.3), we obtain

The first part of the theorem is proved for g = 0. The case g = 0 is obtained by a lifting argument. Let us define

where

and where, according to Theorem 3.2 in Ref. 17 or Lemma 3.

Thus, defining u 0 := uu g and

Regularity of the solution. For the second part of Appendix A.1 concerning the regularity of (u, p), we refer to Section 5.3 in Ref. 9 and Ref. [START_REF] Grubb | Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods[END_REF] for the homogeneous case g = 0 and f b = 0. To obtain the result for g = 0 and f b = 0 let us define

where (ϕ g ) τ is the tangential component of ϕ g and H is the mean curvature of ∂Ω.

According to Theorem A in Ref. [START_REF] Amrouche | Problèmes généralisés de Stokes[END_REF], there exists u ∈ H k+2 (Ω) such that div u = g in Ω, u = ϕ g and ν∂ n u = -(f b ) τ + Ψ(g)n + νgn on ∂Ω, and which satisfies the following estimate:

Moreover, with a continuous right inverse of the trace operator, we define a pressure function p ∈ H k+1 (Ω) such that p = f b • n + Ψ(g) + g on ∂Ω and which depends continuously on (g, f b ) in H k+1 (Ω) × H k+1/2 (Ω). Then we have constructed a pair ( u, p) such that

and which obeys the following estimate:

Finally, we check that the vector field

and, since regularity results in the homogeneous case (see Ref. 9 and Ref. [START_REF] Grubb | Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods[END_REF]) ensures that the solution to:    ν∆w + ∇q = F in Ω, div w = 0 on Ω, -ν∂ n w + q n = 0 on ∂Ω, belongs to H k+2 (Ω) × H k+1 (Ω) and satisfies:

the conclusion follows by remarking that (u, p) = (w + u, q + p) and by combining (A.6) with (A.7).

To enunciate the following result concerning the local regularity of the solution of the Stokes problem, we recall some notations introduced in section 3: for k, m ∈ N, k < m and for two open sets Ω 1 and Ω 2 such that Ω 2 ⊂ Ω 1 , we note X k,m (Ω 1 , Ω 2 ) the space of functions in H k (Ω 1 ) such that their restriction to Ω 2 belongs to H m (Ω 2 ). Similarly, X * ,m (Ω 1 , Ω 2 ) is the space of functions in H 1 (Ω 1 ) ′ such that their restriction to Ω 2 belongs to H m (Ω 2 ). where f := ϕf -νu∆ϕ -2ν∇u∇ϕ + p∇ϕ ∈ L 2 (C ′ ) and g := ϕg + u • ∇ϕ ∈ H 1 (C ′ ). From Appendix A.1, ϕu ∈ H 2 (C ′ ), ϕp ∈ H 1 (C ′ ) and, taking into account of the expression of f and g, we also have

(A.9) Using this inequality and the estimate on u H 1 (Ω\ω) and p L 2 (Ω\ω) given by Appendix A.1, we obtain the announced estimate for k = 0. Proceeding as above, we conclude by induction.