R Germs
email: r.germs@rug.nl

J Riezebos

Workload Balancing Capability of Pull Systems in MTO Production

Keywords: CONWIP, POLCA, m-CONWIP pull, MTO, workload balancing, POLCA, CONWIP, m-CONWIP

Pull systems focusing on throughput time control and applicable in situations with high variety and customisation are scarce. This paper compares three unit-based pull systems that can cope with such situations: POLCA, CONWIP and m-CONWIP. These systems control the shop floor throughput time of orders by limiting the number of orders on the shop floor. However, their effectiveness in terms of reducing total throughput time is questioned. Theory states that an improvement of the average total throughput time will be due to the workload balancing capability of a pull system, but that many pull systems lack this capability. This paper shows this workload balancing capability to exist for POLCA and m-CONWIP, but not for CONWIP. The magnitude of the effect strongly differs, depending on the configuration of the system, the order arrival pattern and the variability of the processing time of the orders.

Nowadays, many make-to-order (MTO) companies focus on realising short throughput times as a competitive edge. Material control is an important part of the chain of tools used in realising short throughput times. A material control system regulates the flow of goods on the shop floor. This includes the authorisation to start an order, release of new material on the shop floor, setting priorities for orders that are waiting to be processed, and initiating the start of succeeding activities, such as transport, quality control, et cetera.

Pull systems are a special type of material control system. They aim to control the throughput times of orders by limiting the amount of work (workload) on the shop floor [START_REF] Hopp | To Pull or not to Pull: What is the question?[END_REF]). The simplest way to limit the workload on the shop floor is by controlling the number of orders on the shop floor. Alternatively, the workload can be limited based on the work content (processing time) of orders. We refer in this paper to pull systems that control the number of orders on the shop floor as unit-based pull systems and to pull systems that limit the workload based on the work content of orders as load-based pull systems. The Kanban material control system [START_REF] Sugimori | Toyota production system and kanban system: Materialization of just-in-time and respect-for-human system[END_REF]) is a well-known unitbased pull system, while WLC (Work Load Control, see [START_REF] Gaalman | Workload control in job shops: an introduction to the special issue[END_REF] for a discussion) is the most sophisticated example of a load-based pull system. We restrict our attention to unit-based pull systems that control the throughput time of orders in an MTO environment. However, unit-based pull systems that are applicable in an MTO environment are scarce. This paper discusses the throughput time performance of three unit-based pull systems that, according to [START_REF] Stevenson | A review of production planning and control: the applicability of key concepts to the make-to-order industry[END_REF], seem suitable in an MTO environment: POLCA, CONWIP and m-CONWIP.

The throughput time performance of a pull system in an MTO environment depends on its capability to create a balanced distribution of the workload among the workstations on the shop floor. This capability of a pull system is known in the literature as the workload balancing capability (see [START_REF] Land | The performance of workload control concepts in job shops: improving the release method[END_REF]. The workload balancing capability of a pull system causes a better control of the arrival moment of orders at the workstations on the shop floor. As a consequence, the average queue length required in front of these workstations to achieve a given utilisation level becomes lower. This reduces the time between the release and completion of an order and might reduce the time between the arrival and completion of an order. We refer to the former as the shop floor throughput time and to the latter as the total throughput time of orders (see Section 2 for a detailed description of these two throughput time measures). We call the workload balancing capability of a pull system effective when the limit on the workload results in both a reduction of the average shop floor throughput time and average total throughput time compared to the non-limited system.

There are very few papers that are able to show the effective workload balancing capability of pull systems. Literature on workload control (e.g. Kanet 1988, Melnyk and[START_REF] Melnyk | Order review/release and its impact on the shop floor[END_REF] even suggests the existence of a paradox related to the absence of this workload balancing capability in pull systems. While practical implementations show significant reductions in the total throughput time of orders, simulation studies show that constraining the workload on the shop floor leads to both shorter shop floor throughput times and longer total throughput times.

There are some studies, such as those of [START_REF] Land | The performance of workload control concepts in job shops: improving the release method[END_REF], [START_REF] Breithaupt | The workload control concept: theory and practical extensions of Load Oriented Order Release[END_REF][START_REF] Land | Workload control in job shops, grasping the tap[END_REF], that show the existence of effective workload balancing capability in loadbased pull systems. However, we are not aware of any study that shows the existence of effective workload balancing capability in unit-based pull systems.

The central question in this paper is whether unit-based pull systems can have effective workload balancing capability in an MTO environment. We introduce an MTO production system that perfectly suits pull systems that are able to balance workload. By means of a simulation study, we analyse the workload balancing capability of POLCA, CONWIP and m-CONWIP in this specific production system. In the simulation study several experimental The structure of this paper is as follows. Section 2 gives attention to pull systems in MTO environments and describes the characteristics of the three unit-based pull systems we consider in this paper. Section 3 presents the research questions and Section 4 the design of the simulation study. Section 5 discusses the results of the simulation study and Section 6 concludes.

Pull systems

As we briefly mentioned in the introduction, literature on unit-based pull systems that are applicable in an MTO environment is scarce. Well-known unit-based pull systems such as Kanban are designed for make-to-stock (MTS) situations, as they use small intermediate stocks. In these pull systems, cards or containers (bins) are directly related to a specific product type. For example, an empty bin signals that it should be filled with exactly the same product type as before. For MTO companies, such a direct relation between signal and product type is not useful. MTO companies face a much higher product variety, which would lead to a very large number of different bins or card loops. Next, the repetition of identical orders is not that frequent, which would lead to long waiting times of the intermediate stock in a bin. The combination of both effects would result in large work-in-process inventories.

There are some unit-based pull systems that are applicable in MTO companies [START_REF] Stevenson | A review of production planning and control: the applicability of key concepts to the make-to-order industry[END_REF]. However, the unit-based pull systems that seem suitable for MTO companies (POLCA, CONWIP and m-CONWIP according to [START_REF] Stevenson | A review of production planning and control: the applicability of key concepts to the make-to-order industry[END_REF] receive only limited attention in performance comparisons. [START_REF] Framinan | The CONWIP production control system: review and research issues[END_REF] provide an overview of 15 comparison studies of CONWIP with other material control systems, but only two of these studies consider the applicability of these systems in an MTO environment. The POLCA system is not included in one of these comparison studies. [START_REF] Fernandes | Generic POLCA-A production and materials flow control mechanism for quick response manufacturing[END_REF] do include the performance of POLCA, but again for an MTS system. Studies that analyse the throughput time performance of unit-based pull systems in an MTO environment are therefore still largely lacking. This paper aims at making a start to fill this gap in the literature by considering the throughput time performance of POLCA, CONWIP and m-CONWIP in an MTO environment.

In order to determine the throughput time performance of these unit-based pull systems, we distinguish in this paper three (complementary) measures of the throughput time of orders that can be influenced by controlling the workload in an MTO production system. Figure 1 illustrates these measures graphically in the flow of an order through an MTO production system. Orders arrive to the production system and the material control system determines when an order is released to the shop floor. Until release of the order to the shop floor it waits in the order pool. As Figure 1 illustrates, we refer to the average time an order spends waiting in the order pool as the average order pool time (OPT) and to the average time between the release and completion of an order as the average shop floor throughput time (STT). The average total throughput time (TTT) is defined as the average time between the arrival and completion of an order and is therefore the sum of the OPT and the STT.

<Insert Figure 1 here> Pull systems can influence the STT by limiting the workload on the shop floor. If the total workload on the shop floor is low, according to Little's law [START_REF] Little | A proof for the queuing formula: L = λ[END_REF], the STT will be short. However, a short STT does not necessarily mean a short TTT. The workload limit blocks the release of orders whenever the workload limit is reached. When this happens, arriving orders have to wait in the order pool until the workload on the shop floor drops below the limit. Limiting the release of orders to the shop floor therefore increases the OPT. As a result, the reduction in the STT can be offset by the increase in the OPT. In order to reduce the TTT, the pull system must improve the balance of the workload on the shop floor, such that the variability of the workload at each workstation decreases. The resulting more balanced arrival pattern of orders at each workstation leads to less blocking of the release of orders and thereby to a shorter OPT for a given STT.

Pull systems can balance the workload on the shop floor by restricting the release of orders to workstations that are busy and by releasing orders to workstations that are waiting for work. Figure 2a and 2b illustrate workload balancing on a shop floor consisting of four workstations (A, B, C and D). Orders can follow two different routings on the shop floor: white orders follow route A B C, whereas black orders follow route A B D. In Figure 2a, the workload on the shop floor is not balanced. The variability of the workload at workstations C and D is large. For example, workstation C is very busy, while workstation D is waiting for orders. In a short while the opposite situation will occur, as a lot of work directed for workstation D is waiting to be processed at workstation A and B.

<Insert Figure 2 here> Figure 2b shows the same shop floor, but this time the workload is balanced among the workstations. The variability of the workload at the workstations in this system is lower than in the unbalanced shop floor of Figure 2a.

Theory leads to our expectation that a reduction in both STT and TTT can only be achieved if the pull system has sufficient workload balancing capability. Pull systems that locate the workload in the production system ineffectively will have a low workload balancing capability. Location of workload is mainly determined by the pull structure, i.e. the specific pattern of control loops that regulates the workload on the shop floor [START_REF] Gaury | Designing Pull Production Control Systems: Customization and robustness[END_REF].

In the next subsections we describe the pull structure of CONWIP, m-CONWIP and POLCA in detail.

CONWIP and m-CONWIP

The simplest unit-based pull system that is also applicable in MTO production is CONWIP [START_REF] Spearman | CONWIP: A pull alternative to kanban[END_REF]. Our description refers to the basic CONWIP system, but several variants of CONWIP have been developed (see e.g. [START_REF] Framinan | The CONWIP production control system: review and research issues[END_REF]. The basic CONWIP system can easily be explained by considering the production system in Figure 3.

<Insert Figure 3 here>

The production system consists of an order pool and four workstations (A, B, C and D).

The flow of orders between workstations A, B, C and D in the system is depicted by the thick arrows. After release, orders can follow two different routings on the shop floor. The dashed loop shows the part of the production system where the workload is controlled by cards. This loop is called the control loop. The CONWIP system has one control loop that limits the workload on the shop floor. This works as follows. An order may only enter the shop floor if a free card can be attached to it. Upon completion, the order leaves the shop floor to fulfil the customers demand while the attached card returns to the entrance of the shop floor, where it waits until it is attached to another order in the order pool. In the CONWIP system, each time an order leaves the shop floor the order pool receives a signal to authorise the release of a new order. Because no order can enter the shop floor without a card attached to it, the number of orders on the shop floor is limited by the number of cards circulating on the shop floor.

CONWIP uses a single control loop covering all workstations on the shop floor. This loop Therefore, the CONWIP system has no workload balancing capability.

Instead of using just one control loop for the whole shop floor, we can introduce a CONWIP loop for every possible routing on the shop floor. We denote such a system as "m-CONWIP", where the m stands for multiple CONWIP loops. Figure 4 gives an illustration of an m-CONWIP system. There are two routings on the shop floor and, therefore, the m-CONWIP system consists of two CONWIP loops. The two loops in this system balance the work among routings A B C and A B D by constraining the amount of orders that are allowed in these routings separately.

<Insert Figure 4

here>

In the CONWIP and m-CONWIP system that we consider in this paper, a free card signals the release opportunity of a new order for the loop to which the card belongs. This means that in the order release decision, the processing time of the new order is not taken into account.

Hence, CONWIP and m-CONWIP constrain the release based on the numbers of orders on the shop floor and are therefore unit-based pull systems.

POLCA

POLCA [START_REF] Suri | Quick Response Manufacturing: A Companywide Approach to Reducing Leadtimes[END_REF][START_REF] Riezebos | Design of POLCA material control system[END_REF]) is a pull system according to the definition of [START_REF] Hopp | To Pull or not to Pull: What is the question?[END_REF] because of its triggering authorisation mechanism. The triggering mechanism is a card system, which can be implemented either physically or electronically [START_REF] Vandaele | Load-based POLCA: An integrated material control system for multiproduct, multimachine job shops[END_REF]. Figure 5 displays a POLCA controlled MTO production system. In the POLCA system each control loop covers two workstations. An order is allowed to start production on a given workstation when a card is available for the loop the order is trying to enter. Similar as in the CONWIP system, the number of cards in a loop limits the number of orders that are allowed in that loop.

<Insert Figure 5 here> POLCA uses overlapping loops for orders that need to visit more than two workstations, as shown in Figure 5. The overlapping loops assure that workstation B will only process orders for which in the near future capacity becomes available in workstations C and/or D downstream. For example, if no card B C is available in workstation B, it means that workstation C is backlogged with work. Working on an order destined for workstation C would only increase the inventory on the shop floor, since workstation C has a lack of capacity to work on this order. It is better to process another order, for example one that needs further processing in workstation D. In this way the POLCA system balances work between workstation C and D [START_REF] Suri | How to plan and implement POLCA: A material control system for high-variety or custom-engineered products[END_REF]. The basic POLCA system is indifferent with respect to the amount of work (in processing time units) represented by a POLCA card. [START_REF] Suri | Quick Response Manufacturing: A Companywide Approach to Reducing Leadtimes[END_REF] notes that each order will have one new POLCA card attached to indicate the workstation to be visited after all operations of this order in the current workstation have been completed. However, if the processing time per order differs too much, the original POLCA system can be transformed such that the number of cards in each loop is replaced by an allowable workload (in processing time units) for that loop [START_REF] Vandaele | Load-based POLCA: An integrated material control system for multiproduct, multimachine job shops[END_REF]. In this paper we consider the basic, unit-based POLCA system.

Research questions

In the previous section we discussed how the pull structures of POLCA, CONWIP and m-CONWIP control (i.e. limit and balance) the workload in an MTO production system. We stated our expectation that a pull system can only reduce the STT and TTT if it has sufficient workload balancing capability. Since CONWIP has no workload balancing capability, we expect that limiting the workload in a CONWIP controlled MTO production system increases the TTT of orders. In Sections 2.1 and 2.2 we explained that the m-CONWIP and POLCA system balance the workload in the system in different ways. m-CONWIP uses multiple CONWIP loops -one for every routing in the production system -to balance the workload among the different routings in the production system. POLCA balances the workload by releasing an order based on the available capacity in the next workstation in the order's routing. Which of these two pull structures has the best performance with respect to workload balancing is one of the research questions we would like to answer in this paper.

Besides the pull structure we expect that the configuration of the pull system has a large influence on the workload balancing capability. The configuration of a pull system is defined as "the set of card numbers to be placed in the control loops" of the pull system (Gaury 2000, p. 12). If the number of cards in each control loop is large, the workload on the shop floor is not limited by any of the control loops of the pull system. This configuration can be used to represent a push system with immediate release, since in a push system there is no explicit limit on the workload that can be on the shop floor [START_REF] Hopp | To Pull or not to Pull: What is the question?[END_REF].

Figure 6 shows an illustrative example of the STT and TTT performance of a pull system that has effective workload balancing capability. The points in the figure represent the STT and TTT performance of different configurations of the pull system. In Figure 6, the point at the right end of the curve shows the throughput time performance of a push system. Note that in this point the TTT and the STT are equal since the OPT is zero in the push system.

When we move to the left of the curve in the figure, the configurations become more constrained and the STT decreases, while the OPT increases. In this example, when the configurations become more constrained the TTT first decreases and after a certain point the TTT increases rapidly. For the configurations below the dashed line in the figure, the reduction in STT, compared to the push system, is greater than the increase in OPT. This means that these configurations show the effective workload balancing capability of the pull system. For the configurations above the dashed line the increase in the OPT overcompensates the decrease in the STT. The lowest point of the curve shows the configuration for which the pull system obtains its optimal TTT performance. The difference between the TTT of this point and the TTT of the push system measures the maximum TTT reduction which can be obtained by the pull system.

<Insert Figure 6 here> We expect that effective workload balancing capability not only depends on the structure and configuration of the pull system. From the literature on queueing theory (see e.g. Buzacott and Shantikumar 1993) we know that factors such as the order arrival pattern and the variability of the processing time of orders have to be taken into account as well. Variability in the order arrival pattern increases the average queue length in front of the workstations on the shop floor and, thereby, the choice of orders in front of the control loops. We expect that this increase in choice improves the balancing capability of the control loops and reduces the blocking of the release of orders.

Variability in the processing time of orders increases the variability of the workload (in terms of processing time units) released into a control loop. This is because the pull systems that we consider in this paper control the workload on the shop floor based on the numbers of orders on the shop floor and not on the processing times of orders. Variability of the workload in a control loop increases the variability of the workload at each workstation and, thereby, decreases the workload balance. Therefore, we expect that processing time variability Note that processing time variability also increases the average queue length in front of the workstations in the system and, thereby, the choice of orders in front of the control loops. We expect, however, that this positive effect on the workload balancing capability will be offset by the negative effect caused by increased variability of the workload in the control loops.

The central question of our paper is whether m-CONWIP and POLCA can have effective workload balancing capability. We mentioned in this section the factors that we expect to influence the effective workload balancing capability of pull systems. Together with our central question, these expectations lead to the following four research questions that we address in this paper:

(1) Does the TTT of orders increase when the workload in an MTO production system is controlled by a CONWIP system?

(2) How do POLCA and m-CONWIP perform with respect to workload balancing?

(3) What influence has the configuration of POLCA and m-CONWIP on the workload balancing capability of these pull systems? (4) How sensitive is the workload balancing capability of POLCA and m-CONWIP to factors such as the order arrival pattern and the variability of the processing time of orders?

Experimental design

In the previous section we formulated our research questions. These questions will be analysed by means of a simulation study. In the next two subsections, we discuss the simulated production system and the performance measurement in detail. Figure 7 shows the topology of the simulated MTO production system. This type of MTO system can be denoted as a divergent segmented MTO system, where orders generally visit more than one operation [START_REF] Hyer | Reorganizing the factory : competing through cellular manufacturing[END_REF]. It perfectly suits pull systems that are able to balance workload. This specific topology of a production system enables us to identify whether m-CONWIP and POLCA can have effective workload balancing capability and whether the various experimental factors have a significant impact on the workload balancing capability of these pull systems.

<Insert Figure 7 here>

The production system consists of seven workstations (A to G) and an order pool. Each workstation can handle one order at a time. The capacity of the workstations is supposed to be constant during the experiments. Each operation requires one specific workstation. Customer orders are handled in an MTO strategy, i.e. production cannot be started until the customer order has arrived. There is no finished good inventory to fulfil demand. The routing of an order is known at the moment the order arrives. There are four different routings. The percentages on the thick arrows give the transition probability of going from one workstation to another.

Workstation processing time is an experimental variable; it is either constant or random (Erlang-2 distributed, squared coefficient of variation is 0.5). The (mean) processing time of workstation A is one time unit, of workstation B and C two time units and of workstation D, E, F and G four time units. Hence, all workstations have the same average utilisation level.

Both the arrival rate and the distribution of the inter-arrival times are experimental variables.

The arrival rate is such that the workstations have an average utilisation level of 80, 85 or 90

Performance measurement

Table 1 summarises the experimental factors and their experimental levels that we consider in our simulation study. The distribution of the inter-arrival time, utilisation level and the batch size of orders are used as intermediate variables to measure the influence of the order arrival pattern on the workload balancing capability of the pull systems. The distribution of the processing time of orders is used as a variable to measure the influence of processing time variability on the workload balancing capability.

<Insert Table 1 here> We use a full factorial design for the combinations of inter-arrival time, utilisation, batch size and processing time. For a given combination of experimental factors, we simulate the POLCA, CONWIP and m-CONWIP system for a decreasing number of cards in the control loops. We start with a large number of cards, such that the release of orders is not constrained by any of the control loops. Then we decrease the number of cards in the control loops gradually, such that the configurations of the pull systems become more constrained. Note that an identical number of cards in the CONWIP, POLCA and m-CONWIP systems does not mean that the STT in these systems is the same. For example, due to the overlapping loops, the number of cards in a POLCA configuration will generally be larger than the number of cards in an m-CONWIP configuration for a given STT level. In each simulation experiment we measure the STT and TTT performance of the pull system. If we plot the STT performance against the TTT performance for different configurations of a pull system, we obtain a performance curve similar to the one in Figure 6.

By comparing the curves of CONWIP, m-CONWIP and POLCA for a given combination of experimental factors, we can determine the throughput time performance difference between these pull systems. For ease of comparison, we will only plot the configurations that use the same number of cards in every control loop of a pull system. However, the total number of cards used in a configuration may differ per pull system.

Naturally, we are interested in the optimal throughput time performance of the pull systems for a given combination of experimental factors. Therefore, we determine for each pull system the configuration for which the pull system obtains the shortest TTT. Determining the optimal configuration of a pull system can be a difficult task, especially when a pull system consists of multiple control loops. [START_REF] Gaury | Designing Pull Production Control Systems: Customization and robustness[END_REF] gives a short review of the techniques that can be used for determining the optimal configuration of a pull system. We will use exhaustive search to determine the optimal configuration of the pull system.

The simulation model is built in DESIMP, a discrete event simulation library within Delphi. DESIMP is very fast, flexible and suitable for this type of research. We use common random numbers to reduce the variance across experiments. Each experiment consists of 100 independent experiments with run length of 100,000 time units. All experiments include a warm-up period of 25,000 time units in order to eliminate the initial transient. If we state that there is a performance difference between two experiments in the following section, the significance can be shown by a paired t-test at a 95% confidence level.

Results

This section presents the results of the simulation experiments. It gives an in-depth analysis of the workload balancing capabilities of CONWIP, POLCA, and m-CONWIP.

In Figure 8 we give a graphical representation of the throughput time performance of the pull systems for three different combinations of experimental factors (a, b and c). In all three combinations the utilisation level is 85% and the batch size is 1. Hence, the combinations (a), (b) and (c) only differ with respect to the distribution of the inter-arrival time and processing time of orders. This allows us to visualise the influence of variability in the inter-arrival time and processing time on the throughput time performance of the pull systems. In Table 2 and POLCA(b), we see that an increase in the variability of the inter-arrival times increases the TTT reductions that can be realised by POLCA and m-CONWIP. This confirms our expectation that an increase in the average queue length in front of the workstations in the production system, due to increased variability in the inter-arrival times, improves the workload balancing capability of the pull systems.

In combination (c), both the inter-arrival time and processing time of orders are random.

Introducing variability in the processing time of orders has a strong negative effect on the workload balancing capability of m-CONWIP and POLCA, as can be seen from a comparison of combinations (b) and (c) in Figure 8. For combination (c), the POLCA system has no workload balancing capability, while the optimal throughput time performance of m-CONWIP is only slightly below that of the push system (see also Table 2). This result confirms our expectation that increased processing time variability has a negative impact on the workload balancing capability of m-CONWIP and POLCA.

In Table 2 and3 in their optimal configuration, relative to, respectively, the TTT and STT of the push system.

Recall from the previous section that the optimal configuration of a pull system is defined as the configuration for which the pull system obtains the shortest TTT. Note that we have omitted the performance of the CONWIP system in these tables since the CONWIP system has no effective workload balancing capability and, therefore, the optimal configuration of the CONWIP system is equal to the push system.

Table 2 contains the optimal throughput time performance of m-CONWIP and POLCA for different combinations of experimental factors, given the restriction that the same number of cards is used in every control loop. The curves in Figure 8 already indicated a relationship between the variability in the inter-arrival and processing time of orders and the effective workload balancing capability of the pull systems. The results in Table 2 confirm that increased variability in the inter-arrival time of orders improves the workload balancing capability of POLCA and m-CONWIP. Table 2 also confirms the negative influence of increased processing time variability on the workload balancing capability of POLCA and m-CONWIP.

<Insert Table 2 here> Table 2 further shows that the other intermediate variables of the order arrival pattern, the utilisation level and batch size, influence the TTT reduction that can be realised by m-CONWIP and POLCA. For instance, given a constant inter-arrival and processing time and a batch size of 1, the TTT reduction for m-CONWIP increases from 1.56% to 6.56% if the utilisation increases from 80% to 90%. In general, we see in Table 2 that the percentage of TTT reduction obtained by m-CONWIP increases with increasing utilisation levels. This means that for m-CONWIP workload balancing has more effect for higher levels of [START_REF] Land | Workload control in job shops, grasping the tap[END_REF] shows that the same relationship between utilisation and workload balancing exists for load-based pull systems. For the POLCA system we see that this relationship does not hold for the combinations with constant inter-arrival and processing time and a batch size of 10. The exception to the rule is caused by the restriction we put on the number of cards that can be used in each control loop. This restriction reduces the set of allowable configurations for POLCA. As we see in Table 3 the relationship between utilisation and workload balancing holds for POLCA if we consider all configurations of POLCA.

Table 2 also shows that the batch size has a large effect on the TTT performance of m-CONWIP and POLCA. For instance, given a constant inter-arrival and processing time and a utilisation level of 80%, the TTT reduction for m-CONWIP increases from 1.56% to 6.37% as a result of the increased batch size. Utilisation and batch size both increase the average queue length in front of the workstations in the system, and thereby the choice of orders in front of the control loops. The results in Table 2 again confirm our intuition that an increase of choice in orders improves the balancing capability of the pull systems and reduces the blocking of the release of orders.

Table 3 contains the optimal throughput time performance of m-CONWIP and POLCA, without any restriction on the number of cards used in the control loops. We have excluded from this table the experiments with random processing times, because for these experiments the POLCA system has no effective workload balancing capability. Note that the optimal configuration for the m-CONWIP system is not changed after relaxing the restriction on the number of cards. For the POLCA system, however, the optimal configuration has changed. In the optimal configuration of POLCA, the number of cards in the loops A B and A C is infinite, which means in fact that the workload released to the shop floor is not limited. Note that the optimal POLCA configuration is therefore not consistent with definition of a pull The pull structure of CONWIP has no workload balancing capability and our simulation results show that limiting the workload in a CONWIP controlled MTO production system, while aiming at the same utilisation level, increases the TTT of orders. The overlapping loops in the POLCA system bring forward some workload balancing capability compared to CONWIP, but they do not perfectly detect and signal an imbalance in workload. As a result POLCA faces a longer TTT for a given STT than m-CONWIP, the system with the best workload balancing capability.

Our results further show that the configuration of the pull system has a large influence on the workload balancing capability. When the configurations of a pull system become more constrained, the STT decreases, while the OPT increases. If the pull system has effective workload balancing capability, limiting the amount of work on the shop floor will first result in a STT reduction (compared to the non-limited system) that outweighs the increase in OPT.

After a certain point the configurations become too constrained and the OPT increases rapidly and finally overcompensates the decrease in STT.

Our simulation studies show that an increase in the choice of orders in front of the control loops of POLCA and m-CONWIP improves the workload balancing capability of these pull systems. Such an increase in the choice of orders can occur, for example, due to an increase in the variability of the inter-arrival times of orders. Variability of the workload in a control loop increases the variability of the workload at each workstation and, thereby, decreases the workload balance. This is because the pull systems that we consider in this paper control the workload based on the number of orders on the shop floor and not on the processing time of orders, i.e. are unit-based instead of load-based. Our simulations results show that processing Although our paper shows that unit-based pull systems can reduce both the TTT and STT, our simulation studies also show that as soon as the manufacturing conditions become more realistic (i.e. when variability in the processing times of orders is introduced), the increase in the OPT is off-setting the decrease in the STT. An important issue that requires additional study is whether the throughput time performance of POLCA and m-CONWIP can be improved if the release of orders is load-based instead of unit-based. A remarkable result of our paper shows that in the optimal configuration of POLCA, the final control loops are fully responsible for the effective workload balancing capability of POLCA. An interesting issue for future research is whether this result can also be found in production systems with different topologies than the one we considered in our paper.

  , such as the processing time of orders and the order arrival pattern, to determine their influence on the magnitude of the workload balancing effect.

  on the shop floor, but does not balance the work across the workstations.

  impact on the workload balancing capability of the unit-based pull systems.

  inter-arrival time is either constant or random (exponentially distributed). The number of orders arriving simultaneously (batch size) is an experimental variable and can be either 1 or 10. Orders are processed on a First Come First Serve (FCFS) basis at each workstation.

  3 we show the throughput time performance of POLCA and m-CONWIP for a broader range of experimental factors. We first consider the simulation experiments of combination (a) in Figure 8. In this combination, both the inter-arrival time and processing time of orders are constant. Hence, the curves CONWIP(a), POLCA(a) and m-CONWIP(a) show the throughput time performance of CONWIP, POLCA and m-CONWIP for the case that there is no randomness in the interarrival and processing time of orders. Figure 8 shows that when the configuration of CONWIP(a) becomes more constrained (i.e. when we move from right to left along the curve), the TTT increases immediately. This result confirms our expectation about the throughput time performance of CONWIP (see Section 3): because the CONWIP system has no workload balancing capability, any reduction in the STT obtained by limiting the workload on the shop floor is offset by an increase in the OPT. When the configurations of m-CONWIP(a) and POLCA(a) become more constrained, the TTT first decreases (only slightly for POLCA(a), see also Table2) and after a certain point the TTT increases rapidly.POLCA(a) reaches this point much earlier than m-CONWIP(a) and this means that m-CONWIP obtains a better performance in terms of STT and TTT than POLCA for ). Note that because both pull systems are able to reduce both the TTT and STT, the workload balancing capability of POLCA and m-CONWIP is effective.<Insert Figure8here> In combination (b), the inter-arrival time of orders is random, while the processing time of orders is constant. The introduction of randomness in the inter-arrival time of orders in combination (b) naturally increases the STT and TTT for all pull systems compared to combination (a). However, the relative performance of the pull systems does not alter. If we compare the curves m-CONWIP(a) and POLCA(a) with the curves m-CONWIP(b) and

  we show the optimal throughput time performance of POLCA and m-CONWIP in terms of the percentage of TTT and STT reduction these two pull system achieve

  infinite number of cards in the loops A B and A C results in a POLCA configuration that does not use overlapping loops (see Section 2.2) to balance the workload on the shop floor. This remarkable result implies that the control loops B D, B E, C F and C G are fully responsible for the effective workload balancing capability of POLCA.<Insert Table3 here>6. ConclusionsFor MTO companies, a short average total throughput time (TTT) is of strategic importance for winning orders. Pull systems try to reduce the throughput times by controlling the workload on the shop floor. Limiting the workload on the shop floor reduces the average time orders spend on the shop floor (STT) compared to the non-limited production system. However, the restricted release of orders onto the shop floor increases the average time orders spend waiting before being released (OPT) due to the blocking of the release of orders. As a result, the reduction in STT obtained by restricting the release of orders might be offset by the increase in the OPT. Literature on workload control shows that a reduction in the TTT can only be obtained if the release mechanism not only reduces the workload but also improves the balance of the workload on the shop floor. This literature, however, shows the existence of effective workload balancing capability only in load-based pull systems. The central question in our research is whether unit-based pull systems can improve the workload balance on the shop floor such that both the STT and TTT are reduced.To answer this question we have used simulation to analyse the throughput time performance of three unit-based pull systems that are considered applicable in MTO environments: POLCA, CONWIP and m-CONWIP. The results of our simulations show that unit-based pull systems can reduce both the TTT and STT, and that the magnitude of the on the pull structure, on the configuration of the pull system and on the order arrival pattern and processing time variability of the orders.

  a large negative impact on the workload balancing capability of the unitbased pull systems considered in our paper.

Figure 1 .

 1 Figure 1. Throughput time measures in an MTO production system.

Figure 2 .Figure 6 .

 26 Figure 2. Shop floor without (a) and with (b) workload balancing.

Figure 7 .

 7 Figure 7. Topology of simulated production system.

Figure 8 .

 8 Figure 8. TTT and STT performance of CONWIP, POLCA and m-CONWIP. Combination

Table 1 .

 1 Experimental factors.

	factor	experimental levels
	order arrival pattern:
	-inter-arrival time	constant, random (exponential)
	-utilisation	80, 85, 90 percent
	-batch size	1, 10
	processing time variability:
	-processing time	constant, random (Erlang-2)
	F o r
	P
	e
	e r
	R
	e
	v i e
		w
		O n l
		y
		33

Table 2 .

 2 Optimal throughput time performance* of m-CONWIP and POLCA.

	constant	constant inter-arrival time	random inter-arrival time
	processing time	POLCA	m-CONWIP	POLCA	m-CONWIP
	batch size	utilisation %TTT %STT %TTT %STT %TTT %STT %TTT %STT
	1	80%	0.47	1.14	1.56	23.09	2.23	8.56	10.13 39.34
		85%	0.71	1.43	3.23	34.30	2.73	9.13	13.40 50.40
		90%	1.09	2.49	6.56	50.12	3.31	11.61 17.31 63.71
	10	80%	1.15	9.17	6.37	35.69	5.22	30.31 19.65 75.63
		85%	1.05	5.10	8.21	44.30	5.47	38.42 21.66 81.35
		90%	1.21	4.81	10.68 56.49	5.67	43.25 23.54 87.09
	random	constant inter-arrival time	random inter-arrival time
	processing time	POLCA	m-CONWIP	POLCA	m-CONWIP
	batch size	utilisation %TTT %STT %TTT %STT %TTT %STT %TTT %STT
	1	80%	0.00	0.00	0.00	0.00	0.00	0.00	1.44	11.56
		85%	0.00	0.00	0.00	0.00	0.00	0.00	1.71	14.80
		90%	0.00	0.00	0.00	0.00	0.00	0.00	2.38	18.79
	10	80%	0.00	0.00	0.00	0.00	0.00	0.00	4.39	47.33
		85%	0.00	0.00	0.00	0.00	0.00	0.00	4.94	48.46
		90%	0.00	0.00	0.00	0.00	0.00	0.00	5.25	52.95

Table 3 .

 3 Optimal throughput time performance* of POLCA and m-CONWIP.

	constant	constant inter-arrival time	random inter-arrival time
	processing time	POLCA	m-CONWIP	POLCA	m-CONWIP
	batch size	utilisation %TTT %STT %TTT %STT %TTT %STT %TTT %STT
	1	80%	3.72	3.72	1.56	23.09	6.12	6.12	10.13 39.34
		85%	5.57	5.57	3.23	34.30	7.76	7.76	13.40 50.40
		90%	8.30	8.30	6.56	50.12	9.69	9.69	17.31 63.71
	10	80%	3.36	3.36	6.37	35.69	6.31	6.31	19.65 75.63
		85%	4.56	4.56	8.21	44.30	6.84	6.84	21.66 81.35
		90%	6.77	6.77	10.68 56.49	7.34	7.34	23.54 87.09

http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.ukInternational Journal of Production Research


Matthieu Bonnivard

On the stability of self-propelled bodies with respect to their shape motion

This paper studies the stability of the trajectories of self-propelled bodies immersed in a fluid at low Reynolds number, with respect to their dynamic shape deformation. We consider both adherence and perfect slip boundary conditions on the moving body. Using shape derivative arguments in association with recent higher order regularity results for the Bogovskiȋ operator, we bring the shape stability question to a finite dimension dynamical system analysis.

1 Introduction.

The locomotion mechanisms of aquatic animals have aroused the interest of zoologists, engineers and mathematicians for a long time. We refer to [START_REF] Munnier | On the self-displacement of deformable bodies in a potential fluid flow[END_REF][START_REF] San Martín | An initial and boundary value problem modeling of fish-like swimming[END_REF] and the references therein for a review of the works on this subject. Recently, researchers have tried to conceive underwater vehicles that move by shape motion and do not require propellers. A pioneering work in this direction was performed by Tryantafyllou and Tryantafyllou [34]. This type of motion is called self-propelled motion. The principle is the following: the deformation of the object causes a reaction of the fluid, which modifies the inertia and angular momenta of the vehicle and sets the object in motion.

The modelling of self-propelled motion requires to couple the equations of motion of the immersed body with the equations of motion of the fluid. The body is modeled as a deformable solid and its movement is described by Newton's laws of motion. Several works consider a fluid in inviscid and irrotationnal flow, which implies that, after solving an elliptic boundary value problem, one can bring the coupled system to a system of ordinary differential equations involving only the degrees of freedom of the solid. This approach is developed in [START_REF] Bressan | Impulsive control of Lagrangian systems and locomotion in fluids[END_REF] and [START_REF] Munnier | On the self-displacement of deformable bodies in a potential fluid flow[END_REF], for instance. If we take into account the vorticity and viscosity effects, one relevant model for the fluid consists in the stationnary Navier-Stokes equations. This case was studied by San Martín et al. [START_REF] San Martín | An initial and boundary value problem modeling of fish-like swimming[END_REF], who proved the global existence and uniqueness of the trajectory of a deformable solid on which a smooth deformation is imposed. Nevertheless, at low Reynolds number, the vorticity effects are negligible with respect to the viscosity effects, so the Stokes equations are well adapted to the description of the flow. As a result, Stokes equations are 1 relevant to study the locomotion of micro-organisms, such as bacteria or certain species of amoebae. Initiated by the pioneer works of Taylor [START_REF] Taylor | Analysis of the swimming of microscopic organisms[END_REF], Berg and Anderson [START_REF] Berg | Bacteria swim by rotating their flagellar filament[END_REF] and Purcell [START_REF] Purcell | Life at low reynolds number[END_REF], the study of swimming at low Reynolds number was the subject of many recent papers.

In [START_REF] Shapere | Geometry of self-propulsion at low Reynolds number[END_REF], Shapere and Wilczek introduced a general geometric framework for low Reynolds number swimming, that was developed by Cherman et al. [START_REF] Cherman | Low Reynolds number swimming in two dimensions[END_REF] in the two-dimensional case.

Let us mention the work of Alouges, DeSimone and Lefebvre [START_REF] Alouges | Optimal strokes for low Reynolds number swimmers: an example[END_REF], who proved the global controllability of a three-sphere swimmer immersed in a Stokes fluid.

In this paper, we consider a simplified model of self-propelled motion at low Reynolds number. This model consists in a deformable solid immersed in a Stokes fluid, in a bounded domain. The displacement of the solid is decomposed into a rigid part corresponding to a global displacement, and a deformation part. We make the assumption that we can impose the deformation and deduce the rigid part from the interaction between the solid and the fluid. The location of the solid-fluid interface is an unknown, so we have to tackle a free boundary problem. This difficulty has been recently investigated in several papers in the case of rigid bodies immersed in a viscous incompressible fluid (see, for instance, Desjardins and Esteban [START_REF] Desjardins | Existence of weak solutions for the motion of rigid bodies in a viscous fluid[END_REF], San Martín et al. [START_REF] Alonso San Martín | Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid[END_REF], Feireisl [START_REF] Feireisl | On the motion of rigid bodies in a viscous incompressible fluid[END_REF] or Gunzburger et al. [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF]), and in the case of elastic or viscoelastic bodies (see, for instance, Coutand and Shkoller [START_REF] Coutand | Motion of an elastic solid inside an incompressible viscous fluid[END_REF][START_REF] Coutand | The interaction between quasilinear elastodynamics and the Navier-Stokes equations[END_REF], Chambolle et al. [START_REF] Chambolle | Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate[END_REF], Desjardins et al. [START_REF] Desjardins | Weak solutions for a fluid-elastic structure interaction model[END_REF], and Boulakia [START_REF] Boulakia | Existence of weak solutions for the motion of an elastic structure in an incompressible viscous fluid[END_REF]).

In this work, we give a new proof of the global existence and uniqueness of the trajectory of a deformable solid on which the nonrigid part of the displacement is imposed, which is immersed in a viscous incompressible fluid at low Reynolds number. The proof given in [START_REF] San Martín | An initial and boundary value problem modeling of fish-like swimming[END_REF] for a general viscous incompressible flow is based on the transformation of a non-cylindrical problem into a cylindrical one, and on a fixed point argument. The principle of our proof is similar to the approach adopted by Munnier [START_REF] Munnier | On the self-displacement of deformable bodies in a potential fluid flow[END_REF] in the case of a potential flow: we use the existence, uniqueness and regularity of the solution to the Stokes problem to bring the coupled system of equations to a system of ordinary differential equations. To prove the regularity of the corresponding operator, we need to establish the differentiability of the weak solution to the Stokes problem with respect to both rigid and nonrigid deformations of the solid, and to the boundary conditions at the solid-fluid interface. To this aim, we use shape derivative methods that were introduced by Murat and Simon [START_REF] Murat | Étude de problèmes d'optimal design[END_REF] and Soko lowski and Zolésio [START_REF] Soko | Introduction to shape optimization[END_REF]. Several difficulties arise here, due to the fact that we treat both classical adherence conditions and perfect slip conditions which are not frequent in the literature. Besides, to deal with boundary conditions that depend on the deformation of the solid, we need high order regularity results for the Bogovski ǐ operator which were recently proved by Mitrea et al. [START_REF] Mitrea | The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains[END_REF].

One major interest of our approach is that, treating the deformation as a parameter is the differential system, we are able to prove the stability of the trajectories of the solid with respect to the imposed deformation. The stability of the trajectories is a crucial step to perform a dynamical shape control of the immersed body. Although this goal is not achieved, it was the first motivation of this work. A way to deal with this problem is to impose a perfect slip condition at the solid-fluid interface, to consider small boundary deformations and to produce dynamic boundary conditions by the rugosity effect (see Casado-Díaz et al. [START_REF] Casado-Díaz | Why viscous fluids adhere to rugose walls: a mathematical explanation[END_REF], Bucur et al. [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF]).

The paper is organized as follows. After presenting the model in Section 2, we state our main results in Section 3. The proof to Theorem 1 is carried out in Section 4, and Theorem 2 is proved in Section 5.

2 Presentation of the model.

All results in this paper are valid in two or three dimensions. For the sake of simplicity we present a two-dimensional model, which is adapted from [START_REF] San Martín | An initial and boundary value problem modeling of fish-like swimming[END_REF]. However, the derivation of the model and the mathematical study we perform in the next sections can be adapted to the three-dimensional case without extra fundamental mathematical difficulties.

2.1 A kinematic model of self-propelled motion in a fluid.

We denote by Ω the domain occupied by the solid-fluid system, which is supposed to be a bounded and regular open set. The state of the system at the initial instant is called the reference configuration. We denote by S 0 ⊂ Ω (resp. F 0 := Ω \ S 0 ) the domain occupied by the solid (resp. the fluid) in the reference configuration. We suppose that ∂S 0 is of class C 3 . We assume that the solid is at positive distance from the boundary of Ω in the reference configuration. Thus, there exists δ > 0 such that if we define the following subset of Ω:

Ω -δ = {x ∈ Ω | dist(x, ∂Ω) > δ} , then S 0 ⊂ Ω -δ .
We consider a reference frame with the origin at the mass center of S 0 , and a mobile frame attached to the solid and which coincides with the reference frame at instant t = 0. In what follows, we use the following convention. We reserve the letters x, y for space variables in the reference frame, while x * stands for a space variable in the mobile frame. We write with a star the vector fields whose components are expressed in the mobile frame. We impose a smooth deformation of the solid, that is, we consider a time-dependant vector field Θ * ∈ C 1 ([0, ∞), C 3 (S 0 , R 2 )) such that the deformation field X * in the solid is defined by

X * (y, t) = y + Θ * (y, t), (y, t) ∈ S 0 × [0, ∞). (1)
We suppose that the deformation satisfies the following hypothesis: (H1) For every t 0, the mapping y → X * (y, t) is a C 3 diffeomorphism from S 0 onto S * (t) := X * (S 0 , t).

For every t 0 we denote by Y * (•, t) the inverse of X * (•, t). We define the deformation velocity of the solid in Eulerian coordinates by

w * (x * , t) = ∂Θ * ∂t (Y * (x * , t), t) t 0, x * ∈ S * (t).
We suppose that the volume of the solid is preserved during its movement, which leads to the following constraint on w * :

(H2)

∂S * (t)
w * (x * , t)ds(x * ) = 0, t 0.

Here we have denoted ds the surface measure on ∂S * (t).

Since we aim to study self-propelled motions, it is natural to assume that the deformation X * does not affect the linear and angular momenta of the solid. The expression of these properties requires to define the density field ρ * of the solid. Let ρ 0 be the density field of the solid in the reference configuration S 0 . The local form of the mass conservation in the solid yields the following expression of ρ * :

ρ * (x * , t) = ρ 0 (Y * (x * , t)) det (∇X * )(Y * (x * , t), t) t 0, x * ∈ S * (t). ( 2
)
Since the origin of the reference system of coordinates coincides with the mass center of S 0 , the conservation of the linear and angular momenta yields the following constraints on w * :

(H3)

S * (t)
ρ * (x * , t)w * (x * , t)dx * = 0, t 0, (H4)

S * (t) ρ * (x * , t)x * ⊥ • w * (x * , t)dx * = 0, t 0,
where for every x = (x 1 , x 2 ) ∈ R 2 we denote by x ⊥ the vector x ⊥ = (-x 2 , x 1 ). Let X denote the motion of the solid. We assume that there exists a rigid displacement X R such that X(y, t) = X R (X * (y, t), t), t 0, y ∈ S 0 .

The rigid motion

X R : R 2 × [0, ∞) → R 2 is given by X R (x * , t) = ξ(t) + R θ(t) x *
where, for every t 0, ξ(t) is the position of the mass center of the solid, θ(t) is the angle giving the orientation of the solid with respect to a fixed axis in the reference frame and R θ(t) ∈ SO 2 (R) is the matrix associated to the rotation of angle θ(t). We denote by S(ξ(t), θ(t), t) the domain occupied by the solid at instant t, that is,

S(ξ(t), θ(t), t) = R θ(t) S * (t) + ξ(t), t 0.
The Eulerian velocity field of the solid is given by

u S (x, t) = ξ (t) + θ (t)(x -ξ(t)) ⊥ + w(x, t), t 0, x ∈ S(ξ(t), θ(t), t), where w(x, t) = R θ(t) w * (R -θ(t) (x -ξ(t)), t), t 0, x ∈ S(ξ(t), θ(t), t).
The Eulerian density field of the body is given by

ρ(x, t) = ρ * (R -θ(t) (x -ξ(t)), t), t 0, x ∈ S(ξ(t), θ(t), t), (3)
with ρ * given by [START_REF] Beirão Da | Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions[END_REF]. The mass m of the body and its moment of inertia I(t) with respect to an axis orthogonal to the plane of motion and passing by the mass center of S(ξ(t), θ(t), t), are given by

m = S(ξ(t),θ(t),t) ρ(x, t)dx, (4)
I(t) = S(ξ(t),θ(t),t) ρ(x, t)|x -ξ(t)| 2 dx, t 0. ( 5
)
Let us notice that from (2), ( 3), ( 4) and ( 5), we have m = S 0 ρ 0 (y)dy,

I(t) = S 0 ρ 0 (y)|X * (y, t)| 2 dy, t 0.
In particular, since

X * ∈ C 1 (S 0 × [0, ∞), Ω), the mapping t 0 → I(t) ∈ R is of class C 1 .

The full system.

If there is no risk of confusion, we denote

S(t) = S(ξ(t), θ(t), t)
the domain occupied by the solid at instant t, and we define

F(t) = Ω \ S(t)
the domain filled by the fluid at instant t. We denote F 0 = F(0). For k = 1, 2 we define

H k (F(t)) = H k (F(t)) 2
where H k (F(t)) is the usual Sobolev space of order k. We denote by L 2 0 (F(t)) the subspace of L 2 (F(t)) consisting of functions with mean value equal to 0.

We consider the following problem: find a real T > 0 and functions

ξ ∈ C 2 ((0, T ), R 2 ) ∩ C 1 ([0, T ), R 2 ) and θ ∈ C 2 ((0, T ), R) ∩ C 1 ([0, T ), R) solutions to the system mξ (t) = - ∂S(t) σ(u, p)n ds, t ∈ (0, T ), (6)
(Iθ ) (t) = - ∂S(t) σ(u, p)n • (x -ξ) ⊥ ds, t ∈ (0, T ), (7)
where for every t ∈ (0, T ), (u,

p) ∈ H 1 (F(t)) × L 2 0 (F(t)
) is solution to the following problem:

-ν∆u + ∇p = 0 in F(t), ( 8
) div u = 0 in F(t), (9)
u = 0 on ∂Ω, (10)
u • n = u S • n on ∂S(t), (11)
[σ(u, p)n] tan = 0 on ∂S(t). (12)
In the above system ν > 0 stands for the viscosity of the fluid, u and p denote the Eulerian velocity field and the pressure field of the fluid, respectively. n stands for the unit normal to ∂S(t) oriented towards the solid. In ( 6)-( 7), a prime stands for the derivation operator with respect to time. The stress tensor field σ is defined by

σ(u, p) = 2νD(u) -pId
where D(u) is the symmetric part of ∇u defined by

D(u) = 1 2 (∇u) + (∇u) T
and Id is the identity matrix in M 2 (R).

Let us point out that the integrals in ( 6)-( 7) require the H 1 regularity of σ in the neighborhood of ∂S(t), so that σn is properly defined in L 2 (∂S(t)), in the sense of trace. This regularity will be discussed in Section 4.

Notice that ( 11)-( 12) express a perfect slip condition at the solid-fluid interface. All the results in this paper hold if we replace ( 11)-( 12) by the adherence condition

u = u S on ∂S(t). (13)
System ( 6)-( 12) is completed by the initial conditions

ξ(0) = 0, θ(0) = 0, ξ (0) = ξ0 , θ (0) = θ0 . ( 14
)
3 Statement of the main results.

We follow the notations in Section 2. Let us first define the set D of admissible deformations Θ * . Let K ∈ (0, 1) be a given constant. Since C 3 (S 0 , R 2 ) is a Banach space, (H1) holds for every Θ * ∈ C 1 ([0, ∞), C 3 (S 0 , R 2 )) satisfying the following condition:

∀t 0 Θ * (•, t) C 3 (S 0 ,R 2 )) K. (15)
Replacing K by a smaller positive value if necessary, we may assume that for every

Θ * ∈ C 1 ([0, ∞), C 3 (S 0 , R 2
)) satisfying [START_REF] Desjardins | Existence of weak solutions for the motion of rigid bodies in a viscous fluid[END_REF], the following property holds:

∀t 0 S * (t) ⊂ Ω -δ/2 . ( 16
)
The set D of admissible deformations is defined by

D := Θ * ∈ C 1 ([0, ∞), C 3 (S 0 , R 2 )) | ∀t 0 Θ * (t) C 3 (S 0 ,R 2 )) K, ∂S * (t) w * (x * , t)ds(x * ) = 0, S * (t) ρ * (x * , t)w * (x * , t)dx * = 0, S * (t) ρ * (x * , t)x * ⊥ • w * (x * , t)dx * = 0. ( 17
)
Theorem 1. Let ( ξ0 , θ0 ) ∈ R 3 and Θ * ∈ D. There exists T max > 0 such that:

• For every T ∈ (0, T max ), system (6)-( 12), completed with initial conditions [START_REF] Demailly | Analyse numérique et équations différentielles[END_REF], admits one unique solution on [0, T ].

• The following alternative holds:

(i) T max = ∞; or (ii) lim t→Tmax dist(S(t), ∂Ω) = 0.

• In the two-dimensional case, if the adherence condition ( 13) is imposed at the solidfluid interface, then T max = ∞.

In the second result, D is endowed with the

C 1 ([0, ∞), C 3 (S 0 , R 2 )) norm.
Theorem 2. Let ( ξ0 , θ0 ) ∈ R 3 and Θ * ∈ D. Let (ξ, θ) be the maximal solution to (6)-( 12) associated to Θ * , with initial conditions [START_REF] Demailly | Analyse numérique et équations différentielles[END_REF]. Let (Θ * n ) n∈N be a sequence of deformations converging to Θ * in D. For every n ∈ N we denote by (ξ n , θ n ) the maximal solution associated to Θ * n with the same initial conditions. Then:

• There exists T 1 > 0 such that (ξ, θ), (ξ n , θ n ) exist on [0, T 1 ] for all n ∈ N. • The sequence (ξ n , θ n ) n∈N converges to (ξ, θ) uniformly in C 1 ([0, T * ], R 3
), for any time T * such that the solution (ξ n , θ n ) is well-defined on [0, T * ] for any n.

4 Proof of Theorem 1.

In this section, the deformation Θ * ∈ D is given. To obtain the local existence and uniqueness of the trajectories, we rewrite problem ( 6)-( 12) as a system of second order differential equations satisfied by the unknown functions ξ, θ, and apply the Cauchy-Lipschitz theorem.

Step 1. For every t 0 and every (ξ, θ) ∈ R 3 we denote by S(ξ, θ, t) the domain

S(ξ, θ, t) = ξ + R θ S * (t).
By [START_REF] Desjardins | Weak solutions for a fluid-elastic structure interaction model[END_REF], there exists an open subset W ⊂ R 3 such that 0 ∈ W and ∀t 0, ∀(ξ, θ) ∈ W, S(ξ, θ, t) ⊂ Ω.

For every t 0 and every (ξ, θ) ∈ W we denote by F(ξ, θ, t) the domain

F(ξ, θ, t) = Ω \ S(ξ, θ, t).
We denote by α a quadruplet (ξ, θ, ξ, θ) in R 6 and by U the admissible set for the parameter α, defined by

U = W × R 3 . ( 18
)
For every (α, t)

∈ U × [0, ∞) we define (u α,t , p α,t ) in H 1 (F(ξ, θ, t)) × L 2 0 (F(ξ, θ, t))
as the solution to the following problem:

           -ν∆u α,t + ∇p α,t = 0 in F(ξ, θ, t), div u α,t = 0 in F(ξ, θ, t), u α,t = 0 on ∂Ω, u α,t • n ξ,θ,t = u S α,t • n ξ,θ,t on ∂S(ξ, θ, t), [σ(u α,t , p α,t )n ξ,θ,t ] tan = 0 on ∂S(ξ, θ, t), (19)
where n ξ,θ,t stands for the unit normal to ∂S(ξ, θ, t) oriented towards S(ξ, θ, t). The function u S α,t is defined by

u S α,t (x) = ξ + θ(x -ξ) ⊥ + R θ w * (R -θ (x -ξ), t), x ∈ ∂S(ξ, θ, t),
and L 2 0 (F(ξ, θ, t)) is the subspace of L 2 (F(ξ, θ, t)) consisting of functions with mean value equal to 0.

Let us precise the sense of [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF]. For every (ξ, θ, t) ∈ W × [0, ∞), we define the following subspace of H 1 (F(ξ, θ, t)):

H 1 tan (F(ξ, θ, t)) = {ψ ∈ H 1 (F(ξ, θ, t)) : ψ • n ξ,θ,t = 0 on ∂F(ξ, θ, t), ψ = 0 on ∂Ω}. ( 20
) (u α,t , p α,t ) ∈ H 1 (F(ξ, θ, t)) × L 2 0 (F(ξ, θ, t)
) is solution to [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF] in the following sense:

div u α,t = 0 in L 2 (F(ξ, θ, t)),
u α,t satisfies the boundary conditions

u α,t = 0 on ∂Ω, u α,t • n ξ,θ,t = u S α,t • n ξ,θ,t on ∂S(ξ, θ, t)
in the sense of trace, and for every test

function ψ ∈ H 1 tan (F(ξ, θ, t)), ν F (ξ,θ,t) ∇u α,t : ∇ψ dx - F (ξ,θ,t) p α,t (div ψ)dx = 0. (21)
To state the local H 2 × H 1 regularity of (u α,t , p α,t ) in the neighborhood of ∂S(ξ, θ, t), we introduce an open subset Ω ⊂ R 2 such that

S 0 ⊂ Ω ⊂ Ω
and ∂Ω is at positive distance from ∂S 0 and ∂Ω. We set

F 0 = F 0 ∩ Ω . ( 22
)
By classic regularity results for the Stokes problem with slip boundary conditions (see for instance [START_REF] Solonnikov | On a boundary value problem for a stationary system of navier-stokes equations[END_REF], [START_REF] Beirão Da | Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions[END_REF]), problem [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF] admits one unique solution (u α,t , p α,t ) in H 1 (F(ξ, θ, t)) × L 2 0 (F(ξ, θ, t)). Furthermore, this solution satisfies

(u α,t , p α,t ) ∈ H 2 (X ξ,θ,t (F 0 )) × H 1 (X ξ,θ,t (F 0 )). ( 23
)
We now define the operators

F : U × [0, ∞) → R 2 , G : U × [0, ∞) → R, (24) by
       F (α, t) = - 1 m ∂S(ξ,θ,t) σ(u α,t , p α,t )n ξ,θ,t ds, G(α, t) = - 1 I(t) ∂S(ξ,θ,t) σ(u α,t , p α,t )n ξ,θ,t • (x -ξ) ⊥ ds + I (t) θ .
With these notations, problem ( 6)- [START_REF] Coutand | Motion of an elastic solid inside an incompressible viscous fluid[END_REF], completed by initial conditions [START_REF] Demailly | Analyse numérique et équations différentielles[END_REF], can be rewritten as follows: find T > 0 and functions ξ

∈ C 2 ((0, T ), R 2 ) ∩ C 1 ([0, T ), R 2 ) and θ ∈ C 2 ((0, T ), R) ∩ C 1 ([0, T ), R) solutions to the system (ξ (t), θ (t)) = (F, G) (ξ(t), θ(t), ξ (t), θ (t), t) ,
t ∈ (0, T ), (ξ(0), θ(0), ξ (0), θ (0)) = (0, 0, ξ0 , θ0 ).

(

) 25
Remark 3. Since the dependance of S(ξ, θ, t), n ξ,θ,t and (u α,t , p α,t ) on the time parameter t appears only through Θ * (t) and (Θ * ) (t) (via S * (t) and the boundary conditions on ∂S(ξ, θ, t) in ( 19)), system (25) could be written as

(ξ (t), θ (t)) = (F, G) (ξ(t), θ(t), ξ (t), θ (t), Θ * (t), (Θ * ) (t)) , t ∈ (0, T ), (ξ(0), θ(0), ξ (0), θ (0)) = (0, 0, ξ0 , θ0 ).
(26) In this formulation, the couple (Θ * (t), (Θ * ) (t)) appears as the natural control variable for system [START_REF] Purcell | Life at low reynolds number[END_REF]. However, since we aim to study the stability of the solid-fluid system with respect to deformations Θ * ∈ D, which is an infinite-dimensional function space, the analyticity of (F, G) with respect to (Θ * (t), (Θ * ) (t)) is not well defined, to our knowledge. For this reason, and to simplify the notations, we keep formulation [START_REF] Murat | Étude de problèmes d'optimal design[END_REF] in the rest of the paper.

Step 2. To obtain the continuity of F, G and the local Lipschitz condition with respect to α, we need to prove the regularity of the solution to [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF] with respect to (α, t). To this end, we introduce a C 3 -diffeomorphism X ξ,θ,t from Ω onto itself, which is defined in the next paragraph.

The principle is to extend the deformation X * up to a neighborhood of S 0 and to combine this deformation with the flow of a Cauchy problem, so that the restriction of X ξ,θ,t to S 0 coincides with the effective deformation of the solid, namely,

X ξ,θ,t (y) = ξ + R θ X * (y, t), y ∈ S 0 . (27)
For every t 0, we define Θ * (t) ∈ C 3 (S 0 , R 2 ) by

Θ * (t) = Θ * (•, t).
We consider an operator of extension

E : C 3 (S 0 , R 2 ) → C 3 (Ω, R 2 ),
which is supposed to be linear and bounded, and satisfies

(EΘ * (t))(y) = Θ * (y, t), Θ * ∈ D, y ∈ S 0 .
The existence of E results from the regularity of ∂S 0 . Let τ be a C ∞ function with compact support in Ω -δ/2 , that is equal to 1 in a neighborhood of Ω -δ . We define the extended deformation X * by

X * (y, t) = y + (EΘ * (t))(y)τ (y) t 0, y ∈ Ω. ( 28
)
By definition of E and τ , and since Θ * ∈ D, the mapping

X * ∈ C 1 ([0, ∞), C 3 (Ω, R 2 )).
Moreover, by continuity of E, supposing that the constant K that appears in the definition of D is small enough, for every t 0, the mapping

X * (•, t) is a C 3 -diffeomorphism from F 0 onto F * (t) := Ω \ S * (t) such that ∀y ∈ F 0 dist(y, ∂Ω) < δ/2 ⇒ X * (y, t) = y. ( 29
)
To deal with the rigid part of the displacement, for every (ξ, θ) ∈ W, we introduce χ ξ,θ the flow of the following Cauchy problem:

dχ ξ,θ ds (z, s) = z ξ,θ (χ ξ,θ (z, s), s), χ ξ,θ (z, 0) = z, z ∈ Ω, ( 30
)
where the function z ξ,θ is defined for every (x,

s) ∈ R 2 × R + by z ξ,θ (x, s) = (ξ + θ(x -sξ) ⊥ )τ (x).
Since z ξ,θ is C ∞ and for every s 0, z ξ,θ (•, s) has a compact support in Ω, problem (30) admits a unique solution on the interval [0, 1]. One can easily check that for every z ∈ S 0 , this solution is equal to

χ ξ,θ (z, s) = sξ + R sθ z, s ∈ [0, 1]. (31)
Since the mapping

(x, s, ξ, θ) ∈ R 2 × R + × W → z ξ,θ (x, s) ∈ R 2 is of class C ∞ , the mapping (z, ξ, θ) ∈ Ω × W → χ ξ,θ (z, 1) ∈ Ω is of class C ∞ . Since z ξ,θ has a compact support in Ω, χ ξ,θ (•, 1) is a C ∞ -diffeomorphism from Ω onto itself. For every (ξ, θ, t) ∈ W × [0, ∞) we now define X ξ,θ,t (y) = χ ξ,θ (X * (y, t), 1) y ∈ Ω. (32)
Applying [START_REF] Soko | Introduction to shape optimization[END_REF] with s = 1, we get that X ξ,θ,t satisfies [START_REF] San Martín | An initial and boundary value problem modeling of fish-like swimming[END_REF]. Thus, for every (ξ, θ, t) ∈ W × [0, ∞), the restriction of X ξ,θ,t to F 0 is a C 3 -diffeomorphism from F 0 onto F(ξ, θ, t). By composition, the mapping

(ξ, θ, t, y) ∈ W × F 0 → X ξ,θ,t (y) ∈ R 2 (33) is of class C ∞ (W, C 1 ([0, ∞), C 3 (F 0 , R 2 ))).
The crucial point in the proof of Theorem 1 is the following:

Proposition 4. The mapping (α, t) ∈ U × [0, ∞) → σ(u α,t , p α,t ) • X ξ,θ,t ∈ H 1 (F 0 ) 4 is of class C 1 .
Proof of Proposition 4. The proof is based on technics developed by Simon [START_REF] Simon | Domain variation for drag in Stokes flow[END_REF] and Bello et al. [START_REF] Bello | The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow[END_REF] to prove the differentiability of the solution to a Stokes (resp. Navier-Stokes) problem, with respect to Lipschitz deformations of the domain. The principle is to transform a problem with constant coefficients posed on a moving domain, into a problem with variable coefficients posed on a fixed reference domain, via an appropriate change of unknown functions. Then the implicit function theorem is applied. The main difference with all previous works is that we impose no-slip boundary conditions, which requires to consider the variation of the normal on ∂S 0 induced by the diffeomorphism X ξ,θ,t . Besides, the boundary conditions are inhomogeneous and depend on the parameters α and t. To deal with these boundary conditions we use a lift operator to transform the nonhomogeneous problem into an homogeneous problem with a source term. Another difficulty arises here: in order to preserve the H 2 × H 1 regularity of the solution in a neighborhood of the solid, we need a lift operator involving Sobolev spaces of higher order that the well-known Bogovski ǐ operator (see for instance [START_REF] Bogovskiȋ | Solution of the first boundary value problem for an equation of continuity of an incompressible medium[END_REF] or [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]). The existence of such operators, defined on a subspace of H 3/2 with value in H 1 , was proved recently by Mitrea, Mitrea and Monniaux [START_REF] Mitrea | The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains[END_REF].

For every (ξ, θ, t) ∈ W × [0, ∞) we denote by J ξ,θ,t ∈ C 2 (F 0 , R) 4 the Jacobian matrix of the restriction of X ξ,θ,t to F 0 . Using properties ( 29) and ( 33) and the Taylor formulas, we obtain the following Lemma 5. The mapping

(ξ, θ, t) ∈ W × [0, ∞) → J ξ,θ,t ∈ W 2,∞ (F 0 ) 4 is of class C 1 .
In the next paragraph, we use the restriction of X ξ,θ,t to F 0 to write problem [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF] in the reference domain F 0 . We denote by J ξ,θ,t the determinant of J ξ,θ,t . We can assume without loss of generality that

J ξ,θ,t (y) > 0 ∀(ξ, θ, t, y) ∈ W × [0, ∞) × F 0 .
Recall that F 0 is defined by [START_REF] Inoue | On existence of solutions of the Navier-Stokes equation in a time dependent domain[END_REF]. The change of functions that we use is motivated by the following result:

Lemma 6. Let (ξ, θ, t) ∈ W ×[0, ∞) and (u, p) ∈ H 1 (F(ξ, θ, t))×L 2 (F(ξ, θ, t)). Let (U, P ) ∈ H 1 (F 0 ) × L 2 (F 0 ) defined by U = J ξ,θ,t J -1 ξ,θ,t (u • X ξ,θ,t ), P = J ξ,θ,t (p • X ξ,θ,t ).
The following results hold:

(i) p ∈ L 2 0 (F(ξ, θ, t)) if and only if P ∈ L 2 0 (F 0 ), (ii) (u, p) ∈ H 2 (X ξ,θ,t (F 0 )) × H 1 (X ξ,θ,t (F 0 )) if and only if (U, P ) ∈ H 2 (F 0 ) × H 1 (F 0 ), (iii) div u = 0 in L 2 (X ξ,θ,t (F 0 )) if and only if div U = 0 in L 2 (F 0 ).
Proof of Lemma 6. To prove (i), notice that by change of variable x = X ξ,θ,t (y),

F (ξ,θ,t) p(x)dx = F 0 J ξ,θ,t (y)(p • X ξ,θ,t )(y)dy = F 0 P (y)dy.
Point (ii) results from the fact that the coefficients of J ξ,θ,t and J -1 ξ,θ,t are in W 2,∞ (F 0 ) and that X ξ,θ,t is in C 2 (F 0 , R 2 ). To prove (iii), we perform the following calculation. For every ζ ∈ C 1 c (F(ξ, θ, t)), using Green's formula, change of variable x = X ξ,θ,t (y) and the chain rule formula

∇(ζ • X ξ,θ,t )(y) = [J ξ,θ,t (y)] T ∇ζ(X ξ,θ,t (y)), y ∈ F 0 , (34)
we obtain that

F (ξ,θ,t) (div u)(x)ζ(x) dx = - F (ξ,θ,t) u(x) • ∇ζ(x) dx = - F 0 (u • X ξ,θ,t )(y) • J ξ,θ,t (y) -1 T ∇(ζ • X ξ,θ,t )(y) J ξ,θ,t (y) dy = - F 0 J ξ,θ,t (y) J ξ,θ,t (y) -1 (u • X ξ,θ,t )(y) • ∇(ζ • X ξ,θ,t )(y) dy = F 0 (div U )(y)(ζ • X ξ,θ,t )(y) dy. Since the mapping ζ ∈ C 1 c (F(ξ, θ, t)) → ζ • X ξ,θ,t ∈ C 1 c (F 0
) is bijective, this proves (iii) by density in L 2 of the C 1 functions with compact support.

For every α = (ξ, θ, ξ, θ) ∈ U and every t 0, we define the following change of functions:

U α,t = J ξ,θ,t J -1 ξ,θ,t (u α,t • X ξ,θ,t ), P α,t = J ξ,θ,t (p α,t • X ξ,θ,t ), ( 35
)
where (u α,t , p α,t ) ∈ H 1 (F(ξ, θ, t)) × L 2 0 (F(ξ, θ, t)) is the solution to [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF]. This change of functions was introduced by Inoue and Wakimoto [START_REF] Inoue | On existence of solutions of the Navier-Stokes equation in a time dependent domain[END_REF] in the case J ξ,θ,t ≡ 1. By [START_REF] Mitrea | The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains[END_REF] and Lemma 6, (U α,t , P α,t ) has the following regularity:

(U α,t , P α,t ) ∈ H 2 (F 0 ) ∩ H 1 (F 0 ) × H 1 (F 0 ) ∩ L 2 0 (F 0 ). (36)
We will prove that (u α,t , p α,t ) is solution to [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF] if and only (U α,t , P α,t ) is solution to the following problem:

                       i -ν div (L i (ξ, θ, t)U α,t ) + j ∂ j (M ij (ξ, θ, t)P α,t ) ∇X (i) ξ,θ,t = 0 in F 0 , div U α,t = 0 in F 0 , U α,t = 0 on ∂Ω, U α,t • n 0 = U S α,t • n 0 on ∂S 0 , i [(L i (ξ, θ, t)U α,t ) • n 0 ] ∇X (i) ξ,θ,t tan = 0 on ∂S 0 , (37)
in a sense that will be made precise below. In the above system, n 0 stands for the unit normal to ∂S 0 oriented towards S 0 . For every (α, t)

∈ U × [0, ∞), U S α,t ∈ C 3 (S 0 , R 2 ) is defined by U S α,t (y) = J ξ,θ,t (y)(DX * t (y)) -1 R -1 θ ξ + θX * (y, t) ⊥ + ∂X * ∂t (y, t) , y ∈ S 0 ,
where DX * t stands for the Jacobian matrix of X * (•, t). Notice that

J ξ,θ,t (y) = R θ DX * t (y), y ∈ S 0 . M (ξ, θ, t) ∈ W 2,∞ (F 0 ) 4 is defined by M (ξ, θ, t)(y) = J ξ,θ,t (y) -1 T , y ∈ F 0 . ( 38
) For i = 1, 2, L i (ξ, θ, t) ∈ L(H 2 (F 0 ) ∩ H 1 (F 0 ), H 1 (F 0 ) ∩ L 2 (F 0 ) 2 )
is the bounded operator defined by

L i (ξ, θ, t)U = J ξ,θ,t J -1 ξ,θ,t M (ξ, θ, t)∇(J -1 ξ,θ,t ∇X (i) ξ,θ,t • U ) ∀U ∈ H 2 (F 0 ) ∩ H 1 (F 0 ). ( 39
)
Let us precise the sense of (37). We define the following subspace of H 1 (F 0 ):

H 1 tan (F 0 ) = {φ ∈ H 1 (F 0 ) : φ • n 0 = 0 on ∂F 0 , φ = 0 on ∂Ω}. (U α,t , P α,t ) ∈ H 1 (F 0 ) × L 2 0 (F 0 ) satisfies (37) in the following sense: div U α,t = 0 in L 2 (F 0 ),
U α,t satisfies the boundary conditions

U α,t = 0 on ∂Ω, U α,t • n 0 = U S α,t • n 0 on ∂S 0
in the sense of trace, and for every test function φ ∈ H 1 tan (F 0 ),

ν F 0 i (L i (ξ, θ, t)U α,t ) • ∇ ∇X (i) ξ,θ,t • φ (y) dy - F 0 P α,t (y)
i M (ξ, θ, t)∇(∇X (i) ξ,θ,t • φ) (y) (i) dy = 0. (40)
To prove that problems [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF] and (37) are equivalent via transformation (35), we need the following result involving the spaces of test functions H 1 tan (F 0 ) and H 1 tan (F(ξ, θ, t)) (defined by [START_REF] Henrot | Variation et optimisation de formes[END_REF], p. 8).

Lemma 7. For every ψ ∈ H 1 tan (F(ξ, θ, t)), if we define the function φ :

F 0 → R 2 by φ(y) = J ξ,θ,t (y) -1 (ψ • X ξ,θ,t )(y), y ∈ F 0 , (41)
then φ ∈ H 1 tan (F 0 ), and the mapping

ψ ∈ H 1 tan (F(ξ, θ, t)) → φ ∈ H 1 tan (F 0 )
is one-to-one and onto.

Proof of Lemma 7. Let ψ ∈ H 1 (F(ξ, θ, t)). Clearly, by the regularity of X ξ,θ,t , the function φ defined by (41) belongs to H 1 (F 0 ) and vanishes on ∂Ω in the sense of trace. We need to prove the following equivalence:

ψ • n ξ,θ,t = 0 on ∂S(ξ, θ, t) ⇐⇒ φ • n 0 = 0 on ∂S 0 . (42)
By density of C 1 (F(ξ, θ, t)) in H 1 (F(ξ, θ, t)), we may assume that ψ ∈ C 1 (F(ξ, θ, t)). Introducing a parametrisation γ of ∂S 0 and the corresponding parametrisation X ξ,θ,t • γ of ∂S ξ,θ,t , one can express the unit normal n ξ,θ,t as a function of n 0 and of the first order space derivatives of X ξ,θ,t as follows:

n ξ,θ,t (X ξ,θ,t (y)) = [J ξ,θ,t (y) -1 ] T n 0 (y) | [J ξ,θ,t (y) -1 ] T n 0 (y)| , y ∈ F 0 . (43)
Consequently, for every y ∈ ∂S 0 ,

(ψ • X ξ,θ,t )(y) • n ξ,θ,t (X ξ,θ,t (y)) = 0 ⇐⇒ (ψ • X ξ,θ,t )(y) • ( J ξ,θ,t (y) -1 T n 0 (y)) = 0 ⇐⇒ J ξ,θ,t (y) -1 (ψ • X ξ,θ,t )(y) • n 0 (y) = 0 ⇐⇒ φ(y) • n 0 (y) = 0.
This proves (42).

Let us prove that for every φ ∈ H 1 tan (F 0 ), if we define ψ ∈ H 1 tan (F 0 ) by (41), then (40) is equivalent to [START_REF] Hillairet | Lack of collision between solid bodies in a 2D incompressible viscous flow[END_REF]. Noticing that for i = 1, 2,

ψ (i) • X ξ,θ,t = ∇X (i) ξ,θ,t • φ (44)
and applying formule (34) to ζ = ψ (i) • X ξ,θ,t , we obtain the following chain rule: ∇ψ (i) (X ξ,θ,t (y)) = (M (ξ, θ, t)∇(∇X

(i) ξ,θ,t • φ))(y), y ∈ F 0 .
By change of variables x = X ξ,θ,t (y), we obtain

F (ξ,θ,t) i ∇u (i) α,t (x) • ∇ψ (i) (x)dx = F 0 i J ξ,θ,t M (ξ, θ, t)∇(∇X (i) ξ,θ,t • (J -1 ξ,θ,t U α,t )) (y) • M (ξ, θ, t)∇(∇X (i) ξ,θ,t • φ) (y) dy = F 0 i (L i (ξ, θ, t)U α,t ) • ∇ ∇X (i) ξ,θ,t • φ (y) dy and F (ξ,θ,t) p α,t (x)(div ψ)(x)dx = F 0 P α,t (y) i M (ξ, θ, t)∇(∇X (i) ξ,θ,t • φ) (y) (i)
dy.

By Lemma 7, we conclude that problems [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF] and (37) are equivalent.

Remark 8. System (37) can be obtained from equation (40) by the following formal computations. If (U α,t , P α,t ) is regular enough, then for every φ ∈ H 1 tan (F 0 ),

F 0 i (L i (ξ, θ, t)U α,t ) • ∇ ∇X (i) ξ,θ,t • φ (y)dy = - F 0 i (div (L i (ξ, θ, t)U α,t )) (y)∇X (i)
ξ,θ,t (y) • φ(y)dy

+ ∂S 0 i ((L i (ξ, θ, t)U α,t )(y) • n 0 (y)) ∇X (i)
ξ,θ,t (y) • φ(y)ds(y)

and -

F 0 P α,t (y) i M (ξ, θ, t)∇(∇X (i) ξ,θ,t • φ) (y) (i) dy = F 0 ij ∂ j (M ij (ξ, θ, t)P α,t ) (y)∇X (i) ξ,θ,t (y) • φ(y)dy - ∂S 0 P (y) ij M ij (y)n (j) 0 (y)∇X (i) ξ,θ,t (y) • φ(y)ds(y). (45)
Using definition (38), we obtain To prove the regularity of the solution to problem (37) with respect to (α, t), we need to lift the boundary condition. To this end, we introduce the following subspace of H 3/2 (∂S 0 ) 2 :

∂S 0 P (y) ij M ij (y)n (j) 0 (y)∇X
G(∂S 0 ) = Γ ∈ H 3/2 (∂S 0 ) 2 | ∂S 0 Γ • n 0 ds = 0 .
We consider a bounded operator

Π : G(∂S 0 ) → H 2 (F 0 )
such that for every Γ ∈ G(∂S 0 ), if we define K = ΠΓ, then K satisfies the following constraints:

   div K = 0 in F 0 , K = 0 on ∂Ω, K = Γ on ∂S 0 .
The existence of Π is proved in [START_REF] Mitrea | The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains[END_REF] (Corollary 1.4). For every (α, t) ∈ U × [0, ∞), we define

K α,t ∈ H 2 (F 0 ) by K α,t = ΠU α,t .
Since Π is bounded and since the mapping

(α, t) ∈ U × [0, ∞) → U S α,t ∈ H 3/2 (∂S 0 ) 2 is of class C 1 ,
by composition, we get the following Lemma 9. The mapping

(α, t) ∈ U × [0, ∞) → K α,t ∈ H 2 (F 0 ) is of class C 1 .
We define the new unknown function

V α,t := U α,t -K α,t . V α,t ∈ H 2 (F 0 )∩H
[-ν div(L i (ξ, θ, t)[V α,t + K α,t ]) + j ∂ j (M ij (ξ, θ, t)P α,t )] ∇X (i) ξ,θ,t = 0 in F 0 , div V α,t = 0 in F 0 .
(46) Using (46) we can write (V α,t , P α,t ) as a zero of a bounded operator depending on the parameter (α, t) ∈ U × [0, ∞). The definition of this operator requires to introduce the following Banach spaces:

A = H 2 (F 0 ) ∩ H 1 tan (F 0 ) × H 1 (F 0 ) ∩ L 2 0 (F 0 ), B = L 2 (F 0 ) 2 ∩ (H 1 tan ) (F 0 ) × H 1 (F 0 ) ∩ L 2 0 (F 0 ), endowed with the norms: (V, P ) A = V H 2 (F 0 ) + V H 1 (F 0 ) + P H 1 (F 0 ) + P L 2 (F 0 ) , (f, g) B = f L 2 (F 0 ) 2 + f (H 1 tan ) (F 0 ) + g H 1 (F 0 ) + g L 2 (F 0 ) . In the definition of B, (H 1
tan ) (F 0 ) stands for the dual space of H 1 tan (F 0 ). We now define the operator

H : U × [0, ∞) × A → B by    H(α, t; V, P ) = (Q(α, t; V, P ), R(α, t; V, P )), Q(α, t; V, P ) = i [-ν div(L i (ξ, θ, t)[V + K α,t ]) + j ∂ j (M ij (ξ, θ, t)P )] ∇X (i) ξ,θ,t , R(α, t; V, P ) = div V. ( 47
)
Remark 10. For every V ∈ H 1 tan (F 0 ), div V ∈ L 2 0 (F 0 ), which justifies the definition of B. Indeed, by Stokes formula,

F 0 div V dx = ∂S 0 V • n 0 ds = 0.
With these notations, system (46) can be rewritten in the following implicit form:

H(α, t; V α,t , P α,t ) = 0. ( 48
)
The final step is to prove that the mapping

(α, t) ∈ U × [0, ∞) → (V α,t , P α,t ) solution to (48) is of class C 1 .
For this purpose, we apply the implicit function theorem. Let (α * , t * ) be a given point in U × [0, ∞), with α * = (ξ * , ξ * , θ * , θ * ). We introduce the following notations:

X = X ξ * ,θ * ,t * , J = J ξ * ,θ * ,t * , J = J ξ * ,θ * * , M * = M (ξ * , θ * , t * ), L * i = L i (ξ * , θ * , t * ), F = X(F 0 ), F = X(F 0 ), (V * , P * ) = (V α * ,t * , P α * ,t * ).
We will apply the implicit function theorem to the function H defined by (47), at point

(α * , t * , V * , P * ) ∈ U × [0, ∞) × A. • H is a C 1 mapping from U × [0, ∞) × A into B. Indeed, -since the mappings (ξ, θ, t) ∈ W × [0, ∞) → J ξ,θ,t ∈ W 2,∞ (F 0 ) 4 and (ξ, θ, t) ∈ W × [0, ∞) → J ξ,θ,t ∈ W 2,∞ (F 0 ) are of class C 1 , by Lemma 9 and by continuity of the product W 2,∞ (F 0 ) × H 2 (F 0 ) ∩ H 1 (F 0 ) → H 2 (F 0 ) ∩ H 1 (F 0 ), for i = 1, 2, the mapping (α, t, V ) ∈ U × [0, ∞) × H 2 (F 0 ) ∩ H 1 (F 0 ) → J -1 ξ,θ,t ∇X (i) ξ,θ,t • [V + K α,t ] ∈ H 2 (F 0 ) ∩ H 1 (F 0 ) is of class C 1 .
Recall that for i = 1, 2, L i (ξ, θ, t) is defined by (39) (p. 13). Since the matrix inversion is a C ∞ operation from GL 2 (R) onto itself and since the derivation in Sobolev spaces is a bounded operator, we obtain that for i = 1, 2, the mapping

(α, t, V ) ∈ U ×[0, ∞)×H 2 (F 0 )∩H 1 (F 0 ) → L i (ξ, θ, t)[V +K α,t ] ∈ H 1 (F 0 )∩L 2 (F 0 ) 2
is of class C 1 , and consequently, that the mapping

(α, t, V ) ∈ U × [0, ∞) × H 2 (F 0 ) ∩ H 1 (F 0 ) → (div (L i (ξ, θ, t)[V + K α,t ])) ∇X (i) ξ,θ,t ∈ L 2 (F 0 ) 2 ∩ (H 1 tan ) (F 0 )
is of class C 1 . By the same arguments, that the mapping

(α, t, P ) ∈ U × [0, ∞) × H 1 (F 0 ) ∩ L 2 (F 0 ) → j ∂ j (M ij (ξ, θ, t)P ) ∇X (i) ξ,θ,t ∈ L 2 (F 0 ) 2 ∩ (H 1 tan ) (F 0 ) is of class C 1 . Consequently, Q is a C 1 mapping from U ×[0, ∞)×A into L 2 (F 0 ) 2 ∩ (H 1 tan ) (F 0 ); -R is independant on the variable (α, t) and induces a bounded operator from A into H 1 (F 0 ) ∩ L 2 0 (F 0 ). In particular, R is of class C 1 .
• By definition, (V * , P * ) satisfies H(α * , t * ; V * , P * ) = 0.

• Let us prove that the operator Λ := D (V,P ) H(α * , t * , V * , P * ) is a bicontinuous isomorphism from A onto B. Λ is defined for every (V, P ) ∈ A by

   Λ(V, P ) = (Λ 1 , Λ 2 )(V, P ), Λ 1 (V, P ) = i [-ν div(L * i V ) + j ∂ j (M * ij P )] ∇X (i) , Λ 2 (V, P ) = div V.
Clearly Λ ∈ L(A, B). We need to prove that Λ is invertible with continuous inverse. Let (f, g) ∈ B. We need to prove that there exists one unique pair (V, P ) ∈ A such that (f, g) = Λ(V, P ), that is, that there exists one unique solution to the problem i [-ν div(L * i V ) + j ∂ j (M * ij P )]

∇X (i) = f in F 0 , div V = g in F 0 . (49)
Besides, we need to prove that this solution satisfies natural estimates with respect to (f, g). To this end, we perform the opposite change of functions to (35) which transforms problem (49) posed on the reference domain F 0 into a Stokes problem posed on F. The existence and uniqueness of the solution to problem (49) result from the existence and uniqueness of the solution to the Stokes problem on F, via transformation (35). Then we use classical estimates on the solution to the Stokes problem with slip boundary condition and the regularity of the diffeomorphism X to control the norm of (V, P ) in A by the norm of the source term (f, g) in B.

We define the functions v, p, f , g by the following relations:

       v • X = J -1 JV, p • X = J -1 P, f • X = J -1 M * f, g • X = J -1 g.
Proposition 4 results directly from this statement.

Step 3. Continuity of F , G and local Lipschitz condition with respect to α.

Recall that F, G are defined by (24) (p. 8). Using the change of variable formula in the curvilinear integrals (see for instance [START_REF] Henrot | Variation et optimisation de formes[END_REF] p. 190), we transform the integrals on ∂S(ξ, θ, t) into integrals on the fixed boundary ∂S 0 . We denote by J tan (X ξ,θ,t ) the tangential jacobian of X ξ,θ,t on ∂S 0 , defined by J tan (X ξ,θ,t ) = J -1 ξ,θ,t T n 0 J ξ,θ,t .

For i = 1, 2, we obtain

F (i) (α, t) = - 1 m ∂S(ξ,θ,t)
(σ(u α,t , p α,t )) (i) • n ξ,θ,t ds = -1 m ∂S 0 (σ(u α,t , p α,t ) • X ξ,θ,t ) (i) • (n ξ,θ,t • X ξ,θ,t )J tan (X ξ,θ,t )ds.

By formula (43) (p. 14), last equality can be written as follows:

F (i) (α, t) = - 1 m ∂S 0 J ξ,θ,t J -1 ξ,θ,t (σ(u α,t , p α,t ) • X ξ,θ,t ) (i) • n 0 ds. ( 54
)
As a consequence of Lemma 5 (p. 11) and Proposition 4, the mapping (α, t) ∈ U × [0, ∞) → J ξ,θ,t J -1 ξ,θ,t (σ(u α,t , p α,t ) • X ξ,θ,t

) (i) ∈ H 1 (F 0 )
is of class C 1 . Since the trace operator from H 1 (F 0 ) into L 2 (∂S 0 ) 2 and the inner product with the vector field n 0 in L 2 (∂S 0 ) 2 are bounded and linear, we obtain that for i = 1, 2, the mapping (α, t) ∈ U × [0, ∞) → F (i) (α, t) ∈ R is of class C 1 . In particular, F is continuous and satisfies a local Lipshitz condition with respect to α. The continuity of G and the local Lipschitz condition result from the same argument, and from the facts that the mappings

ξ ∈ R 2 → (• -ξ) ⊥ ∈ L 2 (∂S 0 ) and t ∈ [0, ∞) → I(t) ∈ R are of class C 1 .
Step 4. Rewriting of problem [START_REF] Murat | Étude de problèmes d'optimal design[END_REF] as a first order differential system. Problem (25) (p. 9) is equivalent to the following: find T > 0 and a function α ∈ C 2 ((0, T ), R 6 ) ∩ C 1 ([0, T ), R 6 ) such that α (t) = Aα(t) + H(α(t), t) ∀t ∈ (0, T ), α(0) = α 0 , (55)

0 P 0 P

 00 ,t (y) • φ(y)ds(y) = ∂S (y)(M (ξ, θ, t)(y)n 0 (y)) • (J ξ,θ,t (y)φ(y))ds(y) = ∂S (y)n 0 (y) • φ(y)ds(y) = 0.

  1 tan (F 0 ) and (U α,t , P α,t ) is solution to (37) if and only if (V α,t , P α,t ) is solution to

	i

Acknowledgements. The author gratefully acknowledge the anonymous referee for his careful reading and for a number of useful suggestions that have been incorporated in the current version of the manuscript.

( f , g) ∈ (H 1 tan ) (F) ∩ L 2 (F ) 2 × H 1 (F ) ∩ L 2 0 (F) and (v, p) is solution to the following problem: find (v, p) ∈ H 1 tan (F) × L 2 0 (F) such that -ν∆v

By the results in [START_REF] Solonnikov | On a boundary value problem for a stationary system of navier-stokes equations[END_REF], problem (50) admits one unique solution, this solution has the regularity (v, p) ∈ H 2 (F ) ∩ H 1 tan (F) × H 1 (F ) ∩ L 2 0 (F) and there exists a constant C > 0 depending only on (α * , t * ) such that

By construction of X, the partial derivatives of X (resp. X -1 ) up to order 2 are uniformly bounded on F 0 (resp. F). The domains F 0 and F being bounded, there exist two constant C 1 , C 2 > 0 (depending only on (α * , t * )) such that

Finally, by inequalities (51)-( 53), there exists a constant C > 0 such that

which proves that Λ is invertible in L(A, B).

By the implicit function theorem, the mapping

Using Lemma 5 (p. 11), formula (35) (p. 12) and Lemma 9, we obtain Proposition 11. The mapping

where (u α,t , p α,t ) is the solution to problem [START_REF] Gunzburger | Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions[END_REF], is of class C 1 . of the result by Hillairet [START_REF] Hillairet | Lack of collision between solid bodies in a 2D incompressible viscous flow[END_REF].

5 Proof of Theorem 2.

To study the stability of the trajectories with respect to the deformation Θ * , we treat Θ * as a parameter in the Cauchy problem (55). We use the same notations as in Section 4.

We consider the open subset U ⊂ R 6 defined by (18) (p. 7) and define the following Cauchy problem with parameter Θ * :

where α 0 ∈ U is a fixed initial datum and L is defined by

Here we have treated Θ * as a variable in the definition of H given in Section 4. (Recall that X ξ,θ,t depends on Θ * through formulas ( 28) and [START_REF] Solonnikov | On a boundary value problem for a stationary system of navier-stokes equations[END_REF].) Adapting the method in Section 4 one can easily prove that L is continuous on U × [0, ∞) × D. Now we can prove Theorem 2. Let Θ * ∈ D be a given deformation and (Θ * n ) n∈N be a sequence of deformations converging to Θ * in D. We denote by (CP ) Θ * the Cauchy problem associated to Θ * with initial datum α 0 , and by α the maximal solution to (CP ) Θ * . For every n ∈ N, we denote by (CP ) Θ * n the Cauchy problem associated to Θ * n with the same initial datum α 0 , and by α n the maximal solution to (CP ) Θ * n . By a standard topology argument, the set

Let us fix τ > 0 and define

By continuity of L, there exists M > 0 such that

By ( 56)-( 57) and by the Cauchy-Peano theorem, for every n ∈ N, (CP ) Θ * and (CP ) Θ * n admit a solution defined on [0, T 1 ]. By the results in Section 4, these solutions are unique on [0, T 1 ], and by maximality of α, α n , these solutions necessarily coincide with α, α n on [0, T 1 ], which yields

This proves the first point in Theorem 2. Now let T * > 0 such that the solutions (ξ n , θ n ) are well-defined on [0, T * ] for all n ∈ N. Since L is continuous on the compact K, there exists a continuous function

and lim u→0 + η(u) = 0. As a result, for every n ∈ N, α n is an ε n -approximate solution to (CP Θ * ), with

. By the results in Section 4, we can suppose that there exists k > 0 (depending on Θ * ) such that the mapping (α, t) ∈ B(α 0 , r 0 ) × [0, T * ] → L(α, t, Θ * ) ∈ B(α 0 , r 0 ) is k-lipschitzian with respect to α. As a result, by Gronwall lemma (see for instance [START_REF] Demailly | Analyse numérique et équations différentielles[END_REF] α n (t) -α(t) e kT * -1 k η ( Θ * n -Θ * D ) .

Taking the limit when n → ∞, we get the second point in Theorem 2.