Moritz Fieback
email: m.c.r.fieback@tudelft.nl

Guilherme Cardoso Medeiros

Anteneh Gebregiorgis

Hassen Aziza

Mottaqiallah Taouil

Said Hamdioui
email: s.hamdioui@tudelft.nl

Intermittent Undefined State Fault in RRAMs

Keywords: RRAM test, defect modeling, device-aware test

Industry is prototyping and commercializing Resistive Random Access Memories (RRAMs). Unfortunately, RRAM devices introduce new defects and faults. Hence, high-quality test solutions are urgently needed. Based on silicon measurements, this paper identifies a new RRAM unique fault, the Intermittent Undefined State Fault (IUSF); this fault causes the RRAM device to intermittently change its switching mechanism from bipolar to complementary switching, resulting in undefined state faults. First, we characterize the IUSF by analyzing RRAM devices, and demonstrate that a single RRAM device can suffer from the IUSF up to 1.068 % of its switching cycles; we relate the IUSF to two defects: capping layer doping, and over-forming. This clearly shows the importance of detecting this fault. Second, we develop a device-aware defect model that accurately describes the physical behavior of these defects and gives essential insights into the IUSF's behavior and its detection. Third, we perform fault modeling by applying the device-aware defect model, and the results are used to develop high-quality test solutions for the IUSF. The contributions in this work improve the overall RRAM test quality, which enables mass commercialization of RRAMs.

I. INTRODUCTION

Resistive Random Access Memories (RRAMs) are a promising alternative to replace traditional memory technologies such as Flash and Dynamic RAM (DRAM) [START_REF] Wong | Metal-Oxide RRAM[END_REF][START_REF] Yu | Emerging Memory Technologies: Recent Trends and Prospects[END_REF]. RRAM devices are non-volatile and consume no static power, which makes them suitable for embedded applications. Furthermore, RRAM devices can be read and written in few nanoseconds, making them significantly faster than Flash [START_REF] Luo | 8-Layers 3D vertical RRAM with excellent scalability towards storage class memory applications[END_REF]. They are relatively simple to manufacture and enable dense crossbar structures and can store multiple bits per cell [START_REF] Wong | Metal-Oxide RRAM[END_REF][START_REF] Yu | Emerging Memory Technologies: Recent Trends and Prospects[END_REF][START_REF] Luo | 8-Layers 3D vertical RRAM with excellent scalability towards storage class memory applications[END_REF][START_REF] Wu | 14.3 A 43pJ/Cycle Non-Volatile Microcontroller with 4.7µs Shutdown/Wake-up Integrating 2.3-bit/Cell Resistive RAM and Resilience Techniques[END_REF] and even enable new computing paradigms [START_REF] Nguyen | Memristive devices for computing: Beyond CMOS and beyond von Neumann[END_REF][START_REF] Chi | PRIME: A Novel Processing-in-Memory Architecture for Neural Network Computation in ReRAM-Based Main Memory[END_REF]. Although they have many benefits, some obstacles still need to be overcome. One of these is that RRAM devices introduce new defects and faults that have not been seen in traditional memories, e.g., forming defects [START_REF] Fieback | Testing Resistive Memories: Where are We and What is Missing?[END_REF]. Hence, appropriate fault modeling and high quality and efficient test solutions are needed.

Several RRAM defects and faults have been described in the literature [START_REF] Ginez | Design and Test Challenges in Resistive Switching RAM (ReRAM): An Electrical Model for Defect Injections[END_REF][START_REF] Haron | On Defect Oriented Testing for Hybrid CMOS/Memristor Memory[END_REF][START_REF] Kannan | Sneak-Path Testing of Crossbar-Based Nonvolatile Random Access Memories[END_REF][START_REF] Chen | Fault modeling and testing of 1T1R memristor memories[END_REF][START_REF] Liu | Defect Analysis and Parallel March Test Algorithm for 3D Hybrid CMOS-Memristor Memory[END_REF][START_REF] Fieback | Device-Aware Test: A New test Approach Towards DPPB Level[END_REF]. The majority of the studied defects are modeled as linear resistors in (and at) the terminals of the RRAM device, i.e., opens, short and bridges [START_REF] Ginez | Design and Test Challenges in Resistive Switching RAM (ReRAM): An Electrical Model for Defect Injections[END_REF][START_REF] Haron | On Defect Oriented Testing for Hybrid CMOS/Memristor Memory[END_REF][START_REF] Kannan | Sneak-Path Testing of Crossbar-Based Nonvolatile Random Access Memories[END_REF][START_REF] Chen | Fault modeling and testing of 1T1R memristor memories[END_REF][START_REF] Liu | Defect Analysis and Parallel March Test Algorithm for 3D Hybrid CMOS-Memristor Memory[END_REF]. This defect model led to the identification of several RRAM fault models, e.g., undefined write or read faults [START_REF] Haron | On Defect Oriented Testing for Hybrid CMOS/Memristor Memory[END_REF][START_REF] Kannan | Sneak-Path Testing of Crossbar-Based Nonvolatile Random Access Memories[END_REF], and deep faults [START_REF] Kannan | Sneak-Path Testing of Crossbar-Based Nonvolatile Random Access Memories[END_REF]. However, a linear resistor does not accurately describe defects in the RRAM device itself, e.g., forming defects [START_REF] Fieback | Device-Aware Test: A New test Approach Towards DPPB Level[END_REF]. The RRAM device is non-linear by nature, and the linear resistor approach models the defects in the surrounding interconnections rather than the defects in the device itself; using this inappropriate defect model leads to test escapes and yield loss. A solution to this is device-aware defect modeling, in which the defect's impact on the RRAM device's electrical parameters is incorporated in the RRAM device model [START_REF] Fieback | Device-Aware Test: A New test Approach Towards DPPB Level[END_REF]. The resulting defect model leads to realistic fault models that can be used to develop high-quality test solutions. However, since RRAM is a new technology, unique defects and failure mechanisms will evolve, e.g., defects caused by miniaturization of the device, or complex faulty behaviors such as SET switching failure [START_REF] Balatti | Set Variability and Failure Induced by Complementary Switching in Bipolar RRAM[END_REF]. Hence, it is of prime importance to identify such defects (in an industrial environment), characterize, and accurately model them for optimal and high-quality test development.

This paper identifies and characterizes, based on measurement data, a new RRAM fault, the Intermittent Undefined State Fault (IUSF). This fault causes the RRAM device to intermittently change its switching mechanism from bipolar to complementary, which affects write operations and causes undefined state faults. We apply the device-aware approach to model the defects that cause IUSFs. Subsequently, we use this model to develop appropriate fault models and test solutions. Summarizing, in this paper we:

• Identify the IUSF in RRAMs based on silicon data.

• Determine RRAM defects that could lead to IUSFs.

• Demonstrate that the conventional defect modeling approach fails to appropriately describe these defects.

• Perform device-aware defect modeling to accurately model these defects.

• Perform device-aware fault modeling to develop realistic fault models and thereafter optimal test solutions.

The remainder of the paper is structured as follows. Section II briefly presents RRAM background information. Section III presents experimental evidence, characterization, and underlying defects of the IUSF. Section IV demonstrates the failure of the linear resistor defect model, and presents the defect model that accurately describes the behavior of a defective RRAM device. This model is used in Section V to perform fault modeling and analysis. Section VI describes a test algorithm to detect the IUSF. Section VII discusses and concludes the paper.

-2 -1 0 1 -100 0 100 V SET V RESET V TE [A.U.] I TE [A.U.]
(a) Bipolar switching (BS)

-1 -0.5 0 0.5 1 -100

0 100 V SET+ V RESET+ V SET- V RESET- V TE [A.U.] I TE [A.U.] (b) Complementary switching (CS)
Fig. 2: Resistance switching mechanisms in RRAM.

II. BACKGROUND

Fig. 1a illustrates a RRAM device; it consists of two electrodes, the top electrode (TE) and bottom electrode (BE), in between which a metallic oxide is sandwiched (commonly, HfO x , TaO x , or TiO x ) with an additional capping layer [START_REF] Wong | Metal-Oxide RRAM[END_REF][START_REF] Yu | Emerging Memory Technologies: Recent Trends and Prospects[END_REF][START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF]. After fabrication, a voltage V forming is applied to the oxide to break some of the metal and oxygen ions bonds [START_REF] Wong | Metal-Oxide RRAM[END_REF][START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF]. Fig. 1a shows how the (negatively charged) oxygen ions (white circles) move towards the positive electrode and enter into the capping layer (cap) under the influence of the electric field and bond there [START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF][START_REF] Celano | Imaging the three-dimensional conductive channel in filamentary-based oxide resistive switching memory[END_REF]. This leaves behind a chain of conducting vacancies (black circles) known as Conductive Filament (CF), as shown in Fig. 1b. When the voltage is removed, the CF remains, making the RRAM device non-volatile. The shape (e.g., radius) of the CF determines the resistance of the device [START_REF] Celano | Imaging the three-dimensional conductive channel in filamentary-based oxide resistive switching memory[END_REF]; it can be changed by applying voltages to the oxide. Note that the ability to bind oxygen ions in the capping layer determines the resistance switching mechanism of the RRAM device [START_REF] Nardi | Complementary switching in oxide-based bipolar resistive-switching random memory[END_REF][START_REF] Chen | Complementary resistive switching behaviors evolved from bipolar TiN/HfO2/Pt device[END_REF]; if (almost) all of oxygen ions can be bound, then the device will be Bipolar Switching (BS). Conversely, if many oxygen ions remain unbound, then the device will be Complementary Switching (CS).

The BS mechanism relies on the formation and dissolution of the CF by movement of oxygen ions [START_REF] Wong | Metal-Oxide RRAM[END_REF][START_REF] Yang | Probing nanoscale oxygen ion motion in memristive systems[END_REF]. Its electrical behavior is illustrated in Fig. 2a. When a negative voltage V TE less than a threshold V RESET is applied to the TE, then some of the oxygen ions move back into the oxide and re-oxidize the CF leaving a gap in the CF, as shown in Fig. 1c. This movement increases the resistance of the device and is called a RESET operation. When a positive voltage higher than a threshold V SET is applied, the bonds between the metal and oxygen ions break again, and the CF regrows, as shown in free oxygen ions [START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF]. This process is called a SET operation and results in a lower device resistance. The exact value of the resistance in SET (R SET ) and in RESET (R RESET ) varies per write cycle due to the random nature of the filament formation and dissolution; it is known as cycle-to-cycle variation [START_REF] Wong | Metal-Oxide RRAM[END_REF][START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF].

The CS mechanism relies on the formation and dissolution of the CF by movement of oxygen vacancies [START_REF] Nardi | Complementary switching in oxide-based bipolar resistive-switching random memory[END_REF][START_REF] Chen | Complementary resistive switching behaviors evolved from bipolar TiN/HfO2/Pt device[END_REF]. Its electrical behavior is illustrated in Fig. 2a. After initial forming, a normal BS RESET is performed, resulting in the CF shown in Fig. 1c. Now, by applying a voltage exceeding the switching threshold V SET+ , a SET operation will take place (similar to BS) that results in the breaking of the bonds between the oxygen and metal ions and the forming of the CF as shown in Fig. 1b. However, when the voltage is further increased and exceeds the threshold V RESET+ , the negative oxygen ions cannot all be bound in the capping layer. Instead, they will remain in the oxide and the positive oxygen vacancies in the CF will start to move along the electric field towards the negative electrode. This disrupts the just created CF and brings the device in a RESET state, as shown in Fig. 1d. Note that both SET and RESET occur at the same voltage direction, they complement each other. When a negative voltage is applied less than the threshold V SET-, first the oxygen vacancies will move back into the oxide and form a CF, see Fig. 1e. With further decreasing of the voltage below threshold V RESET-, the CF will be broken again by the oxygen ions that move back into the oxide from the capping layer, leading to the state shown in Fig. 1c.

Fig. 3 shows two commonly used cell designs that consist of only one RRAM device (1R, Fig. 3a), or of one transistor and one RRAM device (1T1R, Fig. 3b); each of these cell designs can be used to build a crossbar memory cell array. In the figures, BL, WL, and SL refer to bit line, word line, and select line, respectively. The RRAM devices can be written by setting these lines to appropriate voltages. A cell is read out by applying a read voltage and sensing the resulting current through the device using a sense amplifier (SA).

III. CHARACTERIZATION OF THE INTERMITTENT UNDEFINED STATE FAULT

First the results of the characterization of RRAM devices under investigation (based on which the IUSF was identified) are presented. Thereafter, a brief overview on the underlying physics and potential defects behind the IUSF is given.

A. Characterization

We measured the electrical characteristics of 7 × 7 = 49 1T1R BS RRAM devices on a single wafer during 936 RESET-SET cycles. The RRAM devices are manufactured at ST Microelectronics and have the following structure (BE, oxide, cap, TE) = (TiN, HfO 2 (10 nm), Ti(10 nm), TiN). An ). An illustrative measured I-V graph of a defect-free device is shown in Fig. 4a. Typical nominal values for V SET and V RESET are 0.82 V and -0.88 V.

2 -1 0 1 -100 0 100 '1' V TE [V] I TE [µA] (a) Defect-free device -2 -1 0 1 -100 0 100 'U' 1 2 V TE [V] I TE [µA] (b) Faulty device -2 -1 0 1 -100 0 100 disturbed '1' V TE [V] I TE [µA] (c) Weak device
While analyzing the measurement data, we noticed that some of the devices showed a strange switching characteristic! After a number of cycles, the switching resembles CS when performing SET, even though the devices are BS. The observed undesired CS in all faulty devices can be classified into two groups: faulty and weak devices. Fig. 4b represents the I-V graph of the faulty devices. A SET event occurs when the voltage increases from 0 V. However, as the voltage increases even further, the current through the device suddenly decreases at 1 , leading to an undesired RESET putting the device in an undefined state of R SET =33 kΩ. The subsequent RESET operation at 2 also differs from a nominal operation, as the current first increases before decreasing, which again resembles CS. Hence, in this group of devices, a SET operation ends in undefined state. Contrarily, Fig. 4c represents the group of weak devices that shows a similar switching behavior, but instead the operation ends in a disturbed '1' state.

Table I lists the occurrence probability (P CS .) of undesired CS in the measured devices (using the WL and BL to indicate them); this is the percentage of cycles in which CS occurs given a total of 936 cycles, and includes both groups of faulty It can be seen that depending on the defect strength, the failure rate varies from 0.107 % up to 1.068 % of the cycles, all resulting in an IUSF. Also, the probability of weak CS can be easily derived using P weak =P CS -P IUSF , and can thus be a high as 5.021 %. Note that the weak devices do not result in logic faulty behavior at time=0, but can result in reliability concerns as the weak devices may become logic faults in field, e.g., due to aging, the weak fault may become an IUSF.

B. Physical Explanation and Possible Defects

In Section II, we already distinguished between BS and CS in RRAM devices; BS devices rely on the movement of oxygen ions to grow and dissolve the CF, while CS devices rely on the movement of the oxygen vacancies due to the shortage of ions in the capping layer to bind the free oxygen ions [START_REF] Nardi | Complementary switching in oxide-based bipolar resistive-switching random memory[END_REF][START_REF] Chen | Complementary resistive switching behaviors evolved from bipolar TiN/HfO2/Pt device[END_REF]. Hence, the physics behind the IUSF must involve a reduced oxygen binding capability of the device with respect to the nominal BS case. Two manufacturing defects may lead to this, individually or combined: 1) low-doping of the capping layer, and 2) over-forming; both are explained next.

1) Low doping of capping layer: The capping layer is not doped with enough positive ions that can bind the negative oxygen ions originating from the CF growth. This low binding capacity prevents the widening of the CF and thus favors CS.

Since the formation and dissolution of a CF is a random process, the exact amount of oxygen ions varies per write cycle, and hence CS will not occur in every cycle [START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF].

2) Over-forming: Over-forming causes the filament to grow wider than in the nominal case. Therefore, more oxygen ions need to bind in the capping layer [START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF], leading more quickly to a shortage of binding capacity, thus resulting in CS. The random nature of CF growth and dissolution causes this behavior not to occur in every cycle.

IV. DEFECT MODELING FOR IUSF

Traditionally, defects in RRAMs are modeled using linear resistors injected in the netlist at or in the terminals of the defective device, e.g., a bridge between the device terminals. However, these models are unable to actually describe the faulty behavior in the defective device [START_REF] Fieback | Device-Aware Test: A New test Approach Towards DPPB Level[END_REF]. Next, we first show that linear resistors fail to sensitize the IUSF. Then, we use device-aware defect modeling to model the two defects discussed in the previous section with a single defect model able to sensitize IUSFs.

A. Linear Resistor Defect Models

When modeling defects in a RRAM device with linear resistors, there are only two possible options; the resistor can either be in parallel or in series with the device. Fig. 5a and Fig. 5b show how these defect models affect the behavior of the RRAM device for varying defect strength (i.e., the resistance of the resistor). Clearly none of these two defect models can describe the IUSF properly; a parallel resistor shows that the defective device behaves as a low ohmic resistor for low defect values, while a series resistor shows that the device behaves as a high ohmic resistor for high defect sizes.

B. Device-Aware Defect Model

The device-aware approach appropriately models a defect by incorporating the impact of the affected device technology parameters (e.g., filament dimensions, oxide thickness) on the electrical behavior of the defective device [START_REF] Fieback | Device-Aware Test: A New test Approach Towards DPPB Level[END_REF]; it consists of three steps: 1) physical defect modeling, 2) electrical defect modeling, and 3) fitting and model optimization.

• 10 -3 K -1 R th,eff 6 • 10 6 K•W -1 4
1) Physical Defect Modeling: In this step, the effects of the defect on the physical parameters of the RRAM device are analyzed and modeled. Section III showed that a shortage of ions that can bind free oxygen ions causes CS in BS devices, which leads to the IUSF. To appropriately model this phenomenon, we use the physics-based HfO x RRAM model for CS, JART VCM v2, from [START_REF] Torre | Compact modelling of resistive switching devices based on the valence change mechanism[END_REF]; the model describes CS as the exchange of oxygen vacancies between two regions (region 1 and 2, see Fig. 1a) in the oxide; the paper also describes how the model can be adapted to perform BS by introducing asymmetry between these regions via their lengths, barrier height, and electron mobility. Based on this, we adapt the model so that region 1 becomes the BS CF, and that the switching depends solely on this region. Then, we include the unwanted CS by changing two of the model parameters: the initial oxygen vacancy concentration in region 1 (parameter N init1 ), and the maximal vacancy concentration in the oxide (parameter N max ). The former controls the number of oxygen vacancies that form the CF in region 1, while the latter controls the overall number of oxygen vacancies in the oxide. Since the model is based on the exchange of vacancies between region 1 and 2, the ratio N init1 /N max determines the number of vacancies that can be in region 2. The lower this ratio, the more vacancies can move to region 2 and thus the stronger the CS effect.

2) Electrical Defect Modeling: The next step is to incorporate the affected physical parameters of the device (e.g., the filament radius) into its electrical parameters (e.g., V SET , R RESET ). JART VCM v2 is written in Verilog-A. Hence, it can directly be integrated in a SPICE simulator to derive the electrical behavior of the memory in the presence of the modeled defects.

3) Fitting and Model Optimization:

The model is fitted to match the measurements from Section III. The parameter values used for the JART VCM v2 model are listed in Table III. Fig. 6a shows the simulation results for the BS device model as well as the measurements of defect-free devices for different SET-RESET cycles. The model matches the measurements well, except for the RESET current decrease, which is steeper than the measured current decrease. This is because the model does not include the randomness of the filament growth and rupture, and it is based on the CS model assuming that vacancies always shift between the two switching regions; in a real RRAM device, the exchange is between the capping layer and the oxide, leading to a less abrupt switching.

-2 -1.5 -1 -0.5

0 0.5 1 1.5 -100 0 100 V TE [V] I TE [µA]
Defect-free BS measurements Defect-free BS simulation (a) Simulation vs. measurements for defect-free device.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 -100

0 100 1 S 2 V TE [V] I TE [µA]
Defective BS measurements Defective BS simulation (b) Simulation vs. measurements for defective device Fig. 6: Comparing defect-free and defective device simulation model.

Fig. 6b shows the simulation results (of the calibrated model) and the measurement data for a defective BS RRAM device suffering from an IUSF; we use the same parameters as in Table III but with N init1 =1.3 • 10 27 m -3 and N max =8.0 • 10 27 m -3 so that the ratio N init1 /N max is rather small while still being physically realistic. It can be seen that the simulation and the measurements match well, and the the simulation can predict the switching moments ( 1 , 2 ), and the final resistance states ( S ). Note that that the simulation has marginal deviations compared with the data, e.g., some shifts in SET and RESET voltages. This is again caused by the used physical model that favors the abrupt switching over the more gradual switching. Nonetheless, the obtained device model is accurate enough (e.g., accurate in the prediction of the intermediate state S ) to be used for fault modeling.

The model is also calibrated for weak devices modeling such as that shown in Fig. 4c, e.g., using N init1 =1.8 • 10 27 m -3 and N max =8.0 • 10 27 m -3 . The results (not shown here) show a very good matching between measurements and simulations.

V. FAULT MODELING AND ANALYSIS

To describe the faults, we will use the fault primitive (FP) notation as S/F/R [START_REF] Hamdioui | An experimental analysis of spot defects in SRAMs: realistic fault models and tests[END_REF]. Here, S denotes the sensitizing operation that is applied; e.g., S=1w0 denotes writing a '0' to a cell that previously stored a '1', and S=1r1 denotes reading a '1' from a cell that stores a '1'. F denotes the faulty state of the cell after the operation has completed; i.e., F ∈ {0, 1, U}. Finally, R denotes the output if the final operation in S is a read operation, where R ∈ {0, 1, ?, -}. '?' indicates a random read output (i.e., it is unpredictable if '1' or '0') that might occur when the input of the SA is to close to its reference; '-' is used if there is no read output expected.

In order to analyze the IUSF effects on RRAM, a SPICEbased simulation is built; it consists of a 1T1R cell (with same dimensions as those devices used during the characterization) and the required circuitry to drive appropriate voltages in the three control lines WL, BL and SL (see Fig. 3b). The defect-free device is now replaced with the model of the defective RRAM device obtained in the previous section. The defect strength is governed by the ratio N init1 /N max ; N init1 is varied from 1.2 • 10 27 m -3 to 2.0 • 10 27 m -3 , and N max from 3.0 • 10 27 m -3 to 8.0 • 10 27 m -3 to fit within realistic physical limits [START_REF] Bengel | Variability-Aware Modeling of Filamentary VCM based Bipolar Resistive Switching Cells Using SPICE Level Compact Models[END_REF] and ensure proper functioning of the model. Similar to the characterization, the voltages are applied via the BL and are swept from 0 V to 1.2 V back to 0 V for SET, as this is the only operation that can sensitize the fault. While performing SPICE simulations, we inspect both the final resulting resistance of the defective device (e.g., R SET after SET operation) and the I-V graph; these are used to derive the behavior of the memory in the presence of the modeled defects. For example, if the final device resistance is outside of the R SET range, then F=U, and if the undesired CS occurs but only disturbs the R SET value without putting it outside the spec, then it is a weak fault.

Table IV shows the obtained results. Note that observed faults can be classified into two types: strong and weak faults. Strong faults are faults causing functional errors and can always be sensitized by applying a sequence of operations. In contrast, weak faults do not cause functional errors; instead, they cause parametric deviations (e.g., disturbances in the SET state). In the table, strong faults are described with FP notation. It can be seen that faults are sensitized when the ratio N init1 /N max decreases; strong faults IUSFs are sensitized for ratios below 0.2, while weak faults are caused for ratio's between 0.20 and 0.25. A lower ratio gives more room for vacancies to move from region 1 into region 2. This makes it easier for the device to perform the undesired CS.

VI. TEST DEVELOPMENT

As shown in Table IV, the defect causes strong and weak faults. A straightforward test for the strong fault IUSF would be a march test. However, due to the nature of the faults (being intermittent and causing the cell to switch into 'U'), a march test cannot guarantee the detection of such a fault. It will rather probabilistically detect the fault; reading this state will sometimes result in '1' and sometimes in '0'. The following march algorithm can be used:

March-IUSF = (w0, w1, r1) k where indicates any memory addressing order, w0 denotes a RESET operation, w1 a SET operation, r1 a read 1 operation, and k indicates the number of times the sequence is applied.

If we assume that reading a cell in 'U' state results the same probability of getting '1' or '0' (i.e., 50 %), and that the occurrence probability of IUSF is P IUSF , then the detection probability is:

P d =1 -(1 -(P IUSF • 50 %)) k .
Assuming that P IUSF =1.068 % (see Section III) results in k = 560 to realize a fault coverage (FC) of FC=95 %, and in k=1291 to realize FC=99 %. Hence, realizing high FC needs long test time; not to mention the potential impact of repeating memory accesses on the endurance. Note that the detection capabilities might be further improved by adding additional stress factors, e.g., by using back-to-back operations and special data backgrounds.

To reduce the test time while providing high fault coverage, design-for-test (DfT) schemes can be used. For example, the authors in [START_REF] Chen | RRAM defect modeling and failure analysis based on march test and a novel squeeze-search scheme[END_REF] present a DfT scheme that deploys different references in the SA to perform a binary search to find the resistance of the RRAM device. This scheme can be modified to detect the IUSF by setting the reference of the SA directly at the boundary between the '1' and 'U' state, making the read operation deterministic; i.e., if IUSF occurs, then the SA will always output '0', and detect it. This results into P d =1 -(1 -P IUSF )

k . In this case realizing a FC=95 % requires k=279 and FC=99 % requires k=644; a reduction in test time of about ≈2X as compared with march test only.

To further decrease the test time while keeping high FC, specialized DfTs are required. Such schemes could aim at increasing P IUSF (and thus decreasing k) by e.g., increasing the current through the device during SET. This will lead to a wider CF which increases P IUSF and thus decreases P weak [START_REF] Nardi | Complementary switching in oxide-based bipolar resistive-switching random memory[END_REF]. This could be done by boosting the WL or BL voltage during SET in test mode. A drawback of this scheme is that it may lead to lower R SET , which will increase power consumption. Alternatively, additional DfT schemes could be introduced that continuously monitor and verify write operations, for example by performing a read operation using the above mentioned DfT from [START_REF] Chen | RRAM defect modeling and failure analysis based on march test and a novel squeeze-search scheme[END_REF] after every SET operation. This will increase the write latency and energy consumption, but it will significantly boost the FC.

VII. DISCUSSION AND CONCLUSION

This work has demonstrated the existence of some unique defects in RRAM that can not be modeled neither detected using traditional memory test approaches; these defects cause the Intermittent Undefined State Fault (IUSF). The Device-Aware Test approach has been put in place in order to appropriately model the defects and develop efficient test solutions. Given the nature of IUSF being intermittent, it is worth to note the following:

• Prevention versus detection: As the detection of this fault is hard and could be expensive, it is worth to invest in preventing it. This can be done by tightly controlling the SET current via the access transistor or by optimizing the production process of the capping layer so that always enough oxygen ions can be bound and CS is prevented.

• Intermittent behavior modeling: The defect model developed in this work does not consider the intermittent nature of the fault. Additional effort is now put in further extension of the model to include this behavior and cycleto-cycle variations similar to [START_REF] Bengel | Variability-Aware Modeling of Filamentary VCM based Bipolar Resistive Switching Cells Using SPICE Level Compact Models[END_REF]. Finally, this work has shown the importance of identifying and modeling new unique defects in RRAMs in an appropriate manner. The space of RRAM manufacturing defects is not explored completely yet, and even new defects may be introduced with further downscaling. Hence, further analysis of such space is needed especially now that the traditional memory approach is not able to guarantee the detection of unique RRAM faults, as demonstrated in this work.

  This work was supported by the European Unions Horizon 2020 Research and Innovation Program through the project MNEMOSENE (Grant 780215).

Fig. 1 :

 1 Fig. 1: Evolution of the conductive filament

Fig. 3 :

 3 Fig.1aillustrates a RRAM device; it consists of two electrodes, the top electrode (TE) and bottom electrode (BE), in between which a metallic oxide is sandwiched (commonly, HfO x , TaO x , or TiO x ) with an additional capping layer[START_REF] Wong | Metal-Oxide RRAM[END_REF][START_REF] Yu | Emerging Memory Technologies: Recent Trends and Prospects[END_REF][START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF]. After fabrication, a voltage V forming is applied to the oxide to break some of the metal and oxygen ions bonds[START_REF] Wong | Metal-Oxide RRAM[END_REF][START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF]. Fig.1ashows how the (negatively charged) oxygen ions (white circles) move towards the positive electrode and enter into the capping layer (cap) under the influence of the electric field and bond there[START_REF] Degraeve | Causes and consequences of the stochastic aspect of filamentary RRAM[END_REF][START_REF] Celano | Imaging the three-dimensional conductive channel in filamentary-based oxide resistive switching memory[END_REF]. This leaves behind a chain of conducting vacancies (black circles) known as Conductive Filament (CF), as shown in Fig.1b. When the voltage is removed, the CF remains, making the RRAM device non-volatile. The shape (e.g., radius) of the CF determines the resistance of the device[START_REF] Celano | Imaging the three-dimensional conductive channel in filamentary-based oxide resistive switching memory[END_REF]; it can be changed by applying voltages to the oxide. Note that the ability to bind oxygen ions in the capping layer determines the resistance switching mechanism of the RRAM device[START_REF] Nardi | Complementary switching in oxide-based bipolar resistive-switching random memory[END_REF][START_REF] Chen | Complementary resistive switching behaviors evolved from bipolar TiN/HfO2/Pt device[END_REF]; if (almost) all of oxygen ions can be bound, then the device will be Bipolar Switching (BS). Conversely, if many oxygen ions remain unbound, then the device will be Complementary Switching (CS).The BS mechanism relies on the formation and dissolution of the CF by movement of oxygen ions[START_REF] Wong | Metal-Oxide RRAM[END_REF][START_REF] Yang | Probing nanoscale oxygen ion motion in memristive systems[END_REF]. Its electrical behavior is illustrated in Fig.2a. When a negative voltage V TE less than a threshold V RESET is applied to the TE, then some of the oxygen ions move back into the oxide and re-oxidize the CF leaving a gap in the CF, as shown in Fig.1c. This movement increases the resistance of the device and is called a RESET operation. When a positive voltage higher than a threshold V SET is applied, the bonds between the metal and oxygen ions break again, and the CF regrows, as shown in Fig.1b. The oxygen vacancy-rich capping layer collects the

-

Fig. 4 :

 4 Fig. 4: Comparing defect-free, faulty, and weak devices.

Fig. 5 :

 5 Fig. 5: Linear resistor defect models.

TABLE I :

 I Occurrence probability of CS (P CS ) in % and maximal duration suffering from CS (D max ) in absolute numbers.

			WL 0	WL 1	WL 2	WL 3	WL 4	WL 5	WL 6
	BL 0 BL 1 BL 2 BL 3 BL 4 BL 5 BL 6	P CS Dmax P CS Dmax P CS Dmax P CS Dmax P CS Dmax P CS Dmax P CS Dmax	5.983 3 0 0 0 0 0 0 0 0 0 0 0.427 1	0.427 2 1.282 2 0 0 0.427 1 0 0 0 0 0.107 1	0 0 0 0 0 0 0 0 0.427 1 2.03 2 0.427 1	0 0 1.175 1 0.107 1 0 0 0.214 1 0 0 0.321 1	0.427 1 0.107 1 0.107 1 0 0 2.564 3 0 0 1.389 3	0 0 1.709 2 0 0 0.107 1 0.214 1 0.107 1 0 0	0 0 0.641 2 1.282 1 0 0 0.855 3 0 0 0 0

ST Microelectronics 130 nm technology NMOS high-voltage thick oxide transistor is placed in series to control the current through the device. The switching in a nominal defect-free device is bipolar, where logic '1' is represented by the SET state with R SET <25 kΩ, and logic '0' by the RESET state with 100 kΩ<R RESET <1 MΩ. The range [25 kΩ, 100 kΩ] is called an undefined state ('U'

TABLE II :

 II Occurrence probability (P IU SF ) of IUSF in % Additionally, the table lists the maximal duration (D max ) of the undesired behavior, expressed in the maximal number of consecutive cycles in which the device shows this behavior. For example, if the undesired CS occurs first in cycles 54, 55 and 56, and later in cycles 141 and 142, then P CS =5/936=0.534 % and D max =3 cycles. From the table it follows that undesired CS occurs in 25 devices, in up to 5.983 % of the cycles, and with a maximal duration of 3 cycles. Hence, the faulty behavior is intermittent, and it is a serious concern for BS RRAM devices. As the first group of faulty devices ends in undefined state after a SET operation, we will refer to this fault as Intermittent Undefined State Fault (IUSF).TableIIlists the occurrence probability (P IUSF ) of the IUSF in the 49 investigated devices over all 936 RESET-SET cycles.

		WL 0	WL 1	WL 2	WL 3	WL 4	WL 5	WL 6
	BL 0 BL 1 BL 2 BL 3 BL 4 BL 5 BL 6	0.962 0 0 0 0 0 0	0.427 0.107 0 0.107 0 0 0	0 0 0 0 0.107 0.321 0.321	0 0.107 0 0 0 0 0	0.107 0 0 0 1.068 0 0.748	0 0.427 0 0 0 0 0	0 0 0.107 0 0 0 0
	and weak devices.

TABLE III :

 III Model parameters for JART VCM v2[START_REF] Torre | Compact modelling of resistive switching devices based on the valence change mechanism[END_REF]

	Symbol	Value	Symbol	Value
	l1	8.75 nm	a	0.6 nm
	l cell	10 nm	R series,0	2 kΩ
	l2	l cell -l1	Nmax	3 • 10 27 m -3
	N init1	2 • 10 27 m -3	µn0	1.8 • 10 -5 m 2 /(Vs)
	N init2	3.5 • 10 25 m -3	ν0	4 • 10 12 Hz
	eφ Bn0,1	0.35 eV	∆W A	0.9 eV
	eφ Bn0,1	0.1 eV		17 • 0
	r fil	18 nm	φ B	5.5 • 0
	∆Eac	0.04 eV	α T,series

TABLE IV :

 IV Fault Analysis Results for IUSF.

	N init1 /Nmax	Nmax	N init1	Fault
	[-]	[10 27 m -3 ]	[10 27 m -3 ]
	0.15 0.18 0.20 0.23 0.25 0.27 0.30 0.53 0.67	8.0 8.0 8.0 8.0 8.0 6.0 6.0 3.0 3.0	1.2 1.4 1.6 1.8 2.0 1.6 1.8 1.6 2.0	0w1/U/-0w1/U/-Weak fault Weak fault Weak fault Fault free Fault free Fault free Fault free

ACKNOWLEDGMENT

We thank Christopher Bengel from RWTH Aachen for his input on modeling the defects.


Emmanuel Moulay

Vincent Léchappé

Emmanuel Bernuau

Franck Michael Defoort

Franck Plestan

Fixed-time sliding mode control with mismatched disturbances

Keywords: Sliding mode control, fixed-time stability, mismatched disturbances, fixed-time differentiator

published or not. The documents may come

Introduction

Sliding mode control (SMC) has been developed since the seminal work of Utkin [START_REF] Utkin | Variable structure systems with sliding modes[END_REF] in the 70s, see for instance [START_REF] Shtessel | Sliding mode control and observation[END_REF] for more details. Recently, the problem of robust fixed-time SMC was treated in [START_REF] Corradini | Nonsingular terminal slidingmode control of nonlinear planar systems with global fixedtime stability guarantees[END_REF][START_REF] Zuo | Non-singular fixed-time terminal sliding mode control of non-linear systems[END_REF] where fixed-time stability developed in [START_REF] Andrieu | Homogeneous approximation, recursive observer design and output feedback[END_REF][START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF][START_REF] Polyakov | Generalized homogeneity in systems and control[END_REF] ensures that all solutions converge to the equilibrium point in a fixed-time. However, the proposed controllers were rather complex to implement due to a problem of singularity avoidance coming from the fixed-time stability strategy. In [START_REF] Moulay | Robust fixed-time stability: application to sliding mode control[END_REF], simple robust controllers are proposed by using state-dependent variable exponent coefficient in the sliding surfaces and the controllers.

All the above mentioned articles deal with matched disturbances which are disturbances acting on the control input channel and are the most common disturbances encountered in control applications. Mismatched disturbances acting on other channels than the control channel also appear in automatic control for instance in [START_REF] Ginoya | Sliding mode control for mismatched uncertain systems using an extended disturbance observer[END_REF][START_REF] Han | Sliding-mode control of uncertain systems in the presence of unmatched disturbances with applications[END_REF][START_REF] Kayacan | Feedback linearization control for systems with mismatched uncertainties via disturbance observers[END_REF][START_REF] Li | Generalized extended state observer based control for systems with mismatched uncertainties[END_REF][START_REF] Li | Continuous finite-time output regulation for disturbed systems under mismatching condition[END_REF][START_REF] Moreno | Strict lyapunov functions for the super-twisting algorithm[END_REF][START_REF] Sun | Composite adaptive disturbance observer based control and back-stepping method for nonlinear system with multiple mismatched disturbances[END_REF][START_REF] Sun | Non-linear disturbance observer-based back-stepping control for airbreathing hypersonic vehicles with mismatched disturbances[END_REF][START_REF] Tian | Fixed-time stabilization of high-order integrator systems with mismatched disturbances[END_REF][START_REF] Yang | Non-linear disturbance observer-based robust control for systems with mismatched disturbances/uncertainties[END_REF][START_REF] Yang | Continuous nonsingular terminal sliding mode control for systems with mismatched disturbances[END_REF][START_REF] Zhang | Disturbance observer-based integral sliding-mode control for systems with mismatched disturbances[END_REF][START_REF] Zhang | Disturbance rejection for nonlinear systems with mismatched disturbances based on disturbance observer[END_REF][START_REF] Zhang | A nonsmooth composite control design framework for nonlinear systems with mismatched disturbances: Algorithms and experimental tests[END_REF][START_REF] Zuo | Robust fixedtime stabilization control of generic linear systems with mismatched disturbances[END_REF] Email addresses: emmanuel.moulay@univ-poitiers.fr (Emmanuel Moulay), vincent.lechappe@insa-lyon.fr (Vincent Léchappé), emmanuel.bernuau@agroparistech.fr (Emmanuel Bernuau), michael.defoort@uphf.fr (Michael Defoort), franck.plestan@ec-nantes.fr (Franck Plestan).

and it can model the DC-DC buck power converter systems [START_REF] Wang | Extended state observer-based sliding mode control for PWM-based DC-DC buck power converter systems with mismatched disturbances[END_REF]. In the previous works, the control problems are solved by using observers to estimate mismatched disturbances. In our article, we use a fixed-time SMC strategy adapted from [START_REF] Moulay | Robust fixed-time stability: application to sliding mode control[END_REF] which is known to be robust in order to reject the mismatched disturbance of a secondorder system without using disturbance observers. This renders this new strategy easier to tune. To the best of the authors' knowledge, this is the first time that a mismatched disturbance is rejected by using a fixed-time sliding surface known for its robustness properties. Thus, we adapt the sliding-mode controller developed in [START_REF] Moulay | Robust fixed-time stability: application to sliding mode control[END_REF] in order to be used with a mismatched disturbance. The results are tested in simulation on the double integrator in order to show the efficiency of the proposed method. Finally, the proposed control strategy is applied to develop a new fixed-time differentiator, the reader may refer to [START_REF] Zhang | Tracking Differentiator Algorithms: Theories, Implementations and Applications[END_REF] for more details on differentiators.

The article is organized as follows. After some preliminaries given in Section 2, the main results are presented in Section 3 and tested on the double integrator with matched and mismatched disturbances. They are also applied to develop a new fixed-time differentiator. Finally, a conclusion is addressed in Section 4.

Preliminaries

In the following, denote e the constant such that lnpeq " 1. Let us recall some results on finite-time stability and fixed-time stability. Consider the following ordinary differential equation

9 xptq " f pxptqq, xptq P R n (1) xp0q " x 0
with f a locally essentially bounded function.

Definition 1 [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF][START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF] The origin is globally finite-time stable for system (1) if it is Lyapunov stable and for all x 0 P R n there exists T px 0 q ě 0 such that, for any xp¨q solution of (1) with xp0q " x 0 , lim tÑT px0q }xptq} " 0, i.e. }xptq} " 0 for all t ě T px 0 q. Such a function T is called a settling-time function.

Definition 2 [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] System (1) is globally fixed-time stable if:

(1) it is globally finite-time stable;

(2) there exists a globally upper bounded settling-time function T , i.e. there a constant T ą 0, independent of the initial condition, such that for all x 0 P R n , T px 0 q ď T.

Main results

Consider the following uncertain nonlinear second-order system # 9

x 1 " x 2 `d1

9 x 2 " f pxq `gpxqu `d2 (2)
with x " px 1 , x 2 q P R 2 the state, u P R the control input, f and g continuous functions such that f p0q " 0, gpxq ‰ 0 for all x P R 2 , d 1 a mismatched disturbance such that |d 1 ptq| ă δ 1 and d 2 a matched disturbance such that |d 2 ptq| ă δ 2 . The equilibrium trajectory of system (2) is given by p0, ´d1 ptqq. The second order systems with mismatched disturbances have been studied for instance in [START_REF] Yang | Continuous nonsingular terminal sliding mode control for systems with mismatched disturbances[END_REF].

The main objective of this subsection is to design a sliding variable leading to global robust fixed-time stabilization of system (2) towards the equilibrium trajectory. Consider the sliding variable with a state-dependent variable exponent coefficient given by

spxq " x 2 `β|x 1 | λ 1 x 2 1 1`µ 1 x 2 1 sgnpx 1 q (3) with λ 1 ą 0, µ 1 ą 0 such that θ 1 " λ1 1`µ1 ą 1, β ą δ 1 e λ 1 2e
and the following new controller inspired from Proposi-tion 2 in [START_REF] Moulay | Robust fixed-time stability: application to sliding mode control[END_REF] upxq " ´gpxq

´1" f pxq `k1 sgnpsq|s| λ 2 s 2 1`µ 2 s 2 `ϕp|x 1 |qpx 2 `k2 sgnpsqq ı (4)
with λ 2 ą 0, µ 2 ą 0 such that θ 2 " λ2 1`µ2 ą 1 and

ϕpzq " βλ 1 1 `µ1 z 2 ˆ2 lnpzq 1 `µ1 z 2 `1˙z λ 1 z 2 1`µ 1 z 2 `1.
Let us state a first technical lemma.

Lemma 1 For z ě 0, we have ϕpzq ą 0 when z ą 1 and |ϕpzq| ď 2βλ 1 p1 `µ1 q when 0 ď z ď 1.

Proof. The first statement is obvious. When 0 ď z ď 1, we write

ϕpzq " βλ 1 p1 `µ1 z 2 q 2 `2z lnpzq `z `µ1 z 3 ˘z λ 1 z 2 1`µ 1 z 2 .
We see that 0 ď βλ1 p1`µ1z 2 q 2 ď βλ 1 and 0 ď z

λ 1 z 2 1`µ 1 z 2 ď 1. Now denoting ψpzq " 2z lnpzq `z `µ1 z 3
we see that ψ 1 pzq " 2 lnpzq `3 `3µ 1 z 2 . Denoting z 0 the unique solution of ψ 1 pzq " 0, we see that ψ is decreasing on p0, z 0 q and increasing on pz 0 , 1q. Therefore it leads to |ψpzq| ď maxt1 `µ1 , |ψpz 0 q|u. But, given that ψ 1 pz 0 q " 0, we have

ψpz 0 q " z 0 `2 lnpz 0 q `1 `µ1 z 2 0 " z 0 `´3 ´3µ 1 z 2 0 `1 `µ1 z 2 0 " ´2z 0 p1 `µ1 z 0 q.
Finally, because 0 ď z 0 ď 1, we see that |ψpz 0 q| ď 2p1 μ1 q and we get the result.

We can now state the first theorem. 3)-( 4) reaches the equilibrium trajectory p0, ´d1 ptqq in a fixed time satisfying

Theorem 1 If k 1 ą δ 3 " δ 2 `2βλ 1 p1 `µ1 qpδ 1 `k2 q and k 2 ą δ 1 , the closed-loop system (2)-(
T px 0 q ď 1 pk 1 ´δ3 qpθ 2 ´1q `1 k 1 e ´λ2 2e ´δ3 `1 pβ ´δ1 qpθ 1 ´1q `1 βe ´λ1 2e ´δ1 . (5) s "f pxq `gpxqupxq `βλ 1 |x 1 | px 2 `d1 q 1 `µ1 x 2 1 ˆ2 ln |x 1 | 1 `µ1 x 2 1 `1˙| x 1 | λ 1 x 2 1 1`µ 1 x 2 1 `d2 " ´k1 |s| λ 2 s 2 1`µ 2 s 2 sgnpsq `d2 `ϕp|x 1 |qpd 1 ´k2 sgnpsqq (6)
Now, considering the Lyapunov function V psq " s 2 , we see that

9 V psq ď 2|s| ˆ´k 1 |s| λ 2 s 2 1`µ 2 s 2 `δ2 `ϕp|x 1 |qpsgnpsqd 1 ´k2 q ˙. (7) When |x 1 | ą 1, we have ϕp|x 1 |q ą 0, therefore 9 V psq ď 2|s| ˆ´k 1 |s| λ 2 s 2 1`µ 2 s 2 `δ2 `ϕp|x 1 |qpδ 1 ´k2 q ˙. ( 8
)
But given that k 2 ą δ 1 , we get ϕp|x 1 |qpδ 1 ´k2 q ă 0 which in turn yields

9 V psq ď 2|s| ˆ´k 1 |s| λ 2 s 2 1`µ 2 s 2 `δ2 ˙. (9)
When |x 1 | ď 1, we have |ϕp|x 1 |q| ď 2βλ 1 p1 `µ1 q and thus

9 V psq ď 2|s| ˆ´k 1 |s| λ 2 s 2 1`µ 2 s 2 `δ2 `2βλp1 `µ1 qpδ 1 `k2 q ˙.
(10) Finally, we see that

9 V psq ď 2|s| ˆ´k 1 |s| λ 2 s 2 1`µ 2 s 2 `δ3 ˙(11)
where δ 3 " δ 2 `2βλ 1 p1 `µ1 qpδ 1 `k2 q. By using Theorem 2 in [START_REF] Moulay | Robust fixed-time stability: application to sliding mode control[END_REF], one deduces that system (6) starting at sp0q " s 0 reaches the sliding surface ts " 0u in a fixedtime satisfying T ps 0 q ď

1 pk1´δ3qpθ2´1q `1 k1e ´λ2 2e ´δ3
. From

(3), one has

9 x 1 " ´β|x 1 | λ 1 x 2 1 1`µ 1 x 2 1 sgnpx 1 q `d1 . ( 12
)
By using one more time Theorem 2 in [START_REF] Moulay | Robust fixed-time stability: application to sliding mode control[END_REF], it is deduced that x 1 ptq starting at x 1 p0q " x 10 reaches the origin in a fixed-time satisfying T px 10 q ď

1 pβ´δ1qpθ1´1q `1 βe ´λ1 2e ´δ1
.

Finally, the closed-loop system (2)-( 3)-( 4) reaches the equilibrium trajectory p0, ´d1 ptqq in a fixed time satisfying T px 0 q " T ps 0 q `T px 10 q.

Remark 1 Let us remark that if we want to use the controller given in Proposition 2 of [START_REF] Moulay | Robust fixed-time stability: application to sliding mode control[END_REF] upxq " ´gpxq

´1« f pxq `k|s| λ 2 s 2 1`µ 2 s 2 sgnpsq `βλ 1 |x 1 |x 2 1 `µ1 x 2 1 ˆ2 ln |x 1 | 1 `µ1 x 2 1 `1˙| x 1 | λ 1 x 2 1 1`µ 1 x 2 1 ff ( 13
)
with θ 2 " λ2 1`µ2 ą 1, k ą δ 2 e λ 2
2e then we have to prove that βλ1|x1|d1

1`µ1x 2 1 ´2 ln |x1| 1`µ1x 2 1 `1¯| x 1 | λ 1 x 2 1 1`µ 1 x 2 1 is bounded, i.e. lim x1Ñ8 βλ 1 |x 1 |d 1 1 `µ1 x 2 1 ˆ2 ln |x 1 | 1 `µ1 x 2 1 `1˙| x 1 | λ 1 x 2 1 1`µ 1 x 2 1 ă `8. ( 14
)
It leads to the condition λ 1 ď µ 1 because |x 1 | λ 1 x 2 1 1`µ 1 x 2
1 behaves as |x 1 | λ µ far from the origin x 1 " 0. However, we have the condition 1 `µ1 ă λ 1 coming from Theorem 2 in [START_REF] Moulay | Robust fixed-time stability: application to sliding mode control[END_REF] which is incompatible with the condition λ 1 ď µ 1 . This is the reason why we developed the new controller (4) which is discontinuous in the sliding variable but able to manage the mismatched disturbance.

Remark 2 With Theorem 1, we obtain that x 2 ptq tracks ´d1 ptq in a fixed time. Thus, we obtain a new fixed-time estimator of the mismatched disturbance without using observers.

Consider now the following new controller inspired from Proposition 3 in [START_REF] Moulay | Robust fixed-time stability: application to sliding mode control[END_REF] upxq " ´gpxq

´1" f pxq `k1 sgnpsq `k2 |s| α sgnpsq `ϕp|x 1 |qpx 2 `k3 sgnpsqq ı (15)
with α ą 1.

Theorem 2 If k 1 ą δ 3 " δ 2 `2βλ 1 p1 `µ1 qpδ 1 `k3 q, k 2 ą δ 3 and k 3 ą δ 1 then the closed-loop system (2)-( 3)-( 15) reaches the equilibrium trajectory p0, ´d1 ptqq in a fixed time satisfying

T px 0 q ď 1 pk 2 ´δ3 qpα ´1q `1 k 1 ´δ3 `1 pβ ´δ1 qpθ 1 ´1q `1 βe ´λ1 2e ´δ1 . ( 16
)
Proof. Similarly to the proof of Theorem 1, we get, with V psq " s 2 :

9 V psq ď 2|s| p´k 1 ´k2 |s| α `δ3 q ( 17
)
where δ 3 " δ 2 `2βλ 1 p1 `µ1 qpδ 1 `k3 q. If |s| ą 1, we see that 9

V ď 2|s|pδ 3 ´k2 |s| α q. But given that |s| ą 1 and δ 3 ą 0, we get 9 V psq ď 2|s|pδ 3 |s| α ´k2 |s| α q (18)

ď 2pδ 3 ´k2 qV psq 1`α 2 (19)
which yields, as long as |sptq| ą 1, 1 ě V psptqq 1´α 2 ě p1 ´αqpδ 3 ´k2 qt. This inequality shows that sptq reaches the set tV ď 1u in a time T 1 ď

1 p1´αqpδ3´k2q . If |s| ď 1 we get 9 V psq ď 2|s|pδ 3 ´k1 q ď 2pδ 3 ´k1 qV 1 2 . ( 20
)
Therefore, as long as sptq ą 0, V psptqq 1 2 ď 1 `pδ 3 ḱ1 qt, which yields that sptq reaches 0 in a time T 2 ď 1 k1´δ3 . Finally, we conclude that, regardless of its initial condition, sptq reaches 0 in a fixed-time T ps 0 q ď T 1 `T2 . The end of the proof is similar to the one of Theorem 1. Some guidelines are provided below in order to tune the parameters of controller ( 15): 1. choose δ 1 and δ 2 according to the maximum values of disturbances d 1 and d 2 ; 2. take λ 1 ą 0 and µ 1 ą 0 such that θ 1 " λ1 1`µ1 ą 1; 3. take β such that β ą δ 1 e λ 1 2e ; 4. take k 3 ą δ 1 ; 5. take k 1 ą δ 3 and k 2 ą δ 3 with δ 3 " δ 2 `2βλ 1 p1 `µ1 qpδ 1 `k3 q; 6. take α ą 1. The guidelines for the tuning of controller (4) can be obtained in a similar way.

Example 1 The results of Theorem 1 and Theorem 2 have been tested in simulation on a double integrator, i.e. taking f " 0 and g " 1 in system (2). The following disturbances have been chosen d 1 ptq " sinp5tq and d 2 ptq " 2 sinp10tq so δ 1 " 1 and δ 2 " 2. The Euler solver with a fixed step of 10 ´4s has been used. The parameters of both controllers are as follows: λ 1 " 3, µ 1 " 1, β " 5.5, k 1 " 30, λ 2 " 4, µ 2 " 1, k 2 " 2 for controller (4) and λ 1 " 3, µ 1 " 1, β " 5.5, k 1 " 39, k 2 " 20, k 3 " 2, α " 3 for controller [START_REF] Sun | Composite adaptive disturbance observer based control and back-stepping method for nonlinear system with multiple mismatched disturbances[END_REF]. This set of parameters verifies the conditions of Theorem 1 and Theorem 2. The results of the simulations with the initial condition xp0q " r1, 3s T are displayed on Figure 1(a)-(b). For comparison purpose, one has implemented the fixed-time controller given in [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]. One can observe that after a reaching phase, the state x 1 converges to 0. Note that the dynamics of x 2 tracks the unknown mismatched disturbance ´d1 . The upper bound of the settling-time [START_REF] Andrieu | Homogeneous approximation, recursive observer design and output feedback[END_REF] given in Theorem 1 is equal to 30s and the upper bound of the settling-time [START_REF] Sun | Non-linear disturbance observer-based back-stepping control for airbreathing hypersonic vehicles with mismatched disturbances[END_REF] given in Theorem 2 is equal to 8.5s. One can see that this settling-time is much smaller in the simulation, around 1s for controller (4) and around 2s for controller [START_REF] Sun | Composite adaptive disturbance observer based control and back-stepping method for nonlinear system with multiple mismatched disturbances[END_REF]. Let us notice that the fixed-time controller for the second order system proposed in [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] and displayed on Figure 1(c) is Example 2 Let us consider a differentiable signal hptq defined on r0, 8q which is differentiable with bounded time derivative, i.e. | 9 hptq| ď δ 1 . The objective here is to estimate in fixed-time the time derivative of this signal. To achieve this objective, let us consider the following fixed-time differentiator:

9 x1 " x2 9 x2 " upx 1 ´h, x2 q (21)
where upx 1 ´h, x2 q is defined in (4) or (15) with f " 0 and g " 1. Defining the error e " x1 ´h, one obtains

9 e " x2 ´9 h 9 x2 " upe, x2 q (22)
It yields system (2) with d 2 " 0 and d 1 " ´9 h. In the following simulations, the signal has been chosen as hptq " 0.2 cosp 1 5 tq so δ 1 " 1 and δ 2 " 0. The Euler solver with a fixed step of 10 ´5s has been used. The parameters of u are as follows: λ 1 " 2.5, µ 1 " 1, β " 2, k 1 " 150, λ 2 " 2.2, µ 2 " 1, k 2 " 3 for scheme (4) and λ 1 " 2.5, µ 1 " 1, β " 2, k 1 " 150, k 2 " 45, k 3 " 1.1, α " 3 for scheme [START_REF] Sun | Composite adaptive disturbance observer based control and back-stepping method for nonlinear system with multiple mismatched disturbances[END_REF]. This set of parameters verifies the conditions of Theorem 1 and Theorem 2. Furthermore, an additional uniformly bounded random measurement noise of magnitude 10 ´2 is considered in the following simulations. The results of the simulations with the initial condition xp0q " r1, 1s T are displayed on Figure 2 and3. One can observe that after a reaching phase, the state x1 converges to h. Note that the dynamics of x2 converges to given in Theorem 1 is equal to 8s and the upper bound of the settling-time [START_REF] Sun | Non-linear disturbance observer-based back-stepping control for airbreathing hypersonic vehicles with mismatched disturbances[END_REF] given in Theorem 2 is equal to 8s.

One can see that this settling-time is much smaller in the simulation, around 0.9s for both schemes.

It is worth mentioning that there exist several finite time differentiators (see for instance [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF]). Nevertheless, the convergence time, grows unboundedly with respect to the initial differentiation error. Recently, an arbitrary order fixed-time differentiator, which can been seen as a generalization of the Levant's differentiator, has been introduced in [START_REF] Moreno | Arbitrary order fixed-time differentiators[END_REF]. For comparison purpose, the root mean square error (RMSE) after the reaching phase for our proposed differentiator [START_REF] Zhang | A nonsmooth composite control design framework for nonlinear systems with mismatched disturbances: Algorithms and experimental tests[END_REF] and [START_REF] Moreno | Arbitrary order fixed-time differentiators[END_REF] is as follows: 2.84e0 5 when using [START_REF] Zhang | A nonsmooth composite control design framework for nonlinear systems with mismatched disturbances: Algorithms and experimental tests[END_REF] with u given in Theorem 1, 3.06e´05 when using [START_REF] Zhang | A nonsmooth composite control design framework for nonlinear systems with mismatched disturbances: Algorithms and experimental tests[END_REF] with u given in Theorem 2, 4.72e ´05 when uisng [START_REF] Moreno | Arbitrary order fixed-time differentiators[END_REF]. It can be seen that our proposed differentiators achieve better accuracy in the presence of measurement noise than the first order fixed-time differen-tiator proposed in [START_REF] Moreno | Arbitrary order fixed-time differentiators[END_REF].

Conclusion

A new sliding mode controller is proposed in order to get fixed time-convergence of second order systems in presence of unknown but bounded matched and mismatched perturbations. It is applied to develop a new fixed-time differentiator. The fixed-time controller is based on the design of a robust sliding surface and does not require any disturbance observer which simplifies the solution.

For future works, a high order SMC or implicit SMC strategy could be used for reducing the chattering when dealing with robust fixed-time stabilization.

Fig. 1 .

 1 Fig. 1. (a) Double integrator with sliding surface (3) and controller (4); (b) Double integrator with sliding surface (3) and controller (15); (c) Double integrator with controller [6]. not able to manage the mismatched disturbances d 1 , contrary to the proposed controllers (4) and (15) displayed on Figures 1(a)-(b) respectively.

Fig. 2 .

 2 Fig. 2. (a) Fixed-time observer with sliding surface (3) and scheme (4).

Fig. 3 .

 3 Fig.3. Fixed-time observer with sliding surface (3) and scheme[START_REF] Sun | Composite adaptive disturbance observer based control and back-stepping method for nonlinear system with multiple mismatched disturbances[END_REF] 9 h in fixed time. The upper bound of the settling-time[START_REF] Andrieu | Homogeneous approximation, recursive observer design and output feedback[END_REF] given in Theorem 1 is equal to 8s and the upper bound of the settling-time[START_REF] Sun | Non-linear disturbance observer-based back-stepping control for airbreathing hypersonic vehicles with mismatched disturbances[END_REF] given in Theorem 2 is equal to 8s. One can see that this settling-time is much smaller in the simulation, around 0.9s for both schemes. It is worth mentioning that there exist several finite time differentiators (see for instance[START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF]). Nevertheless, the convergence time, grows unboundedly with respect to the initial differentiation error. Recently, an arbitrary order fixed-time differentiator, which can been seen as a generalization of the Levant's differentiator, has been introduced in[START_REF] Moreno | Arbitrary order fixed-time differentiators[END_REF]. For comparison purpose, the root mean square error (RMSE) after the reaching phase for our proposed differentiator[START_REF] Zhang | A nonsmooth composite control design framework for nonlinear systems with mismatched disturbances: Algorithms and experimental tests[END_REF] and[START_REF] Moreno | Arbitrary order fixed-time differentiators[END_REF] is as follows: 2.84e0 5 when using[START_REF] Zhang | A nonsmooth composite control design framework for nonlinear systems with mismatched disturbances: Algorithms and experimental tests[END_REF] with u given in Theorem 1, 3.06e´05 when using[START_REF] Zhang | A nonsmooth composite control design framework for nonlinear systems with mismatched disturbances: Algorithms and experimental tests[END_REF] with u given in Theorem 2, 4.72e ´05 when uisng[START_REF] Moreno | Arbitrary order fixed-time differentiators[END_REF]. It can be seen that our proposed differentiators achieve better accuracy in the presence of measurement noise than the first order fixed-time differen-


Gilles Le Moullac
email: gilles.le.moullac@ifremer.fr

Jerome Tiapari

Hinano Teissier

Elodie Martinez

Jean-Claude Cochard

Jérôme Tiapari

G Le

Growth and gonad development of the tropical black-lip pearl oyster, Pinctada margaritifera (L.), in the Gambier archipelago (French Polynesia)

Keywords:

published or not. The documents may come

Introduction

Pearl oyster farming in French Polynesia is spread over a large geographical area.

Environmental conditions vary between islands in this part of the South Pacific, particularly due to differences in their latitudinal position. Seasonal cycles are more marked in southern Polynesia than further north. The annual reproductive cycle of Pinctada margaritifera was studied in the Takapoto atoll of the northern Tuamotu Islands, showing that reproduction occurs throughout the year due to a high average temperature with low seasonal variation [START_REF] Thielley | Etude cytologique de la gamétogenèse, de la sex-ratio et du cycle de reproduction chez l'huître perlière Pinctada margaritifera (L) var cumingii (Jameson)[END_REF], Pouvreau et al. 2000a). Adequate data on the marine environment and the reproduction of pearl oysters are, however, not yet available for other islands with pearl oyster farming. The largest thermal amplitude in French Polynesia [START_REF] Salvat | Le suivi de l'état des récifs coralliens de Polynésie française et leur récente évolution[END_REF]) is observed in the Gambier Islands (134°52' W, 23°07' S). Although reproduction of P. margaritifera still occurs in these islands, its reproductive cycle has not yet been described.

In the southern part of its distribution (Australia), the reproductive cycle of Pinctada albina has been observed to be modulated by seasonal temperature variations [START_REF]Latitudinal variation in reproductive behaviour in the pearl oyster, Pinctada albina sugillata[END_REF]. The reproductive cycle of the pearl oyster Pinctada mazatlanica is also affected by seasonal variations in temperature in Baja California [START_REF] Saucedo | Seasonal change in the histological and biochemical profile of the gonad, digestive gland and muscle of the calafia mother-ofpearl oyster, Pinctada mazatlanica (Hanley, 1856) associated with gametogenesis[END_REF], where a period of intense sexual activity is observed in September when water temperature is high. Temperature changes due to El Nino and La Nina events have also been reported to affect the reproductive cycle of P. mazatlanica [START_REF] Garcia-Cuellar | El Nino and La Nina effects on reproductive cycle of the pearl oyster Pinctada mazatlanica (Hanley, 1856) (Pteriidae) at Isla Espiritu Santo in the Gulf of California[END_REF].

The trophic environment is also known to act on the reproductive status of P. margaritifera (Pouvreau, et al. 2000b). In French Polynesia, due to low chlorophyll a concentrations, the trophic environment is considered oligotrophic. Data on phytoplanktonic pigments have been obtained in the Takapoto atoll in northern Tuamotu (southern hemisphere), where concentration in chlorophyll a ranged between 0.05 and 0.46 μg •L -1 [START_REF] Delesalle | The phytoplankton of Takapoto Atoll (Tuamotu Archipelago, French Polynesia): time and space variability of biomass, primary production and composition over 24 years[END_REF].

Very little information on these parameters has been collected for the southern islands (Gambier Islands).

The objective of the present study was to examine the effect of the annual cycle of temperature and chlorophyll a concentration on the reproductive cycle of P. margaritifera, to determine whether local environmental conditions in the Gambier Islands induce any marked seasonality in the reproductive cycle of this tropical pearl oyster species.

Material and methods.

Gambier Islands

Pearl culture has developed in several sites in the Gambier Islands (134°52' W & 23°07' S), an Archipelago located in southeast French Polynesia (South Pacific).

Environmental data

Temperature was recorded hourly in situ using an automatic sensor (TidbiT temp Logger-Onset ® ). Chlorophyll a concentration (Chl a) data were derived from radiometric ocean colour provided by the Sea viewing Wide Field of view Sensor (SeaWiFS) onboard the SeaStar spacecraft [START_REF] Mcclain | An overview of the SeaWiFS project and strategies for producing a climate research quality global ocean bio-optical time series[END_REF]. Chl a data were provided weekly from September 2002 to August 2003.

Sampling

Growth and reproduction of P. margaritifera were studied in pearl oysters reared in the Gambier lagoon (134°52' W, 23°07' S). The cultivation technique was similar to that commonly used in commercial farms. Suspended long-lines were immersed at a depth of 7 m and pearl oysters were 'ear hung' on downlines in groups of ten. The reproductive cycle was studied over a year (from September 2002 to August 2003) by systematic sampling of 30 randomly selected individuals every 15 days. These pearl oysters were collected, cleaned of fouling organisms and sent by air to the Ifremer laboratory on Tahiti Island.

Biometry

For all specimens (n = 720), total weight and height of the shell were measured prior to dissection, weighing, sexing and histology. Sex was determined by rapid microscopic observation of fresh gonad smears. Glycogen content was then measured on 5 dried oysters for each sampling and the visceral mass of 15 other oysters was placed in Davidson's fixative for gonad histology.

Total wet flesh weights were obtained after 15 min of draining. After flesh dissection, the same operation was conducted separately on adductor muscle, gills + mantle and the visceral mass (gonad + digestive gland). Condition index (CI) was computed as the ratio of visceral mass weight to shell weight.

Glycogen measurement

Dry tissues of oyster were powdered and a 10-mg sample was then homogenized in 1 ml of de-ionized water. Changes in the glycogen composition of tissue over the year were quantified using the phenol-sulphuric acid method of [START_REF] Dubois | Colorimetric method for determination of sugars and related substances[END_REF].

Histology

After dissection, gonadal tissue samples were placed in Davidson's fixative for five days, after which they were preserved in 70 % ethanol. Samples were then dehydrated through a graded ethanol series, embedded in paraffin, sectioned at 3-4 µm on a rotary microtome, stained with Giemsa dye and, finally, mounted on microscope slides. Sections were made in the gonad area, between the proximal end of the gut loop and the base of the foot. Slide preparations were examined with a light microscope to scan the entire gonadal area and assess follicle stages. Reproductive stages of gonad development are based on the description by Pouvreau et al. (2000a) shown in table 1.

Statistics

The growth parameters shell height, total weight, flesh and tissue weights and glycogen content in dried tissue are presented on the figures, with vertical bars representing the 95 % confidence intervals. Correlations between environmental and physiological variables were tested using the critical value table for Pearson's correlation coefficient at the 5 % alpha level and 108 df. The null hypothesis was rejected when r < 0.195.

Results

Environmental data

Water temperature gradually increased from the end of September up to a peak in March and then declined to its lowest value in August. Mean annual water temperature was 25.42 ± 1.75 °C, with a minimum of 22.3 °C and a maximum of 27.8 °C (Fig. 1a). The annual mean value of Chl a was 0.188 ± 0.075 µg L -1 , while the maximum value of 0.436 µg L -1 was reached in August 2003 (Fig. 1b). The gaps in the data are due to cloud cover.

Pearl oyster growth

Shell height and total weight increased regularly over the sampling period (Fig. 2a,2b). The growth rates in terms of shell height and total weight were 87.40 µm.d -1 and 176.1 g.d -1 , respectively. From September 2002 to February 2003, flesh dry weight remained stable (Fig. 2c). A rapid increase then occurred from February 2003 to mid March 2003, followed by a period during which mean flesh dry weight stabilised once more. Muscle dry weight showed large variations over the year, with the highest value in mid-March (Fig. 3). Changes in weight of the visceral mass were recorded twice, in mid-April and mid-August. (Fig. 3).

Gametogenic cycle

The condition index (CI) showed three significant decreases, the first in December, the second in March and the third in June, while two non-significant changes of CI occurred in May and August (Fig. 4). Three significant increases of CI were recorded during this study; the first time was in October, one month before the first significant decrease, the second time was in April, just after the second significant decrease, and the third in July, just after the third significant decrease (Fig. 4).

The temporal distribution of maturity stages over the year is shown in Figure 5. The three significant CI decreases coincided with increases in the frequency of inactive (0) and early gametogenic (1) stages, and the decrease in frequency of stages 2 to 5, thus meaning that three significant spawning events had occurred. The first spawning event in December was smallest, since the frequency of the stage 0 and 1 was low. The spawning event in March affected > 70 % of oysters, since these were found to be at stage 1 of the maturity index, while in May and June, spawning only affected > 30 %. In August, the non-significant decrease of CI coincided with a 7 % frequency of stage 1.

Sex ratio

The sex of all individuals was determined during this study. Figure 6a illustrates the change in of this protandrous alternate hermaphrodite sex ratio over the year, while figure 6b represents the change in sex ratio according to the size of individuals. On average, 7 % of individuals were female, the highest value being 14 %. Only animals larger than 80 mm were female. The proportion of females increased up to 22 % for individuals larger then 140 mm.

Glycogen content

Glycogen content in muscles significantly increased to 200 mg. g -1 between February and May. In February, glycogen content increased significantly in the mantle, while it remained unchanged in the visceral mass (Fig. 7).

Regression analysis

Growth parameters (shell, flesh and tissue) were positively (p < 0.05) correlated with Chl a (Table 2). Glycogen content in VM and gill-mantle were negatively with Chl a, but glycogen content in all tissues was positively correlated with water temperature (Table 2). Data related to reproductive processes (CI) were not correlated with either chlorophyll or water temperature (Table 2).

Discussion

Environmental data

Temperature recordings during this study (2002)(2003) revealed an annual thermal amplitude (6 °C) that was smaller than the one observed in 1998-1999 (8 °C) during the strong El Niño-La Niña events. In comparison, the thermal amplitudes observed in the atolls of northern French Polynesia were 2 °C in 1990-1991 [START_REF] Buestel | La matière particulaire des eaux du lagon de Takapoto : nourriture potentielle pour les élevages d'huîtres perlières[END_REF] and 4 °C in 1998-1999 [START_REF] Pouvreau | Étude et modélisation des mécanismes impliqués dans la croissance de l'huître perlière[END_REF]. Radiometric ocean colour derived from satellite observations was extracted for the eastern part of the Gambier archipelago in 2002 and 2003. Two difficulties arose. One was due to cloud cover, which interrupted the continuity of the time series. The second was related to the fact that ocean color is derived from radiance leaving the water, which may be influenced by water depth and bottom albedo [START_REF] Maritorena | Diffuse reflectance of oceanic shallow waters: Influence of water depth and bottom albedo[END_REF].

Interpretation of Chl a variability must therefore to be considered with caution. However, the annual mean (0.19 µg. L -1 ) and amplitude range of Chl a reported here are in agreement with values and seasonal patterns reported in several atolls in French Polynesia from in situ measurements [START_REF] Charpy | Phytoplankton biomass and production in two Tuamotu atoll lagoons (French Polynesia)[END_REF][START_REF] Intes | Les lagons d'atoll en Polynésie française: Bilan des travaux du programme "Atoll[END_REF]). When considering the Chl a time series extended to the beginning of 2002 (data not shown), the maximum value of Chl a that we report here in August appears to be related to the Chl a seasonal cycle. This Chl a maximum in austral winter is also a typical seasonal pattern of Chl a in Polynesian lagoons [START_REF] Charpy | Phytoplankton biomass and production in two Tuamotu atoll lagoons (French Polynesia)[END_REF][START_REF] Charpy | Quality and quantity of available food for pearl oyster: a key parameter for a successful culture[END_REF]. The large variations of Chl a recorded during this study are also consistent with those measured in water samples from another Polynesian lagoon, Takapoto [START_REF] Buestel | La matière particulaire des eaux du lagon de Takapoto : nourriture potentielle pour les élevages d'huîtres perlières[END_REF]. These previous authors attributed this high variability to local meteorology (wind and precipitation), which had short-term, and night/day rhythms affecting some parameters (temperature and chlorophyll a).

Growth

Growth rate of pearl oysters seemed to be faster during the first six months of this study since the shell height and the total weight increased by 25 mm and 63 g, respectively, in this period.

No further significant changes in growth were then observed from April to August. In the size class considered (2-year-old oysters), shell growth was equivalent to that of oysters studied by [START_REF] Buestel | Ecophysiologie de l'huître perlière. Approche des relations entre la croissance de l'huître Pinctada margaritifera et le milieu dans le lagon de Takapoto[END_REF] in Takapoto, where shell growth was found to be 30 mm over a whole year. Shell growth dynamics of pearl oysters observed in the present work were similar to those found in a study conducted in 1998-1999 in the Gambier Islands [START_REF] Pouvreau | Growth of the black-lip pearl oyster, Pinctada margaritifera, at nine culture sites of French Polynesia: synthesis of several sampling designs conducted between 1994 and 1999[END_REF].

Our results showed that pearl oyster growth in the Gambier Islands was correlated with Chl a resources, which correspond to food supply, but not with temperature. The comparison of P. margaritifera growth between oysters reared in Gambier (this study; [START_REF] Pouvreau | Growth of the black-lip pearl oyster, Pinctada margaritifera, at nine culture sites of French Polynesia: synthesis of several sampling designs conducted between 1994 and 1999[END_REF] and those reared in Pioneer Bay, north Queensland (18°37' S, 146°30' E), Australia (Yukihira et al. 2006), reveals a similar annual growth pattern. The two sites have similar temperatures, and a similar conclusion was drawn: that temperature had no significant effect on growth rates of P. margaritifera over the annual cycle.

Reproductive activity

Pearl oyster reproduction was studied by [START_REF] Thielley | Etude cytologique de la gamétogenèse, de la sex-ratio et du cycle de reproduction chez l'huître perlière Pinctada margaritifera (L) var cumingii (Jameson)[END_REF], and the reproductive cycle was described by Pouvreau et al. (2000a) in the Takapoto atoll, northern Tuamotu. These authors showed that, in a fairly stable tropical environment, P. margaritifera is a multiple spawner with an opportunistic reproductive strategy that allows investment of any surplus energy into gamete production throughout the year. CI variations in the present study were related to quantitative losses of weight caused by gamete releases, while the histological analysis revealed a degree of synchronicity within the population. Three spawning periods were characterized by significant decreases of CI; these occurred in December 2002, March 2003 and June 2003. The decreases of CI corresponded precisely to periods of gamete release, as confirmed by histological examination of gonadic stages. The first gamete emissions can be regarded as resulting from the accumulation of gametes during winter. The second spawning, which occurred when temperature was maximal, involved most of the oysters since histological analysis showed that 80 % of them were at stage 1 or 2. This spawning was followed by rematuration. A third spawning occurred in June, when the temperature descended below 24 °C, as shown by a significant decrease in CI and the stages shown in the histological analysis. After this event, however, 30-40 % of individuals remained at histological stage 1 of maturity and a similar proportion remained at stage 2 until the beginning of August. At the end of August, a non-significant decrease in CI occurred when 7 % of the pearl oysters were at histological stage 1; these observations could indicate a spawning event, despite the fact the temperature reached 22 °C.

Histology validated the interest of CI as spawning indicator for the pearl oyster. Spawnings were triggered when CI was around 0.08 %. Before each significant spawning event, the CI values were 0.084, 0.070 and 0.076 respectively in December, March and June. This study has shown that CI was not correlated with temperature and did not offer any new evidence to support the hypothesis that spawnings were influenced by temperature. The greatest variations in CI corresponded to the spawning in December, when temperature increased up to 25 °C; the one in March, when the temperature was maximal (28 °C); and the one in June, when the temperature decreased below 24 °C.

Sex ratio

The proportion of females (7 %) did not differ from the observations made by Pouvreau et al. (2000a) in Takapoto. Indeed, for the age/size class considered here (2-year-old oysters, 100-110 mm height), the sex-ratio in Takapoto was 0.1 female/1 male. The evolution noted over the year of our study was typical of this age group, in which the first inversions of sex are normally known to occur [START_REF] Thielley | Etude cytologique de la gamétogenèse, de la sex-ratio et du cycle de reproduction chez l'huître perlière Pinctada margaritifera (L) var cumingii (Jameson)[END_REF]. The population raised in the Gambier Islands, therefore, did not differ from populations grown elsewhere in this respect.

Glycogen storage

Marine bivalves show cycles of energy storage and utilization that are closely related to gametogenic cycles. Most species are able to store nutrient reserves during periods of food abundance. These reserves are mobilized during periods of food depletion and high energy demand [START_REF] Bayne | Aspects of the reproduction in bivalve mollusks of reproduction[END_REF][START_REF] Barber | Energy storage and utilization in relation to gametogenesis in Argopecten irradians concentricus (Say)[END_REF][START_REF] Heude Berthelin | Storage metabolism in the pacific oyster (Crassostrea gigas) in relation to summer mortalities and reproductive cycle (west coast of France)[END_REF]. During gametogenesis, high-energy demands are made due to maintenance costs and gamete synthesis; these have to be met by the food supply and/or stored reserves. In P. mazatlanica, growing on the west coast of the Gulf of California, glycogen stored in the muscle and digestive gland were actively used for gametogenesis [START_REF] Saucedo | Seasonal change in the histological and biochemical profile of the gonad, digestive gland and muscle of the calafia mother-ofpearl oyster, Pinctada mazatlanica (Hanley, 1856) associated with gametogenesis[END_REF]. In P. margaritifera, our results show that glycogen obtained from ingested food was mainly stored in the adductor muscle and secondarily in the digestive gland, to be later used for ATP needs and to produce gametes. We studied the relationship between the glycogen storage cycle and gametogenesis of the pearl oyster P. margaritifera over the annual cycle (September 2002to August 2003). Glycogen content increased drastically, showing a 20-fold increase in muscle from February 2003 to May 2003. The energy stored in the muscle could also be used at the time of spawning to achieve vigorous valve movement and to increase the pressure on the gonad to expel the gametes. Our result shows that the maximum glycogen content in the muscle corresponded to the spawning period from March to June. Changes in glycogen content were correlated with temperature in the muscle, mantle and visceral mass (digestive gland and gonad), and with chlorophyll in the visceral mass. Our study confirms that mantle tissue has a negligible role in storage of nutrients during reproduction in P. margaritifera. Few changes were observed in the visceral mass over the study period. The digestive gland considered as a short term storage organ [START_REF] Saucedo | Seasonal change in the histological and biochemical profile of the gonad, digestive gland and muscle of the calafia mother-ofpearl oyster, Pinctada mazatlanica (Hanley, 1856) associated with gametogenesis[END_REF]Vite-Garcia et al. 2008) did not accumulate glycogen over a long period.

Conclusion

The main goal of this work was to examine the hypothesis that the highest thermal amplitude recorded in southern French Polynesia could generate seasonal variation of reproduction in P. margaritifera. This study was conducted in the Gambier Islands, where pearl farming is an important industry, and has shown that a seasonal cycle in the reproduction of the pearl oysters does not exist in this area. This study showed that CI (wet weight of gonad/shell weight) could be a good indicator of reproductive events in the pearl oyster. It thus appears that the development of gonads goes on throughout the year in the Gambier Islands, without any detectable phase of sexual rest. Gambier Islands). We also thank Helen McCombie-Boudry for improving the English in this paper.

  Somatic growth pattern showed two phases. From September 2002 to February 2003, the flesh dry weight did not change significantly, but in March 2003 it doubled. Flesh dry weight then remained stable until August 2003. This change was mainly due to muscle increase, with visceral mass and mantle accounting for less. This growth event coincided with the maximum temperature observed during the study, but the examination of correlations only indicated a significant relationship with chlorophyll a.

FiguresFig. 1

 1 Figures

Acknowledgments

This work was supported by logistic and financial aid from the Service de la Perliculture (Grant N°021435: Reproduction of the black-lip pearl oyster Pinctada margaritifera in the

 Table 1 Criteriafor histological scoring of gametogenic stages according to Pouvreau et al. (2000a)

Histological criteria Score

Undetermined or inactive 0

Early gametogenesis, numerous gonia, small follicles 1

Actively developing without mature gametes 2

Near ripe follicles with mature gametes 3

Spawning ripe, distended and confluent follicles, entirely filled 4

Partially spawned, partially empty lumen 5