B Hernandez

T Vanhoorne

P R Schöne

Thompson

Stathis C Stiros

T Aarup

G Wöppelmann
email: guy.woppelmann@univ-lr.fr

P L Woodworth

F Hernandez

B Vanhoorne

T Schöne

P R Thompson

Comments on the article "Uncertainty and bias in electronic tide-gauge records: evidence from collocated sensors" by Stella Pytharouli, Spyros Chaikalis

Keywords: Sea level database, Tide gauge records, Tide gauge benchmarks, Openly

accessible data; GLOSS programme

We refer to a recent article "Uncertainty and bias in electronic tide-gauge records: evidence from collocated sensors" published in Measurement by Stella Pytharouli, Spyros Chaikalis and Stathis C. Stiros. This comment aims at correcting some incorrect statements in that paper. In addition, we hope to clarify some important aspects to do with the source of the data used in their study, so as to enable any reader to better understand the findings presented.

The authors looked at sea level data obtained by two or three collocated sensors at 10 sites around the world, and made comparisons between the different sensors using statistical methods akin to the buddy-checking methods that are often used by tide gauge operators [START_REF] Ioc) | Manual on Sea Level Measurement and Interpretation -Radar Gauges[END_REF].

(1) They mention that their comparisons were limited by the number of sites with collocated sensors. They state (page 496) that "Currently, the percentage of sites with collocated sensors is of the order of a few percent of the total sites covered with tide-gauges."

Most tide gauge sites from which data are provided to the Intergovernmental Oceanographic Commission (IOC) Global Sea Level Observing System (GLOSS), the Permanent Service for Mean Sea Level (PSMSL), and the University of Hawai'i Sea Level Center (UHSLC) have multiple sensors. For example, most stations in the UK National Network (with which, given the address, one would have assumed the lead author to be familiar) contain dual bubbler systems. Data from both primary and secondary channels can be downloaded readily from the British Oceanographic Data Centre (BODC, https://www.bodc.ac.uk).

Other examples include the modern US (NOAA) network and the international network maintained by the UHSLC. The former has been based on acoustic sensors supplemented by pressure sensors, although that network is now transitioning to radar sensors [START_REF] Park | Water level and wave height estimates at NOAA tide stations from acoustic and microwave sensors[END_REF]). The latter is primarily based on radar sensors (often two radars at each site) supplemented by pressure sensors when possible. Detailed descriptions of sea level recording in many other countries can be obtained from the national reports to regular meetings of the GLOSS programme (http://www.gloss-sealevel.org/) and those reports show that most countries employ two or more sensors at each site.

All of the data from duplicate or multiple sensors will be available for inspection by the owners of the sites and the local technical specialists. However, it is not necessarily the case that data from all sensors are passed to the IOC Sea Level Station Monitoring Facility (SLSMF) at the Flanders Marine Institute (http://www.ioc-sealevelmonitoring.org/), which is where Pytharouli et al. obtained their data. For example, in the case of the UK, data from only the so-called primary bubbler channel are provided to the SLSMF.

(2) Section 2.1 states that the data used in this paper were obtained from the IOC SLSMF.

The IOC SLSMF was established to provide near-real time displays of data from those tide gauges around the world which are part of GLOSS or the several regional IOC tsunami networks (IOC 2012). The facility was provided primarily for the operational benefit of tide gauge owners who do not have their own display systems so that faults could be identified and fixed as soon as possible.

All data are displayed as received, archived and are accessible to interested users. However, the web site makes clear that the data has not undergone any quality control, and there is a clear disclaimer to that effect. The data providers for each station are mentioned in order to facilitate contact in case of a user's concerns or requests (e.g., if data are required for commercial purposes).

The data from the SLSMF have occasionally been used in scientific research. However, such research has usually been made by people who are aware of the difficulties of dealing with such data and who have been able to make additional quality control. Pytharouli et al. state correctly that the "policy of IOC is to check data in time due and automatically discard or repair sections of time series, so that final records are free of errors and of significant inconsistencies". For operational or research applications, users are directed by the SLSMF to dedicated IOC/GLOSS data centres which differ in the type of data quality control applied and the delay in making data available (IOC 2012).

(3) Section 2.1 rightly mentions the need for access to metadata as well as to tide gauge data itself.

All tide gauge stations experience problems in their supply of data from time to time. Some stations are remote and will have had power outages, communication problems etc. as well as technical faults in the gauge hardware itself. Occasionally, it will have been necessary to replace the sensors. Over a period of a decade or more (this paper seems to consider data from 2007 onwards) it is inevitable that there will have been gaps in the recording or data which appears anomalous in some way.

When inspecting time series of real-time tide gauge data from the SLSMF (or a local display), the only people who are in a position to understand anything that looks anomalous are the owners and operators of the equipment who will have access to installation reports and maintenance logs. Such detailed information is not available to a random user of the SLSMF web site.

For example, Figure 4 of the paper shows flat values near low tide for pressure sensor PRS1 at Aden. The installation report of the Aden station shows that PRS1 was installed as a 'halftide sensor' (IOC 2006) for which the observed bottoming-out at low tide was in fact intentional. (PRS2 was installed much deeper than PRS1 and may also occasionally bottomout during very low tidal levels.) This is something that cannot be appreciated without some knowledge of the hardware and the station layout. A thoughtless downloading of data from all channels, and subjecting them to a statistical comparison, is pointless. The radar data from this station have proved to be very reliable (e.g. [START_REF] Woodworth | Insight into long term sea level change based on new tide gauge installations at Takoradi, Aden and Karachi[END_REF]). (We can further point out that the location of Aden shown in Figure 2 is incorrect.) Section 3.2.2 refers to zero values for some or all sensors at certain sites. These can sometimes arise from local hardware malfunction but are as likely to occur from transmission errors between the station, the satellite (e.g. Meteosat), the Global Telecommunications System (GTS) and the SLSMF. Meteosat and other meteorological satellites are not 100% reliable for data transmission, with occasional bit-error corruptions and short data gaps which meteorological data users can usually tolerate. Nevertheless, these satellites are made available freely to tide gauge owners in GLOSS, which explains why they have been adopted as one of the main data transmission methods for the global network. In practice, zero values can be spotted instantly and rejected from any record, with small gaps interpolated. We do not see why the authors felt the need to discuss such a trivial issue.

Another example we can mention is to do with the Alexandria pressure sensors in Figure 6.

Alexandria was installed as part of modernisation of the African sea level network [START_REF] Woodworth | Sea level monitoring in Africa[END_REF]) for which there were many problems with pressure sensors, explaining the time series shown in this figure. Sensors were replaced from time to time, hence the jumps in datum. Bad data are never removed from the SLSMF archive in case they are needed again in the future, and neither are they flagged as bad, so there is always a danger that they can be spotted and remarked on by authors such as the present ones who are not in a position to know all the historical details. At this station the radar sensor provided good data for many years (2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016) with monthly mean sea level values included in the PSMSL data set.

A final example is the "sudden step" in the difference between radar and pressure sensors from Kaumalapau, Hawai'i shown in Figure 7. The discussion of this step is rightly attributed to station maintenance, but it is a naïve and misleading perspective without noting that this offset is not carried forward into the quality-controlled sea level datasets provided by GLOSS data centres. When a radar (or any other) sensor is serviced or replaced, it must be unmounted and then remounted. Station technicians perform an engineering-grade survey to find the offset of the reinstalled sensor relative to the station benchmarks. This offset is then applied to the observed levels prior to distributing the data for oceanographic and geodetic applications. As discussed elsewhere in this response, data from the SLSMF is for real-time monitoring purposes and not appropriate for other applications. Thus, the offset should not be classified as an "error" as the authors present. Rather, the offset reflects water level measurements before they have been converted to sea level. By definition, 'sea level' is water level relative to land, or more specifically, relative to geodetic benchmarks.

Variability in SLSMF data is not guaranteed to represent sea level in this strict sense due to an inability to apply levelling offsets in real time.

Section 2.1 implies that the authors made "inquiries to tide-gauge or data basis curators", from which they might have learned more about the histories of recording at their 10 sites.

That would indeed have been sensible. However, there is no record of them having made enquiries for information from IOC, SLSMF, National Oceanography Centre (NOC, which provided the Alexandria gauge), UHSLC (which provided the Hawai'i and Tern gauges), GFZ Potsdam (which provided the Aden gauge) or from any of the other station contacts given on the SLSMF web site.

(4) Different sensors for different purposes.

This leads to a further aspect of sea level recording that appears to have been ignored in the paper, which is that different sensors are installed for different purposes. The paper seems to treat all channels as equally worthy of statistical comparison.

The GLOSS Implementation Plan (IOC 2012) and IOC Manuals (IOC 2006;[START_REF] Ioc) | Manual on Sea Level Measurement and Interpretation -Radar Gauges[END_REF] make it clear that new stations must have multiple sensors (as mentioned above, most of them already have multiple sensors). For example, radar gauges have been shown to be capable of providing time series with little or no instrumental drift, unlike pressure sensors which do tend to drift. However, both are desirable at a site, given that pressure sensors can be sampled at high frequency, and so be useful for detecting processes such as seiches and tsunamis (and even waves if sampled at 1 Hz or similar). There is little point, therefore, in deriving statistical conclusions as in this paper from sensors that were installed for different purposes.

We can furthermore point out that channels of data into the SLSMF could be occasionally from experimental setups and therefore provide information that is completely meaningless to anyone else.

(5) Their abstract states "Pressure gauges were found significantly more sensitive to noise than radar-type sensors, and with higher chances of long-term transient bias".

The finding of a 'long-term transient bias' with pressure sensors is something that has been known for years [START_REF] Woodworth | A one year comparison of radar and bubbler tide gauges at Liverpool[END_REF][START_REF] Martin Míguez | Performance of modern tide-gauges: towards mm-level accuracy[END_REF], and which is now discussed at length in manuals and textbooks (IOC 2006;[START_REF] Pugh | Sea-level science: Understanding tides, surges, tsunamis and mean sea-level changes[END_REF]. In brief, pressure tide gauges are based on the principle of measuring sub-surface pressure, which is converted to height based on knowledge of the water density and local acceleration due to gravity. However, all pressure gauges are sensitive to temperature effects and are prone to instrumental ageing. Some sensors have an in-built temperature sensor to allow temperature compensation. However, large instrumental drifts have been reported in excess of 1 mm/year, even when using the best possible (and most costly) sensors [START_REF] Martin Míguez | Performance of modern tide-gauges: towards mm-level accuracy[END_REF]. This is clearly a major concern for long-term sea level studies. (There is an additional source of datum error if pressure sensors are taken out of the water for annual maintenance and replaced at a slightly different height, see IOC 2006).

Pressure gauge designs exist which attempt to circumvent these problems (e.g., see discussion of 'B gauges' in IOC 2006). However, they can be expensive and have not been adopted by most groups. As for different noise levels, one should note that measurements with the same nominal sampling period (e.g. one or several minutes) are not necessarily equivalent, those by pressure sensors sometimes being spot (instantaneous) measurements while radar values will be averages over manufacturer-defined periods.

Why then are pressure sensors still so widely used in coastal tide gauge stations? The reason is that they are particularly well suited for installation and operation in environmentally hostile areas at sites with minimal infrastructure and harsh weather conditions, where the exposed structures may be subject to extreme winds and waves. Hence, they are more likely to resist and record a tsunami wave or a storm surge (identification of which does not require rigorous datum stability). Consequently, pressure gauges are important components of multi-application sea level networks such as the GLOSS core network of sea level stations, in which they complement the primary tide gauge (typically a radar sensor nowadays). We suggest that the authors read IOC (2016) for more information.

(6) The abstract further says that on average, over the 1-4 year timespan of data considered, "only 9% of the sensors analysed were found to satisfy the 1-cm accuracy criterion" imposed by the GLOSS programme (IOC 2012).

This finding is consistent with our comments above on the way that the authors selected their data for analysis, especially since their study considered multi-year periods of operation. It is interesting that, of the ten stations with co-located sensors analysed in their paper, only Punta Cana, Dominican Republic had two co-located sensors which were both radar gauges. The radar technique has progressively replaced the previous tide gauge technologies over the past decade (e.g., Martin Míguez et al 2008; [START_REF] Park | Water level and wave height estimates at NOAA tide stations from acoustic and microwave sensors[END_REF], even though additional research is required concerning their longer-term suitability at some locations (IOC 2016). It is thus comforting to read that a 1.8 cm rms (root mean square) difference between sensors was found at Punta Cana, and no substantial bias reported over a 518-days long recording time span.

(7) Section 4.9 states that "Errors usually found in raw tidal data are removed and not displayed in data bases. Such modification tends to shadow important metrological information. On the contrary, raw and corrected data sets should be fully shown, as is the case in disciplines, for example in data bases used in Climatology" This is a misleading statement. When raw data are passed for further quality control to the other centres connected with the GLOSS programme (IOC 2012), they are processed to modern data management standards, with all data flagged and capable of recovery and not simply deleted. As the lead author is based in the UK, we suggest that she discusses with BODC how they process and archive sea level data.

(8) Section 2.2.1 states that to provide records with a common datum one needs 19 years of data. This is a tangential point but is worth clarifying. The text is alluding to the traditional requirement for 18.6 years of data for the definition of hydrographic datums such as Lowest Astronomical Tide (a requirement that in practice is almost never met). However, in the context of a common datum discussed here, there is no data requirement at all. Provision of a common datum is a simple technical issue.

In conclusion, we believe that this paper is misleading in several respects and that the statistical analyses undertaken have little value. In our opinion, if the authors intend to continue investigating tide gauge data, we encourage them to engage properly with IOC, SLSMF, NOC or UHSLC and to understand properly the data that they are dealing with. Blind data mining studies such as this are of little value. At least we can agree with the paper ending on a positive note, recognising that it is becoming increasingly common to have multiple sensors at tide gauge sites and that there is much to be learned about systematic errors from their inter-comparison.

Acknowledgements

We are grateful for discussions about the Pytharouli et al. paper with Peter Foden, Jeff Pugh and other colleagues.


Jakub Orlowski
email: jakub.orlowski@l2s.centralesupelec.fr.

Antoine Chaillet
email: antoine.chaillet@centralesupelec.fr.

Mario Sigalotti
email: mario.sigalotti@inria.fr.

Jakub Orłowski

Alain Destexhe
email: destexhe@unic.cnrs-gif.fr.

Adaptive scheme for pathological oscillations disruption in a delayed neuronal population model

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

term in the gain adaptation law, as well as the "λ-tracking", proposed in [START_REF] Ilchmann | Universal λ-tracking for non-linearlyperturbed systems in the presence of noise[END_REF] and applied to linear infinite-dimensional systems in [START_REF] Ilchmann | Adaptive λ-tracking for a class of infinite-dimensional systems[END_REF], which increases robustness with respect to external noise by guaranteeing that the tracking error converges to a set of a specified radius. In the context of parkinsonian brain oscillations, the strong uncertainty and possible evolution in time of the model parameters naturally lead to the application of the aforementioned σ-modification.

A key feature to be taken into account in the control design is the presence of delays in the dynamics, representing the non-instantaneous communication between neurons due to axonal propagation. Delays are believed to play a crucial role in the onset of pathological oscillations [START_REF] Nevado-Holgado | Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network[END_REF]. This peculiarity requires to adapt existing adaptive control strategies to timedelay systems. To that aim, we take advantage of another specificity of neuronal dynamics, namely their evolution over a bounded set, as neuronal activity cannot overpass certain limits due to neurophysiological considerations.

In order to prove the efficiency of the σ-modification adaptative strategy for nonlinear time-delay systems, we rely on the framework of input-to-output stability (IOS), originally introduced in [START_REF] Sontag | Notions of input-to-output stability[END_REF] in a finite-dimensional context and extended to delay systems in [START_REF] Karafyllis | Input-to-Output Stability for systems described by retarded functional differential equations[END_REF]. We show that, for systems evolving on a bounded set, this property boils down to asymptotic stability in the absence of disturbances. This observation allows us to reduce the problem to controlling a disturbance-free system with an adaptive control whose gain never decreases in time (unlike the σ-modification). After presenting a general adaptive control strategy for nonlinear time-delay systems with bounded dynamics, we apply this methodology to the neuronal population model of [START_REF] Nevado-Holgado | Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network[END_REF], which already served as a basis for the developments in [START_REF] Haidar | A firing-rate regulation strategy for closed-loop deep brain stimulation[END_REF], and assess its performance through numerical simulations. Notation. Given x ∈ R n , |x| denotes its Euclidean norm. Given a compact set I ⊂ R and a measurable signal u :

I → R m , ∥u∥ := ess sup t∈I |u(t)|. Given b > a, u [a;b] : [a; b] → R m denotes the function defined as u [a;b] (t) = u(t) for all t ∈ [a; b]. Given X ⊂ R n , C(I, X) denotes the set of all continuous functions ϕ : I → X and, given δ > 0, C := C([-δ; 0], R). Given a signal x : [-δ, +∞) → R n and a time t ≥ 0, x t ∈ C n denotes the history function: x t (s) := x(t + s) for all s ∈ [-δ; 0]. A function α : R ≥0 → R ≥0 is said to be of class PD if it is continuous and positive definite. α ∈ K if α ∈ PD and it is increasing. α ∈ K ∞ if α ∈ K and it is unbounded. A function β : R ≥0 × R ≥0 → R ≥0 is of class KL if β(•, t) ∈ K for each t ∈ R ≥0 and, for each s ∈ R ≥0 , β(s,
•) is continuous, non-increasing and tends to zero as its argument tends to infinity.

II. y-GAS AND IOS

We start by recalling and adapting results on the stability of input-free systems and their robustness with respect to disturbances, in a time-delay context. Let X be an open subset of C n for the topology of the uniform convergence. Let U be an open subset of R containing the origin, and U be the set of all measurable and locally essentially bounded u : R ≥0 → U . Consider the perturbed time-delay system:

ẋ(t) = f (x t , u(t)), y(t) = h(x t ), (1)
where x t ∈ X is the state, u ∈ U m is the applied input, and

y ∈ C(R ≥0 , R l
) is an output of interest. We start by recalling Driver's formulation of the upper-right Dini derivative [START_REF] Driver | Existence and stability of solutions of a delay-differential system[END_REF].

Given any continuous function V : X → R, its Driver's derivative D + V along the solutions of ( 1) is defined, for all ϕ ∈ X and all v ∈ U m , as

D + (1) V (ϕ, v) := lim sup h→0 + V (ϕ ⋆ h,v ) -V (ϕ) h , (2)
where, for all h ∈ (0; δ), ϕ ⋆ h,v ∈ X is defined as

ϕ ⋆ h,v (s) := { ϕ(s + h) if s ∈ [-δ; -h) ϕ(0) + f (ϕ, v)(s + h) if s ∈ [-h; 0]. (3)
It was shown in [START_REF] Pepe | On Liapunov-Krasovskii functionals under Carathéodory conditions[END_REF] that, if V is Lipschitz on any bounded set of C n , then Driver's derivative of V computed at (x t , u(t)) coincides, at almost every t where x t is defined, with the upper-right Dini derivative of t → V (x t ), that is,

D + (1) V (x t , u(t)) = lim sup τ →0 + V (x t+τ ) -V (x t ) τ , a.e.

A. Characterizations of y-GAS

Let us first consider an input-free version of (1):

ẋ(t) = f (x t ), y(t) = h(x t ), (4)
where y(t) ∈ R l denotes an output for which we want to ensure a desirable behavior. We assume the following.

Assumption 1 There exists a bounded open forwardinvariant set X ⊂ C n for (4), the functionals f : C n → R n and h : C n → R l are Lipschitz on X , and f (0) = 0.

The above regularity assumptions are standard and ensure existence and uniqueness of solutions [8, Section 2.6, p. 55]. We also impose that solutions evolve on a bounded invariant set as this assumption greatly simplifies the analysis and it is naturally satisfied in the neuroscience applications considered here (see Section IV). Note that this boundedness assumption in turn ensures forward completeness of (4).

For the needs of this paper, we will study global asymptotic output-stability as defined in the following sense.

Definition 1 (y-GAS) We say that system (4) is globally asymptotically y-stable (y-GAS) on X if there exists β ∈ KL such that, for all x 0 ∈ X , its solution satisfies

|h(x t )| ≤ β(∥x 0 ∥, t), ∀t ≥ 0.
This stability notion is strongly related to "stability with respect to two measures" employed in [START_REF] Lakshmikantham | Stability analysis in terms of two measures[END_REF], [START_REF] Teel | A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions[END_REF], to "partial stability" studied in [START_REF] Vorotnikov | Partial stability and control[END_REF], as well as to output stability concepts [START_REF] Sontag | Notions of input-to-output stability[END_REF] that were more recently extended to timedelay systems [START_REF] Karafyllis | Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations[END_REF], [START_REF] Karafyllis | Input-to-Output Stability for systems described by retarded functional differential equations[END_REF], [START_REF] Kankanamalage | On Lyapunov-Krasovskii Characterizations of Input-to-Output Stability[END_REF]. It imposes that the output y eventually converges to the origin, with a transient overshoot that may depend on the magnitude of whole initial state x 0 .

This output-stability can be established through the following Lyapunov-Krasovskii characterizations.

Theorem 1 (y-GAS characterizations) Under Assumption 1, the following statements are equivalent:

1) System (4) is y-GAS on X .

2) There exist a Lipschitz functional V : X → R ≥0 and α, α ∈ K ∞ such that, for all ϕ ∈ X ,

α(|h(ϕ)|) ≤ V (ϕ) ≤ α(∥ϕ∥) (5) D + (4) V (ϕ) ≤ -V (ϕ). (6)
3) There exist a Lipschitz functional V : X → R ≥0 and α, α, α ∈ K ∞ such that, for all ϕ ∈ X ,

α(|h(ϕ)|) ≤ V (ϕ) ≤ α(∥ϕ∥) (7) D + (4) V (ϕ) ≤ -α(|h(ϕ)|). ( 8
)
Proof: The equivalence between Items 1) and 2) is a direct consequence of [START_REF] Karafyllis | Input-to-Output Stability for systems described by retarded functional differential equations[END_REF]Theorem 3.3]. More precisely, all the requirements (S1)-(S8) in that paper are satisfied under Assumption 1, and the property referred to as "almost Lipschitz" in that paper turns out to be "Lipschitz on bounded sets" in the present time-invariant context. Moreover, Item 2) readily implies Item 3) as ( 5)-( 6) imply that D + (4) V (ϕ) ≤ -α(|h(ϕ)|). Finally, the fact that Item 3) implies Item 1) was proven in [START_REF] Vorotnikov | Partial stability and control[END_REF]Theorem 6.2.1]. Although that reference makes only use of "ε -δ" stability definitions, the KL formulation employed in Definition 1 can be recovered using similar reasonings as in the proof of [START_REF] Karafyllis | The non-uniform in time small-gain theorem for a wide class of control systems with outputs[END_REF]Lemma 3.4].

B. IOS for 0-y-GAS systems evolving on bounded sets

We now go back to the perturbed system (1) to study the impact of disturbances on output-stability. To that aim, recall the notion of input-to-output stability (IOS), originally introduced in [START_REF] Sontag | Notions of input-to-output stability[END_REF] in a finite-dimensional context and extended in [START_REF] Karafyllis | Input-to-Output Stability for systems described by retarded functional differential equations[END_REF], [START_REF] Kankanamalage | On Lyapunov-Krasovskii Characterizations of Input-to-Output Stability[END_REF] to time-delay systems.

Definition 2 (IOS) System (1) is said to be input-to-output stable (IOS) on X and U m if there exist β ∈ KL and γ ∈ K ∞ such that, for all x 0 ∈ X and all u ∈ U m , its solution satisfies

|h(x t )| ≤ β(∥x 0 ∥, t) + γ(∥u [0;t] ∥), ∀t ≥ 0. (9)
IOS not only imposes y-GAS in the absence of disturbances, but also that, in their presence, the output eventually converges to a neighborhood of the origin whose size is "proportional" to their magnitude.

Here also, we assume that the system evolves on a bounded set for all considered initial states and disturbances.

Assumption 2 There exists a bounded open set X ⊂ C n which is forward-invariant for (1) for any u ∈ U m . Moreover, the functionals f :

C n × U m → R n and h : C n → R l are
Lipschitz on X × U m and X respectively, and f (0, 0) = 0.

The above assumption requires that solutions of the perturbed system (1) evolve on the bounded set X no matter what the applied input u ∈ U m . Here again this assumption is naturally satisfied for the relevant neuroscience applications.

Definition 3 (0-y-GAS) System (1) is said to be 0-y-GAS if the following input-free system is y-GAS:

ẋ(t) = f (x t , 0), y(t) = h(x t ). (10)
Clearly, IOS implies 0-y-GAS. Based on Theorem 1, we can show that these two properties are actually equivalent for systems evolving in a bounded set, as stated next.

Theorem 2 (0-y-GAS ⇔ IOS, on bounded sets) Under Assumption 2, system (1) is IOS on X and U m if and only if the input-free system (10) is y-GAS on X .

In the above statement, the fact that solutions evolve in a bounded set is crucial. Without this assumption, IOS is strictly more conservative that 0-y-GAS. A similar result was obtained in [START_REF] Yeganefar | Input-to-state stability of time-delay systems: a link with exponential stability[END_REF] for systems that are globally exponentially stable in the absence of disturbances. That result does not require that solutions evolve on a bounded set, but rather imposes that f is globally Lipschitz and that there exist c ≥ 0 and p [0; 1) such that |f (ϕ, v) -f (ϕ, 0)| ≤ c max{∥ϕ∥ p , 1}|v| for all ϕ ∈ C n and all v ∈ R m .

Due to space limitation, the proof of this theorem is not included but proceeds as follows. We only prove that 0-y-GAS implies IOS, as the converse is straightforward. From Theorem 1 we get the existence of a Lyapunov-Krasovskii functional V satisfying ( 5)-( 6) along the disturbance-free system. Using the properties of the Driver's derivative, Lipschitz continuity of V and f (Assumption 4), and the fact that the system evolves on a bounded set, we compute the derivative of V along the solutions of the perturbed system [START_REF] Benabid | Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus[END_REF]

to get that D + (1) V (ϕ, v) ≤ -V (ϕ)+ℓ V ℓ f |v| for all ϕ ∈ X and all v ∈ U m , where ℓ V is the Lipschitz constant of V on X and ℓ f is the Lipschitz constant of f on X × U m . This fulfills condition d) of [16, Thm. 3.3], establishing IOS.

III. ADAPTIVE GAIN FOR TIME-DELAY SYSTEMS

We now make use of these results for adaptive control. Consider the following class of time-delay control systems:

ẋ(t) = f (x t , µ(t), u(t)) (11a) z(t) = g(x t ), (11b)
where x t ∈ C n is the state and u ∈ U m represents a disturbance acting on the system. µ(t) ∈ R denotes the scalar control input. z(t) ∈ R denotes the scalar output available for measurements. The adaptive control we propose here is based on the "σ-modification" originally introduced in [START_REF] Ioannou | Instability analysis and improvement of robustness of adaptive control[END_REF] for linear systems with no delay: µ is defined in terms of an additional scalar variable θ as follows:

µ(t) = -θ(t)z(t) (12a) θ(t) = κ(z(t)) -εθ(t), ( 12b
)
where ε ≥ 0 is a tunable parameter and κ : R ≥0 → R ≥0 is a locally Lipschitz function to be chosen. As illustrated in Section IV-B, the choice of ε and κ can be guided by the sought closed-loop performance. We assume the following.

Assumption 3 There exists a bounded open set X ⊂ C n such that for every u ∈ U m , every continuous µ : R ≥0 → R, and every ε ≥ 0, the set X is forward invariant for equation (11a). The functionals f : C n ×R×U m → R n and g : C n → R are Lipschitz on bounded sets, and f (0, 0, 0) = 0.

We also assume that the origin of this plant is globally exponentially stabilizable by high-gain proportional feedback in the absence of disturbances u, as stated in the following.

Assumption 4 (High-gain stabilizability) There exist a Lipschitz functional V : X → R ≥0 , positive constants a, a and a, and a positive gain θ * such that, for all ϕ ∈ X ,

a|ϕ(0)| 2 ≤ V (ϕ) ≤ a∥ϕ∥ 2 , ( 13
)
and, for all fixed θ ≥ θ * , Driver's derivative of V satisfies

D + (15) V (ϕ, θ) ≤ -a|ϕ(0)| 2 (14)
along the solutions of the closed-loop system

ẋ(t) = f (x t , -θz(t), 0), z(t) = g(x t ). (15)
Our main result is as follows.

Theorem 3 (Robust stability with adaptive gain) Let κ : R → R ≥0 be any locally Lipschitz function satisfying

κ(s) ≥ κ 0 |s|, ∀s ∈ R, (16)
for some κ 0 > 0 and κ(g(0)) = 0. Then, under Assumptions 3 and 4, there exist β ∈ KL, γ ∈ K ∞ , and ϑ > θ * such that, for all (x 0 , θ 0 ) ∈ X × [0; θ * ], all u ∈ U m , and all ε ≥ 0, the solution of [START_REF] Ilchmann | Universal λ-tracking for non-linearlyperturbed systems in the presence of noise[END_REF] in closed-loop with [START_REF] Ioannou | Instability analysis and improvement of robustness of adaptive control[END_REF] satisfies

|x(t)| ≤ β(∥x 0 ∥ + |θ 0 -θ * |, t) + γ(ε + ∥u [0;t] ∥) (17) θ(t) ∈ [0; ϑ], ∀t ≥ 0. ( 18
)
The above statement contains several ingredients. First, in the absence of disturbances (u ≡ 0), practical stability is achieved in the sense that the steady-state error on x can be arbitrarily reduced by picking ε sufficiently small. The transient overshoot is "proportional" to the magnitude of the initial state and to the distance between θ 0 and θ * (this distance being small if we initialize the control algorithm with a good guess of θ * ). For ε = 0, this steady-state error is zero, meaning that x(t) converges to the origin as t tends to infinity. The use of a positive ε allows to avoid parameter drift in presence of disturbances [START_REF] Ioannou | Instability analysis and improvement of robustness of adaptive control[END_REF], as certified by [START_REF] Mareels | A simple selftuning controller for stably invertible systems[END_REF]. More crucially, ε > 0 allows to react to possible variations of the model parameters, as detailed in Section IV-B. Finally, in the presence of disturbances, the state is guaranteed to converge to a neighborhood of the origin, whose size is "proportional" to the magnitude of the applied disturbance and the chosen value of the parameter ε.

Proof: We see the system as a perturbed version of

ẋ(t) = f (x t , -θ(t)g(x t ), 0), θ(t) = κ(g(x t )). (19)
In other words, we see both u and ε as perturbations. In view of Theorem 1, IOS of the closed-loop system can be derived based on the output-stability of [START_REF] Nevado-Holgado | Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network[END_REF]. Let θ * be as in Assumption 4 and let θ := θ -θ * . Then [START_REF] Nevado-Holgado | Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network[END_REF] reads

ẋ(t) = f ( x t , -( θ(t) + θ * )g(x t ), 0 ) (20a) θ(t) = κ(g(x t )). ( 20b
)
The results of Section II do not readily apply to this system since θ does not necessarily evolve in a bounded set (because κ(g(x t )) ≥ 0). However, we can show that there exists θ > 0 such that θ(t) ∈ [-θ * , θ] for all t ≥ 0 and all θ0 ∈ [-θ * , 0]. Hence, we can consider the modified dynamics:

ẋ(t) = f ( x t , -( θ(t) + θ * )g(x t ), 0 ) (21a) θ(t) = κ(g(x t )) -max{ θ(t) -θ; 0}( θ(t) -θ). (21b)
This modified system (21) does possess an invariant set, even in presence of disturbances, as stated next. We express the invariance in terms of the history function for θ (even though no delays apply to it) in order to apply later to the augmented state variable (x, θ) the results of Section II.

Claim 1 Let κ be any positive constant such that κ•g(ϕ) ≤ κ for all ϕ ∈ X , as ensured by Assumption 3, and let

Θ := C ( [-δ; 0], (-2θ * ; 2 θ + √ κ) ) ⊂ C.
Then, for every u ∈ U m and ε ≥ 0, the bounded open set X × Θ is invariant for

ẋ(t) = f ( x t , -( θ(t) + θ * )g(x t ), u ) (22a) θ(t) = κ(g(x t )) -ε( θ(t) + θ * ) -max{ θ(t) -θ; 0}( θ(t) -θ). ( 22b
)
Proof: In view of Assumption 3, the claim readily follows by observing that θ(t) ≤ 0 whenever θ(t) ≥ θ + √ κ (and in particular for θ(t) ≥ 2 θ + √ κ) and that θ(t) ≥ 0 whenever θ(t) ≤ θ (and in particular for θ(t) ≤ -2θ * ).

The claim in particular proves [START_REF] Mareels | A simple selftuning controller for stably invertible systems[END_REF] with ϑ = 2 θ+ √ κ+θ * . Based on this, we now proceed to show that ( 21) is IOS on X ×Θ with output y(t) = x(t). To that end, let V be as in Assumption 4 and consider the functional W : X ×C → R ≥0 defined as

W (ϕ, ψ) = V (ϕ) + p 2 min{ψ(0); 0}ψ(0), (23)
where p denotes a positive constant to be chosen later. Then W is Lipschitz on X × Θ and, for all (ϕ, ψ) ∈ X × C,

a|ϕ(0)| 2 ≤ W (ϕ, ψ) ≤ ( a + p 2 ) ∥(ϕ, ψ)∥ 2 . ( 24
)
Moreover, if ψ(0) ≥ 0 (corresponding to θ(t) ≥ θ * ), then W (ϕ, ψ) = V (ϕ) and it holds from Assumption 4 that

D + (20) W (ϕ, ψ) ≤ -a|ϕ(0)| 2 , ∀ψ(0) ≥ 0. ( 25
)
Consider now the case when ψ(0

) < 0. Then W (ϕ, ψ) = V (ϕ) + p 2 ψ(0) 2 . It follows that D + (20) W (ϕ, ψ) = D + (20) V (ϕ, ψ) + pψ(0)κ(g(ϕ)). ( 26
)
Moreover, from (2),

D + (20) V (ϕ, ψ) = lim sup τ →0 + 1 τ ( V (ϕ ⋆ τ,ψ ) -V (ϕ) ) , ( 27
)
where, in view of (3), ϕ ⋆ τ,ψ (s) = ϕ(s + τ ) for all s ∈ [-δ; -τ ) and, for all s ∈ [-τ ; 0],

ϕ ⋆ τ,ψ (s) = ϕ(0) + f ( ϕ, -(ψ(0) + θ * )g(ϕ), 0 ) (s + τ ).
Observe that, for all (ϕ, ψ) ∈ X ×C and all τ > 0 sufficiently small, it holds that ϕ ⋆ τ,ψ ∈ X . Let ℓ V denote the Lipschitz constant of V over the bounded set X . Then, for all (ϕ, ψ) ∈ X × C, it holds from (27) that

D + (20) V (ϕ, ψ) ≤ D + (20) V (ϕ, 0)+lim sup τ →0 + 1 τ V (ϕ ⋆ τ,ψ ) -V (ϕ ⋆ τ,0 ) ≤ -a|ϕ(0)| 2 + lim sup τ →0 + ℓ V τ ∥ϕ ⋆ τ,ψ -ϕ ⋆ τ,0 ∥, ( 28
)
where we used ( 14) by observing that

D + (20) V (ϕ, 0) is D + (19) V (ϕ, ψ) computed for θ = θ * . Furthermore, since ϕ ⋆ τ,ψ and ϕ ⋆ τ,0 coincide over [-δ; -τ ], it holds that ∥ϕ ⋆ τ,ψ -ϕ ⋆ τ,0 ∥ = sup s∈[-τ ;0] |ϕ ⋆ τ,ψ (s) -ϕ ⋆ τ,0 (s)| = sup s∈[-τ ;0] f ( ϕ, -(ψ(0) + θ * )g(ϕ), 0 ) -f ( ϕ, -θ * g(ϕ), 0 ) (s + τ ).
Let G := {-(ψ(0) + θ * )g(ϕ) : ϕ ∈ X , ψ ∈ Θ} and let ℓ f > 0 denote the Lipschitz constant of f on X × G × {0}, as ensured by Assumption 3. Then we get that

∥ϕ ⋆ τ,ψ -ϕ ⋆ τ,0 ∥ = sup s∈[-τ ;0] ℓ f |ψ(0)g(ϕ)|(s + τ ) = ℓ f |ψ(0)g(ϕ)|τ.
Plugging this into (28), we obtain that

D + (20) V (ϕ, ψ) ≤ -a|ϕ(0)| 2 + ℓ V ℓ f |ψ(0)g(ϕ)|.
Thus, recalling that ψ(0) < 0 and κ(g(ϕ)) ≥ 0, we get from (26) that

D + (20) W (ϕ, θ) ≤ -a|ϕ(0)| 2 - ( pκ(g(ϕ)) -ℓ V ℓ f |g(ϕ)| ) |ψ(0)| ≤ -a|ϕ(0)| 2 - ( pκ 0 -ℓ V ℓ f ) |g(ϕ)||ψ(0)|,
where the last bound comes from [START_REF] Karafyllis | Input-to-Output Stability for systems described by retarded functional differential equations[END_REF]. Therefore, by picking p = ℓ f ℓ V /κ 0 , we obtain that, for all (ϕ, ψ) ∈ X × Θ,

D + (20) W (ϕ, ψ) ≤ -a|ϕ(0)| 2 , ∀ψ(0) < 0. Using (25), we conclude that, for all (ϕ, ψ) ∈ X × Θ, D + (20) W (ϕ, θ) ≤ -a|ϕ(0)| 2 .
Based on this, [START_REF] Vorotnikov | Partial stability and control[END_REF], and Claim 1, Theorem 1 ensures that there exists β ∈ KL such that, for all x 0 ∈ X and all θ0 ∈ (-2θ * ; 2 θ + √ κ), the solution of (21) satisfies |x(t)| ≤ β(∥x 0 ∥ + | θ0 |, t) for every t ≥ 0. Theorem 2, in turn, guarantees the existence of β ∈ KL and γ ∈ K ∞ such that, for all x 0 ∈ X , all θ0 ∈ (-2θ * ; 2 θ + √ κ) and all u ∈ U m , the solution of (21) satisfies

|x(t)| ≤ β(∥x 0 ∥ + | θ0 |, t) + γ(ε + ∥u [0;t] ∥), ∀t ≥ 0.
We conclude that the same bound holds for the solutions of (20) starting within X and θ0 ∈ [-θ * ; 0]. In particular, the solutions of (24) satisfy

|x(t)| ≤ β(∥x 0 ∥ + |θ 0 -θ * |, t) + γ(ε + ∥u [0;t] ∥), ∀t ≥ 0,
for all x 0 ∈ X , all θ 0 ∈ [0; θ * ], and all u ∈ U m .

IV. APPLICATION TO ADAPTIVE STIMULATION OF PARKINSONIAN NEURONAL POPULATIONS

All the results presented so far have been designed with neuroscience applications in mind. More precisely, in [START_REF] Haidar | A firing-rate regulation strategy for closed-loop deep brain stimulation[END_REF], a proportional stimulation strategy has been proposed to disrupt brain oscillations that are linked to Parkinsonian symptoms. This result was obtained based on a linearized version of the following model:

τ 1 ẋ1 (t) = -x 1 (t) (29a)
+ S 1 ( c 11 x 1 (t -δ 11 ) -c 12 x 2 (t -δ 12 ) + µ(t) + u 1 (t)
)

τ 2 ẋ2 (t) = -x 2 (t) (29b)
+ S 2 ( c 21 x 1 (t -δ 21 ) -c 22 x 2 (t -δ 22 ) + u 2 (t)
) .

This model was originally proposed in [START_REF] Nevado-Holgado | Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network[END_REF] (we recentered the model at 0) to explain a possible origin of parkinsonian brain oscillations in deep brain regions known as subthalamic nucleus (STN) and external globus pallidus (GPe). In this model, x 1 (t) and x 2 (t) represent the instantaneous activity of STN and GPe respectively, in number of spikes per second. τ 1 , τ 2 > 0 are decay time constants. S 1 , S 2 : R → R are called the activation functions. For each i, j ∈ {1, 2}, the constant c ij ≥ 0 represents the synaptic strength between x j and x i and δ ij ≥ 0 describes the delay due to axonal propagation from x j to x i . u 1 (t) ∈ R models activity from the motor cortex, and u 2 (t) ∈ R represents striatal activity. Finally, µ(t) ∈ R represents the stimulation that can be introduced through implanted electrodes.

In the absence of stimulation (µ ≡ 0), this model exhibits sustained activity in the β-band (13-30Hz) that arises from a too strong coupling between STN and GPe. These oscillations correlate with parkinsonian motor symptoms [START_REF] Hammond | Pathological synchronization in Parkinson's disease: networks, models and treatments[END_REF]. A spatio-temporal extension of this model was proposed and analyzed in [START_REF] Detorakis | Closed-loop stimulation of a delayed neural fields model of parkinsonian STN-GPe network: a theoretical and computational study[END_REF], [START_REF] Chaillet | Robust stabilization of delayed neural fields with partial measurement and actuation[END_REF]. In those works, it was shown that a proportional stimulation strategy, relying on STN measurements only (meaning on x 1 (t)) successfully disrupts these pathological oscillations provided that c 22 < 1, meaning that the GPe internal synaptic coupling is weak (which is in line with physiologic evidence).

A. Theoretical analysis

In practice, the parameters involved in (29) suffer high imprecision and variability. Hence, it would be desirable to adapt the value of the stimulation gain in real time. Since only x 1 (t) is available to measurements, we see (29) as a particular case of [START_REF] Ilchmann | Universal λ-tracking for non-linearlyperturbed systems in the presence of noise[END_REF] with output z(t) := x 1 (t). We also let δ := max i,j∈{1,2} δ ij . In line with the results of Section III, we propose the adaptive stimulation strategy

µ(t) = -θ(t)x 1 (t) (30a) θ(t) = 1 T ( |x 1 (t)| -ε ′ θ(t) ) , ( 30b
)
where ε ′ ≥ 0 and T > 0 denote tuning parameters. With this closed-loop stimulation policy, we have the following. The main steps of the proof are as follows. Assumption 3 is fulfilled due to the properties of the functions S i . Assumption 4 can be checked by creating a Lyapunov-Krasovskii functional for (29) in a closed loop with µ(t) = -θx 1 (t), for some constant θ > 0, and in the absence of disturbances (u ≡ 0). This is done along the proof steps of the main result in [START_REF] Chaillet | Robust stabilization of delayed neural fields with partial measurement and actuation[END_REF], which establishes the high-gain stabilizability of a spatio-temporal extension of (29). Finally, we use Theorem 3 to obtain robust stability. All initial states are considered because, due to the boundedness of S i , every solution eventually enters the invariant set of Assumption 3. Like the high-gain stimulation policies proposed in [START_REF] Haidar | A firing-rate regulation strategy for closed-loop deep brain stimulation[END_REF], [START_REF] Detorakis | Closed-loop stimulation of a delayed neural fields model of parkinsonian STN-GPe network: a theoretical and computational study[END_REF], [START_REF] Chaillet | Robust stabilization of delayed neural fields with partial measurement and actuation[END_REF], the proposed adaptive stimulation successfully disrupts pathological oscillations in this model. As we detail in Section IV-B, the adaptive nature of this control law offers interesting advantages as compared to those policies, specifically in presence of uncertain or varying parameters.

The result is based on limited information on the model, namely the maximal slope of the activation function S 2 (normalized here to 1) and the GPe internal synaptic weight c 22 . In particular, the proposed strategy is valid for any synaptic weights between STN and GPe and any transmission delays δ ij . Moreover, no measurement of GPe activity is needed.

B. Simulation results

We ran the simulations of system (29) in Python 2.7, using the pydelay library. The simulation parameters were chosen as follows: the time constants τ 1 = 6 ms, τ 2 = 14 ms; the delays δ 11 = 0 ms, δ 12 = δ 21 = 6 ms, δ 22 = 4 ms; the connection strength coefficients c 11 = 0, c 12 = 20, c 21 = 10.7, c 22 = 0.2. The activation functions S i were picked as

S i (s) = m i b i /(b i + (m i -b i ) exp(-4(s + m i ln((m i -b i )/b i )/4)/m i )) -m i /2,
with m 1 = 300 spk/s, m 2 = 400 spk/s, b 1 = 17 spk/s, b 2 = 75 spk/s. All the above parameters, except c 22 (chosen small, to satisfy the stabilizability condition in Theorem 4), correspond to the parkinsonian state in [START_REF] Nevado-Holgado | Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network[END_REF]. The control parameters were picked as ε ′ = 10 -4 and T = 0.01 ms. The disturbance u 1 was picked as a random noise drawn from a uniform distribution on (-2; 2). u 2 was set at zero.

Figure 1 illustrates the efficacy of the proposed adaptive stimulation. For t ∈ [0; 0.1], the feedback is off and the systems exhibits sustained oscillations. At time t = 0.1, the stimulation is switched on. The adaptive gain θ increases (at a rate essentially given by T ) and reaches a value that leads to significantly decreased oscillations (whose amplitude depends on the disturbance magnitude and the choice of ε ′ ). At t = 0.5 s, the connection weights c 12 and c 21 were artificially diminished by a factor 2, for which a weaker proportional gain is needed. Accordingly, θ automatically adjusts to a weaker value, giving rise to a stimulation signal µ of lower amplitude.

V. CONCLUSIONS

We conclude that the adaptive feedback explored here constitutes a promising approach to control pathological oscillations such as in Parkinson's disease or epilepsy. Such paradigms should be explored by future studies.

Fig. 1 :

 1 Fig. 1: Simulation results for the closed-loop adaptive system (29) in closed loop with (30). Top: STN activity x1(t). Middle: applied stimulation µ(t) = -θ(t)x1(t). Bottom: adaptive control gain θ(t).

Figure 2

 2 Figure2represents the magnitude of steady-state oscillations of x 1 and x 2 as a function of the chosen parameter ε ′ . In line with Theorem 4, this steady-state amplitude is an increasing function of ε ′ and vanishes for ε ′ = 0 with u 1 and u 2 set to zero.

Fig. 2 :

 2 Fig. 2: Steady state amplitude (max{xi(t)} -min{xi(t)} for t ∈ [19, 20]s) for different values of ε ′ := εT . The remaining parameters were kept identical as in the simulation performed to obtain Figure 1 but with u = 0. The results obtained are in accordance with Theorem 4 which states that the steady-state amplitude of the system is upper bounded by a K∞ class function of ε + ∥u∥. The x-axis is shifted by 10 -4 to allow displaying values for ε ′ = 0.

  Theorem 4 (Adaptive brain stimulation) Assume S 1 , S 2 : R → R are bounded Lipschitz nondecreasing functions satisfying S 1 (0) = S 2 (0) = 0. Assume that the Lipschitz constant of S 2 is 1 and let T > 0 be any constant parameter. Under the condition c 22 < 1 there exists γ ∈ K ∞ such that system (29) in closed loop with (30) satisfies, for all ε ≥ 0, all x 0 ∈ C 2 , and all bounded u : R ≥0 → R 2 ,

	lim sup

t→+∞ |x(t)| ≤ γ(ε + ∥u∥).

Supported by the E.C. (Human Brain Project H2020-785907) to A.D., and by the iCODE project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.


Theocharis Baxevanis

Gilles Pijaudier-Cabot

Frédéric Dufour

Bifurcation and size effect in a viscoelastic non-local damageable continuum

BIFURCATION AND SIZE EFFECT IN A VISCOELASTIC NON-LOCAL DAMAGEABLE CONTINUUM

Th. Baxevanis, G. Pijaudier-Cabot and F. Dufour R&DO, Institut de recherches en Genie Civil Ecole Centrale de Nantes, 1 rue de la Noe BP 92101, F-44321 Nantes cedex 3, FRANCE theocharis@tem.uoc.gr

The growth of microcracks in progressively fracturing rate-independent materials is a fairly distributed process which is associated with stable material response. However, for certain stress trajectories such as tertiary creep, a different deformation mode may prevail, consisting of formation of discrete failure planes (macrocracks). In the latter case, the mechanical response as observed on the macroscale becomes, in general, unstable. The inception of such a localized mode may be considered as a bifurcation problem related to the loss of positive definiteness of the tangent material stiffness operator governing the homogeneous deformation (Rudnicki and Rice [START_REF] Rudnicki | [END_REF]). In mathematical terms, the elliptic character of the set of governing differential equations, in quasi-static problems, is lost (ill-posedness of the related boundary value problem). This result was derived for the linearized rate equation problem, considering what is commonly denoted as a linear comparison solid (Hill [5]).

In common practice, it is usually assumed that for low stress levels linear viscoelasticity takes place and that the instantaneous mechanical behavior is elastic. Viscoelasticity of the material causes strong time dependence of fracture growth. Bazant and Gettu [START_REF] Bazant | [END_REF] conducted size effect tests on geometrically similar specimens loaded at different rates and observed that the slower the loading rate, the more brittle the response. Pijaudier-Cabot et al [START_REF] Pijaudier-Cabot | Proccedings of ICF 11[END_REF] performed also a series of three point bending beam tests (notched specimens) applying a sustained load for 3 months up to 85 percent of the maximum load prior to the fracture test. They arrived to a similar conclusion, namely, that upon size effect tests, the specimens subjected to creep exhibited a more brittle response compared to the ones that were not. This property yields a shift to the right of the data set on the size effect plot.

For high stress levels the linearity hypothesis fails. Nevertheless, creep cannot be captured merely by nonlinear generalization of viscoelastic stress-strain relation (Bazant [3]). The experimental evidence associates creep strains at high stress levels to microcracking nucleation and growth -cracks grow and interact with visco-elasticity -(Rüsch [12] was the first one to conduct such experiments). Some experimental and analytical results concerning non-linear creep can be found in the literature (see among others Bazant [START_REF] Bazant | Mathematical modeling of creep and shrinkage of concrete[END_REF], Bazant and Gettu [START_REF] Bazant | [END_REF], Mazzotti and Savoia [START_REF] Mazzotti | [END_REF]). The coupling of the mechanical behaviour and the visco-elastic behaviour of the material assumed as a continuum, and its nonlinear instantaneous behaviour has been tackled by many authors using a linear creep model (viscoelastic behaviour) coupled with a plasticity model (De Borst et al [START_REF] De Borst | the Proccedings of the 5th Int. RILEM Symp. on Creep and Shrinkage of Concrete[END_REF]), or a smeared crack model (Rots et al [START_REF] Rots | [END_REF]), or a damage model (Mazzotti and Savoia [START_REF] Mazzotti | [END_REF]) (nonlinear instantaneous behaviour).

The aim of this paper is to investigate the localization properties of a rate-independent material described by a non-local damage-based constitutive relation coupled with a creep relation of the Kelvin type. It is proved that the coupled model preserves the properties of localization limiters as they were pointed out in the paper of Pijaudier-Cabot and Benallal [8] and reproduces the experimentally observed rate-effects of creep. Namely, the minimum wavelength of the localization modes can not be zero in accordance with energy considerations and the material response due to creep becomes more brittle, at least under a certain condition on the evolution of the yield function of damage. Moreover, the normal n to the localization band depends only upon the elastic state of strain at the onset of bifurcation and a shift on the size effect plot occurs towards the strength criterion when the material parameter (0 1), which is the fraction of creep release energy which contributes to damage, is decreasing. The motivation for introducing this parameter into the constitutive law is that for low stress levels although creep strain can be large (even larger than that corresponding to peak stress for short term loading) there is no significant variation of the elastic modulus; that is no significant damage (Mazzotti and Savoia [START_REF] Mazzotti | Proceedings of Concreep 6[END_REF]). The proposed model fails to predict creep rate-effects when damage is supposed, as it is common practice, to be driven solely from the elastic release energy (=0).


Pierre Bayard

Caroline Julliot

How to investigate books like you've never read them before

How to investigate books like you've never read them before.

[1 ère partie : Pierre : Roger Ackroyd/Hamlet/Baskerville/10 petits nègres] You've now understood it : As in real life, you can't trust anyone within fictionand, above all, you can't trust any narrator. The cases my honorable colleague just explained here are far from being isolated. To show you that writers from Perfidious Albion are not the only ones who, deliberately or accidentally, hide the truth, let's take an example closer to you, geographically and historically speaking : Top of the Lake, Jane Campion's latest TV series.

This police thriller was produced on Australian, English and American budget, and the second season, released on 2017, takes place here in Sydney. I want to talk to you about the 2013 case, main plot of the First season of the show. We follow the investigations led by detective Robin Griffin, a police inspector recently returned from Sydney, in Laketop, her home town, small city lost in New Zealand mountains. A pregnant twelve-year-old girl, Tui Mitcham, is found half-drowned in the lake, and then, gone missing. I won't get into details because I don't wan't to spoil completely the pleasure for those who haven't watched it yet, but I would like to point out how obviously unsatisfying the solution seems. One of the main avenues of the investigation is discovering who's the father of Tui's unborn child, presumably the result of rape. Suspicion quickly falls on Tui's father, Matt, a local mafia godfather who runs the area with expeditious and violent methods. In the final episode, Al Parker, detective sergeant of the city, declares that « a double-crossed DNA test » confirmed that Matt is indeed the father of the baby Tui had just given birth to. Can we really believe that ?

There's no doubt that Matt's character has everything to be the ideal culpritboth from an internal and an external perspective. In the eyes of the local community, he's a wellknown offender ; in addition, even if the matter is never directly adressed in the dialogues, he's clearly a foreigner (actor Peter Mullan kept his scottish accent for playing that character, whereas Elizabeth Moss, the american actress who plays the part of Robin, pursued a specific training to talk with a « kiwi » accent) ; and many scenes in the series prove that, in a backwater such as Laketop, not being part of this closed community equals to suffer prejudice and hatred : Robin's credibility, for instance, is repeatedly damaged by the fact that she now lives in Sydney.

Even more serious, Matt has a full chance to be also the ideal culprit in the eyes of the audience : as spectators of an artistically demanding TV-crime series, directed by an awardwinning filmaker, our collective expectations are largely determined by the founding work of the genre : Twin Peaks, created by David Lynch and Mark Frost (1990). No doubt that Top of the Lake follows in the footsteps of this previous show : as in Twin Peaks, the scene takes place in a small isolated town, lost in a postcard moutain landscape ; there, the drowning of a young and (too) pretty girl reveals the hidden and shameful secrets of the local community, and we find out it all thanks to the investigation of a police inspector recently arrived from a big city. And so, from the beginning, following Twin Peaks' model plot, we are implicitely encouraged to consider that the evil villain can only be the incestuous father.

However, we all know that ideal culprits in the eyes of society may be perfectly innocent of the charges they're accused of. If the numerous detective novels that base their solution on this kind of twist can't convince you enough, you should watch the remarkable multi-awarded documentaries Jean-Xavier de Lestrade made about american trials who covered the front pages, in particular Murder on a sunday morning (2003) and The Staircase (2004/2017). Matt surely is a scoundrel, but, given what we learn troughout the show, the least we can say is that he's not the only one around ; putting all the blame on him as scapegoat is a convenient way to protect other potential suspects, probably a crooked community leader. Al, the one who supervised the DNA test procedure, eventually turns out to be an unscrupulous and depraved bastard ; and, as Chief of police, it would have been very easy for him to tamper with evidence, by surreptitiously switching Matt's DNA sample with the real father's, or by blackmailing both DNA labs. Especially if Tui's rapist is none other than himself : such evidence, if he didn't dispose of it, could prove conclusively the charges of child abuse he has to face (and stubbornly denies) in Season 2.

As you can see with this example, detective criticism can't narrow its focus to English crime novels, for there are lies and unpunished criminals everywhere : in Europe, Africa, America, Asia and Oceania ; in books and in films ; in pop culture as well as in classics of worldwide literature. If we want Justice to prevail, we need every single fiction, in every country around the world, to be thouroughly examined.

Apart from hunting the unpunished murderers, because an innocent was unfairly convicted or because the crime was disguised in suicide or accident, detective criticism gives itself many other tasks :

 Search for missing characters : Some characters all of a sudden vanish from the book or film in which they were playing an important part, and no one seems to notice itor much more likely, have shameful reasons to act like they don't. Pierre has already carried out investigations in Shakespeare's As you like it, and in Hitchcock's Vertigo. A small start compared to the immense task which awaits us.

 Characters on the run :

Just like real criminals, who start from scratch a new life in other countries under a false name and live there with perfect impunity, some fiction characters are very shrewd, and you can't confront them unless you prove that they are hiding in other books. As one can easily cross the border between two real countries, some characters move from a fiction to another, inside a shared universe. Fiction being multidimensional, detective criticism needs to prove first that such links exist and that the passage is possible, but, once it's done, crossexamination of various fictions linked together may shed a new light on unsolved mysteries. I've been personnally in charge of various Fairy tale filesand all I can say is that the spotless reputation of various characters really takes a hit, in particular Prince Charming's and Fairy Godmother's. Prince Charming, if you consider, as in Bill Willingham's comicbook Fables, that it's the same character that seduced all princesses (Snow White, Cinderella and the others...), it changes it all. A character who seems absolutely unimpeachable within one book becomes clearly shady if you consider all the plots in which he appears. On that matter, moreover, we need to get at the root on the talebecause, as Bruno Bettelheim already suggested, there may be political reasons why the latest versions of the story are watereddown : For instance, if we stick to Perrault's or, worse, Disney's version, we can't see what really happens in Sleeping Beauty : as the original narration tells, the prince, already married, abuses the unconscious princess, and then leaves her alone for several yearsbefore eventually going back to her and the children she gave birth to, pretending to be a foremost gentleman.

 Gumshoe criticism :

Detective criticism can also be mandated to reveal secret affairs or relationships between characters. About his science-fiction cartoon Valerian and Laureline, French illustrator Jean-Claude Mézières once answered to his frustrated fans, wondering why there were no romance between the protagonists, that the two were actually kissing a lotbut between the « boxes » of the comic-strip... Pierre has already challenged the reputation of marital fidelity of Ulyses' et Othello's wifes, Penelope and Desdemona. These investigations about the characters' private lifes can't be reduced to a voyeuristic activity : sometimes it helps a lot to understand what is really going on, and is the only way to untangle strings. For instance, unmasking the secret family links between The Beauty, the Beast and Donkey Skin (it turns out that they are all relatives) has allowed me to unveil the Lilac Fairy's machiavelic agenda, extending over three generationsand leading to her marriage, showed in Jacques Demy's 1970 film, with the King.

 Conspiracy criticism :

There's an another line research that I'd like to highlight today. I've initiated it working on Michel Houellebecq's latest book, Submission (2015) ; in France that novel has been repeatly accused of playing into the game of extremists, by disseminating the racist conspiracy theory of « the Great replacement » (stating that there's a worlwide plot to make the European white, christian race disappear by replacing it with some massive immigration). I've tried to experiment to what extend such an interpretation was possible in the novel. At the age of conspiracy theories and alternative facts, detective criticism stands as a bulwark to sort out the true from the false. An approach that may be useful, especially for spy fictions and more generally for all plots involving major manipulations and state secrets.

It's been a year now that Pierre and I founded InterCriPol, the International criminal criticism organization (ICCPO), according to the model of InterPol (ICPO), international organization which works toward federating, troughout the world, the police forces, in order to make the search for criminals more efficient. The inaugural event took place in May 2017 : we held in Le Mans University, in which I teach, the first detective criticism symposium, published online the following Qctober. On this occasion, we re-opened several cold cases about suspicious deathsamong others, Emma Bovary's, Bartleby's, or the protagonists' in Goethe's Elective Affinities. Alistair Rolls, profesor in Newcastle University, thanks to whom we're talking to you right now, presented a paper about Agatha Christie's last novel, Curtain where the murderer may not be the one the Duchess of crime pretended. During these last months, about fifty researchers in fiction, specialized in various fields of expertise, have joined our network, and we've been able to open offices abroad : in Argentina, Canada, England, Greece, Lebanon, Switzerland, USA, etc... and, thanks to Alistair, in Australia. Two weeks ago, returning to the scene of the primal crime, we organized the first ICCPO international summit, in Le Mans again, when we discussed about our action list for the future.

Meanwhile, in order to conduct joint investigations, we developed our digital lab, responding to the Code name MATRIOCHCAopen to our agents, and, more broadly, to any person eager to help our public utility mission. You'll find on our website a request form, which allows you to submit to us your doubts or hypotheses, based on the clues or inconsistencies you may have noticed. The website, ready soon, is available both in French and English (and maybe shortly in Spanish) ; you can already fill in the form, and participate in an ongoing investigation or request the opening of a new one. http://perso.univ-lemans.fr/%7Ealphar/index.html

The first collective inquiry will be about Agatha Christie's classic, And Then they were none. As you can guess, Pierre has already completed his investigations about that case, and will publish his conclusions when the time comes ; so you can just wait for the truth to be revealedor search it yourself or with us. Remember, you can't trust any narrator ; and, I hate to tell it, especially not Pierre's narrators. As far as I know him, my esteemed colleague seems to be the mostle gentle and honnest person on earth ; but I'm not sure we can say the same about his characters. As he himself confesses, in spite of the fact that he writes his essays in the first person, he can't master the various partially fictional characters who talk inside each of his worksto the extend that, when people ask him about the positions we can find in his books, he often feels like (I quote) « a court-appointed lawyer obliged to defend a criminal whose innocence he's not convinced of ». How could we not suspect Pierre's narrators to manipulate us all and hide the truth from everyonemy dear colleague included ?

We need to face it : like all others, Pierre Bayard's books are likely to be full of liersmaybe even full of unpunished murderers ; and eventually InterCriPol will have to investigate on them too. This legitimate suspicion doesn't question the outstanding skills Pierre's avatars show, and their remarkable capacity of deduction : it woudn't be the first time that an evil genious turns out to be the best to hunt machiavellian cold-blooded serial-killers that the police fail to track down (see Hannibal Lecter and Dexter Morgan) ; but, just to be sure, we'd better lead counter-inquiries on their counter-inquiries, and don't take for granted what they assert. It will take a lot of us, and a lot more, to achieve a project as ambitious. So...

INTERCRIPOL NEEDS YOU !


Intoduction

The French and most of European civil society date back to the Middle Ages. The French Monarchy unified early the nation by joining regions through wars and marriages to the early centre and this is why France was, and still is despite two Decentralization Acts (1983 and 2003) a centralized state while most European countries are decentralized where most of the decisions on education, social and cultural fields are the local governments responsibility

The article highlights first the historical background: two long-term trends have most strongly influenced the development of the non-profit sector1 throughout its millennial history: -the systematic restrictions on non-profit organizations imposed by a centralized and interventionist state during the 1791-1901 period; -the early secularization of the non-profit sector in an old Catholic country Then it outlines the evolution of civil society after WW2 to match its position now.

Then an analysis the French Civil society nowadays is made using the most recent statistics on this sector ignored by the statistical system two decades ago. The number, size and human resources of civil society organizations (CSOs) come first, then their financial resources by public and private origin and their economic contribution and social impact

The third part deals with recent trends and issues. The evolution of the French non-profit sector during the Millennium decade is outlined. Then the impact of the financial, economic and social crisis on the French third sector is highlighted.

The Historical Background

1.1. The first long-term trend: Statism, centralization and the non-profit sector.

Statism, state control is no doubt the most important feature in French history. For a millennium the central State has fought against any form of local power. The kings fought against feudal order and against urban citizens' organizations during the Middle Ages, then they fought against regional governments and religious minorities such as Protestants and Jews during the 17 th and 18 th centuries. The French Revolution is the great break in the history of the non-profit sector. It suppressed the guilds, and their social subsidiaries, the brotherhoods, as brake to free enterprise and fair competition in the Loi Le Chapelier (1791) that stipulates "No one shall be allowed to arouse in any citizen any kind of intermediate interest and to separate him from the public weal through the medium of so-called common interests". The struggle of the new Republic against the Church and its congregations had important consequences on their charitable organizations, hospitals, and schools mainly, which were either closed or nationalized while the Church's property and real estate was seized. During the 19th century the successive monarchs and republics authorized some non-profit organizations but they fought against the emerging labor movement, the Opposition's political clubs and the authorised organizations that were supposed to hide them.

The liberal laws at the end of the 19th were the end of these restrictions to the freedom of association that existed earlier in most European countries : labor unions were authorized in 1884, mutual societies in 1898 and all types of associations in 1901. The 1901 Act is the legal consecration of the freedom of association. It defines an association as a "convention according to which two or more individuals permanently put in a common knowledge or activity with another aim than sharing profit". When it has been created, an association may be declared but it is not compulsory Undeclared associations have no legal rights. Declared associations have only limited legal rights: they are not allowed to own real estate except for their operation or to receive legacies. "State-approved" associations have a full legal capacity and can own real estate and receive legacies. They have to be acknowledged by the Conseil d'Etat2 after a rather long and restrictive procedure.

The beginning of the 20th century marks a turning point: the non-profit sector is no longer illegal. New forms of social or economic concerns are becoming public interest. A corporatist social security inspired by the German one began later, in the inter-war period while a more extended welfare state emerged after WWII in France; as in many European countries. and many non-profit institutions delivering health or social services to the elderly, the disabled and the poor received grants or third party payments from the state, the local governments or the social security.

Since the 1960s, many non-profit institutions delivering health or social services to the elderly, the disabled and the poor created before began to receive grants or third party payments from the state, the local governments or the social security. This is the beginning of a government/civil society partnership in contrast with the historical fight of the state against all intermediaries. The trend is also to less state control and more attention has being paid to the tradition of decentralization of most European continental countries. The first Decentralization Act (1983) is a way to strive towards a more European political structure; decentralization is recent but it seems to have been a strong incentive to the non-profit sector development. Reducing the prerogatives of the central government to the benefit of regions, departments and local communities, decentralization have given way to a new kind of partnership between non-profit organizations and local authorities.

Another long-term trend impacting Civil society : the rivalry between State and the

Catholic church

France has always been a Catholic country: The beginning of France as a nation is considered to date back to the conversion of Clovis, the leader of the Franks, to the Roman catholic faith at the end of the 5 th century. As the "eldest daughter of the Church", the kingdom of France adopted Catholicism as the State religion. Parishes and congregations were at the origin of the most ancient charities: relief to the poor families and orphan defence and support, sick and elderly people care; schools and other education institutions. However, unlike Germany, Belgium or Netherlends, France did not develop a long lasting tradition of private action in spite of the strong influence of the Church. Instead, we observe that very early, especially during and after the 1789 Revolution, the State replaced the Church in hospitals and asylums and later in schools.

The "eldest daughter of the Church" was indeed also the first to emancipate. Since the Enlightenment period, atheism or free thinking grew more quickly than anywhere.Throughout the past, other religions were either tolerated or banned. The emigration of most Protestants after the Edit de Nantes Revocation (1685), the nearly permanent restrictions imposed to the Jews and the WWII genocide have led to the quasi-monopoly of Catholicism. The dominant religion was threatened, not by other religions, but by a growing atheism and by a widespread indifference toward religions in general. Even the recent rise of Islam due to the recent waves of immigration has challenged the French society more than the Church. The competition among Catholic and Protestant religions has not been an incentive to the development of the non-profit sector, as it has been in most European countries However, the competition between the Catholic church and an active atheism or anticlericalism was an essential feature throughout the 19th century and still has an impact on the non-profit sector. The "school war" between Catholic and "without God" schools has indeed been constant in France throughout the 19th and the 20th centuries. Though declining since the 1980s, it contrasts with the peaceful relationship between the Church and the state in the field of health and social welfare. Since the Disestablishment of the Church in 1905, religion is a private concern. This is why the French non-profit sector is secular, except for religious schools, in contrast with the more religious system of Germany, Belgium or Netherlands.

The evolution of the French civil society since WW2

After the 1945-50 period of reconstruction the non-profit sector was booming catching up his historical gap in the millennium decade. In the 1960s, this associative boom was mainly a development of organizations born in the interwar period: the non-profits created in the interwar period to advocate the rights of civil handicapped persons to benefit the same rights than the disabled by war became providers of the quasi totality of health and welfare services and residential facilities. In the same way, "social tourism" associations who provide holydays establishments and camps for the working class, created when the first paid holydays were voted in 1936, took off after the WW2. During the 1960s other non-profits enhancing the democratization of sports and culture were encouraged by the government as well as multipurpose associations disseminating a popular culture to those who had not a first opportunity at school because they had worked early : these Maisons de la Culture, Maisons des Jeunes et de la Culture were promoted by André Malraux, a novelist who was the first Minister of Culture. Before and after the cultural break of 1968 and its criticism of "consumption society", new concerns appeared and gave rise to new fields of advocacy non-profits run by the baby-boom generation: feminism, birth control, environment defence, aid to Third world countries, Human Rights defence.

The 1980 and 1990 decades are earmarked by the growing unemployment and especially the beginning of long-term unemployment. So many new WISE (work integration social enterprises) were created to cope the major social issue of social exclusion and to give insertion through work to the unemployed. Other CSOs advocated against racism and all kind of kind of gender, sexual orientation or ethnic discrimination. During these decades many professions created a "without border"associations oriented towards international emergency and development, following the pattern of Medecins sans frontieres, the famous "French doctors" awarded by the Peace Nobel Price.. During the beginning 2000 decade, cultural and sports associations mushroomed, especially those oriented towards performing arts or music and other oriented towards integration of young dropouts or long-term unemployed through sport and culture. The ageing of the population joint to specific tax exemptions or subsidies gave rise to many home care services provided mainly by the non-profit sector. The Millenium decade ended with the 2008 great recession that we will examine in the last part of this article.

The French civil society nowadays

Number, size and human resources of French Civil Society

The French non-profit sector is large and growing. It has caught up its historical gap and is now in a high rank among other European countries. In 2017, there are 1.5 million civil society organizations, operating mainly in two unequal legal status: 1.500.000 associations and 2300 foundations. Most associations are grassroots organizations run by volunteers active in the field of culture, sports and recreation mainly but advocacy and social services as well. Only 160.000 associations are managed by a professional staff. 2.000 associations have been granted by the government the "Reconnue d'Utlité Publique" label, but many other associations are public interest without having this label. Foundations are obviously a very small part of the French non-profit sector but it is growing quickly since a favourable law in 2003.

The French civil society is a major employer, with 8% f the total full-time equivalent employment (FTE) 3 . Of course associations are from far the main employer: their 185 000 establishments hire 1 852 000 employes (1 600 000 FTE), while the staff of the 2300 foundations is 86 000 (78 000 FTE). Non-profit employment is higher than the employment in construction or transport. Most of CSOs are small-sized: more than half of them hire only one or two employees and 70% hire less than five. However 11% hire 20 employees or more and the largest organizations may have several establishments to cover many regions.

The first field where non-profits are active is social services with nearly half of non-profit employment and 60% of total employment of this activity (Table 1) For social services, according to the subsidiarity principle, government does not provide a service if a NPI nearer the beneficiary can provide it, but government affords the bulk of the NPI's income. Conversely the non-profit sector provides a little percentage of health and education services (10% and 18%) because the largest part of these services is public. In the culture sports and recreation field, 40% of total employment in the non-profit sector. Employees are mainly women (69%) in the whole civil society sector, that reflects the predominance of social, health and education services where women are the major part of the staff. The percentage of part-time, seasonal and other forms of odds jobs is higher than elsewhere and CSOs welcome the greatest part of the "helped jobs" created to cope with the high long-term unemployment of the youth. The level of qualification of non-profit employees is higher than in the private business sector. Despite this fact the average monthly wage is 1650 euros, that is 20% less than the average wage of private sector. This lower wage is due to multiple factors:  the relative weight of part-time employees and of women,  the high part of social services in the non-profit sector, a field among the less paid industries, especially home care services  the fact that employees accept a lower wage in exchange of more flexibility in working hours and a job that makes sense to their lives when the employee agrees with the aim of the non-profit institution.

Besides paid employment, volunteering is a main human resource for the civil society. The official statistical system ran two specific population surveys on participation and volunteering for or through associations, in 2002 and 2010 with the same questionnaire and recently Lionel Prouteau conducted another one with the same questionnaire in 2017. These comparable surveys show a rapid increase of volunteering in the last decades from 12 million in 2002 up to 22 million volunteers in 2017 (43% of adult population). These 22 million volunteers work 2.1 billion hours, that is 1 275 000 FTE jobs, one fifth less than paid employees, but of the same order of magnitude. Volunteering is an important workforce for the CSOs with staff where the tasks of employees and volunteer are different and complementary. It is a vital workforce for the CSOs without staff that disappear if volunteers give no longer their work time. In France, as in most European countries, the main destination of volunteering is the field of culture, sports and recreation (43% of volunteer time. Then come social services (28%), advocacy organizations (16%), education and training (5%).. The characteristics of volunteers are steady in the successive population surveys. Table 2 shows the main results of the 2010 survey. Men volunteer more often than women as usually in Europe. Men volunteer mainly in sports and professional associations while women volunteer preferably in health, social services and education, highly professionalized fields. household of children over 3 is an incentive to volunteering (volunteering in sports, parentsteacher, recreation, disabled children, social tourism CSOs…). Conversely children under 3 are an obstacle to volunteering, especially for their mother. Volunteering grows quickly with the highest level of diploma of the interviewed person; when French diploma are classified according to the international levels of education . The income of the household is significant as well and correlated with education highest diploma : volunteering increases with the income of the household but even in the lower quartile of income distribution it is not so far the average ratio. Volunteering is obviously correlated with the sense of belonging to a religionthe main one, Catholicism, or the minority religions, by declining order : Islam, Protestantism or Jewand more with the degree of practice of this religion. A regular practice of religion is the most significative variable; it enhances not only volunteering for religious CSOs but for any kind of CSO. The example of their parents' volunteering influences teenagers as well, since volunteer work of the interviewed is nearly double when one parent at least was a volunteer when s/he was 18. Finally Volunteering is widespread in rural areas and large cities more than in small and middle-sized towns.

Resources, economic contribution and social impact of the French non-profit sector

The resources of CSOs employing a staff come firstly from public funding, up to 56%, while CSOs without staff are less dependent on public money (26% of their income). Public funding comes mainly from the State, the 101 departements, the 36 000 municipalities and the Social security. However this public funding whatever its formsubsidies, contracts, competitive bids -is concentrated on education, health and social services, the industries near the welfare state. Fees and dues afford 40% of the CSOs with staff and more than half of total resources for the other industries than the three cited above. The income of grassroots CSOs without staff comes firstly from fees and dues but they receive some subsidies from local authorities. Giving (households donations and corporate grants) is a minor but symbolic resource for both kind of CSOs, up to 5% of total resource in average, but some large charities rely mainly on donations.

The average budget of CSOs with staff eceeds €400 000. The total output of the French non-profit sector is about €120 billion, that is a contribution to GDP of 3.5%.. The value added by the non-profit sector exceeds that of the hotels and restaurants (2,6%) and utilities (2.5%) industries and is of the same level than the sum of agriculture and manufacture of food products and beverages industries (3.4%).

The impact of CSOs on their members or beneficiaries and on their environment is difficult to measure and partially not measurable. However CSOs provide nearly all residential facilities for emergency cases, three quarters of the private residential care, with a quasimonopole for the disabled that they help to find a fitted job as well. Their share in residential care for the elderly is less, but they provide the bulk of home care services for this part of the population. They run also one hospital bed on ten and half of day care for children under 3 4 . CSOs provide one fifth of primary and secondary education, mainly in Catholic schools and the main part of post-school culture and sports activities, holydays and summer camps, nature class-rooms… 250 000 sports clubs, ten of thousands of social and affinity clubs , thousands of theatres, museums, concert halls, performing arts or music groups, cine-clubs, libraries and multipurpose cultural facilities mushroomed in the last three decades and it is mainly through them that culture, sports and active recreation have been spread in the French population and become more democratic. CSOs run also tourist offices, transport services and social tourism facilities. They provide vocational training and reintegration and following-up of long-term unemployed and recent immigrants. The impact of the advocacy CSOs is difficult to seize but no doubt that a great part of the change of the mentality and society is partly due to them, especially tolerance attitudes towards the disabled, the sexual minorities and the immigrants.

-Recent trends and issues

3-1 The evolution of the French non-profit sector during the Millennium decade

The period from 1997 to 2007 was a time of slow economic growth with growing income inequalities and high level of unemployment (between 8 and 10%). These inequalities were attenuated by the redistribution of income through the large system of social protection and targeted public policies often using CSOs as providers of social services according to the subsidiarity principle. During this period, there was a tendency to contract out health and social services to CSOs or for-profit companies, through tenders. Therefore, competition among CSOs and standard companies was increasing.

New associations declared each year mushroomed, during this decade and after, as table 3 shows it. Even if half of these new associations die during the first five years, the other half survive, increasing the number of associations and the competition for getting public money and attract volunteers. The foundations historically rare in France have a new trend since 2003, when a law increased dramatically the tax incentives to donations 5 : foundations created by the rich are still few however corporate foundations have multiplied since 2003, as have three new forms of foundations initiated by this Law (scientific cooperation, university and community foundations).

Tableau 3 Annual creations of associations 4 After 3 all French children are at school (école maternelle from 3 to 6). These écoles maternelles , free of charge, are mainly public. Few of them are private kindergarten. 5 50% of donations to CSOs in 1996, then 60% in 2003 and 66% in 2006 were deducible from tax (and not from taxable income as it was the case before this date), with a cap that increased from 1.25% in 1996 to 20% of taxable income in 2003. The donation over the cap can be reported on the following 5 years .

For corporations and other companies the tax deduction increased from 33% up to 66% in 2003.

because of job cuts unseen at such levels since World War II. However the high level of social protection in France played as a security net for most of them. The residual new poor are lengthening lines in front of Restaurants du Coeur and other free meals providers. CSOs dealing directly with these problems, as well as many others are facing a scissor effect in coping with such new issues: resources are decreasing, because raising fees is not a solution when faced with insolvent people. The cut of European provision of food supluses to these CSOs in 2010 provoked a strong reaction and the European Community went backward. . Competition among CSOs to attract donations is becoming harder with the increasing costs and decreasing returns of fund-raising. For the French civil society the hardest part of the crisis is no doubt due to the retrenchment of public money owing to its high level of dependence to this resource. The reduction of subsidies and contracts paid by the State is no longer being compensated by the local communities because most of them also have deficits.

During the early stages of the crisis, public grants or reimbursements were delayed and sometimes suppressed, provoking at best cash flow difficulties, and at worst bankruptcies along with the dismissal of employees. It has been the case for some home care services and culture CSOs. Some associations have merged with other organizations working in the same field or area, or are pooling their infrastructure, equipment or human resources. There is no doubt that the French CSOs are too small and too numerous, and will have to concentrate in the near future. A 2014 law on social and solidarity economy gives them some support for merging and changing the legal statute. The more counter-cyclical nature of French CSOs, along with the lag with which France is responding to the crisis explain why employment in the non-profit sector continued to grow during 2008 and 2009, while it was decreasing in the rest of the economy. But, for the first time that it is measured, employment in the non-profit sector was decreasing slightly in 2010 and since this date it is increasing but slower than during the three preceding decades..

Structural differences remain and make sense:

There are still about 2300 foundations in France, even if they have multiplied since the 2003 Law, and if the new legal status of "endowment fund" (created in 2008) may make a difference in the future, as it has been successful so far. However, the assets of French foundations are low comparing to their European counterparts and associations have restrictions on their capacity to own interestbearing capital., excepting those associations that are reconnues d'utilité publique, some 2000 only but the largest ones. This limited legal capacity prevents the bulk of CSOs to smooth their financial difficulties by selling a part of their assets. Foundations have to be more selective in the choice of the projects they fund if they are grant-making. Foundations and associations have to merge, compress their overhead costs and reduce or suppress their programs if they are service providing to the less privileged part of the population. If their services are intended to the population at large (culture, sports, social tourism, training, environment…) they can raise their fees, but by this way they become more middle-class oriented, they loose their capacity to mix all categories of the society by eliminating the population with less purchasing power and they loose a part of their raison d'être.

Conclusion

The sharing of educational; health and social services provision between the state, the local governments and the civil society obviously depends on functions that have been devolved to local governments as well as on the level of externalization or subcontracting which authorities consider to be relevant. Recourse to the non-profit sector makes it possible to reduce public employment, which is very high in France. Grassroots associations provide local services adapted to the local population through voluntary work. That is the reason why the central and local governments and social security affords an almost total financing to the welfare CSOs and a partial financing to the others while the public sector regulates the CSOs and guarantees universal access to the services provided as well as the equal treatment of all users.

Associations are presently dynamic institutions in France and much appreciated by the public, whereas political institutions, unions and religious organizations are subject to mistrust or indifference. Foundations are less popular and have been discredited for a long time but they experience a spectacular growth since 2003. The whole civil society sector is currently looking for its place in the public sphereeither in close partnership with public authorities, or within the social and solidarity economy with the cooperatives and the mutual societies -Civil society organizations have demonstrated their ability to answer to the new needs of a more diverse population and they have proved their resilience during less favourable times

French

Tableau 1 Non-profit employment by activity, 31/12/2015

	Activity	Employees FTE	% of total
			Employees	FTE
				salaried
				employment
	Health	160 000	137 000	9.6%
	Social services	950 000	781 000	60.2%
	Culture and recreation	124 000	106 000	40.1%
	Education, and research	341 000	319 000	18.5%
	Other	363 000	330 000
	TOTAL	1 869 000	1 533 000	8.0%

INSEE, Tableaux harmonisés de l' Economie sociale

Tableau 2 Main results on volunteering, Population survey 2017 Characteristics of volunteers

  Table2shows nearly one half of adult population volunteer between 55 and 75 years; it is lower after 75 for evident health reasons. According to familial status, volunteering is less for single persons single and the presence in the

	Number and age of children less	None : 43% ; one or several children <3 :37% ;
	than 18	1 child >3 : 42% ; 2 children>3: 50% ;
		At least 3 children >3 ans : 48%
	Level of education	None, VI or V :36% ; IV :33% ; III :46% ; II or I
		:55%
	Net incomeof the household	Lower quartile : 37% ; 2 d Q : 42% ; 3d Q: 45% ,
		Upper Q: 50%
	Size of the residence city	Rural zone : 41% ; 2000-20 000 : 37% ;
		20 000-100 000 : 39% ; more than 100 000 : 47%
	Religious belonging and practice	Belonging and regular practice : 57%
		Belonging with no or irregular practice : 42%
		No religion, no practice : 41%
	Parents influence	At least one parent was volunteer : 62%
		No parents was volunteer : 39%
	Source : Lionel Prouteau : Enquête CRA/CSA 2017
		Volunteering ratio
		43% of the population of 18 and over
	Gender	Men : 45% ; Women : 41.5%
	Age	18-24 : 43% ; 25-34 : 42% ; 35-44 : 43% ; 45-
		54 :43% ; 55-64 :48% ; 65-74 : 48% ; 75 et plus :34%
	Personal situation	Single :41% ; Couple : 45%

In this article, the term of civil society and non-profit sector are used with the same meaning. They include all private organizations, with a free access and exit, non distributing their profit (if any) and self-governing (SNA

2008)

Conseil d'Etat is the highest court for public law conflicts

Part time jobs are more frequent in CSOs then in standard companies or public agencies. So FTE employment is used here for comparison

Source : Ministère de l'Intérieur ; Journal officiel

During this Millenium decade employment in the non-profit sector continues to grow twice faster than in the public and business sector. Employment progressed especially in social services, culture and advocacy CSOs. Volunteering went up continuously : Table 4 shows that the number of volunteers doubled within 20 years and more hereafter. So the increase of wage-earners did not crowd out volunteers and both became more professional Young volunteers prefer new associations where they have more power than in the older; Therefore the largest CSOs have some difficulties to replace their board members and, in a less measure, their operational volunteers Competition for attracting and retaining volunteers coupled with a competition for public grants and contracts. The partnerships with public powers have multiplied but the bidding competition as well. The share of public funding in non-profit sector resources has increased slowly during the first part of the decade, and was stagnating at the end of the period with a shift from central to local governments. Membership dues, fees and other commercial resources increased while individual and corporate giving stayed at the same level despite the generous tax incentives.

The impact of the financial, economic and social crisis on the French third sector

The financial crisis of course began in early 2008 in the USA and its destroying effects, especially on employment, stretched out over the years 2008 and 2009. France, along with the whole of Europe followed six months later. The crisis has created numbers of new poor