M Asmami

L Wald

A method for an accurate on-flight calibration of AVHRR data for vegetation index calculation

A significant degradation in the Advanced Very High Resolution Radiometer (AVHRR) responsitivity, on the NOAA satellite series, has occurred since the prelaunch calibration and with time since launch. This affect the index vegetation (NDVI), which is an important source of information for monitoring vegetation conditions on regional and global scales. Many studies have been carried out which use the Viewing Earth calibration approach in order to provide accurate calibration correction coefficients for the computation of the vegetation index using the visible and near-infrared spectral channels 1 and 2 of AVHRR. In the work reported here, we present an accurate, simple and promising method using high reflective clouds as a target to calibrate the vegetation index derived from AVHRR data.

The accurate calibration of the Earth observation systems in space is now recognized as a major component of the quality of an image (see e.g. [START_REF] Abel | Report of the workshop on radiometric calibration of satellite sensors of reflected solar radiation[END_REF]). The efficiency of the utilization of remote sensing data by the scientific community for the study of the global changes or by private bodies for land management and other purposes is directly related to the performances of the calibration procedures. Satellite sensors in the solar spectrum are very difficult to calibrate due to the lack of reliable onboard calibration devices. The preflight calibrations are subject to change due to the hostile environment of the sensor: outgasing, deterioration in the optronic system, variation in the spectral filter characteristics, ... Engineering responses have been brought up to this problem. Some are dealing with very sophisticated on-board calibration devices with the following drawbacks: increase in complexity, increase in weight, more demand in energy, increase in cost, possible decrease in reliability of the total system. The Earth viewing calibration approach has been recently developped as a backup solution to the possible failure or unreliability of on-board calibration devices. It is based on the knowledge of physical characteristics of some Earth phenomena as well as upon the processing of the digital imagery flowing down from the sensor itself [START_REF] Koepke | Vicarious satellite calibration in the solar spectral range by means of calculated radiances and its application to Meteosat[END_REF][START_REF] Koepke | Calibration of the VIS-channel of Meteosat 2[END_REF]Fraser, Kaufman 1986;[START_REF] Frouin | Calibration of NOAA 7 AVHRR, GOES 5 and GOES 6 VISSR/VAS solar channels[END_REF][START_REF] Price | Radiometric calibration of satellite sensors in the visible and near infrared: history and outlook[END_REF][START_REF] Price | An update on visible and near infrared calibration of satellite instruments[END_REF][START_REF] Price | Calibration comparison for the Landsat 4 and 5 multispectral scanners and thematic mappers[END_REF][START_REF] Paris | A cloudy-sky radiative transfer model suitable for calibration of satellite sensors[END_REF][START_REF] Teillet | Absolute calibration of the NOAA AVHRR sensors[END_REF][START_REF] Teillet | Three methods for the absolute calibration of the NOAA AVHRR sensors in-flight[END_REF][START_REF] Holben | NOAA-11 visible and near-IR inflight calibration[END_REF][START_REF] Brest | Radiometric calibration and monitoring of NOAA AVHRR data for ISCCP[END_REF]Kaufman, Holben 1990). It has also been proposed as an alternative to the on-board calibration devices for the monitoring of the post-launch calibration. Such an approach has the following advantages: decrease in complexity and weight, less energy consumption, decrease in cost. Each type of approach may be phased from time to time with on-ground radiometric measurements or perfectly calibrated airborne measurements [START_REF] Begni | Absolute calibration of the SPOT-1 HRV cameras[END_REF][START_REF] Hovis | Aircraft measurements for calibration of an orbiting spacecraft sensor[END_REF][START_REF] Smith | Calibration of the solar bands of the NOAA-9 AVHRR using high altitude aircraft measurements[END_REF][START_REF] Gu | Evaluation of measurement errors on the reflectance of "La Crau", the french SPOT calibration area[END_REF][START_REF] Hill | Radiometric comparison and calibration of remotely sensed data from polar orbiting earth observation satellites[END_REF]).

The radiometer AVHRR aboard the NOAA satellite series is now widely used to provide maps of vegetation index which are of high potential in vegetation studies. This index, more exactly called the normalized difference vegetation index (NDVI), is computed from the data of the visible and nearinfrared channels of AVHRR once converted into reflectances. This radiometer is poorly calibrated for these channels and if not corrected, this fact prevents the generation of time-series of accurate vegetation index, which are of the highest interest in the monitoring of the vegetation for global change, for crop survey or disease warning. Many studies have been carried out which use the Earth calibration approach in order to provide accurate calibration correction coefficients for the computation of the vegetation index. Variations in time of these coefficients may be important and should be taken into account. This communication presents an accurate but still simple method to calibrate vegetation index derived from AVHRR data.

If  i is the calibration coefficient, C 0i the deep space count for band i,  s the solar zenith angle and taking into account the solar extraterrestrial flux F 0i , the relation between the reflectance  i derived from the spectral radiance detected by the radiometer and the integer count value on a computer tape C i is:

 i =  i (C i -C 0i ) / cos ( s ) with  i =  i  / F 0i (1)
The change in the sensor calibration r i between the true calibration  i and the preflight calibration coefficient  i * given on the computer tapes supplied by the NOAA is:

r i =  i * /  i (2)
All the studies above-mentioned found that the deep space values did not change substantially with time and were close to the preflight values. Hence, the procedure for calibration correction only consists from the derivation of the coefficients  i or r i . If  i * denotes the reflectance computed using the preflight calibration coefficient:

 i
Hence from here on the calibration procedure for vegetation index consists in the accurate evaluation of the interband calibration by the means of the coefficient r 21 . Some Earth's bodies present spectrally flat reflectances in the solar spectrum and the interband calibration is done by adjusting the coefficient r 21 so that the ratio of the spectral reflectances observed over these targets is equal to 1, if other atmospheric effects can be corrected or do not change this ratio. Clouds, ocean sunglint and some deserts are such bodies. The very reflectance of the cloud is spectrally constant in most of the solar spectrum. Such a property is often used in procedures for cloud detection (see e.g. [START_REF] Saunders | An automated scheme for the removal of cloud contamination from AVHRR radiances over western Europe[END_REF][START_REF] Saunders | An improved method for detecting clear sky and cloudy radiances from AVHRR data[END_REF][START_REF] Wald | Automatic cloud screening in NOAA-AVHRR day-time imagery[END_REF]. Moreover clouds are very frequent in satellite imagery and this renders an operational procedure possible. In the present study emphasis has been put on the clouds over the ocean but work dealing with clouds over desertic areas is under progress.

At satellite level, the reflectance observed over a cloud is a function of the very reflectance of the cloud, the albedo of the underlying surface, the water vapour distribution of the environment in which the cloud is located, the clear-sky layer above the clouds in which molecular and particle scattering take place, and gaseous transmittance.

If the reflectance of the ground is large enough with respect to the optical depth of the cloud, some of the spectral variations in reflectance observed at satellite level are due to the spectral variations of the ground reflectance. Also multiple reflections between the ground surface and a highly reflective cloud base may become important. Reflectances have to be corrected for ozone absorption if necessary. The decrease of the atmospheric diffuse reflectances as the wavelength increases is somewhat balanced by the increase of the diffuse transmittance and of the cloud reflectance. Water vapour absorption is not negligible for AVHRR 2 and helps maintening the reflectance to a more or less constant level.

In order to check the feasability of an accurate interband calibration of AVHRR data over clouds, the ratio of simulated AVHRR 2 reflectances to simulated AVHRR 1 reflectances is computed for different atmospheres and cloud parameters.

Interband calibration is possible if this ratio is equal to 1 or very close to it. The radiative transfer model of Justus, Paris consists of two clear layers sandwiching a plane-parallel cloud layer. Clear-sky optical effects are treated with modified Beer -Bouguer -Lambert's law relationships and cloud optical effects are treated with the delta-Eddington method. It is fully described in [START_REF] Justus | Modeling solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere[END_REF], [START_REF] Paris | A cloudy-sky radiative transfer model suitable for calibration of satellite sensors[END_REF] and [START_REF] Justus | An operational procedure for calibrating and assessing the stability and accuracy of shortwave satellite sensors[END_REF]. This model does not provide bi-directional reflectances and the outputs are albedos. These albedos are equal to the reflectances observed at satellite level for vertical sighting of the scene.

Attention has been paid to the influence of the viewing angle on the ratio. The reflectance of the cloud may be seen as a function of the sun zenithal angle, the viewing angle and the optical thickness defined for a vertical path. As cloud thickness increases, the original direction of the incoming beam is lost due to multiple scattering events and the reflectance of the cloud tends towards isotropy. A simple model (Eq. 8 in [START_REF] King | Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds[END_REF] indicates that for the wavelengths 503, 673, 744 and 866 nm, the variation in reflectance with the viewing angle decreases as the optical thickness increases but still amounts to about 2 % for an optical thickness of 100 and across the field of view of AVHRR. However, the ratios of the reflectances at these wavelengths to the reflectance at 503 nm do not vary in an appreciable way with the viewing angle, even for weak optical depth. Hence the influence of the viewing angle on the interband calibration is negligible.

The Paris, Justus model has been ran for various combinations of the input parameters. Since this model defines an atmosphere by the ozone and water vapour contents, the air temperature at ground level and its lapse rate, the pressures at ground level and tropopause, some modifications of the typical values given for each standard atmosphere are possible. The influence of the error in measuring ozone upon the interband calibration can thus be assessed for example.

It appears that with respect to the scope of the study, the most important among the various parameters are the optical depth and the altitudes of the base and top of the cloud. Under reasonable conditions, some clouds may serve as targets for interband calibration.

In particular the clouds must have a large optical depth but too much reflective clouds (i.e. very large optical depths) are not suitable. Reasonable changes in the granulometry of the cloud have little effects on the interbande calibration for such clouds. These conclusions guide the processing of AVHRR data and also serves as an help to understand the results given by the image processing.

The AVHRR images used in this study are for the satellite NOAA-9. Three sets of data have been processed:

-data from the archives of CNES. A first processing of all the three data sets has been made in the following way:

-calibrate digital counts into reflectances according to the NOAA recommendations and to the corrections proposed by Kaufman, Holben (1990);

-reject non ocean pixels;

-reject pixels which reflectance is lower than 0.65 in channel AVHRR1. This correspond to a typical optical depth of 25 (see e.g. [START_REF] King | Determination of the scaled optical thickness of clouds from reflected solar radiation measurements[END_REF];

-compute at each pixel the ratio of the reflectance in channel 2 to reflectance in channel 1;

-compute the statistical distribution of this ratio for the scene and derive the mode, the mean and the standard deviation.

The detailed results are not given here. In table 1 In that case, the model predicts lower typical values for the ratio which are in accordance with the 0.95 found for Pacific. In that case, too, the ratio increases as the optical depth or the altitude of the cloud-top increases. The different mean value given for France (1.01) can be explained by the fact that the 15 images processed have been selected for the large amount of cloud coverage they exhibit (greater than 70 %) and hence, they may present many clouds with high albedo for which the ratio is close to 1.

As Table 1 clearly states it, an algorithm must be devised which selects within an image the clouds which are appropriate for interband calibration. This algorithm is based upon the results of the numerical simulations. These results can be expressed under the form of criteria dealing with the density of probability of the visible reflectances of clouds. They express the fact that only clouds of medium reflectivity are an ideal target. They have been completely defined by the analysis of the cloudy AVHRR images. Optical thicknesses of appropriate clouds are comprised between 10 and 70. However within a cloud field optical properties may fluctuate strongly and it is necessary to investigate the density of probability of the reflectance of clouds. For example, very bright clouds may have portions of lower reflectance, which will be considered as clouds of medium reflectivity but which are still inappropriate for the calibration procedure. Thus the probability of having very bright clouds must not be too large. Similar reasoning holds with low reflectance clouds which exhibit bright portions. If too much such clouds are present, the sample of pixels representing the medium reflectivity clouds is corrupted. Indeed the density of probability should be fairly large for medium reflectances. In the following, the density of probability is approximated by the means of the histogram of the cloudy pixels.

Therefore the selected algorithm puts conditions on the pixel populations of medium reflectances, and is the following:

-calibrate digital counts into reflectances according to the NOAA recommendations and to the corrections proposed by Kaufman, Holben (1990); -reject non ocean pixels; -reject pixels which reflectance is lower than 0.4 in channel 1. The number of remaining pixels is somewhat arbitrarily called number of cloudy pixels; -define five classes of reflectances: 0.4-0.5; 0.5-0.6; 0.6-0.7; 0.7-0.8; 0.8-0.9. Typical cloud optical thicknesses are respectively 10, 15, 20, 30, 70 and 200. Define a sixth class which lower limit is 0.4 and the higher limit, the highest reflectance a cloud can take (here arbitrarily set to 2.0 for sake of simplicity). Said differently, this class contains all the cloudy pixels; -compute the histogram of the channel 1 reflectances of the cloudy pixels by scanning the image; -the first four classes must contain at least 250 pixels when summed up, an arbitrary number ensuring some kind of statistical significance. Otherwise reject the scene; -each of the second, third and fourth classes must contain at least 10 % of the population of cloudy pixels (ensure that medium reflective clouds are significantly represented). Otherwise reject the scene; -the first class must not contain more than 20 % of the population of cloudy pixels (not too much low reflective clouds). Otherwise reject the scene; -the mean channel 1 reflectance of the cloudy pixels must not be greater than 0.7 and the lower limit of the most populated class must be less than 0.7 (not too much bright clouds). Otherwise reject the scene;

-compute at each pixel entering one of the four first classes, i. e. for which the channel reflectance is comprised between 0.4 and 0.8, the ratio of the reflectance in channel 2 to reflectance in channel 1; -compute the mean value of these ratios and the standard deviation.

Only a few scenes meet these requirements, all for NOAA-09. Unfortunately, it was not possible to process the Benin data set in that way. For the Pacific region, one scene is for 1985 (julian day 270), the other for 1986 (julian day 259). For both dates, the ratio is equal to 1.01. For the France data set (summer 1988), six scenes are suitable. The ratio ranges from 0.98 to 1.00 with a mean value equal to 0.99. Within these scenes, the suitable clouds have been identified as being a very widespread layer of low clouds, stratus and stratocumulus. If temperature observed in AVHRR4 is related to standard vertical profiles of temperature and pressure, the altitudes of the top of these clouds are mostly comprised between 2 and 3 km, and range from 1 to 6 km. Rain may be observed under these clouds.

The difference in ratio between the two data sets of Pacific and France and also within a same data set is slight. It is at maximum equal to 0.03 (from 0.98 to 1.01) and is close to the error claimed by Kaufman, Holben (1990) which is 0.05. Drifts with time in gain of the AVHRR sensor which are not accurately enough corrected may partly explain this difference.

The standard deviation within a single scene ranges between 0.01 and 0.03. Considering the data set composed of the eight mean ratios independently of location and year, its mean is 0.99 and its standard deviation is 0.01.

The effects of viewing angle have been examined by processing portions of images and have been found negligible if any. This confirms the theoretical findings.

CONCLUSION

From these results, it can be concluded that the interband calibration of AVHRR data is possible using clouds as targets. Following the procedure described above (without using the corrections proposed by Kaufman, Holben (1990) and adjusting the coefficient r 21 so that the mean ratio is equal to 1, it has been demonstrated that an accuracy (r.m.s) better than 5 % can be achieved in a simple way.

Table 1

 1 Statistical parameters for the image sets.

	are


Thierry Ranchin

come


Mireille Lefevre

Christelle Rigollier

Sylvain Cros

Michel Albuisson

Lucien Wald
email: lucien.wald@ensmp.fr

A SHORTWAVE RADIATION DATABASE TO SUPPORT GODAE-RELATED ACTIVITIES

A new method Heliosat-II was designed for the conversion of spaceborne observations made in the visible range by geostationnary satellites into SW radiation available at ocean level. It offers several improvements in operation and accuracy with respect to previous methods. Typical bias for irradiance for a month is 3 W m -2 . A database is being produced, covering the Eastern Atlantic Ocean, Europe and Africa from 1985 onwards and for each day. This database is accessible through the SoDa service on a free basis (http://soda.jrc.it).

-Introduction -The method Heliosat-II

Shortwave (SW) radiation is an element of the radiation budget, an essential component in climate studies that will be supported by the GODAE experiment. The network of stations measuring radiation is very scarce in the ocean and coastal areas. [START_REF] Perez | Comparing satellite remote sensing and ground network measurements for the production of site/time specific irradiance data[END_REF] and [START_REF] Zelenka | Effective accuracy of satellite-derived hourly irradiances[END_REF] demonstrate that a proper processing of satellite data provides better results than interpolation techniques. Several methods are available for the conversion of spaceborne observations made in the visible range by geostationnary satellites into SW radiation available at ocean level. Our concern is the series of Meteosat satellites that observe the Eastern Atlantic Ocean and the Mediterranean Sea for several years. When operated on a routine basis, many of these methods exhibit several drawbacks, one of them being the poor accuracy in irradiance [START_REF] Rigollier | The HelioClim Project: from satellite images to solar radiation maps[END_REF]. We designed a new method that is capable of processing long time-series of images acquired by the series of sensors aboard the Meteosat satellites. The method is using the same principle than several methods of proven quality: [START_REF] Pastre | Développement d'une méthode de détermination du rayonnement solaire global à partir des données Meteosat[END_REF] [START_REF] Delorme | Quick use of Wefax images from Meteosat to determine daily solar radiation in France[END_REF]. With respect to these methods, the new one, called Heliosat-II, offers several improvements in operation and accuracy. These improvements are due to several causes: • the Meteosat data are calibrated and converted into radiances [START_REF] Rigollier | The operational calibration of images taken in the visible channel of the Meteosat-series of satellites[END_REF];

[5] [6] [7] [8] [9] [10]
• we use a new database of monthly values of the atmospheric optical turbidity for clear skies available on cells of 5' of arc angle in size (SoDa Web site: http://soda.jrc.it); • we use terrain elevation TerrainBase database using the same cell size (useful for land / ocean separation); • a better modelling of the irradiation under clear-skies and overcast skies was performed [START_REF] Rigollier | On the clear sky model of the 4 th European Solar Radiation Atlas with respect to the Heliosat method[END_REF];

• more physical description of the optical processes was made possible by the calibration step; known proven models are implemented in the method; • observations of [START_REF] Taylor | Reflectance characteristics of uniform Earth and cloud surfaces derived from Nimbus 7 ERB[END_REF] were used to model the spatial distribution of radiances of the very thick clouds;

• changes in ocean albedo due to sun glitter are taken into account.

-Results -Constructing a database

We made comparisons between satellite-derived assessments and measurements performed in the world radiation network in Europe and Africa. The results depend upon several parameters; the type of data (highresolution or B2 format) and the number of pixels whose values are averaged for the comparis on with the irradiation measurements. As for the high-resolution data, assessments were compared to observations made by 60 stations in Europe for one year. The bias and root mean square error (RMSE) for the assessment of the irradiance for a month are equal to respectively 2 and 11 W m -2 on cells of 5' of arc angle in size (approx. 10 km at mid-latitude). The RMSE may decrease down to 4 W m -2 if assessments are averaged over cells of 0.5° of arc angle (see Table ). The performances are worse for the data in B2 format. This format results from a sub-sampling of the highresolution data. Briefly written, one pixel out of six original pixels is kept. Estimates at the geographical locations of the stations are therefore produced by spatial interpolation [START_REF] Lefèvre | An improved distance-based interpolation scheme[END_REF]. Comparisons were performed using 60 stations in Europe and 30 stations in Africa for the same year. The bias and RMSE are better than respectively 3 and 17 W m -2 for one month and a cell of 5'. The RMSE decreases to 9 W m -2 for cells of 0.5° in size. Data in B2 format were collected from Eumetsat. They were quality-controlled and calibrated. The method Heliosat-II is being operated to produce a database of SW downward radiation for the Eastern Atlantic Ocean, Europe and Africa from 1985 onwards and for each day. This database is accessible through the SoDa service on a free basis [START_REF] Wald | SoDa: a project for the integration and exploitation of networked solar radiation databases[END_REF]. Further, tools are available through this service to estimate longwave downward irradiance and net irradiance from the SW downward irradiance. The Heliosat-II method can be operated in real-time. When applied to Meteosat data (MOP and MSG), it produces maps of downward SW irradiance within the hour following the acquisition.

Errors (W m -2 ) in assessing SW downward irradiance for a month Hi-Res images B2 format 5' cell 0.5° cell


Basly Sami

Irgo

CONSERVATISM: AN EXPLANATION OF THE FINANCIAL CHOICES OF THE SMALL AND MEDIUM FAMILY ENTERPRISE

Introduction

The financial behavior of the family firm is a recurring topic in the literature. Hirigoyen's pioneer research (1984) reveals the specificity of the financial behaviors of industrial family medium-sized companies1 compared to those of the other categories of firms. This type of firms set out effective structures enabling them a better financial management because, on the one hand, they have a long term vision and, on the other hand, are not accountable for short-term results [START_REF] Dreux Iv | Financing Family Businesses: Alternatives to Selling out or Going Public[END_REF], especially for unquoted firms. In addition, the desire to transmit the firm to the next generations would more encourage the effective management of capital [START_REF] Gallo | Finance in Family Business[END_REF]. Precisely, the long-term horizons of the family firm make it possible to qualify its capital as "patient financial capital" [START_REF] Reynolds | Changing Mind-sets: Taking the Long View of Investment Planning[END_REF]. Indeed, this capital is invested for long periods without threat of liquidation contrary to "ordinary" financial capital which generally have a due limit corresponding to the end of investment [START_REF] Dobrzynski | Relationship investing[END_REF]. The firms having this type of capital would be able to pursue more creative and innovating strategies.

Besides, family firms, of small and medium size especially, are characterized by the lack of financial resources. [START_REF] Ward | Keeping the Family Business Healthy : How to Plan for Continuing Growth, Profitability and Family Leadership , The Jossey-Bass Management Series[END_REF] observes that poverty in capital which is necessary to finance the needs of the family and the business is a factor which inhibits growth. Two explanations justify the lack of financial resources in SME. The first makes responsible the financial markets. Indeed, investors would be very hesitant as for investing in these firms [START_REF] Lyagoubi | Medium-sized Family Businesses Financial Behaviours : An Empirical Study Over a Five Years Period of Quotation[END_REF]. Many family firms SME are not able to meet the necessary conditions, or sometimes unwilling to set up the adequate organizational answers, to facilitate their access to the external capital (Davis and ali., 2000). The second explanation is about internal obstacles. To achieve its goal of durability, the family firm tries to evolve in a more or less hermetic universe. Accordingly, external financial intervention is avoided because it can deteriorate the independence of the firm.

The small and medium family enterprise is characterized by a strong conservative attitude. How does conservative orientation influence the financial choices of the small and medium family enterprise? This theoretical contribution tries to answer this question. For Kreiser and ali. (2002), the family firm in general adopts a strategy of a conservative growth dedicating the "living company" model. Such a company consecrates long-term survival instead of financial performance as the main objective to be pursued. Accordingly, highly aware of its identity, it privileges financial conservatism and maintains a narrow control on strategic decisions in family hands. The analyzes of [START_REF] Hirigoyen | Les implications de la spécificité des comportements financiers des moyennes entreprises industrielles (M.E.I) familiales[END_REF] join this idea since the author emphasizes that the industrial family SME does not furnish a true effort to increase its market share because it is mainly preoccupied by controlled growth.

The analysis will be done in two steps. After pointing out the main dimensions of the financial conservatism of family SME: internal financing and avoidance of the external financial involvement, the analysis will explain the manifestations of conservatism and its sources. The paper will be concluded by reflections as for the strategies enabling to avoid, limit or even eliminate the impacts of conservatism.

Independence orientation and financial choices

Family SME seeks to be financially independent. The theory of resource dependency provides an explanation to this attitude: the higher the dependence on capital is, the more the potential financier would dispose of an increased influence in the decision-making within the firm (Davis and ali., 2000). As the requested resources are lower than the available resources, the suppliers have the right to exert a considerable control on those which require the supply (Pfeffer and Salancick, 1978). From the agency theory point of view, external financing in family SME, even if it does not involve agency problems between owner and managers may draw its specific batch of agency problems with outsider financier (banks and minority shareholders). Thus conflicts can occur as the outsiders' objectives relating to the control of organizational performance and, for the financial ones in particular, to the liquidity, payback and the debt interest, would be opposed to those of the insiders pursuing growth, value creation, and profitability or growing returns for the shareholder (Davis and ali., 2000). These antagonisms are exacerbated because of information asymmetry which could develop, on the one hand, between the owner (as a borrower) and the bankers and, on the other hand, between the owner (as an issuer of shares) and the purchasers of these shares.

Ultimately, the family firm appears to be resistant to the adoption of financing modes other than internal ones. Conservative and independent, it seems strongly predisposed to implement or at least to adhere to the recommendations of the theory of hierarchical financing.

The assumption of pecking order theory was developed by [START_REF] Myers | Corporate Financing and Investment Decisions when Firms Have Information that Investors Do Not Have[END_REF]. In addition, its origins go back to [START_REF] Donaldson | Corporate Debt Capacity[END_REF] who observed that firms usually abstain from issuing shares and borrow only if the investment requires more funds than the existing cash-flows. Accordingly, there would be an order of adoption of financing modes: internal financing, longterm loans and finally issue of equity. In their empirical study, [START_REF] Belletante | Accès aux marchés financiers et performances des entreprises industrielles[END_REF] corroborate this thesis when observing that the management of quoted SMEs expresses a real reserve to practice capital increase thus translating a hierarchy in their financing preferences. The research carried out by these authors shows that the call for capital stocks is the last solution considered by management of quoted SMEs (even if they could chose it more easily than the management of unquoted SME), although they believe that this resource is less expensive than debt [START_REF] Belletante | Accès aux marchés financiers et performances des entreprises industrielles[END_REF]. Particularly to the family firm, Khan (2000) observes that this entity, when obliged to make evolve its financial structure, should adopt a path formed by three phases of financing: Initially, during the first phases of growth, internal financing is privileged (1.1), then a first external phase through debt is considered (1.2.1) and finally a second external phase, through financial markets or opening of capital could be envisaged (1.2.2).

Internal financing: privileged source of financing

"The cash flow is the first accounting line which I look at, well before the benefit", declares Michel Haag, chairman of Météor, a French family firm. His father and uncle voluntarily pursued the same policy by limiting the distributed benefit. Accordingly, the firm invested each year 12% of its sales turnover while having a minimum recourse to debt. In general, the family firm favors internal financing of its activities by the retention of profits and the constitution of reserves. In addition, since the indebtedness is not always easy because of the mistrust of banks with regard to small and medium firms, most of the time it would be simpler to be self-financed.

Internal financing is a process which consists in financing the needs by means of resources drawn from the firm's activity. Thus, it enables to avoid the recourse to external funds. Two benefits are required. On the one hand, the risk of the firm does not increase contrary to debt. In addition, the firm prevents from creating conflicts of interests between shareholder (owner) and creditor. Moreover, contrary to the issue of equity, internal financing is not accompanied by a dilution effect. Finally, it has the advantage of avoiding revealing information, relating for example to future projects and investments, to investors in case of external financing.

The family firm which decides to internally finance its needs would make it at the expense of other financial decisions. There is an opposed relation between internal financing and distribution of dividends. [START_REF] Hirigoyen | Le comportement financier des moyennes entreprises industrielles familiales[END_REF][START_REF] Hirigoyen | Peut-on parler d'une politique de dividendes dans les moyennes entreprises familiales non cotées ?[END_REF] observes that the majority of unquoted family firms do not distribute dividends. Likewise, the results obtained by [START_REF]Le capitalisme familial, dans un contexte français, induitil moins de dividendes que les autres formes d'actionnariat ?[END_REF] show that family firms are less generous than other categories of firms as for the distribution of dividends2 . Thus, the percentage of capital held by management seems to have a negative impact on the rate of distribution [START_REF]Le capitalisme familial, dans un contexte français, induitil moins de dividendes que les autres formes d'actionnariat ?[END_REF]. Indeed, a generous dividend policy limits, all things being equal, internal financing and then intensifies the need for recourse to financial markets for the realization of investments (Easterbrook, 1984). In addition, it increases the financial risk of the firm [START_REF]Le capitalisme familial, dans un contexte français, induitil moins de dividendes que les autres formes d'actionnariat ?[END_REF] and limits the free cash-flows3 available to the management. In sum, the family firm is able to draw on its incomes in order to constitute internal financing. Therefore, [START_REF] Jenster | Resting on your Laurels : The Plateauing of the Ownermanager[END_REF] observe that many of these firms hold abundant liquidities which do not necessarily find relevant uses.

However, the internal constitution of financial funds is not always easy. Indeed, some family firm's specific events could involve a consumption of its financial resources. The financing of succession planning, retirement or other personal projects can push the owners to quickly harvest the "fruits" of the activity rather than to reinvest them (Ward, 1988). [START_REF] Galbraith | Divorce and the Financial Performance of Small Family Businesses : An Exploratory Study[END_REF] observes, for example, that a particular event like divorce or separation would draw a decrease of short-term financial performance which suggests that a consumption of financial resources occurs. Sometimes also, it is necessary to satisfy the shareholders' major requirement which is distribution of dividends or generally funds to family members. Ward (1988) underlines, nevertheless, that a good growth of the business may sometimes not satisfy the economic needs of a family growing in size and standards of living.

External financing

Family SME is reticent to open up financially to outside. The pecking order theory suggests that between debt and equity issue, the firm primarily chooses the first method, because its cost, i.e. interest, would be generally lower than probable dividends to be distributed. We will discuss the position of family SME with respect to debt and opening of capital.

Debt

Family SME avoids external financing and debt, in particular. [START_REF] Schulze | The Impact of Distribution of Ownership on the Use of Financial Leverage in Family Firms[END_REF] observe that approximately a third of American family firms affirm not to have debt and that the 2/3 asserts to have a "debt/ equity" ratio lower than 25%. A more recent study shows that, on average, family firms have not less debt than nonfamily firms (Hagelin and ali., 2006). However, owners with high "voting rights/capital" ratio are associated with firms with less debt (Hagelin and ali., 2006). For [START_REF] Minton | Financial Conservatism: Evidence on Capital Structure from Low Leverage Firms[END_REF], many firms adopt a conservative financial policy implying less debt than dominant financial theories predict. These firms hold a lot of available funds as well as a balanced treasury which enable them to finance their expenditures in an internal way. Thus, by relying on costly internal capital, there would be a negative impact on growth and financial performance because those firms do not benefit from debt leverage effect. Besides, for Schulze and ali. (2003), the behavior of owner-managers with respect to debt depends on the market state. They would be more favorable to debt in periods of market growth than in periods of stagnation. Moreover, [START_REF] Minton | Financial Conservatism: Evidence on Capital Structure from Low Leverage Firms[END_REF] observe that financial conservatism is temporary. Indeed, 70% of the firms they observed change posture as for debt and 50% do it in the five years. Theoretically, Schulze and ali. (2003) believe that altruism and family firm specific relationships can render owners more disposed to be involved in debt and to assume risk that debt may entail as for their individual wealth (Schulze and ali., 2003).

Opening of capital

The opening of capital has other specificities. Quotation, for example, could involve a major change of the ownership structure and thus the firm's governance because of the entry of external shareholders. Davis and ali. (2000) explain that the family firms are obliged to share governance responsibilities in order to facilitate the acquisition of critical resources while remaining sensitive to the expectations of family members. The influence of outsiders, within the board of directors, for example, would be more intense as the need for an external financing and an access to the money markets increases. For [START_REF] Ehrhardt | The Effect of IPOs on German Family-Owned Firms: Governance Changes, Ownership Structure, and Performance[END_REF], a sale of shareholders' equity by initial public offerings involves a second consequence: because of the importance of the initial returns, it causes a wealth transfer from the current owners to the new shareholders.

The opening of capital, and to the extreme the dependence4 , worry the family firms because it is likely to create an agency relation between at least two unequal poles of shareholders of different nature (Adam-Ledunois and [START_REF] Irgo | Entreprises moyennes : l'indépendance en question[END_REF]. In opposition to the process of opening of the firm by an engagement in a cooperation relation, which can affect only one activity or a function of the firm, the dependence of the family firm has a more global direct effect which limits the firm's independence as for the whole strategic decisions (Adam-Ledunois and [START_REF] Irgo | Entreprises moyennes : l'indépendance en question[END_REF]. Indeed, the capacity and the willingness of the external entity to direct the strategic decisions would involve an attenuation of firm's independence. Generally, the minority shareholder in a medium-sized company is not regarded like a simple holder of a receivable amount indexed on the firm's prosperity and does not agree to give up completely the main prerogatives of his voting rights at the profit of the majority (Adam-Ledunois and [START_REF] Irgo | Entreprises moyennes : l'indépendance en question[END_REF]. Two reasons explain this position. First of all, because the supplier of external capital (the principal) would hardly control the majority's actions (assimilated to the manager: the agent) due to weak information transparency (e.g. non-quotation of the firm, absence of a market of managers), agency costs would be important. Thus, the need for controlling as well as possible these costs, in situation of opening of capital, constitutes an explanatory factor of the necessary implication of the external entity in the decision-making process of medium-sized companies. The second reason is that none non-controlling associate in medium-sized companies can do without a minimal implication because of the weak liquidity of its investment (when there is no quotation).

Risk aversion, search of durability and the will of preservation and transmission of the family heritage underlie the logic of financial independence. Overall, a "state of mind" hostile to change is likely to be disseminated within the organization: conservatism.

Conservatism: a dominating context

Conservatism is the attachment to the choices of past (Timur, 1988). The literature treating about political conservatism5 speaks about ex post conservatism consisting in a high prudence as for the choices preferred and carried out by a group, and about ex ante conservatism consisting in an aversion testing new ideas (Dearden, Ickes and Samuelson, 1990).

For the literature about cultural identities, conservatism is a cultural dimension related to the look which societies have on the individual as an autonomous entity or as a member a social group. This cultural character would be present in societies dedicating the values of harmony and convenience in relations/groups (Johnson and Lenartowicz, 1998 6 ). Values such as moderation, social order, security, tradition and reciprocity of favors are crucial in conservative societies. The maintenance of the status quo and also of the harmonious relations not only within the group, but also within society is crucial. [START_REF] Schwartz | Beyond Individualism/Collectivism: New Cultural Dimensions of Values[END_REF] identifies three components of conservatism as a collective cultural dimension:

-harmonious working relationships and a social harmony: this characteristic implies, inter alia, the dedication of the group interest at the expense of individual's even if the group decisions go against what the individuals prefer.

-safeguarding of public image: this image is likely to be ruined when the individuals do not manage carrying out what is expected from them. Some managers try to preserve their image and thus act in a conservative way.

-security, conformism and tradition: this implies a propensity to devote an autocratic and paternalist style of management (Chui and ali. 2002) which can be reflected, for example, in the firm financing choices.

The impact of conservatism is observed and exerted on the level of firm's governance bodies (2.1), mainly the manager and the board of directors. Even if the family logic and the owner-family's expectations can explain conservative behavior, other explanations are found in past performance and in the founder's age. Often, succession constitutes a crucial opportunity to break free from organizational conservatism (2.2).

The firm's governance: resistance to change

Which effects does conservatism have on the governance of family SME? Conservative organizations and particularly family firms are characterized by the persistence and substantial power of old generations who exert a strong supervision on the ownermanager. Thus, conservative firms would have a risk-averse, a not-innovating, passive and reactive style of management (Covin, 1991). Generally, conservatism is associated with the owner-manager (founder) (2.1.1). Indeed, his role in the strategic orientation adopted by the firm is of primary importance. In addition, board of directors7 exhibits a lack of effectiveness and does not fully play his role (2.2.2).

The owner-manager

The owner-manager of the family firm plays an important role in the strategic posture of his firm. The cultural configuration of the family and the role it gives to the founder explain its disposition to change [START_REF] Jenster | Resting on your Laurels : The Plateauing of the Ownermanager[END_REF]. Indeed, when it is patriarchal, i.e. highly depending on its founder, the organization would be less inclined to change and to challenging values and family relations [START_REF] Moloktos | Change and Transition and Family Businesses[END_REF]. A patriarchal family controlling a paternalist organization is the ultimate case of figure [START_REF] Jenster | Resting on your Laurels : The Plateauing of the Ownermanager[END_REF]: being dependent to a high degree on its founder, the organization would be unable to promote change as it is not instigated by the founder. However, the founder or owner-manager may be unwilling to promote change. [START_REF] Hambrick | Executive Commitment to the Status Quo: Some Tests of its Determinants[END_REF] call this tendency to slow down the change "commitment to the status quo" (CSQ). The management believes in the permanent accuracy of current strategies or organizational behaviors [START_REF] Hambrick | Executive Commitment to the Status Quo: Some Tests of its Determinants[END_REF]. Thus, this type of ownermanager even in fact perceives only one weak need for adjustment of critical changes in the external environment.

A first explanation of the stagnancy of the family firm's owner-manager is psychological. Indeed, the founder depends on his firm in order to be defined and to assert himself. The firm is emotionally charged [START_REF] Moloktos | Change and Transition and Family Businesses[END_REF]. In addition, it seems that owner-manager's conservatism would be stronger as his psychological dependence on his deceased or retired father was high (Miller and &li., 2003). Besides, family firms' founders tend to be identified with their own vision of the organization, an inevitably subjective if not erroneous vision. They conceive the organization as an extension of their own identities, and try to maintain, sometimes within nonreasonable limits, an adequacy between the organization and their personal identity. In this sense, Ranft and O' Neill (2001) explain that the desire to maintain the organization in the founder's personally-preferred state constitute a demonstration of a narcissistic behavior and Hubris8 . The consequence of these observations is that the strong personal implication and the commitment become obstacles to opening up and search of change. This individual's attachment to the organization should increase with age. The founder tends to privilege security by avoiding the use of resources to increase growth and therefore causing the stagnation of the business. As he becomes aged, the founder becomes increasingly conservative and risk-averse.

Besides, the owner-manager's conservatism is explained by his neglect of entrepreneurial initiative. For [START_REF] Jenster | Resting on your Laurels : The Plateauing of the Ownermanager[END_REF], the family firm's founder is likely to reach a state of plateauing manifested by a situation of stagnancy which has negative impacts on the firm. He is not aggressive and proactive any more but rests on his laurels. He is less involved in the firm but more in other social activities. Ranft and O' Neill (2001) observe that founders who were successful in the development of their firms see their managerial responsibilities pushing around and limiting their entrepreneurial responsibilities as the firm grows or opens to more owners. From now on, the needs of business management consume more the time and attention of the founder who will devote less time to creative activities such as research and development. Ultimately, the "mixture" of managerial activity and entrepreneurial activity leads to a reduction of the latter. Accordingly, [START_REF] Rubenson | The Founder's Disease: A Critical Re-examination[END_REF] recommend that the style of management must change as the firm evolves. Concretely, the founder's attitude must evolve from passionate commitment to non passionate objectivity. The neglect of entrepreneurial initiative is all the more serious as the firm lives a delegation crisis and an absence of decentralization to children for example. The owner-manager will be unable or unwilling to give up his prerogatives of control but maintains responsibilities and authority in his own hands. As a consequence, the rigidity of the firm will increase.

The board of directors

The board of directors is strategic source of initiative and relevant information and also source of expertise, consulting and control since it must also correct the trajectory in case of deviating management. Schematically, it achieves two missions: a mission of control and a mission of service. According to agency theory, control relates to the appointment, remuneration, discipline and dismissal of management. It is also about adopting the initiatives suggested by the latter and evaluating their performance (Johnson and ali., 1996). The task of service includes the activities intended to improve the reputation and competitiveness of the firm: it is about consulting given to management, establishment of links with the outside and representation of the firm within the community.

However, its role within family SME needs to be moderated. [START_REF] Mustakallio | Optimal Governance in Family Firms[END_REF] argue that the role of the board of directors, measured by its composition and by the intensity of the control it exerts, would be more significant as the implication of the family members in the management decreases -suggesting at the opposite that the more the family is involved, the less decisive the role of the board would be. In general, the traditional family firm is known to have a board of directors whose members, selected according to their status and influence within the family and not according to their knowledge of the activity or industry, occupy their positions for long periods and have insufficient or inadequate professional competences. According to this description, they constitute a barrier to any attempt of change which potentially threats the stability of the firm. [START_REF] Ranft | Board Composition and High-flying Founders: Hints of Trouble to Come?[END_REF], notice that the founders of high-performing firms are even tempted to weaken deliberately the board of directors of their firms in order to maintain the status quo. The inward orientation is more corroborated in some family firms who simply do not implement such a body [START_REF] Melin | Corporate Governance in Family Firms : The Role of Influential Actors and the strategic Arena[END_REF].

The family: source or remedy against conservatism?

According to the analysis of Harris and ali. (1994), the family firm exhibits some rigidity when a change of paradigm is necessary. These rigidities are due to the fact that:

-It privileges internal succession, which is one of its objectives, and dedicates loyalty, whereas new paradigms are more likely to originate from external employees or management;

-The in-house trained successors have a weak external experience whereas new paradigms can be formed on the basis of the variety of personal experiences;

-The heir of the entrepreneur could suffer from a lack of self-confidence whereas the possibility of emergence of new paradigms generally requires a great confidence in its own judgment.

Insofar, does the family constitute the single source of organizational conservatism?

Conservatism and its multiple origins

What are the explanations of the conservative attitude adopted by the family firm? First of all, the interaction between the family and business systems is the central element which prevents the firm from quickly adapting to new conditions [START_REF] Moloktos | Change and Transition and Family Businesses[END_REF]. For [START_REF] Moloktos | Change and Transition and Family Businesses[END_REF], when the life cycles of these two systems do not evolve at the same pace, risks of crisis would be important. Because of their interdependence, transition and change are issues to be managed all at the same time by the firm and the family. An illustration of this life cycle non-parallelism could be seen in the passage from the entrepreneurial phase to the administrative phase [START_REF] Ward | Keeping the Family Business Healthy : How to Plan for Continuing Growth, Profitability and Family Leadership , The Jossey-Bass Management Series[END_REF]. Indeed, this transition is generally accompanied by a family resistance to change and to accompany the necessary development of the firm. Mustakallio and Autio (2002) advance other reasons influencing family firm's entrepreneurial orientation. First of all, the strategic and operational decision-making processes are often not separated, because the owner-family furnishes at the same time one or more directors and one or more operational managers. The influence exerted by operational decision-making on strategic decision-making combined with the fact that these processes are not explicit would lead to the avoidance of strategic initiative (Mustakallio and Autio, 2002).

In addition, potential conflicts between family members stick on the firm. Thus, the family quarrels can constitute a barrier to the implementation of development plans and then reduce the capacity of the firm to adopt a proactive posture. Moreover, because of the investment of the totality of its wealth in the firm, the family will tend to avoid risk taking. Another explanation of conservatism is about the weak liquidity of the firms' stocks which, in consequence, exempt it from external control of strategies (Mustakallio and Autio, 2002). In addition, because it is difficult to determine the right price of these stocks and to integrate the prospects for future growth into this price, the firm is not interested by its growth and that of the future benefits but rather by the increase in its balance-sheet value (Mustakallio and Autio, 2002).

The former performance plays also an important role in shaping the conservative behavior of the firm. Indeed, various studies observe that the success of the firm creates personal and organizational forces manifested by a certain form of arrogance as for competitive pressures (Ranft and O' Neill, 2001). Personal paradigms which by the past proved their efficacy constitute inhibitors to change. Thus, in spite of the evolution of the environment and performance requirements, the owner-manager could become inflexible and rigid by promoting practices and strategies resulting from past successes and avoiding decisions which can threaten his image or his economic wealth (Ward, 1997). Consequently, he perceives a weak need for adjustment even in case of critical changes in the external environment. The sociologists speak about an impregnation process which occurs during the first stages of existence of the organization. The members of the firm create and learn various routines contributing to firm's performance. However, during time these routines are transformed into habits which in their turn become traditions tending to preserve the initial conditions of success [START_REF] Kelly | Organizational Inertia and Momentum: A Dynamic Model of Organizational Change[END_REF]. In addition, a high performance coupled with the firm's age and the duration of founder's activity would exacerbate the forces leading to impregnation and traditionalism. For Mustakallio and Autio (2002), organizations lose their entrepreneurial orientation as they become aged. The more the firm is aged the more it tends to exploit its initial specific advantages. Moreover, the more it learns how to exploit its initial advantage, the less it will be inclined to explore new advantages. Thus, firms having made successfu breakthroughs tend more to privilege exploitation for longer periods and are characterized by a greater inertia [START_REF] Levinthal | The Myopia of Learning[END_REF]. In sum, giving a high importance to traditions and emotionally attached to the firm and stability of ownership, the family firm shows a greater reduction of its innovativity and proactivity than other firms.

In short, family SME exhibits a high strategic conservatism when the strategy hitherto adopted showed its effectiveness. The feeling of stability and acquired profitability constitute a barrier in front of the willingness to discover new outlets for firms' products. A strategy which functions or which functioned well in the past has all the chances to be set up as the firm's strategic paradigm. Therefore, the management is reticent to seek other performance paths and prefers to stick to the strategy which proved reliable in the past [START_REF] Jenster | Resting on your Laurels : The Plateauing of the Ownermanager[END_REF].

A characteristic of family SME is that it is prone to succession which can take place within the family circle. Since founder's and family's attitudes, paradigms and schemas could not be (or hardly) amended, succession constitutes the major opportunity to release the organization from conservatism.

Succession: a solution to conservatism

The firm's controlling generation is a variable exerting a strong influence on governance and the strategy of the family firm. In spite of the risk which the succession could provoke on the firm since it can imply the fragmentation of control and ownership and the reduction of size if the firm is divided into separate entities (Yeung, 2000), succession would have beneficial effects as for the strategic orientation. Miller and ali. (2003) identify three types of succession for the family firm: The succession can be conservative, hesitant or rebel. In this last configuration, the general strategy should undergo great changes which touch the extent of product/market portfolio and functional marketing strategies (Miller and ali., 2003). [START_REF] Jenster | Resting on your Laurels : The Plateauing of the Ownermanager[END_REF] confirm the correlation between succession and change since the adoption of change depends above all on firm's leadership: Indeed, organizational transitions would have more chance to be carried out if a leadership change occurs through firm's transfer from the founder to the successors. The new owner-manager can choose acquisitions, investment withdrawals, expansions, changes of product or market and changes of firm's general policy. However, it is possible that changes induced by this transmission will not be founded on a real will to seize new opportunities but rather by the desire of the successor to leave his own print and to flee the past. Indeed, it seems that a rebel succession is more likely to occur when there is a conflict between father and son (Miller and ali., 2003). In this case, the new owner-manager rejects the legacy of the former generation and the detachment from past and its practices is total.

It is undeniable that the competence of the successor and his leadership qualities constitute necessary conditions to the success of the strategic revival. In this vein, [START_REF] Ward | Keeping the Family Business Healthy : How to Plan for Continuing Growth, Profitability and Family Leadership , The Jossey-Bass Management Series[END_REF] stresses that if successor's qualities of leadership are weak and its competences limited then the future growth of the family firm would be inhibited [START_REF] Ward | Keeping the Family Business Healthy : How to Plan for Continuing Growth, Profitability and Family Leadership , The Jossey-Bass Management Series[END_REF]. For this author, the weak leadership is explained by security and inherited wealth which would deprive younger generations of the desire, the need and the eagerness required to be good entrepreneurs and thus to be able to assume the management responsibilities. If competence is absent or insufficient, strong leadership can thus be followed by periods of conservative strategies and organizational behavior.

Conclusion

This paper analyzes how organizational's conservatism impacts the financial choices of family SME. Through family SME main governance bodies i.e. the owner-manager and board of directors, conservatism influences decision-making and particularly financial decisions and choices.

The successful and sustainable family firm has to renew its business strategy several times as the market and competitive pressures evolve (Ward, 1988). However, the literature suggests that the family system attempts to create and maintain a cohesiveness that supports the family "paradigm" which is described as the core assumptions, beliefs, and convictions that the family holds in relation to its environment (Gudmundson and ali., 1999). Information that is not consistent with this paradigm is resisted or ignored [START_REF] Davis | Realizing the Potential of the Family Business[END_REF]. The more the family is conservative the less it works for change.

For this reason, Harris and ali. (1994) suggest that the family firm must carry out some critical tasks to strategy development: Reinterpreting the role of the entrepreneurial hero (the founder); challenging old strategic paradigms and promoting strategic development as process of continuous change.

The succession is an event which is favorable to change. Other favorable actions deserve to be mentioned. Thus, the revitalization of the board of directors may permit to struggle against conservative "temptations". In particular, outside directors' contribution would be valuable. This role can be emphasized through the two missions they must achieve: control and consulting. For the first, it is known that outside directors would be more independent and able to defend the shareholders' interests. As for the second, outsiders would prevent from the dominance of a single line of thought by challenging the assumptions underlying the firm's strategies and injecting external knowledge. Empirically, research indicates that boards with high proportion of external directors are more intensely involved in strategic decision-making [START_REF] Judge | Institutional and Strategic Choice Perpectives on Board Involvement in Strategic Decision Process[END_REF], are ready to analyze the firm's forces and weaknesses and act as a change catalyst (Muelle, 1988 9 ).

Under the assumption of attenuated conservatism, [START_REF] Corbetta | Internationalization Strategies in Family Businesses : Implications for Family and Company[END_REF] stress that, in order to receive external financing, family SME must exhibit solid and transparent financial and patrimonial structures. Accordingly, the firm must choose the "rich firm -poor family" model at the expense of the "poor firm -rich family" one. Actually, this latter implies that the family regularly withdraws money from the firm and reinvests it privately thus causing an over-estimation of debt and an inaccurate image of firm's finance. Besides, Yeung (2002) recommends to the family firm seeking financing on the money market to conform to overall governance, banking and accounting standards. In sum, it is necessary to set up adequate organizational responses to the requests and expectations of financial parties in order to increase the legitimacy of the firm and permit access to capital.

The recurring problem of the family firm definition will not be tackled. On this question, see:Allouche and Aman (2000).

Currently, in France the direction of this relation seems to be reversed since the family firms are tending to distribute more dividends because of the ISF (French Tax on Wealth).

Free Cash-Flows are the cash-flows in excess after the distribution of the funds necessary to projects having a positive net present value with a correct cost of capital.[START_REF] Jensen | Agency Costs of Free Cash Flow: Corporate Finance and Takeovers[END_REF] recommends distributing the free cashflows to shareholders in order to reduce the resources controlled by managers and thus their power. Indeed, the disadvantage of such a situation would be that management escapes from market control by carrying out nonprofitable investments.

A firm becomes dependent if its equity is controlled by a coalition of persons (e.g. a family) together with minority shareholders such as financial institutions, banks, or venture capitalists.

Derived from the Latin term conservare and applied to identify the political movements and intellectuals whose purpose is the preservation of social order and the re-establishment of a former order founded in their eyes on natural laws or transcendent data.

Cited by A.Chui and ali. (2002).

For the firms which adopt one.

The hybris is a Greek concept which can be translated by "excessiveness". It is a violent feeling inspired by passions and more particularly, by pride.

Cited by P. SHARMA and ali. (1997).


Gilles Carbou

Stéphane Labbé

STABILIZATION OF WALLS FOR NANO-WIRES OF FINITE LENGTH

Keywords: 1991 Mathematics Subject Classification. Primary: 35B35; Secondary: 35K55 Landau-Lifschitz equation, control, stabilization

come

Introduction

This paper is concerned with stabilization of wall configurations in a monodimensional model of finite length nanowire. This kind of object can be found in nano electronic devices. The three dimensional model is the following (see [START_REF] Brown | Micromagnetics[END_REF], [START_REF] Halpern | Modélisation et simulation du comportement des matériaux ferromagétiques[END_REF] and [START_REF] Wynled | Ferromagnetism. Encyclopedia of Physics[END_REF]). We denote by m : R + t × Ω x -→ R 3 the magnetic moment, defined on the ferromagnetic domain Ω. We assume that the material is saturated, so that m satisfies the constraint:

for almost every (t, x) ∈ R + × Ω, |m(t, x)| = 1.

The magnetic moment links the magnetic induction B with the magnetic field H by the relation B = H + m, where m is the extension of m by zero outside Ω. The behavior of m is governed by the Landau-Lifschitz equation:

(1) We remark that -H d is the orthogonal projection (for the L 2 inner product) onto the curl free vector fields, so that

                 ∂m ∂t = -m ∧ H e -m ∧ (m ∧ H e ),
H d (m) L 2 (R 3 ) ≤ m L 2 (Ω) .
• H a is an applied magnetic field,

• n is the outward unit normal on ∂Ω.

The effective field is derived from the micromagnetism energy:

H e = -∂ m E mic with E mic (m) = A 2 2 Ω |∇m| 2 dω + 1 2 R 3 |H d (m)| 2 dω - Ω H a • m dω.
Existence of weak solutions for (1) is established in [START_REF] Carbou | Time average in micromagnetism[END_REF], [START_REF] Labbé | Simulation numérique du comportement hyperfréquence des matériaux ferromagnétiques[END_REF] and [START_REF] Visintin | On Landau Lifschitz equation for ferromagnetism[END_REF]. Existence of strong solutions is proved in [START_REF] Carbou | Regular solutions for Landau-Lifschitz equation in a bounded domain[END_REF] and [START_REF] Carbou | Regular solutions for Landau-Lifschitz equation in R 3 . Commun[END_REF]. Numerical simulations are performed in [START_REF] Labbé | Microwave polarisability of ferrite particles with non-uniform magnetization[END_REF]. For thin domains, equivalent 2-d models are justified in [START_REF] Alouges | Néel and cross-tie wall energies for planar micromagnetic configurations[END_REF], [START_REF] Carbou | Thin layers in micromagnetism[END_REF], [START_REF] Rivière | Compactness, kinetic formulation, and entropies for a problem related to micromagnetics[END_REF]. For nanowires, 1-d models are discussed in [START_REF] Carbou | Stability for static walls in ferromagnetic nanowires[END_REF], [START_REF] Carbou | Control of travelling walls in a ferromagnetic nanowire[END_REF] and [START_REF] Sanchez | Méthodes asymptotiques en ferromagnétisme[END_REF].

In this paper we deal with ferromagnetic nanowires. We assume that the wire is a cylinder of length 2L and radius η. Taking the limit by Gamma convergence arguments when η tends to zero, we obtain an asymptotic one dimensional model (see Section 2).

After renormalization, the one dimensional wire is assimilated to the segment ] - i.e. -h d is the orthogonal projection onto the plane orthogonal to the wire. In addition we assume that we apply a magnetic field in the direction of the wire axis, and we denote by h a its renormalized intensity. Therefore we deal with the following system:

(3)

                             ∂m ∂t = -m ∧ H e -m ∧ (m ∧ H e ), H e = ∂ 2 m ∂x 2 -m 2 e 2 -m 3 e 3 + h a e 1 , ∂m ∂x (t, - L A √ 2 ) = ∂m ∂x (t, L A √ 2 ) = 0, with m : R + t ×] -L A √ 2 , L A √ 2 [-→ R 3 , |m| = 1, associated to the energy E(m) = L/A √ 2 -L/A √ 2 1 2 |∂ x m| 2 + 1 2 (|m 2 | 2 + |m 3 | 2 ) -h a m 1 dx.
In [START_REF] Carbou | Stability for static walls in ferromagnetic nanowires[END_REF] and [START_REF] Carbou | Control of travelling walls in a ferromagnetic nanowire[END_REF], we studied a similar model of infinite nanowire. After renormalization, the model was the following: the wire is assimilated to the real axis Re 1 , and the one dimensional Landau-Lifschitz equation is in this case:

(4)                  ∂m ∂t = -m ∧ H e -m ∧ (m ∧ H e ), H e = ∂ 2 m ∂x 2 -m 2 e 2 -m 3 e 3 + h a e 1 ,
with m : R + t × R x -→ R 3 , |m| = 1, This system is invariant by translation in the x variable and by rotation around the wire axis, that is, if m is a solution of (4), then for σ and τ in R, (t, x) → ρ τ (m(t, x -σ)) is another solution of (4), where ρ τ is the rotation of angle τ around the e 1 axis:

ρ τ =   1 0 0 0 cos τ -sin τ 0 sin τ cos τ   .
We dealt with wall configurations for this model, that is solutions separating a left hand side domain where the magnetization is closed to -e 1 to a right hand side domain where the magnetization is closed to +e 1 . Such solutions are described by rotations and translations of the canonical profile x → (th x, 1/ch x, 0). In [START_REF] Carbou | Stability for static walls in ferromagnetic nanowires[END_REF] and [START_REF] Carbou | Control of travelling walls in a ferromagnetic nanowire[END_REF], we proved the asymptotic stability and the controlability for these configurations.

Here, in the case of a finite wire, the situation is quite different.

Definition 1.1. We call canonical wall profile a static solution for (3) of the form

M 0 = (sin θ 0 , cos θ 0 , 0) such that θ 0 : [-L A √ 2 , L A √ 2 ] -→ [-π/2, π/2
] is a non decreasing map satisfying θ 0 (-π/2) < 0 < θ 0 (π/2) .

In the following theorem, we claim that the wall profiles exist if and only if the wire is long enough compared to the exchange length: Theorem 1.2. There exists a wall profile if and only if L A √ 2 > π. This profile is unique and is centered in the middle of the wire, that is θ 0 (0) = 0. Remark 1. In the infinite wire case, the corresponding canonical profiles are obtained taking θ 0 (x) = Arcsin th (x/2), and all its translation in the x variable (because of he invariance of (4) by translation). In the finite wire case, we only have invariance by rotation, and we loose the invariance by translation, so we have only one canonical profile.

Concerning the stability, we obtained in [START_REF] Carbou | Stability for static walls in ferromagnetic nanowires[END_REF] the stability of the wall profile. In that case, the invariance by rotation-translation induces that 0 is a double eigenvalue of the linearized version of (4) around the wall profile. In the finite wire case, the lack of invariance by translation induces that the linearized version of (3) around the canonical profile has one negative eigenvalue, and therefore we can prove that the wall profile given by Theorem 1.2 is unstable.

Theorem 1.3. Assume that L A √ 2 > π.
Let θ 0 given by Theorem 1.2. The static solution M 0 = (sin θ 0 , cos θ 0 , 0) in linearly unstable for the Landau-Lifschitz equation (3) with h a = 0.

Remark 2. This phenomenon was expected. Let us consider a small translation of the centered wall. Without energetic cost, the Landau-Lifschitz equation induces then a displacement of the wall and pushes it outside the wire. Then the magnetic moment tends to +π/2 or -π/2 (i.e the minimizers of the ferromagnetism energy E). In the case of an infinite wire, obviously, this translation cannot make the wall desappear, and we have stability.

We prove now a stabilization result. We control the system by an applied field parallel to the wire axis.

Theorem 1.4. Let L and A as in Theorem 1.2, and let M 0 = (sin θ 0 , cos θ 0 , 0) be the canonical profile given by this theorem. We consider the following control:

h(m(t, .)) = - A √ 2L L A √ 2 -L A √ 2 m 1 (t, s)ds .
Then M 0 is stable for the Landau Lifschitz equation controled with the applied field h a = h(m).

Remark 3. The control given here is quite natural: for example when the wall is translated to the right hand side, then the average of the profile first component is negative and our applied field he 1 (with h > 0) pushes the wall to the left hand side.

The paper is organized as follows. In Section 2 we justify the one dimensional model by Gamma convergence arguments. In Section 3 we prove the existence of a canonical wall profile. We address the unstability of this profile in Section 4 by linearization of the Landau-Lifschitz equation. The last section is devoted to the stabilization of the wall by a convenient applied magnetic field.

modelization

In this section, we address the justification of the one dimensional model by Γ convergence arguments. At the beginning we deal with the three-dimensional static case. The finite 3d wire is the cylinder Ω η =] -L, L[×B 2 (0, η), where B d (x, r) is the ball of radius r and center x in R d . We assume that we apply on the wire a magnetic field in the direction e 1 : H a = h a e 1 , where h a ∈ R.

The micromagnetism energy on Ω η is given by

E η (v) = A 2 2 Ωη |∇v| 2 dω + 1 2 R 3 |H d (v)| 2 dω -h a Ωη v • e 1 dω for v ∈ H 1 (Ω η ; S 2 ).
The static configurations satisfy the minimization problem:

   find u in H 1 (Ω η , S 2 ) such that E η (u) = min v∈H 1 (Ωη,S 2 ) E η (v), (5)
In order to work with a fixed domain, we introduce the following rescaling: for

v ∈ H 1 (Ω η ; S 2 ), we define v ∈ H 1 (Ω 1 ; S 2 ) by v(x, y, z) = v(x, ηy, ηz).
In addition, for v ∈ H 1 (Ω 1 ; S 2 ), we denote by H η d (v) the rescaled demagnetizing field:

H η d (v)(x, y, z) = H d (v)(x, ηy, ηz), where v ∈ H 1 (Ω η ; S 2 ) is deduced from v by v(x, y, z) = v(x, y/η, z/η).
We deal with the rescaled energy defined for v ∈ H

1 (Ω 1 ; S 2 ) by E(v) = 1 η 2 E η (v): E(v) = A 2 2 Ω1 (|∂ x v| 2 + 1 η 2 |∂ y v| 2 + 1 η 2 |∂ z v| 2 ) dω+ 1 2 R 3 |H η d (v)| 2 dω-h a Ω1 v•e 1 dω.
and the problem ( 5) is rescaled in the following equivalent problem:

   find u in H 1 (Ω 1 , S 2 ) such that E η (u) = min v∈H 1 (Ω1,S 2 ) E η (v), (6)
Proposition 1. For all (v η ) η∈R + * in H 1 (Ω 1 , S 2
), sequence of minimizers of ( 5),

(E η (v η )) η∈R + * is a bounded sequence of R. Proof. Let (v η ) η∈R +
* , be a sequence of minimizers of ( 5) in H 1 (Ω 1 , S 2 ). We denote by v η the rescaled of v η : v η (x, y, z) = v η (x, y/η, z/η).

In order to exhibit an upper bound for E η (v η ), we write:

∀η ∈ R + * , E η (v η ) ≤ E η (e 1 ), then E η (v η ) ≤ -2h a η 2 πL + 1 2 R 3 |H d (e 1 )| 2 dω ≤ 2η 2 Lπ(1 -h a )
. Then, the lower bound is obtained by canceling the positive contributions of the energy and maximizing the external contribution:

-η 2 2h a πL ≤ E η (v η ). So, we can conclude -2h a πL ≤ E η (v η ) ≤ 2Lπ(1 -h a ).
We introduce the limit energy : we define H 1 (Ω 1 ; S 2 ) by

H 1 (Ω 1 ; S 2 ) = {u ∈ H 1 (Ω 1 ; S 2 ), ∂ y u = ∂ z u = 0}.
For v ∈ H 1 (Ω 1 ; S 2 ), we set:

E(v) = A 2 2 Ω1 |∂ x v| 2 dω + 1 4 Ω1 |v 2 | 2 + |v 3 | 2 dω -h a Ω1 v • e 1 dω = A 2 π 2 ]-L,L[ |∂ x v| 2 dx + π 4 ]-L,L[ |v 2 | 2 + |v 3 | 2 dx -πh a ]-L,L[ v • e 1 dx.
The limit minimization problem is given by:

   find u in H 1 (Ω 1 , S 2 ) such that E(u) = min v∈H 1 (Ω1,S 2 ) E(v), (7)
The main result of this section is the following:

Theorem 2.1. E η gamma-converges to E in sense of H 1 (Ω 1 , R 3 )
, it is to say:

(i) (lower semi continuity) for all sequence (v η ) η∈R + * of H 1 (Ω ε,1 , S 2 ) such that lim η→0 v η = v 0 and (E η (v η )) η∈R + * bounded, the limit v 0 is an element of H 1 (Ω 1 , S 2
) such that:

lim inf η→0 E η (v η ) ≥ E(v 0 ), (ii) (construction) for all u 0 ∈ H 1 (Ω 1 , S 2
), there exists (v η ) η∈R + * , sequence of H 1 (Ω 1 , S 2 ), such that:

lim η→0 v η = u 0 , and lim sup η→0 E η (v η ) ≤ E(u 0 ).
Proof. (i) Lower semi-continuity: let (v η ) η be a sequence in H 1 (Ω 1 ; S 2 ) such that v η tends to v in H 1 (Ω 1 ) and such that there exists c such that for all η

(8) E η (v η ) ≤ c.
Extracting a subsequence, we can assume that v η tends to v almost everywhere, so |v| = 1 a.e.

Using (8), we remark that since Ω1 v η e 1 ≤ meas(Ω 1 ),

∂ y v η 2 L 2 (Ω1) + ∂ z v η 2 L 2 (Ω1) ≤ Cη 2 , so ∂ y v = ∂ z v = 0, that is v ∈ H 1 (Ω 1 ; S 2 ). From straightforward arguments, A 2 2 Ω1 |∂ x v| 2 dω -h Ω1 v • e 1 dω ≤ lim inf A 2 2 Ω1 (|∂ x v η | 2 + 1 η 2 |∂ y v η | 2 + 1 η 2 |∂ z v η | 2 ) dω -h a Ω1 v η • e 1 dω .
Concerning the demagnetizing field, the sequence (

H η d (v η )) η is bounded in L 2 (R 3
). Indeed, we denote by v η the rescalled of v η : v η (x, y, z) = v η (x, y/η, z/η). We have

H η d (v η ) L 2 (R 3 ) = 1 η H d (v η ) L 2 (R 3 ) (by change of variable in the integral) ≤ 1 η v η L 2 (R 3 ) (since -H d is an orthogonal projection in L 2 ) ≤ 1 η (Lπη 2 ) 1 2 (since |v η | = 1) ≤ √ 2π L.
So, up to the extraction of a subsequence, we can assume that H η d (v η ) H for the weak topology in L 2 (R 3 ).

We denote the coordinates of H η d (v η ) and those of H by:

H η d (v η ) =       h 1 η h 2 η h 3 η       and H =       h 1 h 2 h 3       .
Writting (2) in the rescalled coordinates, we obtain first that

η∂ x (h 1 η + v 1 η ) + ∂ y (h 2 η + v 2 η ) + ∂ z (h 3 η + v 3 η ) =
0, so taking the weak limit, we have:

(9) ∂ y (h 2 + v 2 ) + ∂ z (h 3 + v 3 ) = 0.
In addition, from the curl free condition, we obtain that

           ∂ y h 3 η -∂ z h 2 η = 0 η∂ x h 2 η -∂ y h 1 η = 0 η∂ x h 3 η -∂ z h 1 η = 0,
so, taking the weak limit we get

∂ y h 1 = ∂ z h 1 = 0 so h 1 = 0,
and

∂ y h 3 -∂ z h 2 = 0. So H is deduced from v by the relations:            ∂ y (h 2 + v 2 ) + ∂ z (h 3 + v 3 ) = 0, ∂ y h 3 -∂ z h 2 = 0 h 1 = 0,
that is for a fixed x, H(x, •) is defined from v(x, •) by the 2-d demagnetizing field.

In the following lemma we calculate the 2-d demagnetizing field induced by a constantly magnetized 2-d disk.

Lemma 2.2. Let ξ = (ξ 2 , ξ 3 ) ∈ R 2 be a fixed vector. Let V = (V 2 , V 3 ) ∈ L 2 (R 2 ; R 2 ) such that curl 2D V := ∂ y V 3 -∂ z V 2 = 0, div 2D (V + ξ) := ∂ y (V 2 + ξ 2 ) + ∂ z (V 3 + ξ 3 ) = 0,
where ξ(y, z) = ξχ B(0,1) (y, z) (we denote by χ B(0,1) the indicatrix function of B(0, 1)). Then

(10) V (y, z) =              - 1 2 ξ if (y, z) ∈ B 2 (0, 1), 1 2
1 (y 2 + z 2 ) 2   ξ 2 (y 2 -z 2 ) -2ξ 3 yz -ξ 3 (y 2 -z 2 ) -2ξ 2 yz   if (y, z) / ∈ B 2 (0, 1),
and

R 2 |V | 2 dydz = π 2 |ξ| 2 .
Proof. We have (see [START_REF] Carbou | Regularity for critical points of a nonlocal energy[END_REF] or [START_REF] Labbé | Simulation numérique du comportement hyperfréquence des matériaux ferromagnétiques[END_REF]), for X = (y, z) ∈ R 2 :

V (X) = 1 2π Y ∈∂B2(0,1) X -Y |X -Y | 2 ξ • ν(Y )dσ(Y ),
where ν is the unit normal on ∂B 2 (0, 1).

Using complex notations, we write V (X) = V 2 (X) + iV 3 (X), X = y + iz, ξ = ξ 2 + iξ 3 and we have:

V (X) = 1 2iπ z∈C(0,1) ξ + w 2 ξ 2Xw(w -1 X )
dw.

If |X| < 1, the only pole is 0, and by the residu formula, V (X) = -1 2 ξ. If |X| > 1, there are two poles, 0 and 1/ X, and we obtain V (X) = 1 2 ξ X2 which is equivalent to [START_REF] Desimone | Magnetic microstructures -a paradigm of multiscale problems[END_REF].

In addition, by a direct calculation, we obtain that:

R 2 |V (X)| 2 dydz = π 2 |ξ| 2 .
We apply the previous lemma to calculate the weak limit H:

H(x, y, z) =                        - 1 2   0 v 2 (x) v 3 (x)   if (y, z) ∈ B 2 (0, 1), 1 2 1 (y 2 + z 2 ) 2     0 v 2 (x)(y 2 -z 2 ) -2v 3 (x)yz -v 3 (y 2 -z 2 ) -2v 2 yz     if (y, z) / ∈ B 2 (0, 1),
and by a direct calculation,

R 3 |H(x, y, z)| 2 dxdydz = π 2 R (|v 2 (x)| 2 + |v 3 (x)| 2 )dx.
We remark now that since -H d is an orthogonal projection for the L 2 inner product,

R 3 |H η d (v η )| 2 dω = - Ω1 H η d (v η ) • v η dω.
Thus we can take the limit in this integral, and we obtain that:

R 3 |H η d (v η )| 2 dω -→ - Ω1 H • v dω = R 3 |H| 2 dω. So, in fact, H η d (v η ) tends to H strongly in L 2 (R 3
), and we obtain that

π 4 R (|v 2 (x)| 2 + |v 3 (x)| 2 )dx = lim inf 1 2 R 3 |H η d (v η )| 2 dω.
Therefore,

E(v) ≤ lim inf E η (v η ).
(ii) Reconstruction. Let u 0 ∈ H 1 (Ω 1 ; S 2 ). For η > 0, we define u η = u 0 . Then,

E η (u η ) = A 2 2 Ω1 |∂ x u 0 | 2 dω + 1 2 R 3 |H η d (u 0 )| 2 dω -h a Ω1 u 0 • e 1 dω.
As in the previous step, we have:

R 3 |H η d (u 0 )| 2 dω -→ 1 4 Ω1 (|u 2 0 | 2 + |u 3 0 | 2 ) dω, that is E η (u η ) -→ E(u 0 ).
This theorem gives the behavior of minimizers for the one dimensional limit case: the limit energy, for u ∈ H 1 (] -L, L[; S 2 ) is then E(u) = E(u • χ B 2 (0,1) ):

E(u) = πA 2 2 ]-L,L[ | du dx | 2 dx + π 4 ]-L,L[ (|u 2 | 2 + |u 3 | 2 ) dx -hπ ]-L,L[ v • e 1 dx,
then, in order to find out the effective field, we write (see [START_REF] Halpern | Modélisation et simulation du comportement des matériaux ferromagétiques[END_REF]):

H e = - dE du ,
it is to say

H e = πA 2 ∂ 2 u ∂x 2 - π 2 (u 2 e 2 + u 3 e 3 ) + πh e 1 .
The boundary conditions for the minimizers are:

∂u ∂x = 0 at x = -L and x = L,
since we minimize the energy without Dirichlet boundary conditions. Then, the limit dynamic system is obtain using the Landau Lifchitz combined with the new effective field H e .

We perform the following rescaling in time t = π 2 t and in space x = x/A √ 2, we denote by h a = 2h a the rescaled applied field. The resulting rescaled system is then given by (3) as we expected.

Existence of particular equilibrium states: the walls

In this section we are interested in characterizing equilibrium states of the magnetization in a finite nano-wire when h = 0, it is to say when there is no external magnetic field. In this case, we look for solutions which can be written as follows:

M 0 (x) =   sin θ 0 cos θ 0 0   , ∀x ∈] - L A √ 2 , L A √ 2 [, where θ 0 is a non decreasing map from ] -L A √ 2 , L A √ 2 [ into R satisfying θ 0 (-L A √ 2 ) < 0 < θ 0 ( L A √
2 ), and such that M 0 is a stationary solution to (3). In fact, we want M 0 to verify:

-M 0 ∧ H e -M 0 ∧ (M 0 ∧ H e ) = 0, ∀x ∈] - L A √ 2 , L A √ 2 [,
with

H e = ∂ 2 M 0 ∂x 2 -cos θ 0 e 2 ,
then, one has the following relation

-θ 0 -sin θ 0 cos θ 0 = 0, ∀x ∈] - L A √ 2 , L A √ 2 [, (11)
with, on the boundary

θ 0 (- L A √ 2 ) = θ 0 ( L A √ 2 ) = 0. ( 12
)
Setting γ 0 = θ 0 ( L A √ 2 ) (γ 0 is supposed to be non negative, that is we look for a non decreasing solution θ 0 ), we have, integrating the equation [START_REF] Halpern | Modélisation et simulation du comportement des matériaux ferromagétiques[END_REF] and using [START_REF] Kapitula | Multidimensional stability of planar travelling waves[END_REF]:

(θ 0 ) 2 + sin 2 θ 0 = sin 2 γ 0 . ( 13
)
The length of the nano-wire has to be such that the function θ 0 goes from -γ 0 to γ 0 . From formula (13), we deduce the length:

(γ 0 ) = γ0 -γ0 dθ sin 2 γ 0 -sin 2 θ
, Using the length expression computed above, we Prove now Theorem 1.2:

Proof. In order to ensure the existence of an equilibrium state as defined above, we must verify that there exists

γ 0 > 0 such that (γ 0 ) = 2 L A √ 2 .
First of all, one has

(γ 0 ) = 1 -1 γ 0 sin γ 0 dθ 1 -sin 2 θ sin 2 γ0
, we then see that lim

γ0→ π 2 (γ 0 ) = +∞.
The first remark is that γ 0 is in the interval ]0, π 2 [. Now, computing the derivative of (γ 0 ) with respect to γ 0 , we conclude that is strictly increasing on ]0, π 2 ]. Then, the comparison could be done at the limit γ 0 = 0 :

lim γ0→0 (γ 0 ) = lim γ0→0 1 -1 γ 0 sin γ 0 du 1 -sin 2 (u γ0) sin 2 γ0 = 1 -1 du √ 1 -u 2 = 2π, then, we require that 2 L A √ 2 > 2π.

Unstability of walls without applied field

Let us consider L, A and γ 0 > 0 given by Theorem 1.2. We denote by M 0 the wall profile

M 0 (x) =   sin θ 0 (x) cos θ 0 (x) 0   .
This profile is a static solution to (3) with h a = 0. We aim to calculate the linearized equation of ( 3) around M 0 . In order to calculate this linearization taking into account that we are only interested in the perturbations of M 0 satisfying the constraint |m| = 1, we describe the small perturbations of M 0 as follows:

m(t, x) = r 1 M 1 + r 2 M 2 + 1 -r 2 1 -r 2 2 M 0 ,
where the mobile frame (M 0 (x), M 1 (x), M 2 ) is given by ( 14)

M 0 (x) =   sin θ 0 (x) cos θ 0 (x) 0   , M 1 (x) =   -cos θ 0 (x) sin θ 0 (x) 0   , M 2 (x) =   0 0 1   , for x ∈] -L A √ 2 , L A √ 2 [
. The new unknown is now r = (r 1 , r 2 ) taking its values into R 2 . The profile M 0 corresponds to r = 0, so the linearized of (3) around M 0 in the variable r is the following equation:

∂ t r = J (L -cos 2 γ 0 )r 1 Lr 2 , with • J = -1 -1 1 -1 , • γ 0 = θ 0 (L/A √ 2), • Lr = -∂ xx r 2 + g 0 r 2 , where g 0 (x) = sin 2 θ 0 -(θ 0 ) 2 .
The linear unstability of the wall structure computed in the previous section is given by the study of the operator L.

Proposition 2. L is a linear, positive operator. Its first eigenvalue, 0, is associated to the eigenfunction cos θ 0 and its second eigenvalue, 1, is associated to the eigenfunction sin θ 0 .

Proof. We set: f = θ 0 tan θ 0 , then L = * , where = ∂ x + f, then, we can conclude that L is a positive operator and that cos θ 0 is in the kernel of L. Thus 0 is the first eigenvalue of L 2 . Furthermore, we have:

L(sin θ 0 ) = sin θ 0 ,
it is to say that 1 is an eigenvalue of L associated to the eigenfunction sin θ 0 . In addition we remark that sin θ 0 vanishes once in the domain, so by Sturm-Louville theorem, 1 is the second eigenvalue of L 2 .

We can now prove Theorem 1.3.

Proof. From the previous proposition, since -cos 2 γ 0 < 0, we conclude that Lcos 2 γ 0 has one strictly negative eigenvalue, then, zero is unstable for the linearized of (3) around M 0 .

Stabilization of walls

Now, we discuss the stabilization of M 0 by the command H a (the applied field). We recall that we introduced the following command:

h a = h a (m) = -A √ 2L L A √ 2 -L A √ 2 m 1 (t, s) ds.
We want to prove that the profile M 0 is a stable stationary solution for the following system:

                     ∂m ∂t = -m ∧ H e -m ∧ (m ∧ H e ), ∂m ∂x (- L A √ 2 ) = ∂m ∂x ( L A √ 2 ) = 0, H e = ∂ 2 m ∂x 2 -m 2 e 2 -m 3 e 3 + h a (m) e 1 . (15)
Proof. To start with, let us introduce M 0 = (sin θ 0 , cos θ 0 , 0) given in Theorem 1.2. We recall that θ 0 + sin θ 0 cos θ 0 = 0,

θ (-L A √ 2 ) = θ ( L A √ 2 ) = 0, Furthermore, on [-L A √ 2 , L A √ 2 ], cos 2 θ 0 -(θ 0 ) 2 = cos 2 γ 0 , where γ 0 = θ 0 ( L A √ 2 ).
Since h a (M 0 ) = 0, we remark that M 0 is a stationary solution of (15). 5.1. First step: moving frame. As in the previous section, in the spirit of [START_REF] Carbou | Stability for static walls in ferromagnetic nanowires[END_REF], we will describe the problem in the moving frame

(M 0 (x), M 1 (x), M 2 ),
given by [START_REF] Labbé | Microwave polarisability of ferrite particles with non-uniform magnetization[END_REF].

We write the solutions to [START_REF] Rivière | Compactness, kinetic formulation, and entropies for a problem related to micromagnetics[END_REF] as:

m(t, x) = M 0 (x) + r 1 (t, x)M 1 (x) + r 2 (t, x)M 2 + ν(r(t, x))M 0 (x)
where

ν(r) = 1 -r 2 1 -r 2 2 -1.
In this moving frame, we get

H e = (g 0 + a 0 )M 0 + (a 1 + ã1 )M 1 + a 2 M 2 with g 0 = sin 2 θ 0 -(θ 0 ) 2 a 0 = 2θ 0 ∂ x r 1 + r 1 θ 0 -cos θ 0 sin θ 0 r 1 + ∂ xx ν -ν(θ 0 ) 2 + ν sin 2 θ 0 + h(r) sin θ 0 , a 1 = ∂ xx r 1 + cos 2 γ 0 r 1 -S(r 1 ) cos θ 0 , ã1 = -2∂ x νθ 0 -ϕ(r) cos θ 0 , a 2 = ∂ xx r 2 .
where

S(r 1 ) = A √ 2L L A √ 2 -L A √ 2 r 1 cos θ 0 ds, ϕ(r) = - A √ 2L L A √ 2 -L A √ 2
ν(r) sin θ 0 ds, and h(r) = S(r 1 ) + ϕ(r).

Using these coordinates in the Landau-Lifchitz equation [START_REF] Kapitula | Multidimensional stability of planar travelling waves[END_REF] and projecting on M 1 and M 2 yield:

(16) ∂ t r = Λr + F (x, r, ∂ x r, ∂ xx r),
where

Λr = -1 -1 1 -1 L1 (r 1 ) L(r 2 ) with L = -∂ xx + g 0 , L1 = L -cos 2 γ 0 + cos θ 0 S,
and where the non linear part F is given by

F (x, r, ∂ x r, ∂ xx r) = F 1 (r)∂ xx r + F 2 (r)(∂ x r, ∂ x r) + F 3 (x, r)∂ x r + F 4 (r) + F 5 (r), with • F 1 ∈ C ∞ (R 2 ; M 2 (R)): F 1 (r) =   (r 2 ) 2 + (ν(r)) 2 + 2ν(r) ν(r) -r 1 r 2 r 1 r 2 -ν(r) (r 1 ) 2 + (ν(r)) 2 + 2ν(r)   +   -r 1 -r 2 -ν(r)r 1 r 1 -r 2 -ν(r)r 2   ν (r), • F 2 ∈ C ∞ (R 2 ; L 2 (R 2 ; R 2 )): F 2 (r)(∂ x r, ∂ x r) =   -r 1 -r 2 -ν(r)r 1 r 1 -r 2 -ν(r)r 2   ν (r)(∂ x r, ∂ x r), • F 3 ∈ C ∞ ([-L A √ 2 , L A √ 2 ] × R 2 ; L(R 2 ; R 2 )): F 3 (x, r)∂ x r = 2θ 0 (x)   -r 1 -r 2 -ν(r)r 1 r 1 -r 2 -ν(r)r 2   ∂ x r 1 -2θ 0 (x)   (r 2 ) 2 + (ν(r)) 2 + 2ν(r) + 1 r 1 r 2 -ν(r) -1   ν (r)(∂ x r), • F 4 (r) ∈ C ∞ ([-L A √ 2 , L A √ 2 ] × R 2 ; R 2 ): F 4 (r) = (cos 2 γ 0 r 1 )   (r 2 ) 2 + (ν(r)) 2 + 2ν(r) r 1 r 2 -ν(r)   -g 0 ν(r)   r 1 r 2   +((θ 0 -cos θ 0 sin θ 0 )r 1 + g 0 ν(r))   -r 1 -r 2 -ν(r)r 1 r 1 -r 2 -ν(r)r 2   .
• F 5 is given by

F 5 (r) = -h(r) cos θ 0   (r 2 ) 2 + (ν(r)) 2 + 2ν(r) r 1 r 2 -ν(r)   -ϕ(r) cos θ 0   1 -1   +h(r) sin θ 0   -r 1 -r 2 -ν(r)r 1 r 1 -r 2 -ν(r)r 2   .
Remark 4. The command h makes the linear part of ( 16) positive. Indeed, on one hand, we know that L ≥ 0 with Ker L = R cos θ 0 . On the other hand, L1 = L + cos θ 0 S -cos 2 γ 0 . On R cos θ 0 , L1 cos θ 0 = α 0 cos θ 0 with (17)

α 0 = A √ 2L L A √ 2 -L A √ 2 cos 2 θ 0 (x)dx -cos 2 γ 0 > 0 since for x ∈] -L A √ 2 , L A √
2 [, cos 2 θ 0 (x) > cos 2 γ 0 , and on (cos θ 0 ) ⊥ , L1 | (cos θ0) ⊥ = L -cos 2 θ 0 ≥ 1 -cos 2 θ 0 since L ≥ 1 on (cos θ 0 ) ⊥ (see Proposition 2). 5.2. Second step. 5.2.1. New unknown. The Landau Lifschitz equation ( 3) is invariant by rotation around the wire axis, so we can build a one parameter family of static solutions for [START_REF] Sanchez | Méthodes asymptotiques en ferromagnétisme[END_REF].

For τ ∈ R let us introduce the rotation around the x-axis given by

ρ τ =   1 0 0 0 cos τ -sin τ 0 sin τ cos τ   .
We denote M τ (x) = ρ τ M 0 (x), and R τ its projection on the moving frame:

R τ (x) = M τ (x).M 1 (x) M τ (x).M 2 (x) = cos θ 0 (x) sin θ 0 (x)(cos τ -1) cos θ 0 (x) sin τ .
Since M τ is solution to [START_REF] Rivière | Compactness, kinetic formulation, and entropies for a problem related to micromagnetics[END_REF], R τ is a static solution of ( 16), that is

(18) ΛR θ + F (x, R θ , ∂ x R θ , ∂ xx R θ ) = 0.
Now in order to avoid the problems due to the zero eigenvalue of Λ, we describe r in the new coordinates (τ, σ, W ) defined by

(19) r(t, x) = R τ (t) (x) + σ(t) cos θ 0 (x) 0 + W (t, x),
where (τ, σ) ∈ C 1 (R + ; R 2 ) and W ∈ C 1 (R + ; H 2 ) such that both coordinates of W are in (cos θ 0 ) ⊥ . Indeed, as in [START_REF] Kapitula | Multidimensional stability of planar travelling waves[END_REF], we can prove that for a given r ∈ H

2 (-L A √ 2 , L A √
2 ) in a neighbourhood of 0, there exists a unique (τ, σ, W

) ∈ R × R × W such that ρ = R θ + σ cos θ 0 0 + W, where W is the set of the W = W 1 W 2 ∈ H 2 satisfying the homogeneous Neu- man Boundary condition ∂W ∂x (- L A √ 2 ) = ∂W ∂x ( L A √ 2
) = 0 and the orthogonality condition:

L A √ 2 -L A √ 2 W 1 (x) cos θ 0 (x)dx = L A √ 2 -L A √ 2 W 2 (x) cos θ 0 (x)dx = 0.
Remark 5. On (cos θ 0 ) ⊥ , the operators L and L -cos 2 θ 0 are non negative, so we introduce the following norms on W, respectively equivalent to the H 2 and the H 3 norms:

W H 2 = L 1 2 (L -cos 2 γ 0 ) 1 2 W 1 2 L 2 + LW 2 2 L 2 1 2 , W H 3 = L 1 2 (L -cos 2 γ 0 )W 1 2 L 2 + L 3 2 W 2 2 L 2 1 2 .
5.2.2. Equation ( 16) with these unknowns. Plugging the decomposition (19) in ( 16) and using ( 18) yield the following equivalent form for the Landau-Lifschitz equation in the coordinates (τ, σ, W ), valid for small perturbations of M 0 : (20)

dτ dt R τ + dσ dt cos θ 0 0 + ∂ t W = σ -α 0 cos θ 0 α 0 cos θ 0 + ΛW + G(x, τ, σ, W, ∂ x W, ∂ xx W ).
Let us describe the different terms in this equation: Concerning the time derivatives, we have

∂ t r = dτ dt R τ (x) + dσ dt cos θ 0 0 + ∂ t W, where R τ (x) = -sin τ cos θ 0 sin θ 0 cos τ cos θ 0 .
Concerning the linear part, we have Λr = ΛR τ + -α 0 cos θ 0 α 0 cos θ 0 σ + Λ(W ) (see Remark 2 for the definition of α 0 ), Concerning the non linear part, we have

F (x, r, ∂ x r, ∂ xx r) = F (x, R τ , ∂ x R τ , ∂ xx R τ ) + G(x, τ, σ, W, ∂ x W, ∂ xx W ).
The last term G is obtained from F with the Taylor formula around R τ :

G(x, τ, σ, W, ∂ x W, ∂ xx W ) = F 1 (r)(∂ xx w) + F1 (r)(w)(∂ xx R τ ) +2F 2 (r)(∂ x w, ∂ x R τ ) + F 2 (r)(∂ x w, ∂ x w) + F2 (r)(w)(∂ x R τ , ∂ x R τ ) +F 3 (x, r)(∂ x w) + F3 (x, r)(w)∂ x R τ ) + F4 (r)(w) +F 5 (r),
where

• w = W + cos θ 0 0 σ, • for i = 1..4, Fi (r) ∈ L(R 2 ; R 2 ) is given by Fi (r) = 1 0 F i (R τ + sw)ds.
From straightforward calculations, we see that: Therefore with all these estimates, if r H 2 is sufficiently small, we have (21)

F 1 (r) = O(|r| 2 ), F1 (r) = O(|r| 2 ) F 2 (r) = O(|r|), F2 (r) = O(1)
G L 2 ≤ K r L ∞ [|σ| + W H 2 ], ∂ x G L 2 ≤ K r L ∞ [|σ| + W H 3 ]
(see Remark 5.) 5.2.3. Separation of the unknowns. In order to separate the unknowns τ , σ and W , we first take the inner product of (20) with cos θ 0 0 and with 0 cos θ 0 . We remark that both ∂ t W and ΛW are orthogonal to these vectors, so that we obtain:

ρ 0 σ = -α 0 ρ 0 σ + G 1 , g τ τ = α 0 ρ 0 σ + G 2 ,
where

ρ 0 = L A √ 2 -L A √ 2 cos 2 θ 0 , G 1 = L A √ 2 -L A √ 2 G(x, τ, σ, W, ∂ x W, ∂ xx W ) • cos θ 0 0 dx, g τ = L A √ 2 -L A √ 2 R τ (x) • 0 cos θ 0 dx, G 2 = L A √ 2 -L A √ 2 G(x, τ, σ, W, ∂ x W, ∂ xx W ) • 0 cos θ 0 dx.
By subtraction, we have:

∂ t W = ΛW + G with G = G - G 2 g τ R τ - G 1 ρ 0 cos θ 0 0 + α 0 σ 0 cos θ 0 - ρ 0 g τ R τ .
5.2.4. equivalent formulation. We are then led to study the following equation

(22) τ = α 0 ρ 0 g τ σ + 1 g τ G 2 ,
together with the system coupling:

(23) σ = -α 0 σ + 1 ρ 0 G 1 , with (24)
∂ t W = -(L -cos 2 γ 0 )W 1 -LW 2 (L -cos 2 γ 0 )W 1 -LW 2 + G.
From (21), with Remark 5 we have

(25) 1 ρ 0 G 1 (t) ≤ K r L ∞ |σ(t)| + W (t) H 3 .
In addition, since g τ = ρ 0 + O(τ 2 ), since R τ = 0 cos θ 0 + O(τ ), we get:

(26) G H 1 ≤ K r L ∞ |σ(t)| + W (t) H 3 .

5.3.

Last step: Variational estimates. Taking the inner product of (24) with L(L -cos 2 γ 0 )W 1 L 2 W 2 , we obtain using (26) that

d dt W 2 H 2 + W 2 H 3 ≤ K r L ∞ |σ(t)| + W (t) H 3 .
Remark 6. In order to avoid the boundary terms when we integrate by part, we perform the previous estimates on the Galerkin approximation of (24) built on the eigenvectors of L with the homogeneous Neumann boundary condition.

Multiplying (23) by σ, (25) yields:

d dt σ 2 + α 0 σ 2 ≤ K r L ∞ |σ(t)| 2 + W (t) 2 H 3 .
Summing up the previous estimates, we have:

d dt |σ(t)| 2 + W (t) 2 H 2 + α 0 |σ(t)| 2 + W (t) 2 H 3 (1 -K r L ∞ ) ≤ 0.
So there exists δ > 0 such that while r L ∞ ≤

H e = A 2

 2 ∆m + H d (m) + H a , ∂m ∂n = 0 on ∂Ω, where • A 2 is the exchange constant, depending on the material, • the demagnetizing field H d (m) is characterized by (2) curl H d (m) = 0 and div (H d (m) + m) = 0 in R 3 .

2 [

 2 [START_REF] Alouges | Néel and cross-tie wall energies for planar micromagnetic configurations[END_REF] , where (e 1 , e 2 , e 3 ) is the canonical basis of R 3 . The magnetic moment m is then defined on R +t ×] -. The equivalent demagnetizing field (after renormalization) is given byh d (m) = -m 2 e 2 -m 3 e 3 ,

2 2LFigure 1 .

 21 Figure 1. Domain Ω η .

F 3 (

 3 x, r) = O(|r|), F3 (x, r) = O(1)F4 (r)O(|r|).• Concerning F 5 , on one hand we remark thatS(r 1 ) = S(W 1 ) + σS(cos θ 0 ) (since S(R τ ) = 0.)On the other hand, ϕ(R τ ) = 0 andϕ(r) = -ε 2L τ + sw)ds = O(|r|).

The authors were partially supported by the ANR project SICOMAF ("SImulation et COntrôle des MAtériaux Ferromagnétiques").

Now, with equation ( 22), we have

Therefore, if τ 0 , σ 0 and W 0 H 2 are small enough, we remain in the domain { r L ∞ ≤ 1 2K } and all the previous estimates remain valid for all times. This concludes the proof of Theorem 1.4.