E Diler

Dr S Rioual
email: rioual@univ-brest.fr

B Lescop

D Thierry

B Rouvellou

Stability of ZnMgO oxide in a weak alkaline solution

Keywords: Zinc oxide, ZnMgO Solid solution, Dissolution, Photo-dissolution

Zinc oxide (ZnO) is a chemical compound of great interest used, for example, as photocatalyst in the purification of wastewater or polluted air. However, neither dissolution, nor photo-dissolution of ZnO is negligible: indeed, both processes reduce significantly the efficiency of photocatalysis and then lead to a secondary pollution by free Zn 2+ . In the present study, the stability of ZnMgO thin films in weak alkaline solution is investigated. We demonstrates that the replacement of Zn 2+ ion with Mg 2+ ion results in the production of a Zn 0.84 Mg 0.16 O solid solution, whose stability is higher than that of the ZnO sample. This alloy, thus, constitutes an alternative to the use of ZnO in photocatalysis applications. To gain more insights into the higher resistance of such alloys to the dissolution process, X-Ray photoelectron spectroscopy measurements were performed. They highlighted the role of OH group adsorption in the experimentally observed enhancement of ZnMgO stability.

Introduction

Zinc oxide (ZnO) is an n-type wide-band gap semiconductor with a high exciton binding energy. Because of its interesting multi-functionality, over the last few years it was the subject of extensive studies for applications in gas sensors, solar cells, photodetectors, light emitting diodes, magnetic storage and photocatalysis. In some cases, ZnO has displayed a greater efficiency in photocalysis than TiO 2 , which is generally considered as the most suitable photocatalyst [START_REF] Gouvea | Semiconductor-assisted photocatalytic degradation of reactive dyes in aqueous solution[END_REF]. With respect to the other photocatalytic materials, the emitting properties of ZnO are unique and enable one to create catalytic systems to simultaneously detect and destroy environmental organic contaminants [START_REF] Kamat | A Sense and Shoot Approach for Photocatalytic Degradation of Organic Contaminants in Water[END_REF]. However, its use in photocatalysis is limited by two parameters: Firtsly, as it is activated under UV irradiation because of its large band gap of 3.3 eV (λ≈380 nm), only about 5% of the photons received by natural light illumination can be used in the photocatalytic reactions. The second drawback is its instability in aqueous media due to photocorrosion by UV irradiation. Investigations by Spathis and Poulios have dealt with the corrosion of Zn and ZnO [START_REF] Spathis | The corrosion and photocorrosion of zinc and zinc oxide coatings[END_REF]: in particular, these authors observed a quicker formation of corrosion products under illumination. This behaviour was explained by the formation of an electron-hole pair produced by photoexcitation and responsible for an enhancement of dissolution. Recently, a more extensive study about ZnO stability [START_REF] Han | Potential dissolution and photo-dissolution of ZnO thin films[END_REF] demonstrated that the greater stability displayed by ZnO near pH=9.9 was explained by the formation of Zn(OH) 2 hydroxide, which limits Zn dissolution. At such a pH level, the low solubility of Zn(OH) 2 protects ZnO from further reaction with OH -and dissolution.

Different methods were, recently, initiated to suppress the photodissolution process.

They include organic coating [START_REF] Comparelli | UVinduced photocatalytic degradation of azo dyes by organic-capped ZnO nanocrystals immobilized onto substrates[END_REF] and surface hybridisation with carbon [START_REF] Li | Formation of Surface Complex Leading to Efficient Visible Photocatalytic Activity and Improvement of Photostabilty of ZnO[END_REF]. Instead of protecting only the surface that can be oxidised and thus damaged, a solution would be to elaborate compounds with higher photocatalytic properties and better photocorrosion resistance. Recently, doping of ZnO with Mg 2+ proved to lead to an enhancement of photocatalytic efficiency [START_REF] Qiu | Origin of the Enhanced Photocatalytic Activities of Semiconductors: A Case Study of ZnO Doped with Mg2+[END_REF]. Many methods have been developed so far in order to produce ZnMgO thin films, e.g. pulsed laser deposition [START_REF] Yang | Ultraviolet photoconductive detector based on epitaxial Mg0. 34Zn0. 66O thin_films[END_REF], RF and DC sputtering [START_REF] Li | The effects of thermal annealing on properties of MgxZn1-xO films by sputtering[END_REF][START_REF] Li | The influence of ambient conditions on properties of MgxZn1-xO films by sputtering[END_REF] and spray pyrolysis [START_REF] Reddy | Growth and physical behaviour of Zn1-xMgxO films[END_REF][START_REF] Yoshino | Structural, optical and electrical characterization of undoped ZnMgO film grown by spray pyrolysis method[END_REF][START_REF] Zhang | Structural and optical properties of Zn1-xMgxO thin films deposited by ultrasonic spray pyrolysis[END_REF]. In the present work, we use the spray pyrolysis method, a versatile low cost technique which enables deposition onto large areas. However, in contrast to ZnO, no investigation about the stability of this oxide is available in the literature. Here, focus is on the stability of Zn (1-x) Mg (x) O oxides in a weak alkaline solution, where ZnO displays a relatively low dissolution rate, and may, thus, be used as a photocatalyst. After a physico-chemical characterisation of the samples under study, we will show in the present work that ZnMgO is more stable than ZnO, and then discuss the origin of this behaviour.

Experimental details

Preparation

To produce Zn (1-x) Mg (x) O, Zinc acetate dihydrate (Zn(CH 3 COO) 2 .2H 2 O, Alpha Aesar) and Magnesium acetate tetrahydrate (Mg(CH 3 COO) 2 .4H 2 O, Alpha Aesar) were dissolved in demineralised water with magnesium at different concentrations varying from 0% to 16%. For the enhancement of the film growth and to avoid zinc hydroxide precipitation in solution, the pH was adjusted to pH= 4.5 by addition of a small quantity of acetic acid [START_REF] Paraguay | Growth, structure and optical characterization of high quality ZnO thin films obtained by spray pyrolysis[END_REF]. The final solution was pulverised using compressed air at a distance of 50 cm on the 500°C-heated substrate (conductive tin-doped indium oxide or glass). The flow rate was 3.3 ml/min.

Characterisation

The sample film thickness was determined by: i) weighing the samples before and after deposition and ii) inductively-coupled plasma atomic emission (ICP). The glass substrates were weighed 5 times before and after deposition with a precision weighing balance, Genius series from Sartorius. The thickness of the thin film was estimated from the density of the zinc oxide (d = 5.606 g/cm 3 ). For ICP measurements, ZnO films were let to dissolve for 30 min in a saturated solution of glycine placed in an ultrasonic bath, and then analysed.

The crystalline structure of the different films was analysed in (θ-2θ) configuration by X-ray diffraction (XRD) with a Brucker apparatus using the CuKα radiation. The compositions of the different films were determined by depth-profile X-Ray photoelectron spectroscopy (XPS) according to the procedure described elsewhere [START_REF] Salou | Initial oxidation of polycrystalline Permalloy surface[END_REF]. XPS apparatus was also used to measure the secondary electron energy spectra by polarising the sample by 10 V with respect to the analyser. The surface topography was studied by Atomic Force Microscopy (AFM) (Veeco, NanoScope V) in tapping mode using commercial P doped Si tip (Veeco, RTESP). The optical properties in the 200-1100 nm wavelength range were analyzed using UV-visible spectrometry with a Jasco apparatus. The resistivity was measured by the four-point probe technique in the air and in a vacuum chamber at about 10 -1 Pa.

The stability of the thin films was investigated in a solution of borax at pH 8.4 to keep the pH at a fixed value, between 7 and 10, where dissolution is expected to be rather low.

After immersion of the thin film in this solution, the open circuit potential (OCP) was

measured with a leak-free saturated Ag/AgCl electrode as reference and a multi channel data logger (EPC8 from NKE). These measurements were made over several days under the direct illumination of the sun.

Results and discussion

Characterisation of the Zn 1-x Mg x O alloys

Characterisation of the Zn 1-x Mg x O films for magnesium concentration, x, in the range 0 to 16% led to thickness of 200 ± 10 nm. Figure 1 displays the transmittance measurements.

For the direct band gap materials, the relationship between the absorption coefficient, α, and the photon energy, hν, can be expressed as follows [START_REF] Moss | Semiconductor Opto-electronics[END_REF]: αhν = A(hν-E g ) 1/2 , where E g is the band gap. The determination of the band gap of the Zn 1-x Mg x O oxide films showed a linear variation from 3.22 to 3.42 eV with x. This relatively weak increase with Mg concentration should not lead to a deterioration of the photocatalytic efficiency of ZnMgO with respect of ZnO.

The XRD patterns of the films displayed an intense (002) peak at about 34.5° with additional small (100) and ( 101) features located at 31.8° and 36.3° respectively. These lines (not shown here) are indicative of the polycrystalline würtzite structure of the films. This crystallographic structure was still observed even at the 16% Mg concentration with no MgOassociated pattern. This is in agreement with other studies that have shown the presence of würtzite structure up to 40% Mg concentration [START_REF] Bendersky | Microstructural study of epitaxial ZnMgO composition spreads[END_REF]. The higher (002) peak displayed on Fig. 2 indicates that the preferential orientation for the film is along the c-axis. One should also note the Mg-doping-induced shift of this peak towards high angles: it corresponds to a lattice compression caused by the replacement of Zn 2+ by Mg 2+ with its associated decrease in the ion radius from 0.60 to 0.57 Å. This behaviour consistent with band gap results is characteristic of a solid solution and has been previously reported on ZnMgO thin films produced by either spray pyrolysis [START_REF] Yoshino | Structural, optical and electrical characterization of undoped ZnMgO film grown by spray pyrolysis method[END_REF][START_REF] Zhang | Structural and optical properties of Zn1-xMgxO thin films deposited by ultrasonic spray pyrolysis[END_REF] or pulse laser deposition (PLD) [START_REF] Lu | The effects of thermal annealing on ZnO thin films grown by pulsed laser deposition[END_REF]. It is worth noting that this shift was neither visible on XRD patterns from powders [START_REF] Kim | Evolution of local structures in polycrystalline Zn1-xMgxO (0=x=0.15) studied by Raman spectroscopy and synchrotron x-ray pair-distribution-function analysis[END_REF] nor predicted by recent electronic structure calculations [START_REF] Maznichenko | Structural phase transitions and fundamental band gaps of MgxZn1-xO alloys from first principles[END_REF]. This disagreement is likely explained by the number of defects within the samples and, thus, by the method in use for their preparation.

The composition of Zn 1-x Mg x O for x = 0 and 0.16 was determined by XPS on using an AlKα source. Zn(2p), Mg(1s) and O(1s) spectra were measured and fitted by CasaXPS software to calculate their integral. By correcting these values by the Scofield factors and the inelastic mean free paths of the detected photoelectrons, the stoechiometry of the compounds was determined. The dopant concentrations in the solution and within the film were found to be in good agreement. Figure 3 displays the O(1s) spectrum of ZnO and Zn 0.84 Mg 0.16 O. For ZnO, the two components contributing to the spectrum at 529.9 and 531.8 eV correspond to ZnO and OH groups at the surface, respectively. The latter feature was previously observed on PLD-produced ZnO and explained by the dissociation of adsorbed water at the surface [START_REF] Lu | The effects of thermal annealing on ZnO thin films grown by pulsed laser deposition[END_REF]. Thorough investigations by Almeida et al. [21] showed that this process is governed by the number of defects, e.g. oxygen vacancies, within the oxide under study.

The electrical resistivity was measured by the following equation:

2 ln π ρ t R s = (1)
Where ρ is the resistivity, R S is the sheet resistance and t the sample thickness. The resistivity values of undoped ZnO measured in the air and at 10 -1 Pa were 820 and 4 Ωcm, respectively. Such a vacuum dependence of the resistivity can be explained by the capture of the electrical charges at the surface by hydroxyl groups as explained in Ref. [START_REF] Ahn | Origin of the slow photoresponse in an individual sol-gel synthesized ZnO nanowire[END_REF]. It, thus, confirms the importance of water adsorption in this study. The comparison of the resistivity values of ZnO and of Zn 0.84 Mg 0.16 O samples measured in air showed an increase from 820 to 9.10 5 Ωcm. For intermediate Mg doping ratios (x=0, 0.02, 0.04, 0.08, 0.16), one observes a continuous increase of the resistivity with the x doping. This increase is in agreement with Yoshino et al. [START_REF] Yoshino | Structural, optical and electrical characterization of undoped ZnMgO film grown by spray pyrolysis method[END_REF] and is consistent with the stronger contribution of the OH peak observed in our XPS measurements for Zn 0.84 Mg 0.16 O with respect to ZnO. One should note that, in Ref. [START_REF] Lu | The effects of thermal annealing on ZnO thin films grown by pulsed laser deposition[END_REF], the significant contribution of the OH peak in XPS spectrum was explained by the porous character of the PLD-produced ZnO. In the present study, as shown in figure 4 by AFM measurements, the strong water adsorption can be explained by the microstructure of the surface composed of numerous grains of about 60 nm: OH adsorption is, thus, promoted by the large surface/bulk ratio. Both surfaces display similar roughness and microstructure leading to the hypothesis that the chemical nature of the thin films plays also an important role in the OH adsorption.

Stability of the Mg-based solid solution

The recent use of open-circuit potential (OCP) measurement as an indicator of the damage of photoactive materials in aqueous solutions [START_REF] Straka | Open-circuit potential as an indicator of damage of atomic layer deposited TiO2 on AISI 304 stainless steel[END_REF] To investigate the modification of the thin films after borax immersion, XPS measurements were carried out. An argon ion sputtering at 1.5 keV with a sputtering rate of about 0.5 nm/minute was used to investigate the chemical environment of thin film elements throughout its thickness. Figure 6(a) illustrates the O(1s) spectra obtained for different sputtering times. The difference observed in the shape of the outer surface layer O(1s) spectrum measured at t=0 min with respect to the other ones is explained by strong carbon contamination of the surface. In the inner part of the film, two components at about 530.2 and 532.4 eV contribute to the O(1s) spectrum. As discussed above, they are explained by Zn-O-Zn bonds and OH species, respectively. A comparison of figures 6(a) and 3 shows that, after a stay in borax, the contribution of OH species is: i) definitely stronger than previously and ii) not limited to the surface region.

For both films, the OH peak presents a decreasing amplitude from the surface to the inner part of the film where the ZnO and ZnMgO oxides are localised. As shown in figure 6(b), this decrease of the OH contribution is confirmed by an increase of the Zn and Mg contents in the inner part of the film with respect to the surface. The measurements of the secondary electrons spectra of ZnO presented in figure 6(c) show an increase of the work function of the surface with respect to the inner part of the film; this is indicative of the changes in the composition of the film observed in figure 6(a) and 6(b). The observed high value of the work function induced by the hydroxide formation in the first nanometer range reduces the charge transfer rate at the liquid/solid interface and then prevents the dissolution of the underlying substrate. Furthermore, in figure 6(a), as found already before immersion of the samples in borax, the inclusion of Mg in ZnO leads to an enhancement of hydroxide concentration. This observation could be correlated to the slight increase of the work function of ZnMgO with respect to ZnO as displayed in figure 6(c).

Immersion of a sample in borax for a certain time permits the development of several reactions at its surface. Among them, the creation of Zn(OH) 2 film on ZnO is suspected to represent the predominant reaction in the dark [START_REF] Han | Potential dissolution and photo-dissolution of ZnO thin films[END_REF]. This is explained by the low solubility of Zn(OH) 2 in a weak alkaline solution. This scenario is confirmed by our XPS results. The results presented above deal with the relative stability of ZnMgO with respect to ZnO. OCP measurements (Fig. 5) showed indeed that ZnMgO thin films were definitely more stable than ZnO in a weak alkaline solution. However, the exact role of Mg in the protection of ZnMgO against corrosion is a rather complex phenomenon and needs further clarification. The higher band gap of Mg-doped ZnO compared to ZnO (Fig. 1) suggests the implication of a lower number of photons in the creation of electron/hole pair with its consecutive reduction of photocorrosion in the doped Mg alloy. However, the amplitude of OCP being almost alike for the two kinds of materials, the numbers of holes created by irradiation are rather similar. It is likely that this parameter is not the main factor at the origin of the greater stability of the Mgdoped ZnO. The second difference observed in figure 3 and6(a) between both materials concerns the OH enrichment of the ZnMgO surface with respect to ZnO after immersion in the borax solution. To further discuss this point, we present in Table 1 the OH/O ratio measured from the decomposition of the O(1s) spectra for both ZnO and ZnMgO samples; and finally the increase of the OH content in the last column. As shown in this table, this increase is definitely higher than the increase of 16% due to magnesium doping. This is explained by the complex nature of the hydroxide layer which is not a simple mixture of Zn(OH) 2 and Mg(OH) 2 but which should contains hydroxides such as Zn 1-x Mg x (OH) 2 . The different nature of these hydroxides combined with their higher content are thus at the origin of the observed better stability of ZnMgO. The strong OH adsorption may result from the stabilization of the ZnMgO surface. A theoretical investigation of the chemistry of ZnMgO oxide surface with respect to ZnO would certainly give further information on this point.

Finally, one should note that the observed enhancement of OH amount and the resulting greater stability of ZnMgO, are both in agreement with studies made on the corrosion resistance of ZnMg metallic alloys, where their greater resistance to corrosion compared to Zn was explained by the formation of hydroxide [START_REF] Hosking | Corrosion resistance of zincmagnesium coated steel[END_REF][START_REF] Volovitch | Understanding corrosion via corrosion product characterization: I. Case study of the role of Mg alloying in Zn-Mg coating on steel[END_REF].

Conclusion

Solid solutions of Zn 1-x Mg x O with x lying within 0 and 16% were produced by spray pyrolysis. Increase of Mg content led to both the keeping of the würtzite structure for ZnO and the decrease of the lattice parameter. OCP measurements allowed us to investigate the stability of the samples in borax at pH = 8.4, and ZnMgO proved to be more stable than ZnO. This behaviour is attributable to the important contributions of OH species in the very first nanometers just below the surface and proves that ZnMgO is a better candidate for catalytic applications than ZnO.

  drove us to make such measurements of ZnO and Zn 0.84 Mg 0.16 O in a borax solution at pH = 8.4 and under natural night/day cycles. The results presented in Fig. 5(a) are about samples exposed to cyclic illuminations, and those in Fig. 5(b) deal with samples placed under dark conditions for 5 days and then exposed to natural night/day cycles. For short exposure times, both figures show equilibrium between the Fermi level and the redox couple of the electrolyte at about 35 mV vs Ag/AgCl. Light exposure causes a shift of OCP by about 330 mV towards negative values. This can be explained by the light-induced creation of the electron-hole pairs and the subsequent creation of an additional photovoltage. The latter is due to the separation of both charges and to the low recombination rate within the material. In the dark, the OCP comes back immediately to its initial magnitude (35 mV vs Ag/AgCl). After few days in the borax solution, Figure 5(a) clearly shows a reduction of ZnO-OCP amplitude with respect to that of ZnMgO. Figure 5(b) presents the same behaviour, but with a delay of about 5 days since the samples was exposed later to the light, showing that the integrated time of illumination is the key parameter of the ZnO degradation. These results give thus evidence of both: i) the relevance of ZnO films degradation by light irradiation and ii) the greater stability of ZnMgO with respect to ZnO.

Figures 3 . 1

 31 Figures

3

 3

Fig. 2 .Fig. 3 .

 23 Fig. 2. XRD patterns of ZnO and Zn (1-x) Mg (x) O (002) main peak with Mg doping for x = 0,

Fig. 4 .

 4 Fig. 4. AFM pictures of ZnO (a) and Zn 0.84 Mg 0.16 O (b).

Fig. 5 .Fig. 6 .

 56 Fig. 5. Open circuit potential of ZnO (full line) and Zn 0.84 Mg 0.16 O (dot line) under daylight

  αhν² versus hν, optical band gap evolution with Mg doping for Zn (1-x) Mg (x) O (x = 0, 0.02, 0.04, 0.08 and 0.16).

		6	ZnO
	(αhν)² (eV² cm -2	2 4	Zn 0.98 Mg 0.02 O Zn 0.92 Mg 0.08 O Zn 0.84 Mg 0.16 O Zn 0.96 Mg 0.04 O
		0	.2	3.3	3.4	3.5
			Energy (eV)
			ZnO
	Normalized Intensity		Zn 0.92 Mg 0.08 O Zn 0.84 Mg 0.16 O
			34.0	34.5		35.0
			2θ angle (deg)

) Fig. 1.

Acknowledgment

The authors acknowledge C. Calvarin for his contribution in the design of the spray pyrolysis device and P. Elies for AFM measurements.

represents the Mg(1s) spectra; (c) secondary electrons spectra measurements of ZnO (dashed line) and ZnMgO (full line).


Salima Bouayad-Agha

Lionel Vedrine
email: lionel.vedrine@cemagref.fr

Estimation strategies for Spatial Dynamic Panel using GMM. A new approach to the convergence issue of European regions 1

Keywords: Spatial Econometrics, Dynamic Panel Model, GMM, Regional Convergence JEL Classification: C21, C23, O52, R11, R15

The recent progress in spatial econometrics offers a number of estimators for models that treat spatial dependence explicitly but techniques for handling spatial dependence appear to be essentially confined to crosssectional studies. Despite the fact that dynamic panel models have been the object of recent important developments, the econometric analysis of spatial and dynamic panel models is at an early stage.

In this context, the aim of this paper is to review the different methods used in the literature, for such space-time data sets, and to suggest two strategies in order to estimate spatial dynamic panel using GMM. The first is to extend the moment restrictions of Arellano and Bond estimator for spatial autoregressive dynamic panel. The second allows for spatial dependence in the error process in calculating the optimal weighting matrix at the second step of the Arellano and Bond estimator.

These strategies are carried out to estimate the convergence of European regions during the last 25 years.

Introduction

While econometric analysis of dynamic panel models is now fairly standard [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF][START_REF] Arellano | Another look at the instrumental variable estimation of errorcomponents models[END_REF][START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] and spatial econometric literature is well documented either on cross-sections [START_REF] Anselin | Testing for spatial error autocorrelation in the presence of endogenous regressors[END_REF][START_REF] Anselin | Spatial Econometrics[END_REF] or on static panel [START_REF] Elhorst | Specification and Estimation of Spatial Panel Data Models[END_REF], econometric analysis of spatial and dynamic panel models is still at an early stage of development. Modelling space-time data is quite complex and several problems must be tackled: obviously, leaving correlation on each spatial unit over time and spatial dependence on the spatial units at each date aside would lead to misspecification but the main difficulty is that we have a priori no reason to believe that one problem is more important than the other is.

Empirically, there are several examples where the presence of a dynamic process with spatial dependence might occur. This is the case with the regional economic convergence/divergence issue.

Many convergence studies stems from neoclassical growth model [START_REF] Solow | A contribution to the theory of economic[END_REF][START_REF] Swan | ECONOMIC GROWTH and CAPITAL ACCUMULATION[END_REF] and most of them rely on a beta convergence model: regions (countries) do not have the same structural characteristics and thus converge towards different steady states relative per capita incomes. Factor mobility, trade relations and geographical spillovers (like technology spreading) can matter to understand how economic development of a region is likely to be influenced by neighbouring regions.

Various recent convergence studies have found evidence for model misspecifications if spatial interdependencies of regional growth are ignored [START_REF] Arbia | Does evidence on regional economic convergence depend on the estimation strategy? Outcomes from analysis of a set of NUTS2 regions[END_REF]. Working on a dynamic panel specification, [START_REF] Badinger | Regional convergence in the European Union, 1985-99: a spatial dynamic panel analysis[END_REF] applied a GMM estimator to spatially filtered variables; [START_REF] Elhorst | Unconditional Maximum Likelihood Estimation of Linear and Log-Linear Dynamic Models for Spatial Panels[END_REF] suggests a maximum likelihood estimation of models that are dynamic both in space and time for regional analysis; [START_REF] Piras | Convergence in per-capita GDP across EU NUTS2 regions using panel data models extended to spatial autocorrelation effects[END_REF] extend panel-data models with spatial error autocorrelation for a convergence analysis of EU regions. More precisely the main argument of applying the extended GMM in a spatial context is that it corrects for the endogeneity of the spatial lagged dependent variable and other potentially endogenous explanatory variables.

In this context, the aim of this paper is to review the different methods used in the literature, for such space-time data sets, and to suggest an estimation strategy that consider together the dynamic specification and the spatial dimension of the panel with an illustration on regional conditional convergence framework. Estimating a per capita GDP conditional convergence model that incorporates an explicit consideration of spatial dependence effects, we rely on dynamic panel generalised method of moment estimations that control for endogeneity, variable omission and spatial dependence problems.

The reminder of the paper is organised as follows. In the next section, we address the econometric issues on spatial dynamic panel model and develop an empirical estimation strategy. Section 3 reports the main results of our empirical estimation strategy for the convergence approach on panel data.

Section 4 concludes and suggests areas for further research.

Econometric issues on spatial dynamic panel data model

Dynamic panel model

A general dynamic panel model can be described as follows:

(1)

( ) T t ,N , i y y t i i t i it , , 2 ; 1 1 , 1 , K K = = < + + = - α ν η α where t i
y , is an observation on some series for individual i in period t, 1 ,t i y is the observation on the same series for the same individual in the previous period and

t i i , ν η +
is the usual "error components" decomposition of the error term which allows for unobserved heterogeneity ( it ε hereafter). These specific effects are supposed to be stochastic, which here implies that they are necessarily correlated with the lagged dependent variable. With the additional assumption that t i, ν is serially uncorrelated, the Ordinary Least Squares (OLS) estimator of α in the levels equations is inconsistent. Standard results for omitted variable bias indicate that, at least in large samples, the OLS levels estimator is biased upwards. The Within estimator for panel data that allows to control for time invariant characteristics lead to substantial gains in robustness (compared with cross-section estimator) but it is not without costs if one do not pay attention to the dynamic of adjustment. Standard results for omitted variable bias indicate that, at least in large samples, the Within Group estimator is biased downward [START_REF] Nickell | Biases in dynamic models with fixed effects[END_REF]. Thus, we might hope that a candidate consistent estimator will lie between the OLS and Within estimates. The most widely-used alternative strategy is to difference the model to eliminate the fixed effects and then applies the GMM using a set of appropriate instruments to address the correlation between the differenced lagged dependent variable and the induced MA(1) error term (see [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF]. The GMM estimator is a two-step estimator. In the first step, an initial positive semi definite weight matrix is used to obtain consistent estimates of the parameters. Given this consistent estimates, a weight matrix can be constructed that is consistent for the efficient weight matrix, which is used for the asymptotically efficient two-step estimates. It is common practice to use the inverse of the moment matrix of the instruments as an initial weight matrix. More precisely the first step in the estimation procedure consists in eliminating the individual effects via a first difference transformation (GMM-DIFF) or a forward orthogonal deviation [START_REF] Arellano | Another look at the instrumental variable estimation of errorcomponents models[END_REF].

The dynamic panel-data specification has become frequent in growth convergence empirical studies.

Since the inclusion of the time-lagged dependent variable in the equation might lead to inconsistent estimates, instrumental variable estimators are needed. A commonly employed estimation procedure to estimate the parameters in a dynamic panel data model with unobserved individual specific heterogeneity is to transform the model into first differences. Sequential moment conditions are then used where lagged levels of the variables are instruments for the endogenous differences and the parameters estimated by GMM (see [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF]adopted by Caselli et al., 1996 in the growth context). The first step in the estimation procedure consists in eliminating the individual effects via a first difference transformation (GMM-DIFF) or a forward orthogonal deviation 2 (as suggested by [START_REF] Arellano | Another look at the instrumental variable estimation of errorcomponents models[END_REF]. Assuming that the error terms it ε are serially uncorrelated, the lagged difference of the endogenous variable is instrumented with the lagged difference of the endogenous

            = y y 0 0 0 . . . . . 0 0 y y 0 0 0 0 0 2 - iT i1 i2 i1 1 L L L L L L L L i i y Z
2 First differencing and forward orthogonal deviation transformation involve the same procedure (similar instrument matrix) to estimate dynamic panel data specification.

where rows correspond to the first-differenced equations for period t=3,4,…,T for individual i and exploits the moments conditions (

)

0 = ∆ it is y E ε for for s=1,…,T-2 and t=2,…,T .
The overall validity of instruments can be checked by a Sargan-Hansen test of over-identifying restrictions.

This GMM estimator extends in a natural way to autoregressive-distributed lag models of the form:

(2)

( ) T t ,N , i x y y t i i t i t i it , , 2 ; 1 1 , , 1 , K K = = < + + + = - α ν η β α
where it x can be a vector of current and lagged values of additional explanatory variables. Different moment conditions will be available depending on what is assumed about the correlation between it x and the two components of the error term [START_REF] Bond | Dynamic panel data models: a guide to micro data methods and practice[END_REF]:

• If it x is strictly exogenous, ( ) 0 = ∆ it is x E ε for s=1,…,T and t=2,…,T. • If it x is weakly endogenous, ( ) 0 = ∆ it is x E ε for s=1,…,T-1 and t=2,…,T. • If it x is strictly endogenous, ( ) 0 = ∆ it is x E ε for s=1,…,T-2 and t=2,…,T.
Unfortunately in the case of persistent data and for a small number of time series observations, lagged levels are only weak instruments for subsequent first differences and the GMM-DIFF can have very poor finite sample properties in terms of bias, and precision. [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] proposed the system GMM estimator (GMM-SYS) that combines moment conditions for the model in first differences with moment conditions for the model in levels. It has been shown to correspond to the model in levels, with lagged differences of the endogenous variables as instruments. [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] argued that the GMM-SYS estimator performs better than the GMM-DIF estimator because the instruments in the model remain good predictors for the endogenous variables in this model even when the series are very persistent.

The choice between the two estimators (GMM-DIFF, GMM-SYS) is conducted according to a common test of over identifying restrictions: the validity of the additional instruments used by GMM-SYS for the level equation can easily be tested using difference Hansen tests.

Spatial dynamic panel model

Spatial econometric methods deal with the incorporation of spatial interaction and spatial structure into regression analysis. Spatial error model and spatial lag model are two different approaches to address the issue of spatial dependencies [START_REF] Anselin | Spatial Econometrics[END_REF]. The first one is a nuisance form of spatial dependence and incorporates a spatial autoregressive process in the error term (as in time series analysis). Ignoring this spatial autocorrelation may lead to inefficient estimates. The second method incorporates spatial dependence in the form of a spatial lag variable, often considered as a spatial autoregression model. This specification is more appropriate for explicitly specifying as completely as possible the impact of nearby observations on the dependent variable so that explicit inferences can be made for spatially lagged variables [START_REF] Beck | Space Is More than Geography: Using Spatial Econometrics in the Study of Political Economy[END_REF][START_REF] Blonigen | FDI in space: Spatial autoregressive relationships in foreign direct investment[END_REF].

The design of these two specifications relies on a spatial weight matrix W describing the spatial arrangement of the cross-section units. As it is standard in spatial econometrics, for ease of interpretation, this weighting matrix W is row-standardized so that each row in W sums to one.

Throughout the rest of the paper, we assume that W remains constant over time.

First, we consider a "time-space simultaneous" model [START_REF] Anselin | Spatial panel econometrics[END_REF]:

(3) ( ) T t ,N , i Wy x y y t i i t i t i t i it , , 2 ; 1 1 , , , 1 , K K = = < + + + + = - α ν η ρ β α where t i
Wy , is a first order spatial lag. The coefficient ρ stands for the intensity of spatial effects.

Thus when ρ =0, the model remains as section 2.1.

According to [START_REF] Anselin | Spatial Econometrics[END_REF] and [START_REF] Abreu | Space and Growth: a Survey of Empirical Evidence and Methods[END_REF], including a spatially lagged dependent variable causes simultaneity and endogeneity problems, which in turn means that this variable must be treated as endogenous and thus proper estimation methods must account for this endogeneity. There are only a limited number of available estimators for Dynamic Spatial lag Model on Panel Data. Assuming all explanatory variables are exogenous beside the spatial lag term3 , dynamic spatial lag models are usually estimated using the GMM estimator (see for example, [START_REF] Madriaga | FDI in Chinese Cities: Spillovers and Impact on Growth[END_REF]. Spatial lag term and autoregressive term are instrumented by their lagged values, by lagged values of the dependent variables as well as by spatially weighted explanatory variables.

If the spatial lag is strictly endogenous, the moment restrictions of section 2.1 are not sufficient to provide an unbiased and consistent estimation. However, a natural way to estimate (3) is to assume further moment restrictions of 2.1 by considering t i Wy , as endogenous variable:

( ) 0 = ∆ it is Wy E ε for t=2,…,T and s=1,…,T-2.
Moreover, we can use spatially weighted explanatory variables t i Wx , to instrument the spatial lag term. In other terms, we identify the exogenous part of the spatial lag variability by a spatially weighted model.

The validity of this strategy requires the following moment restrictions:

If it Wx is strictly exogenous, ( ) 0 = ∆ it it Wx E ε for t=2,…,T.
We suggest checking the robustness of this extended GMM estimator by proceeding successively to cross sections, Least Squares Dummy Variables (LSDV), GMM dynamic panel and GMM spatial dynamic panel estimation. Spatial error model in dynamic panel context is more complicated to estimate than spatial lag model. Nevertheless, [START_REF] Elhorst | Unconditional Maximum Likelihood Estimation of Linear and Log-Linear Dynamic Models for Spatial Panels[END_REF] propose maximum likelihood estimation for dynamic panel including spatial error autocorrelation. In a static panel model, a consistent estimator can be obtained from a set of moment conditions as demonstrated in [START_REF] Kapoor | Panel data models with spatially correlated error components[END_REF]. [START_REF] Mutl | Dynamic panel data models with spatially correlated disturbances[END_REF] extends this approach to a dynamic panel case.

We consider the following specification:

(4) T t ,N , i x y y t i t i t i it , , 2 ; 1 1 , , 1 , K K = = < + + = - α ε β α ( ) 1 , , < + + = ρ ν η ε ρ ε t i i t i it W
where the error term consists of the spatially lag simultaneous error and the previous error component ( )

,t i i ν η + .
We assume that is t i, ν independently distributed, with a constant variance 2 ν σ .

We estimate a GMM estimator using the same moment as the Arellano Bond estimator but with optimal weight matrix that allows for spatial error autocorrelation. According to [START_REF] Mutl | Dynamic panel data models with spatially correlated disturbances[END_REF], this weighting matrix is constructed using first step estimation of 2 ν σ , 2 1 σ and ρ based on the [START_REF] Kapoor | Panel data models with spatially correlated error components[END_REF] spatial GM estimator4 .

Empirical application: β-convergence of European regions

Modelling β-convergence on spatial dynamic panel specification

Many convergence studies stems from traditional neoclassical growth model [START_REF] Solow | A contribution to the theory of economic[END_REF][START_REF] Swan | ECONOMIC GROWTH and CAPITAL ACCUMULATION[END_REF]. Following the "model based" specification of [START_REF] Mankiw | A Contribution to the Empirics of Economic Growth[END_REF] [START_REF] Barro | Convergence[END_REF], we rely on a β-convergence model where the GDP per capita (hereafter GDP p.c.) growth depends not only on the initial GDP level, but also on other conditioning variables. Regions do not have the same structural characteristics and thus converge towards different steady-state income levels. The further a region finds itself from its own steady state, the faster its growth rate will be. In this case, convergence is conditional: economies converge towards the same growth rate. Accordingly, the following general model is in line with the empirical growth literature:

(5)

t i g t i g g k t i k k i i t i Z X y y y , , , , , 0 , 1 0 0 , , ln ln ε π β β β + + + + =         ∑ ∑ where ( ) T t n i y t i , 1 ; , , 1 , K K = =
is the GDP p.c. of region i at time t=; 0 β is an intercept term, 1 β is the convergence coefficient, X is a set of k explanatory variables related to the growth model of Solow [START_REF] Solow | A contribution to the theory of economic[END_REF]) and Z is a set of variables that may affect the convergence process but are not directly related to the model of [START_REF] Solow | A contribution to the theory of economic[END_REF] [START_REF] Durlauf | Handbook of Economic Growth[END_REF].

We assume that the variables sets in Z i and X i are independent (see [START_REF] Mankiw | A Contribution to the Empirics of Economic Growth[END_REF] for a discussion on this point). It is now possible to extend [START_REF] Mankiw | A Contribution to the Empirics of Economic Growth[END_REF] model towards a dynamic panel-data specification of the convergence model, current output is regressed on lagged output and control variables [START_REF] Durlauf | Handbook of Economic Growth[END_REF]):

(6) ( )

t i t i t i t i t i t i t i t i t i t i pop pop pop I pop Y pop Y , 1 , , 3 , , 2 1 , 1 , 1 , , ln ln ln 1 ln ε µ α β β β + + +         +         +         + =         - - - where t i t i pop Y , , , t i t i pop I , ,
are respectively the gross domestic product and the investment per capita and

        -1 , , ln t i t i pop pop
is the demographic growth rate. We introduce individual and time specific intercepts (respectively i α and t µ ) in order to control for unobserved heterogeneity. Thereby, 1 β measures the GDP convergence conditionally of investment per capita and population growth rate.

However, for both econometrics and economic theory, it is too restrictive to assume no cross-sectional dependence in testing convergence hypothesis. The restriction implies that the economies are closed, which is obviously inappropriate to understand regional convergence process. Regions are units naturally open to economic flows like trade, technology diffusion and factor mobility, which may affect regional convergence process.

Following the model specification of (3) to address with spatial dependence, we estimate successively:

(7) ( )

t i t i t i t i t i t i t i t i t i t i t i t i pop pop pop I pop Y W pop Y pop Y , 1 , , 3 , , 2 , , 1 , 1 , 1 , , ln ln ln ln 1 ln ε µ α β β ρ β + + +         +         +         +         + =         - - - and (8) ( ) t i t i t i t i t i t i t i t i t i t i pop pop pop I pop Y pop Y , 1 , , 3 , , 2 1 , 1 , 1 , , ln ln ln 1 ln ε µ α β β β + + +         +         +         + =         - - - ( ) , , t i i t i it W ν η ε ρ ε + + =

Data description

We use a panel dataset of 191 regions in 14 member states of EU-15 (see appendix A, which describes the set of regions, included and excluded in the sample) over a period to 25 years .

Data variables come from the Cambridge Econometrics database5 . The gross domestic product (GDP) and investment (provided by Cambridge Econometrics in 1995 constant euro) have been transformed into logarithm of per capita term (

        t i t i pop Y , , ln ,         t i t i pop I , , ln
) in order to consider the scale effect. The demographic growth rate is measured from the total population data dynamics (

        -1 , , ln t i t i pop pop ).
For the estimation, we consider five aggregated time-periods (1980-84, 1985-89, 1990-94, 1995-99 and 2000-2005) to avoid short run variations in GDP growth rates due to business-cycle effects. The accurate number of years required to avoid short-run variations is still under discussion in the literature (see [START_REF] Temple | The New Growth Evidence[END_REF], for an analysis). [START_REF] Temple | The New Growth Evidence[END_REF] recommends 5 or 10 years long periods, but we preferred to follow the approach from [START_REF] Badinger | Regional convergence in the European Union, 1985-99: a spatial dynamic panel analysis[END_REF] and chose quinquennial time periods to collect information on at least 5 periods. Thereby, we have a panel on 955 observations of 191 regions during 5 periods. Of course, the dynamic panel specification restricts this panel to 4 periods because of the autoregressive term (

        - - 1 , 1 , ln t i t i pop Y ).
The spatial lag term

        t i t i pop Y W , ,
ln is constructed from a weighted average of neighbouring regions' income level. Thus, we have chosen a geographical definition of neighbourhood based on the Euclidean distance between regions in order to construct the spatial weight matrix (W ). More precisely we have chosen a k -nearest neighbours weight specification, ) (k w ij represents the element of W matrix in row i and column j :

0 ) ( * = k w ij if j i = 1 ) ( * = k w ij if ) (k d d i ij ≤ 0 ) ( * = k w ij if ) (k d d i ij ≥ ij d
is the distance between the regions i and j centroids, and ) (k d i is a cut-off distance based on the distance of k -nearest neighbour for region i . The interactions are assumed to be negligible above this distance. Although we have constructed W with k =10, the results are similar with k =5, 15 and 20.

So, the matrix is row-standardised

∑ = j ij ij ij k w k w k w ) ( ) ( ) ( * *
to provide easier interpretation (each weight may be interpreted as the region's share in the total spatial effect of the sample) and make parameter estimates more comparable (see [START_REF] Kelejian | Spatial correlations: a suggested alternative to the autoregressive model[END_REF] for a good discussion on the spatial weighting matrix).

k -nearest neighbours' weight matrix has the most fitted to representing spatial interaction of our sample: this specification lead to each region has the same number of neighbouring regions ( k ) including islands on our sample and reduce the heterogeneity problem of regional superficies [START_REF] Anselin | Under the hood issues in the specification and interpolation of spatial regression models[END_REF]. during the whole period. In the same time, the decreasing of GDP p.c. standard deviation points out the reduction of GDP p.c. disparities across time. This evolution is confirmed by the analysis of the distribution of that variable (Figure 1). The next section will enable us to assess the strength of this process and to test the various factors which affect it.

Econometric analysis

Table 2 and 3 report respectively estimates of conditional convergence model for the period 1980-2005. This time lag has been split into 5 periods (1980-84, 1985-89, 1990-94, 1995-99 and 2000-05).

It has been shown that ignorance of the spatial correlation leads to potentially misleading estimates and tests, so we will first analyse the spatial properties of the residuals, and then present the estimation results and validity tests.

Validity tests

The presence of spatial correlation has been tested using the Moran's statistics on the GMM residuals (Z moran , Table 2 and3). This test is the most commonly used to detect spatial correlation and its application has been extended to residuals regression in [START_REF] Anselin | Testing for spatial error autocorrelation in the presence of endogenous regressors[END_REF].

Strong significant spatial correlation can be detected on GMM residuals of model ( 1) that ignore spatial dependence (Table 2). This result confirms the presence of spatial effects in the European regional convergence. As expected, the introduction of a spatial lag term (model ( 2)) reduces significantly the spatial correlation of residuals for the GMM estimator. Moran's test fail to reject the null hypothesis of no spatial correlation for 1990-1994, 2000-05 and for 1985-89, 1995-99 respectively with a 10% and 5% significance level (Table 3). Similar results are obtained with a spatial error specification (Table 3). These results confirm the presence of spatial dependencies which leads to inconsistent results and inefficient validity tests in model ( 1). Hence, we are not looking on the remaining tests in Table 2 and we therefore proceed with the spatial lag model, concentrating on Table 3 tests results.

The consistency of the GMM estimator depends on whether lagged values of the autoregressive and spatial autoregressive terms are valid instruments for the regression. In the estimation process, we are using the orthogonality conditions between the error term in first difference and lagged values of the dependent variables. In order to test these conditions, we report tests for first and second-order serial correlation [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF] and we consider three specification tests (AR(1), AR(2) and

Hansen tests).

AR(1) and AR( 2) tests provides further support to the model and its estimation since this statistics fails to reject this hypothesis of no second order correlation while reject no first-order serial correlation (Table 3). The overall validity of the instruments can be tested by the Hansen test of over-identifying restrictions. This test confirms the overall validity of the instrumental variables (Table 3) at the 10% level of significance in the spatial lag model. Nevertheless, the result of Hansen test on spatial error specification points out the weakness of the instruments in this context (0.023). Hence, the results must be interpreted carefully.

The Hansen test can be also informative on the validity of additional instruments by comparing the difference of Hansen statistic between two sets of instruments. The Hansen-diff reported in Table 3 check the validity of additional instruments used by system-GMM. We report first-differenced GMM results because Hansen-diff (21.01) rejects the validity of additional instruments used by system-GMM for the level equation.

We assume that the explanatory variables are exogenous using the Hansen-diff test between exogenous and predetermined set of instruments (as suggested by [START_REF] Bond | GMM estimation of empirical growth models[END_REF]. Overall validity tests

do not indicate problems with instrument validity and orthogonality conditions used by firstdifferenced GMM estimators. We do not use system-GMM because additional instruments of the level equation are not valid.

Results

Within the framework of a dynamic panel specification that considers the spatial dimension of GDP per capita, we find empirical evidence on conditional convergence of European regions. This process is strongly affected by spatial dimension.

As explained above, the GMM estimator controls for both endogeneity and other econometric problems; it is expected to address the inconsistency of POLS and LSDV estimators which provide however upper and lower bounds for the autoregressive parameter [START_REF] Bond | GMM estimation of empirical growth models[END_REF] 6 . For Model ( 6) and ( 7), LSDV values are respectively lies between 0.48 and 0.39, and around 0.89 when the POLS estimator is used. As expected, for these two models, the estimated autoregressive parameters fall between these bounds: close to 0.8 for the model (6) and 0.5 for the model ( 7).

The estimated autoregressive parameter (Table 2) are close to those obtained in other studies [START_REF] Caselli | Reopening the convergence debate: A new look at cross-country growth empirics[END_REF]. this result points out that there is a significant European regional convergence, conditionally to investment per capita (which have a significant positive impact on regional development). The non significant impact of demographic growth suggests that the evolution of labour force does not affect regional development.

        1 , 1 , ln i i pop Y ,         2 , 2 , ln i i pop Y ,...,         - - 2 , 2 , ln T i T i pop Y
) and the overall set of other explanatory variables (considered as exogenous) .

The results from the dynamic spatial panel data models are displayed in Table 3. The introduction of spatially lagged variable leads to significantly different results. The autoregressive parameter (0.453) in model ( 7) drops sharply regarding to model ( 6) indicating that we measure a faster regional conditional convergence when we consider the impact of neighbouring income on regional development. Spatial lagged income coefficient (0.417) suggests a strong significant impact of spillover effect between European regions on their development dynamics.

We can explain this drop by the fact that a part of y it variability previously explained by the variability of the lagged term y it-1 in ( 6) is due to spatial interdependence Wy it variability in (7). The sum of ( )

1 1 β + and ρ in (7) is close to ( ) 1 1 β + in (6).
ρ reflects how spillover affect income growth rate for all regions with the same initial income. The development of the European regions is strongly affected by their spatial interdependence, thus suggesting that the convergence process is not only a "temporal" process but also a spatial process. In the framework of the Solow model, we can assume that speed of convergence to the steady state is affected by regional spillover. The rate at which the regions converge towards their steady state depends on the assumption of decreasing marginal product. Regional spillovers may induce this decline, so, the dynamics toward the steady state doesn't depend only of the gap between 0 , i Y and its steady state value but also on the strength of spatial spillover. Thus, spatial concentration of GDP p.c. must impede the convergence process. However, [START_REF] Egger | Spatial convergence[END_REF] explain that spatial spillover implies that the convergence speed depends on the region's location. Thus, they suggest amending the computation of the convergence speed in order to allow for this issue. we can decompose (As suggested by [START_REF] Egger | Spatial convergence[END_REF] the convergence speed into its "classical" part, a remoteness effect and the impact of initial gap.

The spatial error model provides results similar to the other studies. The convergence speed (4.8%) is in line with the values computed by [START_REF] Badinger | Regional convergence in the European Union, 1985-99: a spatial dynamic panel analysis[END_REF].

The introduction of a spatial lag does not affect the impact of investment per capita spending on regional development The demographic growth rate is not significant for any specification.

Table 3

Estimation results of the Spatial Dynamic model ( 7) and ( 8

Conclusion

The aim of this paper is to review strategies to estimate Spatial Dynamic panel data model using GMM.

Then, we estimate a conditional convergence model of European regions which allows both space and dynamic dimension to illustrate these strategies. Using a panel dataset of 191 regions over 1980-2005, we estimate a Spatial Dynamic Panel using spatial lag strategy estimation in addition to standard dynamic model on panel data techniques [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF][START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF].

We find empirical evidence on conditional convergence of European regions. European convergence is conditional to the investment rate. In analysing the spatial dimension, we find that convergence is significantly affected by the spatial disparities.

Note that other drivers of regional development can affect convergence process of European regions, as European policies [START_REF] Esposti | Impact of Objective1 funds on regional growth convergence in the European Union: a panel-data approach[END_REF][START_REF] Bouayad-Agha | Fostering the endogenous potential development of European regions: a panel data analysis of the cohesion policy on regional convergence over the period 1980-2005[END_REF].

The Spatial Dynamic Panel data model is at early stage. In this paper, we concentrate on studying strategies to estimate spatial dynamic panel data. But this approach needs further development.

First, the usual tests on spatial specification (LM-test) must be extended to time dynamic dimension.

Furthermore, there are several possibilities to crossing Dynamic and spatial literature. In the spatial error model, using GMM system estimator as suggested by [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] in time dynamic can improve its efficiency.

Figure 2

 2 Figure2graphs the GDP p.c. geographic pattern relative to the EU-14 average GDP level for the 5

Figure 2

 2 Figure 2Geographic pattern of the GDP per capita relative to the EU-14 average GDP level for the 5 periods (adapted fromBouayad-Agha et al, 2009, Cambridge Econometrics database)

  still significant around 0.16. Its value increases to 0.236 when we allow for spatial error dependence.

  ) Notes: *, ** and *** indicate significance at the 1%, 5% and 10% level. Robust standard errors are displayed in parentheses. SGMM-DIFF estimations are obtained by a forward orthogonal deviation transformation with instrument set composed by ( set of other explanatory variables (considered as exogenous) .

Table 1 Descriptive statistics Table 1

 1statistics1

depicts the dynamics of GDP p.c., investment per capita, demographic growth rate and spatially lagged GDP for European regions. One can see the regular evolution of each variable. The sample mean of regional GDP pc increases from 9.44 to 9.81, what represents a growth of 3.92%

Table 2

 2 Estimation results of Dynamic model (6)

Notes: POLS: Pooled Ordinary Least Squares; LSDV: Least Squares Dummy Variable; GMM-DIFF: Generalized Method of Moments on forward orthogonal deviation equation. *, ** and *** indicate significance at the 1%, 5% and 10% level. Robust standard errors are displayed in parentheses. GMM-DIFF estimations are obtained by a forward orthogonal deviation transformation with instrument set composed by (

As underlined by[START_REF] Kukenova | Spatial dynamic panel model and system GMM: a Monte-Carlo investigation[END_REF] there is no currently available estimator to consider this simultaneity problem in line with the potential endogeneity of other explanatory variables.

[START_REF] Mutl | Dynamic panel data models with spatially correlated disturbances[END_REF] demonstrates the consistency of the spatial GM in a dynamic panel data model.

The Cambridge Econometrics database is available at http://www.camecon.com.

With fixed T, POLS gives an estimate of the coefficient of the lagged income that is biased upward in the presence of individual specific effects[START_REF] Hsiao | Analysis of Panel Data[END_REF] and LSDV gives an biased downwards estimate for the same coefficients[START_REF] Nickell | Biases in dynamic models with fixed effects[END_REF].

APPENDIX A

Code

Name Code Name Code Name Code Name Included regions (190)


Laurent Arcese
email: laurent.arcese@bourges.univ-orleans.fr

Matthieu Fruchard
email: matthieu.fruchard@bourges.univ-orleans.fr

Antoine Ferreira
email: antoine.ferreira@ensi-bourges.fr

Nonlinear modeling and robust controller-observer for a magnetic microrobot in a fluidic environment using MRI gradients

This paper reports the use of a MRI device to pull a magnetic microrobot inside a vessel and control its trajectory. The bead subjected to magnetic and hydrodynamic forces is first modeled as a nonlinear control system. Then, a backstepping approach is discussed in order to synthesize a feedback law ensuring the stability along the controlled trajectory. We show that this control law, combined with a high gain observer, provides good tracking performances and robustness to measurement noise as well as to some matched uncertainties.

I. INTRODUCTION

Microrobots designed to perform targeted therapy by navigating in the cardiovascular system are a prolific research area for minimally invasive surgery purposes [START_REF] Abbott | Robotics in the small: Part 1 microrobotics[END_REF]. Indeed, these untethered robots have the distinctive feature of being less invasive than catethered methods, and thus may reach remote parts of the human body that remain inaccessible to present tools without operating. On such a scale, it proves difficult to embed actuators sufficiently powerful to propel the robot, especially when swimming against the tide. Therefore, the great majority of swimming approaches considers magnetic fields to wirelessly transmit power to the robot [START_REF] Gillies | Magnetic manipulation instrumentation for medical physics research[END_REF], [START_REF] Honda | Micro swimming mechanisms propelled by external magnetic fields[END_REF].

Untethered devices have been mainly developed according to three different designs: magnetic beads pulling, biomimetic flagellated robots and magnetotactic bacteria. The former propelling scheme relies on generating thrust on the microrobot using the magnetic force related to the gradients of the external field. This concept has first been studied using magnets [START_REF] Quate | Goniometric motion controller for the superconducting coil in amagnetic stereoaxis system[END_REF] or superconducting magnets [START_REF] Takeda | Development of magnetically targeted drug delivery system using superconducting magnet[END_REF]. The last innovation in this domain has been provided by Martel [START_REF] Mathieu | Method of propulsion of a ferromagnetic core in the cardiovascular system through magnetic gradients generated by an mri system[END_REF], [START_REF] Mathieu | Magnetic microparticle steering within the constraints of an mri system: proof of concept of a novel targeting approach[END_REF], where the basic idea is the use of the magnetic gradients coils of a clinical MRI to pull the robot. Since medical applications are the underlying theme of these works, using a clinical MRI device to actuate the robot combines several advantages. MRI devices are widely implanted and provide both a permanent magnetic field and magnetic gradient that can be used for propelling purposes. Besides, the MRI imager can provide fine observation of the scene for navigation and observation, using multiplexing for controlling and imaging [START_REF] Tamaz | Real-time mri-based control of a ferromagnetic core for endovascular navigation[END_REF]. In-vivo experiments have also been carried out [START_REF] Martel | Automatic navigation of an untethered device in the artery of a living animal using a conventional clinical magnetic resonance imaging system[END_REF]. The second class of robots, using respectively oscillating and rotating magnetic fields, imitates either the motion of a beating flagella [START_REF] Dreyfus | Microscopic artificial swimmers[END_REF] provided by bending and pivoting a magnetic particles made elastic rod, or the corkscrew effect of an helical coil attached to a magnetic core [START_REF] Behkam | Design methodology for biomimetic propulsion of miniature swimming robots[END_REF] to swim in Stokes flow. These techniques are well suited for nanorobots since they move at low Reynolds number, and are proved more efficient than bead pulling for robots whose radius is down to a few micrometers [START_REF] Abbott | How should microrobots swim?[END_REF]. However, they will prove difficult to implement on classical MRI devices, since their propulsion requires a time-varying external field whereas actual MR imaging requires a uniform DC field. In [START_REF] Martel | Flagellated magnetotactic bacteria as controlled mri-trackable propulsion and steering systems for medical nanorobots operating in the human microvasculature[END_REF], the flagella of magnetotactic bacteria have been exploited to provide propulsion, while steering has been achieved by inducing a torque on a chain of nanoparticles (magnetosomes) synthesized in the cell of each bacterium. Despite promising, this recent design faces some unsolved problems leading to reduced efficiency as reported in [START_REF] Martel | Flagellated magnetotactic bacteria as controlled mri-trackable propulsion and steering systems for medical nanorobots operating in the human microvasculature[END_REF].

Efforts have mostly focused on fine modeling, feasibility and sizing of these various designs, but few works deal with advanced control and observation of these nonlinear systems. To our knowledge, the more advanced feedback controller scheme designed in the scope of microrobotics facing nonlinear drag is a PID approach developed in [START_REF] Tamaz | Real-time mri-based control of a ferromagnetic core for endovascular navigation[END_REF], [START_REF] Mathieu | In-vivo validation of a propulsion method for untethered medical microrobots using a clinical magnetic resonance imaging system[END_REF]. Besides, there is no consideration about observation issues. Authors report in [START_REF] Tamaz | Real-time mri-based control of a ferromagnetic core for endovascular navigation[END_REF] instabilities and important oscillations around the equilibrium, especially when the blood stream is modeled as a pulsatile flow. They also indicate a lack of robustness to noise and unmodeled dynamics. However, it is clear that nonlinear control techniques can improve the tracking, both at the control and at the observation point of view.

The paper is organized as follows. Section II is devoted to the 2D-state space nonlinear model for a microrobot subjected to hydrodynamic and magnetic forces. In Section III, we synthesize a backstepping controller based on the previous model. Then Section IV is devoted to the high gain observer used to reconstruct the state, and especially the unmeasured robot's velocity that is needed to implement the backstepping control law. In Section V, simulation experiments are presented for both swimming with and against the flow and for bifurcating in an Y-shaped vessel. These simulations point out the robustness of the observer-controller pair to noise measurements and to blood's velocity uncertainties. Finally, conclusion and future works are detailed in the last section.

II. MODELING

Modeling at micro and nanoscale has been widely investigated. The purpose of this section is to get a basic state space model at microscale. Since the control is designed to make the robot levitate, it is not necessary to take friction forces into account in a first approximation. Moreover, we assume the microrobot is big enough to be unaffected by specific nanoforces.

A. Forces Balance

The microrobot immersed in a blood vessel is modeled by a ferromagnetic sphere with a high saturation magnetization. In this paper, the core is supposed to be made of permendur. Assuming the robot is spherical, with a some hundreds micrometers radius r, we now detail the forces acting on it, see Figure 1.

ρ ∂ -→ v ∂t + ρ ( -→ v .∇) . -→ v = -→ f -∇p + ∇.σ (1)
where ρ denotes the density of the fluid, -→ v the fluid velocity, -→ f the body forces acting on the fluid, p the pressure and σ the stress tensor. However, equation ( 1) is very complex and, nowadays, we neither know how to solve it analytically, nor whether a solution exists. A numerical approach is possible but heavily computation consuming, which is not to be desired for direct and real-time experiments.

Assuming that the flow is a Newtonian incompressible fluid, hydrodynamic forces are referred as drag force :

-→ F d = - 1 2 ρ (u r -v) 2 AC d ----→ u r -v ----→ u r -v (2)
where u r -v is the relative velocity of the robot with respect to the fluid, A is the frontal area of the core and C d is the drag coefficient.

The drag coefficient represents the overall effect of the robot's shape on the drag force. In the literature, one can find various expressions for the drag coefficient C d as a function of the Reynolds number Re. Re is a dimensionless positive number characterizing the flow regime as laminar or turbulent; its expression for a spherical core is given by:

Re = 2ρ (u r -v) r η
where η is the fluid's viscosity.

In our study case, we assume the flow to be pulsatile and laminar, i.e. we neglect the turbulences generated by heart pumping just downstream from the aortic valve. This assumption is not too restrictive if we consider a blood vessel quite far from the heart. In a low Reynolds number, the drag coefficient is approximated by C d = 24

Re . For higher Reynolds numbers, a more precise drag coefficient in laminar flow for a spherical core is given by [START_REF] White | Viscous Fluid Flow[END_REF]:

C d = 24 Re + 6 1 + √ Re + 0.4 (3)
In addition to the drag force, apparent weight (combined action of weight and buoyancy) is acting on the microrobot:

-→ W a = V (ρ r -ρ) -→ g (4)
where V and -→ g respectively denote the volume of the robot and the gravitational acceleration.

2) Magnetic Force: On the one hand, the permanent magnet of a clinical MRI device commonly generates a strong and uniform magnetic field (B 0 ≥ 1.5T ), so that we can assume the robot's magnetization reaches the saturation magnetization. This magnetic field induces a torque that tends to align the magnetization of the robot along the field. However, since we consider a spherical bead, it is useless to take it into account.

On the other hand, the gradient coils of the MRI system provide magnetic gradients which produce a magnetic force on the robot. This magnetic force can be expressed as:

-→ F m = µ 0 V m -→ M .∇ -→ H (5)
where µ 0 = 4π.10 -7 T.m.A -1 is the permeability of free space, V m is the volume of the magnetic material in the core, -→ M denotes the magnetization of the core and -→ H the external magnetic field, ∇ being the gradient operator.

3) Forces balance: The blood vessel is considered as a tube which divides in two with an angle θ (Figure 2). Let (x, z) denote the location of the robot in the blood vessel with respect to a given frame F(0, i, k). If we consider that the drag force is distributed linearly between the i-axis and the k-axis, applying Newton's second law leads to:

mẍ = -→ F dx + -→ F mx mz = -→ F dz + -→ W a + -→ F mz (6)
where m = 4 3 πr 3 ρ r is the mass of the robot, and indexes x (respectively z) denote the projections on i ( k ).

If we substitute (2) with ( 3), ( 4) and ( 5) in ( 6), we get:

         ẍ = a 1 ( ẋ -v x ) + a 2 ( ẋ -v x ) 2 +a 3 ( ẋ-vx) 2 1+α √ ẋ-vx + a 4 u 1 z = b 1 ( ż -v z ) + b 2 ( ż -v z ) 2 + g( ρ ρr -1) +b 3 ( ż-vz) 2 1+α √ ż-vz + b 4 u 2 (7)
with the following parameters α, a i , b i , and magnetic gradients considered as control inputs u 1 and u 2 :

           a 1 = -4.5 η sin θ

B. State Space Representation

Let x 1 , x 2 , x 3 , x 4 denote respectively the robot's position and velocity along i-axis, and the position and velocity along k-axis. Assuming that positions x 1 and x 3 can be measured thanks to the MRI imaging system, let y denote the state's measure. Then, system (7) can be written as:

(S)            ẋ1 = x 2 ẋ2 = f 2 (x 2 ) + u 1 (S 1 ) ẋ3 = x 4 ẋ4 = f 4 (x 4 ) + u 2 (S 2 ) y = (x 1 x 3 ) T (8)
where functions f and controls u are obtained using:

         u 1 = a 2 v 2 x -a 1 v x + a 4 u 1 u 2 = b 2 v 2 z -b 1 v z + g( ρ ρr -1) + b 4 u 2 f 2 (x 2 ) = a 1 x 2 + a 2 x 2 2 -2v x x 2 + a 3 (x2-vx) 2 1+α √ x2-vx f 4 (x 4 ) = b 1 x 4 + b 2 x 2 4 -2v z x 4 + b 3 (x4-vz) 2 1+α √ x4-vz
We can notice that system (S) can be divided into two subsystems (S 1 ) and (S 2 ), what allows to define two independent control laws Lyapunov-stabilizing trajectories for each subsystem.

III. CONTROL APPROACH

Each of the two subsystems (S 1 ) and (S 2 ) can be straight expressed in a triangular form, so that backstepping control approach easily applies [START_REF] Kanellakopoulos | A toolkit for nonlinear feedback design[END_REF], [START_REF] Krstić | Nonlinear and Adaptative Control Design[END_REF]. Since the two subsystems expressions are similar, we can focus on the first one. The control law synthesis for the first subsystem can thus be applied mutatis mutandis to the second one.

First, we set x1 = x rx 1 as the error between any C 2 reference trajectory x r and the real position x 1 of the core. A Lyapunov Control Function (CLF) is given by:

V 1 (x 1 ) = 1 2 x2 1 ≥ 0 (9)
Differentiating ( 9) leads to:

V1 (x 1 ) = ∂V 1 (x 1 ) ∂ x1 ẋ1 = x1 ( ẋr -x 2 )
To ensure V1 (x 1 ) is definite negative, for instance:

V1 (x 1 ) = -k 1 x2 1 , k 1 ≥ 0 a possible issue is hence to set -k 1 x1 = ẋr -x 2r .
Second, setting x2 = x 2r -x 2 implies ẋ2 = ẍr +k 1 ẋ1 -ẋ2 . Yet we have ẋ1 = ẋrx 2 = -k 1 x1 and x2 = ( ẋrx 2 ) + k 1 x1 , hence it follows that ẋ1 = x2k 1 x1 , and we get:

ẋ2 = ẍr + k 1 x2 -k 2 1 x1 -ẋ2
the CLF (9), a Lyapunov function is given by

V 2 (x 1 , x2 ) = V 1 (x 1 ) + 1 2 x2 2 ≥ 0. We obtain: V2 (x 1 , x2 ) = x1 ẋ1 + x2 ẋ2 = -k 1 x2 1 + x2 [(1 -k 2 1 )x 1 + k 1 x2 + ẍr -ẋ2 ] Likewise, to ensure V2 (x 1 , x2 ) is definite negative, for instance V2 (x 1 , x2 ) = -k 1 x2 1 -k 2 x2
2 with k 2 ≥ 0, a possible issue is hence to set:

-k 2 x2 = (1 -k 2 1 )x 1 + k 1 x2 + ẍr -ẋ2
Substituting expressions of x1 and x2 , and using (8) in the previous expression finally suggests assigning the following control law for subsystem (S 1 ) of ( 8):

ū1 = ẍr + (k 1 + k 2 ) ẋr + (1 + k 1 k 2 )x r -(1 + k 1 k 2 )x 1 -(k 1 + k 2 )x 2 -f 2 (x 2 ) (10)
IV. HIGH GAIN OBSERVER For many applications, appropriate control requires the knowledge of unmeasured parameters, hence the interest in an observer to estimate the state. In our case, we can measure the position of the core but we have also to estimate its velocity, since the control law [START_REF] Dreyfus | Microscopic artificial swimmers[END_REF] requires it.

To study the observability of nonlinear system ẋ = f (x) + g(x)u, y = h(x), with x ∈ R n , u ∈ R m and y ∈ R p , a necessary and sufficient condition is to have a full ranked observability matrix O i,j =

∂L i-1 f h(x) ∂xj
, where L f h(x) denotes the Lie derivative of h(x) along the vector field f (x). In system [START_REF] Tamaz | Real-time mri-based control of a ferromagnetic core for endovascular navigation[END_REF], it is clear that the observability is ensured for all x.

In our study case, we decided to use a high gain observer. This one requires the system to have canonical form, that is for a second order system:

   ẋ1 = x 2 + ϕ 1 (u, x 1 ) ẋ2 = ϕ 2 (u, x 1 , x 2 ) y = x 1 According to [18], if |ϕ k (u, x 1 , ..., x k ) - ϕ k (u, x1 , ..., xk )| ≤ c (x 1 -x1 ) 2 + ... + (x k -xk ) 2
where c is a locally Lipschitz constant, then an observer is given by:

ẋ1 = x2 + ϕ 1 (u, x1 ) + Lg 1 (x 1 -x 1 ) ẋ2 = ϕ 2 (u, x1 , x2 ) + L 2 g 2 (x 1 -x 1 ) , H = g 1 1 g 2 0
where L is the high gain and with H a Hurwitz matrix.

V. SIMULATIONS Simulations are conducted within the scope of actual commonly spread MRI system abilities. At the moment, MRI systems are able to generate magnetic gradients with an intensity of some tens of mT.m -1 . Let us note that this limitation is additionally affected by the gradient coils' duty cycle and by the multiplexing needed both for controlling and observing. That is the reason why some prospects are actually led in order to design additional powerful gradient coils [START_REF] Yelin | Modeling and control of untethered biomicrorobots in a fuidic environment using electromagnetic fields[END_REF], which is a difficult task since they must not affect the imaging capability and quality. In order to make sure that the amplitude of the control inputs remains bounded by physical actuators limits u i,max and to protect the system, we perform a simple time scaling. Thus, the control is first computed using [START_REF] Dreyfus | Microscopic artificial swimmers[END_REF], and the applied control law is given by ui k(t) , with k(t) = max 1, ui ui,max . We suppose that a pre-planned trajectory has been obtained, and the control aims at keeping a null error between desired and real trajectory. The following simulations illustrate two different backgrounds. In the first one, we consider that the reference trajectory requires the robot to move with and against the flow and study effects of noise measurements on tracking. In the latter, a 2D Y-shaped reference trajectory is considered to validate our approach, with uncertainties considerations. Simulation parameters are given in Table I.

A. 1D trajectory with and against the flow

This simulation is performed only on the i-axis, which does not affect the 2D-approach we lead since the two subsystems are similar and so for the results. The only difference is that, on k-axis, the gravity must be counterbalanced in order to levitate the ferromagnetic core.

Reference trajectory is here a sinusoidal function x r (t) = 0.01 sin 5.5 2 t along the i-axis, thus illustrating the case of swimming longitudinally with and against the flow. The noise applied on the output is a white gaussian noise with a 10 -4 amplitude, i.e. about 10% of the measured signal. The blood's velocity v = 0.035(1 + 1.15 sin 2πt) is supposed to be known. Since blood's velocity is periodic, using the previous expression is relevant for simulation's purpose, even if not realistic. Figures 3 and4 illustrate that the position tracking is good with respect to noise and the observer's convergence is efficient. Globally, tracking is not too much affected by the noise, since position error is below 0.1% of the reference position, and the observer-controller pair ensures an efficient stabilization after a short transient phase. Note that notable degradation of tracking performance occurs at times t ∈ {1; 3; 8}, because the control input reaches low saturation at these times (Figure 7).

Figures 5 and6 also show a rather good tracking of Fig. 6. Error between actual and reference speed velocity, yet not as efficient as the position tracking, since the noise induced on speed error is about 2% of desired speed. This result should be expected, as the estimate velocity highly depends on the noised measures on position. We can also notice the three peaks on speed error related with control's saturation.

B. 2D Y-shaped trajectory

In order to illustrate any case of Y-shaped bifurcation, simulation is led so as to work in every quadrant using a circular reference trajectory (Figure 8). Experiment in the 2D space defined by the frame F(0, i, k) of Figure 2 is of particular interest, since the special issue of levitating along k-axis is encompassed as well as less demanding control on i (or j) axis.

Moreover, the blood's velocity v = 0.035(1+1.15 sin 2πt) is unknown at the controller-observer level. This matched additive uncertainty on the physical system is motivated by the difficulty to measure the pumping blood stream in the vessels, whereas it is a decisive parameter in the force balance. We consider here the worst case scenario, since this parameter is considered as null when designing the control as well as the observer.

Figures 8 and9 show that the position tracking performance is quite robust to unmodeled dynamics. After convergence of both controller and observer, tracking error is fewer than 4mm for each of the considered axis, suggesting that bifurcating can be performed even with important fluctuating uncertainties, at least for vessels whose diameter is up to some 5mm. However, to navigate efficiently in smaller vessels, this result suggests it would be necessary either to reject this unknown perturbation, or to rebuilt it.

Figure 10 shows the gradient inputs on i and k-axis. During transient phase, control input on k-axis reaches its actuator saturation, which explains the peaks at the beginning of the simulation. It can be noticed that input on the vertical magnetic gradient is mainly negative: this is due to the imposed constraint of levitating, which is directly related to a force balance between apparent weight versus magnetic force. For the parameters values given in Table I, levitating requires about ∇B lev = -(ρ sphere -ρ)g M ≈ -36mT m -1 , which is at the merge of control saturation.

VI. DISCUSSION

The previous simulations show that it is possible to control a some hundred micrometers radius ferromagnetic core in a blood vessel in 1D and 2D (with no particular difficulty to extend it to 3D), with an improved efficiency compared to PID controllers. In the first case, we show that our controller-observer is robust to output noise. In fact, actual MRI devices have a precision around some hundreds of micrometers, besides the microrobot being ferromagnetic, it may induce artifacts on the MRI imager. That is why output noise robustness is a key point for implementation on such process.

The second simulation points out the relative robustness of the proposed controller-observer with respect to some unmodeled dynamics. Since microrobots are mainly devoted to medical applications, controllers designed for such robots have to deal with well-known uncertain biological systems.

VII. CONCLUSION AND FUTURE WORK

In this paper, we have presented a state space representation of a magnetic microrobot in a fluidic environment and have developed both a nonlinear control law and an observer to improve actual experiments. We have studied the effect of noise on the output and its influence on the position and speed tracking as well as the convergence of the high gain observer. We have remarked that the position is not too much affected by the noise, unlike the velocity of the core. We also have showed that the proposed design is robust with respect to some matched uncertainties. However, we noticed that tracking efficiency is very sensitive to some biological parameters uncertainties. Besides, robustness to blood's velocity unknowing reaches its limits in the case of small vessels.

In near future, we can expect better performances by improving the controller, using adaptative rejection of matched uncertainties based on [START_REF] Chakrabortty | Time-scale separation redesigns for stabilization and performance recovery of uncertain nonlinear systems[END_REF]. Another promising improvement may be obtained using the varying high gain observer proposed in [START_REF] Andrieu | High gain observers with updated gain and homogeneous correction terms[END_REF], which allows, under some conditions, to take into account these perturbations and to ensure the practical convergence of the observer.

Beyond theoretic and simulating developments we are working on, we expect to soon implement these nonlinear tools on the benchmark MR-SUB of the Nanorobotics Laboratory through afoot collaboration with the Ecole Polytechnique of Montreal.

Fig. 1 .

 1 Fig. 1. Forces exerted on the core

FFig. 2 .

 2 Fig. 2. Scheme of a blood vessel with bifurcation

r 2 ρr a 2 =b 1 = - 4 .5 η cos θ r 2 ρr b 2 = -0.15 ρ cos θ r ρr α = 2ρr η b 3 =

 21423 -0.15 ρ sin θ rρr -2.25 cos θ rρr b 4 = M ρr

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Longitudinal position: reference position (grey dots), actual position (black dots) and estimated position (grey solid line)

Fig. 7 .

 7 Fig. 7. Control input (magnetic field gradient on i-axis)

Fig. 8 .Fig. 9 .Fig. 10 .

 8910 Fig. 8. XY trajectory: reference trajectory (grey) and actual trajectory (black dotted)

VIII. ACKNOWLEDGMENTS This work was supported by European Union's 7th

Framework Program and its research area ICT-2007.3.6 Micro/nanosystems under the project NANOMA (Nano-Actuactors and Nano-Sensors for Medical Applications.)