Jean-Roch R Mouret
email: mouretj@supago.inra.fr

Robert David
email: robert.david@uclouvain.be

Francisco Zamorano

Alain Van De Wouwer
email: alain.vandewouwer@umons.ac.be

Denis Dochain
email: denis.dochain@uclouvain.be

Jean-Marie M Sablayrolles

A Vande Wouwer

Closed carbon balance in calculation of metabolic fluxes -Application to the central metabolism of Saccharomyces cerevisiae in wine-making fermentation

Keywords: metabolic flux analysis, Saccharomyces cerevisiae, carbon balance, wine-making fermentation. 

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

INTRODUCTION

Metabolic Flux Analysis (MFA) is a mathematical method for analyzing metabolism of bacterial or eukaryotic cells. It is a powerful methodology for the determination of metabolic pathway fluxes. In this approach, the intracellular fluxes are calculated by using a stoichiometric matrix (metabolic pathway map) for the intracellular reactions and applying mass balances around intracellular metabolites. A set of measured extracellular fluxes, typically uptake rates of substrates and secretion rates of metabolites, is used as input to the calculations [START_REF] Vallino | Metabolic flux distributions in Corynebacterium glutamicum during growth and lysine overproduction[END_REF][START_REF] Jorgensen | Metabolic flux distribution in Penicillium chrysogenum during fed-batch cultivations[END_REF].

This modelling approach has been used to study the metabolism of Saccharomyces cerevisiae for almost two decades [START_REF] Nissen | Flux distributions in anaerobic, glucose-limited continuous cultures of Saccharomyces cerevisiae[END_REF][START_REF] Gombert | Network identification and flux quantification in the central metabolism of Saccharomyces cerevisiae under different conditions of glucose repression[END_REF][START_REF] Forster | Genome-scale reconstruction of the Saccharomyces cerevisiae metabolic network[END_REF]. Nevertheless, studies concerning the modelling of the wine-making fermentation process have been performed only recently [START_REF] Varela | Biomass content governs fermentation rate in nitrogen-deficient wine musts[END_REF][START_REF] Pizarro | Coupling kinetic expressions and metabolic networks for predicting wine fermentations[END_REF][START_REF] Charnomordic | Two modelling approaches of winemaking: first principle and metabolic engineering[END_REF][START_REF] Celton | A constraint-based model analysis of the metabolic consequences of increased NADPH oxidation in Saccharomyces cerevisiae[END_REF].

The final objective of the EC CAFÉ project (www.cafeproject.org) concerning the wine-making process is to design control tools aimed at optimising the fermentation to obtain a well-defined aromatic profile. During the alcoholic fermentation, hexoses (glucose and fructose) are mainly converted to ethanol and carbon dioxide. A large set of other by-products (higher alcohols, esters, sulphur compounds) affecting the organoleptic properties of the wine are also formed during this process [START_REF] Swiegers | Yeast and bacterial modulation of wine aroma and flavour[END_REF]; but they represent less than 1 % of the consumed sugar. Nevertheless, before considering these aroma compounds, a relevant description of the central carbon metabolism is needed. MFA appears to be an interesting tool to model this part of the yeast metabolism.

Based on the assumption that the internal metabolites are in steady state (no accumulation), the system of linear equations resulting from the stoichiometric model is usually underdetermined when only external measurements are used. In the literature, various suggestions are made to overcome this problem [START_REF] Bonarius | Flux analysis of underdetermined metabolic networks: the quest for the missing constraints[END_REF][START_REF] Zamorano | A detailed metabolic flux analysis of an underdetermined network of CHO cells[END_REF]: (i) the introduction of additional metabolic theoretical constraints; (ii) the exploitation of linear optimization tools and the definition of suitable objective functions; or (iii) the use of isotopic tracer experiments to determine some intracellular fluxes. An objective function is usually used in metabolic models concerning the yeast S. cerevisiae [START_REF] Forster | Genome-scale reconstruction of the Saccharomyces cerevisiae metabolic network[END_REF][START_REF] Celton | A constraint-based model analysis of the metabolic consequences of increased NADPH oxidation in Saccharomyces cerevisiae[END_REF]. Recently, a method consisting in using no additional constraint and therefore solving the undetermined system of linear equations by obtaining intervals for the different metabolic fluxes was developed for CHO cells [START_REF] Provost | Dynamic metabolic modelling under the balanced growth condition[END_REF][START_REF] Zamorano | A detailed metabolic flux analysis of an underdetermined network of CHO cells[END_REF].

In the present study, a novel aspect is the Cmol conversion of the reaction schemes (detailed in Section 2). This impacts the coefficients into the reaction schemes, i.e., those of the stoichiometric matrix, and allows a consistent carbon balance of the system through data reconciliation. We compare the results of two methods for calculating the intracellular fluxes of the balanced carbon central metabolism of S. cerevisiae: (i) determination of intervals for the metabolic fluxes, (ii) use of an objective function.

The next two sections give a description of the metabolic network and of the experimental data respectively. MFA is presented in Section 4 and the objective function strategy (FBA) in Section 5. Discussion is provided in Section 6.

METABOLIC NETWORK DESCRIPTION

The central carbon metabolism of S. cerevisiae considered in this paper is represented by a set of 79 biochemical reactions listed in Table 1. The metabolic network involves the following pathways: glycolysis -pentose phosphate pathway (PPP) -TCA cycle -ethanol, glycerol and acetate synthesistransport reactions (sugar uptake and product excretion)amino acid synthesis -DNA and RNA synthesis -lipid synthesis -carbohydrate synthesis -biomass formation. It has to be stressed that this metabolic network corresponds to a metabolism of exponentially-growing cells in wine-making conditions. This network would be different if for example the stationary phase of the yeast was considered.

Knowing that the alcoholic fermentation is an anaerobic process, the TCA pathway does not operate as a cycle but as two branches (v15 and v16 in Table 1) leading both to succinate synthesis [START_REF] Camarasa | Investigation by 13 C-NMR and tricarboxylic acid (TCA) deletion mutant analysis of pathways for succinate formation in Saccharomyces cerevisiae during anaerobic fermentation[END_REF]. The biomass composition in carbohydrates, lipids and proteins was experimentally measured (as described in [START_REF] Celton | A constraint-based model analysis of the metabolic consequences of increased NADPH oxidation in Saccharomyces cerevisiae[END_REF]. The cellular contents of RNA and DNA were standard values available in the literature [START_REF] Lange | Statistical reconciliation of the elemental and molecular biomass composition of Saccharomyces cerevisiae[END_REF]. It has to be noted that to simplify the network the energetic cofactors (NAD and NADP) were not taken into account. Nevertheless, the chosen reactions correctly described the central carbon metabolism of S. cerevisiae.

The metabolic network involves 70 internal metabolites and 9 external metabolites: glucose, ethanol, glycerol, acetate, succinate, CO 2 , biomass, exogenous fatty acids [A_exo] and exogenous lipids [lipid_exo]. A_exo and lipid_exo are not measured data, they correspond to nutrient requirements associated with growth and are thus proportional to the production of biomass [START_REF] Celton | A constraint-based model analysis of the metabolic consequences of increased NADPH oxidation in Saccharomyces cerevisiae[END_REF].

The metabolic fluxes will be expressed in Cmol/L.h.g cell . The Cmol unit corresponds to a normalization of the elements contained in a molecule by the number of carbon atoms C present in the considered compound. For example, the formula of glucose in mol is C 6 H 12 O 6 and turned into Cmol it becomes CH 2 O. When considering reaction schemes this normalization affects the stoichiometric coefficients as illustrated in the following example with the reaction v7: 1.

-in mol: PYR  ACAL + CO2, i.e. '3 C  2 C + 1 C'; -in Cmol: '1 C  2/3 C + 1/3 C', see v7 in Table
On the other hand, the sum of inlet fluxes must be equal to the sum of the outlet fluxes, as presented in equation ( 1):

   j j out i i in v v ) ( ) ( or % 100 ) ( ) (    i i in j j out v v (1)
where v in(i) corresponds to the inlet flux of substrate i and v out(j) corresponds to the outlet flux of product j. If the equation ( 1) is satisfied, the "carbon balance" is closed.

Moreover, there is only one substrate: glucose. The rates of metabolite synthesis can be divided by the glucose consumption rate to be normalized. This fluxes expression in percent can be used within the metabolic-network model because the reaction schemes are defined in Cmol unit and not in mol unit.

Using Cmol instead of mol in addition with the calculation of carbon recovery in percentage provides a straightforward assessment of the use of carbon through the reactions network. It is also an easy and highlighting indicator of the precision of the measurements (as detailed in Section 3). The experimental data used in the current work correspond to the monitoring of the growth of the Saccharomyces cerevisiae commercial strain EC1118 (Lallemand SA) at 20°C on a synthetic medium (SM300) simulating the grape juice [START_REF] Bely | Description of alcoholic fermentation kinetics: its variability and significance[END_REF] where glucose was the sole carbon source.

The extracellular measurements of glucose, ethanol, glycerol, acetate, succinate, biomass and CO 2 used in this study are listed in Table 2. Using these values, the carbon balance is equal to 102%, meaning that the measurements are accurate. Nevertheless, these raw data cannot be used directly. Indeed, if the incoming and outgoing fluxes are not equal, the system of equations cannot be solved. So, it was decided to distribute the error on the ethanol and CO 2 (because they are the main produced metabolites) keeping the ratio ethanol/CO 2 constant. Finally, the rates of metabolite synthesis were divided by the glucose consumption rate to be normalized and ready to use for the model. MFA is a methodology for the quantification of the pathway fluxes from limited experimental data. In the present work, we consider the special case where extracellular measurements in the culture medium are the only available data (using a method described in [START_REF] Provost | Dynamic metabolic modelling under the balanced growth condition[END_REF][START_REF] Provost | Dynamic metabolic modelling under the balanced growth condition[END_REF][START_REF] Zamorano | A detailed metabolic flux analysis of an underdetermined network of CHO cells[END_REF]. On the basis of the metabolic network, the flux distributions are found by applying steadystate mass balances around the internal balanced metabolites. Each admissible flux distribution is represented by a vector v = (v 1 , v 2 , …,v m ) T whose entries are the rates (or fluxes) at which the reactions proceed. The steady-state balance around the internal metabolites is expressed by the algebraic set of equations:

0   v N ; 0  v (2)
where the (m×n) matrix N is the stoichiometric matrix deduced from the metabolic network (m is the number of internal balanced metabolites and n is the number of fluxes).

In our case, the stoichiometric matrix N has dimensions 70 × 79. It is a rather sparse matrix (4.4% of filling). An admissible flux distribution v must satisfy the steady state balance (2) and be compatible with the experimental measurements. The specific uptake and excretion rates of the measured external species are collected in a vector v m and are by definition linear combinations of the unknown fluxes v i . This is expressed as follows:

v N v m m   (3)
Where N m is a proper (p×n) full-rank matrix with p the number of available measurements.

The aim of MFA is to compute the set of admissible flux distributions v, i.e., the set of non-negative vectors v that satisfies the equations (2) (3). The problem is said to be wellposed if the solution set is not empty and if all the solutions are bounded. Otherwise, the system is said to be ill-posed.

When the problem is well-posed, the solution set is a polytope in the positive orthant and each admissible flux distribution v can be expressed as a convex combination of a set of non-negative basis vectors f i which are the vertices of this polytope and form therefore a unique convex basis of the solution space. In other words, the solution set of the MFA problem is the set of admissible flux vectors defined as:

  i i i f v  with 0  i  and   i i 1  (4)
The basis vectors f i are obtained by applying the toolbox METATOOL [START_REF] Pfeiffer | Metatool: for studying metabolic networks[END_REF][START_REF] Schuster | Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering[END_REF] to the matrix

0 m m N N        
Once the basis vectors are known, we can compute the limiting values of the flux interval for each metabolic flux [START_REF] Provost | Dynamic metabolic modelling under the balanced growth condition[END_REF]:

min max i i i v v v   with     min max min , 1,..., , max , 1, ..., i ki i ki v f k m v f k m    
where f ki denotes the i-th element of the basis vector f k .

Calculating the range of possible values for each metabolic flux is relevant for underdetermined systems which do not have a unique solution.

Results

We perform a flux analysis for yeast cells on the basis of the underlying metabolic network presented in Table 1. The first purpose is to characterize the feasible set of solutions using extracellular measurements only. Our system is well-posed (i.e. the sets of equations ( 2) (3) are reliably satisfied), therefore the number of basis vectors f i and the size of the flux intervals will depend on the extracellular measurements that are considered.

The size of the intracellular flux intervals is analyzed under different assumptions. In the first place the 7 experimental rates of the external metabolites are used to compute the flux intervals. In this case, most of the fluxes have fixed values; the values of the undetermined fluxes (internal fluxes) are within a fairly narrow interval. This observation makes sense because a system of 79 linear equations (metabolic reactions) constrained with 70+7 values (internal+external metabolites) is still underdetermined.

Figure 1 illustrates the distribution of G6P between the glycolysis pathway (v2) and the PPP (v18). It stresses solution intervals that are between 59 and 61% for v2 and between 35 and 38% for v18. The reaction v20 corresponding to the outlet of the PPP is between 17 and 18%. These results are satisfactory because the narrow intervals of variation of the metabolic fluxes make possible a good understanding of the carbon distribution within the yeast metabolism.

If we use 6 rates of external metabolites, results are similar because the rate of the "missing" external compound is equal to the experimental value (see Figure 2 for example). This observation is not surprising: there is no carbon accumulation within the cell (the sum of the rates of external metabolites is equal to zero), therefore when only one external metabolite is not taken into account, its value must be equal to its experimental value (knowing that, for our experimental values, the carbon balance is kept).

If we use 5 rates of external metabolites, the intervals of variation that are defined for the two last measured rates are large and include the corresponding measurement value. The ranges of variation of some internal fluxes are consequently large too, making difficult the interpretation of the obtained results. In Figure 1, where CO 2 and biomass fluxes are considered as unknown, the fluxes v2 and v18 corresponding to the distribution of G6P between the glycolysis pathway and the PPP are almost undetermined as they nearly varies between 0 and 100%. The reaction v20 corresponding to the outlet of the PPP is between 0 and 50%. This is obviously a too large interval and the maximum value is far from a standard flux value for the considered strain of yeast (personal communication). Figure 2 also illustrates this result with the calculated fluxes of CO 2 and biomass: the mean value of the interval of variation respectively deviates from 5 and 18 % compared to the experimental value. This is reasonable for CO 2 but not for the biomass.

These last observations show that if the number of constraints is low, the span of solution highly increases and it becomes difficult to analyze and understand the results. On the contrary, using 6 or 7 measurements provides satisfactory results demonstrating the reliability of these measurements and the relevance of the data reconciliation.

USE OF AN OBJECTIVE FUNCTION

A FBA method [START_REF] Lee | Flux balance analysis in the era of metabolomics[END_REF] based on an objective function was tested on the metabolic network. The assumption considered for defining the objective function is the maximization of the biomass production as the measurements were taken during the growth phase of the yeast (see Section 3). This corresponds to the maximization of the reaction v79.

If we use the rates of the 6 external metabolites except biomass, we observe that the calculated biomass production rate is equal to the experimental value (see Figure 2). Once again, this observation makes sense; it is due to the carbon balance. Using the objective function, a single value is obtained for each metabolic flux. An interesting observation is that this value is equal to the central value of the interval obtained with the previous solution method when considering internal fluxes (example in Figure 1). This shows that the methods consisting either in only solving the underdetermined system or in using an objective function give close results as long as sufficient external metabolites are measured.

When we use 5 rates of external compounds, several observations are made:

1) if glycerol, acetate or succinate are among the unused measurements their calculated values are equal to zero. Indeed, the objective function consists in maximizing the biomass production; so no carbon is sent or assigned to the production of a metabolite that is not directly involved in biomass synthesis such as the before-mentioned compounds.

2) CO 2 is involved in different metabolic pathways (glycolysis, ethanol production, biomass synthesis) so its value cannot be equal to zero; nevertheless, when its rate is not fixed, the calculated value is underestimated by 20% compared to the experimental one (see Figure 2). The results in Figure 1 provide an explanation. v2 value is maximum whereas v18 and v20 values are close to 0 meaning that most of the carbon is transferred to glycolysis to the detriment of PPP. As v18 produces CO 2 , it can explain the underestimation of the CO 2 flux.

Consequently the biomass synthesis is systematically overestimated. Figure 2 stresses an overestimation up to 47% for instance.

It can be concluded that the maximization of biomass production is a validated assumption as long as the needed measurements are available. Fig. 2. Calculated fluxes for CO 2 (v26) and biomass X (v79) depending on the available measurements and the solution method.

DISCUSSION

The Cmol conversion of the reaction schemes in addition with the calculation of carbon recovery in percentage provides a quite convenient overview of the use of carbon through the metabolic network. This formulation also eases data reconciliation prior to the use of MFA tools.

The two solution methods used in this work provide reliable and comparable results. As a matter of fact, when all the rates of external metabolites are used to solve the system of equations, the flux calculated for each reaction of the network using an objective function is equal to the central value of the calculated interval provided by the toolbox METATOOL. A good reliability is also obtained when only one external measurement is missing.

With less measurements the convex analysis also provides solutions but with larger intervals making difficult any metabolic interpretation. Results obtained with the objective function do not account for key metabolites (succinate, acetate, and glycerol) if they are not included into the measurements. Indeed, in the considered metabolic network, these compounds are not directly linked to the biomass production and consequently to the objective function. This last observation gives insight about the selection of the metabolic network: the production of these products is associated with the regeneration of energetic cofactors necessary to biomass synthesis, i.e. linking the production of succinate, acetate, glycerol, to the production of biomass. So it would be interesting to establish this link in the next version of the metabolic network so as to overcome this issue. Moreover, taking into account these metabolic intermediates is also of interest because the intensity of some internal fluxes involved in the regeneration of cofactors appears to be unappropriate (for example the flux values in PPP).

In the future, the conversion of reaction schemes into Cmol and Nmol will be considered so as to close carbon and nitrogen balances. This will allow studying an extended version of the yeast metabolism including the nitrogen metabolism and some volatile compounds having an organoleptic interest.

Fig. 1 .

 1 Fig. 1. Calculated fluxes for reactions v2, v18 and v20 depending on the available measurements and the solution method.

Table 1 . Metabolic reactions for the metabolism of S. cerevisiae

 1

	Flux	Reaction
		Glycolysis
	v1	Glucose  G6P
	v2	G6P  F6P
	v3	F6P  G3P
	v4	G3P  PEP
	v5	PEP  PYR
		Ethanol, glycerol and acetate synthesis
	v6	G3P  Glycerol
	v7	PYR  2/3 ACAL + 1/3 CO2
	v8	ACAL  Ethanol
	v9	ACAL  Acetate
	v10	Acetate  ACCOA
		TCA cycle
	v11	PYR  2/3 ACCOA + 1/3 CO2
	v12	3/4 PYR + 1/4 CO2  OAA
	v13	4/6 OAA + 2/6 ACCOA  ICIT
	v14	ICIT  5/6 AKG + 1/6 CO2
	v15	MAL  Succinate
	v16	AKG  4/5 Succinate + 1/5 CO2
	v17	OAA  MAL
		Pentose phosphate pathway (PPP)
	v18	G6P  5/6 R5P + 1/6 CO2
	v19	R5P  4/10 E4P + 6/10 F6P
	v20	5/9 R5P + 4/9 E4P  6/9 F6P + 3/9 G3P
		Transport reactions
	v21	Glucose_ext  Glucose
	v22	Glycerol  Glycerol_ext
	v23	Ethanol  Ethanol_ext
	v24	Acetate  Acetate_ext
	v25	Succinate  Succinate_ext
	v26	CO2  CO2_ext
	v27	A_exo_ext  A_exo
	v28	Lipid_exo_ext  Lipid_exo

Table 2 . Extracellular measurements

 2

			Specific rates
		Experimental	modified to get	Normalized
	Compound	specific rate	a closed	specific
		(Cmol/L.h.gX)	carbon balance	rate (%)
			(Cmol/L.h.gX)
	Glucose	4.47E-02	4.47E-02	100
	Glycerol	2.19E-03	2.19E-03	4.89
	Ethanol	2.30E-02	2.25E-02	50.46
	Acetate	1.36E-04	1.36E-04	0.30
	Succinate	8.34E-05	8.34E-05	0.19
	CO 2	1.45E-02	1.42E-02	31.86
	Biomass	5.70E-03	5.70E-03	12.75
		4. METABOLIC FLUX ANALYSIS
	4.1 Principle of the method
	The yeast metabolism considered in this analysis is described

IFAC CAB 2013 December 16-18, 2013. Mumbai, India

ACKNOWLEDGEMENTS This paper includes results of the CAFÉ project that is supported by the Food, Agriculture and Fisheries, and Biotechnology program of the European Community (Contract number KBBE-212754, www.cafe-project.org). It also presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science, Technology and Culture. The scientific responsibility rests with its authors.


Mario Sigalotti

Jean-Claude Vivalda

Controllability properties of a class of systems modeling swimming microscopic organisms

Controllability properties of a class of systems modeling swimming microscopic organisms

Mario Sigalotti, Jean-Claude

Introduction

The mathematical description of the motion of swimming organisms and of the mechanisms of their propulsion is a challenging and wide field of research. This paper aims at giving a contribution to such a domain in a comparatively narrow direction.

The object of our study are microscopic organisms. The most prominent aspect of the models describing their motion is the very high level of viscosity ( [START_REF] Berg | Bacteria swim by rotating their flagellar filaments[END_REF][START_REF] Purcell | Life at low raynolds numbers[END_REF][START_REF] Taylor | Analysis of the swimming of microscopic organisms[END_REF]). The mechanism of their propulsion depends heavily on the different species and admits a great variety of models (see, for instance, [START_REF] Alouges | Optimal strokes for low raynolds number swimmers: an example[END_REF][START_REF] Childress | Mechanics of swimming and flying[END_REF][START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF][START_REF] Lighthill | Mathematical biofluiddynamics[END_REF] and references therein).

Among microscopic organisms we will further restrict our attention to the class of ciliata, whose propulsion is determined by cilia, hair-like organelles covering their surface and whose size is very small compared with that of the organism (see [START_REF] Blake | A finite model for ciliated micro-organisms[END_REF][START_REF] Brennen | An oscil lating-boundary-layer theory for ciliary propulsion[END_REF][START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]). Recently San Martín, Takahashi and Tucsnak [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF] proposed a control theory approach to the description of ciliata. Following the former literature (and in particular [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF][START_REF] Keller | A porous prolate-spheroidal model for ciliated micro-organisms[END_REF]) they assume that the organism is a rigid body and that the motion of the cilia is described by a set of controlled velocity fields on a surface enclosing the layer of cilia. They are therefore faced to an infinite-dimensional control system coupling Navier-Stokes equations, which describe the motion of the fluid, and the equations for the rigid body arising form Newton's laws. The hypothesis of high viscosity (ie, of low Raynolds number) mentioned above is exploited to reduce such a control system to a finite-dimensional version of it. The state-space is given by the coordinates of the center of mass of the organism, its orientation and its angular and linear velocities.

Summarizing, San Martín, Takahashi and Tucsnak single out a 12-dimensional nonlinear control system which is affine in the controls. The control system depends on a number of parameters which are explicitly derived as functions of the shape of the organism, of its mass distribution, and of the surface velocity fields describing the motion of the cilia. The authors study the control properties of the system proving that it is generically controllable when the number of controlled vector fields is at least six (see Section 5 for details). They also provide some example of controllable systems in the case in which the organism has a spherical shape. In this special case they also study the controllability properties of the linearization of the system at its rest position.

Our scope is to enhance the results of [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF] basically in three directions: firstly we want to give sharper results on the generic controllability of the system; secondly we want to obtain a complete characterization of controllable systems in the case in which the organism is spherical; thirdly we want to provide a complete picture of controllable and non-controllable systems in the simplified situation which -following again [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF]-comes into effect in the case in which the organism and the fluid have densities of the same order of magnitude. The overall aim is to provide a better understanding of the dependence of the controllability properties of the model proposed in [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF] on the physical quantities characterizing it. The hard aspect of the model is indeed the difficulty to obtain reliable expressions for the surface velocity fields describing the motions of the cilia. In this regard, it is important to provide as precise as possible parametric analysis of the controllability properties of the system.

The paper is organized as follows. Section 2 introduces the control theoretical language adopted throughout the paper. Section 3 presents the control system obtained in [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF] and recalls the relations between the physical quantities characterizing the organism and the parameters appearing in the model. Section 4 is technical and contains an abstract controllability result used in the later sections. Section 5 is devoted to the analysis of the genericity of the controllability of the system introduced in Section 3. Genericity is initially formulated in terms of the parameters appearing in the finite-dimensional system and then expressed in terms of the physical objects they depend on (Section 5.3). The main result is that the system is generically controllable if the number of controlled vector fields is at least three and that one controlled vector field suffices for its generic accessibility (Theorem 5.1). Section 6 analyzes the case in which the organism is spherical and provides a complete characterization of the sets of parameters that make the system controllable (Theorem 6.1). Finally, Section 7 establishes a characterization of controllable systems when the organism can be assumed to have the same density as the fluid (Theorem 7.2). The result follows from a further reduction of the system that transforms it into a six-dimensional one. The reduction procedure, suggested in [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF], is carried out in full details.

Notations and definitions

Let us first introduce some notations: in the following Id denotes the 3 × 3 identity matrix, 0 m×n is the zero m × n matrix, 0 m = 0 m×1 is the zero vector in R m . We denote by M m×n the set of m × n real matrices. The interior of a set W is denoted by Int(W ) and its closure by Clos(W ). For simplicity of notation, when no confusion is possible, row and column vectors are identified. The canonical orthonormal basis of R 3 is denoted by {e 1 , e 2 , e 3 }. For every ω ∈ R 3 we denote by S(ω) the skew symmetric matrix such that S(ω)z = ω × z for every z ∈ R 3 , that is,

S(ω) =   0 -ω 3 ω 2 ω 3 0 -ω 1 -ω 2 ω 1 0   .
In what follows we deal with control systems of the type

q = f (q, u), q ∈ M, u ∈ R m , (1)
where M is a smooth (C ∞ ) manifold and f : M ×R m → T M is smooth with respect to both variables. Moreover, for every T ≥ 0 the endpoint mapping E T which associates to an initial condition q 0 ∈ M and a control function u ∈ L ∞ ([0, T ], R m ) the final point of the corresponding trajectory of (1) is well defined and continuous on M × L ∞ ([0, T ], R m ), where the

L 1 norm is chosen on L ∞ ([0, T ], R m ).
(For these and finer properties of the endpoint mapping see, for instance, [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF].) Let F = {f (•, u) | u ∈ R m } be the family of vector fields characterizing [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]. For every X = f (•, u) ∈ F we denote by e tX the one-parameter group generated by X, that is, e tX (q) = E t (q, u1 [0,t] ). The attainable set from q ∈ M is the set

A(q, F ) = ∪ T ≥0 E T (q, L ∞ ([0, T ], R m )).
When no confusion is possible we write A(q) = A(q, F ).

We say that (1) (or, equivalently, F ) is controllable if for every q ∈ M the set A(q) is equal to M . We say that (1) is approximatively controllable if A(q) is dense in M .

Give two vector fields X and Y on M , the Lie bracket between X and Y is defined, in a local system of coordinates, by the relation

[X, Y ] = (DY ) X -(DX) Y
where DX denotes the derivative of X. Denote by L the Lie algebra generated by F . System (1) is Lie-bracket generating if for every q ∈ M the set

L (q) = {V (q) | V ∈ L } is equal to T q M .

The dynamics of swimming microscopic organisms

In [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF] the authors describe the swimming of a microscopic organism immersed in an infinite volume of fluid by coupling the dynamics of a rigid body (representing the organism) with the Navier-Stokes equations describing the behavior of the fluid outside the body. Exploiting the hypothesis of very low Reynolds number, they derive a simplified version of the coupled system. The system obtained in this way is finite-dimensional and nonlinear; more precisely, it has the following expression:

ż = Az + E(z) + Bu , (2) ζ
= Rξ , (3)
Ṙ = RS(ω) , (4)
where

z = (ξ, ω) ∈ R 3 × R 3 , ζ ∈ R 3 , R ∈ SO(3), and
E(z) = ξ × ω J -1 ((Jω) × ω) .
Here J ∈ M 3×3 denotes the inertia matrix of the rigid body representing the organism. Recall that J is symmetric and positive definite. The matrix A ∈ M 6×6 is a function of the rigid body and the viscosity of the fluid and its expression is given at the end of the section. The control function u takes values in R m , m ≥ 1, and B is a 6-by-m matrix, which we write as

B = B 1 B 2
where both B 1 and B 2 belong to M 3×m . An important feature of the system above is that (2) is a well-defined control system on R 6 , since it does not depend on ζ nor R.

Denote by X 0 the vector field on R 6 × R 3 × SO(3) which is the drift of the control system (2)-(4), that is,

X 0 (z, ζ, R) =   Az + E(z) Rξ RS(ω)   .
Let moreover X 1 , . . . , X m be the controlled vector fields, so that system (2)-( 4) can be written as q = X 0 (q) + m i=1 u i X i (q), q ∈ R 9 × SO(3).

Define L as the Lie algebra generated by X 0 , X 1 , . . . , X m . We say that a vector field X on R 9 ×SO(3) is a constant vector field if there exist v 1 , v 2 ∈ R 3 such that X : (ξ, ω, ζ, R) → (v 1 , v 2 , 0 6 ) and we write X = (v 1 , v 2 , 0 6 ). Clearly, X 1 , . . . , X m are constant vector fields. Notice that if V = (v 1 , v 2 , 0 6 ) and W = (w 1 , w 2 , 0 6 ) are constant vector fields, then

[[X 0 , V ], W ] =   v 2 × w 1 + w 2 × v 1 J -1 (Jw 2 × v 2 + Jv 2 × w 2 ) 0 6   (5)
is a constant vector field as well.

Remark 3.1. For every constant vector field

V = (v 1 , v 2 , 0 6 ), the bracket [X 0 , V ] is of the form [X 0 , V ](z, ζ, R) = (G(z, V ), -Rv 1 , -RS(v 2 ))
for some smooth map G with values in R 6 . Therefore, if the constant vector fields contained in L span R 6 × {0 6 }, then system (2)-( 4) is Lie-bracket generating.

We conclude the section by recalling the analytic characterization of the coefficients appearing in A, B and J. Let Ω ⊂ R 3 be an open bounded connected set whose boundary is of class C 2 , representing the (rigid) shape of the organism. Then the coefficients J ij of J are given by

J ij = Ω δ(x)(e i × (x -ξ)) • (e j × (x -ξ))dx, (6)
where ξ denotes the mass center of Ω and δ : Ω → (0, ∞) is the (possibly non-constant) density function.

Denote by F the set R 3 Clos(Ω) and, for l ∈ N, 1 ≤ q ≤ ∞, let

D l,q (F ) = {ϕ ∈ L 1 loc | ∂ α ϕ ∈ L q (F ) for all α ∈ N 3 , |α| = l}.
According to [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF]Lemma 3.1], for every i ∈ {1, 2, 3} there exists a unique solution (h (i) , p (i) ) of

       -∆h (i) + ∇p (i) = 0 on F divh (i) = 0 on F h (i) = e i on ∂Ω lim |y|→∞ h (i) (y) = 0 and a unique solution (H (i) , P (i) ) of        -∆H (i) + ∇P (i) = 0 on F divH (i) = 0 on F H (i) (y) = e i × y on ∂Ω lim |y|→∞ H (i) (y) = 0 such that h (i) , H (i) ∈ L s (F ) ∩ D 1,r (F ) ∩ D 2,θ (F ) ∩ C ∞ (F ) and p (i) , P (i) ∈ L r (F ) ∩ D 1,θ (F ) ∩ C ∞ (F ) for s ∈ (3, ∞], r ∈ (3/2, ∞] and θ ∈ (1, ∞).
The Cauchy stress is the tensor σ defined by the relation

σ(v, p) = -pId + µ ∂v k ∂y l + ∂v l ∂y k 1≤l,k≤3 (7)
where µ is the Reynolds number of the fluid. For i = 1, 2, 3, define

g (i) = σ(h (i) , p (i) )n| ∂Ω , G (i) = σ(H (i) , P (i) )n| ∂Ω ,
where n is the unit inner normal to ∂Ω. Denote by Θ 1 , Θ 2 , Υ 1 , Υ 2 the matrices in M 3×3 defined by

Θ 1 ij = - ∂Ω g (i) j ds, Θ 2 ij = - ∂Ω x × g (i) j ds, Υ 1 ij = - ∂Ω G (i) j ds, Υ 2 ij = - ∂Ω x × G (i) j ds,
where ds is the surface element of ∂Ω. It turns out (see [START_REF] Happel | Low Reynolds number hydrodynamics with special applications to particulate media[END_REF]) that Υ 1 = (Θ 2 ) T and that

A = m-1 Θ 1 m-1 Θ 2 J -1 Υ 1 J -1 Υ 2 , ( 8
)
where m denotes the mass of the organism. Moreover, with respect to the inner product in R 6 defined by

a, b J = m 3 l=1 a l b l + 3 l,k=1 J lk a 3+l b 3+k , (9)
A is self-adjoint and negative-definite.

The matrix B is defined by

B = m-1 κ J -1 K (10)
where the entries of κ and K are

κ ij = - ∂Ω g (i) • ψ j ds, K ij = - ∂Ω G (i) • ψ j ds,
and ψ 1 , . . . , ψ m are fixed functions in C 2 (∂Ω, R 3 ), each one associated to a component of the control u, that describe the admissible propulsive actions of the organism.

4

A controllability result in a more general setting

The following proposition provides sufficient (and necessary) conditions for the controllability of systems that generalize those introduced in the previous section, in the sense that a more general structure for the dynamics in the coordinates z is allowed.

Proposition 4.1. Consider the following control system

ż = f (z, u), z ∈ R 6 , u ∈ R m . ( 11
)
Then (3),(4), [START_REF] Happel | Low Reynolds number hydrodynamics with special applications to particulate media[END_REF] is controllable if and only if it is Lie-bracket generating at (z, ζ, R) = (0 9 , Id) and system [START_REF] Happel | Low Reynolds number hydrodynamics with special applications to particulate media[END_REF] is controllable.

Proof. One direction of the equivalence being obvious, let us assume that system [START_REF] Happel | Low Reynolds number hydrodynamics with special applications to particulate media[END_REF] is controllable and that (3),( 4), [START_REF] Happel | Low Reynolds number hydrodynamics with special applications to particulate media[END_REF] is Lie-bracket generating at (z, ζ, R) = (0 9 , Id). Notice that

(z ′ , ζ ′ , R ′ ) ∈ A(z, ζ, R) =⇒ (z ′ , ζ + Rζ ′ , RR ′ ) ∈ A(z, ζ + Rζ, RR) (12)
for every ( ζ, R) ∈ R 3 × SO(3), where the letter A is used here to denote attainable sets for system (3),( 4), [START_REF] Happel | Low Reynolds number hydrodynamics with special applications to particulate media[END_REF]. Indeed, if t → u(t) is an admissible control steering (z, ζ, R) to (z ′ , ζ ′ , R ′ ) and t → (z(t), ζ(t), R(t)) is the corresponding trajectory, then t → (z(t), ζ + Rζ(t), RR(t)) is an admissible trajectory corresponding to the same control u.

Fix (z 0 , ζ 0 , R 0 ), (z 1 , ζ 1 , R 1 ) ∈ R 9 × SO(3). Since (11) is controllable, then there exist η 0 , η 1 ∈ R 3 and Q 0 , Q 1 ∈ SO(3) such that (0 6 , η 0 , Q 0 ) ∈ A(z 0 , ζ 0 , R 0 ) and (z 1 , ζ 1 , R 1 ) ∈ A(0 6 , η 1 , Q 1 ). We are left to prove that A(0 6 , η 0 , Q 0 ) contains {0 6 } × R 3 × SO(3).
Since (3),( 4), [START_REF] Happel | Low Reynolds number hydrodynamics with special applications to particulate media[END_REF]

V of R 3 × SO(3) such that {0 6 } × V is contained in G.
Notice the following consequence of (12

): if (0 6 , ζ, R) ∈ A(0 6 , η 0 , Q 0 ) and (0 6 , ζ ′ , R ′ ) ∈ G, then (0 6 , ζ + Rζ ′ , RR ′ ) ∈ Int(A(0 6 , η 0 , Q 0 )) . (13)
Indeed, taking (z, ζ, R) = (0 9 , Id) in ( 12), we have that for every (z ′′ , ζ ′′ , R ′′ ) in a neighborhood of (0 6 , ζ ′ , R ′ ), the point (z ′′ , ζ + Rζ ′′ , RR ′′ ) belongs to A(0 6 , ζ, R) and, since (0 6 , ζ, R) ∈ A(0 6 , η 0 , Q 0 ), we are done. Notice that for every nonempty open subset O of SO(3)

W = {P 1 P 2 • • • P k | k ∈ N, P 1 , . . . , P k ∈ O} (14)
is equal to SO(3). Indeed, since O contains at least one element P 0 of finite order, ie, an axial rotation of angle commensurable with π, then Id belongs to the interior of W . The completeness of SO(3) guarantees that W = SO(3). Take as O the projection of V on SO(3). Then for every R ∈ SO(3) there exists ζ ∈ R 3 such that (0 6 , ζ, R) belongs to G; this can be seen by noticing that there exist k elements (0 6 , ζ 1 , P 1 ), . . . , (0 6 , ζ k , P k ) in V such that P i ∈ O and P 1 P 2 • • • P k = R and by applying repeatedly [START_REF] Jurdjevic | Control systems subordinated to a group action: accessibility[END_REF] in the special case η 0 = 0 3 and Q 0 = Id.

As a consequence, without loss of generality,

Q 0 = Id, Q 1 = Id.
Indeed, since there exists

ζ 0 ∈ R 3 such that (0 6 , ζ 0 , Q -1 0 ) ∈ G, equation (13) with ( ζ, R) = (η 0 , Q 0 ) shows that (0 6 , η 0 + Q 0 ζ 0 , Id) ∈ A(0 6 , η 0 , Q 0 ); similarly, the existence of ζ 1 ∈ R 3 such that (0 6 , ζ 1 , Q 1 ) ∈ G implies that (0 6 , η 1 , Q 1 ) ∈ A(0 6 , η 1 -ζ 1 , Id).
Moreover, for every element p of RP 2 , the Grassmannian of one-dimensional subspaces of R 3 , there exists ζ p ∈ R 3 such that (0 6 , ζ p , R p ) lies in G, where R p denotes the rotation of angle π around the axis p. Applying [START_REF] Jurdjevic | Control systems subordinated to a group action: accessibility[END_REF]

to ( ζ, R) = (ζ ′ , R ′ ) = (ζ p , R p ) in the special case η 0 = 0 3 , we obtain that (0 6 , ζ p + R p ζ p , Id) lies in G. Notice that ζ p + R p ζ p = 2 ζ p , p p belongs to the axis p. Therefore, G intersects {0 6 } × p × {Id} for every p ∈ RP 2 .
Of special interest will be for us the expression of ( 13) when R = R ′ = Id, namely

(0 6 , ζ, Id) ∈ A(0 6 , η 0 , Id), (0 6 , ζ ′ , Id) ∈ G =⇒ (0 6 , ζ + ζ ′ , Id) ∈ Int(A(0 6 , η 0 , Id)) . ( 15
) Let S 2 ± = {v ∈ S 2 | {0 6 } × R ≥0 (±v) × {Id} ∩ G = ∅}
where S 2 denotes the unit sphere in R 3 . As we remarked above, S 2 + and S 2 -cover S 2 and, by construction, they are open. Moreover, both are nonempty, since v ∈ S 2 ± implies that -v ∈ S 2 ∓ . Since S 2 is connected we have S 2 + ∩ S 2 -= ∅, ie, there exist v ∈ R 3 and two non-negative constants λ 1 , λ 2 such that both (0 6 , λ 1 v, Id) and (0 6 , -λ 2 v, Id) lie in G; moreover, due to the openness of G, these constants can be assumed to be positive and commensurable and so there exist two positive integers n 1 and n 2 such that n 1 λ 1n 2 λ 2 = 0. Applying repeatedly [START_REF] Jurdjevic | Controllability properties of affine systems[END_REF] in the special case η 0 = 0 3 , we obtain that G contains (0 9 , Id) and, therefore,

{0 6 } × R 3 × {Id}.
Applying [START_REF] Jurdjevic | Controllability properties of affine systems[END_REF] again with ζ = η 0 we deduce that (0 6 , η 1 , Id) is attainable from (0 6 , η 0 , Id).

Generic properties

The expression generic is commonly used to denote a property of a system that is, in a suitable sense, stable under small perturbations and that, even when it fails to apply, can be forced to hold by applying to the system an arbitrary small perturbation.

In order to define precisely what a generic property means in the present context, define

Ξ m 0 = {(A, B, J) ∈ M 6×6 × M 6×m × M 3×3 | J = J T > 0, A ∈ Sym - J }, where Sym - J = {A ∈ M 6×6 |
A is symmetric and negative definite with respect to •, • J } and •, • J is defined as in [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF].

We say that a property is generic for system (2)-( 4) when m = m 0 if there exists an open and dense set O in Ξ m 0 0 such that the property holds for every system defined by a triple in O. Such a definition of genericity, given in terms of the entries of the matrices A, J and B, is adapted to the finite-dimensional formulation (2)-( 4) but can appear unsatisfactory from the point of view of the physical model. Section 5.3 discusses how to define the genericity directly in the space of configurations of the organism and provides a physically justified counterpart of the following result.

Theorem 5.1. (a) Generically when m ≥ 1 the rank of L is maximal at every point. (b) Generically when m ≥ 3 system (2)-( 4) is controllable.

The generic conditions ensuring the Lie-bracket generating condition and the controllability of ( 2)-( 4) are given explicitly in the Propositions 5.2 and 5.3. Notice that the techniques applied in this section are independent of the structure imposed on A by the physical motivations of the model, namely, the fact that A ∈ Sym - J . Indeed, the conditions thatare obtained in the following sections define an open and dense set in 5.1 Generic Lie-bracket generating condition for m = 1

Ξ m 1 = {(A, B, J) ∈ M 6×6 × M 6×m × M 3×3 | J = J T > 0} for m ≥ 1 (
When m = 1 the two matrices B 1 and B 2 are column vectors that we denote by b 1 and b 2 respectively. Then X 1 = (b 1 , b 2 , 0 6 ) and L is the Lie algebra generated by X 0 and X 1 .

Proposition 5.2. Let i = 0 or i = 1.

There exists an open and dense set of triples (A, B, J) in Ξ 1 i such that system (2)-( 4) is Lie-bracket generating. More precisely, if J, b 1 , and b 2 satisfy the three following conditions

det(b 1 , Jb 2 , J 2 b 2 ) = 0, ( 16
) det(b 2 , Jb 2 , J 2 b 2 ) = 0, ( 17
) det J - Jb 2 2 Jb 2 , b 2 Id = 0, ( 18
)
then L is maximal at every point.

Proof. Let us prove, first of all, that if C ∈ M (3 × 3) is symmetric and invertible, and if b ∈ R 3 is such that Cb and C 2 b are linearly independent, then

C -1 (Cb × b) = det(C -1 )(C 2 b × Cb). ( 19
) Notice that C -1 (Cb × b), Cb = 0 = C -1 (Cb × b), C 2 b
, and therefore

C -1 (Cb × b) = k(C 2 b × Cb), (20)
for some k ∈ R. For every

x ∈ R 3 we have det(C -1 x, Cb, b) = (det C -1 ) det(x, C 2 b, Cb) and also det(C -1 x, Cb, b) = C -1 x, Cb × b = x, C -1 (Cb × b) = k x, C 2 b × Cb = k det(x, C 2 b, Cb). Therefore (det C -1 ) det(x, C 2 b, Cb) = k det(x, C 2 b, Cb)
for every x ∈ R 3 , proving that (19) holds true. In particular, taking J = C and b = b 2 , we have

J -1 (Jb 2 × b 2 ) = det(J -1 )(J 2 b 2 × Jb 2 ). ( 21
) Denote by V 1 the Lie bracket [[X 0 , X 1 ], X 1 ]. According to (5) we have V 1 = 2 ((b 2 × b 1 ), J -1 (Jb 2 × b 2 ), 0 6 ).
Similarly, the definition

V 2 = 1 4 [[X 0 , V 1 ], V 1 ] leads to the expression V 2 = (-J -1 (Jb 2 × b 2 ), b 2 b 1 + J -1 (Jb 2 × b 2 ), b 1 b 2 , -J -1 (Jb 2 × b 2 ), b 2 b 2 , 0 6 ). Since J -1 (Jb 2 × b 2 ), b 2 X 1 + V 2 = ( J -1 (Jb 2 × b 2 ), b 1 b 2 , 0 9 ) we have that Z 1 = (b 2 , 0 9 )
belongs to L , as it follows from [START_REF] Keller | A porous prolate-spheroidal model for ciliated micro-organisms[END_REF] and from [START_REF] Sussmann | Some properties of vector field systems that are not altered by small perturbations[END_REF].

Let Z 2 = 1 2 [[X 0 , Z 1 ], V 1 ].
Applying again (5), we have that

Z 2 = (-b 2 × J -1 (Jb 2 × b 2 ), 0 9 ).
Equation [START_REF] Sussmann | Some properties of vector field systems that are not altered by small perturbations[END_REF] implies that

Z 2 = ((det J -1 ) b 2 × (Jb 2 × J 2 b 2 ), 0 9 ). Letting b 3 = b 2 × (Jb 2 × J 2 b 2 ) we can represent Z 3 = 1 2 [[X 0 , Z 2 ], X 1 ] as Z 3 = (b 2 × b 3 , 0 9 ).
We have shown that the Lie algebra L contains the vector fields

Z 1 = (b 2 , 0 9 ), Z 2 = (b 3 , 0 9 ), Z 3 = (b 2 × b 3 , 0 9 ). Notice that b 3 = b 2 , J 2 b 2 Jb 2 -b 2 , Jb 2 J 2 b 2
is not in span(b 2 ), since b 2 , Jb 2 > 0 and because of [START_REF] Lighthill | Mathematical biofluiddynamics[END_REF]. As a consequence, the vectors b 2 , b 3 and b 2 × b 3 are linearly independent. Therefore, L contains every constant vector field of the type (v, 0 9 ), with v in R 3 . It follows that

W 1 = (0 3 , b 2 , 0 6 ) is in L . As a consequence of (5), [[X 0 , W 1 ], W 1 ] = 2(0 3 , J -1 (Jb 2 × b 2 ), 0 6
) lies in L as well. Due to [START_REF] Sussmann | Some properties of vector field systems that are not altered by small perturbations[END_REF], we deduce that 4 ).

W 2 = (0 3 , J 2 b 2 × Jb 2 , 0 6 ) belongs to L . Let b 4 = J 2 b 2 × Jb 2 and let W 3 = [[X 0 , W 1 ], W 2
D = det(b 2 , b 4 , J -1 (Jb 4 × b 2 + Jb 2 × b 4 )) = (det J -1 ) det(Jb 2 , Jb 4 , Jb 4 × b 2 + Jb 2 × b 4 ) = (det J -1 ) Jb 2 , Jb 4 × (Jb 4 × b 2 ) + Jb 4 × (Jb 2 × b 4 ) = (det J -1 ) Jb 2 , Jb 4 , b 4 Jb 2 -Jb 4 , Jb 2 b 4 + Jb 4 , b 2 Jb 4 -Jb 4 2 b 2 . Since Jb 4 , Jb 2 = b 4 , J 2 b 2 = 0 and Jb 4 , b 2 = b 4 , Jb 2 = 0 we have D = (det J -1 ) Jb 2 , Jb 4 , b 4 Jb 2 -Jb 4 2 b 2 = (det J -1 )( Jb 2 2 Jb 4 , b 4 -Jb 4 2 Jb 2 , b 2 ). Now, taking C = J -1 and b = J 2 b 2 in (19), we have Jb 4 = (det J)(Jb 2 × b 2 ), from which we deduce Jb 4 , b 4 = (det J)( Jb 2 , b 2 Jb 2 , J 2 b 2 -Jb 2
We also have

Jb 4 2 = (det J) 2 Jb 2 × b 2 2 = (det J) 2 ( Jb 2 2 b 2 2 -Jb 2 , b 2 2 ). Defining λ = Jb 2 2 / Jb 2 , b 2 we get D = Jb 2 2 ( Jb 2 , b 2 J 2 b 2 , Jb 2 -Jb 2 4 ) -(det J) Jb 2 , b 2 ( Jb 2 2 b 2 2 -Jb 2 , b 2 2 ) = Jb 2 , b 2 3 J 3 b 2 , b 2 Jb 2 , b 2 λ -λ 3 -(det J) b 2 2 Jb 2 , b 2 λ -1 .
Cayley-Hamilton theorem implies that

J 3 = a 2 J 2 + a 1 J + (det J)Id, a 1 , a 2 ∈ R,
and thus

J 2 b 2 , Jb 2 Jb 2 , b 2 = a 2 J 2 b 2 , b 2 Jb 2 , b 2 + a 1 + (det J) b 2 2 Jb 2 , b 2 = a 2 λ + a 1 + (det J) b 2 2 Jb 2 , b 2 .
Therefore,

D = Jb 2 , b 2 3 a 2 λ + a 1 + (det J) b 2 2 Jb 2 , b 2 λ -λ 3 -(det J) b 2 2 Jb 2 , b 2 λ -1 = Jb 2 , b 2 3 (-λ 3 + a 2 λ 2 + a 1 λ + det J) = Jb 2 , b 2 3 det(J -λId)
which is different from zero because of (18).

Generic controllability in the case m = 3

The aim of this section is to prove the genericity of the controllability of (2) in the case m = 3. Let us introduce the notation

A = A 11 A 12 A 21 A 22
with each A ij belonging to M 3×3 .

Proposition 5.3. Let i = 0 or i = 1.

There exists an open and dense set of triples (A, B, J) in Ξ 3 i such that system (2) is controllable. More precisely, if B 2 and Ã11 + ÃT 11 are invertible with

Ã11 = A 11 -B 1 B -1
2 A 21 and if at least one eigenvector of J is not an eigenvector of B 1 B -1 2 , then system (2) is controllable. Propositions 5.2 and 5.3 lead to the following statement, which follows from Proposition 4.1.

Corollary 5.4. For m ≥ 3 and i = 0 or i = 1, there exists an open and dense set of triples (A, B, J) in Ξ 3 i such that system (2)-( 4) is controllable. The proof of Proposition 5.3 will be split in several steps. The following lemma allows us to study the controllability of (2) by investigating the controllability of an equivalent system in R 3 (instead of R 6 ). Lemma 5.5. Assume that B 2 is invertible and define B = B 1 B -1 2 and Ã11 = A 11 -BA 21 . Then, the controllability of (2) is equivalent to the controllability of the system

ẋ = Ã11 x -v × x + ( Ã11 B + A 12 -BA 22 )v -v × Bv -BJ -1 (Jv × v), x ∈ R 3 , (22)
with the control v taking values in R 3 .

Proof. Suppose that system (2) is controllable. Let x = ξ -Bω and rewrite system (2) as

ẋ = Ã11 x -ω × x + ( Ã11 B + A 12 -BA 22 )ω -ω × Bω -BJ -1 (Jω × ω), ω = A 21 x + (A 21 B + A 22 )ω + J -1 (Jω × ω) + B 2 u.
Given two points (x 0 , ω 0 ) and (x 1 , ω 1 ) in R 6 , there exists a control u : [0, T ] → R 3 that steers (x 0 , ω 0 ) to (x 1 , ω 1 ). Thus, denoting by (x(•), ω(•)) the corresponding trajectory, the control v(t) = ω(t) steers system (22) from x 0 to x 1 .

Conversely, let system (22) be controllable and fix two pairs (ξ 0 , ω 0 ) and (ξ 1 , ω 1 ) in R 6 . Notice that ( 22) can be controlled by smooth controls v :

[0, T ] → R 3 satisfying v(0) = ω 0 , v(T ) = ω 1 . (23)
(The result follows from the controllability of ( 22) and the density of

{v ∈ C ∞ ([0, T ], R 3 ) | v(0) = ω 0 , v(T ) = ω 1 } in L ∞ (0, T ) with respect to the L 1 -norm.
The proof can be deduced from the general results in [START_REF] Grasse | Global controllability by nice controls[END_REF] or easily adapted from [START_REF] Brunovský | Controlabilité Bang Bang, controlabilité différentiable, et perturbation des systèmes non linéaires[END_REF] and [START_REF] Jurdjevic | Geometric control theory[END_REF]Theorem 4,p. 110].) Therefore, if v ∈ C ∞ ([0, T ], R 3 ) satisfies (23) and steers ( 22) from ξ 0 -Bω 0 to ξ 1 -Bω 1 , then

u(t) = B -1 2 v(t) -A 21 x(t) -(A 21 B + A 22 )v(t) -J -1 (Jv(t) × v(t))
steers ( 2) from (ξ 0 , ω 0 ) to (ξ 1 , ω 1 ).

The following lemma establishes the controllability of system [START_REF] Taylor | Analysis of the swimming of microscopic organisms[END_REF] in a first case.

Lemma 5.6. Assume that the matrix Ã11 + ÃT 11 has two nonzero eigenvalues with opposite signs. Then there exists an open and dense subset

O of {(B, J) ∈ M 6×3 × M 3×3 | J = J T > 0} such that system (22) is controllable if (B, J) ∈ O. More precisely, if B 2 is invertible and if at least one eigenvector of J is not an eigenvector of B 1 B -1
2 , then system (22) is controllable. For the proof of this lemma we shall use the following result, proven in [START_REF] Jurdjevic | Controllability properties of affine systems[END_REF], about the controllability of a family of affine vector fields in R n . An affine vector field X is a mapping from R n to R n of the form X : x → Cx + c where C ∈ M n×n and c is a constant vector in R n ; the linear part of X, denoted by Λ(X), is the linear vector field x → Cx. If F is a family of affine vector fields, we denote by Λ(F ) the set of linear parts of the vector fields in F and we say that F has no fixed point if there does not exist a point x 0 ∈ R n such that X(x 0 ) = 0 for every X ∈ F . Theorem 5.7 (Jurdjevic and Sallet). Let F be a family of affine vector fields given on R n . Assume that F has no fixed point.

If Λ(F ) is controllable on R n {0} then F is controllable on R n .
Proof of Lemma 5.6. From Lemma 5.5, we know that it is sufficient to prove the controllability of the system defined by the family of affine vector fields

F = { C(v)x + c(v) | v ∈ R 3 } where C(v) = Ã11 -S(v), c(v) = ( Ã11 B + A 12 -BA 22 )v -v × Bv -BJ -1 (Jv × v). Choose v 0 in such a way that S(v 0 ) = ( Ã11 -ÃT 11 )/2. Then Ã11 -S(v 0 ) = Ã11 + ÃT 11 2 .
Define the family of linear vector fields

G = { Ã11 -S(v 0 ), S(v) | v ∈ R 3 }.
The two closed convex cones generated by the families Λ(F ) and G are identical, since for every finite family (α i ) 0≤i≤N of positive numbers and every choice of v 1 , . . . , v N ∈ R 3 ,

α 0 ( Ã11 -S(v 0 )) + N i=1 α i S(v i ) = α 0 ( Ã11 -S(v 0 )) + lim r→+∞ 1 r N i=1 α i Ã11 -S(-r v i ) and N i=1 α i ( Ã11 -S(v i )) = N i=1 α i Ã11 -S(v 0 ) + N i=1 α i S(v 0 -v i ).
Therefore, the controllability of Λ(F ) on R 3 {0} is equivalent to the one of G (see [START_REF] Jurdjevic | Control systems subordinated to a group action: accessibility[END_REF][START_REF] Jurdjevic | Control systems on semi-simple lie groups and their homogeneous sapces[END_REF]). The trajectories of the vector field S(v) are circles contained in planes orthogonal to v and whose centers are at the intersections of these planes with the line Rv. Let x ∈ R 3 {0}; thanks to the vector fields S(v), the attainable set from x for G contains the sphere of center 0 3 passing through x. Now, as Ã11 -S(v 0 ) has two eigenvalues with opposite sign, we can move (thanks to this vector field) along a direction towards the origin and along an half-line exiting the sphere and going to infinity. Therefore, using again the fact that the family {S(v) | v ∈ R 3 } is transitive on every sphere, we proved that G , and thus Λ(F ), is controllable on R 3 {0}.

Assume now that we can find a fixed point, denoted by x 0 , which is common to all the vector fields in F . For every vector v ∈ R 3 , we have

C(v)x 0 + c(v) = C(-v)x 0 + c(-v) = 0 3 from which we deduce Ã11 x 0 -v × Bv -BJ -1 (Jv × v) = 0. (24)
Taking v = 0 3 gives Ã11 x 0 = 0 3 and therefore (24) can be rewritten as

Bv × v = BJ -1 (Jv × v) (25)
for every v ∈ R 3 . In particular, if v is an eigenvector of J but not of B, then the right-hand side of (25) is equal to zero, while the right-hand side is not, leading to a contradiction.

We are left to deal with the case where the eigenvalues of

Ãs 11 = Ã11 + ÃT 11 2 ,
the symmetric part of Ã11 , are all positive or all negative. Although in this case the linear part of the family F is not controllable on R 3 {0}, we can nevertheless adapt the method introduced in [START_REF] Jurdjevic | Controllability properties of affine systems[END_REF] in order to prove the controllability of ( 22).

Lemma 5.8. Assume that the matrix Ãs 11 is invertible and that its eigenvalues have all the same sign. Then there exists an open and dense subset

O of {(B, J) ∈ M 6×3 × M 3×3 | J = J T > 0} such that system (22) is controllable if (B, J) ∈ O.
More precisely, if B 2 is invertible and if at least one eigenvector of J is not an eigenvector of B, then system (22) is controllable.

Notice that, together with Lemma 5.6, Lemma 5.8 concludes the proof of Proposition 5.3. In order to show Lemma 5.8 we need some definitions and have to prove some intermediate results.

Let F and Λ(F ) be defined as above. In the proof of Lemma 5.6 we pointed out that starting form a point x ∈ R 3 and following all possible linear vector fields of the type S(v) one can attain the entire sphere of center 0 3 and radius x . If the eigenvalues of Ãs 11 are negative (resp. positive), the vector field Ãs 11 points towards the interior (resp. the exterior) of this sphere, from which we deduce that

A(x, Λ(F )) = {x} ∪ {y ∈ R 3 | 0 < y < x } if Ãs 11 < 0, {x} ∪ {y ∈ R 3 | y > x }
if Ãs 11 > 0. In the sequel, we shall need the notion of normal accessibility, which is recalled below. Definition 5.9. Let V be a family of complete vector fields on a manifold M . The point y is said to be normally V -accessible from x if there exist X 1 , . . . , X p in V and t 1 , . . . , t p > 0 such that

y = e t 1 X 1 • • • • • e tpX p (x)
and the mapping

(τ 1 , . . . , τ p ) -→ e τ 1 X 1 • • • • • e τpX p (x),
defined in a neighborhood of (t 1 , . . . , t p ), is of rank equal to dim M at (t 1 , . . . , t p ).

In [START_REF] Jurdjevic | Controllability properties of affine systems[END_REF] the authors exploit the fact that if a family of vector fields is controllable, then every point y is normally accessible from every point x. As the family Λ(F ) is not controllable on R 3 {0}, we are lead to prove directly the following normal accessibility property. Lemma 5.10. Assume that the eigenvalues of Ã11 + ÃT 11 have all the same sign. Then we can extract from F a finite family F 0 such that there exists a sphere of center 0 3 and radius r of points normally Λ(F 0 )-accessible from every point of the unit sphere.

Proof. Let v 0 ∈ R 3 be such that Ãs 11 = Ã11 -S(v 0 ) = Ãs 11 . Let, moreover, w 1 be an eigenvector of Ãs 11 and take w 2 , w 3 ∈ R 3 such that the family (w 1 , w 2 , w 3 ) is an orthogonal basis of R 3 . Consider now the family

Λ(F 0 ) = { Ãs 11 , Ãs 11 -k 2 S(w 2 ), Ãs 11 -k 3 S(w 3 ) } extracted from Λ(F ).
Since S(w 2 ) and S(w 3 ) are nonzero skew-symmetric matrices, each of their spectra contains the value zero and two nonzero purely imaginary eigenvalues. Notice that lim

k j →+∞ Ã11 -k j S(w j ) k j = -S(w j ).
As the roots of a characteristic polynomial depend continuously on the coefficients of the related matrix, it follows that for k 2 and k 3 large enough Ãs 11k 2 S(w 2 ) and Ãs 11k 3 S(w 3 ) have a pair of non-real eigenvalues. By projecting the vector fields Ãs 11k 2 S(w 2 ) and Ãs 11k 3 S(w 3 ) on the unit sphere, it is easy to see that the family Λ(F 0 ) is transitive on the directions, that is, from any half-line starting from the origin, one can reach any other half-line. Moreover, thanks to the eigendirection w 1 of the vector field Ãs 11 we can go as far as we want (case Ãs 11 > 0) or as close as we want to the origin (case Ãs 11 < 0). Hence, for every x ∈ R 3 {0 3 }, A(x, Λ(F 0 )) contains a set of the form R 3 B(0, r) (case Ãs 11 > 0) or B(0, r) {0 3 } (case Ãs 11 < 0). The interior of A(x, Λ(F 0 )) being nonempty, there exists at least one point, denoted by y, that can be normally Λ(F 0 )-accessed from x (see [START_REF] Sussmann | Some properties of vector field systems that are not altered by small perturbations[END_REF]). Notice that since y is normally Λ(F 0 )-accessible from x, then y is also normally Λ(F 0 )-accessible from every point in a (sufficiently small) neighborhood of x. Moreover, all the points in A(y, Λ(F 0 )) are normally Λ(F 0 )-accessible from x.

We conclude by using the compactness of the unit sphere: for every x ∈ S 2 , there exists a neighborhood V x of x and a set R 3 B(0, r x ) (case Ãs 11 > 0) or B(0, r x ) {0 3 } (case Ãs 11 < 0) whose points are normally Λ(F 0 )-accessible from every point of V x . As we can include the unit sphere in a finite union of neighborhoods V x , we can claim the existence of a sphere of radius r whose points are normally Λ(F 0 )-accessible from every point of the unit sphere.

The following lemma guarantees the unboundedness of the sets A(x, F ) and A(x, -F ).

Lemma 5.11. If B 2 is invertible and if at least one eigenvector of J is not an eigenvector of B, then for every x ∈ R 3 both sets A(x, F ) and A(x, -F ) are unbounded.

Proof. Take v ∈ R 3 such that q(v) = -v × Bv -BJ -1 (Jv × v) = 0 3 ; this is possible since at least one eigenvector of J is not an eigenvector of B. For every α ∈ R the vector field x → C(αv)x + c(αv) is in F and

lim α→∞ C(αv)x + c(αv) α 2 = q(v).
This proves that, given a time T > 0, the solution of ẋ = C(αv)x + c(αv) on the interval [-T α -2 , T α -2 ] is as close as we want to the solution of ẋ = q(v) on [-T, T ] (with the same initial condition) provided that α is large enough. Now, the solution of ẋ = q(v), x(0) = x 0 , leaves any fixed bounded set both in time T and in time -T , provided that T is large enough.

We consider now, as in [START_REF] Jurdjevic | Controllability properties of affine systems[END_REF], the family {h λ,w | λ > 0, w ∈ R n } of affine diffeomorphisms of R n defined by h λ,w (x) = w + λ(xw). An easy computation shows that, for every X ∈ F ,

(h λ,w * X)(x) = Λ(X)(x -w) + λ X(w)
where h λ,w * X denotes the pushforward of the vector field X by h λ,w . In particular

lim λ→0 (h λ,w * X)(x) = Λ(X)(x -w). ( 26
)
Denote by F λ,w the image of the family F by h λ,w * , that is, the transformation of F under the change of coordinates h λ,w .

Proof of Lemma 5.8. Let Λ(F 0 ) and r be as in the statement of Lemma 5.10. Consider the finite family Λ(F 0 w ) = { Λ(X)(xw) | X ∈ F 0 } and notice that each point of the sphere of center w and radius r is normally Λ(F 0 w )-accessible from every point of the sphere of center w and radius 1. Let F λ,w = {h λ,w * X | X ∈ F }. Thanks to formula (26) we can assert that, if λ is chosen sufficiently small, the sphere of center w and radius r is contained in the reachable set for F λ,w from every point of the sphere of center w and radius 1 (see [START_REF] Sussmann | Some properties of vector field systems that are not altered by small perturbations[END_REF]Lemma 3.2]). Fix such a λ > 0.

Given w in R 3 we claim that A(w, F ) contains a neighborhood of w. Indeed, since the sets A(x, F ) and A(x, -F ) are unbounded, the same is true for the sets A(x, F λ,w ) and A(x, -F λ,w ), for every x ∈ R 3 . This implies in particular, because of the arc-connectedness of attainable sets, that there exists y ∈ A(w, F λ,w ) such that yw = 1. Let ρ < r and fix z ∈ B(w, ρ). Since A(z, -F λ,w ) is unbounded, then, again by arc-connectedness, there exists z ′ ∈ A(z, -F λ,w ) such that z ′ -w = r. Since from every point of the sphere of center w and radius 1, we can reach any point of the sphere of center w and radius r, the point z ′ is reachable from y by the family F λ,w . Finally, every point z ∈ B(w, ρ) belongs to A(w, F λ,w ) (see Figure 1). Since F λ,w is the transformation of F by a diffeomorphism preserving w, we deduce that A(w, F ) contains a neighborhood of w. Therefore, every attainable set A(x, F ) is open.

Since the control system defined by -F has the same form as that defined by F and verifies the hypotheses of both Lemmas 5.10 and 5.11, the reasoning above proves also that every attainable set A(x, -F ) is open. Take now y in the closure of A(x, F ). The set A(y, -F ) being open, there exists z ∈ A(y, -F ) ∩ A(x, F ) which proves that y is reachable from x by the family F . Every attainable set A(x, F ) is therefore both open and closed; thus it is equal to R 3 .

Generic controllability: physical interpretation

The scope of this section is to provide a better physical insight of Theorem 5.1. More precisely, we want to check that for m ≥ 3 there exists an open and dense set in the (suitably defined) space of microscopic organisms modeled here such that the corresponding control system (2)-( 4) is controllable.

The space of organisms, denoted by Σ m , will be identified with the set of pairs (Ω, Ψ), where Ω ⊂ R 3 is open, nonempty, connected, bounded, and of class C 2 , while Ψ is the m-uple of functions in C 2 (∂Ω, R 3 ) determining the action of the controls (see Section 3). For simplicity we assume that the density δ of the organism is constant, so that the inertia matrix J is determined (up to the positive multiplicative constant δ) by the shape Ω. (For the general case see Remark 5.14.) The topology on Σ m can be defined assigning the basis of open subsets defined by

N m,ε (Ω, Ψ) = ((Id + υ)(Ω), Φ) | υ ∈ W 3,∞ (Ω, R 3 ), Φ ∈ (C 2 (∂Ω, R 3 )) m , υ 3,∞ < ε, Φ • (Id + υ) -Ψ C 2 (Ω) < ε
for all ε ∈ (0, 1/2) and all (Ω, Ψ) ∈ Σ m (see [START_REF] Simon | Différentiation de problèmes aux limites par rapport au domaine[END_REF] or, for a different approach, [START_REF] Chitour | On conditions that prevent steadystate controllability of certain linear partial differential equations[END_REF]). By extension Σ 0 denotes the set of C 2 , open, nonempty, connected, bounded subsets of R 3 endowed with the topology whose basis is given by all

N 0,ε (Ω) = (Id + υ)(Ω) | υ ∈ W 3,∞ (Ω, R 3 ), υ 3,∞ < ε , ε ∈ (0, 1/2), Ω ∈ Σ 0 .
Theorem 5.12. Assume that m ≥ 3.

There exists an open and dense set O in Σ m such that (2)-( 4) is controllable if (Ω, Ψ) belongs to O. Moreover, there exists an open and dense set

P in Σ 0 such that O ∩ ({Ω} × (C 2 (∂Ω, R 3 )) m ) is dense in {Ω} × (C 2 (∂Ω, R 3
)) m for every Ω in P.

Proof. Denote by F (Ω, Ψ) the triple (A, B, J) associated to an element (Ω, Ψ) of Σ 0 through ( 6), ( 8) and [START_REF] Grasse | Global controllability by nice controls[END_REF]. We want to show that each condition appearing in the statement of Propositions 5.2 and 5.3 is satisfied by the elements of the image through F of an open and dense subset of Σ 0 . Notice that the map Ψ → B is onto when considered from (C 2 (∂Ω, R 3 )) m to M 6×m (Ω fixed). Indeed, according to [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF]Lemma 4.1], g (1) , g (2) , g (3) , G (1) , G (2) , G (3) are linearly independent in L 2 (∂Ω, R 3 ). Therefore, since the orthogonal to a smooth function in C 2 (∂Ω, R 3 ) with respect to the L 2 -product has infinite codimension, the relation (10) defines a surjective map. In particular, the pre-image of an open and dense set in M 6×m is open and dense in (C 2 (∂Ω, R 3 )) m .

Let us now take into account the dependence on Ω. We claim that the map from Σ 0 to R 3 that associates to a domain Ω the spectrum of the corresponding inertia matrix -with eigenvalues repeated according to their multiplicity and with δ normalized to one-is locally open at every point. In order to check it, fix Ω ∈ Σ 0 and a system of coordinates such that 0 3 is the center of mass of Ω and e 1 , e 2 , e 3 its principal axes of inertia. Then the elements of the normalized inertia matrix J 0 = J/δ of Ω are

J 0 ij = ± Ω x i x j dx = 0 if i = j, Ω ( k =i x 2 k )dx if i = j. (27)
Apply the transformation Id + υ to Ω, where υ is the diagonal matrix diag(υ 1 , υ 2 , υ 3 ), and denote by J 0 (υ) the normalized inertia matrix of (Id + υ)(Ω). Then

J 0 ij (υ) = ±(Π 3 k=1 (1 + υ k ))(1 + υ i )(1 + υ j ) Ω x i x j dx = 0 if i = j, ±(Π 3 k=1 (1 + υ k )) Ω ( k =i (1 + υ k ) 2 x 2 k )dx if i = j.
The spectrum of J 0 (υ) is therefore given by the components of the vector σ 0 (υ) = (±(

Π 3 k=1 (1 + υ k )) Ω ( k =i (1 + υ k ) 2 x 2 k )dx) 3 i=1 .
A straightforward computation shows that the determinant of the Jacobian matrix of σ 0 with respect to (υ 1 , υ 2 , υ 3 ), computed at υ 1 = υ 2 = υ 3 = 0, is different form zero. The map υ → σ 0 (υ) is therefore a submersion at υ = 0.

In particular, the eigenvalues of J are simple for Ω in an open and dense subset of Σ 0 , independently of Ψ. If b 2 does not belong to any of the three planes generated by pairs of eigenvectors of J, then ( 17) is automatically verified.

Similarly, one notices that all the assumptions appearing in the statements of Propositions 5. Remark 5.13. As recalled above it follows from [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF] that, once Ω is fixed, the linear map associating to the m-uple Ψ the matrix B through [START_REF] Grasse | Global controllability by nice controls[END_REF] is onto as a map from (L 2 (∂Ω, R 3 )) m to M 6×m . Thanks to this result, and to the remark that (2)-( 4) is controllable if m = 6 and B is invertible, the authors prove that for m ≥ 6, for an open and dense subset of m-uples in C 2 (∂Ω, R 3 ) (Ω fixed) the corresponding system is controllable (see [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF]Theorem 1.1]). In the language adopted here the result of [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF] says that, for every fixed Ω and for m ≥ 6, controllability is a generic property with respect to Ψ. Remark 5.14. In order to introduce a reasonable notion of genericity in the case where the density δ of the organism is not constant, it is necessary to include δ in the definition of Σ m . Let Σm be the set of triples (Ω, Ψ, δ) where (Ω, Ψ) ∈ Σ m and δ ∈ L ∞ (Ω, (0, +∞)) and endow Σm with the topology whose basis is given by all

Nm,ε (Ω, Ψ, δ) = ((Id + υ)(Ω), Φ, γ) | υ ∈ W 3,∞ (Ω, R 3 ), Φ ∈ (C 2 (∂Ω, R 3 )) m , γ ∈ L ∞ (Ω, (0, +∞)), υ 3,∞ < ε, Φ • (Id + υ) -Ψ C 2 (Ω) < ε, γ • (Id + υ) -δ L ∞ (Ω) < ε , ε ∈ (0, 1/2), (Ω, Ψ, δ) ∈ Σm .
Following the same arguments of proof as above, one can prove that if m ≥ 3 there exists an open and dense set O in Σm such that (2)-( 4) is controllable if (Ω, Ψ, δ) belongs to O. The only difference consists in verifying that the map associating to (Ω, δ) the spectrum of J is locally open, and this can be done by taking the same perturbation (Id + diag(υ 1 , υ 2 , υ 3 ))(Ω) as above for Ω and the perturbation δ • (Id + diag(υ 1 , υ 2 , υ 3 )) for δ.

Swimming spherical microscopic organisms

When the microscopic organism happens to be spherical, the equations presented in the previous sections have a very special form. As described in [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF], general results in hydrodynamics (see, e.g., [START_REF] Happel | Low Reynolds number hydrodynamics with special applications to particulate media[END_REF]) show that the matrix A corresponding to a spherical organism is diagonal and, more precisely, of the form

A = -ρ 1 Id 0 0 -ρ 2 Id , with ρ 2 > ρ 1 > 0. ( 28
)
Moreover, since the inertia matrix J is proportional to the identity (assuming that the mass distribution is homogenous inside Ω), the nonlinear term E appearing in ( 2) is given by

E(z) = ω × ξ 0 3 .
Notice that in the spherical case ω satisfies a well-defined control subsystem, ie,

ω = -ρ 2 ω + B 2 u. (29)
We are going to prove the following result.

Theorem 6.1. Let Ω be a ball and assume that the mass distribution of Ω is homogenous. Then the control system (2)-( 4) is controllable if and only if the rank of B 2 is equal to three and B 1 is different from 0 3×m .

The easier part of the proof is to show that the controllability of ( 2)-( 4) implies that rank

B 2 = 3, B 1 = 0 3×m . (30)
Indeed, the linear control system (29) is, as it is well known, controllable if and only if rank B 2 = 3. Moreover, if B 1 = 0 3×m , then the space {0 3 } × R 3 is invariant for the dynamics of the control system (2), which is therefore non-controllable.

The converse implication will be proven in several steps.

6.1

The case where B 1 and B 2 are linearly independent

In this section we study the case where the organism is spherical and

rank B 2 = 3, B 1 ∈ span(B 2 ). ( 31
)
Let us prove the following technical result.

B 1 Γv = 3 i=1 v i c i , B 2 Γv = 3 i=1 v i d i , c 1 × d 1 = 0 3 . (32)
Proof. Firstly, let us consider the case m = 3. Then B 2 is invertible and for every orthonormal basis {d 1 , d 2 , d 3 } of R 3 , we have the j-th column of B i . The second case we consider is when there exist 1 ≤ j 1 , j 2 , j 3 ≤ m such that B

B 2 Γv = 3 i=1 v i d i , for Γ = B -1 2 (
(j 1 ) 2 , B (j 2 ) 2 , B (j 3 ) 2
are linearly independent and (B

(j 1 ) 1 |B (j 2 ) 1 |B (j 3 ) 1 ) ∈ span(B (j 1 ) 2 |B (j 2 ) 2 |B (j 3 )
2 ). Then, replacing v by Γ ′ v = (v j 1 , v j 2 , v j 3 ), we fulfill the hypotheses of the case m = 3 and we are done.

Finally, consider the case in which for every triple 1 ≤ j 1 , j 2 , j 3 ≤ m such that B

(j 1 ) 2 , B (j 2 ) 2 , B (j 3 ) 2
are linearly independent, we have that (B

(j 1 ) 1 |B (j 2 ) 1 |B (j 3 ) 1 ) is proportional to (B (j 1 ) 2 |B (j 2 ) 2 |B (j 3 ) 2 ). Let {j 1 , . . . , j k } = {j | 1 ≤ j ≤ m, B (j)
2 = 0 3 }.

Then (B

(j 1 ) 1 | • • • |B (j k ) 1 ) is proportional to (B (j 1 ) 2 | • • • |B (j k )
2 ). Assumption (31) implies that there exists 1 ≤ j ≤ m such that B (j) 1 = 0 3 and B (j) 2 = 0 3 . It suffices then to replace v by Γ ′ v = (v j 1 + v j , v j 2 , . . . , v j k ) in order to satisfy the hypotheses of the previous case.

Notice now that for every Q ∈ SO(3), the change of coordinates ξ ′ = Qξ, ω ′ = Qω, ζ ′ = ζ, R ′ = RQ -1 preserves the dynamics of ( 2)-( 4). This is equivalent to say that, up to a change of coordinates, we can replace (B 1 , B 2 ) by (QB 1 , QB 2 ), for every choice of Q ∈ SO(3). Let d 1 , d 2 , d 3 be as in (32) and choose Q such that Qd i = e i for i = 1, 2, 3 (where we recall that (e i ) i=1,2,3 denotes the canonical basis in R 3 ). The above transformations show that proving the controllability of the control system (2)-( 4) under assumption (31) is equivalent to proving it under the hypotheses that m = 3, B 2 = Id, and that the first column of B 1 is not proportional to e 1 .

Denote by b 1 , b 2 , b 3 the columns of B 1 . Accordingly, the controlled vector fields are the three constant vector fields X i = (b i , e i , 0 6 ), i = 1, 2, 3. Lemma 6.3. If (31) holds then system (2)-( 4) is Lie-bracket generating.

Proof. Applying (5) in the case J = Id we obtain that for every v 1 , v 2 ∈ R 3 the constant vector field

V = (v 1 , v 2 , 0 6 ) satisfies [[X 0 , X i ], V ] = (e i × v 1 -b i × v 2 , 0 9 ). (33)
In particular, taking i = 1 and V = X 1 in (33), we deduce that Z = (e 1 × b 1 , 0 9 ) belongs to L . Applying again (33) to V = Z we get that, for i = 1, 2, 3,

Z i = (e i × (e 1 × b 1 ), 0 9 ) belongs to L . Since Z, Z 1 , Z 2 , and Z 3 span R 3 × {0 9 }, it follows that L contains span(X 1 , X 2 , X 3 ) + R 3 × {0 9 } = R 6 × {0 6 }.
As noticed in Remark 3.1 this proves the lemma. The controllability of system (2) follows from Proposition 5.3: indeed, Ã11 = ρ 1 Id is symmetric and invertible and hypothesis (31) implies that B 1 = B 1 B -1 2 has a smaller set of eigenvectors than J = Id. Theorem 6.1 is therefore proven, because of Proposition 4.1, under the stronger assumption that (31) holds true.

The proportional case

In order to complete the proof of Theorem 6.1, we have to prove that system (2)-( 4) is controllable when rank

B 2 = 3, B 1 = λB 2 , λ = 0. ( 34
)
Using the same normalization argument as in the previous section, we can assume that m = 3 and B 2 = Id. Moreover, we can take λ = 1 by performing the change of coordinates ξ ′ = ξ/λ, ω ′ = ω, ζ ′ = ζ/λ, R ′ = R. Therefore, without loss of generality, X i = (e i , e i , 0 6 ) for i = 1, 2, 3.

Let us prove the Lie-bracket generating condition. We have

[X 0 , X i ] = (-ρ 1 e i , -ρ 2 e i , Re i , RS(e i ))
and [[X 0 , X i ], [X 0 , X j ]] = (0 6 , 2R(e j × e i ), RS(e j × e i )).

A further computation yields

[[[X 0 , X i ], [X 0 , X j ]], [X 0 , X k ]] = (0 6 , 3R(e k × (e j × e i )), RS(e k × (e j × e i ))).
Therefore, the Lie algebra L contains the following vector fields

    e i e i 0 3 0 3     ,     -ρ 1 e j -ρ 2 e j Re j RS(e j )     ,     0 3 0 3 2Re k RS(e k )     ,     0 3 0 3 3Re l RS(e l )     , for i, j, k, l = 1, 2, 3. As a consequence, L contains also     (ρ 2 -ρ 1 )e i 0 3 Re i RS(e i )     ,     0 3 (ρ 1 -ρ 2 )e j Re j RS(e j )     ,     0 3 0 3 Re k 0 3     ,     0 3 0 3 0 3 RS(e l )     ,
for i, j, k, l = 1, 2, 3. These vector fields form a moving frame on the manifold R 9 × SO(3) (since

ρ 1 = ρ 2 ).
In order to conclude the proof of Theorem 6.1 we need to show that (2) is controllable. Applying Lemma 5.5 this turns out to be equivalent to the controllability of the following system,

ẋ = -ρ 1 x -v × x + (ρ 2 -ρ 1 )v, x ∈ R 3 , (35)
with the control v taking values in R 3 . Decomposing the control as v = wx + W with w ∈ R and W orthogonal to x, equation (35) rewrites as

ẋ = (-ρ 1 + (ρ 2 -ρ 1 )w)x -W × x.
Clearly, choosing W = 0 3 allows to move as desired along span(x), while the choice w = ρ 1 /(ρ 2ρ 1 ) permits to attain the sphere of radius x centered at the origin. The controllability of (2) is thus proven and the proof of Theorem 6.1 completed.

Let Y 3 = [Y 1 , Y 2 ] whose expression can be obtained using (46). Define the vector field W 1 as the bracket between Y 3 and Y 1 . Applying again (46) we obtain that

W 1 (ζ, R) = [Y 3 , Y 1 ](ζ, R) = R 2 β 1 , γ 1 γ 2 -( β 1 , γ 2 + β 2 , γ 1 )γ 1 -γ 1 , γ 2 β 1 + γ 1 2 β 2 R S(-γ 1 , γ 2 γ 1 + γ 1 2 γ 2 ) .
Let W 2 be the vector field obtained adding γ 1 , γ 2 Y 1γ 1 2 Y 2 to W 1 , ie,

∆ = β 1 β 2 γ 2 × β 1 -γ 1 × β 2 γ 1 × γ 2 γ 1 × (γ 1 × γ 2 ) γ 2 × (γ 1 × γ 2 ) γ 1 γ 2 γ 2 × c 1 0 3 0 3 0 3
is full-rank. The non-degeneracy of ∆ follows easily from the assumption that the vectors γ 1 and γ 2 are linearly independent. We shall prove now that the hypotheses 1. and 2. imply the existence of two vector fields Y 1 and Y 2 as above.

Let r = rank(L 2 ). Notice that for every matrix Γ ∈ GL(m) the reparameterization u → Γu transforms the matrix (L 1 , L 2 ) into (L 1 Γ, L 2 Γ). The condition (L 1 )

T L 2 + (L 2 ) T L 1 = 0 m×m is preserved by this transformation, since, (L 1 Γ)

T L 2 Γ + (L 2 Γ) T L 1 Γ = Γ T ((L 1 ) T L 2 + (L 2 )
T L 1 )Γ. Thus, without loss of generality, we can assume that L 2 = (L 2,1 |0 3×(m-r) ) where L 2,1 ∈ M 3×r is of rank r.

Let us write L 1 as (L 1,1 | L 1,2 ) where L 1,1 ∈ M 3×r and L 1,2 ∈ M 3×(m-r) .

Suppose that 1. holds. If (L 1,1 ) T L 2,1 + (L 2,1 ) T L 1,1 = 0 r×r , then we can find 1 ≤ j 1 , j 2 ≤ r such that b j 1 , c j 1 = 0, or b j 2 , c j 2 = 0, or b j 1 , c j 2 + b j 2 , c j 1 = 0. If b j 1 , c j 1 = b j 2 , c j 2 = 0, then the inner product b j 1 + b j 2 , c j 1 + c j 2 = b j 1 , c j 2 + b j 2 , c j 1 is nonzero, and thus the vector fields Y 1 = X j 1 + X j 2 and Y 2 = X j 2 fulfill the required conditions. Otherwise, (L 2,1 ) T L 1,2 = 0 r×(m-r)

and we can fix 1 ≤ j 1 , j 2 ≤ r and j 3 > r such that j 1 = j 2 , b j 1 , c j 1 = 0 and b j 3 , c j 1 = 0. Then Y 1 = X j 1 + X j 3 and Y 2 = X j 2 fulfill the required conditions. Assume now that 2. holds and 1. does not. Then c 1 and c 2 are linearly independent and As c 2 , c 1 × (c 1 × c 2 ) = 0 the vector fields Y 1 = X 1 and Y 2 = X 2 + [X 1 , X j ] fulfill the required conditions. Let us prove now the "only if" part of the statement. First notice that if r ≤ 1 then there exists a vector x 0 ∈ R 3 {0 3 } such that L 2 u × x 0 = 0 for every control u. Multiplying equation (44) by x 0 , we get that Ṙx 0 = 0 and so R(t)x 0 = R(0)x 0 for every time t. Therefore, system (44) is not controllable.

] = -c 1 , c 2 X 1 + c 1 2 X 2 [Y, X 2 ] = -c 2 2 X 1 + c 1 , c 2 X 2 .
Therefore, the Lie algebra L , which is generated by X 1 and X 2 , is equal to the linear space of vector fields spanned by X 1 , X 2 and Y and cannot be of full rank. Let now r = 3 and (L 1 ) T L 2 + (L 2 ) T L 1 = 0 m×m . The condition (L 2,1 ) T L 1,2 = 0 3×(m-3) implies that the columns of L 1,2 are orthogonal to all the elements of a basis of R 3 . Therefore, L 1,2 = 0 3×(m-3) . As for the columns of L 1,1 , we easily obtain from (L 1,1 ) T L 2,1 + (L for some α 12 , α 13 , α 23 ∈ R. Without loss of generality we can assume that (c 1 , c 2 , c 3 ) is a positively oriented orthonormal basis and an easy computation gives

[X 1 , X 2 ] =X 3 , [X 1 , X 3 ] = -X 2 , [X 2 , X 3 ] =X 1 ,
which proves that L is equal to the linear space of vector fields spanned by X 1 , X 2 and X 3 and cannot be of full rank.

  for the Lie-bracket generating condition) or m ≥ 3 (for the controllability) whose intersection with Ξ m 0 is open and dense in Ξ m 0 . In other words, Lie-bracket generating condition and controllability are generic properties (for m = 1 and m = 3, respectively) also in the class of systems of the form (2)-(4) for which the triple (A, B, J) is taken in Ξ m 1 .

  ]; we have W 3 = (0 3 , b 5 , 0 6 ) with b 5 = J -1 (Jb 4 × b 2 + Jb 2 × b 4 ). We are left to prove that the vectors b 2 , b 4 and b 5 are linearly independent. Let D = det(b 2 , b 4 , b 5 ) and notice that

Figure 1 :

 1 Figure 1: z belongs to A(w, F λ,w ) (the case r < 1).

  2 and 5.3 can be represented in the form G(Ω, Ψ) ∈ O where O is an open and dense subset of a finite-dimensional vector space V , the map G : Σ m → V is continuous and for Ω in an open and dense subset of Σ 0 we have that G(Ω, Ψ) belongs to O if B belongs to an open and dense subset of M 6×m (possibly depending on Ω). Therefore, G -1 (O) is open and dense in Σ m .

Lemma 6 . 2 .

 62 If assumption (31) holds true, then there exist c 1 , c 2 , c 3 ∈ R 3 , an orthonormal basis {d 1 , d 2 , d 3 } of R 3 , and a m × 3 matrix Γ such that

W 2 ( 3 .

 23 ζ, R) = R (2 β 1 , γ 1 γ 2 -( β 1 , γ 2 + β 2 , γ 1 ) γ 1 ) 0 Since W 2 is in L , then the vector field Y 4 = [W 2 , Y 1 ]belongs to L as well and we have thatY 4 (ζ, R) = (R(2 β 1 , γ 1 γ 1 × γ 2 ), 0 3 ). Consider now the vector fields Y 5 = [Y 4 , Y 1 ] and Y 6 = [Y 4 , Y 2 ],both belonging to L , and whose expressions areY 5 (ζ, R) = (R(2 β 1 , γ 1 γ 1 × (γ 1 × γ 2 )), 0 3 ) and Y 6 (ζ, R) = (R(2 β 1 , γ 1 γ 2 × (γ 1 × γ 2 )), 0 3 ). Since each matrix R in SO(3)is invertible, and because β 1 , γ 1 = 0, the vector fields Y 1 , . . . , Y 6 span T (ζ,R) R 3 × SO(3) at every (ζ, R) ∈ R 3 × SO(3) if and only if the matrix

b 1 , c 1 =

 11 b 2 , c 2 = b 1 , c 2 + b 2 , c 1 = 0.Moreover, there exists j > r = 2 such that b j = 0 3 and b j , c 1 = b j , c 2 = 0. Therefore, there exists a real number α = 0 such that b 3 = α(c 1 × c 2 ). Then, according to (46),[X 1 , X j ](ζ, R) = (R(α c 1 × (c 1 × c 2 )), 0 3 ).

22 Assume

 22 now that r = rank(L) = 2 and (L 1 )T L 2 + (L 2 ) T L 1 = 0 m×m . Let Y = [X 1 , X 2 ],whose expression, according to (46), is given byY (ζ, R) = (R(c 1 × b 2c 2 × b 1 ), R S(c 1 × c 2 )). Since b i , c i = 0 for i = 1, 2 and b 1 , c 2 + b 2 , c 1 = 0, we obtain that [Y, X 1

2 , 1 ) T L 1 , 1 = 0 3×3 that b 1 = -α 12 c 2 -α 13 c 3 ,b 2

 21111232 =α 12 c 1α 23 c 3 , b 3 =α 13 c 1 + α 23 c 2 ,

  Since[START_REF] Happel | Low Reynolds number hydrodynamics with special applications to particulate media[END_REF] is controllable, there exists a control law t → u(t), defined on an interval [0, T ] and such that the solution of (11) with initial condition z(0) = z * satisfies z(T ) = 0 6 . The flow corresponding to the control law u, evaluated at time T , is a diffeomorphism sending (z * , ζ * , R * ) to a point of the form (0 6 , ζ * * , R * * ). In particular, since the image of G by such diffeomorphism is contained in G, there exists an open nonempty subset

	is Lie-bracket generating at (z, ζ, R) = (0 9 , Id), then the set
	G = Int(A(0 9 , Id))
	is nonempty, as it follows from Krener's theorem (see, for instance, [12, Theorem 1, p. 66]). Fix
	(z

* , ζ * , R * ) ∈ G.

  d 1 |d 2 |d 3 ), where (d 1 |d 2 |d 3 ) denotes the matrix whose columns are d 1 , d 2 and d 3 . Since B 1 B -1 2 is not proportional to Id, then d 1 can be chosen such that d 1 × (B 1 B -1 2 d 1 ) = 0 3 . It suffices to complete d 1 to any orthonormal basis {d 1 , d 2 , d 3 }, since B 1 Γv = 3 i=1 v i c i with c i = B 1 B -1 2 d i . Let now m > 3. Denote by B

	(j)
	i

 [START_REF] Childress | Mechanics of swimming and flying[END_REF]The case where the densities of the microscopic organism and of the fluid have the same order of magnitude

As pointed out in [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF]Section 2], the assumption that the densities of the microscopic organism and of the fluid have the same order of magnitude leads to stronger simplifications than in the general case studied in the previous sections. In particular, according to the remarks of page 6 the control system (2.18)-(2.27) presented in [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF] reduces in this case to

Physically, v represents the field of velocities of the fluid in F (the coordinates are attached to the body), p is the pressure, ξ and ω are the linear and angular velocities of the organism and finally ζ and R give its position and orientation with respect to a fixed frame. Recall that ψ 1 , . . . , ψ m are the functions characterizing the control actions and that σ is the Cauchy stress defined in [START_REF] Childress | Mechanics of swimming and flying[END_REF]. [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF][START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF]).

The finite-dimensional reduction of system (36)-( 43) can be obtained following the procedure proposed in [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF]. To this extent, as in Section 3, define the fundamental solutions (h (i) , p (i) ) and (H (i) , P (i) ) of the Stokes system and associate to them the matrices Θ k , Υ k , κ, K. Let L be the n × m matrix

For almost every t ∈ (0, T ) we have

Proof. According to [START_REF] San Mart Ín | A control theoretic approach to the swimming of microscopic organisms[END_REF]Lemma 3.4] we have

where the equalities in the right-hand sides follow from (40) and (41).

On the other hand, for almost every t ∈ (0, T ),

and, similarly ∂Ω

which proves the lemma.

System (36)-( 43) therefore reduces to the control system

where

By homogeneity, since the set of admissible controls is the whole R m , the controllability in arbitrary small time of system (44) is equivalent to the controllability of the control system whose admissible velocities are given by the family of vector fields

where the b i 's denote the columns of L 1 and the c i 's those of L 2 . Moreover, since F is symmetrical, the controllability of this family is equivalent to the fact that the Lie algebra generated by F, denoted by L , is of rank 6 at every point of R 3 × SO(3) (see, eg, [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]Corollary 5.11]).

Theorem 7.2. The control system (44) is controllable in arbitrary small time if and only if one of the following conditions is satisfied:

Proof. First of all notice that the Lie bracket between two vector fields of the form

Let us prove the "if" part of the theorem. Suppose that we can find in L two vector fields Y 1 , Y 2 of the form

such that the vectors γ 1 and γ 2 are linearly independent and that the inner product β 1 , γ 1 is nonzero. We prove here below that, in this case, the Lie algebra L is of full rank at every point of R 3 × SO(3). Then we show that both 1. and 2. guarantee that, without loss of generality, such Y 1 and Y 2 can be found.