Thomas Chambrion
email: thomas.chambrion@iecn.u-nancy.fr

Alexandre Munnier
email: alexandre.munnier@iecn.u-nancy.fr

GENERIC CONTROLLABILITY OF 3D SWIMMERS IN A PERFECT FLUID

Keywords: Locomotion, Biomechanics, Ideal fluid, Geometric control theory AMS subject classifications. 74F10, 70S05, 76B03, 93B27

We address the problem of controlling a dynamical system governing the motion of a 3D weighted shape changing body swimming in a perfect fluid. The rigid displacement of the swimmer results from the exchange of momentum between prescribed shape changes and the flow, the total impulse of the fluid-swimmer system being constant for all times. We prove the following tracking results: (i) Synchronized swimming: Maybe up to an arbitrarily small change of its density, any swimmer can approximately follow any given trajectory while, in addition, undergoing approximately any given shape changes. In this statement, the control consists in arbitrarily small superimposed deformations; (ii) Freestyle swimming: Maybe up to an arbitrarily small change of its density, any swimmer can approximately tracks any given trajectory by combining suitably at most five basic movements that can be generically chosen (no macro shape changes are prescribed in this statement).

Introduction.

1.1. Context. Researches on bio-inspired locomotion in fluid have now a long history. Focusing on the area of Mathematical Physics, the modeling leads to a system of PDEs (governing the fluid flow) coupled with a system of ODEs (driving the rigid motion of the immersed body). The first difficulty mathematicians came up against was to prove the well-posedness of such systems. This task was carried out in [START_REF] San Martín | An initial and boundary value problem modeling of fish-like swimming[END_REF] (where the fluid is assumed to be viscous and governed by Navier Stokes equations), in [START_REF] Munnier | On the self-displacement of deformable bodies in a potential fluid flow[END_REF] (for an inviscid fluid with potential flow) and in [START_REF] Maso | An existence and uniqueness result for the motion of self-propelled micro-swimmers[END_REF] (for low Reynolds numbers swimmers, the flow being governed by the stationary Stokes equations).

After the well-posedness of the fluid-swimmer dynamics were established, the following step was to investigate its controllability. On this topic, still few theoretical results are available: In [START_REF] Alouges | Optimal strokes for low Reynolds number swimmers: an example[END_REF], the authors prove that a 3D three-sphere mechanism, swimming along a straight line in a viscous fluid is controllable. In [START_REF] Chambrion | On the locomotion and control of a self-propelled shape-changing body in a fluid[END_REF], we prove that a generic 2D example of shape changing body swimming in a potential flow can track approximately any given trajectory.

Some authors are rather interested in describing the dynamics of swimming in terms of Geometric Mechanics (within the general framework presented for instance in [START_REF] Marsden | Introduction to mechanics and symmetry[END_REF]). We refer to [START_REF] Kanso | Locomotion of articulated bodies in a perfect fluid[END_REF] and the very recent paper [START_REF] Kelly | The mechanics and control of driftless swimming[END_REF] for references in this area.

In this article, we consider a 3D shape changing body swimming in a potential flow. Under some symmetry assumptions (the swimmer is alone in the fluid and the fluid-swimmer system fills the whole space) we prove generic controllability results, generalizing and improving what has been obtained for a particular 2D model in [START_REF] Chambrion | On the locomotion and control of a self-propelled shape-changing body in a fluid[END_REF].

Modeling.

Kinematics. We assume that the swimmer is the only immersed body in the fluid and that the fluid-swimmer system fills the whole space, identified with R 3 . Two frames are required in the modeling, the first one E := (E 1 , E 2 , E 3 ) is fixed and Galilean and the second one e := (e 1 , e 2 , e 3 ) is moving with its origin lying at any time at the center of mass of the swimming body. At any moment, there exist a rotation matrix R ∈ SO(3) and a vector r ∈ R 3 such that, if X := (X 1 , X 2 , X 3 ) t and x := (x 1 , x 2 , x 3 ) t are the coordinates of a same vector in respectively E and e, then the equality X = Rx + r holds. The matrix R is supposed to give also the orientation of the swimmer. The rigid displacement of the swimmer, on a time interval [0, T ] (T > 0), is thoroughly described by the functions t : [0, T ] → R(t) ∈ SO(3) and t : [0, T ] → r(t) ∈ R 3 , which are the unknowns of our problem. Denoting their time derivatives by Ṙ and ṙ, we can define the linear velocity v := (v 1 , v 2 , v 3 ) t ∈ R 3 and angular velocity vector Ω := (Ω 1 , Ω 2 , Ω 3 ) t ∈ R 3 (both in e) by respectively v := R t ṙ and Ω := R t Ṙ, where for every vector u := (u 1 , u 2 , u 3 ) t ∈ R 3 , û is the unique skew-symmetric matrix satisfying ûx := u × x for every x ∈ R 3 . Vectors of R 6 will sometimes be decomposed in the form f := (f 1 , f 2 ) t ∈ R 3 × R 3 . For every f := (f 1 , f 2 ) ∈ R 6 and g := (g 1 , g 2 ) ∈ R 6 , we can define f ⋆ g :

= (f 1 × f 2 , f 1 × g 2 -g 1 × f 2 ) t ∈ R 6 .
Shape Changes. Unless otherwise indicated, from now on all of the quantities will be expressed in the moving frame e. In our modeling, the domains occupied by the swimmer are images of the closed unit ball B by C 1 diffeomorphisms, isotopic the the identity, and tending to the identity at infinity, i.e. having the form Id + ϑ where ϑ belongs to D 1 0 (R 3 ) (definitions of the function spaces are given in the appendix, Section A). With these settings, the shape changes over a time interval [0, T ] can be simply prescribed by means of an absolutely continuous function t ∈ [0, T ] → ϑ t ∈ D 1 0 (R 3 ). Then, denoting Θ t = Id + ϑ t , the domain occupied by the swimmer at the time t ≥ 0 is the closed, bounded, connected set Bt := Θ t ( B) (do not forget that we are working in the frame e). We still require some notation: the unit ball's boundary is Σ := ∂B, Σ t := Θ t (Σ) stands for the body-fluid interface, n t is the unit normal vector to Σ t directed toward the interior of B t and the fluid fills the exterior open set F t := R 3 \ Bt .

So-called self-propelled constraints are necessary to ensure that the deformations result from the work of internal forces (they avoid for instance translations to be considered as allowable shape changes). Let a function ̺ ∈ C 0 ( B) + be given. The density of the deformed body at the instant t, denoted by ̺ t ∈ C 0 ( Bt ), is defined by:

̺ t (x) := ̺(Θ -1 t (x))/J t (Θ -1 t (x)), (x ∈ Bt , t ≥ 0), (1.1)
where J t := | det ∇Θ t | = det ∇Θ t (we can drop the absolute values here because Θ t (x) → x as x R 3 → +∞ and det ∇Θ t (x) = 0 for all x ∈ R 3 and t ≥ 0). The self-propelled constraints read:

Bt ̺ t (x)x dx = 0 and Bt ̺ t (x)∂ t Θ t (Θ -1 t (x)) × x dx = 0 (t ≥ 0). (1.2a)
The former identity means that, as already mentioned before, the center of mass of the swimmer lies at any time at the origin of the moving frame. The latter relation tells us that the angular momentum (in e) has to remain constant as the swimmer undergoes shape changes. Equivalent formulations can be obtained up to a change of variables:

B ̺(x)Θ t (x) dx = 0 and B ̺(x)∂ t Θ t (x) × Θ t (x) dx = 0 (t ≥ 0). (1.2b)
The Flow. The fluid is assumed to be inviscid and incompressible. We denote by ̺ f > 0 its constant density. The flow is governed by Euler equations. According to Helmholtz's third theorem, if the flow is irrotational at the initial time, it remains irrotational for all times. In this case, the Eulerian velocity is equal at any time to the gradient of a potential function. According to Kirchhoff's law, the potential can be decomposed into a linear combination of elementary potentials, each one connected to a degree of freedom of the system (they consist here in the 6 degrees of freedom of the rigid motion of the body plus those connecting to the deformations). These ideas have been thoroughly described in a series of papers [START_REF] Kanso | Locomotion of articulated bodies in a perfect fluid[END_REF][START_REF] Munnier | On the self-displacement of deformable bodies in a potential fluid flow[END_REF][START_REF] Munnier | Locomotion of deformable bodies in an ideal fluid: Newtonian versus lagrangian formalism[END_REF][START_REF] Chambrion | On the locomotion and control of a self-propelled shape-changing body in a fluid[END_REF], to which we refer for further explanations.

The elementary potentials ψ i t , (i = 1, . . . , 6) corresponding to the rigid motion of the swimmer are harmonic in F t , tend to zero as infinity and satisfy the Neumann boundary conditions

∂ n ψ i t = (e i × x) • n t (i = 1, 2, 3) and ∂ n ψ i t = e i-3 • n t (i = 4, 5, 6
) on Σ t . They are well defined in the weighted Sobolev space W 1 (F t ) (defined in the appendix, Section A; see also [START_REF] Amrouche | Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator: an approach in weighted Sobolev spaces[END_REF] for details). The overall potential connecting to the rigid displacement is

ψ t := 3 i=1 Ω i ψ i t + 5 i=4 v i-3 ψ i t (t ≥ 0).
On the other hand, the elementary potential ϕ t associated to the shape changes, harmonic as well in F t , satisfies the boundary condition

∂ n ϕ t = w t • n t on Σ t (i = 1, . . . , n), where w t (x) := ∂ t Θ t (Θ -1 t (x)) (x ∈ R 3
). Like the functions ψ i t (i = 1, 2, 3), ϕ t belongs to W 1 (F t ) for all t > 0.

Dynamics. The modeling of moving rigid (or shape changing) bodies in an ideal fluid classically involves the notion of mass matrices. The mass of the body is m := B ̺ dx and its inertia tensor at the time t ≥ 0 is defined by:

I(t) := Bt ̺ t x 2 R 3 Id -x ⊗ x dx = B ̺ Θ t (x) 2 R 3 Id -Θ t (x) ⊗ Θ t (x) dx. (1.3)
We introduce M r b (t) := diag(I(t), mId) (a 6 × 6 symmetric bloc diagonal matrix), M r f (t) (a 6 × 6 symmetric matrix as well) whose entries read:

̺ f Ft ∇ψ i t • ∇ψ j t dx, (1 ≤ i, j ≤ 6), (1.4)
and we denote M r (t) := M r b (t)+M r f (t). We also need the 6×1 column vector N(t), homogeneous to a momentum, whose elements read:

̺ f Ft ∇ψ i t • ∇ϕ t dx, (1 ≤ i ≤ 6). (1.5)
If we neglect the buoyancy force, it has been proved in a series of papers (we refer for instance to the already mentioned articles [START_REF] Kanso | Locomotion of articulated bodies in a perfect fluid[END_REF][START_REF] Munnier | Locomotion of deformable bodies in an ideal fluid: Newtonian versus lagrangian formalism[END_REF] or [START_REF] Chambrion | On the locomotion and control of a self-propelled shape-changing body in a fluid[END_REF]) that the swimming motion is governed by the equation:

Ω v = -M r (t) -1 N(t), (t ≥ 0). (1.6a)
At this point, we can identified ̺ t (or more simply ̺ since they are linked through relation (1.1)) as a parameter and the control as being the function t ∈ [0, T ] → ϑ t ∈ D 1 0 (R 3 ). Notice that the dependance of the dynamics in the control is strongly nonlinear. Indeed ϑ t describes the shape of the body and hence also the domain of the fluid in which are set the PDEs of the potential functions involved in the expressions of the mass matrices M r (t) and N(t).

To determine the rigid motion, Equation (1.6a) has to be supplemented with the ODE:

d dt R r = R Ω R v , (t > 0), (1.6b)
together with Cauchy data for R(0) and r(0). Remark that the dynamics does not depend on ̺ and ̺ f independently but only on the relative density ̺/̺ f . So we can assume, without loss of generality, that ̺ f = 1 in the sequel.

Main results.

The first result ensures the well posedness of System (1.6) and the continuity of the input-output mapping:

Proposition 1.1. For any T > 0, any ̺ ∈ C 0 ( B) + , any absolutely continuous function (1.6) with control θ and Cauchy data (R 0 , r 0 ). Then, the unique solution (R j , r j ) to System (1.6) with control ϑ j and Cauchy data (R 0 , r 0 ) converges in AC([0, T ], SO(3) × R 3 ) to ( R, r) as j → +∞.

t ∈ [0, T ] → ϑ t ∈ D 1 0 (R 3 ) (respectively of class C p , p = 1, . . . , +∞ or analytic) and any inital data (R(0), r(0)) ∈ SO(3) × R 3 , System (1.6) admits a unique solution t ∈ [0, T ] → (R(t), r(t)) ∈ SO(3) × R 3 (in the sense of Carathéodory) absolutely continuous on [0, T ] (respectively of class C p or analytic). Let t ∈ [0, T ] → ϑ j t ∈ D 1 0 (R 3 ) (for j = 1, . . . , +∞) be a sequence of controls in AC([0, T ], D 1 0 (R 3 )) (see Section A for a definition of this space) which converges in this space to a function t ∈ [0, T ] → θt ∈ D 1 0 (R 3 ). Let a pair (R 0 , r 0 ) ∈ SO(3) × R 3 be given and denote t ∈ [0, T ] → ( R(t), r(t)) ∈ SO(3) × R 3 the solution in AC([0, T ], SO(3) × R 3 ) to System
We denote by M(3) the Banach space of the 3 × 3 matrices endowed wit any matrix norm. The main result of this article addresses the controllability of System (1.6):

Theorem 1.2. (Synchronized Swimming) Assume that the following data are given: (i) A function ̺ in C 0 ( B) + (the target density of the swimmer); (ii) A C 1 function t ∈ [0, T ] → θt ∈ D 1 0 (R 3 ) (the target shape changes) such that the pair (̺, θ) satisfies the self-propelled constraints (1.2); (iii) A C 1 function t ∈ [0, T ] → ( R(t), r(t)) ∈ SO(3) × R 3 (the target trajectory to be followed). Then, for any ε > 0, there exists a function ̺ ∈ C 0 ( B) + (the actual density of the swimmer) and a function t ∈ [0, T ] → ϑ t ∈ D 1 0 (R 3 ) (the actual shape changes that can be chosen of class C p for any p = 1, . . . , +∞ or even analytic) such that the pair (̺, ϑ) 3 is the unique solution to system (1.6) with initial data (R(0), r(0)) = ( R(0), r(0)) and control ϑ.

satisfies (1.2), ̺ -̺ C 0 ( B) < ε, ϑ 0 = θ0 , ϑ T = θT and sup t∈[0,T ] θt -ϑ t C 1 0 (R 3 ) 3 + R(t) -R(t) M(3) + r(t) -r(t) R 3 < ε where the function t ∈ [0, T ] → (R(t), r(t)) ∈ SO(3) × R
This theorem tells us that, maybe up to an arbitrarily small change of its density, any weighted 3D body undergoing approximately any prescribed shape changes can approximately track by swimming any given trajectory. It may seem surprising that the shape changes, which are supposed to be the control of our problem, can also be prescribed. Actually, be aware that they are only approximately prescribed. We are going to show precisely that arbitrarily small superimposed shape changes suffice for controlling the swimming motion. This result improves what has been done in the article [START_REF] Chambrion | On the locomotion and control of a self-propelled shape-changing body in a fluid[END_REF] for a particular 2D model. When no macro shape changes are prescribed we have: Theorem 1.3. (Freestyle Swimming) Assume that the following data are given: (i) A pair (̺, θ) ∈ C 0 ( B) + ×D 1 0 (R 3 ) satisfying (1.2) (the target density and shape at rest) (ii) A C 1 function t ∈ [0, T ] → ( R(t), r(t)) ∈ SO(3) × R 3 (the target trajectory). Then, for any ε > 0 there exists a pair (̺, ϑ) ∈ C 0 ( B) + × D 1 0 (R 3 ) (the actual density and shape at rest) such that (i) B ̺ Θ dx = 0 (where [START_REF] Maso | An existence and uniqueness result for the motion of self-propelled micro-swimmers[END_REF], there exists a function t ∈ [0, T ] → s(t) ∈ R 5 (that can be chosen of class C p for any p = 1, . . . , +∞ or even analytic) such that, using

Θ := Id + ϑ) (ii) ̺ -̺ C 0 ( B) + θ -ϑ D 1 0 (R 3 ) < ε and (iii) for almost any 5-uplet (V 1 , . . . , V 5 ) ∈ (C 1 0 (R 3 ) 3 ) 5 satisfying B ̺ V i dx = 0, B ̺ Θ × V i dx = 0 and B ̺ V i × V j dx = 0 (i, j = 1, . . . ,
ϑ t := ϑ + 5 i=1 s i (t)V i ∈ D 1 0 (R 3 ) as control in the dynamics (1.6), we get sup t∈[0,T ] R(t) -R(t) M(3) + r(t) -r(t) R 3 < ε where the function t ∈ [0, T ] → (R(t), r(t)) ∈ SO(3) × R 3 is the unique solution to ODEs (1.6) with initial data (R(0), r(0)) = ( R(0), r(0)).
Differently stated, we claim in this Theorem that any weighted 3D body (maybe up to an arbitrarily small change of its density) is able to swim by means of allowable deformations (i.e. satisfying the self-propelled constraints) obtained as a suitable combination of pretty much any given five basic movements.

The proofs rely on the following main ideas: First, we shall identify a set of parameters necessary to thoroughly characterize a swimmer and its way of swimming (these parameters are its density, its shape and a finite number of basic movements, satisfying the self-propelled constraints (1.2)). Any set of such parameters will be termed a swimmer configuration (denoted SC in short). Then, the set of all of the SC will be shown to be an (infinite dimensional) analytic connected embedded submanifold of a Banach space.

The second step of the reasoning will consist in proving that the swimmer's ability to track any given trajectory (while undergoing any given shape changes) is related to the vanishing of some analytic functions depending on the SC. These functions are connected to the determinant of some vector fields and their Lie brackets (we will use here some classical result of Geometric Control Theory). Eventually, by direct calculation, we will prove that at least one swimmer (corresponding to one particular SC) has this ability. An elementary property of analytic functions will eventually allow us to conclude that almost any SC (or equivalently any swimmer) has this property.

Remark 1.4. The authors conjecture that in both Theorem 1.2 and Theorem 1.3, the actual density ̺ can be chosen equal to the target density ̺. At this point however and although it is very unlikely, it can not be excluded that all of the swimmers with a particular density might be unable to swim. This issue also appeared in [START_REF] Chambrion | On the locomotion and control of a self-propelled shape-changing body in a fluid[END_REF].

1.4. Outline of the paper. The next Section is dedicated to the notion of swimmer configuration (definition and properties). In Section 3 we show that the mass matrices are analytic functions in the SC (seen as a variable) and in Section 4 we will restate the control problem in order to fit with the general framework of Geometric Control Theory. A particular case of swimmer will be shown to be controllable. In Section 5 the proof of the main results will be performed. Section 6 contains some words of conclusion. Many technical results are gathered in the appendix to avoid overloading the rest of the paper.

Swimmer Configuration.

A swimmer configuration is a set of parameters characterizing swimmers whose deformations consist in a combination of a finite number of basic movements. Definition 2.1. For any positive integer n, we denote

C(n) the subset of C 0 ( B) + ×D 1 0 (R 3 )× (C 1 0 (R 3 ) 3 ) n consisting of all of the triplets c := (̺, ϑ, V) such that, denoting Θ := Id+ϑ and V := (V 1 , . . . , V n ), the following conditions hold (i) the set {V i | B • e k , 1 ≤ i ≤ n, k = 1, 2, 3} is a free family in C 1 0 ( B) (ii) every pair (V, V ′ ) of elements of {Θ, V 1 , . . . , V n } satisfies B ̺ V dx = 0 and B ̺ V × V ′ dx = 0. We call swimmer configuration (SC in short) any element c of C(n). By definition, D 1 0 (R 3 ) is open in C 1 0 (R 3 ) 3 (see appendix, Section A). We deduce that for any c ∈ C(n), the set {s := (s 1 , . . . , s n ) t ∈ R n : ϑ + n i=1 s i V i ∈ D 1 0 (R 3
)} is open as well in R n and we denote S(c) its connected component containing s = 0.

Definition 2.2. For any positive integer n, we call extended swimmer configuration (ESC in short) any pair c := (c, s) such that c ∈ C(n) and s ∈ S(c). We denote C X (n) the set of all of these pairs.

Restatement of the problem in terms of SC and ESC. Pick a SC in C(n) (for some integer n). The characteristics of the corresponding swimmer can be deduced from c as follows:

If c is equal to (̺, ϑ, V) with V := (V 1 , . . . , V n ), the shape of the swimmer at rest is B := Θ( B)
where Θ := Id + ϑ. When swimming, it can occupy the domains Bc := Θ s ( B) for all s ∈ S(c)

(c := (c, s) ∈ C(n) is hence an ESC), where Θ s := Id + ϑ + n i=1 s i V i . Still for any s ∈ S(c), its density is the function ̺ s ∈ C 0 ( Bc ) + defined by ̺ s (x) := ̺(Θ -1 s (x))/J s (Θ -1 s (x)) with J s := det ∇Θ s .
Notice that within this construction, the shape changes on a time interval [0, T ] (T > 0) are merely given through an absolutely continuous function t

: [0, T ] → s(t) ∈ S(c). If t ∈ [0, T ] → ṡ(t) ∈ R n stands for its time derivative, the Lagrangian velocity at a point x of B is n i=1 ṡi (t)V i (x) while the Eulerian velocity at a point x ∈ Bc is n i=1 ṡi (t)w i s (x) where w i s (x) := V i (Θ -1 s (x)
). Due to assumption (ii) of Definition 2.1, the self-propelled constraints (1.2) are automatically satisfied.

The harmonic elementary potential functions of the fluid corresponding to the rigid motions depend only on the ESC. Therefore, they will be denoted in the sequel ψ i c instead of ψ i t . The same remark holds for the inertia tensor I(t) and the mass matrices M r (t), M r b (t) and M r f (t) whose notation is turned into I(c), M r (c), M r b (c) and M r f (c) respectively. The elementary potential connected to the shape changes can be decomposed into n i=1 ṡi ϕ i c . In this sum, each potential function

ϕ i c is harmonic in F c := R 3 \
̺ f Bc ∇ψ i c •∇ϕ j c dx (1 ≤ i ≤ 6, 1 ≤ j ≤ n) (recall that ̺ f
can be chosen equal to 1), the dynamics (1.6a) can now be rewritten in the form:

Ω v = -M r (c) -1 N(c) ṡ, (t ≥ 0). (2.1)
Let us focus on the properties of C(n) and C X (n). Theorem 2.3. For any positive integer n, the set

C(n) is an analytic connected embedded submanifold of C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 ) 3 ) n of codimension N := 3(n + 2)(n + 1)/2.
The definition and the main properties of analytic functions in Banach spaces are summarized in the article [START_REF] Whittlesey | Analytic functions in Banach spaces[END_REF].

Proof. For any

c := (̺, ϑ, V) ∈ C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 ) 3 ) n , denote V 0 := Id + ϑ and V := (V 1 , . . . , V n ). Then, define for k = 0, 1, . . . , n, the functions Λ k : C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 ) 3 ) n → R 3(n+1-k) by Λ k (c) := B ̺ V k dx, B ̺ V k × V k+1 dx, . . . , B ̺ V k × V n dx t .
Every function Λ k is analytic and we draw the same conclusion for Λ :

= (Λ 0 , . . . , Λ n ) t : C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 ) 3 ) n → R N (N := 3(n + 2)(n + 1)/2). In order to prove that ∂ c Λ(c) (the differential of Λ at the point c) is onto for any c ∈ C(n), assume that there exist (n + 2)(n + 1)/2 vectors α j i ∈ R 3 (0 ≤ i ≤ j ≤ n) such that: n i=0 α i • ∂ c Λ(c), c h = 0, ∀ c h ∈ C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 )) 3 , (2.2)
where

α i := (α i i , α i+1 i , . . . , α n i ) t ∈ R 3(n+1-i) (j = 0, . . . , n) and c h := (̺ h , ϑ h , V h ) ∈ C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 ) 3 ) n with V h 0 := Id + ϑ h and V h := (V h 1 , . . . , V h n ).
Reorganizing the terms in (2.2), we obtain that:

B ̺ h n k=0 α k k • V k + 0≤i<j≤n α j i • (V i × V j ) dx+ n k=0 B ̺V h k • k-1 j=0 α k j × V j + α k k - n j=k+1 α j k × V j dx = 0.
Since this identity has to be satisfied for any

(̺ h , ϑ h , V h ) ∈ C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3
)) 3 , we deduce that, for every k = 0, . . . , n:

n p=0 α p p • V p + 0≤i<j≤n α j i • (V i × V j ) = 0 and k-1 j=0 α k j × V j + α k k - n j=k+1 α j k × V j = 0. (2.3)
Multiplying the latter equality by ̺ and integrating over B, we get that α k k = 0 (k = 0, . . . , n). The first equality in (2.3) is now just a linear combination of the second ones (for k = 0, . . . , n), so we can drop it. Taking into account Hypothesis (ii) of Definition 2.1, latter identity in (2.3) with k = 0 leads to α j 0 = 0 for every j = 1, . . . , n. There are no more terms involving V 0 in the other equations and invoking again Hypothesis (ii) we eventually get α j i = 0 for 1 ≤ i < j ≤ n. So, equality (2.2) entails that α i = 0 for all i = 0, . . . , n and the mapping

∂ c Λ(c) is onto for all c ∈ C(n).
The linear space

X = Ker ∂ c Λ(c) is closed since Λ is analytic. Let Y be an algebraic supplement of X in C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 ) 3
) n , and denote by Π Y the linear projection onto Y along X. A crucial observation is that the linear space Y is isomorphic to R N and hence it is finite dimensional and closed in

C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 ) 3 ) n . Define the analytic mapping f : X × Y → R N by f (x, y) = Λ(c + x + y). The mapping ∂ y f (0, 0) = ∂ c Λ(c) • Π Y
being onto, the implicit function theorem (analytic version in Banach spaces, see [START_REF] Whittlesey | Analytic functions in Banach spaces[END_REF]) asserts that there exist an open neighborhood O 1 of 0 in X, an open neighborhood O 2 of 0 in Y , and an analytic mapping g : O 1 → Y such that g(0) = 0 and, for every (x, y) in O 1 × O 2 , the two following assertions are equivalent: (i) f (x, y) = 0 (or, in other words, c + x + y belongs to C(n)), and (ii) y = g(x). The analytic mapping g provides a local parameterization of C(n) in a neighborhood of c.

In order to prove that C(n) is path-connected, consider two elements c † := (̺ † , ϑ † , V † ) and

c ‡ := (̺ ‡ , ϑ ‡ , V ‡ ) of C(n) and denote Θ † := Id + ϑ † , V † := (V † 1 , . . . , V † n ) and Θ ‡ := Id + ϑ ‡ , V ‡ := (V ‡ 1 , . . . , V ‡ n ). According to Definition A.2, D 1 0 (R 3
) is open and connected. It entails that it is always possible to find a continuous, piecewise linear path t : [0, 1] → θt ∈ D 1 0 (R 3 ) such that θt=0 = ϑ † and θt=1 = ϑ ‡ . We introduce 0 = t 0 < t 1 < . . . < t k = 1, a subdivision of the interval [0, 1] such that t → θt is linear on every subinterval [t j , t j+1 ] (j = 0, . . . , k-1) and we denote Θt := Id+ θt , θj := θt=tj , Θj := Id+ θj (j = 0, . . . , k). Let us introduce as well the continuous functions

t ∈ [0, 1] → ̺ t := t̺ ‡ + (1 -t)̺ † ∈ C 0 ( B) and t ∈ [0, 1] → u(t) := -B ̺ t Θt dx/ B ̺ t dx ∈ R 3 . The set ∪ t∈[0,T ] Θt ( B) + u(t) being compact, it
is contained in a large ball Ω. We introduce Ω ′ an even larger ball containing Ω and a cut-off function χ defined in R 3 such that 0 ≤ χ ≤ 1, χ = 1 in Ω and χ = 0 in R 3 \ Ω′ . The derivative u of u exists everywhere on ]0, T [ excepted maybe at the points t 1 , . . . , t k-1 . The flow associated with the Carathéodory's solutions of the Cauchy problem: Ẋt (x) = u(t)χ(X t (x)), (t > 0), X t=0 (x) = x is well defined (see [11, Theorem 1A, page 57]). Moreover, for every fixed t ∈ [0, 1], the mapping

x ∈ R 3 → X t (x) ∈ R 3 is C ∞ . Consider now the mappings t ∈ [0, 1] → ϑ t := X t • Θt -Id and Θ t := Id + ϑ t . If x ∈ R 3 \ Ω′ , Θ t (x) = Θt (x) for all t ∈ [0, T ] and if x ∈ B then ϑ t (x) = θt (x) + u(t) and Θ t (x) = Θt (x) + u(t). Notice that ϑ t ∈ D 1 0 (R 3 ) for all t ∈ [0, T ] and B ̺ t (x)Θ t (x)dx = 0 for all t ∈ [0, T ]. Since C 1 0 (R 3 ) 3 is an infinite dimensional Banach space, it is always possible to find by induction W 1 , W 2 , . . . , W n in C 1 0 (R 3 ) 3 such that (i) both families {W 1 | B •e k , . . . , W n | B •e k , V † 1 | B •e k , . . . , V † n | B •e k , k = 1, 2, 3} and {W 1 | B • e k , . . . , W n | B • e k , V ‡ 1 | B • e k , . . . , V ‡ n | B • e k , k = 1, 2, 3} are free in C 1 0 (R 3
) and (ii) for any pair of elements V, V ′ ( both picked in the same family,

B ̺ † Vdx = 0, B ̺ ‡ Vdx = 0, B ̺ † Θj × Vdx = 0, B ̺ ‡ Θj × Vdx = 0 (for all j = 1, . . . , k), B ̺ † V × V ′ dx = 0 and B ̺ ‡ V × V ′ dx = 0. Define the function t ∈ [0, 1] → V i t ∈ C 1 0 (R 3 ) 3 by V i t := (1 -2t)V † i + 2tW i if 0 ≤ t ≤ 1/2 and V i t := (2 -2t)W i + (2t -1)V ‡ if 1/2 < t ≤ 1 and denote V t := (V 1 t , . . . , V n t ) ∈ (C 1 0 (R 3 ) 3 ) n . Eventually, a continuous function linking c † to c ‡ is given by t ∈ [0, 1] → c t ∈ C(n) with c t := (̺ † , ϑ † , V 3t/2 ) if 0 ≤ t ≤ 1/3, c t := (̺ 3t-1 , ϑ 3t-1 , V 1/2 ) if 1/3 < t ≤ 2/3 and c t := (̺ ‡ , ϑ ‡ , V 3t/2-1/2 ) if 2/3 < t ≤ 1.
We omit the proof of the following corollary, similar to that of the theorem above: Corollary 2.4. For any positive integer n, the set

C X (n) is an analytic connected embedded submanifold of C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 ) 3 ) n × R n of codimension N := 3(n + 2)(n + 1)/2. We denote π the projection of C(n) onto C 0 ( B) × D 1 0 (R 3 ) defined by π(c) = (̺, ϑ) for all c := (̺, ϑ, V) ∈ C(n).
The proof of the following corollary is a straightforward consequence of arguments already used in the proof of Theorem 2.3:

Corollary 2.5. For any positive integer n and for any (̺, ϑ) ∈ π(C(n)), the section

π -1 ({(̺, ϑ)}) is an embedded connected analytic submanifold of {̺}×{ϑ}×(C 1 0 (R 3 ) 3 ) n (identified with (C 1 0 (R 3 ) 3 ) n ) of codimension 3n(n + 3)/2.
3. Sensitivity Analysis of the Mass Matrices. For any positive integers k and p, we denote M(k, p) the vector space of the matrices of size k × p (or simply M(k) when k = p).

Theorem 3.1. For any positive integer n, the mappings

c ∈ C X (n) → M r (c) ∈ M(6) and c ∈ C X (n) → N(c) ∈ M(6, n) are analytic.
The method followed in this proof is inspired from [START_REF] Henrot | Variation et optimisation de formes : une analyse géométrique[END_REF]. The result already appeared in [START_REF] Munnier | On the self-displacement of deformable bodies in a potential fluid flow[END_REF], in a slightly different form though. Due to its crucial importance for our purpose, we recall here the main ideas.

Let us begin with a preliminary lemma of which the statement requires introducing some material. Thus, we denote F := R 3 \ B (recall that B is the unit ball, Σ := ∂B and n is the unit normal to Σ directed toward the interior of B). For all ξ ∈ D 1 0 (R 3 ), we set Ξ := Id + ξ, B ξ := Ξ(B), F ξ := Ξ(F ), Σ ξ := Ξ(Σ) and n ξ stands for the unit normal to Σ ξ directed toward the interior of B ξ . We denote q := (ξ, W), with

W := (W 1 , W 2 ) ⊂ (C 1 0 (R 3 ) 3 ) 2 , the elements of Q := D 1 0 (R 3 ) × (C 1 0 (R 3 ) 3 ) 2 and w i ξ := W i (Ξ -1 ) (i = 1, 2)
. Finally, for every q ∈ Q, we define:

Φ(q) := F ξ ∇ψ 1 q (x) • ∇ψ 2 q (x) dx, (3.1)
where, for every i = 1, 2, the function ψ i q ∈ W 1 (F ξ ) (recall that the function spaces are defined in Section A) is solution to the Laplace equation -∆ψ i q = 0 in F ξ with Neumann boundary data

∂ n ψ i q = w i ξ • n ξ on Σ ξ .
The solution has to be understood in the weak sense, namely:

F ξ ∇ψ i q (x) • ∇ϕ(x) dx = Σ ξ (w i ξ • n ξ )(x)ϕ(x) dσ, ∀ ϕ ∈ W 1 (F ξ ). (3.2) Lemma 3.2. The mapping q ∈ Q → Φ(q) ∈ R is analytic.
Proof. We pull back relation (3.2) onto the domain F using the diffeomorphism Ξ. We get:

F ∇Ψ i q (x)A ξ (x) • ∇(ϕ • Ξ)(x) dx = Σ (W i (x) • n ξ (Ξ(x))ϕ(Ξ(x))J σ ξ (x) dσ, ∀ ϕ ∈ W 1 (F ξ ),
where Ψ i q := ψ i q • Ξ, J ξ := det(∇Ξ), A ξ := (∇Ξ t ∇Ξ) -1 J ξ and J σ ξ := (∇Ξ -1 ) t n R 3 J ξ (usually referred to as the tangential Jacobian). In (3.2), if we specialize the test function to have the form ϕ := φ • Ξ -1 with φ ∈ W 1 (F ), we obtain

F Ψ i q (x)A ξ (x) • ∇φ(x) dx = Σ b i q (x)φ(x) dσ for all φ ∈ W 1 (F ), where b i q := (W i • n ξ (Ξ))J σ ξ (i = 1, 2)
. We now claim that the mapping

ξ ∈ D 1 0 (R 3 ) → A ξ -Id ∈ E 0 0 (Ω, M(3)) is analytic. Indeed, the mappings ξ ∈ D 1 0 (R 3 ) → ∇Ξ t ∇Ξ -Id ∈ E 0 0 (Ω, M(3)), A ∈ E 0 0 (Ω, M(3)) → (Id + A) -1 -Id ∈ E 0 0 (Ω, M(3)) and ξ ∈ D 1 0 (R 3 ) → J ξ -1 ∈ C 0 0 (R 3 ) are analytic.
Reasoning the same way, we can show that the mapping ξ ∈ D 1 0 (R 3 ) → J σ ξ ∈ C 0 (Σ) is analytic as well (notice that for all ξ ∈ D 1 0 (R 3 ), the function (∇Ξ -1 ) t n never vanishes on Σ). It is more complicated to prove that ξ 2 is analytic, so we refer to [START_REF] Munnier | On the self-displacement of deformable bodies in a potential fluid flow[END_REF] for the details. This last result entails the analyticity of q ∈ Q → b i q ∈ C 0 (Σ). Then, denoting by W 1 (F ) ′ the dual space to W 1 (F ), we consider the mapping Γ : (q, u)

∈ D 1 0 (R 3 ) → n ξ • Ξ ∈ C 0 (Σ)
∈ Q × W 1 (F ) → A ξ , u, • -b i q , • ∈ W 1 (F ) ′ , where A ξ , u, φ := F ∇u(x)A ξ (x) • ∇φ(x) dx and b i q , φ := Σ b i q (x)φ(x) dσ (φ ∈ W 1 (F )).
We wish now to apply the implicit function theorem (analytic version in Banach spaces, as stated in [START_REF] Whittlesey | Analytic functions in Banach spaces[END_REF]) to the analytic function Γ . Observe, though, that we are only interested in the regularity result and not in the existence and uniqueness. Indeed, we already know that for every q ∈ Q, there exists a unique function Ψ i q ∈ W 1 (F ) such that Γ (q, Ψ i q ) = 0. The function Ψ i q is equal to ψ i q • Ξ where ψ i q is the unique solution to the well posed variational problem (3.2). The partial derivative ∂ u Γ (q, Ψ i q ) is defined by:

∂ u Γ (q, Ψ i q ), u, φ = F ∇u(x)A ξ (x) • ∇φ(x) dx, ∀φ ∈ W 1 (F ). (3.3)
For all ξ ∈ D 1 0 (R 3 ), the matrix A ξ is uniformly elliptic in R 3 (there exists α ξ > 0 such that X t A ξ (x)X > α ξ X 2 R 3 for all X ∈ R 3 and all x ∈ R 3 ). We deduce that the right hand side of (3.3) can be chosen as the scalar product in W 1 (F ) and hence that ∂ u Γ (q, Ψ i q ) is a continuous isomorphism from W 1 (F ) onto its dual space according to the Riesz representation theorem. The implicit function theorem applies and asserts that the mapping q ∈ Q → Ψ i q ∈ W 1 (F ) is analytic.

To conclude the proof, it remains only to observe that the function Φ(q) introduced in (3.1) can be rewritten, upon a change of variables as Φ(q) = F ∇Ψ 1 q (x)A ξ (x) • ∇Ψ 2 q (x) dx, which is indeed analytic as a composition of analytic functions.

We can now give the proof of Theorem 3.1.

Proof. Recall that the elements of the matrix M r f (c) are defined in (1.4) and those of N(c) in (1.5). For any c := (c, s) ∈ C X (n), where c := (̺, ϑ, V), we apply the lemma with ξ : 6) is analytic. To prove the analyticity of the elements of N(c), we apply the lemma again with ξ : an ordered orthonormal basis of R n , we can seek the function t ∈ [0, T ] → s(t) ∈ S(c) as the solution of the ODE ṡ(t) = n i=1 λ i (t)f i where the functions λ i : t ∈ [0, T ] → λ i (t) ∈ R are the new controls, and rewrite once more the dynamics (2.1) as:

= ϑ + n i=1 s i V i and W 1 , W 2 ∈ {e i ×Ξ, e i , i = 1, 2, 3} to get that the mapping c ∈ C X (n) → M r f (c) ∈ M(
= ϑ + n i=1 s i V i , W 1 ∈ {e i × Ξ, e i , i = 1, 2, 3} and W 2 ∈ {V 1 , . . . , V n }.
  Ω v ṡ   = n i=1 λ i (t) -M r (c, s) -1 N(c, s)f i f i , (t ≥ 0). (4.1)
It is worth remarking that in this form, s is no more the control but a variable which is meant to be controlled and c ∈ C(n) is a parameter of the dynamics. Considering (4.1), we are quite naturally led to introduce, for all c ∈ C X (n), the vector fields 6+n) is a bloc diagonal matrix. The dynamics (4.1) and the ODE (1.6b) can be gathered into a unique differential system:

X i (c) := -M r (c) -1 N(c)f i ∈ R 6 , Y i (c) := ( X1 i (c), X 2 i (c), f i ) t ∈ T Id SO(3) × R 3 × R n (we have used here the notation X i := (X 1 i , X 2 i ) t ∈ R 3 ×R 3 ) and Z i c (R, s) := R R Y i (c) ∈ T R SO(3)×R 3 ×R n where R R := diag(R, R, Id) ∈ SO(
d dt   R r s   = n i=1 λ i (t)Z i c (R, s), (t ≥ 0). ( 4.2)
For every i = 1, . . . , n, the function (R, r, s)

∈ SO(3)×R 3 ×S(c) → Z i c (R, s) ∈ T R SO(3)×R 3
×R n can be seen as an analytic vector field (constant in r) on the analytic connected manifold M(c) := SO(3) × R 3 × S(c). We denote ζ any element (R, r, s) ∈ M(c) and we define Z(c) as the family of vector fields (Z i c ) 1≤i≤n on M(c).

[Z i c (R, s), Z j c (R, s)] = R R    (X 1 i × X 1 j )(c) (X 1 i × X 2 j -X 1 j × X 2 i )(c) 0    + R R    (∂ si X 1 j -∂ sj X 1 i )(c) (∂ si X 2 j -∂ sj X 2 i )(c) 0    . (4.3)
By induction, we can similarly prove that the Lie brackets of any order at any point ζ ∈ M(c) have the same general form, namely the matrix R R multiplied by an element of T (Id,0,s) M(c). We deduce that the dimension of the Lie algebra at any point of M(c) depends only on s. According to the Orbit Theorem (see [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]), the dimension of the Lie algebra is constant along any orbit. But according to the particular form of the vector fields Z i c (whose last n components form a basis of R n ), the projection of any orbit on S(c) turns out to be the whole set S(c) (or, in other words, for any s ∈ S(c) and for any orbit, there is a point of the orbit for which the last component is s). Assume now that dim Lie ζ * Z(c) = 6 + n at some particular point ζ * := (R * , r * , s * ) ∈ M(c). according to the Orbit Theorem, for any s ∈ S(c), there exists at least one point (R s , r s , s) ∈ M(c) such that dim Lie (Rs,rs,s) Z(c) = 6 + n. But since the dimension of the Lie algebra does not depend on the variables R and r, we conclude that dim Lie ζ Z(c) = 6 + n for all ζ ∈ M(c).

Definition 4.2. We say that c, a SC in

C(n) (for some integer n) is controllable if there exists ζ ∈ M(c) such that dim Lie ζ Z(c) = 6 + n.
It is obvious that for a SC to be controllable, the integer n has to be larger or equal to 2. The following result is quite classical (a proof can be found in [START_REF] Chambrion | On the locomotion and control of a self-propelled shape-changing body in a fluid[END_REF]):

Proposition 4.3. Let c ∈ C(n) (for some integer n) be controllable (with the usual notation c := (̺, ϑ, V), V := (V 1 , . . . , V n ) and ϑ s := ϑ + n i=1 s i V i for every s ∈ S(c)). Then for any given continuous function t ∈ [0, T ] → ( R(t), r(t), s(t)) ∈ SO(3) × R 3 × S(c) and for any ε > 0, there exist n C 1 functions λ i : [0, T ] → R (i = 1, . . . , n) such that:

1. sup t∈[0,T ] R(t) -R(t) M(3) + r(t) -r(t) R 3 + ϑ s(t) -ϑ s(t) C 1 0 (R 3 ) 3 < ε; 2. R(T ) = R(T ), r(T ) = r(T ) and s(T ) = s(T ); where t ∈ [0, T ] → (R(t), r(t), s(t)) ∈ M(c) is the unique solution to the ODE (4.2) with Cauchy data R(0) = R(0) ∈ SO(3), r(0) = r(0) ∈ R 3 , s(0) = s(0) ∈ S(c).
Let us mention some other quite elementary properties that will be used later on: Proposition 4.4.

1. If c := (̺, ϑ, V) ∈ C(n) (n ≥ 2) is a controllable SC with V := (V 1 , . . . , V n ) ∈ (C 1 0 (R 3 ) 3 ) n then any c + := (̺, ϑ, V + ) ∈ C(n + 1) such that V + := (V 1 , . . . , V n , V n+1 ) ∈ (C 1 0 (R 3 ) 3 ) n+1 (for some V n+1 ∈ C m 0 (R 3 ) 3 ) is a controllable SC as well. 2. If c := (̺, ϑ, V) ∈ C(n) (n ≥ 2) is a controllable SC, then for any ϑ * ∈ {ϑ+ n i=1 s i V i , s ∈ S(c)} the element c * := (̺, ϑ * , V) belongs to C(n) and is a controllable SC as well. 3. If c := (̺, ϑ, V) ∈ C(n) (n ≥ 2) is a controllable SC, then all of the controllable SC c * := (̺, ϑ, V * ) in C(n) (V * ∈ (C 1 0 (R 3 ) 3 ) n
) form an open dense subset of the section π -1 ({(̺, ϑ)}) (for the induced topology). [START_REF] Chambrion | On the locomotion and control of a self-propelled shape-changing body in a fluid[END_REF]. If there exists a SC in C(n) for some n ≥ 2 then, for any k ≥ n, all of the controllable SC in C(k) form an open dense subset of C(k) (for the induced topology). Proof. The two first assertions are obvious so let us address directly the third point. Denote E k (k positive integer) the set of all of the vectors fields on M(c) obtained as Lie brackets of order lower or equal to k from elements of Z(c). Then, consider the determinants of all of the different families of 6 + n elements of E k as analytic functions in the variable V (the other variables ̺, ϑ and s = 0 being fixed). Since c is controllable, there exist at least one k and one family of 6 + n elements in E k whose determinant is nonzero. According to Corollary 2.5 and basic properties of analytic functions (see [START_REF] Whittlesey | Analytic functions in Banach spaces[END_REF]), the determinant may vanish only in a closed subset with empty interior of the section π -1 ({(̺, ϑ)}) (for the induced topology). The proof of the last point is similar.

Building a controllable SC.

In this subsection, we are interested in computing the Lie brackets of first order [Z i c (R, s), Z j c (R, s)] at (R, s) = (Id, 0), for a particular SC.

General computations. Starting from identity (4.3) and focusing on the second term in the right hand side, we have, for all c ∈ C(n) (n ≥ 2) , all s ∈ S(c) and all i, j = 1, . . . , n:

∂ si X j (c) -∂ sj X i (c) = M r (c) -1 (∂ sj M r (c)X i (c) -∂ si M r (c)X j (c)) + (∂ sj N(c)f i -∂ si N(c)f j ) . (4.4)
From the decomposition M r (c) := M r b (c) + M r f (c), we deduce that:

∂ sj M r (c)X i (c) -∂ si M r (c)X j (c) = (∂ sj M r b (c)X i (c) -∂ si M r b (c)X j (c)) + (∂ sj M r f (c)X i (c) -∂ si M r f (c)X j (c)). (4.5)
We can easily compute the first term in the right hand side when s = 0. Thus, for all i, j = 1, . . . , n, we have:

[∂ sj M r b (c)X i (c) -∂ si M r b (c)X j (c)] s=0 = η j Id 0 0 0 X i (c)| s=0 -η i Id 0 0 0 X j (c)| s=0 , (4.6)
where

η i := 2 B ̺(x) V i (x) • Θ(x) dx.
Rigid shell's deformation. We consider now shape changes that reduce to rigid displacements on the swimmer's boundary Σ (but obviously not inside the body for the self-propelled constraints not to be violated). The idea of using such deformations stemmed from the reading of the article [START_REF] Kozlov | Motion of a body with undeformable shell and variable mass geometry in an unbounded perfect fluid[END_REF]. However, rigid deformations of the shell can not be considered within the modeling described in Section 2 (because rigid deformations do not fit with the general form of the diffeomorphisms described in Section 2). Let us explain how to deal with this difficulty.

Let c := (̺, ϑ, V) ∈ C( 6) be given such that

V := (V 1 , . . . , V 6 ) with V i | Σ (x) := e i ×(x+ϑ(x)) (i = 1, 2, 3) and V i | Σ (x) := e i-3 (i = 4, 5, 6) (Proposition C.
1 in the appendix guarantees the existence of such vector fields satisfying furthermore η i = 0 for all i = 1, . . . , 6). Thus, the shape changes we are considering read Θ s := Id + ϑ + 6 i=1 s i V i (s ∈ S(c)). Let us define also Ξ s := R s (Id + ϑ) + τ s where R s := exp(s 1 ê1 ) exp(s 2 ê2 ) exp(s 3 ê3 ) ∈ SO(3) and 3 . Thus, Θ s is a diffeomorphism which can be achieved within our modeling while Ξ s is a true rigid deformation.

τ s := 3 i=1 s i+3 e i ∈ R
In order to determine the expressions of the terms

∂ si M r f (c)| s=0 (i = 1, . . . , 6
) arising in the computation of the Lie brackets, we have to compare (using the notation of Lemma 3.2) [START_REF] Henrot | Variation et optimisation de formes : une analyse géométrique[END_REF] where:

∂ si Φ(q 1 s )| s=0 and ∂ si Φ(q 2 s )| s=0 (i = 1, . . . ,
• q 1 s := (ξ 1 s , W 1 s ) with ξ 1 s := Ξ s -Id, W 1 s := (W 1,1 s , W 1,2 s ) and W 1,1 s , W 1,2 s ∈ {e i × Ξ s , e i , i = 1, 2, 3} (settings corresponding to a true rigid deformation of the shell); • q 2 s := (ξ 2 s , W 2 s ) with ξ 2 s := Θ s -Id = ϑ + n i=1 s i V i , W 2 s := (W 2,1 s , W 2,2 s ) and W 2,1 s , W 2,2 s ∈ {e i × Θ s , e i , i = 1, 2, 3} (settings allowed in our modeling). Remark that ∂ si Θ s | s=0 = ∂ si Ξ s | s=0 on
Σ for all i = 1, . . . , 6, so the deformations are tangent at s = 0. It entails that ∂ si q 1 s | s=0 = ∂ si q 2 s | s=0 . Since, in addition, q 1 s=0 = q 2 s=0 and ∂ si Φ(q k s )| s=0 = ∂ q Φ(q k s ), ∂ si q k s | s=0 for k = 1, 2 and i = 1, . . . , 6, we deduce that ∂ si Φ(q 1 s )| s=0 = ∂ si Φ(q 2 s )| s=0 for all i = 1, . . . , 6.

We apply the diffeomorphism Ξ s := R s (Id+ϑ)+τ s to B to obtain the domain of the deformed swimmer. We denote B ⋄ := (Id + ϑ)(B) (which can be seen as the shape of the swimmer at rest, i.e. when s = 0), Σ ⋄ := ∂B ⋄ , F ⋄ := R 3 \ B⋄ and B s := Ξ s (B), F s := R 3 \ Bs , Σ s := ∂B s . We seek the potential ψ 1 c , defined and harmonic in F s in the form ψ 1 c (x) = ψ1 c (R t s (xτ s )) where the function ψ1 c defined on F ⋄ has to be determined. It is obvious that ψ1 c is harmonic in F ⋄ and we have only to determine the boundary conditions on Σ ⋄ . For all y ∈ Σ ⋄ , we denote

x := R s y + τ s ∈ Σ s . The relation n| Σs (R s y + τ s ) = R s n| Σ ⋄ (y) entails that ∇ ψ1 c (y) • n| Σ ⋄ (y) = ∇ψ 1 c (R s y+τ s )•n| Σs (R s y+τ s ) = ∇ψ 1 c (x)•n(x)
, and this quantity has to be equal to (e 1 ×x)•n(x). We deduce after some elementary algebra that

∂ n ψ1 c (y) = (R t s e 1 × y)•n(y)+ (R t s e 1 × R t s τ s )•n(y)
. Proceeding similarly and with obvious notation, we obtain more generally that:

∂ n ψi c (y) = (R t s e i × y) • n(y) + (R t s e i × R t s τ s ) • n(y), (i = 1, 2 , 3), and
∂ n ψi c (y) = R t s e i-3 • n(y), (i = 4, 5 , 6).
Denoting M ⋄ f := M r f (c)| s=0 , we get the relations:

M r f (c, s) = Id τ s 0 Id R s 0 0 R s M ⋄ f R t s 0 0 R t s Id 0 -τ s Id ,
and then, differentiating with respect to s i :

∂ si M r f (c, s)| s=0 = êi 0 0 êi M ⋄ f -M ⋄ f êi 0 0 êi , (i = 1, 2 , 3), (4.7a)
∂ si M r f (c, s)| s=0 = 0 êi-3 0 0 M ⋄ f -M ⋄ f 0 0 êi-3 0 , (i = 4, 5 , 6). (4.7b)
The reasoning is quite similar for the entries of the matrix N(c). We only have to be careful that the vector fields W 2,2 s can actually not depend on s (once more, this is due to our modeling). The elements of the matrix [START_REF] Maso | An existence and uniqueness result for the motion of self-propelled micro-swimmers[END_REF][START_REF] Henrot | Variation et optimisation de formes : une analyse géométrique[END_REF]. The only difference with the elements of M r f (c) being that W i,2 does not depend on s for i = 1, 2, 3, we wish to reuse the preceding calculations. Invoking the chain rule, we have to subtract to

∂ sj N(c)| s=0 f i -∂ si N(c)| s=0 f j read (still with the notation of Lemma 3.2) ∂ sj Φ(q i s )| s=0 -∂ si Φ(q j s )| s=0 (i, j = 1, . . . , 6) with this time q i s := (ξ s , W i s ), ξ s := Ξ s -Id, W i s := (W 1 s , W i,2 ), W 1 s ∈ {e k × Ξ s , e k , k = 1, 2, 3} and W i,2 := e i × (Id + ϑ) (i = 1, 2, 3) or W i,2 = e i-3 (i = 4,
∂ sj Φ(q i s )| s=0 -∂ si Φ(q j s )| s=0 the quantity ∂ W 2 Φ(q i s ), ∂ sj W i,2 s | s=0 - ∂ W 2 Φ(q j s ), ∂ si W j,2
s | s=0 which is quite easy to determine because Φ is linear in the variable W 2 . Thus, this last expression is merely equal to Φ(q i,j ) (1 ≤ i, j ≤ 6) with q i,j := (ϑ, W i,j ),

W i,j := (W 1 , W 2 i,j ), W 1 ∈ {e k × (Id + ϑ), e k , k = 1, 2, 3} and
W 2 i,j :=          (e i × e j ) × (Id + ϑ) if 1 ≤ i, j ≤ 3, (e i × e j ) if 4 ≤ i ≤ 6 , 1 ≤ j ≤ 3, (e i × e j ) if 1 ≤ i ≤ 3 , 4 ≤ j ≤ 6, 0 if 4 ≤ i, j ≤ 6.
We eventually obtain that:

∂ sj N(c)| s=0 f i -∂ si N(c)| s=0 f j = ∂ sj M r f (c)| s=0 f i -∂ si M r f (c)| s=0 f j -N ⋄ (f i ⋆ f j ), (4.8)
where

N ⋄ := N(c)| s=0 = M ⋄ f and f i ⋆ f j := (f 1 i × f 1 j , f 1 i × f 2 j -f 1 j × f 2 i ) t with the notation f i = (f 1 i , f 2 i ) t ∈ R 3 × R 3 .
Specifying the density and shape. The expression of the 6 × 6 symmetric added mass matrix M ⋄ f depends only on the domain F ⋄ or equivalently on B ⋄ . As stated in Proposition D.1 in the appendix, this matrix is positive definite if we choose ϑ in such a way that B ⋄ has no axis of symmetry. It entails that, up to a change of frame e at the initial time, M ⋄ f can be assumed to be diagonal with positive eigenvalues µ j (j = 1, . . . , 6). On the other hand, denoting by S(3) + the set of the 3×3 symmetric matrices that are positive definite, we can quite easily prove that for any B ⋄ (which means for any ϑ ∈ D 1 0 (R 3 )), the mapping ̺ ∈ C 0 ( B⋄ ) + → B ⋄ ̺( x R 3 Id -x ⊗ x)dx ∈ S(3) + is onto. We deduce that for any (I 1 , I 2 , I 3 ) ∈ R 3 such that I i > 0 for i = 1, 2, 3, there exists ̺ ∈ C 0 ( B) + such that the inertia tensor I(c)| s=0 is diagonal, equal to diag(I 1 , I 2 , I 3 ). In this case, the matrix M r b (c)| s=0 is diagonal as well, equal to diag(I 1 , I 2 , I 3 , m, m, m, ). We deduce that the vector fields X i (c)| s=0 read -µ i /(I i + µ i )f i if i = 1, 2, 3 and -µ i /(m + µ i )f i if i = 4, 5, 6. Summarizing (4.3), (4.4), (4.5), (4.6) (recall that η i = 0 for all i = 1, . . . , 6), (4.7) and (4.8), we obtain the following expressions for the Lie brackets at (R, s) = (Id, 0):

[Z 1 c , Z 2 c ] = I 1 I 2 (-µ 1 -µ 2 + µ 3 ) + µ 1 µ 2 (-I 1 -I 2 + I 3 ) (µ 1 + I 1 )(µ 2 + I 2 )(µ 3 + I 3 )   ê3 0 0   , [Z 1 c , Z 3 c ] = I 1 I 3 (µ 1 -µ 2 + µ 3 ) + µ 1 µ 3 (I 1 -I 2 + I 3 ) (µ 1 + I 1 )(µ 2 + I 2 )(µ 3 + I 3 )   ê2 0 0   , [Z 2 c , Z 3 c ] = I 2 I 3 (µ 1 -µ 2 -µ 3 ) + µ 2 µ 3 (I 1 -I 2 -I 3 ) (µ 1 + I 1 )(µ 2 + I 2 )(µ 3 + I 3 )   ê1 0 0   , [Z 2 c , Z 4 c ] = I 2 m(µ 4 -µ 6 ) (µ 2 + I 2 )(µ 4 + m)(µ 6 + m)   0 e 3 0   , [Z 3 c , Z 4 c ] = I 3 m(µ 5 -µ 4 ) (µ 3 + I 3 )(µ 4 + m)(µ 5 + m)   0 e 2 0   , [Z 3 c , Z 5 c ] = I 3 m(µ 5 -µ 4 ) (µ 3 + I 3 )(µ 4 + m)(µ 5 + m)   0 e 1 0   .
Since, on the other hand, when (R, s) = (Id, 0) we also have

Z i c =   -µ i êi /(µ i + I i ) 0 f i   if i = 1, 2, 3, and Z i c =   0 -µ i e i /(µ i + m) f i   if i = 4, 5, 6,
we deduce that a sufficient condition ensuring that dim Lie (Id,0,0) Z(c) = 12 is that

I 1 I 2 (-µ 1 -µ 2 + µ 3 ) + µ 1 µ 2 (-I 1 -I 2 + I 3 ) I 1 I 3 (µ 1 -µ 2 + µ 3 ) + µ 1 µ 3 (I 1 -I 2 + I 3 ) I 2 I 3 (µ 1 -µ 2 -µ 3 ) + µ 2 µ 3 (I 1 -I 2 -I 3 ) I 2 m(µ 4 -µ 6 )I 2 3 m 2 (µ 5 -µ 4 ) 2 = 0. (4.9)
According to [10, pages 152-155], if we specialize now B ⋄ to be an ellipsoid, the length of the axes can be chosen in such a way that (i) it has no axis of symmetry (and hence µ i > 0 for i = 1, . . . , 6), (ii) µ 4 = µ 5 and (iii) µ 4 = µ 6 . Since I i > 0 (i = 1, 2, 3) and obviously m > 0, the condition (4.9) reduces to:

I 1 I 2 (-µ 1 -µ 2 + µ 3 ) + µ 1 µ 2 (-I 1 -I 2 + I 3 ) I 1 I 3 (µ 1 -µ 2 + µ 3 ) + µ 1 µ 3 (I 1 -I 2 + I 3 ) I 2 I 3 (µ 1 -µ 2 -µ 3 ) + µ 2 µ 3 (I 1 -I 2 -I 3 ) = 0. (4.10)
As already mentioned, it is always possible to achieve any triplet of positive numbers (I 1 , I 2 , I 3 ) with a suitable choice of density, so whatever the values of µ i (i = 1, 2, 3) are, it is always possible to find ̺ ∈ C 0 ( B) + such that (4.10) holds. It entails, according to the forth point of Proposition 4.4: Proposition 4.5. For any integer n ≥ 5, the set of all the controllable SC is an open dense subset in C(n).

Notice that n = 5 in this Proposition (instead of n = 6), because we did not use the vector field Z 6 c in the computation of the Lie brackets.

Proofs of the Main Results.

Proof of Proposition 1.1. Let a control function ϑ be given in AC([0, T ], D 1 0 (R 3 )) and denote Θ := Id + ϑ. With the notation of Lemma 3.2, at any time t the entries of the matrix M r f (t) read Φ(q) with q := (ϑ, W) and

W := (W 1 , W 2 ), W i ∈ {e i × Θ t , e i , i = 1, 2, 3}. We deduce that t ∈ [0, T ] → M r f (t) ∈ M(3) is in AC([0, T ], M (3)
). To get the expression of the elements of the vector N(t) we only have to modify W 2 which has to be equal to ∂ t ϑ t . It entails that t ∈ [0, T ] → N(t) ∈ R 6 is in L 1 ([0, T ]) 6 . It is quite easy to verify that the inertia tensor I(t) is in AC([0, T ], M(3)). Existence and uniqueness of solutions is now straightforward because

t ∈ [0, T ] → M r (t) -1 N(t) ∈ R 6 is in AC([0, T ], R 6 ).
Let us address the stability result. With the same notation as in the statement of Proposition 1.1, denote (Ω j , v j ) t the left hand side of identity (1.6a) with control ϑ j and ( Ω, v) t when the control is θ. As j → +∞, it is clear that (Ω j , v j ) t → ( Ω, v) t in L 1 ([0, T ], R 6 ). Then, integrating (1.6b) between 0 and t for any 0 ≤ t ≤ T , we get the estimate R(t 3)) as j → +∞ and we use again the ODE to prove that Ṙj → Ṙ in L 1 ([0, T ]). Next, it is then easy to obtain the convergence of r j to r and to conclude the proof.

) -R j (t) M(3) ≤ t 0 R(s) -R j (s) M(3) Ω(s) R 3 + Ω j (s) -Ω(s) R 3 ds. Applying Grönwall inequality, we con- clude that R j → R in C([0, T ], M(
Proof of Theorems 1.2 and 1.3. We shall focus on the proof of Theorem 1.2 because it will contain the proof of Theorem 1.3. For any integer n, we shall use the notation c

C(n) := ̺ C 0 ( B) + ϑ C 1 0 (R 3 ) 3 + n i=1 V i C 1 0 (R 3 ) 3 for all c ∈ C 0 ( B) × C 1 0 (R 3 ) 3 × (C 1 0 (R 3 ) 3 ) n with, as usual, c := (̺, ϑ, V) and V := (V 1 , . . . , V n ). Let ε > 0 and the functions ̺ ∈ C 0 ( B), t ∈ [0, T ] → θt ∈ D 1 0 (R 3 ) and t ∈ [0, T ] → ( R(t), r(t)) ∈ SO(3) × R 3 be
given as in the statement of the theorem.

Set now ̺1 := ̺, θ1 := θt=0 and V1

1 := ∂ t θt=0 ∈ C 1 0 (R 3 ) 3 .
According to the self-propelled constraints (1.2), it is always possible to find four elements V1 j (j = 2, . . . , 5) in C 1 0 (R 3 ) 3 such that the element c1 := (̺ 1 , θ1 , V1 ) be a SC which belongs to C(5), with V1 := ( V1 1 , . . . , V1 5 ). Then, Proposition 4.5 guarantees that for any δ > 0 it is possible to find a SC controllable in C(5), denoted by c

1 := (̺ 1 , ϑ 1 , V 1 ) where V 1 := (V 1 1 , . . . , V 1
5 ), such that c 1 -c1 C(5) < δ/2. Since t → ∂ t θt is continuous on the compact set [0, T ], it is uniformly continuous. For any ν > 0, there exists δ ν > 0 such that ∂t ϑ t -∂t ϑ t ′ C 1 0 (R 3 ) 3 < ν providing that |t -t ′ | ≤ δ ν . We then divide the time interval [0, T ] into 0 = t 1 < t 2 < . . . < t k = T such that |t j+1 -t j | < δ ν for j = 1, . . . , k -1. For any t 1 ≤ t ≤ t 2 , we have the estimate:

θt -(ϑ 1 + (t -t 1 )V 1 1 ) C 1 0 (R 3 ) 3 ≤ θt -( θ1 + (t -t 1 ) V1 1 ) C 1 0 (R 3 ) 3 + θ1 -ϑ 1 C 1 0 (R 3 ) 3 + (t -t 1 ) V1 1 -V 1 1 C 1 0 (R 3 ) 3 .
On the one hande, we have, for all

t ∈ [t 1 , t 2 ], θt -( θ1 + (t -t 1 ) V1 1 ) C 1 0 (R 3 ) 3 < ν|t -t 1 |.
On the other hand, still for t 1 ≤ t ≤ t 2 and if we assume that δ ν < 1, we get: θ1 -ϑ 1

C 1 0 (R 3 ) 3 + (t - t 1 ) V1 1 -V 1 1 C 1 0 (R 3 ) 3 ≤ δ/2.
We introduce now ̺2 := ̺1 and θ2 := θt=t2 . It is always possible to supplement V2

1 := ∂ t θt2 with vector fields V2 j (j = 2, . . . , 5) in such a way that c2 := (̺ 2 , θ2 , V2 ) be in C( 5) with the obvious notation V2 := ( V2 1 , . . . , V2 5 ). We define ̺ 2 := ̺ 1 and ϑ

2 := ϑ 1 +(t 2 -t 1 )V 1 1 . For any t 1 ≤ t ≤ t 2 , Proposition 4.4 guarantees that the SC c 1 t := (̺ 1 , ϑ 1 + (t -t 1 )V 1 1 , V 1
) is controllable. In particular, for t = t 2 , there exists an integer k and a family of 11 vector fields in E k (the set of all the Lie brackets of order lower or equal to k) such that the determinant of the family is nonzero. But this determinant can be thought of as an analytic function in V 1 . The set π -1 ({(̺ 1 , ϑ 1 )}) being an analytic connected submanifold of (C 1 0 (R 3 ) 3 ) 5 (see Corollary 2.5), the determinant is nonzero everywhere on this set but maybe in a closed subset of empty interior (for the induced topology). Therefore, it is possible to find V 2 ∈ (C 1 0 (R 3 ) 3 ) 5 such that c2 -c 2 C(5) < (δ/2 + ν(t 2 -t 1 )) + δ/4, and c 2 := (̺ 2 , ϑ 2 , V 2 ) is controllable.

By induction, we can build cj and c j (j = 1, 2, . . . , k) such that cj -c j ≤ δ/2 + k i=2 δ/2 i + ν(t i -t i-1 ) < δ + νT . We choose δ and ν in such a way that δ + νT < ε/2 and we define t : [0, T ] → θt ∈ D 1 0 (R 3 ) and t ∈ [0, T ] → ct ∈ C(5) as continuous, piecewise affine functions by θt := ϑ j + (t -t j )V j 1 and ct := (̺, θt , Ṽt ) with ̺ := ̺ j = ̺ 1 , Ṽt := V j if t ∈ [t j , t j+1 ] (j = 1, . . . , k -1). Notice that for any t ∈ [0, T ], (i) θtθt C 1 0 (R 3 ) 3 < ε/2 and (ii) ct is controllable.

Definition 4.2 and Proposition 4.3 ensure that, on every interval [t j , t j+1 ] (j = 1, . . . , k -1), there exist five C 1 functions λ j i : [t j , t j+1 ] → R (i = 1, . . . , 5) such that the solution (R j , r j , s j ) : [t j , t j+1 ] → SO(3) × R 3 × R 5 to the ODE (4.2) with vector fields Z i c j (R j , s) and Cauchy data R j (t j ) = R(t j ), r j (t j ) = r(t j ) and s j (t j ) = 0 satisfy:

1. sup t∈[tj ,tj+1] R(t) -R j (t) M(3) + r(t) -r j (t) R 3 + θt -ϑ j t C 1 0 (R 3 ) 3 < ε/2 with ϑ j t := ϑ j + 5 i=1 s j i (t)V j i ; 2. R j (t j+1 ) = R(t j+1 ), r j (t j+1 ) = r(t j+1
) and s j (t j+1 ) = (t j+1 -t j , 0, 0, 0, 0) t ; With these settings, the functions t ∈ [0, T ] → θt ∈ D 1 0 (R 3 ), Ȓ : [0, T ] → SO(3) and ȓ : [0, T ] → R 3 defined by θt := ϑ j t , Ȓ(t) := R j (t) and ȓ(t) := r j (t) if t ∈ [t j , t j+1 ] (j = 1, . . . , k -1) are continuous, piecewise C 1 . We extend also every functions s j on [0, T ] by setting s j (t) := 0 if t ∈ [0, t j [ and s j (t) := (t j+1 -t j , 0, 0, 0, 0) t if t ∈]t j+1 , T ]. They are continuous, piecewise C 1 as well.

It remains to smooth the function θt (and hence also Ȓ and ȓ). We can extend the functions λ j i on the whole interval [0, T ] by merely setting where T j i ( Ȓ, S) := ( Ȓ X1 i (c j , s j ), ȒX 2 i (c j , s j ), F j i ) ∈ T R SO(3) × R 3 × (R 5 ) k . Let λj i denote analytic approximations of the functions λ j i in L 1 ([0, T ]) and denote ( Ř, ř, Š) the corresponding analytic solution to system (5.1) with Š := (š 1 1 , . . . , š1 5 , š2 1 , . . . , š2 5 , . . . , šk 1 , . . . , šk 5 ) and θt := ϑ 1 0 + k j=1 5 i=1 šj i (t)V j i which is analytic from [0, T ] to C 1 0 (R 3 ) 3 . According to Lemma D.2, θtθt can be made arbitrarily small in AC([0, T ], C 1 0 (R 3 ) 3 ), providing that the functions λj i are close enough to λ j i in L 1 ([0, T ]). Notice that θ does probably not satisfy the self-propelled constraints (1.2) (especially at the times t j , j = 2, . . . , k -1). It remains to invoke Proposition B.1 and the continuity of the input-output function (second point of Proposition 1.1) to conclude that there exists t ∈ [0, T ] → ϑ t ∈ D 1 0 (R 3 ) analytic, satisfying (1.2) and such that sup

λ j i (t) = 0 if t / ∈ [t j , t j+1 ]. Then, denoting (F j i ) 1≤i≤5
t∈[0,T ] R(t) -Ȓ(t) M(3) + r(t) -ȓ(t) R 3 + ϑ t -θt C 1 0 (R 3 ) 3 < ε/2,
where (R, r) : [0, T ] → SO(3) × R 3 is the solution to System (1.6) with initial data (R(0), r(0)) = ( R(0), r(0)) and control ϑ. The proof is then complete.

Conclusion.

In this paper, we have proved that, for a 3D shape changing body, the ability of swimming (i.e. not only moving but tracking any given trajectory) in a vortex free environment is generic. The genericity refers to the shape of the body, its density and the basic movements (at most five) required for swimming. This result is part of a series of articles [START_REF] Munnier | On the self-displacement of deformable bodies in a potential fluid flow[END_REF][START_REF] Munnier | Locomotion of deformable bodies in an ideal fluid: Newtonian versus lagrangian formalism[END_REF][START_REF] Munnier | Locomotion of articulated bodies in an ideal fluid: 2d model with buoyancy, circulation and collisions[END_REF][START_REF] Chambrion | On the locomotion and control of a self-propelled shape-changing body in a fluid[END_REF] studying locomotion in a potential flow and the next step of this study will be to investigate whether this controllability result can be extended to a flow with vortices.

Denoting u := 3 i=1 α i e i and v := 6 i=4 α i e i • n = 0, this condition reads (u × x + v) • n = 0 on Σ. The case u = 0 is obviously not possible, otherwise we would have v•n(x) = 0 for all x ∈ Σ so let us assume from now on that u = 0. For all x ∈ Σ, (u × x + v) ∈ T x Σ (the tangent space to Σ at x). Let us set w := (u× v)/|u| 2 . Then (u× x+ v) = u× (xw)+ λu with λ := (v •u)/|u| 2 . In this form, we can explicitly compute the expression of the flow connecting to the ODE Ẋ(t, x) = u × (X(t, x)w) + λu, X(0, x) = x. We obtain X(t, x) = exp(tû)(xw) + w + tλu for all t ∈ R. Since X(t, x) has to remain on Σ for all t ∈ R and Σ is bounded, we deduce that λu = 0. For all x ∈ Σ and all t ∈ R, the point X(t, x) lies on Σ. It means that Σ is globally invariant under a rotation whose axis has u as direction vector and goes through w.

Let M be a smooth embedded submanifold of a Banach space E and X := (X i ) 1≤i≤n be set of smooth vector fields on M.

Lemma D.2. Let α i be in L ∞ ([0, T ]) (i = 1, . . . , n) and x be a Carathéodory's solution defined on the time interval [0, T ] to the ODE ẋ(t) = n i=1 α i (t)X i (x) (0 < t ≤ T ), with Cauchy data x(0) = x 0 ∈ M. Let the functions α k i , (i = 1, . . . , n, k ∈ N) be in L 1 ([0, T ]), such that α k i → α i in L 1 ([0, T ]) as k → +∞. Then, for any sequence of Carathéodory's solutions (x k ) k∈N satisfying ẋk (t) = n i=1 α k i (t)X i (x) with Cauchy data x k (0) = x 0 , the functions x k can be continued on the whole interval [0, T ] for k large enough, x k → x uniformly on [0, T ] and ẋk → ẋ in L 1 ([0, T ], M).

Proof. For any δ > 0 small enough, denote by K δ the compact {x ∈ M : x-x(t) E ≤ δ, t ∈ [0, T ]} and denote k δ > 0 the Lipschitz constant such that n i=1 X i (x) -X i (y) E < k δ x -y E for all x, y ∈ K δ and all i = 1, . . . , n. Let M := max x∈K δ i=1,...,n X i (x) E and m := sup t∈[0,T ] i=1,...,n |α i (t)|.

Any function x k is defined at least on a small interval [0, t k [ and we can choose t k small enough such that x k (t) ∈ K δ for all t ∈ [0, t k [. We get the estimate, for all t ∈ [0, t k [: For any ε > 0, we can choose k large enough such that n i=1 t 0 |α i (s) -α k i (s)|ds < εe -mk δ T /M . Applying Grönwall inequality to (D.1), we obtain that x(t) -x k (t) E < ε for all t ∈ [0, t k [. We deduce first that if ε < δ, the solution x k can be continued on the whole interval [0, T ] and then that x k → x uniformly as k → +∞. Writing now that:

x(t) -x k (t) E ≤ M

Fig. 1 . 1 .

 11 Fig. 1.1. Kinematics of the model: The Galilean frame E := (E j ) 1≤j≤3 and the moving frame e := (e j ) 1≤j≤3 with e j = R E j (R ∈ SO(3)). Quantities are usually expressed in the moving frame. The domain of the body is Bt at the time t and Bt is the image of the unit ball B by a diffeomorphism Θt. The open set Ft := R 3 \ Bt is the domain of the fluid. The center of mass of the body is denoted r (in E), v := R t ṙ is its translational velocity (in e) and Ω its angular velocity.

  Bc and satisfies on Σ c := ∂B c the Neumann boundary conditions ∂ n ϕ i c = w i s • n c , n c being the unit normal to Σ c directed toward the interior of B c . Introducing the mass matrix N(c), whose elements are

2 R 3

 23 Eventually, the analyticity of the elements of M r b (c) is straightforward after rewriting the inertia tensor I(c) defined in (1.3) in the form (upon a change of variables) I(c) = B ̺[ Ξ Id -Ξ ⊗ Ξ] dx, still with Ξ := Id + ξ and ξ := ϑ + n i=1 s i V i . 4. Control. 4.1. Controllable SC. Let us fix c ∈ C(n) (for some positive integer n) and recall that S(c) is the connected open subspace of R n such that (c, s) ∈ C X (n). Introducing (f 1 , . . . , f n )

Lemma 4 . 1 .

 41 Let c be a SC fixed in C(n) (n a positive integer). If there exists ζ ∈ M(c) such that dim Lie ζ Z(c) = 6 + n, then the orbit of Z(c) through any ζ ∈ M(c) is equal to the whole manifold M(c). Proof. Rashevsky Chow Theorem (see [1]) applies: If Lie ζ Z(c) = T ζ M(c) for all ζ ∈ M(c) (or more precisely, for all (R, s) ∈ SO(3) × S(c) since Z i c does not depend on r) then the orbit of Z(c) through any point of M(c) is equal to the whole manifold. Let us compute the Lie bracket [Z i c (R, s), Z j c (R, s)] for 1 ≤ i, j ≤ n and (R, s) ∈ SO(3) × S(c). We get:

1≤j≤k

  the canonical basis of (R 5 ) k and S := k j=1 5 i=1 s j i F j i ∈ (R 5 ) k , we get that ( Ȓ, ȓ, S) is a Carathéodory's solution to the following equation on [0, T ]:

  i (s) -α k i (s)|ds + mk δ t 0 x k (s) -x(s) E ds. (D.1)

GCM, ERC Boscain and INRIA color CUPIDSE, and second author by ANR CISIFS, GAOS and MOSICOB.

Appendix A. Function spaces.

Classical function spaces.

• For any compact set K ⊂ R n (n a positive integer), the space C 0 (K) consists of the continuous functions in K endowed with the norm u When Ω = R 3 , we get

is the Banach space of all of the vector fields in R 3 whose every component belongs to C 1 0 (R 3 ).

Id + ∇ϑ(x), e • e ∈ R (S 2 stands for the unit 2 dimensional sphere) is well defined and continuous. For any ϑ 0 ∈ D1 0 (R 3 ), we have δ ϑ0 > 0 and for all x, y ∈ R 3 and e := (y -x)/|y -x| the following estimate holds: (y + ϑ(y) -x -ϑ(x)) • e = |y -x| 1 0 Id + ∇ϑ(x + te), e • e dt > |y -x|δ ϑ . We deduce that Id + ϑ is one-to-one if ϑ is close enough to ϑ 0 . Further, still for ϑ close enough to ϑ 0 , Id + ϑ is a local diffeomorphism (according to the local inversion Theorem) and hence it is onto.

Definition A.2. We denote

, +∞, be a sequence of absolutely continuous functions (respectively of class

Notice that, at this point, we probably have θj / ∈ D 1 0 (R 3 ). Let now Ω be a large ball containing ∪ t∈[0,T ] j∈N Θj t ( B) and Ω ′ be an even larger ball containing Ω. Consider χ a cut-off function such that 0 ≤ χ ≤ 1, χ = 1 in Ω and χ = 0 in R 3 \ Ω′ . Define Θj as the flow associated with the Cauchy problem Ẋ(t,

) and θj := Θj -Id. Since θj t=0 = ϑ † t=0 , we deduce that θj t ∈ D 1 0 (R 3 ) for all t ∈ [0, T ] and the sequence t ∈ [0, T ] → θj t ∈ D 1 0 (R 3 ) complies with the requirements of the Proposition.

Arguing like in the proof of Theorem 2.3, we can easily show that the mapping

R is onto with infinite dimensional kernel. Hence, it is always possible to find a vector field

We can continue with the same idea: The mapping

Appendix D. Added mass matrix for particular shaped swimmers. Let B be an open, bounded, connected, C 1 subset of R 3 . Denote Σ its boundary, n the unit vector to Σ directed toward the interior of B, F := R 3 \ B and consider the 6 × 6 symmetric matrix M f of which the entries are defined by F ∇ψ i • ∇ψ j dx (1 ≤ i, j ≤ 6) where the functions ψ i (i = 1, . . . , 6) are harmonic in F and satisfy the Neumann boundary conditions ∂ n ψ i = (e i ×x)•n on Σ if i = 1, . . . , 6 and ∂ n ψ i = e i-3 • n if i = 4, 5, 6.

Proposition D.1. The matrix M f is always positive and it is positive definite if and only if B has no axis of symmetry.

Proof. Denote α := (α 1 , . . . , α 6 ) t any element in R 6 and ψ := 6 i=1 α i ψ i . Then, we have α t M f α = F ∇ψ 2 R 3 dx ≥ 0 which proves that M f is indeed positive. Let now α be in R 6 such that α t M f α = 0. It means that ψ = 0 and hence ∂ n ψ =


J Sadefo Kamdem
email: sadefo@lameta.univ-montp1.fr

INTEGRAL TRANSFORMS WITH THE HOMOTOPY PERTURBATION METHOD AND SOME APPLICATIONS

Keywords: He's homotopy method, integral transforms, linear equations, Type G and spherical distributions

This paper applies He's homotopy perturbation method to compute a large variety of integral transforms. As illustration, the paper gives special attention to the Esscher transform, the Fourier transform, the Hankel transform, the Mellin transform, the Stieljes transform and some applications.

introduction

Approximation theory covers some important topics in applied analysis, and its application serves many fields in science and engineering such as fluid mechanics, electromagnetism, diffraction theory, statistics and economics. Although it is an old subject, dating back to Laplace, new methods and applications continue to appear in various publications.

There is now a need to provide new methods in the other main area of asymptotic theory, namely, the asymptotic approximation of integrals. In fact, complicated integrals are difficult to solve, and cannot be expressed in terms of elementary functions or analytical formulae. The purpose of this paper is to fulfil this need.

The application of the homotopy perturbation method (HPM) in mathematical problems is highly considered by scientists, because without demanding a small parameter in equations, HPM continuously transforms a complex problem which is not easy to solve into a simple problem. The homotopy perturbation method [START_REF] He | Approximate analytical solution for seepage flow with fractional derivatives in porous media[END_REF] was first proposed by He in 1998. It is in fact a coupling of the traditional perturbation method and homotopy in topology. The HPM was further developed and improved by He [START_REF] He | Approximate analytical solution for seepage flow with fractional derivatives in porous media[END_REF]- [START_REF] He | New interpretation of homotopy pertubation method[END_REF] and applied to asymptotology [START_REF] He | Asymptotology by homotopy perturbation method[END_REF], bifurcation for non-linear problems [START_REF] He | Homotopy perturbation method for bifurcation of nonlinear problems[END_REF], strongly non-linear equations [START_REF] He | Some asymptotic methods for strongly nonlinar equations[END_REF] and many other subjects. The method yields a very rapid convergence of the solution series in most cases, and usually only a few iterations lead to very accurate solutions. Although the goal of He' s homotopy perturbation method was to find a technique to unify linear and nonlinear, ordinary or partial differential equations for solving initial and boundary value problems, here we deal only with the simple first-order differential equation.

Generally, the solutions of the first-order differential equations can be transformed into integrals. In the case where such integrals are difficult to estimate numerically or analytically, new methods are welcome. For example, Hardar et al. [START_REF] Hardar | Integrations by asymptotic decomposition[END_REF] applied Adomian's decomposition method to calculate certain integrals. In Bobolian et al. [START_REF] Babolian | A new computational method for Laplace transform by decomposition method[END_REF], the so-called Adomian decomposition method is used for the computation of the Laplace transform. The shortcoming of Adomian's method [START_REF] Abbassy | The solution of KDV and mKDV equations using Adomian Pade approximation[END_REF], [START_REF] Sadefo Kamdem | Decomposition method for the Camassa-Holm[END_REF] is a complicated calculation of the socalled Adomian polynomials. Even if Momani et al. [START_REF] Momani | Application of He's variational iteration method to Helmhotz equation[END_REF] showed that the variational iteration method [START_REF] He | Variational iteration method a kind of non-inear analytical technique: some examples[END_REF][START_REF] He | Construction of solitary solution and compacton-like solution by variational iteration method[END_REF] would allow the overcoming of this difficulty, a serious alternative approach to overcoming the shortcoming arising in the Adomian method is the homotopy perturbation method. Note that, [START_REF] He | Some asymptotic methods for strongly nonlinar equations[END_REF] and some references therein found that the shortcomings arising in Adomian method can be completely eliminated by the variational iteration method. Recently, Abbasbandy [START_REF] Abbasbandy | Application of He homotopy perturbation method for Laplace transform[END_REF]- [START_REF] Abbasbandy | A numerical solution of Blasius equation by Adomians decomposition method and comparison with homotopy perturbation method[END_REF] suggested the application of HPM in order to compute the Laplace transform .

In this paper, we use He's homotopy perturbation method to compute the general integral transform (1.1)

I(z, x) = x 0 f (t) exp x t
g(s, z) ds dt.

Note that many important integral transforms, including the Esscher, Fourier, Hankel, Hilbert, Laplace, Mellin and Stieljes transforms, can be put in the form of (1.1). Indeed, consider the first-order differential equation

(1.2) v (x) = g(x, z) v(x) + f (x), x > 0 v(0) = 0,
where the function f (t) on (0, ∞) is locally integrable and the function g(x, z) is defined in (0, ∞) × D and D ⊂ C. After solving the homogeneous equation of (1.2) and using the "method of variation of parameters", the analytic solution of (1.2) is given by

v(x) = x 0 f (t) exp x t g(s, z)ds dt (1.3) = exp x 0 g(s, z)ds x 0 f (t) exp - t 0 g(s, z)ds dt, and therefore v(x) exp - x 0 g(s, z) ds = x 0 f (t) exp - t 0 g(s, z)ds dt.

Homotopy perturbation method

The homotopy perturbation method provides an alternative approach to introducing an expanding parameter.

To illustrate the basic ideas of He's homotopy perturbation method, He [START_REF] He | Approximate analytical solution for seepage flow with fractional derivatives in porous media[END_REF] considers the following differential equation

(2.4) A(v) = F (r), r ∈ Ω with boundary conditions B(v) = B v, ∂v ∂n , r ∈ Γ
where A is a general differential operator, B is a boundary operator, F (r) is a known analytic function, Γ is the boundary of the domain Ω and ∂v ∂n denotes the differentiation along the normal vector drawn outwards from Ω. The operator A can be divided into two parts L and N , where L is a linear operator, while N is a linear or non-linear operator. Therefore (2.4) can be rewritten as follows:

(2.5)

L(v) + N (v) -F (r) = 0.
He [START_REF] He | Approximate analytical solution for seepage flow with fractional derivatives in porous media[END_REF], [START_REF] He | Homotopy perturbation technique[END_REF] constructed a homotopy w(r, p)

: Ω × [0, 1] → R which satisfies (2.6) H(w, p) = (1 -p) [L(w) -L(v 0 )] + p [A(w) -F (r)] = 0
where v 0 is the initial value of (2.4) and p ∈ [0, 1] is an embedding parameter. Hence, it is obvious that

H(w, 0) = L(w) -L(v 0 ) and H(w, 1) = A(w) -F (r).
The changing process of p from zero to unity is simply that of w(r, p) from v 0 to F (r).

In topology, this is called deformation, and L(w) -L(v 0 ), A(w) -F (r) are homotopic.

Applying the perturbation technique, due to the fact that 0 ≤ p ≤ 1 can be considered as a small parameter, we can assume that the solution of (2.6) can be expressed as a series in p, as follows:

(2.7) w = ∞ i=0 p i w i ,
when p → 1 and since A := L + N , (2.7) becomes the approximate solution of (2.5), therefore

(2.8) v = lim p→1 ∞ i=0 p i w i .
The convergence of the series in eq. (2.8) is discussed by He [START_REF] He | Approximate analytical solution for seepage flow with fractional derivatives in porous media[END_REF][START_REF] He | Homotopy perturbation technique[END_REF].

Integral Transforms with the HPM

In this section, we consider the homotopy perturbation method with

L(v(x, z)) = -g(x, z) v(x, z)
and

N (v(x, z)) = ∂v ∂x (x, z).
Following (2.5) and (2.6), we construct the simple homotopy

(3.9) (1 -p) [-g(x, z)[w(x, z) -v 0 (x)]] + p ∂w ∂x (x, z) -g(x, z)w(x, z) -f (x) = 0, or (3.10) -g(x, z)[w(x, z) -v 0 (x)] + p ∂w ∂x (x, z) -g(x, z) v 0 (x) -f (x) = 0.
Substituting (2.7) into (3.10), and equating the terms with the identical powers of p, we have (3.11)

-g(x, z) ∞ i=0 p i w i (x, z) -v 0 (x) -p g(x, z)v 0 (x) + p ∞ i=0 p i ∂w i ∂x (x, z) -f (x) = 0 p 0 : L(w 0 -v 0 ) = 0 p 1 : L(w 1 + v 0 ) + N (w 0 ) -f (x) = 0 p 2 L(w 2 ) + N (w 1 ) = 0 p 3 L(w 3 ) + N (w 2 ) = 0 . . . . . . p n+1 L(w n+1 ) + N (w n ) = 0.
In other words

p 0 : -g(x, z) [w 0 (x, z) -v 0 (x)] = 0 p 1 : -g(x, z) (w 1 (x, z) + v 0 (x)) + ∂w 0 ∂x (x, z) -f (x) = 0 p 2 -g(x, z) w 2 (x, z) + ∂w 1 ∂x (x, z) = 0 p 3 -g(x, z) w 3 (x, z) + ∂w 2 ∂x (x, z) = 0 . . . . . . p n+1 -g(x, z) w n+1 (x, z) + ∂wn ∂x (x, z) = 0.
Theorem 3.1. The solution provided by the HPM is

v(x) = x 0 f (t) exp x t g(s, z) ds dt = lim p→1 ∞ i=0 p i w i (x, z). (3.12) Since exp - x 0 g(s, z)ds x 0 f (t) exp x t g(s, z) ds dt = x 0 exp - t 0 g(s, z)ds f (t) dt,
therefore, according to (1.4) and (2.8), we introduce the following definition theorem: Definition 3.2. If f is a locally integrable function on [0, ∞(, the so-called incomplete Sadefo transform IS g of f is the function of z defined by

IS g (z) = x 0 exp - t 0 g(s, z)ds f (t)dt (3.13) = exp - x 0 g(s, z)ds lim p→1 ∞ i=0 p i w i (x, z)
and the so-called Sadefo transform S g of f is the function of z defined by

S g (z) = ∞ 0 exp t 0 g(s, z) ds f (t) dt (3.14) = lim x→∞ exp - x t g(s, z)ds lim p→1 ∞ i=0 p i w i (x, z) ,
where g is given such that the integrals S X g (z) and IS g (z) converge, and the approximation of IS g (z) and S g (z) are obtained via the application of the homotopy perturbation method to the equation (1.2).

Example 3.3. In the calculation of flux integrals associated with double-scattering waves, Servadio [START_REF] Servadio | Calculation of O(1) interferences of double-scattering waves in 3-3 collision[END_REF] has investigated the asymptotic behavior of the integral

I F (z) = ∞ 0 f (t) F (z t) dt (3.15)
where (3.16)

F (t) = ∞ t e -i τ 2 dτ.
For a given f , the estimation of I F (z) is reduced to the estimation of (1.1) where

g(s, z) = z e -i s 2 z 2 ∞ sz e -i t 2 dt ,
and the components w i (x, z) are given by (3.9). In fact

I F (z) = ∞ 0 f (t) F (z t) dt (3.17) = F (0) ∞ 0 f (t) exp (ln(F (z t)/F (0))) dt = F (0) ∞ 0 f (t) exp - t 0 -z F (z s) F (z s) ds dt = F (0) ∞ 0 f (t) exp - t 0 g(s, z) ds ds dt (3.18) where g(s, z) = -z F (z s) F (z s) = z e -i s 2 z 2 ∞ sz e -i t 2 dt .
Corollary 3.4. Following the example that precedes, it is easy to see that the convolution integral

(3.19) I h (z) = ∞ 0 f (t) h(z t) dt,
and the more symmetric convolution integral

(3.20) f * g(z) = ∞ 0 f (t) g(z t -1 ) t -1 dt
can be obtained via the HPM, when I h (z) and f * g(z) converge.

3.1. The case where g(x, z) is homogeneous. Here we consider the particular case where the function g doesn't depend of x, namely

(3.21) g(x, z) = h(z). So the integral (1.4) becomes v(x) = x 0 f (t) exp(-h(z) (t -x)) dt (3.22) = exp(h(z) x) x 0 f (t) exp(-h(z) t) dt. If f is an homogeneous function of order 1 (ie: ∀u ∈ R, h(u z)= u h(z)), then (3.22) becomes (3.23) v(x) exp(-h(z x)) = x 0 f (t) exp(-h(z t)) dt.
We then have the n-th approximation of the integral (3.23) denoted by I 1 (z) as follows

I 1 (z) = ∞ 0 f (t) exp(-h(z t)) dt (3.24) = lim x→∞ lim p→1 ∞ i=0 p i w i (x, z) exp(-h(z x)). Example 3.5. Let h(z) = z and f (x) = e -x ν
, where x ≥ 0 and the parameter ν ∈ R * + . By choosing v 0 (x) = 0, we have g(s, z) = i z for all s ∈ [t, x], the Fourier transform of f is

w 0 (x, z) = 0 w 1 (x, z) = -f (x) z w 2 (x, z) = -f (x) z 2 w 3 (x, z) = -f (x) z 3 . . . w n (x, z) = -f (n-1) (x) z n . with f (n) is the n-derivative of the function f (x). So for Re(z) > 0, ∞ 0 exp(z t -t ν ) dt = -lim x→∞ lim p→1 ∞ i=0 p i b i,ν (x) z i exp(-z x -x ν )) ≈ -lim x→∞ n i=0 b i,ν (x) z i exp(-z x -x ν )) (3.25) where b 0,ν (x) = 0, b 1,ν (x) = ν x -1+ν , b 2,ν (x) = (1 -ν + ν x ν )x -2+ν ν, b 3,ν (x) = x -3+ν (-2 + ν)(-1 + ν) ν + x -3+2 ν (-1 + ν) ν 2 -x -3+3 ν ν 3 + x -3+2 ν ν 2 (-2 + 2 ν), b 4,ν (x) = 6 -11ν (1 -x ν ) + (6 -18x ν + 6x 2ν )ν 2 + (-1 + 7x ν -6x 2ν )ν 3 + x 3ν x -4+ν ν, . . .
(4.27) F[f, z] = e -i x z ∞ 0 f (-t)e i t z dt + ∞ 0 f (t) e -i t z dt 4.2.
Computation of the Laplace Transform. In the case where we consider h(z) = z which is homogeneous of the order 1, in the integral (3.15), the Laplace transform of the function f (x), denoted by L[f, z], is defined by the integral

L[f, z] = ∞ 0 f (t) exp(-z t) dt (4.28) = lim x→∞ lim p→1 ∞ i=0 p i w i (x, z) exp(-z x),
where Re(z) > 0. The functions f (x) and L[f, z] are called a Laplace transform pair. Even though the approximation of the Laplace transform has been the subject of [START_REF] Abbasbandy | Application of He homotopy perturbation method for Laplace transform[END_REF], we proposed the following theorem: Definition 4.1. For all x ∈ (0, ∞), the incomplete Laplace transform of f as

IL[f, z](x) = x 0 f (t) exp(-z t) dt (4.29) = exp(-z x) lim p→1 ∞ i=0 p i w i (x, z),
where Re(z) > 1.

Theorem 4.2. Using some properties of the Laplace transform, we introduce the following results:

(1) If an integrer n > 0 and lim x→∞ f (x)e -sx = 0, then, for x > 0,

L[f (n) , s] = s n lim x→∞ exp(-s x) lim p→1 ∞ i=0 p i w i (x, z) - n-1 i=0 s n-1-i f (i) (0),
where f (0) ≡ f and f (j) is the j-th derivative of f .

(2) If lim x→∞ e -sx x 0 f (ξ) dξ = 0, then

L x 0 f (ξ)dξ), s = 1 s lim x→∞ exp(-s x) lim p→1 ∞ i=0 p i w i (x, z) , (3)
L e -ax f (x), s = lim x→∞ exp(-(s + a) x) lim p→1 ∞ i=0 p i w i (x, z) , (4)
The Laplace transform of the convolution f * g = g * f of two functions f (x) and g(x), has the following approximation

(4.30) L [f * g(x), s] = lim x→∞ lim p→1 ∞ i=0 p i w i (x, z) ∞ j=0 p j v j (x, z) exp(-2z x)
where

L[f, z] = lim x→∞ lim p→1 ∞ i=0 p i w i (x, z) exp(-z x), L[g, z] = lim x→∞ lim p→1 ∞ j=0 p j v j (x, z) exp(-z x),
and where the convolution function f * g is defined by the integral

f * g(x) = x 0 (f (x -ξ)g(ξ) dξ.
(5) If f is a summable function over all finite intervals, and there is a constant c for which

∞ 0 |f (x)| e -c x dx
is finite, then the Laplace transform is accomplished for the analytic function L[f, z] of order O(z -k ), with k > 1, by means of the inverse integral

f (x) = 1 2 π i γ+i∞ γ-i∞ L[f, z]e x z dz (4.31) = 1 2 π i γ+i∞ γ-i∞ exp(z x) lim x→∞ lim p→1 ∞ i=0 p i w i (x, z) exp(-z x) dz
where γ is a real constant that exceeds the real parts of all the singularities of

L[f, z]. Example 4.3. Consider the integral S(x) = ∞ 0 t 1/2 1 + t sin(x t) dt = Im ∞ 0 t 1/2 1 + t exp(ı x t) dt ,
where ı 2 = -1 and Im(z) denotes the imaginary part of the complex z. To approximate the integral

∞ 0 √ t 1 + t exp(ı z t) dt
via the HPM, it suffices to replace in (1.2), g by If we replace g in (3.22), we get

g(s, z) = -ı z, f (t) = √ t 1 + t .
M[f, z] = ∞ 0 f (t) t z-1 dt (4.32) = lim x→∞ lim p→1 ∞ i=0 p i w i (x, z) x z-1 . (4.33)
Note that a < Re(z) < b is an infinite strip such that the Mellin transform of f (t) denoted by M[f, z] converges (See [START_REF] Wong | Asymptotic approximations of integrals[END_REF] for more details). Proposition 4.4. For all x ∈ (0, ∞), the incomplete Mellin transform of f is defined as

IM[f, z](x) = x 0 f (t) t z-1 dt (4.34) = x z-1 lim p→1 ∞ i=0 p i w i (x, z),
where Re(z) > 1.

Theorem 4.5. In using the basic properties of the Mellin transform, we get the following results:

(1) If lim x→∞ x s-r-1 f (r) (x) = 0, r = 0, 1, . . . , n -1 • M[f (n) , z] = (-1) n Γ(z) Γ(z -n) lim x→∞ lim p→1 ∞ i=0 p i w i (x, z -n) x z-n-1 , • M[x n f (n) , z] = (-1) n Γ(z + n) Γ(z) lim x→∞ lim p→1 ∞ i=0 p i w i (x, z) x z-1 ,
(2) Denoting the n th repeated integral of f (x) by I n (f (x)], where

I n [f (x)] = x 0 I n-1 [f (u)] du, (4.35) M[I n [f (x)], z] = (-1) n Γ(z) Γ(z + n) lim x→∞ lim p→1 ∞ i=0 p i w i (x, z + n) x z+n-1
(3) The transform exists provided the integral

∞ 0 |f (x)| x k-1 dx
is bounded for some k > 0, and then the inversion of the Mellin transform is accomplished by means of the inversion integral

f (x) = 1 2 π i γ+i∞ γ-i∞ M[f, z]x -z dz = 1 2 π i γ+i∞ γ-i∞ x -z lim x→∞ lim p→1 ∞ i=0 p i w i (x, z) x z-1 dz (4.36)
where c > k.

Proposition 4.6. The inversion formula for the Mellin transform is given by

f (t) = 1 2π i c+i ∞ c-i ∞ t -z M[f, z] dz (4.37) = 1 2π i lim x→∞ lim p→1 ∞ i=0 p i c+i ∞ c-i ∞ w i (x, z) x z-1 t -z dz.
where a < c < b.

Remark 4.7. Note that one of the two convolution integral transforms, including the Laplace, Fourier, Hankel and Stieljes transforms, can be put in the form of (4.38)

I(x) = ∞ 0 f (t) h(x t) dt. Since M[I, z] = M[f, 1 -z] M[h, z],
then by inversion we obtain

I(x) = 1 2π i c+i ∞ c-i ∞ x -z M[h, z] M[f, 1 -z] dz (4.39)
where c > k and the Mellin transform in the integral (4.39) can be expressed as in (4.32).

Computation of the Hankel Transform. The Hankel transform (of order zero)

is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel and is also called the Fourier-Bessel transform. It is defined as

g(u, v) = F[f (r)](u, v) = ∞ -∞
f (r)e -2π i(ux+vy) dx dy, (4.40) where r = x 2 + y 2 . Let x = rcos(θ), y = rsin(θ), q = √ u 2 + v 2 , u = q cos(φ) and v = q sin(φ), then

g(q) = ∞ 0 2π 0 f (
r)e -2π i(cos(θ)cos(φ)+sin(θ)sin(φ)) r dr dθ

= ∞ 0 2π 0 f (r)e -2π icos(θ)) dθ r dr = 2π ∞ 0 f (r)J 0 (2πqr)rdr, (4.41)
where J 0 is a zero-order Bessel function of the first kind. Therefore, the Hankel transform pairs are

g(q) = 2π ∞ 0 f (r)J 0 (2πqr)r dr f (r) = 2π ∞ 0 g(q)J 0 (2πqr)q dq.
The Hankel transform of n-th order is defined by

(4.42) H n [f (t), φ] = ∞ 0 t J n (φ t) f (t) dt
where J n is a Bessel function of the first kind (Bronshtein et al. [START_REF] Bronshtein | Handbook of Mathematics[END_REF], p. 706).

We continue and propose the following definition Definition 4.8. For all u ∈ (0, ∞) and for a given function f , the incomplete Hankel transform of order n is

IH n [f, z](u) = u 0 t J n (z t) f (t) dt (4.43)
where the Bessel function of the first kind J n (x) is defined as the solution of the Bessel differential equation

(4.44) x 2 ∂ 2 y ∂x 2 + x ∂y ∂x + (x 2 -n 2 )y = 0
which is non-singular at the origin and n is a fix real or complex number.

Remark 4.9. If we replace the function g(x, z) in (1.4) by

(4.45) g(x, z) = - 1 s -s J n (s z) J n (s z) , we get v(x) = 1 x J n (x z) u 0 t J n (z t) f (t) dt = 1 x J n (x z) IH n [f, z](x). (4.46)
Since the solution v of (1.2) has been approximate via the homotopy perturbation method, we then propose the following theorem: Theorem 4.10. For a given function f and ∀x ∈)0, ∞(, the incomplete Hankel transform is given by

(4.47) IH n [f, z](x) = 1 x J n (x z) lim p→1 ∞ i=0 p i w i (x, z),
where the Bessel function of the first kind J n (x) is defined as the solution of (4.44).

We also propose the following corollary Corollary 4.11. For a given function f , the Hankel transform is given by

H n [f, z] = lim x→∞ 1 x J n (x z) lim p→1 ∞ i=0 p i w i (x, z) (4.48) = ∞ 0 t J n (z t) f (t) dt
where the Bessel function of the first kind J n (x) is defined as the solution of (4.44).

Example 4.12. There are some problems of high energy nuclear physics (see Glauber [10] and Gabutti et al. [START_REF] Gabutti | A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of strongly decreasing even functions[END_REF]) that involve integrals of the form (4.49)

I g (x) = ∞ 0 e -t 2 J 0 (xt) g(t 2 ) t dt,
where J 0 (t) is the Bessel function of order zero and g is a continuous function in (0, ∞).

Clearly these integrals are special cases of the Hankel transform formula (4.48), where

f (t) = e -t 2 g(t 2 )
and n = 0.

(4.51) g(x, z) = p m t m-1 x m + z m , we get v(x) = (x m + z m ) GS m,p [f, z, u]. (4.52)
We therefore obtain the following definition theorem: Definition 4.14. For all x ∈ [0, ∞), the incomplete generalized Stieljes transform of a locally integrable function f on [0, ∞) is defined by

IGS m,p [f, z](x) = x 0 f (t) (t m + z m ) -p dt = (x m + z m ) lim p→1 ∞ i=0 p i w i (x, z), (4.53)
where z is a complex variable in the cut plane | arg z| < π.

The generalized Stieljes transform of a locally integrable function f is defined by

GS m,p [f, z] = ∞ 0 f (t) (t m + z m ) -p dt = lim x→∞ (x m + z m ) lim p→1 ∞ i=0 p i w i (x, z) , (4.54)
where m > 0, p ≥ 1 and z is a complex variable in the cut plane | arg z| < π. 4.6. Computation of the Esscher Transform. The Esscher transform was developed to approximate the aggregate claim amount distribution around a point of interest x 0 , by applying Edgeworth series to the transformed distribution with the parameter h chosen such that the mean is equal to x 0 .

We defined the generalized Esscher transform of an appropriate chosen function f : )0, ∞(→ R by Proposition 4.15.

Es[f, z] = 1 M z ∞ 0 x exp(z x) f (x)dx (4.55) = 1 M z L[g, -z] = 1 M z lim x→∞ lim p→1 ∞ i=0 p i w i (x, -z) exp(z x) (4.56) where M z = ∞ 0 exp(z t) f (t)dt = L[f, -z], g(x) = x f (x)
, L denotes the Laplace transform defined in (4.28), and the real constant Re(z) > 0.

Example 4.16. In the following example we choose

f (x) = e -x 3
where x ∈ R + . Using Gradshteyn and Ryzhik [START_REF] Gradshteyn | Table of integrals, Series and Products[END_REF], we get

M z = ∞ 0 exp(z t) exp(-t 3 ) dt = √ z 18 4π J -1 3 2z 3/2 3 √ 3 -J 1 3 2z 3/2 3 √ 3 + 3z 3/2 1 F 2 1; 4 3 , 5 3 ; - z 3 27 , (4.57)
∞ 0 xe z x e -x 3 dx = z 27 2π √ 3 J -2 3 2z 3/2 3 √ 3 -J 2 3 2z 3/2 3 √ 3 -9 1 F 2 1; 2 3 , 4 3
; -z 3 27 , and

(4.58) Es[f, z] = 18 √ z 2π √ 3 J -2 3 2z 3/2 3 √ 3 -J 2 3 2z 3/2 3 √ 3 -9 1 F 2 1; 2 3 , 4 3 ; -z 3 27 27 4π J -1 3 2z 3/2 3 √ 3 -J 1 3 2z 3/2 3 √ 3 + 3z 3/2 1 F 2 1; 4 3 , 5 3 ; -z 3

27

.

We then deduce that

lim x->∞ lim p→1 ∞ i=0 p i w i (x, -z) e z x = z 27 2π √ 3 J -2 3 2z 3/2 3 √ 3 -J 2 3 2z 3/2 3 √ 3 -9 1 F 2 1; 2 3 , 4 3 ;
- z 3 27
where, for n = 0, 1, . . ., we set

w n (x, -z) = e -x 3 a n (x)
z n , where a n (x) is a polynomial function of order 2(n -1) that admits the following ten first terms, obtained via Mathematica software:

a 0 (x) = 0, a 1 (x) = 1, a 2 (x) = 3x 2 , a 3 (x) = (6x -9x 4 ), a 4 (x) = (6x -54x 3 + 27x 6 ), a 5 (x) = -180x 2 + 324x 5 -81x 8 , a 6 (x) = -360x + 2160x 4 -1620x 7 + 243x 10 , a 7 (x) = -360 + 9720x 3 -1780x 6 + 243x 10 + 7290x 9 -729x 12 , a 8 (x) = 30240x 2 -136080x 5 + 119070x 8 -30618x 11 x 10 + 2187x 14 a 9 (x) = 640480x -771120x 4 + 1360800x 7 -694008x 10 + 122472x 13 -6561x 16
Example 4.17. It is very difficult to get the exact analytic Esscher transform of the function

f (x) = e -x ν ,
where the parameter ν ∈ R * + . In this example we propose to find the Esscher transform of f . Since M z = L[f, -z], by using (4.28), we have

(4.59) M z = ∞ 0 e s z e -s ν ds = lim x→∞ lim p→1 ∞ i=0 p i w i (x, -z) exp(z x),
where, for n = 0, 1, . . ., we set

w n (x, -z) = e -x ν b ν n (x) z n ,
where b ν n (x) is a polynomial function that admits the following five first terms, obtained via Mathematica software:

b ν 0 (x) = 0, b ν 1 (x) = ν x -1+ν , b ν 2 (x) = (1 -ν + ν x ν )x -2+ν ν, b ν 3 = x -3+ν (-2 + ν)(-1 + ν) ν + x -3+2 ν (-1 + ν) ν 2 -x -3+3 ν ν 3 + x -3+2 ν ν 2 (-2 + 2 ν), b ν 4 (x) = 6 -11ν (1 -x ν ) + (6 -18x ν + 6x 2ν )ν 2 + (-1 + 7x ν -6x 2ν )ν 3 + x 3ν
x -4+ν ν, . . .

Some applications for probability

5.1. Approximation of type G and spherical distributions. The Type G family of processes is a subclass of the Lévy processes, which allows retaining some of the Gaussian properties of the Wiener process and makes it possible to incorporate processes with jumps and infinite variance. Marcus [START_REF] Marcus | Ξ-radial processes and random Fourier series[END_REF] initially introduced the concept of Type G random variables and processes. Following the theorem 1 of Fotopoulos [START_REF] Fotopoulos | Type G and spherical distribuions on R d[END_REF], we propose the following corollary:

Corollary 5.1. Let ν 0 be a σ-finite measure on B 0 (R d ) and X ∈ R d a random vector of type G with no Gaussian component such that

(5.60) E (i(x, X)) = exp R d (exp(-x, Σ 1/2 y > 2 /2) -1 ν 0 (dy) = exp(-Φ(|x| Σ )),
where |x| Σ =< x, Σx > with Σ being the positive definite symmetric matrix, and Φ : R d → R + is a continuous function such that Φ(0) = 0. Then there exists a positive σ-finite measure on B 0 (R + ) denoted by µ such that the cumulant transform of X ∈ R d is

Φ(|x| Σ ) = C 1 Γ d 2 Γ d-1 2 2π d-1 4 2 ∞ 0 1 -1 F 1 1 2 , d 2 ; - 1 2 |x| 2 Σ r 2 µ(dr) (5.61)
where C = S d-1 λ(ds)and λ is a σ-finite measure on B 0 (S d-1 ).

Proof. Theorem 1 of Fotopoulos [START_REF] Fotopoulos | Type G and spherical distribuions on R d[END_REF] proposes

Φ(|x| Σ ) = C Γ d-1 2 4π d/2 ∞ 0 µ(dr) 1 0 (1 -e t |x| 2 Σ r 2 /2 t -1/2 (1 -t) d-1
2 -1 dt. (5.62) Since, after some calculations, we have

1 0 (1 -e t |x| 2 Σ r 2 /2 t -1/2 (1 -t) d-1 2 -1 dt = √ π Γ d-1 2 Γ d 2 1 -1 F 1 1 2 , d 2 ; - 1 2 |x| 2 Σ r 2 ,
if we replace the preceded integral in (5.62), we have proved corollary (5.61). QED Example 5.2. Consider the polar decomposition of the measure ν 0 is expressed as ν 0 (dy) = r -1-α λ(ds) dr, where 0 < α < 2 and λ is the Borel measure on S d-1 , the unit sphere of R d . Substituting ν 0 in (5.61) gives

Φ(|x| Σ ) = C 1 Γ d 2 Γ d-1 2 2π d-1 4 2 ∞ 0 r -1-α 1 -1 F 1 1 2 , d 2 ; - 1 2 |x| 2 Σ r 2 dr = 1 √ π M f (u), - α 2 (5.63) = lim y→∞ y -α 2 -1 lim p→1 ∞ i=0 p i w i (y) (5.64) = 2 -2-α 2 √ π Γ d 2 -1 Γ -α 2 Γ 1+α 2 Γ d+α 2 |x| α Σ (5.65)
where

f (u) = 1 -1 F 1 1 2 , d 2 ; - 1 2 |x| 2 Σ u , λ(ds) = 2π d 2 Γ( d 2 )
and M is the Mellin transform as defined in (4.32) .

Theorem 5.3. Suppose that the characteristic function of the random vector X ∈ R d satisfies the

E(exp(i < x, X >)) = exp(-Φ(|x| Σ ))
where |x| 2 Σ =< x, Σx >, exp(-Φ(•)) ∈ L(R d ), and Φ satisfies the same conditions as in theorem 1. Then, the joint probability distribution function f of X ∈ R d has the following expression

f (x) = |x| -d-2 2 Σ (2π) d/2 Hd-2 2 [h, |x| Σ ] (5.66) = |x| -d-2 2 Σ (2π) d/2 lim y→∞ 1 y J n (y |x| Σ ) lim p→1 ∞ i=0 p i w i (x, y) (5.67) where h(z) = z d 2 -1 e -Φ(z)
and Hd-2 2 is the Hankel transform as defined in (4.48).

Proof. Under the conditions of (5.4),we have the joint probability distribution of the random vector X ∈ R d as follows

f (x) = |x| -d-2 2 Σ (2π) d/2 ∞ 0 ρ d 2 e -Φ(ρ) J d-2 2
(|x| Σ ρ) dρ.

Following (4.48), we have

Hd-2 2 [h, |x| Σ ] = ∞ 0 ρ d 2 e -Φ(ρ) J d-2 2 (|x| Σ ρ) dρ, where
h(z) = z d 2 -1 e -Φ(z) . Therefore f (x) = |x| -d-2 2 Σ (2π) d/2 Hd-2 2 [h, |x| Σ ] = |x| -d-2 2 Σ (2π) d/2 lim y→∞ 1 y J n (y |x| Σ ) lim p→1 ∞ i=0 p i w i (x, y) . (5.68)
We have then proved theorem (5.4). QED Example 5.4. When the characteristic function has the form (5.63), in view of theorem (5.4), [START_REF] Fotopoulos | Type G and spherical distribuions on R d[END_REF] proposes the following joint probability density function of X ∈ R d :

f (x) = |x| -d-2 2 Σ (2π) d/2 ∞ 0 ρ d 2 e -c ρ α J d-2 2 (|x| Σ ρ) dρ = |x| -d-2 2 Σ (2π) d/2 Hd-2 2 [f, |x| Σ ]. (5.69) where f (u) = u d 2 -1 e -c u α ,
and the positive constant

c = 2 -2-α 2 √ π Γ d 2 -1 Γ -α 2 Γ 1+α 2 Γ d+α 2 .
Because it is difficult to get the explicit expression of the joint density probability function, we propose to write this as a series, by using He's homotopy perturbation method. Since the density function can be estimated via the Hankel transform, it suffices to find w i as indicated in (2.7), as the term of the series that approximate the solution of (1.2), with the function g given in (4.51) and f (u) = u where E denotes the expectation probability of X. We assume that g is given such that the integral S X g (z) converges.

Example 5.7. Consider a random variable X with the probability density function f (x) = C(ν) exp(-x ν ), if;x ≥ 0 0, if x ≤ 0 Remark 5.8. Here it is understood that the interval is closed and may be replaced by ) -∞, ∞(. Whenever, without loss of generality, we speak of the integral transforms (e.g. the Mellin transform of a distribution F , it is tacitly understood that F is concentrated on [0, ∞(.

Remark 5.9. To estimate the n-th moment E (X n ) of the random variable X with the distribution function F , where the integer n > 1, it suffices to choose the function g as follows g(s, t) = -n s ,

where s > 0. If the and the n-th moment exists, its expression is given by (5.73)

E (X n ) = lim x→∞ lim p→1 ∞ i=0
p i w i (x) x n .

Conclusion

In this work, we apply HPM to estimate a large variety of integral transforms (e.g. so called incomplete Sadefo transform). We also show that some known integral transforms such as the Esscher transform, the Fourier transform, the Hankel transform, the Mellin transform and the Stieljes transform are particular cases of our integral transforms family. The method is interesting in order to derive new integration formulae to approximate certain difficult integrals, to calculate the expectation of certain nonlinear functions of random variables and to approximation of type G and spherical distributions. The HPM requires only simple differentiation in order to deduce the integral formulae. In the next paper, we will suggest an application of the HPM in order to serve probability, statistics and mathematical finance.

4 .

 4 Application for the Esscher, Fourier, Hankel, Mellin and Stieljes transforms In this section, with a change of suitable variables, we show how to compute the Esscher, Fourier, Hankel, Laplace, Mellin and Stieljes transforms.

4. 1 .

 1 Computation of the Fourier Transform. Chosing (4.26)

4. 3 .

 3 Computation of the Mellin Transform. To get the Mellin transform, we choose the function g(s, z) = 1 -z s .

4. 5 .Remark 4 . 13 .

 5413 Computation of the Stieljes Transform. We need to approximate the following integral(4.50) GS m,p [f, z](u) = u 0 (t m + z m ) -p f (t) dtwhere u ∈ (0, ∞), m > 0 and p ≥ 1. Note that in the case where p = 1, m = 1 and u → ∞, we have the Stieljes transform of f . If we replace the function g(x, z) in (1.4) by

d 2 - 1 e -c u α . 5 . 2 .Definition 5 . 6 .

 215256 Integral transform of random variable. Definition 5.5. If F is a proper or defective probability distribution concentrated on [0, ∞(, for a given function g, the so-called incomplete Sadefo transform IS g of F is the function defined by IS g (z, x) = , z)ds dF (t) (5.70) and the so-called Sadefo transform S g of F is the function defined by S g (z) = The Sadefo transform S X g of a non-negative random variable X ≥ 0 with the probability density function f (x) is defined as , z)ds f (t) dt,

The author wishes to thanks Prof. Ji-Huan He for his valuable references.