CONTROLLABILITY OF LOW REYNOLDS NUMBERS SWIMMERS OF CILIATE TYPE

. We study the locomotion of a ciliated microorganism in a viscous incompressible ﬂuid. We use the Blake ciliated model: the swimmer is a rigid body with tangential displacements at its boundary that allow it to propel in a Stokes ﬂuid. This can be seen as a control problem: using periodical displacements, is it possible to reach a given position and a given orientation? We are interested in the minimal dimension d of the space of controls that allows the microorganism to swim. Our main result states the exact controllability with d = 3 generically with respect to the shape of the swimmer and with respect to the vector ﬁelds generating the tangential displacements. The proof is based on analyticity results and on the study of the particular case of a spheroidal swimmer.


Introduction
The aim of this article is to analyze the controllability of a system associated with a model of micro-swimmers.The swimmers considered here are ciliated microorganisms immersed in a viscous incompressible fluid.We use the Blake ciliated model [4,5]: we assume that the shape of the swimmer is fixed, and we replace the propelling mechanism of the cilia by time periodic tangential displacements.Due to the micro-scale of the swimmer (very low Reynolds numbers), the inertial forces are neglected and in particular, the fluid motion is governed by the steady-state Stokes system.For more details about this model, we refer the reader to [4,5,14,26,27,37].An important property of the corresponding system is that it can be rewritten as a finite-dimensional nonlinear control problem and this permits the use of the geometric controllability theory.Such an approach is classical and comes back to [2,33].In the case of very high Reynolds numbers, one can assume that the fluid is potential and this leads also to a finite-dimensional nonlinear controlled system that can again be studied with the geometric controllability theory: see [7] for one of the first results in that direction.
The study done here follows the works of San et al. [33], Sigalotti and Vivalda [36], where a similar model is considered.In this first model, the swimming mechanism is modeled by a tangential velocity which is unrelated to a tangential displacement.If we impose that this tangential velocity comes from a boundary displacement, the problem is more complicated and is only tackled in San et al. [34].In this last work, only axi-symmetric

Notation
We first give some notations used throughout the article.
-| • | stands for the Euclidean norm on R d or on M 3 (R).
-Given k ∈ N, C k 0 (R 3 ) is defined by This is a Banach space when endowed with the norm: We also set containing 0. -For a C ∞ -manifold M, TM is the tangent bundle of M and C k (M, TM) is the set of k-differentiable tangent vector fields of M.

The swimmer mechanism
Before entering in the core of this section, let us say some words about the model construction.First of all, we assume that the shape of the swimmer is diffeomorphic (by a diffeomorphism Id +Ψ 0 ) to the unit ball B 0 of R 3 , and its boundary is thus diffeomorphic to the unit sphere S 2 of R 3 .
Then, we consider boundary displacements of the swimmer associated with a tangential velocity.To simplify, we consider the tangential velocity on S 2 and construct the corresponding boundary displacements also on the sphere.Such a displacement should be a diffeomorphism and this leads to some constraints on the tangential velocity.Then we apply Id +Ψ 0 to obtain a boundary displacement of the swimmer, and we immerse it in the Stokes fluid where it can move through rigid motions.This mechanism is summarized in Figure 1.
Note that in the case of an axi-symmetric swimmer moving along its axis of symmetry, a similar model is considered in [26] and [34].Nevertheless, in these articles the diffeomorphism Id +Ψ 0 is explicit (prolate spherical coordinates) and it is easier to write the boundary displacement from the tangential velocity.
For any k ∈ N * ∪ {∞} and any Θ ∈ C k (S 2 , TS 2 ), we can consider the mapping Here, we recall that sinc is the cardinal sine function.If Θ ≡ 0, then X = Id S 2 .Formula (2.1) comes from the exponential formula X = exp(Θ) in the case of S 2 (see for instance [22,23,28]).Expanding the sine and cosine functions, one can see using the above expression and [38] that for every Figure 1.Swimmer model and swimming mechanism.
is analytic.In fact, we have, and we consider the mapping X δ (s) obtained from (2.1) with Θ = Θ δ (s): We denote by J (δ) the set of s ∈ R d such that X δ (s) is a diffeomorphism of S 2 and by J (δ) the connected component of J (δ) containing 0. We have the following standard result which proof is postponed in Appendix B.
Remark 2.2.In particular, there exists ε = ε(δ) > 0 such that for every For every d ∈ N * and every k ∈ N * ∪ {∞}, we write (see Sect. 2.1 for the definition of This is a subset of the Banach space The elements c = (Ψ 0 , δ) of SC k (d) characterize the swimmer: Ψ 0 corresponds to the shape of the swimmer and δ corresponds to the swimming mechanism.Our main result will be generic with respect to these swimmer characteristics in the topology of for any given swimmer characteristics c, one can find swimmer characteristics c arbitrary close to c that allows the swimmer to control its trajectory.
For every c = (Ψ 0 , δ) ∈ SC k (d), we define the global boundary displacement of the swimmer X c by For every s ∈ J (δ), X c (s) is a C k -diffeomorphism from S 2 onto S c .Our aim is to find a time dependent function s : R + → J (δ) so that the boundary displacement X c (s(t)), t ∈ R + can control the position of the swimmer.
In what follows, we add the following constraint on s: there exists T > 0 so that Such a constraint is natural for the swimming mechanism and allows us to focus on a "cycle", that is t ∈ [0, T ].We are thus lead to consider function s such that s(T ) = s(0).We can see that this constraint does not play any role in our controllability result.
In what follows, in order to establish our generic result with respect to the characteristics of the swimmer, we need to introduce the subset of SC k (d) × R d corresponding to the points (c, s) such that X c (s) is a C kdiffeomorphism from S 2 onto S c .To this end, we set where d ∈ N * and where k ∈ N * ∪ {∞}.We have the following result on A k (d) (we postpone the proof in Appendix B).
for all t ∈ [0, T ].Then from the above proposition and from a compactness argument, there exists ε > 0 such that for every c ∈ SC k (d) and for every s

Fluid-structure interactions and motion of the swimmer
Immersed into a viscous incompressible fluid, the swimmer described in the above section can translate and rotate.We write for Q ∈ SO(3) and h ∈ R 3 , We also denote by . These correspond to fluid domains.
A point on the surface of the swimmer can be parametrized as follows Assume that (h, Q, s) is a C 1 function with respect to the time.Then the velocity of the above point x is: Here and in what follows, • denotes the matrix transposition and the dot above a function means its time derivative.
The system describing the motion of the swimmer is given by the following system: ) where n is the unit outer normal to ∂F † (h, Q) and where we have used the notation The functions u † and p † are respectively the velocity and the pressure of the fluid.Equations (2.8a) and (2.8b) are the Stokes system, (2.8c) corresponds to the no-slip boundary condition.Finally, (2.8e) and (2.8f) are the Newton laws with the hypotheses that the inertial effects can be neglected.
We then perform a change of variable to work in a referential attached to the swimmer: we set where for any ω ∈ R 3 , After some calculation (see, for instance, [35]), we obtain the following system: ) )

2.9h)
In what follows, we will rewrite the above system as a finite-dimensional dynamical system with state (h, Q, s) and control input ṡ, see Section 3, equations (3.5) and (3.9).

Main results
In order to state our main result, we first give the definition of a solution: and if they satisfy the equations of (2.9) almost everywhere or in the trace sense.
We have used the notation D l,q for the homogeneous Sobolev spaces (see [17]).In particular We refer to Proposition 3.3 for the well-posedness of the system.Our main result (Thm.2.8) ensures that generically with respect to the swimmer characteristics c, the system (2.9) is controllable.Let us precise our definition of controllability: Definition 2.6.We say that (2.9) is controllable in time T > 0 for c = (Ψ 0 , δ) ∈ SC k (d) if for any and such that the solution of (2.9) with Remark 2.7.This says in particular that for a swimmer of shape S c given by (2.4), there exist periodic boundary displacements associated with the tangential velocities δ so that the swimmer can control its position h and its orientation Q.The invertibility of X c (s(t)) can be written as s(t) ∈ J (δ).The condition s(0) = s(T ) corresponds to the idea that the swimmer is repeating this mechanism periodically (see (2.6)).
The main result states a better property (tracking) that the controllability: for a given trajectory, there exist boundary displacements associated with the tangential velocities δ so that the position of the swimmer can remain arbitrary close to the trajectory. There such that the corresponding solution (h, Q) of (2.9) with initial conditions The proof of this theorem in given in Section 4.5.
Remark 2.9.Let us point out that we obtain a tracking property (and the controllability) not only for the position (h, Q) but also for the boundary displacement (that is for s ∈ J (δ)).
We have in particular the following corollary.such that (2.9) is controllable in time T for the swimmer characteristics c.
In particular the set of swimmer characteristics c such that the system (2.9) is controllable is an open dense set of SC 2 (d).
Remark 2.11.In our opinion, the most important point of our above results is that the controllability can be achieved with only d = 3 controls.As already mentioned in the introduction, the existing swimming controllability results were obtained for d 4 controls (see for instance [3,24]).
Based on the above theorem, we can also derive the existence of optimal controls.We refer to [13] for similar optimal control problems.
Theorem 2.12.Given d 3 and c = (Ψ 0 , δ) ∈ SC 2 (d) such that the system (2.9) is controllable and set Λ a compact of R d containing 0 in its interior and K a compact set of J (δ) which is connected by C 1 -arcs and has a nonempty interior.Let g 3) and s 0 , s 1 ∈ K, we have: 1. there exists T * > 0 such that for every T > T * , the optimal control problem min (2.10) admits a solution; 2. the time optimal control problem (2.11) admits a solution.
Proof.Let us scratch the proof for the first optimal control problem, that is (2.10).
The only hypothesis that needs to be checked carefully is the existence of an admissible control, i.e. that there exists a triplet (h, Q, s) satisfying the constraints of (2.10).To this end, we are going to construct a trajectory on the time interval [0, 1] satisfying the constraint on s.First of all, since K is connected by C 1 -arcs and since the interior of K is nonempty, there exist a point s in the interior of K and two C 1 -arcs s 0 : [0, 1/3] → K and Let us then define ( h 0 , Q 0 ) ∈ R 3 × SO(3) the final value of the solution of (2.9) in [0, 1/3] with initial condition (h 0 , Q 0 ) and control s 0 .Similarly, we define ( h 1 , Q 1 ) ∈ R 3 × SO(3) the initial condition such that the solution of (2.9) in [2/3, 1] with initial condition (at 2/3) ( h 1 , Q 1 ) and control s 1 reaches (h 1 , Q 1 ) at the final time (such a construction can be obtained by time reversion).We conclude, using Theorem 2.8, together with the fact that s is in the interior of K, that there exists a control All in all, by concatenation of s 0 , s 1/2 and s 1 , we have built a control Nevertheless, the property d s(t)/dt ∈ Λ may not hold.For T > 0, we take the control s(t) = s(t/T ) and we see that s(t) ∈ K for every t ∈ [0, T ] and this control steers (h ) and since 0 is an interior point of Λ, we conclude that for T larger than some T * (depending on s and Λ), s is an admissible control.

Rewriting the system
This section is devoted to rewrite system (2.9) as a nonlinear finite-dimensional control problem (system (3.9) below) and to compute Lie brackets that will be useful to apply the Rashevsky-Chow theorem.
From now on, we assume k 2. It is used in the regularity of the solution of the Stokes system.

Decomposition of the system
In this paragraph, we follow the classical decomposition of low Reynolds number swimmers, see for instance the pioneer work [2].Given d ∈ N * , k 2 and c ∈ SC k (d), let us first expand (2.9c).To this end, we define so that for any solution (h, Q, s) of (2.1), the relation (2.9c) writes where (e 1 , e 2 , e 3 ) is the canonical basis of R 3 .This leads us to consider the following Stokes systems Notice that v i c and q i c are also functions of s.In (3.2), the pairs (u i c , p i c ) and (v i c , q i c ) are well-defined in 3 .We refer to ( [17], Lem.V.1.1,p. 305, Thm.V.1.1,p. 306) for the well-posedness of the exterior Stokes problem. Then satisfies (2.9a)-(2.9c).In that case, (2.9e) and (2.9f) can also be rewritten.Indeed, after an integration by parts and using ( [17], Thm.V.3.2, p. 314), we find so that relations (2.9e) and (2.9f) are equivalent to The following result holds (see [24]).
Lemma 3.1.Given k 2, the mapping (c, s) R) is analytic and for every c, K c is positive definite.
We recall that A k (d) is defined by (2.7).We refer to [38] for the definitions and properties of analytic functions in Banach spaces.
Finally, (2.9) writes ) In the above set of equations, we have introduced the new control variable λ = ṡ.In fact, since we also want to impose some conditions on s, we put s in the state of the system and the control of this extended system is λ.This also shows that (2.9) is a finite-dimensional nonlinear dynamical system with control λ.

Formulation of the system in a Lie group
Let us define: and Notice that the map P : d) is a bijection.In addition, endowed with the matrix product, E(3, d) is a Lie group whose Lie algebra is: and with q( , M, s) Clearly, p : R Let us finally define: so that we have: Let us define for every j ∈ {1, . . ., d}, Let us also define: with T (h,Q,s) E(3, d), the tangent space of E(3, d) at the point P (h, Q, s).
Based on Lemma 3.1 we obtain the following lemma.
is analytic.Using that p and P are linear maps, we deduce the result.

Lie brackets computations
compute the Lie brackets only for h = 0 and Q = I 3 .The second property comes from the definition (3.11) of p.We see that the Lie brackets are always included in the following subspace of e(3, d): of dimension 6.If we can generate this subspace with the Lie brackets of f 1 c , • • • , f d c , then, using the form (3.7) of f j c , we obtain that the Lie algebra generated by To obtain the dimension of the Lie algebra generated by f 1 c , • • • , f d c , we will compute the associated Lie brackets.To this end, one has to compute the derivatives of s → V i c (s), where V i c is defined by (3.6).That is to say that we have to compute: In the above expression, v j c (s) and q j c (s) are the solutions of (3.2c).In particular, (∂ α s v j c (s), ∂ α s q j c (s)) is solution of the following system with D i c defined by (3.1).In general, it is not possible to obtain an explicit formula for ∂ α s N (s)e j , but this can be done in the case of the sphere and for particular boundary conditions (see Sect. 4).

The case of the unit sphere
In this section, we consider the situation where S c = S 2 and namely the case where Ψ 0 = 0.

Derivation of boundary conditions
In this paragraph, we compute the expressions of D i c given by (3.1) for Ψ 0 = 0 at s = 0.In that case, we have.
In the above relations, the differential operator G Γ is defined by The proof of the above result is obtained by combining Lemmas 4.2 and 4.3.
Proof.Let us first notice that for Ψ 0 = 0, we have Proof.To simplify the notation, we set X = X δ and Θ = Θ δ (s) = For every n ∈ N * , set A n = ((2n + 1) Id S 2 + Θ), so that, according to (2.2), we have X = Then, for every n ∈ N * , we have: -1st derivative: and hence, and hence, -3rd derivative: and hence, We are now in position to give the proof of Proposition 4.1.
Proof of Proposition 4.1.According to Lemmas 4.2 and 4.3 and (4.1), we deduce We also have Symmetrizing this expression with respect to j and k, we obtain the result.

Stokes solutions on the exterior of a sphere
The results given here are borrowed from [6].In this section, we use spherical coordinates (r, θ, ϕ We recall that a spherical harmonic of degree n 0 is defined by and a rigid spherical harmonic of degree −(n + 1) by with γ m n any real coefficient, and where Y m n is defined by with and with P m n is the associated Legendre polynomial of degree n and order m, that is to say that We recall that the family forms an orthonormal basis of L 2 (∂S 0 ), see for instance ( [15], Chap.VII, Sect.5.3, p. 513).More precisely, this family is orthonormal for the scalar product and hence, According to Lamb [21], see also Brenner ([6], Eq. 2.13), the solution (v, q) of the Stokes equation in an exterior domain can be expressed in spherical coordinates (see Appendix A for the related definition of spherical coordinates and expression of the usual operators ∇, div and rot) as where χ −(n+1) , φ −(n+1) and p −(n+1) are rigid spherical harmonics of degree −(n + 1) defined as in (4.3).Furthermore, the drag and torque exerted on the immersed domain by the fluid can be expressed as Let us mention that F and T are constant vectors of R 3 .In fact, we have, When the exterior domain is the exterior of the unit ball of R 3 , v • e r , div Γ v and rot Γ v for r = 1 can be expressed as a sum of spherical harmonics (see (A.1) for the definitions of div Γ and rot Γ ), with X n , Y n and Z n spherical harmonics of degree n.
where Y 1 and Z 1 are defined from (4.7) with v = ∂ α s D j c (s).More precisely, we obtain Let us also recall that for a spherical body, the matrix K c introduced in (3.3) is (see Sects.5.2 and 5.3 of [19]) 2π 3I 3 0 0 4I 3 and hence, .9)

Particular choices of δ
In order to fully define the swimmer configuration, c = (0, δ), we introduce some explicit choices of δ i 's.The first type of δ i that we consider is sin θ (−m∂ θ (P m n (cos θ)) − m sin θ(P m n ) (cos θ)) sin(mϕ) = 0 .
In the above relations, we have used the property of the associated Legendre polynomials, see for instance ( [15], Chap.V, Sect.10.3, p. 327) Consequently, by orthogonality of spherical harmonics, we obtain V i c (0) = 0, for n 2. -If δ i = ξ m n .Assume m 0, the case m 0 is similar.Similarly, we have to compute the solution v = v r e r + v θ e θ + v ϕ e ϕ of the Stokes equation set on the exterior of the unit ball with the Dirichlet boundary condition: and similarly to the previous case, we obtain Consequently, for n 2, we have V i c (0) = 0.

Explicit computations
In this section, we combine (4.9) and 4.1 in order to compute explicitly (4.12).This computation has been made by using the computer algebra system maxima.
In conclusion of the above computations, we have the following results.
-For every d 4, the analytic maps ) and (at s = 0) are non identically 0.
Remark 4.6.We tried to prove a similar result for d = 2 but our numerical simulations seem to indicate that it is not possible.More precisely, we went up to the computation of Lie brackets of fifth order.In all the computations, we performed the maximal rank of the Lie algebra evaluated at s = 0 was 3. In these computations, we have considered all possible choices of δ given by (4.10) and (4.11) up to spherical harmonics of order 6.We have also taken the parameters m and n in (4.10) and (4.11) randomly, using a Poisson law for n, and again the maximal rank obtained was 3.However, we believe that the generic result, Theorem 2.8 is still valid for d = 2 but probably the spherical swimmers are too symmetric to be controllable with only two elementary deformations.

Lemma 4 . 5 .
For every d 2 and every