Enhanced controllability of low Reynolds number swimmers in the presence of a wall
EEnhanced controllability of low Reynolds number swimmersin the presence of a wall
François Alouges ∗ , Laetitia Giraldi † November 21, 2018
Abstract : Swimming, i.e., being able to advance in the absence of external forcesby performing cyclic shape changes, is particularly demanding at low Reynolds num-bers which is the regime of interest for micro-organisms and micro-robots. We focuson self-propelled stokesian robots composed of assemblies of balls and we prove thatthe presence of a wall has an effect on their motility. To rest on what has been donein [1] for such systems swimming on R , we demonstrate that a controllable swimmerremains controllable in a half space whereas the reachable set of a non fully controllableone is affected by the presence of a wall. Keywords : low Reynolds motion, control theory, Lie brackets.
Swimming at low Reynolds number is now a well established topic of research whichprobably dates back to the pioneering work of Taylor [25] who explains how a micro-organism can swim without inertia. Later on, Purcell [19] formalized the so-called“scallop theorem” which states that, due to the reversibility of the viscous flow, areciprocal deformation of the body cannot lead to a displacement of the swimmer.However, this obstruction can be circumvented using many swimming strategies [19].Swimmers can be distinguished with respect to their ability to change their shape or toimpose rotational motions of some parts of their body in order to create viscous frictionforces on the fluid, and produce by reaction, the propulsion.Many applications are concerned by this problem as for example, the conception ofmedical micro devices. The book by J.P. Sauvage [21] presents a lot of engine modelsadapted for tiny devices while the design and fabrication of such engines have beenrecently investigated by e.g. B. Watson, J. Friend, and L. Yeo [26]. As an example,let us quote the toroidal swimmer, first introduced by Purcell [19] and which has beensubsequently improved by A.M Leshansky and O. Kenneth [12], Y. Or and M. Murray[18], A. Najafi and R. Zargar [17] among others.The strategy for swimming consists in a cyclic deformation of body with a non-reciprocal motion. The first swimmer prototype belonging to this class is the three link ∗ Centre de Mathématiques Appliquées de l’Ecole Polytechnique, CMAP, Ecole Polytechnique,Plateau de Saclay, 91128 Palaiseau, France, Tel.: +33 1 69 33 46 31, [email protected] † Centre de Mathématiques Appliquées de l’Ecole Polytechnique, CMAP, Ecole Polytechnique,Plateau de Saclay, 91128 Palaiseau, France, Tel.: +33 1 69 33 46 09, [email protected] a r X i v : . [ m a t h . O C ] O c t wimmer also designed by Purcell [19]. More recently, R. Golestanian and A. Ajdari[10] introduced the Three-sphere swimmer which is geometically simpler and allows forexact calculations of motion and speed [2], or even explicit in some asymptotic regimes[10].In the continuation of [2], F. Alouges, A. DeSimone, L. Heltai, A. Lefebvre, and B.Merlet [1] showed that the trajectory of the Three-sphere swimmer is governed by adifferential equation whose control functions correspond to the rate of changing shape.The swimming capability of the device now is recast in terms of a control problem towhich classical results apply.Of particular importance for applications is the issue of the influence of any bound-ary on the effective swimming capabilities of micro-devices or real micro-swimmers.Indeed, boundaries clearly affect the hydrodynamics and may have an influence on theswimmer’s capabilities. In that direction, an biological study of Rothschild [20] claimedfor instance that spermatozoids tend to accumulate on walls. More recently, H. Winet,G. S. Bernstein, and J. Head [27] proved this related boundary effect for the sperm ofhumans which evolves in a narrow channel. Swimming in a geometrically confined envi-ronment then became a subject of major interest, in particular to model this attractionphenomenon (see [23], [9],[4]). D. J Smith, E. A. Gaffney and J. R. Blake [24] havedescribed the motion of a stylized bacterium propelled by a single flagellum and theyshow that the attraction by the wall is effective. Later, H. Shum, E.A Gaffney, and J.Smith [22] investigated to which extent this attraction effect is impacted by a changein swimmer’s morphology.On a more theoretical side, other approaches provide results that show an attractioneffect by the wall. A. P. Berke and P. Allison [3], modelling the swimmer with a simpledipole, put in evidence an attraction due to the presence of the wall. Y. Or and M.Murray [18] derived the swimmer dynamics near a wall for three various swimmers, butwith unvarying shapes. The case of a changing shape swimmer has been studied byR. Zargar and A. Najafi [28], where the dynamics of the Three-sphere swimmer in thepresence of a wall is given. However, some fundamental symmetry are not satisfied intheir swimmer’s motion equation.The aim of this paper is to attack the same problem (the influence of a plane wall inthe motion of the swimmer) by means of control theory. Several recent works present acontrollability results for a self-propelled micro-swimmers in a space without boundary,as example, let us quote the paper of J. Lohéac, J. F. Scheid and M. Tucsnak [14] and thestudy of J. Lohéac and A. Munnier [13] made of the spherical swimmer in the whole space(see also [6] for the same kind of results in a perfect fluid). Furthermore, F. Alouges, A.DeSimone, L. Heltai, A. Lefebvre, and B. Merlet [1] deal with the controllability on R for the Three-sphere swimmer and others specific swimmers. The question that we wantto address now is whether the presence of the plane wall modifies the controllabilityresults. We here prove two results in that direction. Namely, considering the fullycontrollable Four sphere swimmer proposed in [1], we show that in the half space, theswimmer remains fully controllable, while a Three-sphere swimmer enriches its reachableset, at least generically which seems at first sight contradictory with earlier results.Indeed, although previous works show an attraction from the boundary, the set ofreachable points could be of higher dimension. In other words, if the dynamics issomehow more constrained due to the presence of the wall, the set of points that theswimmer may reach could be larger than what it was without the wall.2he rest of this paper is organized as follow. In Section 2, we describe the twomodel swimmers to which our analytical and numerical tools are later applied. Section 3presents the main controllability results associated with the introduced swimmers. InSection 4, we show that swimming is indeed an affine control problem without drift byusing a similar approach than F. Alouges, A. DeSimone, L. Heltai, A. Lefebvre, andB. Merlet in [1]. The controllability result is proved in Section 5 for the Four-sphereswimmer and in Section 6 for the Three-sphere swimmer. Concluding remarks are givenin Section 7. We carry on the study of specific swimmers that were considered in [1] in R . In orderto fix notation, the wall is modeled by the plane W = { ( x, y, z ) ∈ R s. t. y = 0 } , andthe swimmers, which consist of N spheres ( B i ) i =1 ..N of radii a connected by thin jacks,are assumed to move in the half space R = { ( x, y, z ) ∈ R s. t. y > } . As in [1] , theviscous resistance associated with the jacks is neglected and the fluid is thus assumedto fill the whole set R \ ∪ Ni =1 B i . The state of the swimmer is described by two sets ofvariables : • the shape variables, denoted by ξ (here in R N − or R N ), which define the lengthsof the jacks. A stroke consists in changing the lengths of these jacks in a periodicmanner ; • the position variables, denoted by p ∈ R × SO (3), which define swimmer’sposition and orientation in the half-space.In what follows, we call S ⊂ R M for a suitable M ∈ N the set of admissible states( ξ , p ) that we assume to be a connected nonempty smooth submanifold of R M . Wethereafter focus on two swimmers that have been considered in the literature, the Three-sphere swimmer (see [16], [2], [1]) and the Four sphere swimmer (see [1]). It turns outthat this latter is easier to understand that the former, and we therefore start with it. We consider a regular tetrahedron ( S , S , S , S ) with center O ∈ R . The swimmerconsists on four balls linked by four arms of fixed directions −−→ OS i which are able toelongate and shrink (in a referential associated to the swimmer). The four ball clusteris completely described by the list of parameters ( ξ , p ) = ( ξ , . . . , ξ , c , R ) ∈ S =( q a, ∞ ) × R × SO (3) ⊂ ( q a, ∞ ) × R (a rotation R ∈ SO (3) is uniquelycharacterized by its 3 dimensional rotaion vector). It is known (see [1]) that the Foursphere swimmer is controllable in R . This means that it is able to move to any pointand with any orientation under the constraint of being self-propelled, and when thesurrounding flow is dominated by the viscosity. This swimmer is depicted in Fig. 1.3 here S := ( a √ , + ∞ ) , the lower bound being chosen in order to avoid overlaps ofthe balls, P = R × R , and the functions X i are now defined as X i ( ξ, c , α, r ) = c + R θ ( ξ i t i + r ) ∀ i ∈ { , , } . Notice that the functions X i are still analytic in ( ξ, c , θ ), and we use them to computethe instantaneous velocity on the sphere B i v i = ∂ X i ∂t ( ξ, c , θ, r ) = ˙ c + ˙ θ e × ( ξ i t i + r ) + R θ t i ˙ ξ i , where e is the vertical unit vector. Eventually, due to the symmetries of the system,the swimmer stays in the horizontal plane. We now turn tothe more difficult situation of a swimmer able to move in the whole three dimensionalspace and rotate in any direction. In this case, we fix N = 4 and we consider a regularreference tetrahedron ( S , S , S , S ) with center O ∈ R such that dist( O, S i ) = 1and as before, we call t i = OS i for i = 1 , , , c ∈ R and a rotation R ∈
SO(3), insuch a way that d = 6.We place the center of the ball B i at x i = c + ξ i R t i with ξ i > i = 1 , , , R = Id ) of the swimmer is however allowed.The four ball cluster is now completely described by the list of parameters X =( ξ, c , R ) ∈ S × P , where S := ( , + ∞ ) and P = R × SO(3). Again, the lowerbound for ξ i is chosen in order to avoid overlaps of the balls. x e , x x x r , Fig. 2.3 . The four sphere swimmer (4S).
Furthermore, the function X i are now defined as X i ( ξ, c , R , r ) = c + R ( ξ i t i + r ) ∀ i ∈ { , , , } , which are still analytic in ( ξ, c , R ), from which we compute the instantaneous velocityon the sphere B i v i = ∂ X i ∂t ( ξ, c , R , r ) = ˙ c + ω × ( ξ i t i + r ) + R t i ˙ ξ i ξ ξ ξ ξ Figure 1: The Four-sphere swimmer.
This swimmer is composed of three aligned spheres as shown in Fig. 2. We assume thatat t = 0 the swimmer starts in the vertical half-plane H = { ( x, y, ∈ R s. t. z =0 , y ≥ } , it is clear from the symmetry of the problem that the swimmer stays in H for all time, for whatever deformation of its arms it may carry out. We characterizeswimmer’s position and orientation in H by the coordinates ( c , θ ) ∈ R × [0 , π ], where c ∈ H is the position of one of the three spheres, and θ is the angle between the swimmerand the x − axis. Therefore, in that case, the swimmer is completely described by thevector ( ξ , p ) = ( ξ , ξ , c , θ ) ∈ S = (2 a, ∞ ) × H × [0 , π ) ⊂ (2 a, ∞ ) × R × R / π Z . Inthe three dimensional space R (when there is no boundary), it is obvious by symmetrythat the angle θ cannot change in time, and thus this swimmer is not fully controllable.One of the main contributions of this paper is to understand the modifications of thisbehavior due to the presence of the plane wall. x x x a θξ ξ Figure 2: The Three-sphere swimmer.4
The main results
Consider any of the swimmers described in the previous Sections, and assume it is self-propelled in a three dimensional half space viscous flow modeled by Stokes equations.In this paper, we will establish that both swimmers are locally fully controllable almosteverywhere in S . By this we mean the precise following statements. Theorem 3.1
Consider the Four-sphere swimmer described in Section 2.2, and assumeit is self-propelled in a three dimensional viscous flow modeled by Stokes equations inthe half space R . Then for almost any initial configuration ( ξ i , p i ) ∈ S , any finalconfiguration ( ξ f , p f ) in a suitable neighborhood of ( ξ i , p i ) and any final time T > ,there exists a stroke ξ ∈ W , ∞ ([0 , T ]) , satisfying ξ (0) = ξ i and ξ ( T ) = ξ f and such thatif the self-propelled swimmer starts in position p i with the shape ξ i at time t = 0 , itends at position p f and shape ξ f at time t = T by changing its shape along ξ ( t ) . Theorem 3.2
Consider the Three-sphere swimmer described in Section 2.1, and as-sume it is self-propelled in a three dimensional viscous flow modeled by Stokes equationsin the half space R . Then for almost any initial configuration ( ξ i , p i ) ∈ S such that p i ∈ H , any final configuration ( ξ f , p f ) in a suitable neighborhood of ( ξ i , p i ) with p f ∈ H and any final time T > , there exists a stroke ξ ∈ W , ∞ ([0 , T ]) , satisfying ξ (0) = ξ i and ξ ( T ) = ξ f and such that if the self-propelled swimmer starts in position p i with the shape ξ i at time t = 0 , it ends at position p f and shape ξ f at time t = T bychanging its shape along ξ ( t ) and staying in H for all time t ∈ [0 , T ] . Remark 3.1
The sense of “almost every initial configuration” can be further precisedas everywhere outside a (possibly empty) analytic manifold of codimension 1.
The proof of the controllability of the Four-sphere swimmer is given in Section 5whereas Section 6 is devoted to demonstrate devoted to demonstrate Theorem 3.2.
As for their 3D counterparts, the equation of motion of both swimmers take the formof an affine control problem without drift. In this section, we detail the derivation ofthis system.
The flow takes place at low Reynolds number and we assume that inertia of both theswimmer and the fluid is negligible. As a consequence, denoting by Ω = ∪ Ni =1 B i thespace occupied by the swimmer, the flow in R \ Ω satisfies the (static) Stokes equation − µ ∆ u + ∇ p = 0 in R \ Ω,div u = 0 in R \ Ω, − σ n = f on ∂ Ω, u = 0 on ∂ R , u → ∞ . (1)5ere, we have denoted by σ = µ ( ∇ u + ∇ t u ) − p Id the Cauchy stress tensor, n isthe unit normal to ∂ Ω pointing outward to the swimmer. We also set V = { u ∈ D ( R \ Ω , R ) | ∇ u ∈ L ( R \ Ω) , u ( r ) p | r | ∈ L ( R \ Ω) } . It is well known that V is a Hilbert space when endowed with the norm (and theassociated scalar product) k u k V := Z R \ Ω |∇ u | . We also assume that f ∈ H − / ( ∂ Ω) in order to obtain a unique solution ( u , p ) tothe problem (1) in V × L ( R \ Ω) which can be expressed in terms of the associatedGreen’s function (obtained by the method of “images”, see [5]) as u ( r ) = Z ∂ Ω K ( r , s ) f ( s ) d s , (2)where the matricial Green function K = ( K ij ) i,j =1 , , is given by K ( r , r ) = G ( r − r ) + K ( r , r ) + K ( r , r ) + K ( r , r ) , (3)the four functions G , K , K and K being respectively the Stokeslet G ( r ) = 18 πµ (cid:18) Id | r | + r ⊗ r | r | (cid:19) (4)and the three “ìmages” K ( r , r ) = − πµ (cid:18) Id | r | + r ⊗ r | r | (cid:19) , (5) K ,ij ( r , r ) = 14 πµ y (1 − δ j ) δ ij | r | − r i r j | r | ! , (6) K ,ij ( r , r ) = − πµ y (1 − δ j ) r | r | δ ij − r j | r | δ i + r i | r | δ j − r i r j r | r | ! . (7)Here r = ( x , y , z ) and r = r − ˜ r , where ˜ r = ( x , − y , z ) stands for the “image”of r , that is to say, the point symmetric to r with respect to the wall.Let B be the sphere of radius 1 centered at the origin. We identify the boundaryof the domain occupied by the swimmer, ∂ Ω, with ( ∂B ) N and we represent by f i ∈ H − / ( ∂B ) the distribution of force on the sphere B i . Correspondingly, u i ∈ H / ( ∂B )stands for the velocity distribution on the sphere B i (and of the fluid due to non-slipcontact).Following [1], we denote by T ( ξ , p ) the Neumann-to-Dirichlet map T ( ξ , p ) : H − / → H / ( f , . . . , f N ) ( u , . . . , u N ) (8)where we have denoted by H ± / the space ( H ± / ( ∂B )) N . It is well known that themap T ( ξ , p ) is a one to one mapping onto while its inverse is continuous.6sing (2), we can express u i ( i = 1 , ,
3) by ∀ r ∈ ∂B, u i ( r ) = N X j =1 Z ∂B K ( x i + a r , x j + a s ) f j ( s ) d s := N X j =1 h f j , K ( x i + a r , x j + a · ) i ∂B , (9)where h· , ·i ∂B stands for the duality (cid:0) H − / ( ∂B ) , H / ( ∂B ) (cid:1) . Proposition 4.1
The mapping ( ξ , p )
7→ T ( ξ , p ) is analytic from S into L ( H − / , H / ) .Furthermore, T ( ξ , p ) is an isomorphism for every ( ξ , p ) ∈ S , and the mapping ( ξ , p ) − ξ , p ) is also analytic. Proof:
The proof is identical to the one given in [1], replacing the the Stokeslet by theGreen kernel K which is also analytic outside its singularity. (cid:50) Remark 4.1
As the direct consequence, the mapping T ( ξ , p ) and its inverse dependsanalytically on a . In this section, we use the self-propulsion assumption in order to express the dynamicsof the swimmer as an affine control system without drift.
Proposition 4.2
There exists a family of vectorfields F i ∈ T S , such that the state ofthe swimmer is described by the following ODE, ddt ξ p ! = X i F i ( ξ , p ) ˙ ξ i . (10) Proof:
This equation of motion is by now classical in this context (see [1], [2], [8] or [15]).Let us recall the principle of its derivation.At any time t , the swimmer occupies a domain Ω t (we therefore denote by Ω thedomain occupied by the swimmer at time t = 0). We also define the map Φ whichassociates to the points of ∂ Ω × S , the current point in ∂ Ω t ,Φ : ∂ Ω × S → ∂ Ω t ( x , ξ , p ) x t . (11)When inertia is negligible, self-propulsion of the swimmer implies that the total viscousforce and torque exerted by the surrounding fluid on the swimmer vanish i.e., F := Z ∂ Ω t T − p , ξ (cid:18) ∂ Φ ∂t (cid:19) d x t = Z ∂ Ω t T − p , ξ (cid:16) ( ∂ p Φ) ˙ p + ( ∂ ξ Φ ) ˙ ξ (cid:17) d x t = 0 , T := Z ∂ Ω t x t × T − p , ξ (cid:18) ∂ Φ ∂t (cid:19) d x t = Z ∂ Ω t x t × T − p , ξ (cid:16) ( ∂ p Φ) ˙ p + ( ∂ ξ Φ) ˙ ξ (cid:17) d x t = 0 . (12)7rom the linearity of the Neumann-to-Dirichlet map, we deduce that the system(12) reads as a linear system which depends on ˙ p and ˙ ξ . By inverting it, we get ˙ p linearly in terms of ˙ ξ . Assuming that ξ ∈ R k for some k ∈ N , we thus obtain˙ p = k X i =1 W i ( ξ , p ) ˙ ξ i , (13)which becomes (10) when we call F i := E i W i ! , where E i is the i -th vector of thecanonical basis. (cid:50) Let us recall some notations which are used to study the controllability of suchsystems of ODE (see for instance [11]).Let F and G be two vector fields defined on a smooth finite dimensional manifold M . The Lie bracket of F and G is the vector field defined at any point X ∈ M by [ F, G ]( X ) := ( F · ∇ ) G ( X ) − ( G · ∇ ) F ( X ). For a family of vector fields F on M , Lie ( F ) denotes the Lie algebra generated by F . Namely, this is the smallest algebra- defined by the Lie bracket operation - which contains F (therefore F ⊂
Lie ( F ) andfor any two vectorfields F ∈ Lie ( F ) and G ∈ Lie ( F ), the Lie bracket [ F, G ] ∈ Lie ( F )).Eventually, for any point X ∈ M , Lie X ( F ) denotes the set of all tangent vectors V ( X )with V in Lie ( F ). It follows that Lie X ( F ) is a linear subspace of T X M and is hencefinite-dimensional.Lie brackets and Lie algebras play a prominent role in finite dimensional controltheory. Indeed, we recall Chow’s theorem: Theorem 4.3 (Chow [7])
Let M be a connected nonempty manifold. Let us assumethat F = ( F i ) mi =1 , a family of vector fields on M , is such that F i ∈ C ∞ ( M , T M ) , ∀ i ∈{ , · · · , m } . Let us also assume that
Lie X ( F ) = T X ( M ) , ∀ X ∈ M . Then, for every ( X , X ) ∈ M×M , and for every T > , there exists u ∈ L ∞ ([0 , T ]; R m ) such that the solution of the Cauchy problem, ˙ X = m X i =1 u i F i ( X ) ,X (0) = X , (14) is defined on [0 , T ] and satisfies X ( T ) = X . The theorem 4.3 is a global controllability result, we also recall the one which givesa small-time local controllability.
Theorem 4.4 ([7], p. 135)
Let Ω be an nonempty open subset of R n , that F =( F i ) mi =1 , a family of vector fields, such that F i ∈ C ∞ (Ω , R n ) , ∀ i ∈ { , · · · , m } . Let X e such that Lie X e ( F ) = R n . hen, for every (cid:15) > , there exists a real number η > such that, for every ( X , X ) ∈{ X s. t. k X − X e k < η } , there exists a bounded measurable function u : [0 , (cid:15) ] → R n such that the solution of the Cauchy problem ˙ X = m X i =1 u i F i ( X ) ,X (0) = X , (15) is defined on [0 , (cid:15) ] and satisfies X ( (cid:15) ) = X . When the vector fields are furthermore analytic (and the manifold M is also analytic)one also has the Hermann-Nagano Theorem of which we will make an important use inthe theoretical study of the controllability for our model swimmers. Theorem 4.5 (Hermann-Nagano [11])
Let M be an analytic manifold, and F afamily of analytic vector fields on M . Then1. each orbit of F is an analytic submanifold of M , and2. if N is an orbit of F , then the tangent space of N at X is given by Lie X ( F ) . Inparticular, the dimension of Lie X ( F ) is constant as X varies over N . In our context, the family of vector fields is given by F = ( F i ) ≤ i ≤ k which are definedon the manifold M = S , and the controls u i are given by the rate of shape changes ˙ ξ i .In view of the preceding theorems, the controllability question of our model swimmersraised by Theorems 3.1 and 3.2 relies on the dimension of the Lie algebra generated bythe vectorfields ( F i ) ≤ i ≤ k which define the dynamics of the swimmer. In particular theyare direct consequences of the following Lemma. Lemma 4.6
For almost every point (in the sense of remark 3.1) ( ξ , p ) ∈ S , the Liealgebra generated by the vectorfields ( F i ) ≤ i ≤ k at ( ξ , p ) is equal to T ( ξ , p ) S . The proof of this lemma is developed until the rest of the paper. Several tools areused in order to characterize this dimension among which we mainly use asymptoticbehavior and symbolic computations. As we shall see, although the theory is clear, theexplicit computation (or at least asymptotic expressions) is by no means obvious andrequires a lot of care. In particular, before using symbolic calculations, a rigorous proofof the expansion, together with a careful control of the remainders in the expressionsallowed us to go further.
In this section, we give the proof of the controllability result stated in Theorem 3.1.
Proof:
The argument of the proof is based on the fact that K given by (3) satisfies K ( r , r ) = G ( r − r ) + O (cid:18) y (cid:19) , (16)where r = ( x, y, z ) and r = ( x , y , z ) are two points of R , and G is the Green functionof the Stokes problem in the whole space R , namely the Stokeslet, defined by (4).9s a consequence, we obtain that the Neumann to Dirichlet map given by (8) satisfiesfor a swimmer of shape ξ at position p = ( p x , p y , p z , R ) ∈ R × SO T ( ξ , p ) = T ξ + O p y ! , (17)where T ξ is the Neumann-to-Dirichlet map associated to the Green function G .The system (12) now reads Z ∂ Ω t (cid:16) T ξ (cid:17) − + O p y !! (cid:16) ( ∂ p Φ) ˙ p + ( ∂ ξ Φ) ˙ ξ (cid:17) d x t = 0 , Z ∂ Ω t x t × (cid:16) T ξ (cid:17) − + O p y !! (cid:16) ( ∂ p Φ) ˙ p + ( ∂ ξ Φ) ˙ ξ (cid:17) d x t = 0 . (18)Consequently, the ODE (10) becomes ddt ξ p ! = X i =1 F i ( ξ ) + O p y !! ˙ ξ i , (19)where ( F i ) i =1 , ··· , are the vector fields obtained in the case of the whole space R .In other words, we obtain the convergence F ( ξ , p ) = F ( ξ ) + O p y ! as p y → + ∞ (20)and also for all its derivatives to any order.It has been proved in [1] that dim Lie ξ ( F ) = 10 at all admissible shape ξ , showingthe global controllability in the whole space of the underlying swimmer. We thus obtainthat for p y sufficiently large dim Lie ( ξ , p ) ( F ) = 10 , (21)and therefore due to the analyticity of the vector fields ( F i ) i =1 , ··· , , (21) holds in adense subset of S . This shows that the system satisfies the full rank condition almosteverywhere in S and proves Lemma 4.6 in this context, and thus Theorem 3.1 by asimple application of Chow’s theorem. (cid:50) The preceding proof can be generalized to any swimmer for which the Lie algebrasatisfies the full rank condition in R . We now turn to an example for which this is notthe case, namely the Three-sphere swimmer of Najafi Golestanian [16]. Indeed, whenthere is no boundary, this swimmer is constrained to move along its axis of symmetry.The purpose of the next section is to understand to which extent this is still the casewhen there is a flat boundary. This section details the proof of Theorem 3.2. It is organized in several subsections,each of them focusing on a particular step of the proof. In the subsection 6.1, by10ntroducing some notations, we recall the expression of the equation of motion of theThree-sphere swimmer. Subsection 6.2 deals with the special symmetry which have tobe verify by the vector fields of the motion equation. From this symmetry properties,we deduce the reachable set of the particular case where the swimmer is perpendicularto the wall. In the subsection 6.3, we give an expansion of the Neumann-To-Dirichletmapping associated to the Three-sphere swimmer and its inverse, in the case wherethe radius of the sphere a is small enough and the distance of the arm is sufficientlylarge. In subsection 6.4, we deduce from this previous approximation an expansion ofthe motion equation of the swimmer, for a sufficiently small. Finally, the subsection 6.5presents some formal calculations of the vectors fields of the motion equation and theirLie brackets which leads to obtain, almost everywhere, the dimension of its Lie algebra. From Section 2 .
2, we know that the swimmer’s position is parameterized by the vector( x, y, θ ) where ( x, y ) is the coordinate of the center of B as depicted in Fig. 2 and θ isthe angle between the swimmer and the x − axis. We recall that ξ := ( ξ , ξ ) stands forthe lengths of both arms of the swimmer.The motion equation (10) thus reads, ddt ξ ξ xyθ = F ( ξ , x, y, θ ) ˙ ξ + F ( ξ , x, y, θ ) ˙ ξ . (22)Notice that, from translational invariance of the problem, both F and F actually donot depend on x .In what follows, we denote by d ( ξ ,y,θ ) = dim Lie ( ξ ,y,θ ) ( F , F )the dimension of the Lie algebra Lie ( ξ ,y,θ ) ( F , F ) ⊂ R at ( ξ , y, θ ). It is clear, since F and F are independent one to another and never vanish, that2 (cid:54) d ( ξ ,y,θ ) (cid:54) . (23) Proposition 6.1
Let S be the × matrix defined by S = − − . Then one has for all ξ = ( ξ , ξ , x, y, θ ) ∈ S F ( ξ , ξ , y, θ ) = SF ( ξ , ξ , y, π − θ ) (24)11 nd similarly for the Lie bracket [ F , F ]( ξ , ξ , y, θ ) = S [ F , F ]( ξ , ξ , y, π − θ ) . (25) Proof:
Although the plane breaks the 3D axisymmetry along the swimmer’s axis, wecan still make use of the symmetry with respect to the vertical plane that passes throughthe center of the first sphere B . A swimmer with position ( x, y, θ ) and shape ( ξ , ξ )is transformed to one at position ( x, y, π − θ ) and shape ( ξ , ξ ) (see Fig. 3). Makinguse of the fact that corresponding solutions to Stokes equations are symmetric one toanother, we easily get the proposition. x x x ˜ x ˜ x wall θ − θ Figure 3: The plane symmetry which links the situation at ( ξ , ξ , x, y, θ ) with those at( ξ , ξ , x, y, π − θ ). In both cases, solutions to Stokes flow are also symmetric one toanother.Eventually, one deduces the Lie bracket symmetries by applying the former symme-tries on the vectorfields themselves. An easy recurrence shows that the same identitieshold for any Lie bracket of any order of the vectorfields F and F . In particular onehas for instance[ F , [ F , F ]]( ξ , ξ , y, θ ) = S [ F , [ F , F ]]( ξ , ξ , y, π − θ ) . (26) (cid:50) As a direct consequence of proposition 6 .
1, we deduce that the fourth coordinate ofthe Lie bracket [ F , F ] vanishes at ( ξ, ξ, y,
0) and at ( ξ, ξ, y, π ).12 roposition 6.2
Let T be the × matrix defined by T = − − . Then one has for all ξ = ( ξ , ξ , x, y, θ ) ∈ S , and i = 1 , F i ( ξ , ξ , y, θ ) = TF i ( ξ , ξ , y, π − θ ) (27) and similarly for the Lie bracket [ F , F ]( ξ , ξ , y, θ ) = T [ F , F ]( ξ , ξ , y, π − θ ) . (28) Proof:
The two identities readily come from the symmetry which transforms a swimmerwith position ( x, y, θ ) and a shape ( ξ , ξ ) to one at position ( x, y, θ ) with the same shape(see Fig. 4). x x x ˜ x ˜ x wall θπ − θ Figure 4: The plane symmetry which links the situation at ( ξ , ξ , x, y, θ ) with those at( ξ , ξ , x, y, π − θ ). In both cases, solutions to Stokes flow are also symmetric one toanother.Eventually, one deduces the Lie bracket symmetries by applying the former symme-tries on the vectorfields themselves. An easy recurrence shows that the same identitieshold for any Lie bracket of any order of the vectorfields F and F . (cid:50) As a result, in the case where θ = ± π , we get the dimension of the Lie algebra ofthe vector field F and F . 13 orollary 6.3 The dimension of the Lie algebra
Lie ( ξ ,ξ ,y,π/ ( F , F ) is less than orequal to . Proof:
We deduce from the preceding proposition that for i = 1 , j = 3 , F ji ( ξ , ξ , y, ± π/
2) = 0. This simply means that a swimmer starting in the verticalposition cannot change its angle θ and its abscissa x by changing the size of its arms.As a matter of fact, the same holds true for any Lie bracket of F and F at any order,and we can deduce from this that d ( ξ ,ξ ,y,π/ ≤ , since any vector of the Lie algebra Lie ( ξ ,ξ ,y,π/ ( F , F ) has a vanishing third and fifthcomponent.Moreover, by using the argument introduced in Section 5, we get, for almost every y , that the dimension of the Lie algebra is almost equal to the one without boundary(see [1]). (cid:50) Remark 6.1
We denote, for all x ∈ R , the set N x := (cid:26) ( ξ , ξ , x , y, ± π s. t. ( ξ , ξ ) ∈ (2 a, ∞ ) y > (cid:27) which corresponds to the case where the swimmer is perpendicular to the wall and A x the set of states where the the dimension of the Lie algebra generated by F and F isequal to two, i.e., A x := (cid:26) (( ξ , ξ , x, y, ± π s. t. d ( ξ ,ξ ,y,π/ = 2 (cid:27) . By using the property of analyticity 4.1, A x is a finite union, A x = [ y ∈ F y (cid:26) ( ξ , ξ , x, y, ± π s. t. ( ξ , ξ ) ∈ (2 a, ∞ ) (cid:27) . We deduce, that for all x , N x \A x defines a set of three-dimensional orbits strictly in-cluded in the manifold S . Furthermore, the proof of the corollary 6.3 can be applied to all generic positionsthen, it implies that the dimension of the Lie algebra is almost equal to 3, almosteverywhere, i.e., 3 (cid:54) d ( ξ ,ξ ,y,θ ) (cid:54) . For the general case ( θ = π/ a (the radius of the balls) near 0.This part is devoted to the proof of the expansion of the Neumann to Dirichlet map(34) together with its inverse (35) at large arms’ lengths. Let us first define for all( i, j ) ∈ { , , } , the linear map T i,j as 14 i,j : H − / ( ∂B ) → H / ( ∂B ) f j Z ∂B K ( x i + a · , x j + a s ) f j ( s ) d s . We recall that the Green kernel K writes (following (3)) as K ( r , r ) = G ( r − r ) + K ( r , r ) + K ( r , r ) + K ( r , r ) , where G is the Stokeslet (see (4)) and each kernel is given by the corresponding coun-terpart in (3). Eventually, we call T the Neumann to Dirichlet map associated to G T : H − / ( ∂B ) → H / ( ∂B ) f Z ∂B G ( a ( · − s )) f ( s ) d s . Proposition 6.4
Let ( i, j ) ∈ { , , } . We have the following expansions, valid for a (cid:28) : • if i = j then T i,j = K ( x i , x j ) h f j , Id i ∂B + R (29) where || R || L ( H − / ,H / ) = O ( a ) , • otherwise T i,i = T + X k =1 K k ( x i , x i ) h f i , Id i ∂B + R (30) where || R || L ( H − / ,H / ) = O ( a ) . Proof:
Let ( i, j ) ∈ { , , } be such that i = j , and f j ∈ H − / ( ∂B ). We define ∀ r ∈ ∂B , u i ( r ) := ( T i,j f j )( r ) = Z ∂B K ( x i + a r , x j + a s ) f j ( s )d s , (31)and v i ( r ) = u i ( r ) − K ( x i , x j ) Z ∂B f j ( s ) d s = Z ∂B ( K ( x i + a r , x j + a s ) − K ( x i , x j )) f j ( s )d s . Our aim is to estimate the H / ( ∂B ) norm of v i . But k v i k H / ( ∂B ) ≤ k v i k H ( B ) , and since K ( x , y ) is a smooth function in the neighborhood of x = x i and y = x j , onehas ∀ r , s ∈ B | K ( x i + a r , x j + a s ) − K ( x i , x j ) | = O ( a ) , (32) Here and in the sequel, we use the definition for the H / ( ∂B ) norm: k v k H / ( ∂B ) = min w ∈ H ( B , R ) , w = v on ∂B k w k H ( B ) . r and s |∇ r K ( x i + a r , x j + a s ) | = O ( a ) , |∇ s K ( x i + a r , x j + a s ) | = O ( a ) , |∇ r ∇ s K ( x i + a r , x j + a s ) | = O (cid:16) a (cid:17) . Therefore, we obtain ∀ r ∈ B | v i ( r ) | ≤ k K ( x i + a r , x j + a · ) − K ( x i , x j ) k H k f j k H − ≤ O ( a ) k f j k H − , and similarly |∇ r v i ( r ) | ≤ k∇ r ( K ( x i + a r , x j + a · )) k H k f j k H − ≤ O ( a ) k f j k H − . This enables us to estimate the H norm of v i on ∂B k v i k H ( B ) ≤ k v i k H ( B ) = (cid:16) k v i k L ( B ) + k∇ v i k L ( B ) (cid:17) ≤ O ( a ) k f j k H − . which proves (29).In order to prove (30), we use the decomposition (3) where none of the kernels( K i ) i =1 , , is singular. Therefore ∀ r ∈ ∂B u i ( r ) := ( T i,i f i )( r ) = Z ∂B K ( x i + a r , x i + a s ) f i ( s )d s = Z ∂B G ( a ( r − s )) f i ( s )d s + Z ∂B ( K + K + K )( x i + a r , x i + a s ) f i ( s )d s = T f i + Z ∂B ( K + K + K )( x i + a r , x i + a s ) f i ( s )d s . We finish as before, having remarked that for l = 1 , , K l ( x i + a r , x i + a s ) = K l ( x i , x i ) + O ( a ) . (33) (cid:50) Proposition 6.5
For every f ∈ H − / , ( T x f ) i ( r ) = T f i + X l =1 K l ( x i , x i ) h f i , Id i ∂B + X j = i K ( x i , x j ) h f j , Id i ∂B + R i ( f ) , (34) with kR i k L ( H − / , H / ) = O ( a ) . roof: For all i ∈ , ,
3, and all r ∈ ∂B ( T x f ) i ( r ) := Z ∂B K ( x i + a r , x i + a s ) f i ( s )ds + X i = j Z ∂B K ( x i + a r , x j + a s ) f j ( s )d s = T i,i f i + X j = i T i,j f j and the result follows from the application of (29) and (30) of Proposition 6.4. (cid:50) Proposition 6.6
In the regime a (cid:28) , one has for every u ∈ H / , (cid:0) T − x u (cid:1) i = ( T ) − (cid:16) u i − P k =1 K k ( x i , x i ) h ( T ) − u i , Id i ∂B (cid:17) − ( T ) − X j = i K ( x i , x j ) h ( T ) − u j , Id i ∂B + ˜ R i ( u ) (35) with k ˜ R i k L ( H / , H − / ) = O (cid:16) a (cid:17) . Proof:
We recall that T : H − ( ∂B ) → H ( ∂B ) f Z ∂B G ( a ( · − s )) f ( s ) d s , and define for l = 1 , , S l : H − ( ∂B ) → H ( ∂B ) f Z ∂B K l ( x i , x i ) f ( s ) d s , and eventually S i,j : H − ( ∂B ) → H ( ∂B ) f Z ∂B K ( x i , x j ) f ( s ) d s . That these operators are continuous operators from H − ( ∂B ) into H ( ∂B ) is classical.We hereafter are only interested into the estimation of their norms, and more preciselythe way they depend on a , δ and y in the limit a →
0. Notice that since the kernel G is homogeneous of degree -1, one has kT k L ( H − / , H / ) = O (cid:18) a (cid:19) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) T (cid:17) − (cid:13)(cid:13)(cid:13)(cid:13) L ( H / , H − / ) = O ( a ) . (36)As far as S l is concerned, we get that (since | K l ( x i , x i ) | = O (1)) kS l k L ( H − / , H / ) = O (1) , (37)and similarly kS i,j k L ( H − / , H / ) = O (1) . (38)When a → (cid:50) .4 Self-propulsion We now use the fact that the spheres are non-deformable and may only move following arigid body motion. In other words, the velocity of each point r of the i − sphere expressesas a sum of a translation and a rotation as u i ( r ) = u T i + u R i ( r ) , (39)where u T i is constant on ∂B while u R i ( r ) = ω i × a r for a suitable angular velocity ω i (remember that all quantities are expressed on the unit sphere ∂B ). This is of peculiarimportance for the computation of the total force and the total torque, which, due toself-propulsion, should vanish which implies X i Z ∂B f i = X i Z ∂B (cid:16) T − x u (cid:17) i = 0 . (40)Plugging (39) in (40) and using (35) leads to X i Z ∂B ( T ) − u T i + u Ri − X k =1 K k ( x i , x i ) h ( T ) − ( u T i + u Ri ) , Id i ∂B ! − ( T ) − X j = i K ( x i , x j ) h ( ¯ T ) − ( u T j + u Rj ) , Id i ∂B = O (cid:16) a (cid:17) || u || . (41)It is well known that both translations and rotations are eigenfunctions of the Dirich-let to Neumann map of the three dimensional Stokes operator outside a sphere. Namely (cid:16) T (cid:17) − u T i = λ T u T i and (cid:16) T (cid:17) − u Ri = λ R u Ri . It is well-known that λ T = µa leading in particular to the celebrated Stokes formula Z ∂B (cid:16) T (cid:17) − u T i d s = 6 πµa u T i while λ R = 3 µa . We also remark that due to R ∂B u R i d s = 0 , we have Z ∂B (cid:16) T (cid:17) − u R i d s = 0 . We therefore obtain6 πµa X i u T i − πµa X k =1 K k ( x i , x i ) u T i − πµa X j = i K ( x i , x j ) u T j = O (cid:16) a (cid:17) || u || . (42)We now compute the torque with respect to the center x of the first ball B . Self-propulsion of the swimmer implies that this torque vanishes:0 = Z ∂B a r × f ( r )+ Z ∂B ( x − x + a r ) × f ( r )+ Z ∂B ( x − x + a r ) × f ( r ) = I + I + I , (43)18here the quantities I , I and I are respectively given below. Calling e θ = cos θ sin θ the direction of the swimmer I = Z ∂B ( x − x + a r ) × f ( r ) = Z ∂B ( ξ e θ + a r ) × ( T x ) − u = Z ∂B ( − ξ e θ + a r ) × ( T ) − u T + u R − πµa X l =1 K l ( x , x ) u T − πµa X j =2 K ( x , x j ) u T j + O (cid:16) a (cid:17) || u || = − πµaξ e θ × u T − πµa X l =1 K l ( x , x ) u T − πµa X j =1 K ( x , x j ) u T j + O (cid:16) a (cid:17) || u || . Similarly, we get, I = a Z ∂B r × f ( r ) = a Z ∂B r × ( T x ) − u = a Z ∂B r × ( T ) − u T + u R − πµa X l =1 K l ( x , x ) u T − πµa X j =2 K ( x , x j ) u T j + O (cid:16) a (cid:17) || u || = a Z ∂B r × ( T ) − ( u R ) + O (cid:16) a (cid:17) || u || . But since (cid:0) T (cid:1) − u R = λ R u R = λ R ω × a r , we have a Z ∂B r × ( T ) − ( u R ) = a λ R Z ∂B r × ( ω × r ) d r = 8 π µa ω . This leads to I = 8 π µa ω + O (cid:16) a (cid:17) || u || . Correspondingly, I = Z ∂B ( x − x + a r ) × f ( r )= 6 πµaξ e θ × u T − πµa X l =1 K l ( x , x ) u T − πµa X j =3 K ( x , x j ) u T j + O (cid:16) a (cid:17) || u || . Denoting by A the matrix A = A A A A A A A A A (44)19here for i = 1 , , A ii = Id − πµa X l =1 K l ( x i , x i ) (45)and for i, j = 1 , , i = j A ij = − πµa K ( x i , x j ) (46)and S the matrix S = Id Id Id − ξ e θ × ξ e θ × ! , we can rewrite the self propulsion assumption (42), (43) as (notice that angular velocitiesbeing involved of higher order disappear) SA u T u T u T = O (cid:16) a (cid:17) || u || . (47)We end up by expressing u T , u T , u T and ω , ω , ω in terms of ˙ x, ˙ y, ˙ θ, ˙ ξ and ˙ ξ .But, since u T i is the velocity of the center of the ball B i , one has u T = ˙ x − ˙ ξ cos( θ ) + ˙ θξ sin( θ )˙ y − ˙ ξ sin( θ ) − ˙ θξ cos( θ )0 , u T = ˙ x ˙ y , and u T = ˙ x + ˙ ξ cos( θ ) − ˙ θξ sin( θ )˙ y + ˙ ξ sin( θ ) + ˙ θξ cos( θ )0 . Similarly ω = ω = ω = θ . We rewrite these formulas as u T u T u T = T ˙ x ˙ y ˙ θ + U ˙ ξ (48)with T = Id − ξ e ⊥ θ Id Id ξ e ⊥ θ , where e ⊥ θ = − sin θ cos θ and U = − e θ
00 00 e θ . SA + R ) T ˙ x ˙ y ˙ θ + U ˙ ξ = 0 (49)where the residual matrices have a norm which is estimated as || R || = O (cid:16) a (cid:17) . Rewriting from (44), (45) and (46) A = Id + a A , we can expand in power series of a thesolution of (49). This enables us to write an expansion (still in a ) of the two vectorfields F and F . To this end, we have used the software MAPLE to symbolically computethose expressions and the Lie brackets [ F , F ], [ F , [ F , F ]], and [ F , [ F , F ]]. Writingthe vectorfields in components as F ( ξ , ξ , y, θ ) := F + O (cid:0) a (cid:1) F + O (cid:0) a (cid:1) F + O (cid:0) a (cid:1) , F ( ξ , ξ , y, θ ) := F + O (cid:0) a (cid:1) F + O (cid:0) a (cid:1) F + O (cid:0) a (cid:1) , (50)we find, after having furthermore expanded the abovementioned components in powerseries of 1 y , F = 13 cos( θ ) + a θ ) K ( ξ , ξ , θ ) + 3 a y (sin( θ ) cos( θ ) ( ξ + 2 ξ ))+ a y (cid:16) cos( θ ) K ( ξ , ξ , θ ) (cid:17) + a y (cid:16) sin( θ ) cos( θ ) K ( ξ , ξ , θ ) (cid:17) + O (cid:18) ay (cid:19) , F = −
13 cos( θ ) − a θ ) K ( ξ , ξ , − θ ) + 3 a y (sin( θ ) cos( θ ) (2 ξ + ξ )) − a y (cid:16) cos( θ ) K ( ξ , ξ , − θ ) (cid:17) + a y (cid:16) sin( θ ) cos( θ ) K ( ξ , ξ , − θ ) (cid:17) + O (cid:18) ay (cid:19) F = 13 sin( θ ) + a θ ) K ( ξ , ξ , θ ) − a y K ( ξ , ξ , θ )+ a y sin( θ ) K ( ξ , ξ , θ ) − ay K ( ξ , ξ , θ ) + O (cid:18) ay (cid:19) F = −
13 sin( θ ) − a θ ) K ( ξ , ξ , − θ ) − a y K ( ξ , ξ , θ ) − a y sin( θ ) K ( ξ , ξ , − θ ) − ay K ( ξ , ξ , − θ ) + O (cid:18) ay (cid:19) F = 3 a y sin( θ ) cos( θ ) K ( ξ , ξ , θ ) − a y cos( θ ) K ( ξ , ξ , θ ) + O (cid:18) ay (cid:19) F = 3 a y sin( θ ) cos( θ ) K ( ξ , ξ , − θ ) + 9 a y cos( θ ) K ( ξ , ξ , − θ ) + O (cid:18) ay (cid:19) .
21n those expressions the remaining functions are respectively given by K ( ξ , θ ) = (cid:0) ξ ξ − ξ ξ − ξ + 2 ξ ξ + 2 ξ (cid:1)(cid:0) ξ + ξ ξ + ξ (cid:1) ξ ξ ( ξ + ξ ) K ( ξ , θ ) = − ξ cos( θ ) + 12 cos( θ ) ξ + 184 ξ + 24 cos( θ ) ξ ξ − ξ ξ − θ ) ξ ξ − ξ + 105 ξ cos( θ ) − θ ) ξ K ( ξ , θ ) = 1 (cid:0) ξ + ξ ξ + ξ (cid:1) (cid:16)
12 cos( θ ) ξ + 24 ξ cos( θ ) − ξ cos( θ ) − ξ cos( θ ) +112 ξ + 72 ξ ξ − ξ ξ − ξ ξ + 224 ξ − ξ ξ cos( θ ) − ξ ξ cos( θ ) − ξ ξ cos( θ ) + 48 ξ ξ − ξ ξ cos( θ ) −
24 cos( θ ) ξ ξ + 9 cos( θ ) ξ ξ − ξ cos( θ ) ξ (cid:17) K ( ξ , θ ) = (cid:0) ξ ξ − ξ ξ − ξ + 2 ξ ξ + 2 ξ (cid:1)(cid:0) ξ + ξ ξ + ξ (cid:1) ξ ξ ( ξ + ξ ) K ( ξ , θ ) = 6 cos( θ ) ξ + 3 cos( θ ) ξ − ξ − ξ K ( ξ , θ ) = − ξ cos( θ ) + 6 cos( θ ) ξ + 56 ξ + 12 cos( θ ) ξ ξ − ξ ξ − θ ) ξ ξ − ξ + 66 ξ cos( θ ) − θ ) ξ K ( ξ , θ ) = 1 (cid:0) ξ + 512 ξ ξ + 512 ξ (cid:1) (cid:16) −
210 cos( θ ) ξ − ξ cos( θ ) + 232 ξ cos( θ ) +24 cos( θ ) ξ − ξ + 12 cos( θ ) ξ − ξ ξ − ξ ξ − ξ + 104 ξ ξ cos( θ ) − ξ ξ cos( θ ) − ξ ξ + 216 ξ ξ cos( θ ) −
66 cos( θ ) ξ ξ − ξ cos( θ ) ξ − ξ cos( θ ) ξ −
24 cos( θ ) ξ ξ −
21 cos( θ ) ξ ξ + 464 ξ cos( θ ) − ξ ξ +264 ξ ξ cos( θ ) −
204 cos( θ ) ξ ξ + 9 cos( θ ) ξ ξ (cid:17) ,K ( ξ , θ ) = − ξ − ξ + 2 cos( θ ) ξ + cos( θ ) ξ K ( ξ , θ ) = 1 (cid:0) ξ + ξ ξ + ξ (cid:1) (cid:16)
20 cos( θ ) ξ −
40 cos( θ ) ξ − θ ) ξ + 8 ξ cos( θ ) − ξ ξ − ξ ξ + 32 ξ ξ − ξ + 32 ξ cos( θ ) ξ − θ ) ξ ξ + cos( θ ) ξ ξ −
40 cos( θ ) ξ ξ + 8 ξ cos( θ ) ξ + 32 ξ − ξ ξ cos( θ ) (cid:17) . As one can see, the use of a software for symbolic computation seems unavoidable.Subsequently, we get the expansion of the Lie bracket [ F , F ]( ξ , ξ , y, θ )[ F , F ]( ξ , y, θ ) := F , F ] + O (cid:0) a (cid:1) [ F , F ] + O (cid:0) a (cid:1) [ F , F ] + O (cid:0) a (cid:1) , (51)22here the components are given by the following expressions[ F , F ] = − a θ ) ( ξ + 2 ξ ξ + ξ ξ + 2 ξ ξ + ξ )( ξ + ξ ) ξ ξ − a y a cos( θ ) sin( θ ) ξ ξ (cid:0) cos( θ ) − θ ) + 8 (cid:1) (cid:0) − ξ + ξ (cid:1)(cid:0) ξ + ξ ξ + ξ (cid:1) + O (cid:18) ay (cid:19) , [ F , F ] = − a θ ) (cid:0) ξ + 2 ξ ξ + ξ ξ + 2 ξ ξ + ξ (cid:1) ( ξ + ξ ) ξ ξ + 27 a y cos( θ ) ξ ξ (cid:0) cos( θ ) − θ ) + 8 (cid:1) (cid:0) − ξ + ξ (cid:1) ( ξ + ξ ξ + ξ )+ O (cid:18) ay (cid:19) , [ F , F ] = 81 a y cos( θ ) ξ ξ ( ξ + ξ ) (cid:0) cos( θ ) − θ ) + 8 (cid:1) ( ξ + ξ ξ + ξ ) + O (cid:18) ay (cid:19) . Notice that since the two first coordinates of F and F are constant, the correspondingfirst coordinates of the Lie bracket vanish. Similarly, the asymptotic expansion for thesecond order Lie bracket [ F , [ F , F ]] ( ξ , y, θ ) reads[ F , [ F , F ]] ( ξ , y, θ ) := F , [ F , F ]] + O (cid:0) a (cid:1) [ F , [ F , F ]] + O (cid:0) a (cid:1) [ F , [ F , F ]] + O (cid:0) a (cid:1) , (52)where [ F , [ F , F ]] = − a θ ) ξ (cid:0) ξ + 3 ξ ξ + ξ (cid:1) ξ ( ξ + ξ ) + 27 a y cos( θ ) sin( θ ) L ( ξ , θ ) + O (cid:18) ay (cid:19) , [ F , [ F , F ]] = − a θ ) ξ (cid:0) ξ + 3 ξ ξ + ξ (cid:1) ξ ( ξ + ξ ) − a y cos( θ ) L ( ξ , θ ) + O (cid:18) ay (cid:19) , [ F , [ F , F ]] = − a y cos( θ ) L ( ξ , θ ) + O (cid:18) ay (cid:19) . There, L and L are respectively given by L ( ξ , θ ) = ξ (cid:0) − θ ) + cos( θ ) (cid:1) (cid:0) ξ − ξ ξ − ξ (cid:1) ( ξ + ξ ξ + ξ ) ,L ( ξ , θ ) = ξ (cid:0) − θ ) + cos( θ ) (cid:1) (2 ξ + ξ )( ξ + ξ ξ + ξ ) . F , [ F , F ]] is given by[ F , [ F , F ]] ( ξ , ξ , y, θ ) := F , [ F , F ]] + O (cid:0) a (cid:1) [ F , [ F , F ]] + O (cid:0) a (cid:1) [ F , [ F , F ]] + O (cid:0) a (cid:1) , (53)where [ F , [ F , F ]] = − a cos( θ ) ξ (cid:0) ξ + 3 ξ ξ + 3 ξ (cid:1) ( ξ ( ξ + ξ ) ) − a y a cos( θ ) sin( θ ) L ( ξ , ξ , − θ ) + O (cid:18) ay (cid:19) , [ F , [ F , F ]] = 2 a θ ) ξ (cid:0) ξ + 3 ξ ξ + 3 ξ (cid:1) ( ξ ( ξ + ξ ) − a y cos( θ ) L ( ξ , ξ , − θ ) + O (cid:18) ay (cid:19) , [ F , [ F , F ]] = − a y cos( θ ) L ( ξ , ξ , − θ ) + O (cid:18) ay (cid:19) . We now can compute an expansion of det ( F , F , [ F , F ] , [ F , [ F , F ]] , [ F , [ F , F ]])which if it does not vanish implies the local controllability of our model swimmer. Itcan be readily checked that we havedet ( F , F , [ F , F ] , [ F , [ F , F ]] , [ F , [ F , F ]]) == (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ F , F ] [ F , [ F , F ]] [ F , [ F , F ]] [ F , F ] [ F , [ F , F ]] [ F , [ F , F ]] [ F , F ] [ F , [ F , F ]] [ F , [ F , F ]] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 81 a ( ξ − ξ )131072 y sin θ (cos θ ) R ( ξ , θ ) + O (cid:18) y (cid:19) , (54)with, R ( ξ , θ ) = (cid:0) ξ + 27 ξ ξ + 50 ξ ξ + 55 ξ ξ + 50 ξ ξ + 27 ξ ξ + 6 ξ (cid:1) ( ξ + ξ ) (cid:0) ξ + ξ ξ + ξ (cid:1) ξ ξ × (cid:16) −
64 cos( θ ) + 32 cos( θ ) − θ ) + cos( θ ) (cid:17) It is easily seen that R never vanishes. Therefore, the previous determinant hasa non-vanishing first coefficient (in 1 y ) which does not vanish for ξ = ξ and θ / ∈{ , π , π, π } . Since it is an analytic function of ( ξ , ξ , y, θ ) we deduce that it does notvanish except at most on a negligible set. This is sufficient to conclude that d ( ξ ,ξ ,y,θ ) = 5almost everywhere and the local controllability of the Three-sphere swimmer aroundsuch points. 24 emark 6.2 Quite strikingly, when ξ = ξ the first term of the expansion vanishesand one has to go one step further. We find in that casedet ( F , F , [ F , F ] , [ F , [ F , F ]] , [ F , [ F , F ]]) ( ξ, ξ, y, θ ) = T ( ξ, y, θ ) 1 y + O (cid:18) y (cid:19) , where T ( ξ, y, θ ) = − a sin( θ ) cos( θ ) ξ (cid:16) cos( θ ) + 8 − θ ) (cid:17) . This coefficient does not vanish unless θ / ∈ { , π , π, π } . The case θ = 0 or π . We already know from symmetry that when θ = π or θ = π , one has d ( ξ ,ξ ,y,θ ) ≤
3. Therefore, it remains to understand the case θ = 0 (or π by symmetry). The preceding computation does not allow us to conclude aboutthe dimension of the Lie algebra at such points. Indeed, the 2 first coefficients ofthe expansion of the determinant vanish, and it might well be the case at all orders.Nevertheless, in that case, we can expand the subdeterminant∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ F , F ] [ F , [ F , F ]] [ F , F ] [ F , [ F , F ]] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) in order to obtain informations. Indeed, one gets∆ = 45512 y a ( ξ − ξ ) ξ R ( ξ ) + O (cid:18) y (cid:19) , with, R ( ξ ) = 1( ξ + ξ ) ξ (cid:0) ξ + ξ + ξ ξ (cid:1) (cid:16) ξ + 11 ξ ξ + 16 ξ ξ + 19 ξ ξ + 12 ξ ξ + 3 ξ (cid:17) . As the direct consequence, we get that the dimension of the Lie algebra, d ( ξ ,y, ≥ ξ , y ) ∈ ( R + ) .This finishes the proof of Lemma 4.6 and thus of Theorem 3.2. (cid:50) Remark 6.3
As usual, it is possible to pass from local to global controllability on eachof the connected components where the determinant given by (54) does not vanish. Moreprecisely, let A := n ( ξ , p ) s. t. d ( ξ , p ) ≤ o , we define by S ( ξ , p ) the connected componentof the subset S \ A which contains ( ξ , p ) . Applying Chow’s Theorem 4.3 on S ( ξ , p ) , givesthat for every initial configuration ( ξ i , p i ) , any final configuration ( ξ f , p f ) in S ( ξ , p ) ,and any final time T > , there exists a stroke ξ ∈ W , ∞ ([0 , T ]) , satisfying ξ (0) = ξ i and ξ ( T ) = ξ f and such that the self-propelled swimmer starting in position p i with theshape ξ i at time t = 0 , ends at position p f and shape ξ f at time t = T by changing itsshape along ξ ( t ) and staying in S ( ξ , p ) for all time t ∈ [0 , T ] . In other words, S ( ξ , p ) isexactly equal to the orbit of the point ( ξ , p ) . Conclusion
The aim of the present paper was to examine how the controllability of low Reynoldsnumber artificial swimmers is affected by the presence of a plane boundary on the fluid.The systems are those classically studied in the literature (see [1] for instance) but areusually not confined. This is the first in-depth control study of how the presence of theplane wall affects the reachable set of a peculiar micro-swimmer.Firstly, the Theorem 3.1 shows that the controllability on the whole space impliesthe controllability in the half space. Although the proof is applied on the Four-sphereswimmer, it is based on general arguments which can be appropriate for any finitedimensional linear control systems.Secondly, the Theorem 3.2 deals with the controllability of the Three-sphere swim-mer in the presence of the plane wall. We prove that, at least for this example, thehydrodynamics perturbation due to the wall surprisingly makes the swimmer more con-trollable. This result is not in contradiction with the several scientific studies whichshow that the wall seems to attract the swimmer (see [20], [27], [23], [9], [4]). Although,the Theorem 3.2 leads to the fact that the wall contributes to increase the swimmer’sreachable set, we can conjecture that some of them are easier to reach than others.The quantitative approach to this question together than the complete understand-ing of the situation in view of controllability of the underlying systems is far beyondreach and thus still under progress as is, in another direction, the consideration of morecomplex situations like, e.g. rough or non planar wall. This is the purpose of ongoingwork.
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