Controllability of 3D Low Reynolds Swimmers

Controllability of 3D Low Reynolds Swimmers

Abstract

In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequence of macro shape changes. Our second contribution is to prove that, when no macro deformations are prescribed, tracking is generically possible by means of shape changes obtained as a suitable combination of only four elementary deformations. Eventually, still considering finite dimensional deformations, we state results about the existence of optimal swimming strategies for a wide class of cost functionals.

L

ocomotion, Biomechanics, Stokes fluid, Geometric control theory

{AMS}

74F10, 70S05, 76B03, 93B27

1 Introduction

1.1 Context

Relevant models for the locomotion of microorganisms can be tracked back to the work of Taylor [16], Lighthill [11, 10], and Childress [6]. Purcell explains in [12] that these sort of animals are the order of a micron in size and they move around with a typical speed of 30 micron/sec. These data lead the flow regime to be characterized by a very small Reynolds number. For such swimmers, inertia effects play no role and the motion is entirely determined by the friction forces.

In this article, the swimmer is modeled as a self deforming-body. By changing its shape, it set the surrounding fluid into motion and generates hydrodynamics forces used to propel and steer itself. We are interested in investigating whether the microswimmer is able to control its trajectory by means of appropriate shape deformations (as real microorganisms do). This question has already be tackled in some specific cases. Let us mention [14] (the authors study the motion of infinite cylinders with various cross sections and the swimming of spheres undergoing infinitesimal shape variations) and [2] (in which the 1D controllability of a swimmer made of three spheres is investigated).

Our contribution to this question is several folds. First, we give a definitive answer to the control problem in the general case: the swimmer we consider has any shape at rest (obtained as the image by a diffeomorphism of the unit ball) and can undergo any kind of shape deformations (as long as they can also be obtained as images of the unit ball by diffeomorphisms). With these settings, we prove that the dynamical system governing the swimmer’s motion in the fluid is controllable in the following sense: for any prescribed trajectory (i.e. given positions and orientations of the swimmer at every moment) there exists a sequence of shape changes that make him swim arbitrarily close to this trajectory. A somewhat surprising additional result is that this can be done by means of arbitrarily small shape changes which can be superimposed to any preassigned macro deformations (this is called the ability of synchronized swimming in the sequel). Second, when no macro deformations are prescribed (this is called freestyle swimming in the paper), we prove that the ability of tracking any trajectory is possible by means of shape changes obtained as an appropriate combination of only four elementary deformations (satisfying some generic assumptions). Third, we state a result about the existence of optimal swimming.

Notice that the paper follows the lines of [4] in which the authors study the controllability of a swimmer in a perfect fluid.

1.2 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 . Two frames are required in the modeling: The first one is fixed and Galilean and the second one is attached to the swimming body. At any moment, there exist a rotation matrix and a vector such that, if and are the coordinates of a same vector in respectively and , then the equality holds. The matrix is meant to give also the orientation of the swimmer. The rigid displacement of the swimmer, on a time interval (), is thoroughly described by the functions and , which are the unknowns of our problem. Denoting their time derivatives by and , we can define the linear velocity and angular velocity vector (both in ) by respectively and , where for every vector , is the unique skew-symmetric matrix satisfying for every .

Shape Changes

Unless otherwise indicated, from now on all of the quantities will be expressed in the body frame . In our modeling, the domains occupied by the swimmer are images of the closed unit ball by diffeomorphisms, isotopic the identity, and tending to the identity at infinity, i.e. having the form where belongs to (the definitions of all of the function spaces are collected in the appendix, Section A). With these settings, the shape changes over a time interval can be simply prescribed by means of functions lying in . Then, denoting , the domain occupied by the swimmer at every time is the closed, bounded, connected set (keep in mind that we are working in the frame ) and is the swimmer’s Eulerian velocity of deformation. We shall denote the unit ball’s boundary while will stand for the body-fluid interface. The unit normal vector to directed toward the interior of is and the fluid fills the exterior open set .

The Flow

The flow is governed by the stationary Stokes equations. They read (in the body frame ):

where is the viscosity, the Eulerian velocity of the fluid and the pressure. These equations have to be complemented with the no-slip boundary conditions: on . The linearity of these equations leads to introducing the elementary velocities and pressures () and , defined as the solutions to the Stokes equations with the boundary conditions (), () and on . Then, the velocity and the pressure can be decomposed as and . Notice that the pairs () and are well-defined in the weighted Sobolev spaces (see the Appendix, Section C).

Figure 1: Kinematics of the model: The Galilean frame and the body frame with (). Quantities are mostly expressed in the body frame. The domain of the body is at the time and is the image of the unit ball by a diffeomorphism . The open set is the domain of the fluid. The position of the swimmer is given by the vector (in ) and its orientation by . The vector is the translational velocity (in ).

Dynamics

As already pointed out before, for microswimmers, the inertia effects are neglected in the modeling. Newton’s laws reduce to (balance of angular momentum) and (balance of linear momentum) where is the stress tensor of the fluid, with . The stress tensor is linear with respect to so it can be decomposed into . In order to rewrite Newton’s laws in a short compact form, we introduce the matrix whose entries are

and , the vector of whose entries are

With these settings, Newton’s laws take the convenient form . Upon an integration by parts, the entries of the matrix can be rewritten as , whence we deduce that is symmetric and positive definite. We infer that the swimming motion is governed by the equation:

(1a)
To determine the rigid motion in the fixed frame , Equation (1a) has to be supplemented with the ODE:
(1b)

together with Cauchy data for and . At this point, we can identify the control as being the function . Notice that the dependence of the dynamics in the control is strongly nonlinear. Indeed describes the shape of the body and hence also the domain of the fluid in which are set the PDEs of the elementary velocity fields involved in the expressions of the matrices and .

Considering (1), we deduce as a first nice result: {proposition} The dynamics of a microswimmer is independent of the viscosity of the fluid. Or, in other words, the same shape changes produce the same rigid displacement, whatever the viscosity of the fluid is. {proof} Let be an elementary solution (as defined in the modeling above) to the Stokes equations corresponding to the viscosity , then is the same elementary solution corresponding to the viscosity . Since the Euler-Lagrange equation (1) depends only on the Eulerian velocities , the proof is completed.

As a consequence of this Proposition we will set in the sequel.

Self-propelled constraints

For our model to be more realistic, the swimmer’s shape changes, instead of being preassigned, should be resulting from the interactions between some internal forces and the hydrodynamical forces exerted by the fluid on the body’s surface. To do so, the dynamics (1) should be supplemented with a set of equations (for instance PDEs of elasticity) allowing the shape changes to be computed from given internal forces. However, this would make the problem of locomotion much more involved and is beyond the scope of this paper. For weighted swimmers, this issue can be circumvented by adding constraints ensuring that the body’s center of mass and moment of inertia are deformation invariant in the body frame. Unfortunately, massless microswimmers have no center of mass and their moment of inertia is always zero.

To highlight the fact that constraints have still to be imposed to the shape changes for the control problem to make sense, consider the following result: {proposition} Let be two control functions such that and differ up to a rigid displacement on the unit sphere (more precisely, for every , there exists such that and ). Then, denoting by a solution (if any) to System (1) with Cauchy data , we get that the function is also a solution with the same Cauchy data but control . In particular for all (i.e. the swimmer’s global motion is the same in both cases). {proof} If we denote by () (respectively ) the elementary velocity fields obtained with the control function (respectively ), it can be verified that for every , every and every . We deduce that where the elements of (respectively ) have been computed with the elementary velocity fields (respectively ) and is the bloc diagonal matrix . On the other hand, denoting respectively by and the boundary velocity of the swimmer in both cases, we get the relation: for all , where and . With obvious notation, we deduce that . If we set now and , we get the identity . It suffices to integrate this relation, taking into account that , to obtain the conclusion of the Proposition and to complete the proof.

If we apply this proposition with constant in time (the boundary of the swimmer is at any time), we deduce that any shape change which reduces to a rigid deformation on the swimmer’s boundary will produce a displacement . But if we compute the global motion of the swimmer, we obtain for every and every time which means that the swimmer is actually motionless (the rigid deformation of the swimmer’s boundary is counterbalanced by its rigid displacement). To prevent this from happening, we add the following constraints to the deformations (inspired by the so-called self-propelled constraints for weighted swimmers, see for instance [5]):

(2)

About the existence of such deformations, we have in particular: {proposition} For every function in such that , there exists a function in satisfying (2) and an unique absolutely continuous rigid displacement such that , and for every .

In other words, the proposition tells us that any function of satisfying the first equality of (2) at , can be made allowable (in the sense that it satisfies (2)) when composed with a suitable rigid displacement on the unit sphere. {proof} Define (an absolutely continuous function on ) and for every . The matrix is always definite positive since for all and all . We can then define as a function of . The absolutely continuous function is obtained by solving the ODE with Cauchy data (we consider here a Carathéodory solution which is unique according to Grönwall’s inequality). Then, we set for all . The function is in , satisfies (2) but does not take its values in because as . Let and be large balls such that and and consider a cut-off function valued in and such that in and in . To complete the proof, define as the flow associated with the Cauchy problem , . {definition} We denote by the non-empty closed subset of consisting of all of the functions verifying (2).

1.3 Main results

The first result ensures the well posedness of System (1) and the continuity of the input-output mapping:

{proposition}

For any , any function (respectively of class , ) and any initial data , System (1) admits a unique solution (in the sense of Carathéodory) absolutely continuous on (respectively of class ).

Let be a sequence of controls converging to a function . Let a pair be given and denote by the solution in to System (1) with control and Cauchy data . Then, the unique solution to System (1) with control and Cauchy data converges in to as .

We denote by the Banach space of the matrices endowed with any matrix norm. The main result of this article addresses the controllability of System (1): {theorem}(Synchronized Swimming) Assume that the following data are given: (i) A function (the reference shape changes); (ii) A continuous function (the reference trajectory to be followed). Then, for any , there exists a function (the actual shape changes) in , which can be chosen analytic, such that , and where the function is the unique solution to system (1) with initial data and control .

This theorem tells us that any 3D microswimmer 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 somehow preassigned. Actually, the trick is that they can only be approximately prescribed. We are going to show that arbitrarily small superimposed shape changes suffice for controlling the swimming motion.

When no macro shape changes are preassigned we have: {theorem}(Freestyle Swimming) Assume that the following data are given: (i) A function such that (the reference shape at rest) (ii) A continuous function (the reference trajectory). Then, for any there exists a function (the actual shape at rest) such that (i) (ii) and (iii) for almost any -uplet satisfying , and (), there exists a function (which can be chosen analytic) such that, using as control in the dynamics (1), we get where the function is the unique solution to ODEs (1) with initial data .

We claim in this Theorem that any 3D microswimmer (maybe up to an arbitrarily small modification of its initial shape) is able to swim by means of allowable deformations (i.e. satisfying the constraints (2)) obtained as a suitable combination of almost any given four basic movements.

If we still seek the control function as a combination of a finite number of elementary deformations, i.e. in the form

(3)

where is in , and is a fixed family of vector fields satisfying , and () we can state the following result: {theorem}(Existence of an optimal control) Let be a continuous function, convex in the third variable and let be a compact of . Let and be two elements of such that there exists a control function (i) having the form (3) with for a.e. , (ii) satisfying , and (iii) steering the dynamics (1) from (at ) to (at ). Then, among all of the control functions satisfying (i-iii), there exists an optimal control realizing the minimum of the cost

The proofs of these results rely on the following leading ideas: First, we shall identify a set of parameters necessary to thoroughly characterize a swimmer and its way of swimming (these parameters are its shape and a finite number of basic movements, satisfying the constraints (2)). Any set of such parameters will be termed a swimmer signature (denoted SS in short). Then, the set of all of the SS 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 approximately any preassigned shape changes) is related to the vanishing of some analytic functions depending on the SS. These functions are connected to the determinant of some vector fields and their Lie brackets (we will invoke classical results of Geometric Control Theory). Eventually, by direct calculation, we will prove that at least one swimmer (corresponding to one particular SS) has this ability. An elementary property of analytic functions will eventually allow us to conclude that almost any SS (or equivalently any microswimmer) has this property.

Eventually, the existence of an optimal control in Theorem 1.3 is a straightforward consequence of Filippov Theorem (see [1, Chap. 10])

1.4 Outline of the paper

The next Section is dedicated to the notion of swimmer signature (definition and properties). In Section 3 we show that the matrix and the vector (in (1a)) are analytic functions in the SS (swimmer signature, 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. In the same Section, a particular case of swimmer will be shown to be controllable. In Section 5 the proof of the main results will be carried out. Section 6 contains some words of conclusion. Technical results and definitions are gathered in the appendix in order to make the paper more readable.

2 Swimmer Signature

A swimmer signature is a set of parameters characterizing swimmers whose deformations consist in a combination of a finite number of basic movements. {definition} For any positive integer , we denote the subset of consisting of all of the pairs such that, denoting and , the following conditions hold (i) the set is a free family in (ii) every pair of elements of satisfies and .

We call swimmer signature (SS in short) any element of .

By definition, is open in (see appendix, Section A). We deduce that for any , the set is open as well in and we denote its connected component containing . {definition} For any positive integer , we call swimmer full signature (SFS in short) any pair such that and . We denote the set of all of these pairs.

Restatement of the problem in terms of swimmer signature (SS) and swimmer full signature (SFS)

Pick a SS, with (for some integer ). Denote and for all , ( is hence a SFS). The body of the swimmer occupies the domain at rest and (for any ) when swimming. Notice that within this construction, the shape changes on a time interval () are merely given through an absolutely continuous function . If stands for its time derivative in , the Lagrangian velocity at a point of is while the Eulerian velocity at a point is with . Due to assumption (ii) of Definition 2, the constraints (2) are automatically satisfied.

The elementary fluid velocities and elementary pressure functions corresponding to the rigid motions depend only on the SFS. Therefore, they will be denoted in the sequel and to emphasize this dependence. The same remark holds for the matrix whose notation is turned into . The elementary velocity and pressure connected to the shape changes can be decomposed into and respectively. In this sum, each pair solves the Stokes equations in with boundary conditions on .

Introducing the matrix , whose elements are

(recall that the viscosity can be chosen equal to 1), the dynamics (1a) can now be rewritten in the form:

(4)

Let us focus on the properties of and .

{theorem}

For any positive integer , the set is an analytic connected embedded submanifold of of codimension .

The definition and the main properties of Banach space valued analytic functions are summarized in the article [17].

{proof}

For any , denote and . Then, define for , the functions by . Every function is analytic and so is (). In order to prove that (the differential of at the point ) is onto for any , assume that there exist vectors () such that:

(5)

where () and with and . Reorganizing the terms in (5), we obtain that:

Since this identity has to be satisfied for any , we deduce that, for every :

(6)

Integrating this equality over , we get that (