3.1. Controllability Analysis
In this section, we examine the controllability properties of the flow-driven swimmer in a local sense around reference trajectories . The idea is to answer the following question, which constitutes an important theoretical feature for a controlled system: can we target any position and orientation within a neighbourhood of the swimmer path using a control of “small” amplitude that stays close to a reference value ? Here, we will focus on the particular and arguably more practical situation , therefore seeking to control the position and orientation of the free swimmer by generating a flow of “small” amplitude with the neighbouring wall.
Consider an initial state for the swimmer, and let be the solution of the system starting at with no flow, on the time interval , which is simply given by : with no external flow, the swimmer swims along a straight line.
Local controllability around
is schematised on
Figure 2 and formally defined as follows [
38]: System (
4) is said to be locally controllable around the trajectory
if, for all
, there exists neighbourhoods
and
of
and
such that, for all
in
, and
in
, there exists a bounded control
u defined on the time interval
, satisfying
for all
, and such that the solution of System (
4) starting at
satisfies
.
Figure 2 provides a graphical illustration of this concept, with an example of controlled trajectory (in a dotted line) linking an initial state in
and a target state in
around the reference trajectory (in red).
To study the local controllability of System (
4), we consider the linearised control system of System (
4) around
, with reference control
, that reads
with
and
. It is well-known (see for instance [
38] (Theorem 3.6)) that the nonlinear system is locally controllable around
if the linearised system (
4) is (globally) controllable. To examine if this is the case, we can apply a simple algebraic “Kalman-type” condition for time-varying linear control systems [
39,
40]: let
and
; then, System (
6) is controllable in time
T if and only if there exists
such that the matrix
is invertible.
The determinant
of
can be obtained through straightforward calculations:
with
. We can see that the determinant vanishes if any of the parameters among
k,
v,
, or
is equal to 0. In those cases, the linearised system is not controllable. Upon further investigation, we can see that two of these cases are actually trivially not controllable, even globally: if
, the swimmer’s orientation
remains always constant regardless of the control, indicating that a simple sliding motion of a wall is unable to control the swimmer, and the wave-like spatial inhomogeneity is necessary for the wall actuation, and if
(passive particle), the particle can only follow a single flow line. Thus, the wave-like wall actuation is only able to control an active particle. For the two other cases
(horizontal swimming speed equal to the velocity of the flow wave front) and
(vertical initial orientation), the controllability analysis alone does not allow for concluding with certainty that the nonlinear system (
4) is not controllable. However, it suggests that these cases are unfavourable for driving the swimmer in an arbitrary direction and call for extra care when trying to steer its trajectory in practice.
If all of the above parameters are nonzero, then the determinant
does not vanish (or, at worst, vanishes on isolated points), and therefore system (
4) is locally controllable around
.
Overall, the controllability analysis shows that, except in a few unfavourable cases described above, the flow generated by the active wall is able to locally influence the trajectory of the swimmer, both in positional and rotational dynamics. We have therefore theoretically validated the usability of this means of control and will carry on towards constructive determination of optimally controlled trajectories.
3.2. Optimal Control
For practical guidance of a swimmer using the active wall flow, we will study optimal control problems minimising the time required to reach a set target. Formally, given an initial state
and a target
, this problem can be stated as follows:
We will focus on three situations (“Problems”) of interest that together allow for fully steering the trajectory of the swimmer:
- 1.
A displacement parallel to the wall, from to at fixed distance : ;
- 2.
A change of orientation, from to at fixed distance : ;
- 3.
A change of distance to the wall, from to : .
A star (∗) denotes the variables for which the final value is not fixed. Problem 1 is a natural goal that one would wish to address in this situation, consisting of enhancing, with the external flow, the transportation of the swimmer along the horizontal direction. Problems 2 and 3 complete this main objective by adding the capacity of steering the swimmer close to or away from the wall, addressing the practical issue of trapping a swimmer near the wall or letting it escape. Note that Problem 3 is less constrained than Problems 1 and 2, as only one variable is fixed at a final time; for that reason, it will be partly contained in Problem 2, as we will see later. As we observe in the following, the capacity to control the swimmer’s orientation via Problem 2 is also useful for guidance along the wall (Problem 1) for a long distance by occasionally helping the swimmer to face back in its goal direction.
Using classical optimal control theory, we address the time-optimal problem (
8) by writing the associated Hamiltonian
H:
where
, and
is non-trivial and a solution to the adjoint system
. By virtue of the Pontryagin maximum principle,
u must maximise
H along the trajectory, from which we deduce that it switches between values
and
, depending on the sign of the term between brackets in (
9), which we will denote by
. Hence the optimal control is simply expressed as a so-called
bang–bang function:
Note that if
vanishes on an interval, one can quickly deduce that
on this interval, which contradicts the non-triviality of
; thus, the third case of (
10) is almost void here, and the control always switches between
and
. This is valid for any optimal problem including the aforementioned objectives 1, 2, and 3; however, different values and constraints on
and
yield different dynamics for the Hamiltonian, requiring ad hoc analysis for each problem in order to determine the optimal control.
Let us now set
and focus on Problem 1: moving the swimmer from
to
in minimal time, while keeping it at a distance
. To further the analysis, we assume that the wave velocity
is sufficiently larger than the swimmer speed
v and, from Equation (
1), the characteristic parallel flow speed at distance
, defined by
and that
is close to
, so that the swimmer is moving towards the target
. These assumptions imply that the dynamics of the adjoint vector
, as well as those of
z and
, change slowly with respect to the timescale
, so the sign changes of the bracketed term in (
9) are approximately given by those of
.
In other words, to maximise the Hamiltonian on a short timescale, the flow amplitude
u must switch between
and
every time
changes sign, which occurs when the swimmer crosses the border between two of the flow cells generated by the wall. The time
elapsed between two switches can then be estimated as the ratio between the wall wavelength
and the difference between the wall flow velocity and swimmer effective horizontal velocity, yielding
In (
12), the flow velocity term
accounts for the average velocity gained by the swimmer while crossing the flow cell.
In order to estimate
, we consider the leading-order approximation of
z and
at the small timescale
, hence simply assuming that the swimmer remains at a constant distance from the wall
and orientation
. We then consider a time interval of length
in which the control is set to the constant value
, which consequently allows us to explicitly solve the dynamics for
x:
with
, and
is defined in (
11). At the lowest order, the displacement contributed by the flow between two switches is therefore given by
, and the average velocity
that the swimmer gains from the wall flow satisfies the equation
and hence we obtain, at the lowest order,
Finally, the minimal time
for the swimmer to reach
from
can be estimated as
The expressions obtained above show a linear relationship between the switch time
and the frequency
. Moreover, for small
,
is minimal when
is maximal, which is achieved at
, suggesting that this distance to the wall is optimal for parallel displacement purposes. There, setting
, we can give an estimation of the optimal horizontal velocity boost
b given by the flow
which amounts to approximately 8.5% if
, and up to a boost of
if
, as in
Figure 3a.
To validate this analysis, we numerically computed the optimal controls for a range of parameters, using the CasADi framework [
41] and the in-built ordinary differential equation solvers in MATLAB
® (The Mathworks, Inc., Natick, MA, USA) [
42] for the solutions of the control system (
4). For this numerical assessment, we set the swimmer speed
v and the wave number
k to be of unit magnitude, which is equivalent to nondimensionalise the system (
4) with respect to
v and the wall wave velocity
; then, in
Figure 3, the value of
,
t, and
respectively represent, in non-dimensional variables, the ratios
,
, and
. The range of
is taken to match the regime in which we carried out our analysis (
sufficiently larger than
v, with a ratio of 4 seeming to be already satisfying when looking at
Figure 3).
Figure 3 displays the results and shows good agreement with the theory.
Figure 3a,b shows the optimal trajectory and associated control function
for different values of
and
, showing a characteristic aspect due to the successive switches in
u, and confirming our prediction that the optimal time
depends very little on the frequency
. The relationships between the switching time
and
and between the optimal time
and the characteristic speed
are shown in
Figure 3c, with remarkable agreement between the numerical values and the curves predicted by Equations (
15) and (
16).
We now move on to our second control problem: changing the orientation of the swimmer from
to
with identical initial and final distance to the wall
. A few numerically computed trajectories realising this objective for various couples
are displayed on
Figure 4. While switches in the control occur with roughly the same period
as for the parallel displacement problem, in line with the short timescale analysis above, the observation of these trajectories in that case shows a more complex structure emerging for this control problem, with three distinctive phases in the swimmer’s trajectory. First, the control makes the swimmer approach the wall and then rotates it once it is close enough to the wall, before driving it away from the wall again to reach the target distance
. During the first and last phases, the swimmer’s orientation almost remains constant, seemingly indicating that an effective change of orientation cannot be obtained while the swimmer is far away from the wall. This can be understood by looking at the rotational component of the flow,
(see Equation (
4)), along the optimal trajectories, plotted on the right-hand side of
Figure 4. Approaching the wall allows
to reach large absolute values during the second phase and to generate net rotation by oscillating with non-zero average. This “approach-rotate-escape” strategy is consistently observed regardless of the initial and target orientation as well as the distance from the wall
. Hence, controlling the swimmer’s orientation requires efficiently controlling its distance to the wall, which is the goal of Problem 3. We can therefore conclude that the solution to Problem 3 is contained in Problem 2, with the approach and escape phases in Problem 2 corresponding to the optimal motion of the swimmer towards or away from the wall. This analysis is confirmed by numerical observations: trajectories solving Problem 3 closely resemble the ones obtained for the approach and escape phases in the trajectories solving Problem 2.
This exploration of controllability and optimal control properties of the wall-driven swimmer shows that the wall-generated flow can be expected to efficiently steer the swimmer’s trajectory, both improving the swimmer’s speed when swimming parallel to the wall and allowing it to change its orientation. However, this analysis assumes idealised control and neglects the interaction of the swimmer with the wall. In the next section, we address these issues by tackling the robustness of the control and studying the influence of added far-field wall interactions.
3.3. Robust Driving Strategies
The optimal control analysis conducted in the previous section showed that a bang–bang periodic control with period
allows for increasing the velocity of the swimmer while driving it parallel to the wall. Thinking of the practical application of this result, one may be tempted to use this control policy in an
open-loop fashion, i.e., without changing the policy depending on the initial or current state of the swimmer, so that
u is given by
This strategy has the obvious advantage of not requiring any measurement of the swimmer’s position or orientation but will typically be sensitive to measurement errors and initial conditions for that same reason. In the case of the wall-controlled swimmer, we saw in the optimal control study that the periodic control needs to be synchronised with the swimmer position, so that it switches when the swimmer reaches the border between two flow cells. If the synchronisation fails because of a small error on setting
, the swimmer’s motion can become far from optimal. This is illustrated by the numerical simulation displayed in
Figure 5a, where the open-loop strategy is applied to the swimmer for a short time and quickly shows desynchronisation, preventing it from gaining extra velocity from the flow.
To recover better synchronisation between the swimmer and the flow, we move on from this naive open-loop strategy to a feedback (or
closed-loop) control policy, where the flow amplitude
u is allowed to depend on the state
. In line with the observation made in the optimal control study that the control should switch sign along with
, we then define
and apply this policy to drive the swimmer parallel to the wall. The resulting trajectory is displayed in
Figure 5b and resembles the optimal one obtained in the previous section with good synchronisation between the flow and the swimmer and a horizontal speed gain close to the optimal one. However, one can also observe in this figure that this simple feedback strategy defined in (
19) tends to rotate the swimmer towards the wall over several periods and reduce its distance to the wall, with the risk of crashing into the wall after some time. Such a phenomenon highlights the necessity of combining both strategies of parallel displacement and change of orientation to compensate for the trajectory deviations emerging even when using a feedback control.
The analysis carried on so far demonstrates the caution needed to use the optimal controls in practice and provides solid grounds for the flow guidance of microswimmers, using a simple model that neglects the influence of the swimmer’s motion on the surrounding fluid in the presence of the neighbouring wall. As briefly explained in
Section 2, this is qualitatively justified by considering that the scale of these wall–swimmer interactions is typically significantly smaller than the free-swimming velocity of the swimmer, which itself is dominated by the background flow in our study; hence, the wall interactions can be neglected as a first approximation. To further validate our results, we will conclude this section by discussing the effect on the controlled trajectories when taking this interaction into account.
The wall–swimmer fluid interaction depends on the swimming strategy of the swimmer itself, which we will model from now on as a standard “squirmer” [
43,
44] of radius
a with one characteristic parameter
.
Following [
36], at the leading order (far-field approximation), the additional effect of the boundary on the squirmer’s motion is given by
with
a being the radius of the swimmer. By linearity, including these effects (
20) in Equation (
4) is simply done by adding them together, yielding the new system
The swimmer is called a
puller, a
pusher, or a
neutral swimmer if the characteristic parameter
respectively satisfies
,
, or
. Note that the wall effects (
20) vanish at the leading order in the case of a neutral swimmer, and we can reasonably neglect the higher order terms in the context of this study.
When
, it is well known [
45] by cell-wall hydrodynamic interactions that pullers face towards neighbouring walls while pushers tend to swim parallel to them. We subsequently investigated, in
Figure 5c–e, the influence of this additional term in the feedback control trajectories. As expected, in the pusher case (
), displayed in
Figure 5c, the repulsive effect bends the trajectory away from the wall, although relatively weakly; a larger (and possibly unrealistic) absolute value of
(
) yields a trajectory looking almost parallel to the wall in
Figure 5d. Hence, this added repulsive effect seemingly provides increased stability for the control. On the contrary, in the case of a puller
, the trajectory bends faster towards the wall, reducing the average horizontal speed of the swimmer and seemingly making the task of guiding it along the wall more challenging.
It is worth highlighting that the values of the parameter
yielding a visible effect on the swimmer’s trajectories in
Figure 5 are notably large with respect to the other parameters of the system, suggesting that the flow controls prescribed above are relatively robust with respect to the wall effect on the squirmer’s motion. This observation also corroborates, as predicted above, the choice of neglecting the boundary terms (
20) for most of our theoretical study.