This enables us to estimate the $H^{\frac{1}{2}}$ norm of $\mathbf{v}_i$ on $\partial B$

$$ 
\begin{aligned}
\|\mathbf{v}_i\|_{H^{\frac{1}{2}}(B)} &\le \|\mathbf{v}_i\|_{H^1(B)} \\
&= (\|\mathbf{v}_i\|_{L^2(B)}^2 + \|\nabla \mathbf{v}_i\|_{L^2(B)}^2)^{\frac{1}{2}} \\
&\le O(a) \|\mathbf{f}_j\|_{H^{-\frac{1}{2}}},
\end{aligned}
$$

which proves (4.24).

In order to prove (4.25), we use the decomposition (4.13) where none of the kernels $(\mathbf{K}_i)_{i=1,2,3}$ is singular. Therefore $\forall \mathbf{r} \in \partial B$

$$ 
\begin{aligned}
\mathbf{u}_i(\mathbf{r}) &:= (\mathcal{T}_{i,i} \mathbf{f}_i)(\mathbf{r}) \\
&= \int_{\partial B} \mathbf{K}(\mathbf{x}_i + a\mathbf{r}, \mathbf{x}_i + a\mathbf{s}) \mathbf{f}_i(\mathbf{s}) \mathrm{d}\mathbf{s} \\
&= \int_{\partial B} \mathbf{G}(a(\mathbf{r} - \mathbf{s})) \mathbf{f}_i(\mathbf{s}) \mathrm{d}\mathbf{s} + \int_{\partial B} (\mathbf{K}_1 + \mathbf{K}_2 + \mathbf{K}_3)(\mathbf{x}_i + a\mathbf{r}, \mathbf{x}_i + a\mathbf{s}) \mathbf{f}_i(\mathbf{s}) \mathrm{d}\mathbf{s} \\
&= \mathcal{T}^0 \mathbf{f}_i + \int_{\partial B} (\mathbf{K}_1 + \mathbf{K}_2 + \mathbf{K}_3)(\mathbf{x}_i + a\mathbf{r}, \mathbf{x}_i + a\mathbf{s}) \mathbf{f}_i(\mathbf{s}) \mathrm{d}\mathbf{s}.
\end{aligned}
$$

We finish as before, having remarked that for $l = 1, 2, 3$

$$ \mathbf{K}_l(\mathbf{x}_i + a\mathbf{r}, \mathbf{x}_i + a\mathbf{s}) = \mathbf{K}_l(\mathbf{x}_i, \mathbf{x}_i) + O(a). \quad (3.33) $$

**Proposition 3.5.6** For every $\mathbf{f} \in \mathcal{H}^{-1/2}$, and $\mathbf{x} = (\xi, \mathbf{p}) \in \mathcal{S}$

$$ (\mathcal{T}_{\mathbf{x}}\mathbf{f})_i(\mathbf{r}) = \mathcal{T}^0\mathbf{f}_i + \sum_{l=1}^3 \mathbf{K}_l(\mathbf{x}_i, \mathbf{x}_i)\langle\mathbf{f}_i, \mathbf{Id}\rangle_{\partial B} + \sum_{j \neq i} \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j)\langle\mathbf{f}_j, \mathbf{Id}\rangle_{\partial B} + \mathcal{R}_i(\mathbf{f}), \quad (3.34) $$

with $\|\mathcal{R}_i\|_{\mathcal{L}(\mathcal{H}^{-1/2}, \mathcal{H}^{1/2})} = O(a)$, and $i = 1, 2, 3$.

**Proof.**

For all $i \in \{1, 2, 3\}$, and all $\mathbf{r} \in \partial B$

$$ 
\begin{aligned}
(\mathcal{T}_{\mathbf{x}}\mathbf{f})_i(\mathbf{r}) &:= \int_{\partial B} \mathbf{K}(\mathbf{x}_i + a\mathbf{r}, \mathbf{x}_i + a\mathbf{s}) \mathbf{f}_i(\mathbf{s}) \mathrm{d}\mathbf{s} + \sum_{i \neq j} \int_{\partial B} \mathbf{K}(\mathbf{x}_i + a\mathbf{r}, \mathbf{x}_j + a\mathbf{s}) \mathbf{f}_j(\mathbf{s}) \mathrm{d}\mathbf{s} \\
&= \mathcal{T}_{i,i} \mathbf{f}_i + \sum_{j \neq i} \mathcal{T}_{i,j} \mathbf{f}_j
\end{aligned}
$$

and the result follows from the application of (4.24) and (4.25) of Proposition 4.4.2.

**Proposition 3.5.7** In the regime $a \ll 1$, one has for every $\mathbf{u} \in \mathcal{H}^{1/2}$, and $\mathbf{x} = (\xi, \mathbf{p}) \in \mathcal{S}$

$$ (\mathcal{T}_{\mathbf{x}}^{-1}\mathbf{u})_i = (\mathcal{T}^0)^{-1} \left( \mathbf{u}_i - \sum_{k=1}^3 \mathbf{K}_k(\mathbf{x}_i, \mathbf{x}_i) \langle (\mathcal{T}^0)^{-1}\mathbf{u}_i, \mathbf{Id} \rangle_{\partial B} \right) - \\ (\mathcal{T}^0)^{-1} \left( \sum_{j \neq i} \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \langle (\mathcal{T}^0)^{-1}\mathbf{u}_j, \mathbf{Id} \rangle_{\partial B} \right) + \tilde{\mathcal{R}}_i(\mathbf{u}) \quad (3.35) $$

with $\|\tilde{\mathcal{R}}_i\|_{\mathcal{L}(\mathcal{H}^{1/2}, \mathcal{H}^{-1/2})} = O(a^3)$, and $i = 1, 2, 3$.
**Proof.** We recall that

$$ \mathcal{T}^0 : H^{-\frac{1}{2}}(\partial B) \to H^{\frac{1}{2}}(\partial B) \\ \mathbf{f} \mapsto \int_{\partial B} \mathbf{G}(a(\cdot - \mathbf{s}))\mathbf{f}(\mathbf{s}) \, d\mathbf{s}, $$

and define for $l = 1, 2, 3$ the operators

$$ S_l : H^{-\frac{1}{2}}(\partial B) \to H^{\frac{1}{2}}(\partial B) \\ \mathbf{f} \mapsto \int_{\partial B} \mathbf{K}_l(\mathbf{x}_i, \mathbf{x}_j) \mathbf{f}(\mathbf{s}) \, d\mathbf{s}, $$

and eventually

$$ S_{i,j} : H^{-\frac{1}{2}}(\partial B) \to H^{\frac{1}{2}}(\partial B) \\ \mathbf{f} \mapsto \int_{\partial B} \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \mathbf{f}(\mathbf{s}) \, d\mathbf{s}. $$

(Notice that for all $\mathbf{f} \in H^{-\frac{1}{2}}(\partial B)$, $S_l\mathbf{f}$ and $S_{i,j}\mathbf{f}$ are in fact constant maps.)

That these operators are continuous operators from $H^{-\frac{1}{2}}(\partial B)$ into $H^{\frac{1}{2}}(\partial B)$ is classical. We hereafter are only interested into the estimation of their norms, and more precisely the way they depend on $a$, $\delta$ and $y$ in the limit $a \to 0$. Notice that since the kernel $\mathbf{G}$ is homogeneous of degree -1, one has

$$ \|\mathcal{T}^0\|_{\mathcal{L}(\mathcal{H}^{-1/2}, \mathcal{H}^{1/2})} = O\left(\frac{1}{a}\right) \quad \text{and} \quad \|(\mathcal{T}^0)^{-1}\|_{\mathcal{L}(\mathcal{H}^{1/2}, \mathcal{H}^{-1/2})} = O(a). \quad (3.36) $$

As far as $S_l$ is concerned, we get that (since $|\mathbf{K}_l(\mathbf{x}_i, \mathbf{x}_j)| = O(1)$)

$$ \|S_l\|_{\mathcal{L}(\mathcal{H}^{-1/2}, \mathcal{H}^{1/2})} = O(1), \quad (3.37) $$

and similarly

$$ \|S_{i,j}\|_{\mathcal{L}(\mathcal{H}^{-1/2}, \mathcal{H}^{1/2})} = O(1). \quad (3.38) $$

When $a \to 0$ this enables us to invert (4.29) leading to (4.30).

### 3.5.4 Asymptotic expansion of the motion equation

We now use the fact that the spheres are non-deformable and may only move following a rigid body motion. In other words, the velocity of each point $\mathbf{r}$ of the $i$-sphere expresses as a sum of a translation and a rotation as

$$ \mathbf{u}_i(\mathbf{r}) = \mathbf{u}_{T_i} + \mathbf{u}_{R_i}(\mathbf{r}), \quad (3.39) $$

where $\mathbf{u}_{T_i}$ is constant on $\partial B$ while $\mathbf{u}_{R_i}(\mathbf{r}) = \omega_i \times a\mathbf{r}$ for a suitable angular velocity $\omega_i$ (remember that all quantities are expressed on the unit sphere $\partial B$). This is of particular importance for the computation of the total force and the total torque, which, due to self-propulsion, should vanish. This implies

$$ \sum_i \int_{\partial B} \mathbf{f}_i = \sum_i \int_{\partial B} (\mathcal{T}_{\mathbf{x}}^{-1} \mathbf{u})_i = 0. \quad (3.40) $$

Plugging (4.39) in (4.40) and using (4.30) leads to

$$ \sum_i \int_{\partial B} (\mathcal{T}^0)^{-1} \left( \mathbf{u}_{T_i} + \mathbf{u}_{R_i} - \sum_{k=1}^3 \mathbf{K}_k(\mathbf{x}_i, \mathbf{x}_k) \langle (\mathcal{T}^0)^{-1}(\mathbf{u}_{T_i} + \mathbf{u}_{R_i}), \mathrm{Id} \rangle_{\partial B} \right) - \\ \qquad (\mathcal{T}^0)^{-1} \left( \sum_{j \neq i} \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \langle (\bar{\mathcal{T}}^0)^{-1}(\mathbf{u}_{T_j} + \mathbf{u}_{R_j}), \mathrm{Id} \rangle_{\partial B} \right) = O(a^3) \|\mathbf{u}\|. \quad (3.41) $$
It is well known that both translations and rotations are eigenfunctions of the Dirichlet to Neumann map of the three dimensional Stokes operator outside a sphere. Namely

$$ (\mathcal{T}^0)^{-1} \mathbf{u}_{T_i} = \lambda_T \mathbf{u}_{T_i} \text{ and } (\mathcal{T}^0)^{-1} \mathbf{u}_{R_i} = \lambda_R \mathbf{u}_{R_i}. $$

It is well-known that $\lambda_T = \frac{3\mu a}{2}$ leading in particular to the celebrated Stokes formula

$$ \int_{\partial B} (\mathcal{T}^0)^{-1} \mathbf{u}_{T_i} \mathrm{d}\mathbf{s} = 6\pi\mu a \mathbf{u}_{T_i} $$

while $\lambda_R = 3\mu a$. We also remark that due to $\int_{\partial B} \mathbf{u}_{R_i} \mathrm{d}\mathbf{s} = 0$, we have

$$ \int_{\partial B} (\mathcal{T}^0)^{-1} \mathbf{u}_{R_i} \mathrm{d}\mathbf{s} = 0. $$

We therefore obtain

$$ 6\pi\mu a \sum_i \left( \mathbf{u}_{T_i} - 6\pi\mu a \sum_{k=1}^3 \mathbf{K}_k(\mathbf{x}_i, \mathbf{x}_k) \mathbf{u}_{T_i} - 6\pi\mu a \sum_{j \neq i} \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \mathbf{u}_{T_j} \right) = O(a^3) \|\mathbf{u}\|. \quad (3.42) $$

We now compute the torque with respect to the center $\mathbf{x}_2$ of the first ball $B_2$. Self-propulsion of the swimmer implies that this torque vanishes:

$$ 0 = \int_{\partial B} (\mathbf{x}_1 - \mathbf{x}_2 + a\mathbf{r}) \times \mathbf{f}_1(\mathbf{r}) + \int_{\partial B} a\mathbf{r} \times \mathbf{f}_2(\mathbf{r}) + \int_{\partial B} (\mathbf{x}_3 - \mathbf{x}_2 + a\mathbf{r}) \times \mathbf{f}_3(\mathbf{r}) = \mathbf{I}_1 + \mathbf{I}_2 + \mathbf{I}_3, \quad (3.43) $$

where, calling $\mathbf{e}_\theta = \begin{pmatrix} \cos \theta \\ \sin \theta \\ 0 \end{pmatrix}$ the direction of the swimmer, the quantities $\mathbf{I}_1$, $\mathbf{I}_2$ and $\mathbf{I}_3$ are respectively given below.

$$ \begin{aligned} \mathbf{I}_1 &= \int_{\partial B} (\mathbf{x}_1 - \mathbf{x}_2 + a\mathbf{r}) \times \mathbf{f}_1(\mathbf{r}) = \int_{\partial B} (\xi_1 \mathbf{e}_\theta + a\mathbf{r}) \times (\mathcal{T}_\mathbf{x})^{-1} \mathbf{u}_1 \\ &= \int_{\partial B} (-\xi_1 \mathbf{e}_\theta + a\mathbf{r}) \times (\mathcal{T}^0)^{-1} \left( \mathbf{u}_{T1} + \mathbf{u}_{R1} - 6\pi\mu a \sum_{l=1}^3 \mathbf{K}_l(\mathbf{x}_1, \mathbf{x}_1) \mathbf{u}_{T1} \right. \\ & \qquad \left. - 6\pi\mu a \sum_{j \neq 2} \mathbf{K}(\mathbf{x}_1, \mathbf{x}_j) \mathbf{u}_{Tj} + O(a^2) \|\mathbf{u}\| \right) \\ &= -6\pi\mu a \xi_1 \mathbf{e}_\theta \times \left( \mathbf{u}_{T1} - 6\pi\mu a \sum_{l=1}^3 \mathbf{K}_l(\mathbf{x}_1, \mathbf{x}_1) \mathbf{u}_{T1} - 6\pi\mu a \sum_{j \neq 1} \mathbf{K}(\mathbf{x}_1, \mathbf{x}_j) \mathbf{u}_{Tj} \right) + O(a^3) \|\mathbf{u}\|. \end{aligned} $$

Similarly, we get,

$$ \begin{aligned} \mathbf{I}_2 &= a \int_{\partial B} \mathbf{r} \times \mathbf{f}_2(\mathbf{r}) = a \int_{\partial B} \mathbf{r} \times (\mathcal{T}_\mathbf{x})^{-1} \mathbf{u}_2 \\ &= a \int_{\partial B} \mathbf{r} \times (\mathcal{T}^0)^{-1} \left( \mathbf{u}_{T2} + \mathbf{u}_{R2} - 6\pi\mu a \sum_{l=1}^3 \mathbf{K}_l(\mathbf{x}_2, \mathbf{x}_2) \mathbf{u}_{T2} \right. \\ & \qquad \left. - 6\pi\mu a \sum_{j \neq 2} \mathbf{K}(\mathbf{x}_2, \mathbf{x}_j) \mathbf{u}_{Tj} + O(a^2) \|\mathbf{u}\| \right) \\ &= a \int_{\partial B} \mathbf{r} \times (\mathcal{T}^0)^{-1} (\mathbf{u}_{R1}) + O(a^4) \|\mathbf{u}\|. \end{aligned} $$
But since $(\mathcal{T}^0)^{-1} \mathbf{u}_{R1} = \lambda_R \mathbf{u}_{R1} = \lambda_R \omega_1 \times a\mathbf{r}$, we have

$$ 
\begin{aligned}
a \int_{\partial B} \mathbf{r} \times (\mathcal{T}^0)^{-1} (\mathbf{u}_{R2}) &= a^2 \lambda_R \int_{\partial B} \mathbf{r} \times (\omega_1 \times \mathbf{r}) \, d\mathbf{r} \\
&= \frac{8\pi}{3} \mu a^3 \omega_1.
\end{aligned}
$$

This leads to

$$ \mathbf{I}_2 = \frac{8\pi}{3} \mu a^3 \omega_1 + O(a^4) \|\mathbf{u}\|. $$

Correspondingly,

$$ 
\begin{aligned}
\mathbf{I}_3 &= \int_{\partial B} (\mathbf{x}_3 - \mathbf{x}_2 + a\mathbf{r}) \times \mathbf{f}_3(\mathbf{r}) \\
&= 6\pi\mu a \xi_2 \mathbf{e}_\theta \times \left( \mathbf{u}_{T3} - 6\pi\mu a \sum_{l=1}^3 \mathbf{K}_l(\mathbf{x}_3, \mathbf{x}_3)\mathbf{u}_{T3} - 6\pi\mu a \sum_{j \neq 3} \mathbf{K}(\mathbf{x}_3, \mathbf{x}_j)\mathbf{u}_{Tj} \right) + O(a^3) \|\mathbf{u}\|.
\end{aligned}
$$

Denoting by $\mathbf{A}$ the matrix

$$ \mathbf{A} = \begin{pmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} & \mathbf{A}_{13} \\ \mathbf{A}_{21} & \mathbf{A}_{22} & \mathbf{A}_{23} \\ \mathbf{A}_{31} & \mathbf{A}_{32} & \mathbf{A}_{33} \end{pmatrix} \qquad (3.44) $$

where for $i = 1, 2, 3$

$$ \mathbf{A}_{ii} = \mathbf{Id} - 6\pi\mu a \sum_{l=1}^{3} \mathbf{K}_l(\mathbf{x}_i, \mathbf{x}_i) \qquad (3.45) $$

and for $i, j = 1, 2, 3$ with $i \neq j$

$$ \mathbf{A}_{ij} = -6\pi\mu a \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \qquad (3.46) $$

and $\mathbf{S}$ the matrix

$$ \mathbf{S} = \begin{pmatrix} \mathbf{Id} & \mathbf{Id} & \mathbf{Id} \\ -\xi_1 \mathbf{e}_\theta \times & 0 & +\xi_2 \mathbf{e}_\theta \times \end{pmatrix}, $$

we can rewrite the self-propulsion assumption (4.43), (4.44) as (notice that angular velocities being involved of higher order disappear)

$$ \mathbf{S} \mathbf{A} \begin{pmatrix} \mathbf{u}_{T_1} \\ \mathbf{u}_{T_2} \\ \mathbf{u}_{T_3} \end{pmatrix} = O(a^2) \|\mathbf{u}\|. \qquad (3.47) $$

We end up by expressing $\mathbf{u}_{T_1}, \mathbf{u}_{T_2}, \mathbf{u}_{T_3}$ and $\omega_1, \omega_2, \omega_3$ in terms of $\dot{x}, \dot{y}, \dot{\theta}, \dot{\xi}_1$ and $\dot{\xi}_2$. But, since $\mathbf{u}_{T_i}$ is the velocity of the center of the ball $B_i$, one has

$$ \mathbf{u}_{T_1} = \begin{pmatrix} \dot{x} - \dot{\xi}_1 \cos(\theta) + \dot{\theta}\xi_1 \sin(\theta) \\ \dot{y} - \dot{\xi}_1 \sin(\theta) - \dot{\theta}\xi_1 \cos(\theta) \\ 0 \end{pmatrix}, \quad \mathbf{u}_{T_2} = \begin{pmatrix} \dot{x} \\ \dot{y} \\ 0 \end{pmatrix}, $$

and

$$ \mathbf{u}_{T_3} = \begin{pmatrix} \dot{x} + \dot{\xi}_2 \cos(\theta) - \dot{\theta}\xi_2 \sin(\theta) \\ \dot{y} + \dot{\xi}_2 \sin(\theta) + \dot{\theta}\xi_2 \cos(\theta) \\ 0 \end{pmatrix}. $$
Similarly

$$ \omega_1 = \omega_2 = \omega_3 = \begin{pmatrix} 0 \\ 0 \\ \dot{\theta} \end{pmatrix}. $$

We rewrite these formulas as

$$ \begin{pmatrix} \mathbf{u}_{T_1} \\ \mathbf{u}_{T_2} \\ \mathbf{u}_{T_3} \end{pmatrix} = \mathbf{T} \begin{pmatrix} \dot{x} \\ \dot{y} \\ \dot{\theta} \end{pmatrix} + \mathbf{U} \dot{\xi} \qquad (3.48) $$

with

$$ \mathbf{T} = \begin{pmatrix} \mathbf{I} \mathbf{d} & -\xi_1 \mathbf{e}_{\theta}^{\perp} \\ \mathbf{I} \mathbf{d} & 0 \\ \mathbf{I} \mathbf{d} & \xi_2 \mathbf{e}_{\theta}^{\perp} \end{pmatrix}, $$

where $\mathbf{e}_{\theta}^{\perp} = \begin{pmatrix} -\sin\theta \\ \cos\theta \\ 0 \end{pmatrix}$ and

$$ \mathbf{U} = \begin{pmatrix} -\mathbf{e}_{\theta} & 0 \\ 0 & 0 \\ 0 & \mathbf{e}_{\theta} \end{pmatrix}. $$

Plugging (3.48) into (4.48) leads to the motion equation

$$ (\mathbf{S}\mathbf{A} + \mathbf{R}) \begin{pmatrix} \dot{x} \\ \dot{y} \\ \dot{\theta} \end{pmatrix} + \mathbf{U}\dot{\xi} = 0 \qquad (3.49) $$

where the residual matrix has a norm which is estimated as

$$ \|\mathbf{R}\| = O(a^2). $$

### 3.5.5 Dimension of Lie algebra under the small spheres hypothesis

Rewriting from (4.45), (4.46) and (4.47) $\mathbf{A} = \mathbf{I} \mathbf{d} + a \mathbf{A}_1$, we can expand in power series of $a$ the solution of (3.49). This enables us to write an expansion (still in $a$) of the two vectorfields $\mathbf{F}_1$ and $\mathbf{F}_2$. To this end, we have used the software MAPLE to symbolically compute those expressions and the Lie brackets $[\mathbf{F}_1, \mathbf{F}_2]$, $[\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]$, and $[\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]$. Writing the vectorfields in components as

$$ \mathbf{F}_1(\xi_1, \xi_2, y, \theta) := \begin{pmatrix} 1 \\ 0 \\ \mathbf{F}_1^3 + O(a^2) \\ \mathbf{F}_1^4 + O(a^2) \\ \mathbf{F}_1^5 + O(a^2) \end{pmatrix}, \quad \mathbf{F}_2(\xi_1, \xi_2, y, \theta) := \begin{pmatrix} 0 \\ 1 \\ \mathbf{F}_2^3 + O(a^2) \\ \mathbf{F}_2^4 + O(a^2) \\ \mathbf{F}_2^5 + O(a^2) \end{pmatrix}, \qquad (3.50) $$

we find, after having furthermore expanded the abovementioned components in power series of $\frac{1}{y}$,
$$
\mathbf{F}_1^3 = \frac{1}{3} \cos(\theta) + \frac{a}{6} \cos(\theta) K_1^3(\xi_1, \xi_2, \theta) + \frac{3a}{16y^2} \left( \sin(\theta) \cos(\theta) (\xi_2 + 2\xi_1) \right) \\
+ \frac{a}{384y^3} \left( \cos(\theta) K_2^3(\xi_1, \xi_2, \theta) \right) + \frac{a}{512y^4} \left( \sin(\theta) \cos(\theta) K_3^3(\xi_1, \xi_2, \theta) \right) + O\left(\frac{a}{y^5}\right),
$$

$$
\mathbf{F}_2^3 = -\frac{1}{3}\cos(\theta) - \frac{a}{6}\cos(\theta)K_1^3(\xi_2, \xi_1, -\theta) + \frac{3a}{16y^2}\left(\sin(\theta)\cos(\theta)(2\xi_2 + \xi_1)\right) \\
-\frac{a}{384y^3}\left(\cos(\theta)K_2^3(\xi_2, \xi_1, -\theta)\right) + \frac{a}{512y^4}\left(\sin(\theta)\cos(\theta)K_3^3(\xi_2, \xi_1, -\theta)\right) + O\left(\frac{a}{y^5}\right),
$$

$$
\mathbf{F}_1^4 = \frac{1}{3} \sin(\theta) + \frac{a}{6} \sin(\theta) K_1^4(\xi_1, \xi_2, \theta) - \frac{3a}{32y^2} K_2^4(\xi_1, \xi_2, \theta) \\
+ \frac{a}{192y^3} \sin(\theta) K_3^4(\xi_1, \xi_2, \theta) - \frac{a}{y^4} K_4^4(\xi_1, \xi_2, \theta) + O\left(\frac{a}{y^5}\right),
$$

$$
\mathbf{F}_2^4 = -\frac{1}{3}\sin(\theta) - \frac{a}{6}\sin(\theta)K_1^4(\xi_2, \xi_1, -\theta) - \frac{3a}{32y^2}K_2^4(\xi_2, \xi_1, \theta) \\
-\frac{a}{192y^3}\sin(\theta)K_3^4(\xi_2, \xi_1, -\theta) - \frac{a}{y^4}K_4^4(\xi_2, \xi_1, -\theta) + O\left(\frac{a}{y^5}\right),
$$

$$
\mathbf{F}_1^5 = \frac{3a}{64y^3} \sin(\theta) \cos(\theta) K_1^5(\xi_1, \xi_2, \theta) - \frac{9a}{512y^4} \cos(\theta) K_2^5(\xi_1, \xi_2, \theta) + O\left(\frac{a}{y^5}\right),
$$

$$
\mathbf{F}_2^5 = \frac{3a}{64y^3} \sin(\theta) \cos(\theta) K_1^5(\xi_2, \xi_1, -\theta) + \frac{9a}{512y^4} \cos(\theta) K_2^5(\xi_2, \xi_1, -\theta) + O\left(\frac{a}{y^5}\right).
$$

In those expressions the remaining functions are respectively given by

$$
K_1^3(\xi, \theta) = \frac{(\xi_2^2 \xi_1^2 - \xi_2^3 \xi_1 - \xi_2^4 + 2 \xi_1^3 \xi_2 + 2 \xi_1^4)}{(\xi_2^2 + \xi_1 \xi_2 + \xi_1^2) \xi_1 \xi_2 (\xi_1 + \xi_2)},
$$

$$
K_2^3(\xi, \theta) = -210\xi_1^2 \cos(\theta)^2 + 12 \cos(\theta)^4 \xi_1^2 + 184\xi_1^2 + 24 \cos(\theta)^2 \xi_1 \xi_2 \\
-32\xi_1 \xi_2 - 6 \cos(\theta)^4 \xi_1 \xi_2 - 92\xi_2^2 + 105\xi_2^2 \cos(\theta)^2 - 6 \cos(\theta)^4 \xi_2^2,
$$

$$
K_3^3(\xi, \theta) = \frac{1}{(\xi_2^2 + \xi_1\xi_2 + \xi_1^2)} \left( \begin{aligned}
& 12 \cos(\theta)^4 \xi_2^5 + 24 \xi_1^5 \cos(\theta)^4 - 168 \xi_2^5 \cos(\theta)^2 - 336 \xi_1^5 \cos(\theta)^2 \\
& + 112 \xi_2^5 + 72 \xi_1 \xi_2^4 - 176 \xi_1^2 \xi_2^3 - 136 \xi_1^3 \xi_2^2 + 224 \xi_1^5 - 156 \xi_1^3 \xi_2^2 \cos(\theta)^2 \\
& - 24 \xi_1^2 \xi_2^3 \cos(\theta)^2 - 240 \xi_1^4 \xi_2 \cos(\theta)^2 + 48 \xi_1^4 \xi_2 - 156 \xi_1 \xi_2^4 \cos(\theta)^2 \\
& - 24 \cos(\theta)^4 \xi_1^2 \xi_2^3 + 9 \cos(\theta)^4 \xi_1 \xi_2^4 - 21 \xi_1^3 \cos(\theta)^4 \xi_2^2 \end{aligned} \right),
$$

$$
K_1^4(\xi, \theta) = \frac{(\xi_2^2 \xi_1^2 - \xi_2^3 \xi_1 - \xi_2^4 + 2 \xi_1^3 \xi_2 + 2 \xi_1^4)}{(\xi_2^2 + \xi_1 \xi_2 + \xi_1^2) \xi_1 \xi_2 (\xi_1 + \xi_2)},
$$

$$
K_2^4(\xi, \theta) = 6 \cos(\theta)^2 \xi_1 + 3 \cos(\theta)^2 \xi_2 - 4 \xi_1 - 2 \xi_2,
$$
$$K_3^4(\xi, \theta) = -132\xi_1^2 \cos(\theta)^2 + 6\cos(\theta)^4 \xi_1^2 + 56\xi_1^2 + 12\cos(\theta)^2 \xi_1 \xi_2 - 16\xi_1 \xi_2 - 3\cos(\theta)^4 \xi_1 \xi_2 - 28\xi_2^2 + 66\xi_2^2 \cos(\theta)^2 - 3\cos(\theta)^4 \xi_2^2,$$

$$K_4^4(\xi, \theta) = \frac{1}{(512\xi_2^2 + 512\xi_1\xi_2 + 512\xi_1^2)} \left( \begin{aligned} & -210 \cos(\theta)^4 \xi_2^5 - 420 \xi_1^5 \cos(\theta)^4 + 232 \xi_2^5 \cos(\theta)^2 \\ & +24 \cos(\theta)^6 \xi_1^5 - 64 \xi_2^5 + 12 \cos(\theta)^6 \xi_2^5 - 96 \xi_1^4 \xi_2^4 - 64 \xi_1^2 \xi_2^3 \\ & -128 \xi_1^5 + 104 \xi_1^3 \xi_2^2 \cos(\theta)^2 - 56 \xi_1^2 \xi_2^3 \cos(\theta)^2 \\ & -96 \xi_1^4 \xi_2 + 216 \xi_1 \xi_2^4 \cos(\theta)^2 - 66 \cos(\theta)^4 \xi_1^2 \xi_2^3 \\ & -318 \xi_1^4 \cos(\theta)^4 \xi_2 - 240 \xi_1^3 \cos(\theta)^4 \xi_2^2 - 24 \cos(\theta)^6 \xi_1^2 \xi_2^3 \\ & -21 \cos(\theta)^6 \xi_1^3 \xi_2^2 + 464 \xi_1^5 \cos(\theta)^2 - 128 \xi_1^3 \xi_2^2 \\ & +264 \xi_1^4 \xi_2 \cos(\theta)^2 - 204 \cos(\theta)^4 \xi_1 \xi_2^4 + 9 \cos(\theta)^6 \xi_1 \xi_2^4 \end{aligned} \right),$$

$$K_1^5(\xi, \theta) = -8\xi_1 - 4\xi_2 + 2\cos(\theta)^2\xi_1 + \cos(\theta)^2\xi_2,$$

$$K_2^5(\xi, \theta) = \frac{1}{(\xi_2^2 + \xi_1\xi_2 + \xi_1^2)} \left( \begin{aligned} & 20 \cos(\theta)^2 \xi_2^4 - 40 \cos(\theta)^2 \xi_1^4 - 4 \cos(\theta)^4 \xi_2^4 + 8 \xi_1^4 \cos(\theta)^4 \\ & -40 \xi_2^3 \xi_1 - 8 \xi_2^2 \xi_1^2 + 32 \xi_1^3 \xi_2 - 16 \xi_2^4 + 32 \xi_2^3 \cos(\theta)^2 \xi_1 - 7 \cos(\theta)^4 \xi_2^3 \xi_1 \\ & + \cos(\theta)^4 \xi_1^2 \xi_2^2 - 40 \cos(\theta)^2 \xi_1^3 \xi_2 + 8 \xi_1^3 \cos(\theta)^4 \xi_2 + 32 \xi_1^4 - 8 \xi_2^2 \xi_1^2 \cos(\theta)^2 \end{aligned} \right).$$

As one can see, the use of a software for symbolic computation seems unavoidable. Subsequently, we get the expansion of the Lie bracket $[\mathbf{F}_1, \mathbf{F}_2](\xi_1, \xi_2, y, \theta)$

$$[\mathbf{F}_1, \mathbf{F}_2](\xi, y, \theta) := \begin{pmatrix} 0 \\ 0 \\ [\mathbf{F}_1, \mathbf{F}_2]_3 + O(a^2) \\ [\mathbf{F}_1, \mathbf{F}_2]_4 + O(a^2) \\ [\mathbf{F}_1, \mathbf{F}_2]_5 + O(a^2) \end{pmatrix}, \quad (3.51)$$

where the components are given by the following expressions

$$[\mathbf{F}_1, \mathbf{F}_2]_3 = -\frac{a}{3} \cos(\theta) \frac{(\xi_1^4 + 2\xi_1^3\xi_2 + \xi_2^2\xi_1^2 + 2\xi_2^3\xi_1 + \xi_2^4)}{(\xi_1 + \xi_2)^2 \xi_2^2 \xi_1^2} - \frac{27a}{512y^4} a \cos(\theta) \sin(\theta) \frac{\xi_1\xi_2(\cos(\theta)^4 - 4\cos(\theta)^2 + 8)(-\xi_2^2 + \xi_1^2)}{(\xi_2^2 + \xi_1\xi_2 + \xi_1^2)} + O\left(\frac{a}{y^5}\right),$$

$$[\mathbf{F}_1, \mathbf{F}_2]_4 = -\frac{a}{3} \sin(\theta) \frac{(\xi_1^4 + 2\xi_1^3\xi_2 + \xi_2^2\xi_1^2 + 2\xi_2^3\xi_1 + \xi_2^4)}{(\xi_1 + \xi_2)^2 \xi_2^2 \xi_1^2} + \frac{27a}{512y^4} \cos(\theta)^2 \frac{\xi_1\xi_2(\cos(\theta)^4 - 4\cos(\theta)^2 + 8)(-\xi_2^2 + \xi_1^2)}{(\xi_2^2 + \xi_1\xi_2 + \xi_1^2)} + O\left(\frac{a}{y^5}\right),$$

$$[\mathbf{F}_1, \mathbf{F}_2]_5 = \frac{81a}{512y^4} \cos(\theta) \frac{\xi_1\xi_2(\xi_1 + \xi_2)(\cos(\theta)^4 - 4\cos(\theta)^2 + 8)}{(\xi_2^2 + \xi_1\xi_2 + \xi_1^2)} + O\left(\frac{a}{y^5}\right).$$

Notice that since the two first coordinates of $\mathbf{F}_1$ and $\mathbf{F}_2$ are constant, the corresponding first coordinates of the Lie bracket vanish. Similarly, the asymptotic expansion for the second order Lie bracket $[\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]](\xi, y, \theta)$ reads
$$[\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]](\boldsymbol{\xi}, y, \theta) := \begin{pmatrix} 0 \\ 0 \\ [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_3 + O(a^2) \\ [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_4 + O(a^2) \\ [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_5 + O(a^2) \end{pmatrix}, \quad (3.52)$$

where

$$[\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_3 = -\frac{2a}{3} \cos(\theta) \frac{\xi_2 (3\xi_1^2 + 3\xi_1\xi_2 + \xi_2^2)}{\xi_1^3 (\xi_1 + \xi_2)^3} + \frac{27a}{512 y^4} \cos(\theta) \sin(\theta) L_3(\boldsymbol{\xi}, \theta) + O\left(\frac{a}{y^5}\right),$$

$$[\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_4 = -\frac{2a}{3} \sin(\theta) \frac{\xi_2 (3\xi_1^2 + 3\xi_1\xi_2 + \xi_2^2)}{\xi_1^3 (\xi_1 + \xi_2)^3} - \frac{27a}{512 y^4} \cos(\theta)^2 L_3(\boldsymbol{\xi}, \theta) + O\left(\frac{a}{y^5}\right),$$

$$[\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_5 = -\frac{81a}{512 y^4} \cos(\theta) L_4(\boldsymbol{\xi}, \theta) + O\left(\frac{a}{y^5}\right).$$

There, $L_3$ and $L_4$ are respectively given by

$$L_3(\boldsymbol{\xi}, \theta) = \frac{\xi_2^3 (8 - 4 \cos(\theta)^2 + \cos(\theta)^4) (2\xi_1^2 - \xi_1\xi_2 - \xi_2^2)}{(\xi_2^2 + \xi_1\xi_2 + \xi_1^2)^2},$$

$$L_4(\boldsymbol{\xi}, \theta) = \frac{\xi_2^3 (8 - 4 \cos(\theta)^2 + \cos(\theta)^4) (2\xi_1 + \xi_2)}{(\xi_2^2 + \xi_1\xi_2 + \xi_1^2)^2}.$$

Eventually, the expansion of the vector field $[\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]$ is given by

$$[\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]](\xi_1, \xi_2, y, \theta) := \begin{pmatrix} 0 \\ 0 \\ [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]_3 + O(a^2) \\ [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]_4 + O(a^2) \\ [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]_5 + O(a^2) \end{pmatrix}, \quad (3.53)$$

where

$$[\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]_3 = -\frac{2a}{3} \cos(\theta) \frac{\xi_1 (\xi_1^2 + 3\xi_1\xi_2 + 3\xi_2^2)}{(\xi_2^3 (\xi_1 + \xi_2)^3)} - \frac{27a}{512 y^4} \cos(\theta) \sin(\theta) L_3(\xi_2, \xi_1, -\theta) + O\left(\frac{a}{y^5}\right),$$

$$[\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]_4 = \frac{2a}{3} \sin(\theta) \frac{\xi_1 (\xi_1^2 + 3\xi_1\xi_2 + 3\xi_2^2)}{(\xi_2^3 (\xi_1 + \xi_2)^3)} - \frac{27a}{512 y^4} \cos(\theta)^2 L_3(\xi_2, \xi_1, -\theta) + O\left(\frac{a}{y^5}\right),$$

$$[\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]_5 = -\frac{81a}{512 y^4} \cos(\theta) L_4(\xi_2, \xi_1, -\theta) + O\left(\frac{a}{y^5}\right).$$

Having an expansion of $\det(\mathbf{F}_1, \mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2], [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]], [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]])$ which does not vanish implies the local controllability of our model swimmer. It can be readily checked that we have
$$
\begin{align}
\det(\mathbf{F}_1, \mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2], [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]], [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]) &= \nonumber \\
&= \begin{vmatrix}
[\mathbf{F}_1, \mathbf{F}_2]_3 & [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_3 & [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]_3 \\
[\mathbf{F}_1, \mathbf{F}_2]_4 & [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_4 & [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]_4 \\
[\mathbf{F}_1, \mathbf{F}_2]_5 & [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_5 & [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]_5
\end{vmatrix} \nonumber \\
&= \frac{81 a^3 (\xi_1 - \xi_2)}{131072 y^9} \sin \theta (\cos \theta)^2 R(\xi, \theta) + O\left(\frac{1}{y^{10}}\right), \tag{3.54}
\end{align}
$$

with,

$$
R(\xi, \theta) = \frac{(6\xi_1^6 + 27\xi_1^5\xi_2 + 50\xi_1^4\xi_2^2 + 55\xi_1^3\xi_2^3 + 50\xi_1^2\xi_2^4 + 27\xi_1\xi_2^5 + 6\xi_2^6)}{(\xi_1 + \xi_2)(\xi_2^2 + \xi_1\xi_2 + \xi_1^2)^2 \xi_1\xi_2} \\
\times (64 - 64 \cos(\theta)^2 + 32 \cos(\theta)^4 - 8 \cos(\theta)^6 + \cos(\theta)^8).
$$

It is easily seen that $R$ never vanishes. Therefore, the previous determinant has a non-vanishing first coefficient (in $\frac{1}{y^9}$) which does not vanish for $\xi_1 \neq \xi_2$ and $\theta \notin \{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\}$.
Since it is an analytic function of $(\xi_1, \xi_2, y, \theta)$ we deduce that it does not vanish except at most on a negligible set. This is sufficient to conclude that

$d_{(\xi_1, \xi_2, y, \theta)} = 5$

almost everywhere and the local controllability of the Three-sphere swimmer around such points.

**Remark 3.5.8** *Quite strikingly, when* $\xi_1 = \xi_2$ *the first term of the expansion vanishes and one has to go one step further. We find in that case*

$$
\det(\mathbf{F}_1, \mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2], [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]], [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]) (\xi, \xi, y, \theta) = T(\xi, y, \theta) \frac{1}{y^{10}} + O\left(\frac{1}{y^{11}}\right),
$$

where

$$
T(\xi, y, \theta) = - \frac{945}{524288} a^3 \sin^2(\theta) \cos^2(\theta) \xi (\cos^4(\theta) + 8 - 4 \cos^2(\theta))^2 .
$$

This coefficient does not vanish unless $\theta \notin \{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\}$.

We already know from symmetry that when $\theta = \frac{\pi}{2}$ or $\theta = \frac{3\pi}{2}$, one has $d_{(\xi_1, \xi_2, y, \theta)} \le 3$.
Therefore, it remains to understand the case $\theta = 0$ (or $\pi$ by symmetry). The preceding
computation does not allow us to conclude about the dimension of the Lie algebra at such
points. Indeed, the 2 first coefficients of the expansion of the determinant vanish, and
it might well be the case at all orders. Nevertheless, in that case, we can expand the
subdeterminant

$$
\Delta = \begin{vmatrix} [\mathbf{F}_1, \mathbf{F}_2]_3 & [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_3 \\ [\mathbf{F}_1, \mathbf{F}_2]_5 & [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]_5 \end{vmatrix}
$$

in order to obtain informations. Indeed, one gets

$$
\Delta = \frac{45}{512 y^4} a^2 (\xi_1 - \xi_2) \xi_1 R'(\xi) + O\left(\frac{1}{y^5}\right),
$$

with,

$$
R'(\xi) = \frac{2\xi_2^5 + 11\xi_1\xi_2^4 + 16\xi_1^2\xi_2^3 + 19\xi_1^3\xi_2^2 + 12\xi_1^4\xi_2 + 3\xi_1^5}{(\xi_1 + \xi_2)^2 \xi_2^2 (\xi_2^2 + \xi_1^2 + \xi_1\xi_2)^2}.
$$
As a direct consequence, we get that the dimension of the Lie algebra, $d_{(\xi, y, 0)} \ge 4$, for almost every $(\xi, y) \in (\mathbb{R}^+)^3$.

This finishes the proof of Lemma 3.3.6 and thus of Theorem 3.2.2.

**Remark 3.5.9** As usual, it is possible to pass from local to global controllability on each of the connected components where the determinant given by (3.54) does not vanish. More precisely, let $\mathcal{A} := \{(\xi, \mathbf{p}) \text{ s. t. } d_{(\xi, \mathbf{p})} \le 4\}$, we define by $S_{(\xi, \mathbf{p})}$ the connected component of the subset $S \setminus \mathcal{A}$ which contains $(\xi, \mathbf{p})$. Applying Chow’s Theorem 3.3.3 on $S_{(\xi, \mathbf{p})}$, gives that for every initial configuration $(\xi^i, \mathbf{p}^i)$, any final configuration $(\xi^f, \mathbf{p}^f)$ in $S_{(\xi, \mathbf{p})}$, and any final time $T > 0$, there exists a stroke $\xi \in W^{1,\infty}([0,T])$, satisfying $\xi(0) = \xi^i$ and $\xi(T) = \xi^f$ and such that the self-propelled swimmer starting in position $\mathbf{p}^i$ with the shape $\xi^i$ at time $t=0$, ends at position $\mathbf{p}^f$ and shape $\xi^f$ at time $t=T$ by changing its shape along $\xi(t)$ and staying in $S_{(\xi, \mathbf{p})}$ for all time $t \in [0,T]$. In other words, $S_{(\xi, \mathbf{p})}$ is exactly equal to the orbit of the point $(\xi, \mathbf{p})$.

## 3.6 Conclusion

The aim of the present paper was to examine how the controllability of low Reynolds number artificial swimmers is affected by the presence of a plane boundary on the fluid. The systems are those classically studied in the literature (see [7] for instance) but are usually not confined. This is the first in-depth control study of how the presence of the plane wall affects the reachable set of a peculiar micro-swimmer.

Firstly, Theorem 3.2.1 shows that the controllability on the whole space implies the controllability in the half space. Although the proof is applied on the Four-sphere swimmer, it is based on general arguments which can be appropriate for any finite dimensional linear control systems.

Secondly, the Theorem 3.2.2 deals with the controllability of the Three-sphere swimmer in the presence of the plane wall. We prove that, at least for this example, the hydrodynamics perturbation due to the wall surprisingly makes the swimmer more controllable. This result is not in contradiction with the several scientific studies which show that the wall seems to attract the swimmer (see [74], [91], [81], [38], [17]). Although, the Theorem 3.2.2 leads to the fact that the wall contributes to increase the swimmer’s reachable set, we can conjecture that some of them are easier to reach than others.

The quantitative approach to this question together with the complete understanding of the situation in view of controllability of the underlying systems is far beyond reach and thus still under progress as is, in another direction, the consideration of more complex situations like, e.g. rough or non planar wall. This is the purpose of ongoing work.
# Chapitre 4

## Rough wall effect on
micro-swimmers

**Abstract**: We study the effect of a rough wall on the controllability of micro-swimmers made of several balls linked by thin jacks: the so-called 3-sphere and 4-sphere swimmers. Our work completes the previous work [11] dedicated to the effect of a flat wall. We show that a controllable swimmer (the 4-sphere swimmer) is not impacted by the roughness. On the contrary, we show that the roughness changes the dynamics of the 3-sphere swimmer, so that it can reach any direction almost everywhere.
## 4.1 Introduction

Micro-swimming is a subject of growing interest, notably for its biological and medical im-
plications: one can mention the understanding of reproduction processes, the description of
infection mechanisms, or the conception of micro-propellers for drug delivery in the body.
As regards its mathematical modeling and analysis, the studies by Taylor [84], Lighthill [54]
and Purcell [70] have been pioneering contributions to a constantly increasing field: we refer
to the recent work of T. Powers and E. Lauga [53] for an extensive bibliography.

Among the many aspects of micro-swimming, the influence of the environment on swimmers dynamics has been recognized by many biological studies (see for instance [17], [68], [81], [82], [74], [91], [92]). One important factor in this dynamics is the presence of confining walls. For example, experiments have shown that some microorganisms, like E. Coli, are attracted to surfaces.

The focus of this paper is the effect of wall roughness on micro-swimming. Such effect has been already recognized in the context of microfluidics, in connection with superhydrophobic surfaces ([94]). Moreover, recent studies have highlighted the role of roughness in the dynamics of passive spherical particles in a Stokes flow: we refer for instance to the study of S. H. Rad and A. Najafi [71] or to the one of D. Gérard-Varet and M. Hillairet [43].

We want here to study the impact of a rough wall on the displacement of micro-swimmers, at low Reynolds number. Our point of view will be theoretical, more precisely based on control theory. Connection between swimming at low Reynolds number and control theory has been emphasized over the last years (see [63], [27], [60], [59]). We shall ponder here on the recent studies [7] and [11], dedicated to the controllability analysis of particular Stokesian robots, in the whole space and in the presence of a plane wall respectively. We shall here incorporate roughness at the wall, and focus on two classical models of swimmers: the 3-sphere swimmer (see [47],[8], [7], [11]) and the 4-sphere swimmer (see [7], [11]). First, we will show that the controllability of the 4-sphere swimmer (already true near a flat wall) persists with roughness. Then, we will prove that the rough wall leads the 3-sphere swimmer to reach any space direction. The underlying mechanism is the symmetry-breaking generated by the roughness.

The paper is divided into three parts. In Section 4.2, we introduce the mathematical model for the fluid-swimmer coupling, and we derive from there an ODE for the swimmer dynamics. In Section 4.3, we show that the force field in this ODE is analytic with respect to the roughness amplitude and swimmer size and position. Combining this property with the results of [11] yields the controllability of the 4-sphere swimmer "almost everywhere". Section 4.4 provides an asymptotic expansion of the Dirichlet-to-Neumann operator, with respect to the roughness amplitude and swimmer's size. This operator is naturally involved in the expansion of the force fields. Eventually, we use this expansion and make it truly explicit in Section 4.5, in the special case of the 3-sphere swimmer. This allows us to show its controllability.

## 4.2 Mathematical setting

In this part, we present our mathematical model for the swimming problem.
## 4.2.1 Swimmers

We carry on the study of specific swimmers that were considered in [7] in $\mathbb{R}^3$ and in [11] in an half plane. These swimmers consist of $N$ spheres $\cup_{l=1}^N B_l$ of radii $a$ connected by $k$ thin jacks which are supposed free of viscous resistance. The position of the swimmer is described by a variable $\mathbf{p} \in \mathbb{R}_+^3 \times SO(3)$, which gives both the coordinates of one point over the swimmer and the swimmer's orientation. Moreover, the shape variable is denoted by a $k$-tuple $\xi$: its $i$th component $\xi_i$ gives the length of $i$th arm, that can stretch or elongate through time. Nevertheless, the directions of the arms are only modified by global rotation of the swimmer. Let us stress that all the variables above depend implicitly on time, through the transport and deformation of the swimmer. Finally, we call $S \subset \mathbb{R}^M$ (for a suitable $M \in \mathbb{N}$) the set of admissible states $(\xi, \mathbf{p})$. Typically, this admissible set encodes geometrical constraints, like a no-contact condition between the swimmer and the confining wall, or the fact that the balls can not touch each other. We assume that $S$ is a connected smooth submanifold of $\mathbb{R}^M$.

Many results of our paper apply to the general class of swimmers just described. Nevertheless, we will pay a special attention to two examples:

* The 4-sphere swimmer. We consider a regular tetrahedron ($\mathbf{S}_1, \mathbf{S}_2, \mathbf{S}_3, \mathbf{S}_4$) with center $\mathbf{O} \in \mathbb{R}_+^3$. The 4-sphere swimmer consists of four balls linked by four arms of fixed directions $\mathbf{O}\vec{\mathbf{S}}_i$ which are able to elongate and shrink (in a referential associated to the swimmer). The four ball cluster is completely described by the list of parameters $(\xi, \mathbf{p}) = (\xi_1, \dots, \xi_4, \mathbf{x}_c, \mathcal{R}) \in S = (\sqrt{\frac{3}{2}}a, \infty)^4 \times \mathbb{R}_+^3 \times SO(3)$. It is known that the 4-sphere swimmer is controllable in $\mathbb{R}^3$ and remains controllable in presence of a plane wall (see [7], [11]). This means that it is able to move to any point and with any orientation under the constraint of being self-propelled, when the surrounding flow is dominated by viscosity (Stokes flow). This swimmer is depicted in Fig. 4.1.

<figure><img src="image_34.png" /><figcaption>Figure 4.1: The Four-sphere swimmer.</figcaption></figure>

* The 3-sphere swimmer (see [66], [8], [7] and [11]). It is composed of three aligned spheres, linked by two arms, see Fig. 3.2. The dynamics of the swimmer is described through the lengths of the two arms $\xi_1, \xi_2$, the coordinates of the center of the middle ball: $\mathbf{x}_c = (x_c, y_c, z_c)$, and some matrix $\mathcal{R} \in SO(3)$ describing the orientation of the swimmer. Thus,

$$ (\xi, \mathbf{p}) = (\xi_1, \xi_2, \mathbf{x}_c, \mathcal{R}) \in S = (2a, \infty)^2 \times \mathbb{R}^3 \times SO(3). $$

As regards the position and elongation of the swimmer, the angle of the rotation $\mathcal{R}$ around the symmetry axis of the 3-sphere is irrelevant. As a matter of fact, we will not show controllability for this angle: our result, Theorem 4.2.4, yields controllability
of the swimmer up to rotation around its axis. Still, the associated angular velocity is not zero, and will appear in the dynamics.

<figure><img src="image_35.png" /><figcaption>Figure 4.2: Coordinates of the 3-sphere swimmer</figcaption></figure>

### 4.2.2 Fluid flow

We consider a fluid confined by a rough boundary. This boundary is modelled by a surface with equation $z = \varepsilon h(x, y)$, for some Lipschitz positive function $h$. Here, $\varepsilon > 0$ denotes the amplitude of the roughness, that is $\|h\|_{\infty} = 1$. The swimmer evolves in the half-space $\mathcal{O} = \{(x, y, z) \in \mathbb{R}^3 \text{ s. t. } z > \varepsilon h(x, y)\}$. The fluid domain is then $\mathcal{F} := \mathcal{O} \setminus \cup_{l=1}^N B_l$, and again it depends implicitly on time. Finally, we assume that the flow is governed there by the Stokes equation. Thus, the velocity $\mathbf{u}^S$ and the pressure $p^S$ of the fluid satisfy:

$$ -\mu \Delta \mathbf{u}^S + \nabla p^S = 0, \quad \text{div } \mathbf{u}^S = 0 \quad \text{in } \mathcal{F}, \qquad (4.1) $$

where $\mu$ is the viscosity of the fluid. We complement the Stokes equation (4.1) by standard no-slip boundary conditions, that read:

$$ \left\{ \begin{array}{ll} \mathbf{u}^S = \Omega \times (\mathbf{x} - \mathbf{x}_c) + \mathbf{v} + \mathbf{u}_d & \text{at } \cup_{l=1}^N \partial B_l, \\ \mathbf{u}^S = 0 & \text{at } \partial \mathcal{O}. \end{array} \right. \qquad (4.2) $$

In other words, we impose the continuity of the velocity both at the fixed wall and at the boundary of the moving swimmer. Note that the velocity field of the swimmer is made of two parts:

* one corresponding to an (unknown) rigid movement, with angular velocity $\Omega$ and linear velocity $\mathbf{v}$. If $\mathbf{x}_c$ is the point over the swimmer encoded in $\mathbf{p}$, the velocity $\mathbf{v}$ is its speed. The vector $(\Omega, \mathbf{v})^t$ can be identified with $\dot{\mathbf{p}}$ (everything will be made explicit in due course).

* one corresponding to the (known) deformation of the jacks, with associated velocity $\mathbf{u}^d$, depending on $\dot{\xi}$.

Introducing the Hilbert space

$$ \mathcal{V} = \left\{ \mathbf{u} \in \mathcal{D}'(\mathcal{F}, \mathbb{R}^3) \mid \nabla \mathbf{u} \in L^2(\mathcal{F}), \frac{\mathbf{u}(\mathbf{r})}{\sqrt{1 + |\mathbf{r}|^2}} \in L^2(\mathcal{F}) \right\}, \qquad (4.3) $$
we get (for any configuration of the swimmer $\cup B_l$ and velocities $(\Omega, \mathbf{v}, \mathbf{u}^d)$) a unique solution $(\mathbf{u}^S, p^S)$ of (4.1)-(4.2) in $\mathcal{V} \times L^2(\mathcal{F})$.

### 4.2.3 Dynamics

Of course, the previous relations describe the equilibrium of the fluid flow at any given instant $t$. To close the model (that is the fluid-swimmer coupling), we still need to specify the dynamics of the swimmer, based on Newton's laws. The description is by now classical (see for instance [7], [59]), and can be expressed by an affine control system without drift. Let us recall the principle of derivation. Neglecting inertia, Newton's laws become

$$ \left\{ \begin{array}{l} \displaystyle \sum_{l=1}^{N} \int_{\partial B_l} \sigma(\mathbf{u}^S, p^S) \cdot \mathbf{n} \, ds = 0, \\[1.5ex] \displaystyle \sum_{l=1}^{N} \int_{\partial B_l} \sigma(\mathbf{u}^S, p^S) \cdot \mathbf{n} \times (\mathbf{x} - \mathbf{x}_c) \, ds = 0, \end{array} \right. \qquad (4.4) $$

where $\sigma(\mathbf{u}, p) = \mu(\nabla\mathbf{u} + \nabla^t\mathbf{u}) - p\mathbf{Id}$ is the Cauchy tensor.

Moreover, if we introduce an orthonormal basis $(\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3)$ and use linearity, $\mathbf{u}^S$ decomposes into

$$ \mathbf{u}^S = \sum_{i=1}^{3} \Omega_i \mathbf{u}_i + \sum_{i=4}^{6} v_{i-3} \mathbf{u}_i + \mathbf{u}^d. \qquad (4.5) $$

Here, the $\mathbf{u}_i$'s and $\mathbf{u}^d$ are solutions of the Stokes equation, with zero Dirichlet condition at the wall, and inhomogeneous Dirichlet conditions at the ball. The Dirichlet data is $\mathbf{e}_i \times (\mathbf{x} - \mathbf{x}_c)$ for $i = 1, 2, 3$, $\mathbf{e}_{i-3}$ for $i = 4, 5, 6$, $\mathbf{u}_d$ for $\mathbf{u}^d$. Note also that the speed $\mathbf{u}^d$ can be expressed as a linear combination of $(\dot{\xi}_i)_{i=1}^k$:

$$ \mathbf{u}^d = \sum_{i=1}^{k} \mathbf{u}_i^d \dot{\xi}_i. \qquad (4.6) $$

Identifying $(\Omega, \mathbf{v})^t$ with $\dot{\mathbf{p}}$ (everything will be made explicit in due course), the system (4.4) reduces to the following ODE:

$$ \mathbf{M}(\xi, \mathbf{p}) \dot{\mathbf{p}} + \mathbf{N}(\xi, \mathbf{p}) = 0 \qquad (4.7) $$

where the matrix $\mathbf{M}(\xi, \mathbf{p})$ is defined by,

$$ \mathbf{M}_{i,j}(\xi, \mathbf{p}) := \left\{ \begin{array}{ll} \displaystyle \sum_{l=i}^{N} \int_{\partial B_l} ((\mathbf{x} - \mathbf{x}_c) \times \mathbf{e}_i) \cdot \boldsymbol{\sigma}(\mathbf{u}_j, p_j) \, \mathbf{n} \, ds & (1 \le i \le 3, 1 \le j \le 6), \\ \\ \displaystyle \sum_{i=i}^{N} \int_{\partial B_l} \mathbf{e}_{i-3} \cdot \boldsymbol{\sigma}(\mathbf{u}_j, p_j) \, \mathbf{n} \, ds & (4 \le i \le 6, 1 \le j \le 6), \end{array} \right. $$

and $\mathbf{N}(\xi, \mathbf{p})$ is the vector of $\mathbb{R}^6$ whose entries are,

$$ \mathbf{N}_i(\xi, \mathbf{p}) := \left\{ \begin{array}{ll} \displaystyle \sum_{l=1}^{N} \int_{\partial B_l} ((\mathbf{x} - \mathbf{x}_c) \times \mathbf{e}_i) \cdot \boldsymbol{\sigma}(\mathbf{u}^d, p^d) \, \mathbf{n} \, ds & (1 \le i \le 3), \\ \\ \displaystyle \sum_{l=1}^{N} \int_{\partial B_l} \mathbf{e}_{i-3} \cdot \boldsymbol{\sigma}(\mathbf{u}^d, p^d) \, \mathbf{n} \, ds & (4 \le i \le 6). \end{array} \right. $$
The matrix $M(\xi, \mathbf{p})$ is checked to be symmetric and negative definite. By inverting it in (4.7), we end up with the following relation for the swimmer's dynamics:

$$ \dot{\mathbf{p}} = -\mathbf{M}^{-1}(\xi, \mathbf{p})\mathbf{N}(\xi, \mathbf{p}). \quad (4.8) $$

By using (4.6), we deduce that there are vector fields $\mathbf{F}_i$, $i = 1..k$, such that the equation (4.8) reads

$$ \dot{\mathbf{p}} = \sum_{i=1}^{k} \mathbf{F}_i(\xi, \mathbf{p})\dot{\xi}_i. \quad (4.9) $$

### 4.2.4 Main results

Before turning to our mathematical analysis, we synthesize here our main results.

The controllability properties of the swimmers will follow from a careful study of the properties of the $\mathbf{F}_i$'s in (4.9). As a first consequence of this study, we will obtain the analyticity of these vector fields with respect to all parameters: the typical height of the roughness $\varepsilon$, the radius of the balls $a$, the vector of arms lengths $\xi$ and the position of the swimmer $\mathbf{p}$. More precisely, defining

$$ \mathcal{A} := \{(\varepsilon, a, \xi, \mathbf{p}) \in \mathbb{R} \times \mathbb{R}_+^* \times (\mathbb{R}_+^*)^k \times (\mathbb{R}^3 \times \mathcal{SO}(3)) \text{ such that} \\ B_i \cap B_j = \emptyset \quad \forall i \neq j, \text{ and } B_i \cap \partial \mathcal{O} = \emptyset \quad \forall i\}, $$

we have the following

**Theorem 4.2.1** For all $i = 1...k$, the field $\mathbf{F}^i(\xi, \mathbf{p})$ (which depends also implicitly on $\varepsilon$ and $a$) is an analytic function of $(\varepsilon, a, \xi, \mathbf{p})$ over $\mathcal{A}$.

Then, as a consequence of Theorem 4.2.1, we will prove that the roughness does not change the controllability of the 4-sphere swimmer. We restrict here to local controllability "almost everywhere": this terminology refers to the following

**Definition 4.2.2** ("almost everywhere") We say that a property holds for almost every $(\varepsilon, a, \xi, \mathbf{p})$ in $\mathcal{A}$ if it holds for all $(\varepsilon, a, \xi, \mathbf{p})$ outside the zero set of a (non-trivial) analytic function over $\mathcal{A}$.

We have

**Theorem 4.2.3** The 4-sphere swimmer is controllable almost everywhere, in the following sense: for almost every $(\varepsilon, a, \xi^i, \mathbf{p}^i)$, with $(\xi^i, \mathbf{p}^i) \in \mathcal{S}$, one has local controllability from the initial configuration $(\xi^i, \mathbf{p}^i)$. This means that for any final configuration $(\xi^f, \mathbf{p}^f)$ in a suitable neighborhood of $(\xi^i, \mathbf{p}^i)$ and any final time $T > 0$, there exists a stroke $\xi \in \mathcal{W}^{1,\infty}([0,T])$, satisfying $\xi(0) = \xi^i$ and $\xi(T) = \xi^f$ and such that if the self-propelled swimmer starts in position $\mathbf{p}^i$ with the shape $\xi^i$ at time $t = 0$, it ends at position $\mathbf{p}^f$ and shape $\xi^f$ at time $t = T$ by changing its shape along $\xi(t)$.

In the last Section 4.5, we shall address the controllability of the 3-sphere swimmer. In the case of a flat boundary, as shown in [11], symmetries constrain the swimmer to move in a plane. Also, it does not rotate around its own axis. As we will see, the roughness at the wall breaks (in general) such symmetries, allowing for local controllability almost everywhere. Let us point here a subtlety regarding our controllability result. To express the dynamics of the swimmer through the equation (4.9), we have included in variable $\mathbf{p}$ (more precisely in its $\mathcal{SO}(3)$ component) an angle describing rotation of the 3-sphere around its own axis. We are not able to show controllability for this angle: we only show controllability for the other components of $\mathbf{p}$. Of course, this is not a problem with regards to the effective movement of the swimmer: this angle is indeed irrelevant with regards to the swimmer's orientation and position. The analysis of Section 4.5 leads to the
**Theorem 4.2.4** There exists a surface $h \in C_c^\infty(\mathbb{R}^2)$ such that the $\beta$-sphere swimmer is locally controllable almost everywhere (up to rotation around its axis).

Refined statements will be provided in Section 4.5. This controllability result requires a careful asymptotic asymptotic expansion of the force fields $\mathbf{F}^i$. This expansion is related to an expansion of a Dirichlet-to-Neumann map, performed in section 4.4. Eventually, the dimension of the Lie algebra generated by the force fields is computed numerically, and the controllability result follows from application of Chow's theorem.

# 4.3 Analyticity of the dynamics

## 4.3.1 Regularity

This paragraph is devoted to the proof of Theorem 4.2.1. Let $\bar{\mathbf{Y}} = (\bar{\varepsilon}, \bar{a}, \bar{\xi}, \bar{\mathbf{p}}) \in \mathcal{A}$. We must prove analyticity of the $\mathbf{F}^i$'s with respect to $\mathbf{Y} = (\varepsilon, a, \xi, \mathbf{p})$, in a neighborhood of $\bar{\mathbf{Y}}$. It will follow from the analyticity of $\mathbf{M}$ and $\mathbf{N}$ defined after (4.7). Their definitions involve functionals of the type

$$ \mathbf{I} := \sum_{l=1}^{N} \int_{\partial B_l} \left( \frac{1}{\mathbf{x}} \right) \otimes \sigma(\mathbf{u}, p) n \, ds $$

where $(\mathbf{u}, p)$ satisfies the Stokes equation in $\mathcal{F}$, with Dirichlet conditions of the type:

$$ \mathbf{u} = 0 \quad \text{at } \partial\mathcal{O}, \quad \mathbf{u} = \mathbf{u}_l \quad \text{at } \partial B_l, \quad l = 1, \dots, N $$

for some family of rigid fields $\mathbf{u}_l$'s taken in the "elementary set" $\{\mathbf{e}_i \times \mathbf{x}, \mathbf{e}_i, i = 1...3\}$.

We denote by $\bar{\mathbf{x}}_l$, resp. $\bar{\mathbf{x}}_l$ the center of the ball $B_l$, resp. the center of the ball $\bar{B}_l$ corresponding to $\bar{\mathbf{Y}}$. We introduce the diffeomorphisms

$$ \varphi_l(\mathbf{x}) := \frac{a}{\bar{a}}(\mathbf{x} - \bar{\mathbf{x}}_l) + \mathbf{x}_l. $$

Then, we have

$$ \int_{\partial B_l} \left( \frac{1}{\mathbf{x}} \right) \otimes \sigma(\mathbf{u}, p) n \, ds = \left( \frac{a}{\bar{a}} \right)^2 \int_{\partial \bar{B}_l} \left( \varphi_{l}(\mathbf{x}) \right) \otimes \sigma(\mathbf{u} \circ \varphi_{l}, p \circ \varphi_{l}) n \, ds. $$

Hence, in order to prove Theorem 4.2.1, it is enough to show that for all $l = 1...N$, for $\delta, \eta > 0$ small enough:

$$ B(\bar{\mathbf{Y}}, \delta) \mapsto H^1(\mathcal{F} \cap B(\bar{\mathbf{x}}_l, a + \eta)) \times L^2(\mathcal{F} \cap B(\bar{\mathbf{x}}_l, a + \eta)), \quad \mathbf{Y} \mapsto (\mathbf{u} \circ \varphi_l, p \circ \varphi_l) $$

is analytic. Indeed, $\mathbf{Y} \mapsto \sigma(\mathbf{u} \circ \varphi_l, p \circ \varphi_l)$ will be analytic with values in $H^{-1/2}(\partial B_l)$, and the surface integral will be analytic as well.

Therefore, we define the change of variable

$$ \varphi(\mathbf{x}) = \mathbf{x} + \sum_{l} \chi(\mathbf{x} - \bar{\mathbf{x}}_l) (\varphi_l(\mathbf{x}) - \mathbf{x}) + (\varepsilon - \bar{\varepsilon}) \chi_h(\mathbf{x}) (0, 0, h(x_1, x_2)) $$

with $\chi, \chi_h \in C_c^\infty(\mathbb{R}^3)$, $\chi = 1$ near $B(0, \bar{a})$, $\chi_h = 1$ near $x_3 = h(x_1, x_2)$. For $\chi$ and $\chi_h$ with small enough supports, and for $\mathbf{Y} \in B(\bar{\mathbf{Y}}, \delta)$, $\delta > 0$ small enough, it is easily seen that $\varphi$ is a smooth diffeomorphism, which depends analytically on $\mathbf{Y}$, and such that $\varphi(\bar{\mathbf{F}}) = \mathcal{F}$. Moreover, one has $\varphi = \varphi_l$ in a small enough $\delta'$-neighborhood of $\bar{B}_l$. Introducing $\mathbf{U} := \mathbf{u} \circ \varphi$
and $P := p \circ \varphi$, it remains to prove the following

*Claim:* $\mathbf{Y} \mapsto \mathbf{U}$ is analytic from $B(\bar{\mathbf{Y}}, \delta)$ to $\bar{\mathcal{V}}_0$, where

$$ \bar{\mathcal{V}}_0 := \left\{ \mathbf{U} \in D'(\bar{\mathcal{F}}, \mathbf{R}^3) \mid \nabla \mathbf{U} \in L^2(\bar{\mathcal{F}}), \frac{\mathbf{U}(\mathbf{r})}{\sqrt{1 + |\mathbf{r}|^2}} \in L^2(\bar{\mathcal{F}}), \mathbf{U}|_{\partial \bar{\mathcal{O}}} = 0. \right\}. $$

To prove this claim, one first needs to write down the system satisfied by $\mathbf{U}, P$. A simple computation yields

$$ \left\{ \begin{array}{ll} - \operatorname{div} (A \nabla \mathbf{U}) + B \nabla P = 0 & \text{in } \bar{\mathcal{F}}, \\ \operatorname{div} (B^t \mathbf{U}) = 0 & \text{in } \bar{\mathcal{F}}, \\ \mathbf{U} = 0 & \text{at } \partial \bar{\mathcal{O}}, \quad \mathbf{U} = \mathbf{U}_l \quad \text{at } \partial \bar{\mathcal{B}}_l, \end{array} \right. \qquad (4.10) $$

where

$$ A = A(\mathbf{x}) := |\det \nabla \phi(\mathbf{x})| (\nabla \phi^{-1})^t (\nabla \phi^{-1})(\phi(\mathbf{x})) $$

$$ B = B(\mathbf{x}) := |\det \nabla \phi(\mathbf{x})| (\nabla \phi^{-1})(\phi(\mathbf{x})), \quad \mathbf{U}_l := \mathbf{u}_l \circ \varphi_l. $$

Note that $A, B, \mathbf{U}_l$ depend analytically on the parameter $\mathbf{Y}$. We now consider the mapping

$$ \mathcal{L}: B(\bar{\mathbf{Y}}, \delta) \times \bar{\mathcal{V}}_0 \times L^2(\bar{\mathcal{F}}) \mapsto (\bar{\mathcal{V}}_0)' \times L^2(\bar{\mathcal{F}}) \times \prod_l H^{1/2}(\partial \bar{\mathcal{B}}_l), $$

$$ (\mathbf{Y}, \mathbf{V}, Q) \mapsto (-\operatorname{div}(A\nabla\mathbf{V}) + B\nabla Q, \operatorname{div}(B^t\mathbf{V}), (\mathbf{V}|_{\partial\bar{\mathcal{B}}_l} - \mathbf{U}_l)_{l=1}^N). $$

$\mathcal{L}$ is clearly well-defined, and it is analytic in $(\mathbf{Y}, \mathbf{V})$: we refer to [89] for the definition of analytic functions over Banach spaces. Moreover, $\mathbf{U} = \mathbf{U}_{\mathbf{Y}}$ and $P = P_{\mathbf{Y}}$ satisfy

$$ \mathcal{L}(\mathbf{Y}, \mathbf{U}, P) = 0 $$

By the analytic version of the implicit function theorem, see again [89], $\mathbf{U}$ and $P$ will be analytic in $\mathbf{Y}$ near $\bar{\mathbf{Y}}$ if

$$ \frac{\partial \mathcal{L}}{\partial(\mathbf{V}, Q)}|_{(\bar{\mathbf{Y}}, \mathbf{U}, P)} \text{ is an isomorphism from } \bar{\mathcal{V}}_0 \times L^2(\bar{\mathcal{F}}) \text{ to } (\bar{\mathcal{V}}_0)' \times L^2(\bar{\mathcal{F}}) \times \prod_l H^{1/2}(\partial \bar{\mathcal{B}}_l). $$

In other words, analyticity of $\mathbf{U}$ and $P$ follows from the existence and uniqueness in $\bar{\mathcal{V}}_0 \times L^2(\bar{\mathcal{F}})$ of a solution $(\mathbf{V}, Q)$ for the Stokes system

$$ -\Delta \mathbf{V} + \nabla Q = \mathbf{F} \quad \text{in } \bar{\mathcal{F}}, $$

$$ \operatorname{div} \mathbf{V} = G \quad \text{in } \bar{\mathcal{F}}, $$

$$ \mathbf{V} = 0 \quad \text{at } \partial \bar{\mathcal{O}}, \quad \mathbf{V} = \mathbf{V}_l \quad \text{at } \partial \bar{\mathcal{B}}_l, \quad l = 1 \dots N $$

where $\mathbf{F} \in (\bar{\mathcal{V}}_0)'$, $G \in L^2(\bar{\mathcal{F}})$ and $\mathbf{V}_l \in H^{1/2}(\partial \bar{\mathcal{B}}_l)$ are prescribed data. Such well-posedness is proved in the appendix. This ends the proof of Theorem 4.2.1.

### 4.3.2 Application to the 4-sphere swimmer

From the analyticity shown above and the results of [11], we can deduce Theorem 4.2.3. First, by (4.9), we can write the swimmer's dynamics as

$$ (\dot{\mathbf{\epsilon}}_{\mathbf{p}}) = \sum_{i=1}^{4} \mathbf{G}_i ((\mathbf{\epsilon}_{\mathbf{p}})) u_i $$
where $(u_i := \dot{\xi}_i)_{i=1}^4$ is the family of controls, and $G_i := (\mathbf{e}_i)$ ($\mathbf{e}_1, ..., \mathbf{e}_4$) is the canonical basis of $\mathbb{R}^4$). By the analyticity of the $G_i$'s and Chow's theorem, it is then enough to prove that for some $(\varepsilon, a, \xi, \mathbf{p}) \in \mathbb{R}_+ \times \mathbb{R}_+^* \times \mathcal{S}$,

$$ \dim \mathrm{Lie}_{(\xi, \mathbf{p})}(\mathbf{G}_1, \dots, \mathbf{G}_{10}) = 10. $$

We write

$$ \partial^{\alpha} \mathbf{G}_i(\xi, \mathbf{p}) = \partial^{\alpha} \mathbf{G}_i^0(\xi, \mathbf{p}) + O(\varepsilon), \quad \forall \alpha \in \mathbb{N}^7, $$

where the $\mathbf{G}_i^0$'s are force fields corresponding to the flat case $h = 0$. In particular, for $\varepsilon$ small enough

$$ \dim \mathrm{Lie}_{(\xi, \mathbf{p})}(\mathbf{G}_1, \dots, \mathbf{G}_{10}) \geq \dim \mathrm{Lie}_{(\xi, \mathbf{p})}(\mathbf{G}_1^0, \dots, \mathbf{G}_{10}^0). $$

But from [11] we know that for almost every $(a, \xi, \mathbf{p})$

$$ \dim \mathrm{Lie}_{(\xi, \mathbf{p})}(\mathbf{G}_1^0, \dots, \mathbf{G}_{10}^0) = 10. $$

This concludes the proof.

## 4.4 Asymptotic expansion of the Dirichlet-to-Neumann

We now turn to the controllability properties of the 3-sphere swimmer. As before, the key point is to determine the dimension of the Lie algebra generated by the force fields $\mathbf{F}^i$. Therefore, we need to derive an asymptotic expansion of the $\mathbf{F}^i$'s, in $a$ and $\varepsilon$.

A preliminary step is to derive an asymptotic expansion of the so-called Dirichlet-to-Neumann map of the Stokes operator. Indeed, the force fields $\mathbf{F}^i$ involve this map: that is, the definition of the coefficients $M_{ij}$ and $N_i$ involves

$$ DN : \prod_{l=1}^{N} H^{1/2}(\partial B_l) \mapsto \prod_{l=1}^{N} H^{-1/2}(\partial B_l), \quad (\mathbf{u}_l) \mapsto (\mathbf{f}_l := \sigma(\mathbf{u}, p)n|_{\partial B_l}), $$

where $(\mathbf{u}, p)$ is the solution of the Stokes equation

$$ -\Delta \mathbf{u} + \nabla p = 0, \quad \mathrm{div} \, \mathbf{u} = 0 \quad \text{in } \mathcal{F}, \quad \mathbf{u}|_{\partial \mathcal{O}} = 0, \quad \mathbf{u}|_{\partial B_l} = \mathbf{u}_l. $$

More precisely, it involves $DN$ in restriction to $N$-uplets of rigid vector fields over $B_l$, $l = 1...N$. We denote by $R$ the (finite-dimensional) space of such $N$-uplets.

Even restricted to $R$, this operator is not very explicit: to derive directly an expansion in terms of the parameters of the swimmer and wall is not easy. Hence, we follow the same path as in [7, 11]: we write that for all $(\mathbf{u}_l)_{l=1}^N \in R$,

$$ DN ((\mathbf{u}_l)) = T^{-1} ((\mathbf{u}_l)) $$

where

$$ T : \prod_{l=1}^{N} H^{-1/2}(\partial B_l) \mapsto \prod_{l=1}^{N} H^{1/2}(\partial B_l), \quad (\mathbf{f}_l) \mapsto (\mathbf{u}_l := \mathbf{u}|_{\partial B_l}) $$

and $\mathbf{u}$ is the solution of the following Stokes system in $\mathcal{O}$:

$$ -\Delta \mathbf{u} + \nabla p = \sum_{l=1}^{N} \mathbf{1}_{\partial B_l} \mathbf{f}_l, \quad \mathrm{div} \, \mathbf{u} = 0 \quad \text{in } \mathcal{O}, \quad \mathbf{u}|_{\partial \mathcal{O}} = 0. $$
Equivalently, this last system can be written:

$$-\Delta \mathbf{u} + \nabla p = 0, \quad \text{div } \mathbf{u} = 0 \quad \text{in } \mathcal{O} \setminus \cup_l \partial B_l, \quad [\mathbf{u}]|_{\partial B_l} = 0, \quad [\sigma(\mathbf{u}, p)n]|_{\partial B_l} = \mathbf{f}_l,$$

where $[\cdot]|_{\partial B_l}$ denotes the jump across $\partial B_l$. Let us remind that $\mathcal{O} = \{z > \varepsilon h(x, y)\}$ is the domain without the balls. In particular, the operator $T$ (associated to a transmission condition) is not the Neumann-to-Dirichlet operator. The latter one would correspond to the Stokes problem

$$-\Delta \mathbf{u} + \nabla p = 0, \quad \text{div } \mathbf{u} = 0 \quad \text{in } \mathcal{O} \setminus \cup_l B_l, \quad \sigma(\mathbf{u}, p)n|_{\partial B_l} = \mathbf{f}_l,$$

associated to a Neumann type condition. However, in restriction to the space $\mathbf{R}$, the operators $DN$ and $T^{-1}$ coincide, due to the fact that a rigid vector field is a solution of the Stokes equation, with zero pressure and zero stress tensor.

The advantage of $T$ over the Neumann-to-Dirichlet operator is its more explicit representation. Indeed, one has for all $i = 1...N$

$$T(\mathbf{f})_i(\mathbf{x}) = \sum_{l=1}^{n} \int_{\partial B_l} \mathbf{K}^{\varepsilon}(\mathbf{x}, \mathbf{y}) \mathbf{f}_l(\mathbf{y}) d\mathbf{y}, \quad \mathbf{x} \in \partial B_i,$$

where the kernel $\mathbf{K}^{\varepsilon}$ is simply the Green function associated to the Stokes equation in $\mathcal{O}$: in other words, $(\mathbf{K}^{\varepsilon}, \mathbf{q}^{\varepsilon})$ is the solution of the problem:

$$\left\{
\begin{array}{l}
-\mu \Delta_{\mathbf{x}} \mathbf{K}^{\varepsilon}(\mathbf{x}, \mathbf{x}_0) + \nabla_{\mathbf{x}} \mathbf{q}^{\varepsilon}(\mathbf{x}) = \delta_{\mathbf{x}_0}(\mathbf{x}) \mathbf{I}, \quad \mathbf{x} \text{ in } \mathcal{O}, \\
\text{div}_{\mathbf{x}} \mathbf{K}^{\varepsilon}(\mathbf{x}, \mathbf{x}_0) = 0, \quad \mathbf{x} \text{ in } \mathcal{O}, \\
\mathbf{K}^{\varepsilon}(\mathbf{x}, \mathbf{x}_0) = 0 \quad \mathbf{x} \text{ on } \partial \mathcal{O},
\end{array}
\right.
\quad (4.11)$$

where $\mathbf{I}$ stands for the identity matrix. This will make easier the derivation of an asymptotic expansion, through an expansion of $T$. Still, there is one little technical difficulty: the domain of definition and range of $T$, that are $\prod_l H^{\pm 1/2}(\partial B_l)$ depend on the parameter $a$ (and also on $(\mathbf{p}, \xi)$). Let us denote $B := B(0, 1)$ the unit ball, and $H_N^{\pm 1/2} := (H^{\pm 1/2}(\partial B))^N$. We introduce

$$\phi : \prod_{l=1}^{N} H^{1/2}(\partial B_l) \to H_N^{1/2}, \quad \mathbf{u} = (\mathbf{u}_l) \mapsto \mathbf{U} = (\mathbf{U}_l : \mathbf{r} \mapsto \mathbf{u}_l(\mathbf{x}_l + a\mathbf{r})),$$

as well as the adjoint map

$$\phi^* : H_N^{-1/2} \to \prod_{l=1}^{N} H^{-1/2}(\partial B_l), \quad \mathbf{F} = (\mathbf{F}_l) \mapsto \mathbf{f} = (\mathbf{f}_l),$$

defined through the duality relation: $\langle \phi^*(\mathbf{F}), \mathbf{u} \rangle = \langle \mathbf{F}, \phi(\mathbf{u}) \rangle$. Finally, we set $\mathcal{T} := \phi \circ T \circ \phi^* : H_N^{-1/2} \to H_N^{1/2}$. We shall use $\mathcal{T}$ rather than $T$ to compute the expansion of the force field in section 4.5.2. Note that $\mathcal{T}$ depends implicitly on $\varepsilon$, $a$ and on $(\mathbf{p}, \xi)$. In what follows, we will always consider configurations in which the swimmer stays away from the rough wall:

$$\mathrm{dist}(B_l, \partial \mathcal{O}) \geq \delta > 0, \quad \forall l = 1...N, \quad (4.12)$$

for some given $\delta$.
### 4.4.1 Expansion for small $\epsilon$

Under the constraint (4.12), we prove

**Proposition 4.4.1**

$$ \mathcal{T} := \mathcal{T}^0 + \epsilon \mathcal{T}^1 + O(\epsilon^2) \quad \text{in} \quad \mathcal{L}(H_N^{-1/2}, H_N^{1/2}) $$

where $\mathcal{T}^0$ and $\mathcal{T}^1$ are defined in (4.20) and (4.21)-(4.22) respectively.

*Proof.* For $\mathbf{f} = (\mathbf{f}_l) \in H_N^{-1/2}$, we can write

$$ \mathcal{T}(\mathbf{f})_i(\mathbf{r}) = \sum_j \int_{\partial B} \mathbf{K}^{\epsilon}(x_i + a\mathbf{r}, x_j + a\mathbf{s})\mathbf{f}_j(\mathbf{s})d\mathbf{s} $$

(with a classical and slightly abusive notation: the integral should be understood as a duality bracket). Thus, the whole point is to expand the kernel $\mathbf{K}^{\epsilon}$ defined in (4.11). Of course, the first term should be $\mathbf{K}^0$, that is the Green function in the flat case. This Green function can be computed in terms of the Stokeslet by the method of images (see [20]): one has

$$ \mathbf{K}^0(\mathbf{r}, \mathbf{r}_0) = \mathbf{G}(\mathbf{r} - \mathbf{r}_0) + \mathbf{K}_1(\mathbf{r}, \mathbf{r}_0) + \mathbf{K}_2(\mathbf{r}, \mathbf{r}_0) + \mathbf{K}_3(\mathbf{r}, \mathbf{r}_0), \quad (4.13) $$

the four functions $\mathbf{G}$, $\mathbf{K}_1$, $\mathbf{K}_2$ and $\mathbf{K}_3$ being respectively the Stokeslet

$$ \mathbf{G}(\mathbf{r}) = \frac{1}{8\pi\mu} \left( \frac{\mathbf{Id}}{|\mathbf{r}|} + \frac{\mathbf{r} \otimes \mathbf{r}}{|\mathbf{r}|^3} \right) \quad (4.14) $$

and the three “images”

$$ \mathbf{K}_1(\mathbf{r}, \mathbf{r}_0) = -\frac{1}{8\pi\mu} \left( \frac{\mathbf{Id}}{|\mathbf{r}'|} + \frac{\mathbf{r}' \otimes \mathbf{r}'}{|\mathbf{r}'|^3} \right), \quad (4.15) $$

$$ K_{2,ij}(\mathbf{r}, \mathbf{r}_0) = \frac{1}{4\pi\mu} z_0^2 (1 - 2\delta_{j3}) \left( \frac{\delta_{ij}}{|\mathbf{r}'|^3} - \frac{3r_i'r_j'}{|\mathbf{r}'|^5} \right), \quad (4.16) $$

$$ K_{3,ij}(\mathbf{r}, \mathbf{r}_0) = -\frac{1}{4\pi\mu} z_0 (1 - 2\delta_{j3}) \left( \frac{r_3'}{|\mathbf{r}'|^3} \delta_{ij} - \frac{r_j'}{|\mathbf{r}'|^3} \delta_{i3} + \frac{r_i'}{|\mathbf{r}'|^3} \delta_{j3} - \frac{3r_i'r_j'r_3'}{|\mathbf{r}'|^5} \right). \quad (4.17) $$

Here $\mathbf{r}_0 = (x_0, y_0, z_0)$ and $\mathbf{r}' = \mathbf{r} - \tilde{\mathbf{r}}_0$, where $\tilde{\mathbf{r}}_0 = (x_0, y_0, -z_0)$ stands for the “image” of $\mathbf{r}_0$, that is to say, the point symmetric to $\mathbf{r}_0$ with respect to the flat wall.

We now consider $\mathbf{u}^{\epsilon}(\mathbf{x}, \mathbf{x}_0) = \mathbf{K}^{\epsilon}(\mathbf{x}, \mathbf{x}_0) - \mathbf{K}^0(\mathbf{x}, \mathbf{x}_0)$, for $\mathbf{x}_0 \in \cup_l B_l$. As a function of $\mathbf{x}$, it satisfies the Stokes equation in $\mathcal{O}$:

$$ -\Delta \mathbf{u}^{\epsilon}(\cdot, \mathbf{x}_0) + \nabla \mathbf{p}(\cdot, \mathbf{x}_0) = 0, \quad \text{div} \, \mathbf{u}^{\epsilon}(\cdot, \mathbf{x}_0) = 0 \quad \text{in} \, \mathcal{O} $$

with Dirichlet condition

$$ \mathbf{u}^{\epsilon}(\cdot, \mathbf{x}_0) = -\mathbf{K}^{0}(\cdot, \mathbf{x}_0), \quad \text{at} \quad \partial \mathcal{O}. $$

We can then expand the boundary data: for $\mathbf{x} = (x, y, \epsilon h(x, y)) \in \mathcal{O}$

$$ -\mathbf{K}^0(\mathbf{x}, \mathbf{x}_0) = -\sum_{k=1}^{n} \epsilon^k \frac{h(x, y)^k}{k!} \partial_z^k \mathbf{K}^0(x, y, 0, \mathbf{x}_0) + O(\epsilon^{n+1}). $$

More precisely, under the constraint (4.12), one has

$$ \| - \mathbf{K}^0(\cdot, \mathbf{x}_0) + \sum_{k=1}^{n} \epsilon^k \left( \mathbf{x} \mapsto \frac{h(x, y)^k}{k!} \partial_z^k \mathbf{K}^0(x, y, 0, \mathbf{x}_0) \right) \|_{H^s(\partial \mathcal{O})} \le C_{\delta,s} \epsilon^{n+1}, \quad \forall s. $$
We deduce from this inequality that

$$ \| \nabla (\mathbf{u}^{\varepsilon}(\cdot, \mathbf{x}_0) - \sum_{k=1}^{n} \varepsilon^k \mathbf{u}^k(\cdot, \mathbf{x}_0)) \|_{L^2(\mathcal{O})} \leq C \varepsilon^{n+1} \quad (4.18) $$

where $\mathbf{u}^k$ is the solution of

$$ - \Delta \mathbf{u}^k(\cdot, \mathbf{x}_0) + \nabla \mathbf{p}(\cdot, \mathbf{x}_0) = 0, \quad \text{div} \, \mathbf{u}^k(\cdot, \mathbf{x}_0) = 0 \quad \text{in} \ \mathcal{O}, \\ \mathbf{u}^k(\mathbf{x}, \mathbf{x}_0) = -\frac{h(x, y)^k}{k!} \partial_z^k \mathbf{K}^0(x, y, 0, \mathbf{x}_0), \quad \mathbf{x} \in \partial \mathcal{O}. $$

The existence of the $\mathbf{u}^k$'s and the estimate (4.18) are obtained by classical arguments (see the appendix for the more difficult case of a rough half-space minus the balls). In particular, we have

$$ \| \nabla (\mathbf{u}^{\varepsilon}(\cdot, \mathbf{x}_0) - \varepsilon \mathbf{u}^{1}(\cdot, \mathbf{x}_0)) \|_{L^2(\mathcal{O})} \leq C\varepsilon^2. \quad (4.19) $$

The last step consists in replacing $\mathbf{u}^1$ by the solution $\mathbf{K}^1$ of

$$ - \Delta \mathbf{K}^1(\cdot, \mathbf{x}_0) + \nabla \mathbf{p}(\cdot, \mathbf{x}_0) = 0, \quad \text{div} \, \mathbf{K}^1(\cdot, \mathbf{x}_0) = 0, \quad z > 0, \\ \mathbf{K}^1(x, y, 0, \mathbf{x}_0) = -h(x, y) \partial_z \mathbf{K}^0(x, y, 0, \mathbf{x}_0), \quad (x, y) \in \mathbb{R}^2, $$

that is replacing the rough half-space by the flat half-space. We claim that

$$ \| \nabla (\mathbf{u}^1(\cdot, \mathbf{x}_0) - \mathbf{K}^1(\cdot, \mathbf{x}_0)) \|_{L^2(\mathcal{O} \cap \{z>0\})} = O(\varepsilon^2). $$

With no loss of generality, we can assume that $h > 0$ (meaning that the flat wall is below the rough wall). Otherwise, we can make an intermediate comparison with the solution $\tilde{\mathbf{K}}^1$ of the same Stokes problem in $\{z > -\varepsilon(\sup|h| + 1)\}$. Now, an easy but important remark is that

$$ \| \mathbf{K}^1(\cdot, \mathbf{x}_0) \|_{H^s(\{0<z<Z\})} \leq C_{s,Z}, \quad \forall s \in \mathbb{N}, \forall Z > 0. $$

This comes from Poincaré inequality and standard elliptic regularity results. Hence,

$$ \mathbf{K}^1(\mathbf{x}, \mathbf{x}_0) = -h(x, y) \partial_z \mathbf{K}^0(x, y, 0, \mathbf{x}_0) + O(\varepsilon) \quad \text{in} \ H^s(\partial \mathcal{O}). $$

By a simple estimate on $\mathbf{u}^1 - \mathbf{K}^1$, we deduce the claim.

Back to the definition of $\mathbf{u}^\varepsilon$, we obtain thanks to standard elliptic regularity in variable $\mathbf{x}$: for all $\alpha \in \mathbb{N}^3$,

$$ |\partial_{\mathbf{x}}^{\alpha} (\mathbf{K}^{\varepsilon}(\mathbf{x}, \mathbf{x}_0) - \mathbf{K}^{0}(\mathbf{x}, \mathbf{x}_0) - \varepsilon \mathbf{K}^{1}(\mathbf{x}, \mathbf{x}_0))| = O(\varepsilon^2), $$

uniformly in $\mathbf{x}, \mathbf{x}_0 \in \cup_l B_l$. The same reasoning as above can then be applied to the fields $\mathbf{u}^\varepsilon = \partial_{\mathbf{x}_0}^\beta (\mathbf{K}^\varepsilon - \mathbf{K}^0)$, for all $\beta \in \mathbb{N}^3$. Hence,

$$ |\partial_{\mathbf{x}}^{\alpha} \partial_{\mathbf{x}_0}^{\beta} (\mathbf{K}^{\varepsilon}(\mathbf{x}, \mathbf{x}_0) - \mathbf{K}^{0}(\mathbf{x}, \mathbf{x}_0) - \varepsilon \mathbf{K}^{1}(\mathbf{x}, \mathbf{x}_0))| = O(\varepsilon^2), $$

uniformly in $\mathbf{x}, \mathbf{x}_0 \in \cup_l B_l$. The theorem follows straightforwardly, considering

$$ \mathcal{T}^0(\mathbf{f})_i(r) := \sum_j \int_{\partial B} \mathbf{K}^0(x_i + a\mathbf{r}, x_j + a\mathbf{s}) \mathbf{f}_j(\mathbf{s}) d\mathbf{s} \quad (4.20) $$

and

$$ \mathcal{T}^1(\mathbf{f})_i(r) := \sum_j \int_{\partial B} \mathbf{K}^1(x_i + a\mathbf{r}, x_j + a\mathbf{s}) \mathbf{f}_j(\mathbf{s}) d\mathbf{s}. \quad (4.21) $$

Expressing $\mathbf{K}^1(\mathbf{x}, \mathbf{x}_0)$ with a Poisson kernel yields

$$ \mathbf{K}^1(\mathbf{x}, \mathbf{x}_0) := - \int_{\partial \mathbb{R}_+^3} h(\mathbf{s}) \frac{\partial}{\partial z} (\mathbf{s} \mapsto \mathbf{K}^0(\mathbf{s}, \mathbf{x})) \frac{\partial}{\partial z} (\mathbf{s} \mapsto \mathbf{K}^0(\mathbf{s}, \mathbf{x}_0)) \, d\mathbf{s}. \quad (4.22) $$
As a simple consequence of the previous propositions, we have

**Proposition 4.4.3** For every $\mathbf{f} \in H_N^{-1/2}$, for all $(\mathbf{x}, \boldsymbol{\xi}) \in \mathcal{S}$,

$$ (\mathcal{T}\mathbf{f})_i(\mathbf{r}) = \mathcal{T}^G \mathbf{f}_i + \sum_{l=1}^4 \mathbf{K}_l(\mathbf{x}_i, \mathbf{x}_l) \langle \mathbf{f}_i, \mathbf{Id} \rangle_{\partial B} + \sum_{j \neq i} \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \langle \mathbf{f}_j, \mathbf{Id} \rangle_{\partial B} + \mathcal{R}_i(\mathbf{f}), \quad (4.29) $$

with $\|\mathcal{R}_i\|_{\mathcal{L}(H_N^{-1/2}, H_N^{1/2})} = O(a + \varepsilon^2)$, and $i = 1...N$.

**Proof:** By Proposition 4.4.1: for all $i = 1...N$, and all $\mathbf{r} \in \partial B$

$$ \begin{aligned} (\mathcal{T}\mathbf{f})_i(\mathbf{r}) &:= \int_{\partial B} \mathbf{K}(\mathbf{x}_i + a\mathbf{r}, \mathbf{x}_i + a\mathbf{s}) \mathbf{f}_i(\mathbf{s}) \mathrm{d}\mathbf{s} + \sum_{i \neq j} \int_{\partial B} \mathbf{K}(\mathbf{x}_i + a\mathbf{r}, \mathbf{x}_j + a\mathbf{s}) \mathbf{f}_j(\mathbf{s}) \mathrm{d}\mathbf{s} + \mathcal{R}^\varepsilon(\mathbf{f}) \\ &= \mathcal{T}_{i,i} \mathbf{f}_i + \sum_{j \neq i} \mathcal{T}_{i,j} \mathbf{f}_j + \mathcal{R}^\varepsilon(\mathbf{f}), \quad \|\mathcal{R}^\varepsilon\|_{\mathcal{L}(H_N^{-1/2}, H_N^{1/2})} = O(\varepsilon^2) \end{aligned} $$

and the result follows from the application of (4.24) and (4.25) of Proposition 4.4.2. $\square$

**Proposition 4.4.4** For every $\mathbf{u} \in H_N^{1/2}$, for all $(\mathbf{p}, \boldsymbol{\xi}) \in \mathcal{S}$, one has

$$ \begin{aligned} (\mathcal{T}^{-1}\mathbf{u})_i ={}& (\mathcal{T}^G)^{-1} \left( \mathbf{u}_i - \sum_{k=1}^4 \mathbf{K}_k(\mathbf{x}_i, \mathbf{x}_k) \langle (\mathcal{T}^G)^{-1}\mathbf{u}_i, \mathbf{Id} \rangle_{\partial B} \right) - \\ & \qquad (\mathcal{T}^G)^{-1} \left( \sum_{j \neq i} \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \langle (\mathcal{T}^G)^{-1}\mathbf{u}_j, \mathbf{Id} \rangle_{\partial B} \right) + \tilde{\mathcal{R}}_i(\mathbf{u}) \end{aligned} \quad (4.30) $$

with $\|\tilde{\mathcal{R}}_i\|_{\mathcal{L}(H_N^{-1/2}, H_N^{1/2})} = O(a^3 + a^2\varepsilon^2)$, $i = 1...N$.

**Proof:** We recall that

$$ \mathcal{T}^G : H^{-\frac{1}{2}}(\partial B) \to H^{\frac{1}{2}}(\partial B), \quad \mathbf{f} \mapsto \int_{\partial B} \mathbf{G}(a(\cdot - \mathbf{s}))\mathbf{f}(\mathbf{s}) \mathrm{d}\mathbf{s}, $$

and define for $l = 1, \dots, 4$ the operators

$$ S_l : H^{-\frac{1}{2}}(\partial B) \to H^{\frac{1}{2}}(\partial B), \quad \mathbf{f} \mapsto \int_{\partial B} \mathbf{K}_l(\mathbf{x}_i, \mathbf{x}_i) \mathbf{f}(\mathbf{s}) \mathrm{d}\mathbf{s}, $$

and eventually

$$ S_{i,j} : H^{-\frac{1}{2}}(\partial B) \to H^{\frac{1}{2}}(\partial B), \quad \mathbf{f} \mapsto \int_{\partial B} \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \mathbf{f}(\mathbf{s}) \mathrm{d}\mathbf{s}. $$

Notice that for all $\mathbf{f} \in H^{-\frac{1}{2}}(\partial B)$, $S_l \mathbf{f}$ and $S_{i,j} \mathbf{f}$ are constant applications.

That these operators are continuous operators from $H^{-\frac{1}{2}}(\partial B)$ into $H^{\frac{1}{2}}(\partial B)$ is classical. We are only interested in estimating their norms, and more precisely in the way they depend on $a$ in the limit $a \to 0$. Notice that since the kernel $\mathbf{G}$ is homogeneous of degree -1, one has

$$ \|\mathcal{T}^G\|_{\mathcal{L}(H^{-1/2}, H^{1/2})} = O\left(\frac{1}{a}\right) \quad \text{and} \quad \|(\mathcal{T}^G)^{-1}\|_{\mathcal{L}(H^{1/2}, H^{-1/2})} = O(a). \quad (4.31) $$
As far as $S_l$ is concerned, we get that (since $|\mathbf{K}_l(\mathbf{x}_i, \mathbf{x}_i)| = O(1)$)

$$ \|S_l\|_{\mathcal{L}(H^{-1/2}, H^{1/2})} = O(1), \qquad (4.32) $$

and similarly

$$ \|S_{i,j}\|_{\mathcal{L}(H^{-1/2}, H^{1/2})} = O(1). \qquad (4.33) $$

When $a \to 0$ this enables us to invert (4.29) leading to (4.30). □

## 4.5 Controllability of the Three-sphere swimmer

We deal in this section with the controllability of the 3-sphere swimmer, namely Theorem 4.2.4.

### 4.5.1 Preliminary remarks on the 3-sphere dynamics

We must first come back to equation (4.7) (4.9), in the particular case of the 3-sphere swimmer. Remember that the writing in this equation was slightly abusive: we had denoted by $\dot{\mathbf{p}}$ the vector $(\frac{\Omega}{V})$ associated to the rigid movement of the swimmer, see (4.5). In our case, $\Omega = \begin{pmatrix} \Omega_1 \\ \Omega_2 \\ \Omega_3 \end{pmatrix}$ and $\mathbf{v} = \dot{\mathbf{x}}_c = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$ are respectively the angular velocity and the linear velocity of the middle sphere, decomposed in an arbitrary orthonormal basis ($\mathbf{e}_i$). Moreover, it is natural to take for $\mathbf{e}_1$ the unit vector of the 3-sphere axis. Let $\theta$ be the angle between the swimmer's axis and $\mathbf{e}_z$, while $\phi$ is the angle between the y-axis and the projection of the swimmer in $Oxy$ plane (see figure 4.2). Then, the unit vector of the 3-sphere axis reads (in

the canonical basis) $\mathbf{e}_1 = \begin{pmatrix} \cos(\phi) \sin(\theta) \\ \sin(\phi) \sin(\theta) \\ \cos(\theta) \end{pmatrix}$. It is completed into an orthonormal basis by defining

$$ \mathbf{e}_2 = \begin{pmatrix} \cos(\phi) \cos(\theta) \\ \sin(\phi) \cos(\theta) \\ -\sin(\theta) \end{pmatrix}, \qquad \mathbf{e}_3 = \begin{pmatrix} -\sin(\phi) \\ \cos(\phi) \\ 0 \end{pmatrix}. $$

Hence, a rigorous writing of (4.7) or (4.9) is

$$ \mathbf{M}(\frac{\Omega}{V}) + \mathbf{N} = 0, \quad \text{or} \quad (\frac{\Omega}{V}) = -\mathbf{M}^{-1}\mathbf{N}. \qquad (4.34) $$

A crucial remark is that $\mathbf{M}$ and $\mathbf{N}$ do not depend on the whole of $\mathbf{p}$. The angle $\theta_1$ of rotation around the swimmer's axis is not involved, as it is irrelevant to the swimmer's position, orientation or elongation. In particular, keeping only the five bottom lines of the last system, we end up with a closed relation of the type

$$ \begin{pmatrix} \dot{\theta}_2 \\ \dot{\theta}_3 \\ \dot{\mathbf{x}}_c \end{pmatrix} = \sum_{i=1}^{2} \tilde{\mathbf{F}}_i \left( \begin{pmatrix} \theta_2 \\ \theta_3 \\ \mathbf{x}_c \end{pmatrix} \right) \dot{\xi}_i \qquad (4.35) $$

where $\theta_2$ and $\theta_3$ are the rotation angles around $\mathbf{e}_2$ and $\mathbf{e}_3$ respectively. Then, by the analyticity of the $\tilde{\mathbf{F}}_i$'s and Chow's theorem, it remains to prove that there exists some $(\varepsilon, a, \theta_2, \theta_3, \mathbf{x}_c)$ such that

$$ \dim \mathrm{Lie}_{(\theta_2, \theta_3, \mathbf{x}_c)} \left( \begin{pmatrix} \frac{1}{0} \\ \frac{1}{0} \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \end{pmatrix} \right) = 7. $$
Actually, we shall not work directly with angles $\theta_2, \theta_3$. We find it more convenient to work with the angles $\theta, \phi$ introduced above (see Figure ??). From the relation $\frac{d}{dt}\mathbf{e}_1 = \Omega \times \mathbf{e}_1$, we infer that

$$ \Omega_2 = -\sin\theta\dot{\phi}, \quad \Omega_3 = \dot{\theta}. $$

Note that in the special case $\sin\theta = 0$, the angle $\phi$ coincides with the useless angle $\theta_1$. Moreover, the mapping $(\theta_2, \theta_3) \to (\theta, \phi)$ is not a diffeomorphism in the vicinity of $\theta = 0[\pi]$. Thus, we shall restrict to orientations of the swimmer for which

$$ |\sin\theta| \ge \delta > 0. \qquad (4.36) $$

We shall establish the maximality of the Lie algebra at points satisfying this condition.

Before entering the computation of this Lie algebra, we state a technical lemma, that will somehow allow us to neglect the rotation around the swimmer's axis. As mentioned before, we assume inequality (4.36). We have

**Lemma 4.5.1** There exists a constant $C$ which does not depend on $a$ and $\epsilon$ such that

$$ |\Omega_1| \le C (|\dot{\theta}| + |\dot{\phi}| + |\dot{x}_c| + |\dot{\xi}|). $$

**Proof:** We go back to the first identity in (4.34). The first line gives

$$ \mathbf{M}_{1,1} \Omega_1 = -\mathbf{N}_1 \dot{\xi} + \mathbf{M}_{1,2} \sin(\theta) \dot{\phi} - \mathbf{M}_{1,3} \dot{\theta} - \mathbf{M}_{1,4} v_1 - \mathbf{M}_{1,5} v_2 - \mathbf{M}_{1,6} v_3. \quad (4.37) $$

We recall that, in the definitions of $\mathbf{M}$ and $\mathbf{N}$, we denoted by $\mathbf{u}_i$ and $\mathbf{u}^d$ some solutions of the Stokes equation, with zero Dirichlet condition at the wall, and inhomogeneous Dirichlet conditions at the ball. The Dirichlet data is $\mathbf{e}_i \times (\mathbf{x} - \mathbf{x}_c)$ for $i = 1, 2, 3$, $\mathbf{e}_{i-3}$ for $i = 4, 5, 6$, and $\mathbf{u}_d$ for $\mathbf{u}^d$. In the case of the 3-sphere swimmer, $\mathbf{u}_d$ is $-\dot{\xi}_1 \mathbf{e}_1$ on the sphere $\partial B_1$, 0 on the middle sphere and $\dot{\xi}_2 \mathbf{e}_2$ on the sphere $\partial B_3$.

Let us first examine

$$ \begin{align} \mathbf{M}_{1,1} &= \sum_{l=1}^{3} \int_{\partial B} (\mathbf{x}_l - \mathbf{x}_c + a\mathbf{r}) \times \mathbf{e}_1 \cdot \mathcal{T}^{-1}(\mathbf{e}_1 \times a\mathbf{r}, \mathbf{e}_1 \times a\mathbf{r}, \mathbf{e}_1 \times a\mathbf{r}) d\sigma \tag{4.38} \\ &= 3 \int_{\partial B} a\mathbf{r} \times \mathbf{e}_1 \cdot \mathcal{T}^{-1}(\mathbf{e}_1 \times a\mathbf{r}, \mathbf{e}_1 \times a\mathbf{r}, \mathbf{e}_1 \times a\mathbf{r}) d\sigma \nonumber \end{align} $$

using that $(\mathbf{x}_l - \mathbf{x}_c) \times \mathbf{e}_1 = 0$. We then use the expansion (4.30). We recall the well-known fact that the rotation are eigenfunctions of $(\mathcal{T}^G)^{-1}$, with associated eigenvalue $3\mu a$. In particular,

$$ (\mathcal{T}^G)^{-1}(\mathbf{e}_1 \times a\mathbf{r}) = 3\mu a \mathbf{e}_1 \times a\mathbf{r}, \quad \text{and} \quad \langle (\mathcal{T}^G)^{-1}(\mathbf{e}_1 \times a\mathbf{r}), \mathbf{Id} \rangle_{\partial B} = 0. $$

We find then easily that $\mathbf{M}_{1,1} = -3\mu a^3 + O(a^5 + \epsilon^2 a^3)$.

Then, we examine

$$ \mathbf{N}_1 = \sum_{l=1}^{3} \int_{\partial B} (\mathbf{x}_l - \mathbf{x}_c + a\mathbf{r}) \times \mathbf{e}_1 \cdot \mathcal{T}^{-1}(-\dot{\xi}_1 \mathbf{e}_1, 0, \dot{\xi}_2 \mathbf{e}_2) d\sigma. $$

Again, we can expand $\mathcal{T}^{-1}$ using (4.30). This time, we use that translations are eigenfunctions of $(\mathcal{T}^G)^{-1}$ with associated eigenvalue $\frac{3}{2}\mu a$. Thus,

$$ (\mathcal{T}^G)^{-1}(\mathbf{e}_1) = \frac{3}{2}\mu a \mathbf{e}_1. $$
It follows that the first terms in the expansion vanish, and we find

$$N_1 = O((a^4 + a^3 \epsilon^2) |\dot{\xi}|)$$

The remaining terms $M_{1,j}$, $j = 2, ..., 4$ can be handled with similar arguments. The lemma follows straightforwardly. □

### 4.5.2 Asymptotics of the 3-sphere dynamics

We shall now provide an accurate description of the 3-sphere dynamics: broadly speaking, the point is to obtain an explicit expansion of the $F_i$'s in (4.35) (with angles $\theta_2, \theta_3$ replaced by $\theta, \phi$, see remark above). We remind that the dynamics (that is the 6x6 system in (4.34)) is governed by self-propulsion: it corresponds to

* The sum of the forces on the swimmer being zero.

* The sum of the torques on the swimmer being zero.

*Forces.* By the definition of the swimmer, each sphere obeys a rigid body motion. More precisely, the velocity of each point $\mathbf{r}$ of the $l$th sphere expresses as a sum of a translation and a rotation as

$$\mathbf{u}_l^S(\mathbf{r}) = \mathbf{u}_{Tl} + \mathbf{u}_{Rl}(\mathbf{r}), \quad (4.39)$$

where $\mathbf{u}_{Tl}$ is constant on $\partial B$ while $\mathbf{u}_{Rl}(\mathbf{r}) = \Omega \times a\mathbf{r}$ (remember that all quantities are expressed on the unit sphere $\partial B$). The vanishing of the total force, due to self-propulsion, reads

$$\sum_l \int_{\partial B} \mathbf{f}_l = \sum_l \int_{\partial B} (\mathcal{T}^{-1}(\mathbf{u}_1^S, \mathbf{u}_2^S, \mathbf{u}_3^S))_l = 0. \quad (4.40)$$

Plugging (4.39) in (4.40) and using (4.30) leads to

$$\sum_l \int_{\partial B} (\mathcal{T}^G)^{-1} \left( \mathbf{u}_{Tl} + \mathbf{u}_{Rl} - \sum_{k=1}^4 \mathbf{K}_k(\mathbf{x}_l, \mathbf{x}_k) \langle (\mathcal{T}^G)^{-1}(\mathbf{u}_{Tl} + \mathbf{u}_{Rl}), \mathrm{Id} \rangle_{\partial B} \right) - (\mathcal{T}^G)^{-1} \left( \sum_{j \neq i} \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \langle (\mathcal{T}^G)^{-1}(\mathbf{u}_{Tl} + \mathbf{u}_{Rl}), \mathrm{Id} \rangle_{\partial B} \right) = (O(a^3) + O(a^2 \epsilon^2)) ||\mathbf{u}||. \quad (4.41)$$

where $||\mathbf{u}|| = ||(\mathbf{u}_i^S)||$ is any norm on the n-uplets of rigid vector fields over the ball. Here,

$$||u|| = O(|\dot{\theta}| + |\dot{\phi}| + |\dot{\mathbf{x}}_c| + |\Omega_1|) = O(|\dot{\theta}| + |\dot{\phi}| + |\dot{\mathbf{x}}_c|) \quad (4.42)$$

where the last equality comes from Lemma 4.5.1. As mentioned earlier, it is well known that both translations and rotations are eigenfunctions of the Dirichlet to Neumann map of the three dimensional Stokes operator outside a sphere. Namely

$$(\mathcal{T}^G)^{-1} \mathbf{u}_{Tl} = \lambda_T \mathbf{u}_{Tl} \quad \text{and} \quad (\mathcal{T}^G)^{-1} \mathbf{u}_{Rl} = \lambda_R \mathbf{u}_{Rl}.$$

It is also well-known that $\lambda_T = \frac{3\mu a}{2}$, $\lambda_R = 3\mu a$, leading in particular to the celebrated Stokes formula

$$\int_{\partial B} (\mathcal{T}^G)^{-1} \mathbf{u}_{Tl} \, ds = 6\pi\mu a \mathbf{u}_{Tl}$$
We also remark that due to $\int_{\partial B} \mathbf{u}_{R_l} \mathrm{d}\mathbf{s} = 0$, we have $\int_{\partial B} (\mathcal{T}^G)^{-1} \mathbf{u}_{R_l} \mathrm{d}\mathbf{s} = 0$. We therefore obtain

$$ 
\begin{aligned}
& 6\pi\mu a \sum_l \left( \mathbf{u}_{Tl} - 6\pi\mu a \sum_{k=1}^4 \mathbf{K}_k(\mathbf{x}_l, \mathbf{x}_l) \mathbf{u}_{Tl} - 6\pi\mu a \sum_{j \neq l} \mathbf{K}(\mathbf{x}_l, \mathbf{x}_j) \mathbf{u}_{Tj} \right) \\
& = (O(a^3) + O(a^2\epsilon^2)) ||\mathbf{u}||. \quad (4.43)
\end{aligned}
$$

Torques. We now compute the torque with respect to the center $\mathbf{x}_c$ of the middle ball $B_2$. Self-propulsion of the swimmer implies that this torque vanishes:

$$ 0 = \int_{\partial B} (\mathbf{x}_1 - \mathbf{x}_2 + a\mathbf{r}) \times \mathbf{f}_1(\mathbf{r}) + \int_{\partial B} a\mathbf{r} \times \mathbf{f}_2(\mathbf{r}) + \int_{\partial B} (\mathbf{x}_3 - \mathbf{x}_2 + a\mathbf{r}) \times \mathbf{f}_3(\mathbf{r}) = \mathbf{I}_1 + \mathbf{I}_2 + \mathbf{I}_3, \quad (4.44) $$

with the quantities $\mathbf{I}_1$, $\mathbf{I}_2$ and $\mathbf{I}_3$ given below.

$$ 
\begin{aligned}
\mathbf{I}_1 &= \int_{\partial B} (\mathbf{x}_1 - \mathbf{x}_2 + a\mathbf{r}) \times \mathbf{f}_1(\mathbf{r}) = \int_{\partial B} (\xi_1 \mathbf{e}_\xi^1 + a\mathbf{r}) \times (\mathcal{T}^{-1}(\mathbf{u}_1^S, \mathbf{u}_2^S, \mathbf{u}_3^S))_1 \\
&= \int_{\partial B} (-\xi_1 \mathbf{e}_1 + a\mathbf{r}) \times (\mathcal{T}^G)^{-1} \left( \mathbf{u}_{T1} + \mathbf{u}_{R1} - 6\pi\mu a \sum_{k=1}^4 \mathbf{K}_k(\mathbf{x}_1, \mathbf{x}_1) \mathbf{u}_{T1} \right. \\
& \qquad \left. - 6\pi\mu a \sum_{j \neq 1} \mathbf{K}(\mathbf{x}_1, \mathbf{x}_j) \mathbf{u}_{Tj} + O(a^2 + a\epsilon^2) ||\mathbf{u}|| \right) \\
&= -6\pi\mu a \xi_1 \mathbf{e}_\xi^1 \times \left( \mathbf{u}_{T1} - 6\pi\mu a \sum_{k=1}^4 \mathbf{K}_k(\mathbf{x}_1, \mathbf{x}_1) \mathbf{u}_{T1} - 6\pi\mu a \sum_{j \neq 1} \mathbf{K}(\mathbf{x}_1, \mathbf{x}_j) \mathbf{u}_{Tj} \right) \\
& \qquad + (O(a^3) + O(a^2\epsilon^2)) ||\mathbf{u}||.
\end{aligned}
$$

Similarly, we get,

$$ 
\begin{aligned}
\mathbf{I}_2 &= a \int_{\partial B} \mathbf{r} \times \mathbf{f}_2(\mathbf{r}) = a \int_{\partial B} \mathbf{r} \times (\mathcal{T}^{-1}(\mathbf{u}_1^S, \mathbf{u}_2^S, \mathbf{u}_3^S))_2 \\
&= a \int_{\partial B} \mathbf{r} \times (\mathcal{T}^G)^{-1} \left( \mathbf{u}_{T2} + \mathbf{u}_{R2} - 6\pi\mu a \sum_{k=1}^4 \mathbf{K}_k(\mathbf{x}_2, \mathbf{x}_2) \mathbf{u}_{T2} \right. \\
& \qquad \left. - 6\pi\mu a \sum_{j \neq 2} \mathbf{K}(\mathbf{x}_2, \mathbf{x}_j) \mathbf{u}_{Tj} + O(a^2 + a\epsilon^2) ||\mathbf{u}|| \right) \\
&= O(a^3 + a^3\epsilon^2) ||\mathbf{u}||.
\end{aligned}
$$

Finally,

$$ 
\begin{aligned}
\mathbf{I}_3 &= \int_{\partial B} (\mathbf{x}_3 - \mathbf{x}_2 + a\mathbf{r}) \times \mathbf{f}_3(\mathbf{r}) \\
&= 6\pi\mu a \xi_2 \mathbf{e}_1 \times \left( \mathbf{u}_{T3} - 6\pi\mu a \sum_{k=1}^4 \mathbf{K}_k(\mathbf{x}_3, \mathbf{x}_3) \mathbf{u}_{T3} - 6\pi\mu a \sum_{j \neq 3} \mathbf{K}(\mathbf{x}_3, \mathbf{x}_j) \mathbf{u}_{Tj} \right) \\
& \qquad + (O(a^3) + O(a^2\epsilon^2)) ||\mathbf{u}||.
\end{aligned}
$$
Denoting by **A** the matrix

$$ 
\mathbf{A} = \begin{pmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} & \mathbf{A}_{13} \\ \mathbf{A}_{21} & \mathbf{A}_{22} & \mathbf{A}_{23} \\ \mathbf{A}_{31} & \mathbf{A}_{32} & \mathbf{A}_{33} \end{pmatrix} \qquad (4.45) $$

where for $i = 1, 2, 3$

$$ \mathbf{A}_{ii} = \mathbf{Id} - 6\pi\mu a \sum_{l=1}^{4} \mathbf{K}_l(\mathbf{x}_i, \mathbf{x}_i) \qquad (4.46) $$

and for $i, j = 1, 2, 3$ with $i \neq j$

$$ \mathbf{A}_{ij} = -6\pi\mu a \mathbf{K}(\mathbf{x}_i, \mathbf{x}_j) \qquad (4.47) $$

and **S** the matrix

$$ \mathbf{S} = \begin{pmatrix} \mathbf{Id} & \mathbf{Id} & \mathbf{Id} \\ -\xi_1 \mathbf{e}_1 \times & 0 & +\xi_2 \mathbf{e}_1 \times \end{pmatrix}, $$

we can rewrite the self-propulsion assumption (4.43), (4.44) as

$$ \mathbf{S} \mathbf{A} \begin{pmatrix} \mathbf{u}_{T_1} \\ \mathbf{u}_{T_2} \\ \mathbf{u}_{T_3} \end{pmatrix} = (\mathcal{O}(a^2) + \mathcal{O}(a\epsilon^2)) \|\mathbf{u}\|. \qquad (4.48) $$

Terms involving the $\mathbf{u}_{R_i}$'s are included in the r.h.s.

We now express $\mathbf{u}_{T_1}$, $\mathbf{u}_{T_2}$ and $\mathbf{u}_{T_3}$ in terms of $\dot{\mathbf{x}}_c, \dot{\theta}, \dot{\phi}$ and $\dot{\xi}$. Since $\mathbf{u}_{T_2}$ is the velocity of the center of the ball $B_2$, one has

$$ \mathbf{u}_{T_2} = \dot{\mathbf{x}}_c = \begin{pmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{pmatrix} \quad \text{in the canonical basis of } \mathbb{R}^3. $$

Then, by using $\frac{d}{dt}\mathbf{e}_1 = \dot{\theta}\mathbf{e}_2 + \sin(\theta)\dot{\phi}\mathbf{e}_3$, we get

$$ \mathbf{u}_{T_1} = \mathbf{u}_{T_2} - \xi_1 (\dot{\theta}\mathbf{e}_2 + \sin(\theta)\dot{\phi}\mathbf{e}_3) - \dot{\xi}_1 \mathbf{e}_1, \quad \mathbf{u}_{T_3} = \mathbf{u}_{T_2} + \xi_2 (\dot{\theta}\mathbf{e}_2 + \sin(\theta)\dot{\phi}\mathbf{e}_3) + \dot{\xi}_2 \mathbf{e}_1. $$

In matrix form, all this reads Then, the speed $\mathbf{u}_{T_i}$ ($i = 1, 2, 3$) is expressed as

$$ \begin{pmatrix} \mathbf{u}_{T_1} \\ \mathbf{u}_{T_2} \\ \mathbf{u}_{T_3} \end{pmatrix} = \mathbf{T} \begin{pmatrix} \Omega_1 \\ \dot{\theta} \\ \dot{\phi} \\ \dot{x} \\ \dot{y} \\ \dot{z} \end{pmatrix} + \mathbf{U} \dot{\xi}. \qquad (4.49) $$

with

$$ \mathbf{T} = \begin{pmatrix} 0 & -\xi_1 \mathbf{e}_2 & -\xi_1 \sin(\theta) \mathbf{e}_3 & \mathbf{Id} \\ \vdots & 0 & 0 & \mathbf{Id} \\ 0 & 0 & 0 & \mathbf{Id} \\ 0 & +\xi_2 \mathbf{e}_2 & +\xi_2 \sin(\theta) \mathbf{e}_3 & \mathbf{Id} \end{pmatrix}, \quad \text{and} \quad \mathbf{U} = \begin{pmatrix} 0 & 0 \\ -\mathbf{e}_1 & 0 \\ 0 & \vdots \\ \vdots & 0 \\ 0 & \mathbf{e}_1 \end{pmatrix}. $$

Combining with (4.48), the motion equation (4.34) becomes

$$ (\mathbf{S}\mathbf{A} + \mathbf{R}_1) \begin{pmatrix} \mathbf{T} \\ \begin{pmatrix} \Omega_1 \\ \dot{\theta} \\ \dot{\phi} \\ \dot{x} \\ \dot{y} \\ \dot{z} \end{pmatrix} \end{pmatrix} + (\mathbf{U} + \mathbf{R}_2) \dot{\xi} = 0 \qquad (4.50) $$
where the residual matrices $\mathbf{R}_1, \mathbf{R}_2$ satisfy

$$|\mathbf{R}_1| + |\mathbf{R}_2| = (O(a^2) + O(a\epsilon^2))$$

using (4.42). Finally, we only keep the five bottom lines of this system. It yields the following 5x5 system

$$(\tilde{\mathbf{S}}\mathbf{A} + \tilde{\mathbf{R}}) \begin{pmatrix} \dot{\mathbf{T}} \\ \dot{\phi} \\ \dot{x} \\ \dot{y} \\ \dot{z} \end{pmatrix} + \tilde{\mathbf{U}}\dot{\mathbf{\xi}} = 0, \quad (4.51)$$

where

$$\tilde{\mathbf{S}} := (S_{i,j})_{2 \le i \le 6, 1 \le j \le 9}, \quad \tilde{\mathbf{T}} := \begin{pmatrix} -\xi_1 \mathbf{e}_2 & -\xi_1 \sin(\theta) \mathbf{e}_3 & \mathbf{Id} \\ 0 & 0 & 0 \\ 0 & 0 & \mathbf{Id} \\ 0 & 0 & 0 \\ +\xi_2 \mathbf{e}_2 & +\xi_2 \sin(\theta) \mathbf{e}_3 & \mathbf{Id} \end{pmatrix}, \quad \tilde{\mathbf{U}} := \begin{pmatrix} -\mathbf{e}_1 & 0 & 0 \\ 0 & \vdots & \vdots \\ \vdots & 0 & 0 \\ 0 & \mathbf{e}_1 & 0 \end{pmatrix},$$

and where the residual matrices still satisfy $|\tilde{\mathbf{R}}_1| + |\tilde{\mathbf{R}}_2| = O(a^2) + O(a\epsilon^2)$. We leave to the reader to check that $\tilde{\mathbf{S}}\mathbf{A}\tilde{\mathbf{T}} = \tilde{\mathbf{S}}\tilde{\mathbf{T}} + O(a)$ is invertible, with $|(\tilde{\mathbf{S}}\mathbf{A}\tilde{\mathbf{T}})^{-1}| = O(1)$ uniformly in $a$ and $\epsilon$. Then, we can write system (4.51) as

$$\begin{pmatrix} \dot{\theta} \\ \dot{\phi} \\ \dot{x} \\ \dot{y} \\ \dot{z} \end{pmatrix} = -(\tilde{\mathbf{S}}\mathbf{A}\tilde{\mathbf{T}})^{-1}\tilde{\mathbf{S}}\mathbf{A}\tilde{\mathbf{U}}\dot{\mathbf{\xi}} + \tilde{\mathbf{R}}\dot{\mathbf{\xi}} \quad (4.52)$$

with $|\tilde{\mathbf{R}}| = O(a^2 + \epsilon^2 a)$.

### 4.5.3 Reachable set

We are now ready to prove Theorem 4.2.4. We drop the tilda in the ODE (4.52) and express it as

$$\dot{\mathbf{X}} = \mathbf{F}_1(\mathbf{X})\dot{\mathbf{\xi}}_1 + \mathbf{F}_2(\mathbf{X})\dot{\mathbf{\xi}}_2, \quad \mathbf{X} := \begin{pmatrix} \xi_1 \\ \xi_2 \\ \theta \\ \phi \\ x \\ y \\ z \end{pmatrix}. \quad (4.53)$$

To expand the $\mathbf{F}_i$'s, we decompose the matrix $\mathbf{A}$ into three matrices: $\mathbf{A} := \mathbf{Id} + \mathbf{A}^1 + \mathbf{A}^2$ where

$$\mathbf{A}_{ii}^1 = -6\pi\mu a \sum_{k=1}^{3} \mathbf{K}_k(\mathbf{x}_i, \mathbf{x}_i) \quad \forall i, \quad \mathbf{A}_{ij}^1 = -6\pi\mu a \left( \mathbf{G}(\mathbf{x}_i, \mathbf{x}_j) + \sum_{k=1}^{3} \mathbf{K}_k(\mathbf{x}_i, \mathbf{x}_j) \right) \quad \forall i \neq j$$

and where

$$\mathbf{A}_{i,j}^2 = -6\pi\mu a \mathbf{K}_4(\mathbf{x}_i, \mathbf{x}_j) \quad \forall i, j.$$

Thanks to (4.51), we get an expansion of the form $\mathbf{F}_i := \mathbf{F}_i^0 + \mathbf{F}_i^1 + \mathbf{F}_i^2 + \mathbf{R}_i$ where $\mathbf{F}_i^0, \mathbf{F}_i^1$ and $\mathbf{F}_i^2$ are respectively the zero order term, the term of order $a$ and the term of order $\epsilon a$. The remainder is $\mathbf{R}_i = (O(a^2) + O(a\epsilon^2))$. These vector fields are given by

$$\mathbf{F}_i^0 = -(\mathbf{S}\mathbf{T})^{-1}(\mathbf{S}\mathbf{U})\mathbf{e}_i,$$

$$\mathbf{F}_i^1 = ((\mathbf{S}\mathbf{T})^{-1}\mathbf{S}\mathbf{A}^1\mathbf{T}(\mathbf{S}\mathbf{T}))^{-1}\mathbf{S}\mathbf{U} - (\mathbf{S}\mathbf{T})^{-1}\mathbf{S}\mathbf{A}^1\mathbf{U})\mathbf{e}_i,$$

$$\mathbf{F}_i^2 = ((\mathbf{S}\mathbf{T})^{-1}\mathbf{S}\mathbf{A}^2\mathbf{T}(\mathbf{S}\mathbf{T}))^{-1}\mathbf{S}\mathbf{U} - (\mathbf{S}\mathbf{T})^{-1}\mathbf{S}\mathbf{A}^2\mathbf{U})\mathbf{e}_i.$$
where $\mathbf{e}_1 = (\begin{smallmatrix} 1 \\ 0 \end{smallmatrix})$ and $\mathbf{e}_2 = (\begin{smallmatrix} 0 \\ 1 \end{smallmatrix})$.

Now, we want to find some $(\varepsilon, a, \mathbf{X})$ in $\mathbb{R}_+ \times \mathbb{R}_+^* \times S$ for which the determinant

$$ \det(\mathbf{X}) := \begin{vmatrix} \mathbf{F}_1, \mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2], [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]], [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]] \\ [\mathbf{F}_1, [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]], [\mathbf{F}_2, [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]] \end{vmatrix} \neq 0. \quad (4.54) $$

As the l.h.s. defines an analytic function of $\mathbf{X}$, it will be non-zero almost everywhere. Thus, the Lie algebra generated by $\mathbf{F}_1$ and $\mathbf{F}_2$ will be maximal (of dimension 7) at almost every $\mathbf{X}$, and local controllability will follow from Chow's theorem, see [51].

For all $\mathbf{G} \in \text{Lie}(\mathbf{F}_1, \mathbf{F}_2)$, let us denote $\mathbf{G}^0$, $\mathbf{G}^1$ and $\mathbf{G}^2$ the zero order term, the term of order $a$ and the term of order $a\varepsilon$ in the expansion of the vector field $\mathbf{G}$ respectively. Thus,

$$ \mathbf{G} = \mathbf{G}^0 + \mathbf{G}^1 + \mathbf{G}^2 + O(a^2) + O(a\varepsilon^2). $$

For instance the expansion of the first Lie bracket reads

$$ [\mathbf{F}_1, \mathbf{F}_2] = [\mathbf{F}_1, \mathbf{F}_2]^0 + [\mathbf{F}_1, \mathbf{F}_2]^1 + [\mathbf{F}_1, \mathbf{F}_2]^2 + O(a^2) + O(a\varepsilon^2). $$

with

$$ [\mathbf{F}_1, \mathbf{F}_2]^0 = [\mathbf{F}_1^0, \mathbf{F}_2^0], \quad [\mathbf{F}_1, \mathbf{F}_2]^1 = [\mathbf{F}_1^1, \mathbf{F}_2^0] + [\mathbf{F}_1^0, \mathbf{F}_2^1], \quad [\mathbf{F}_1, \mathbf{F}_2]^2 = [\mathbf{F}_1^2, \mathbf{F}_2^0] + [\mathbf{F}_1^0, \mathbf{F}_2^2]. $$

Note that for all $\mathbf{G} \in \text{Lie}(\mathbf{F}_1, \mathbf{F}_2)$, $\mathbf{G}^0 + \mathbf{G}^1$ is a "flat wall" expansion, first order in $a$. Meanwhile, $\mathbf{G}^2$ is the first term which takes into account the roughness.

Without including this extra term, the three-sphere swimmer would not be controllable (see [11]), meaning that the determinant would vanish. We have notably

**Lemma 4.5.2** For all $\mathbf{G} \in \text{Lie}(\mathbf{F}_1, \mathbf{F}_2) \setminus \{\mathbf{F}_1, \mathbf{F}_2\}$, $\mathbf{G}^0 = 0$.

**Proof:** A simple calculation yields

$$ \mathbf{F}_1^0(\mathbf{X}) = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ \frac{1}{3} \cos(\phi) \sin(\theta) \\ \frac{1}{3} \sin(\phi) \sin(\theta) \\ \frac{1}{3} \cos(\theta) \end{pmatrix}, \quad \mathbf{F}_2^0(\mathbf{X}) = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \\ -\frac{1}{3} \cos(\phi) \sin(\theta) \\ -\frac{1}{3} \sin(\phi) \sin(\theta) \\ -\frac{1}{3} \cos(\theta) \end{pmatrix}. $$

It implies that $[\mathbf{F}_1^0, \mathbf{F}_2^0]$ is zero. The lemma is proved. $\square$

As regards the $O(a)$ term, we have

**Lemma 4.5.3** Let $\text{Lie}(\mathbf{F}_1, \mathbf{F}_2)^1 := \{\mathbf{G}^1 \text{ s.t. } \mathbf{G} \in \text{Lie}(\mathbf{F}_1, \mathbf{F}_2)\}$. For all $\mathbf{X} \in S$, the dimension of the subspace $\text{Lie}(\mathbf{F}_1, \mathbf{F}_2)^1(\mathbf{X})$ is less than 5.

**Proof:** As said above, for all $\mathbf{G} \in \text{Lie}(\mathbf{F}_1, \mathbf{F}_2)$, the sum $\mathbf{G}^0 + \mathbf{G}^1$ is a $O(a)$ expansion of the "flat wall field", corresponding to the case $h = 0$. But in such flat case, symmetries constrain the swimmer within a plane. Thus, the associated manifold has at most dimension 5 ($\xi_1, \xi_2$, two coordinates for the center of the middle ball, one angle). This implies the result. $\square$
**Remark 4.5.4** Since without roughness the swimmer evolves in a plane, it follows that the angle $\phi$ cannot change with time. Consequently, for all $\mathbf{F}(\mathbf{X}) \in \text{Lie}(\mathbf{F}_1, \mathbf{F}_2)^1(\mathbf{X})$ the fourth component of the vector $\mathbf{F}(\mathbf{X})$ is zero.

**Remark 4.5.5** The lemma also 4.5.3 applies to the vector fields which does not take into account the roughness i.e., the ones which appear in the expansion without $\epsilon$.

From this, we will get that the non-zero leading term in the expansion of $\det$ has power $a^5\epsilon^2$. Theorem 4.2.4 follows directly from

**Proposition 4.5.6** In the regime $1 \gg \epsilon \gg a$, one can find a surface $h \in C_c^\infty(\mathbb{R}^2)$ and a non-trivial analytic function $\mathcal{A}$ on $S$ such that for all $\mathbf{X} \in S$

$$ \det(\mathbf{X}) = a^5 \epsilon^2 \mathcal{A}(\mathbf{X}) + O(a^6 \epsilon^2 + a^5 \epsilon^3). $$

**Proof:** For all vector $\mathbf{G}$, we denote $(\mathbf{G})_{j'}^j := (\mathbf{G}_k)_{j \le k \le j'}$. Since $\mathbf{F}_i$, $i = 1, 2$, is of the type

$$ \begin{pmatrix} \mathbf{e}_i \\ * \\ \vdots \\ * \end{pmatrix}, \text{ we get easily that} $$

$$ \det(\mathbf{X}) = |\mathbf{Z}_1, \mathbf{Z}_2, \mathbf{Z}_3, \mathbf{Z}_4, \mathbf{Z}_5| $$

where

$$ \left\{ \begin{array}{l} \mathbf{Z}_1 := ([\mathbf{F}_1, \mathbf{F}_2])_3^7, \\ \mathbf{Z}_2 := ([\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]])_3^7, \\ \mathbf{Z}_3 := ([\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]])_3^7, \\ \mathbf{Z}_4 := ([\mathbf{F}_1, [\mathbf{F}_1, [\mathbf{F}_1, \mathbf{F}_2]]])_3^7, \\ \mathbf{Z}_5 := ([\mathbf{F}_2, [\mathbf{F}_2, [\mathbf{F}_1, \mathbf{F}_2]]])_3^7. \end{array} \right. \qquad (4.55) $$

From Lemma 4.5.2, $\mathbf{Z}_i^0 = 0$ for all $i = 1...5$. Moreover, by Lemma 4.5.3, any determinant of the type

$$ |\mathbf{Z}_{k_1}^1, \mathbf{Z}_{k_2}^1, \mathbf{Z}_{k_3}^1, *|, \quad k_i \in \{1, ..., 5\} \text{ is zero.} $$

Expanding the function $\det$ by 5-linearity, we obtain

$$ \det(\mathbf{X}) = a^5 \epsilon^2 \mathcal{A}(\mathbf{X}) + O(a^6 \epsilon^2 + a^5 \epsilon^3), $$

where the function $\mathcal{A}(\mathbf{X})$ is defined as follows. Let

$$ \mathcal{I} := \{ \mathbf{k} \in \{1, ..., 5\}^5 \text{ with } k_1 < k_2 \text{ and } k_3 < k_4 < k_5 \text{ distinct of } k_1 \text{ and } k_2 \}. $$

We set

$$ \mathcal{A}(\mathbf{X}) := \sum_{\mathbf{k} \in \mathcal{I}} \pm |\mathbf{Z}_{k_1}^2, \mathbf{Z}_{k_2}^2, \mathbf{Z}_{k_3}^1, \mathbf{Z}_{k_3}^1, \mathbf{Z}_{k_4}^1, \mathbf{Z}_{k_5}^1|, $$

where the $\pm$ is the signature of the permutation $i \to k_i$.

It remains to prove that there exists $\mathbf{X}_0 \in S$ such that $\mathcal{A}(\mathbf{X}_0)$ is non-zero. By calling $\mathbf{K}_4^{\text{int}}$ the function $(\mathbf{s}, \mathbf{r}, \mathbf{r}') \mapsto \frac{\partial}{\partial z}(\mathbf{s} \mapsto \mathbf{K}^0(\mathbf{s}, \mathbf{r})) \frac{\partial}{\partial z}(\mathbf{s} \mapsto \mathbf{K}^0(\mathbf{s}, \mathbf{r}'))$, we have (see (4.22))

$$ \mathbf{K}_4(\mathbf{r}, \mathbf{r}') = -\epsilon \int_{\partial \mathbb{R}_+^3} h(s_1, s_2) \mathbf{K}_4^{\text{int}}(\mathbf{s}, \mathbf{r}, \mathbf{r}') \, \mathrm{d}\mathbf{s}. $$

We then define the 3x3 block matrix $\mathbf{A}_{\text{int}}^2(\mathbf{s})$ through

$$ (\mathbf{A}_{\text{int}}^2(\mathbf{s}))_{ij} = -6\pi\mu a \mathbf{K}_4^{\text{int}}(\mathbf{s}, \mathbf{x}_i, \mathbf{x}_j), \quad i, j = 1...3. $$
By using the linearity of the integral, the vector fields $\mathbf{F}_i^2$, $i = 1, 2$ read

$$ \mathbf{F}_i^2 = \left( -\varepsilon \int_{\partial \mathbb{R}_+^3} h(\mathbf{s}) (\mathbf{F}_{i,int}^2(\mathbf{s})) \, \mathrm{d}\mathbf{s} \right), $$

where,

$$ \mathbf{F}_{i,int}^2(\mathbf{s}) = - \left( -(\mathbf{S}\mathbf{T})^{-1} \mathbf{S} \mathbf{A}_{int}^2(\mathbf{s}) \mathbf{T}(\mathbf{S}\mathbf{T})^{-1} \mathbf{S} \mathbf{U} + (\mathbf{S}\mathbf{T})^{-1} \mathbf{S} \mathbf{A}_{int}^2(\mathbf{s}) \mathbf{U} \right) \mathbf{e}_i. \quad (4.56) $$

Then, denoting

$$ \mathbf{Z}_{1,int}^2(\mathbf{s}) := [\mathbf{F}_{1,int}^2(\mathbf{s}), \mathbf{F}_{2}^0] + [\mathbf{F}_{1}^0, \mathbf{F}_{2,int}^2(\mathbf{s})] $$

leads to

$$ \mathbf{Z}_1^2 = -\varepsilon \int_{\partial \mathbb{R}_+^3} h(s_1, s_2) \mathbf{Z}_{1,int}^2(\mathbf{s}) \, \mathrm{d}\mathbf{s}. \quad (4.57) $$

We can go on with this process and find explicitly functions $\mathbf{Z}_{i,int}^2(\mathbf{s})$ for $i = 2, \dots, 5$ such that

$$ \forall i \in \{2, \dots, 5\}, \quad \mathbf{Z}_i^2 = -\varepsilon \int_{\partial \mathbb{R}_+^3} h(s_1, s_2) (\mathbf{Z}_{i,int}^2(\mathbf{s})) \, \mathrm{d}\mathbf{s}. $$

Finally,

$$ \mathcal{A}(\mathbf{X}) = -\varepsilon^2 \int_{\partial \mathbb{R}_+^3} \int_{\partial \mathbb{R}_+^3} h(s_1, s_2) h(s'_1, s'_2) \sum_{\mathcal{I}} \pm |\mathbf{Z}_{k_1,int}^2(s_1, s_2) \mathbf{Z}_{k_2,int}^2(s'_1, s'_2) \mathbf{Z}_{k_3}^1 \mathbf{Z}_{k_4}^1 \mathbf{Z}_{k_5}^1(\mathbf{X})| \, \mathrm{d}\mathbf{s} \, \mathrm{d}\mathbf{s}'. \quad (4.58) $$

We call $\det_{int}$ the function $(\mathbf{X}, \mathbf{s}, \mathbf{s}') \mapsto \sum_{\mathcal{I}} \pm |\mathbf{Z}_{k_1,int}^2(\mathbf{s}) \mathbf{Z}_{k_2,int}^2(\mathbf{s}') \mathbf{Z}_{k_3}^1 \mathbf{Z}_{k_4}^1 \mathbf{Z}_{k_5}^1(\mathbf{X})|^2$. Clearly, for Theorem 4.5.6 to hold, it is enough that there exists $\mathbf{X}_0 \in \mathcal{S}$ and $(\mathbf{s}, \mathbf{s}') \in (\partial \mathbb{R}_+^3)^2$ such that $\det_{int}(\mathbf{X}_0, \mathbf{s}, \mathbf{s}')$ is not zero for some $(\mathbf{s}, \mathbf{s}') \in \mathbb{R}^4$. Indeed, we can then adjust the function $h$ to make the integral non-zero. The calculation of $\det_{int}$ can be carried out using Maple. More precisely, one can derive an equivalent as $z$ goes to infinity, and check that $\det_{int}(\mathbf{X}_0, \cdot, \cdot) \neq 0$ for $\mathbf{X}_0 = (1, 2, \frac{\pi}{3}, \frac{\pi}{3}, 1, 2, z)$ for $z$ large enough. This concludes the proof.

## 4.6 Conclusions and perspectives

The aim of this present paper was to examine how the controllability of low Reynolds number artificial swimmers is affected by the presence of a rough wall on a fluid. This study generalizes the one made by F. Alouges and L. Giraldi in [11] which deals with the effect of a plane wall on the controllability of this particular swimmers.

Firstly, we show Theorem 4.2.1. It deals with the regularity of the dynamics of the swimmers. Indeed, we prove that the equation of motion of such particular swimmers are analytic with respect to the parameters defining the swimmer (radius of the ball, position and length of the arms) and the typical height of roughness of the wall. Then, we deduce Theorem 4.2.3 which claims that the 4-sphere swimmer remains controllable with the presence of roughness. The proof is based on general arguments which could be used for other models of micro-swimmer.

Secondly, Theorem 4.2.4 examines the controllability of the Three-sphere swimmer in the presence of a rough wall. More precisely, we show that there exists a roughness such that the
swimmer can locally reach any direction. We recall that the previous studies made on the 3-sphere swimmer allow to show that it can reach only one direction (see [7] when it evolves in a whole space and three directions with the presence of a plane wall (see [11]). In our case, the roughness leads to break the symmetry of the system "fluids-swimmer". As a result, it allows the swimmer to reach any direction. The proof is an in-depth study which associates several tools both in hydrodynamics and control theory. The general "idea" emphasizes here is the fact that in the real life all the micro-organism, regardless how symmetric it is, can move in any direction.

The quantitative approach to this question together with the complete understanding in a view of controllability of underlying systems is far beyond reach and thus still under progress as in another direction, the consideration of a confined environment, e.g. when the fluid is bounded. Future work will also explore which are the directions easier to reach than the others by varying the rough wall.

## 4.7 Appendix: A well-posedness result for the Stokes system

We show here the well-posedness of the inhomogeneous Stokes system involved in the proof of Theorem 4.2.1. We refer to this proof for notations, and shall drop here all bars for brevity. What we want to show is

**Proposition 4.7.1** Let $(F, G, V_1, ..., V_N)$ given in $(V_0)' \times L^2(\mathcal{F}) \times \prod_l H^{1/2}(B_l)$. There exists a unique solution $(V, Q)$ in $V_0 \times L^2(\mathcal{F})$ of

$$ 
\begin{array}{l}
-\Delta V + \nabla Q = F \quad \text{in } \mathcal{F}, \\
\operatorname{div} V = G \quad \text{in } \mathcal{F}, \\
V = 0 \quad \text{at } \partial \mathcal{O}, \quad V = V_l \quad \text{at } \partial B_l, \quad l = 1...N.
\end{array}
$$

*Proof of the Proposition.* In the "standard" case where $G = 0$ and all $V_l$'s are zero, the result follows from a direct variational argument. Thus the whole point is to find a $W \in V_0$ satisfying

$$ \operatorname{div} W = G \quad \text{in } \mathcal{F}, \quad V = 0 \quad \text{at } \partial \mathcal{O}, \quad V = V_l \quad \text{at } \partial B_l, \quad l = 1...N. \quad (4.59) $$

Indeed, setting $V' := V - W$, one comes back to the standard case with $F + \Delta W$ replacing $F$.

To build such $W$, a key ingredient is

**Lemma 4.7.2** Given $G \in L^2(\mathcal{O})$, there exists a field $W \in L_{\text{loc}}^2(\mathcal{O})$, $\nabla W \in L^2(\mathcal{O})$ such that

$$ \operatorname{div} W = G, \quad W|_{\partial \mathcal{O}} = 0 \quad \|\nabla W\|_{L^2} \le C \|G\|_{L^2}. $$

Note that this lemma is only about the domain $\mathcal{O}$, that is without the balls. Let us postpone its proof, and show how it implies the existence of a $W$ satisfying (4.59).

* First step: we lift the boundary data $V_l$. One can find $\bar{W} \in H^1(\mathcal{F})$ compactly supported near the balls, such that $\bar{W} = V_l$ at $\partial B_l$. Up to replace $W$ by $W - \bar{W}$ and $G$ by $G - \operatorname{div} \bar{W}$, we can assume $V_l = 0$ for all $l$.

* Second step (assuming now $V_l = 0$ for all $l$): we extend $G$ by 0 in the balls and apply the Lemma: it provides a $\bar{W}$ satisfying $\operatorname{div} \bar{W} = G$, $\bar{W}|_{\partial \mathcal{O}} = 0$. However, the boundary data at the balls is non-zero: $\bar{W}|_{\partial B_l} \ne 0$.
• Third step: we correct this non-zero boundary data. We observe that

$$ \int_{\partial B_l} \tilde{W} \cdot n \, ds = 0 = \int_{B_l} \text{div} \, \tilde{W} = 0, $$

as $G$ was extended by zero inside the balls. Thanks to this "compatibility" condition, we can use a standard result of Bogovskii, see [1, Exercise III.3.5, p176]: for all $l$, there exists a field $W_l$ defined over the annulus $\{a < |x - x_l| < a + \eta\}$, satisfying

$$ \text{div } W_l = 0, \quad W_l|_{\partial B_l} = -\tilde{W}|_{\partial B_l}, \quad W_l|_{\{|x-x_l|=a+\eta\}} = 0. $$

We take $\eta$ small enough so that the annuli do not intersect. Then, we extend the $W_l$'s by 0 outside the annuli and set $W := \tilde{W} + \sum W_l$. This new field $W$ satisfies (4.59), as expected.

*Proof of the Lemma.* In the case where $h = \text{cst}$, that is for a flat half-space, the result is classical: cf [1, Corollary 4.3.1, p261]. In particular, if the support of $G$ is included in $\{x_3 > \sup|h|\}$, the problem is solved: one can take the solution $W$ of

$$ \text{div } W = G \quad \text{for} \quad x_3 > \sup|h|, \quad W|_{\{x_3=\sup|h|\}} = 0 $$

and extend it by zero below $\{x_3 = \sup|h|\}$.

For general $G$, we can decompose $G = G 1_{\{x_3>\sup|h|\}} + G 1_{\{x_3<\sup|h|\}}$, and handle the first part as previously. In other words, it remains to consider the case where $G$ is compactly supported in $x_3$. From there, we proceed in three steps:

• Step 1. Let $R$ such that $G = 0$ for $x_3 \ge R$. We introduce $W^1 := \nabla\psi 1_{\{x_3<R\}}$ where $\psi$ satisfies

$$ \Delta\psi = G \quad \text{for } \varepsilon h < x_3 < R, \quad \partial_n\psi|_{\partial\mathcal{O}} = 0, \quad \psi|_{x_3=R} = 0. $$

This Poisson equation has a unique solution in $H^2(\{\varepsilon h < x_3 < R\})$: note that Poincaré inequality applies thanks to the Dirichlet condition at $x_3 = R$. Hence, $W^1$ satisfies $\text{div } W^1 = G$ in the strip $\{\varepsilon h < x_3 < R\}$, and also trivially in the half-space $\{x_3 > R\}$. However, two problems remain: the normal component of $W^1$ jumps at $x_3 = R$, and it has non-zero boundary data at $\{x_3 = \varepsilon h\}$.

• Step 2. Correction of the jump at $x_3 = R$. We just introduce the field $W^2 := \tilde{W} 1_{\{x_3>R\}}$, where $\tilde{W}$ satisfies

$$ \text{div } \tilde{W} = 0 \quad \text{for} \quad x_3 > R, \quad \tilde{W}|_{\{x_3=R\}} = \nabla\Psi|_{x_3=R}, \\ \|\nabla\tilde{W}\|_{L^2} \le C\|\nabla\psi\|_{H^{1/2}(\{x_3=R\})} \quad (\le C\|G\|_{L^2}). $$

The existence of such $\tilde{W}$ is classical, see [39, Theorem IV.3.3].

• Step 3. Correction of the boundary data. Thanks to the Neumann condition on $\psi$, we have $W^1 \cdot n|_{\partial\mathcal{O}} = 0$. We introduce some partition of unity $(\chi_k = \chi_k(x_1, x_2))_{k \in \mathbb{Z}^2}$ associated to a covering of $\mathbb{R}^2$ by rectangles $R_k$. More precisely, we assume that the lengths of $R_k$ are uniformly bounded in $k$, and that the $C^1$ norms of $\chi^k$ are uniformly bounded in $k$ (we leave the construction of examples to the reader). Thanks to the tangency condition on $W^1$, we can apply the Bogovskii's result seen above on slices $S_k := \{(x_1, x_2) \in R_k, \varepsilon h(x_1, x_2) < x_3 < R\}$, $k \in \mathbb{Z}^2$. Hence, there exists some $W_k \in H^1(S_k)$ such that

$$ \text{div } W_k = 0 \quad \text{in } S_k, \quad W_k = -\chi_k W^1 \quad \text{at } \partial S_k \cap \partial\mathcal{O}, \\ W_k = 0 \quad \text{at } \partial S_k \setminus \mathcal{O}, $$
and $\|\nabla W_k\|_{L^2} \le C \|\chi_k W^1\|_{H^{1/2}(\partial \mathcal{O})}$. Extending all $W_k$'s by 0 outside $S_k$, and setting $W^3 := \sum_{k \in \mathbb{Z}^2} W_k$, we find that

$$ \operatorname{div} W^3 = 0 \quad \text{in } \mathcal{O}, \quad W^3|_{\partial \mathcal{O}} = -W^1|_{\partial \mathcal{O}}, \\ \|\nabla W^3\|_{L^2(\mathcal{O})} \le C \|W^1\|_{H^{1/2}(\partial \mathcal{O})} \le C \|W^1\|_{H^1(\mathcal{O})}. $$

Finally, $W = W^1 + W^2 + W^3$ fulfills all requirements, which concludes the proof of the lemma.
Troisième partie

Problèmes optimaux associés au
déplacement d’un micro-nageur
<figure><img src="image_36.png" /></figure>
# Chapitre 5

## Optimal strokes for driftless swimmers : a geometrical approach

This work is done in collaboration with T. Chambrion and A. Munnier. The aim of this study is to provide a general framework to examine optimal controllability of driftless swimmer. We focus on the analyze on the optimal strokes i.e., periodic shape changes. More precisely, we prove the existence of optimal strokes, minimizing or maximizing various cost functionals (related to the energy of the system, the efficiency, the time). Then, we demonstrate some properties of the optimal strokes. In particular, we show how the optimal controls corresponding to different cost functionals can actually be deduced one from the others. Finally, we analyze the regularity and the monotony of the value functions. We prove that the cost *increases* along with the covered distance. Numerical simulations are presented on a particular swimmer which has an explicit dynamics.
## 5.1 Introduction

Understanding the mechanics of swimming has been an issue in Mathematical Physics for a long time. Aside from improving the academic understanding of locomotion in fluid, this interest growth from the observation that fish and aquatic mammals evolved swimming capabilities far superior to those achieved by human technology and consequently provide an attractive model for the design of biomimetic robots.

Significant contributions to this matter are due to Taylor [84], J. Lighthill [54], E.M. Purcell [70] and T. Y. Wu [93].

Among the many models available in the literature, let us focus on those for which the Reynolds number of the fluid is either very low or very high. The main interest of these cases lies in the fact that the dynamics governing the fluid-swimmer system are simple enough to allow theoretical results to be proved. Theses two cases are usually referred to as “driftless models”. The first one for which the fluid is assumed to be very viscous is called “resistive model”. It is relevant for microswimmers (like microorganisms) and consists in neglecting the inertial effects in the modeling. The second one, called “reactive model”, is obtained by neglecting rather the viscous forces and is supposed to be relevant for swimmers with elongated bodies (like eels). Surprisingly, the dynamics are very close for both cases and their study fall under the same general abstract framework.

The well-posedness of the system of equations for these models was established for instance in [32], [22] and [59] in a Stokesian flow and in [27] in a perfect fluid. The controllability is addressed in [27] and [59] where the authors prove the generic controllability of 3D driftless swimmers in a perfect and Stokesian flow respectively. An earlier result was established in [8] and next improved in [11], in which 3D three or four-sphere mechanisms are shown to be controllable.

To our knowledge, still few theoretical studies have been conducted about optimal swimming (although more numerical approach of the problem are many). In [60], J. Loheac et al. are interested in optimizing the swimming of a 3D slightly deformable sphere in order to minimize its displacement time. In [7], the authors describe an algorithm allowing optimizing the strokes for a three-sphere swimmer, based on the theory of calculus of variations.

### 5.1.1 Contribution

The aim of this work is to provide a general framework to study optimal controllability of driftless swimmers. In particular, every aforementioned paper falls within this framework. After recalling minimal hypotheses ensuring the controllability of the system under consideration, we shall focus on the study of optimal strokes i.e. periodic shape changes. More precisely, we will be interested in the following points:

* Existence of optimal strokes, minimizing or maximizing various cost functionals (related to the energy of the system, the efficiency, the time). As in [60], contraints on the state of the system are taken into account (for instance, the deformations are required to be *not too large*).

* Qualitative properties of the optimal strokes (or, differently stated, of the corresponding optimal controls). In particular, we will show how the optimal controls corresponding to different cost functionals can actually be deduced one from the others.

* Regularity and monotony of the value functions (does the cost increases along with the covered distance?).

Most of the proofs rely on the following arguments:

* The analyticity of the system;
* The Riemannian and sub-Riemannian underlying structure.

For pedagogical purposes, the results will be applied to the toy model introduced in [27] and [65]. The main interest of this model, dealing with a swimmer in a 2D potential flow, is that the governing equations, although not trivial, can be made fully explicit by means of complex calculus.

## 5.1.2 Abstract Framework and Notation

We introduce in this subsection the general framework of our study. In the sequel, we call swimmer any 5-uple $\mathcal{S} = (\mathcal{S}, \mathbf{g}, \mathbf{Q}^S, \mathbf{s}^\dagger, \mathcal{L})$, where:

* $(\mathcal{S}, \mathbf{g})$ is a $N$-dimensional ($N \ge 1$), connected, analytic manifold endowed with an analytic Riemannian structure $\mathbf{g}$. Every element $\mathbf{s}$ of $\mathcal{S}$ stands for a possible shape of the swimmer. The shape changes of the swimmer over a time interval $[0, T]$ will be described by a function $\mathbf{s} : [0, T] \mapsto \mathbf{s}(t) \in \mathcal{S}$.

* The metric $\mathbf{g}$ will be used to measure the cost required to achieve this shape change. The cost of a shape change $\mathbf{s} : [0, T] \mapsto \mathbf{s}(t) \in \mathcal{S}$ could be, for instance, the length of the curve parameterized by the function $\mathbf{s}$, i.e.

$$ \int_{0}^{T} \sqrt{\mathbf{g}_{\mathbf{s}(t)}(\dot{\mathbf{s}}(t), \dot{\mathbf{s}}(t))} dt, \qquad (5.1a) $$

or something more energy-like, usually called the action:

$$ \frac{1}{2} \int_{0}^{T} \mathbf{g}_{\mathbf{s}(t)}(\dot{\mathbf{s}}(t), \dot{\mathbf{s}}(t)) dt. \qquad (5.1b) $$

* The reference shape $\mathbf{s}^\dagger$ is a point of $\mathcal{S}$ which could be thought of as the natural shape of the swimmer, when it is at rest for instance. It will be the starting point for every shape change we will consider.

* The mapping $\mathbf{Q}^S : TS \to \mathbb{R}^n$ is an analytic vector valued 1-form. It accounts for the physical constraints that every shape change has to satisfy to physically make sense. Let us be more specific:

**Definition 5.1.1** An admissible shape change is any absolutely continuous curve $\mathbf{s} : [0, T] \to \mathcal{S}$, with essentially bounded first derivative, and which satisfies for almost every time,

$$ \mathbf{Q}_{\mathbf{s}(t)}^{S} \dot{\mathbf{s}}(t) = 0. \qquad (5.2) $$

This last identity means that for a given shape (i.e. a given element of $\mathcal{S}$), not every direction on $\mathcal{S}$ is admissible. For instance, by self-deforming, the swimmer will not be allowed to modify the position of its center of mass.

Among admissible shape changes, we will mostly focus on strokes:

**Definition 5.1.2** An admissible shape change $\mathbf{s} : [0, T] \mapsto \mathcal{S}$ will be termed a stroke if $\mathbf{s}(0) = \mathbf{s}(T)$.

* We are only interested in the motion of the swimmer in one given direction. The displacement in this direction is measured thanks to the analytic differential 1-form $\mathcal{L}$ on $\mathcal{S}$. When undergoing the admissible shape change $\mathbf{s} : [0, T] \to \mathcal{S}$, the displacement of the swimmer is given by:

$$ \int_{0}^{T} \mathcal{L}_{\mathbf{s}(t)}(\dot{\mathbf{s}}(t)) dt. \qquad (5.3) $$
Most of our results will rest on the following elementary but fundamental observation:

**Remark 5.1.3** The constraint (5.2) as well as the quantities (5.1a) and (5.3) are time repa-rameterization invariant. They depend only on the oriented curve $\Gamma \subset S$, a parameterization of which being $s: [0, T] \to S$.

The famous *Scallop Theorem* (see for instance [70]) can be seen as a straightforward consequence of this remark. Indeed, it states that if the shape change is nothing more than a parameterization back and forth of a curve on $S$, then the resulting displacement is null.

### 5.1.3 Outline and Main Achievements

In Section 5.2, we show that two important cases of locomotion in a fluid (namely the locomotion of a single swimmer in an infinite extent of fluid at infinite and zero Reynolds number) fit within the framework of Section 5.1.2. In Section 5.3, we establish minimal hypotheses (based on the computation of Lie brackets) ensuring the controllability of the system. Section 5.4 is dedicated to the study of optimal strokes. After stating 5 classical optimization problems, we show that every one admits minimizers or maximizers. We prove that most of these problems are actually equivalent (for instance, it is completely equivalent to minimize the time, as in [60] and to minimize the efficiency as in [8]). Then we focus on two values functions: the first one associates to every covered distance the minimal cost necessary to achieve this displacement and the other associates to every given cost, the maximum distance that can be covered with no greater cost. Among others properties, we prove the continuity and study the monotony of these functions. In Section 5.5, we restrict our study to the case where the shape manifold $S$ is 2 dimensional. We show that the stroke optimization problem turns into an isoperimetric problem on the shape manifold. The sub-Riemannian structure becomes a *contact* sub-Riemannian structure and applying results from [1], we give a precise description of all the “small” optimal strokes. Some numerics applications on the example of swimmer in a potential flow are displayed in the last section.

## 5.2 Modeling

In this Section we aim to establish the dynamics governing the motion of low and high Reynolds numbers swimmers and show that they fall within the abstract framework introduced in Section 5.1.2.

Our purpose it to highlight that, although the properties of the fluid are different in both cases, the equations of motion turn out to have the exact same general form. The modeling is carried out in 3D, the 2D case being similar.

We assume that the swimmer is alone in the fluid and that the fluid-swimmer system fills the whole space. The buoyant force is not taken into account.

We consider a Galilean fixed frame ($\mathbf{E}_1, \mathbf{E}_2, \mathbf{E}_3$) and an attached coordinate system $\mathcal{R}_f := (O, \mathbf{E}_1, \mathbf{E}_2, \mathbf{E}_3)$ where $O$ is a fixed point of $\mathbb{R}^3$. In $\mathcal{R}_f$, coordinates will be written with capital letters (as $X := (X_1, X_2, X_3)$). Since we are interested in seeking optimal strokes, we will consider only shape changes that make the center of mass of the swimmer remains on the $X_1$-axis. Thus, the swimmer is compelled to swim along a straight line.

### Kinematics

We denote by $\mathbf{r}$ the center of mass of the swimmer (lying on the $X_1$-axis) and we consider the coordinate system $\mathcal{R}_m := (\mathbf{r}, \mathbf{E}_1, \mathbf{E}_2, \mathbf{E}_3)$, attached to the swimmer.

We assume that every possible shape of the swimmer, when described in $\mathcal{R}_m$, can be characterized by a so-called shape variable $\mathbf{s}$ belonging to some connected analytic hypersurface $S$ of $\mathbb{R}^{N+1}$ (for some integer $N \ge 1$).
Thus, we denote by $\mathcal{A}_m(\mathbf{s})$ the domain of $\mathbb{R}^3$ occupied by the swimmer, in the coordinate system $\mathcal{R}_m$, and hence $\mathcal{A}_f(\mathbf{s}) = \mathbf{r} + \mathcal{A}_m(\mathbf{s})$ is the same domain expressed in $\mathcal{R}_f$. For every $\mathbf{s} \in \mathcal{S}$, the set $\mathcal{A}_m(\mathbf{s})$ is the image of the unit ball $B$ by a $C^1$ diffeomorphism $\chi(\mathbf{s}, \cdot)$ depending on the parameter $\mathbf{s}$. Knowing every diffeomorphism $\chi(\mathbf{s}, \cdot)$ for every shape variable $\mathbf{s}$, the shape changes over a time interval $[0, T]$ can be merely described by means of a (shape) function:

$$t \in [0, T] \mapsto \mathbf{s}(t) \in \mathcal{S}.$$

The Eulerian velocity $\mathbf{W}$ at any point $X \in \mathcal{A}_f(\mathbf{s})$ of the swimmer is the sum of the rigid velocity $\dot{\mathbf{r}} := \dot{r}\mathbf{E}_1(\dot{r} \in \mathbb{R})$ and the velocity of deformation

$$\mathbf{W}_d(\mathbf{s}, \dot{\mathbf{s}}, X) := \nabla_{\mathbf{s}}\chi(\mathbf{s}, \chi(\mathbf{s}, (X - \mathbf{r}))^{-1}) \cdot \dot{\mathbf{s}}.$$

Thus, we get:

$$\mathbf{W} = \dot{\mathbf{r}} + \mathbf{W}_d \quad \text{in } \mathcal{A}_f(\mathbf{s}).$$

In the coordinate system $\mathcal{R}_m$, this equality turns out to be:

$$\mathbf{w} = \dot{\mathbf{r}} + \mathbf{w}_d \quad \text{in } \mathcal{A}_m(\mathbf{s}),$$

where, for every $x \in \mathcal{A}_m(\mathbf{s})$ we have set $\mathbf{w}(x) := \mathbf{W}(x + \mathbf{r})$ and $\mathbf{w}_d(\mathbf{s}, \dot{\mathbf{s}}, x) := \mathbf{W}_d(\mathbf{s}, \dot{\mathbf{s}}, x + \mathbf{r})$.

## Dynamics

In $\mathcal{R}_m$, the density $\varrho(\mathbf{s}, \cdot)$ of the body can be deduced from a given constant density $\varrho_0 > 0$, defined in $B$, according to the conservation of mass principle:

$$\varrho(\mathbf{s}, \chi(\mathbf{s}, x)) = \frac{\varrho_0}{|\det \nabla_x \chi(\mathbf{s}, x)|}, \quad x \in B.$$

The volume of the swimmer is $\text{Vol}(\mathbf{s}) = \int_B |\det \nabla_x \chi(\mathbf{s}, x)|dx$ and its mass $m = \varrho_0 \text{Vol}(B)$.

Although prescribed, the deformations should be interpretable as produced by some internal forces. It means that in the absence of fluid, the swimmer is not able to modify its linear momentum, which reads:

$$\int_{\mathcal{A}_m(\mathbf{s})} \varrho(\mathbf{s}, x) \mathbf{w}_d(\mathbf{s}, \dot{\mathbf{s}}, x) dx = \varrho_0 \int_B \nabla_{\mathbf{s}} \chi(\mathbf{s}, x) \cdot \dot{\mathbf{s}} dx = \mathbf{0}. \quad (5.4)$$

We introduce the $3 \times N$ matrix:

$$\mathbf{Q}^S(\mathbf{s}) := \varrho_0 \int_B \nabla_{\mathbf{s}} \chi(\mathbf{s}, x) dx, \quad (5.5)$$

and we rewrite (5.2) as:

$$\mathbf{Q}^S(\mathbf{s})\dot{\mathbf{s}} = \mathbf{0}. \quad (5.6)$$

This equation has to be understood as a constraint on the shape variable and is referred to as the self-propulsion hypothesis.

The fluid obeys, in the whole generality, to the Navier-Stokes equations for incompressible fluid:

$$\varrho_f \frac{D}{Dt} \mathbf{U}(t, X) - \nabla_X \cdot \mathbb{T}_f(\mathbf{U}, P)(t, X) = 0 \quad t > 0, X \in \mathcal{F}_f(\mathbf{s}(t)); \quad (5.7a)$$

$$\nabla_X \cdot \mathbf{U}(t, X) = 0 \quad t > 0, X \in \mathcal{F}_f(\mathbf{s}(t)); \quad (5.7b)$$

where

1. For every $\mathbf{s} \in \mathcal{S}$, $\mathcal{F}_f(\mathbf{s}) := \mathbb{R}^3 \setminus \overline{\mathcal{A}_f(\mathbf{s})}$ is the domain of the fluid;
2. $\rho_f > 0$ is the fluid's density;

3. $\mathbf{U}(t, X)$ is the Eulerian velocity of the fluid at the time $t > 0$ and the point $X \in \mathcal{F}_f(\mathbf{s}(t))$;

4. $D/Dt := \partial/\partial t + (\mathbf{U}(t, X) \cdot \nabla_X)$ is the convective derivative;

5. $\mathbb{T}_f(\mathbf{U}, P)(t, X) := \mu(\nabla_X \mathbf{U}(t, X) + \nabla_X \mathbf{U}^T(t, X)) - P(t, X)\mathbb{I}_d$ is the stress tensor, $\mu$ the dynamic viscosity and $P$ the pressure.

The rigid displacement of the body is governed by Newton's laws for the linear momentum:

$$m\ddot{r}(t) = - \int_{\partial A_f(\mathbf{s})} \mathbf{E}_1 \cdot \mathbb{T}_f(\mathbf{U}, P)(t, X) \mathbf{n} \, d\sigma_X, \quad (t > 0),$$

where $\mathbf{n}$ is the unit normal vector to $\partial A_f(\mathbf{s})$ directed towards the interior of $A_f(\mathbf{s})$.

These equations have to be supplemented with boundary conditions on $\partial A_f(\mathbf{s})$, which can be either

$$\mathbf{U} \cdot \mathbf{n} = \mathbf{W} \cdot \mathbf{n} \quad \text{on } \partial A_f(\mathbf{s}),$$

known as *slip* or Navier boundary conditions or

$$\mathbf{U} = \mathbf{W} \quad \text{on } \partial A_f(\mathbf{s}),$$

referred to as *no-slip* boundary conditions. Eventually, for the system to be well-posed, initial data are needed:

$$\mathbf{U}(0) = \mathbf{U}_0, \mathbf{r}(0) = \mathbf{r}_0 \text{ and } \dot{\mathbf{r}}(0) = \dot{\mathbf{r}}_0.$$

As mentioned in the introduction, we focus on two limit problems connecting to the value of the Reynolds number $Re := \rho \bar{U} L / \mu$ ($\bar{U}$ is the mean fluid velocity and $L$ is a characteristic linear dimension). The first case $Re \ll 1$ concerns low Reynolds swimmers like bacteria (or more generally so-called micro swimmers whose size is about $1\mu m$). For the second $Re \gg 1$, we will restrain our study to irrotational flows and so it is relevant for large animals swimming quite slowly, a case where vorticity can be neglected.

### 5.2.1 Low Reynolds number swimmers

For micro-swimmers, scientists agree that inertia (for both the fluid and the body) can be neglected in the dynamics. It means that in the modeling, we can set $\rho_0 = \rho_f = 0$. In this case, the Navier-Stokes equations reduce to the steady Stokes equations

$$-\nabla_X \cdot \mathbb{T}_f(\mathbf{U}, P)(t, X) = 0 \quad t > 0, X \in \mathcal{F}_f(\mathbf{s}(t));$$

$$\nabla_X \cdot \mathbf{U}(t, X) = 0 \quad t > 0, X \in \mathcal{F}_f(\mathbf{s}(t));$$

supplemented with no-slip boundary conditions

$$\mathbf{U} = \mathbf{W} \quad \text{on } \partial A_f(\mathbf{s}).$$

Introducing, for all $x \in \mathcal{F}_m(\mathbf{s}) := \mathbb{R}^3 \setminus \overline{\mathcal{A}_m(\mathbf{s})}$,

$$\mathbf{u}(t, x) := \mathbf{U}(t, x + \mathbf{r}(t)) \quad \text{and} \quad p(t, x) := P(t, x + \mathbf{r}(t)),$$

the equations keep the same form when expressed in the coordinate system $\mathcal{R}_m$, namely, with evident notation:

$$-\nabla_x \cdot \mathbb{T}_m(\mathbf{u}, p)(t, x) = 0 \quad t > 0, x \in \mathcal{F}_m(\mathbf{s}(t)); \quad (5.8a)$$

$$\nabla_x \cdot \mathbf{u}(t, x) = 0 \quad t > 0, x \in \mathcal{F}_m(\mathbf{s}(t)); \quad (5.8b)$$

$$\mathbf{u}(t, x) = \mathbf{w}(t, x) \quad t > 0, x \in \partial \mathcal{A}_m(\mathbf{s}(t)). \quad (5.8c)$$

From a mathematical point of view, the advantage is two folds:
1. The equations are now linear;

2. The fluid has no more proper degree of freedom. Indeed, the fluid equations simplify from an initial and boundary value problem into merely a boundary value problem. In particular, no more initial data is required.

Newton's law for linear momentum reads:

$$ \int_{\partial A_m(s)} \mathbf{E}_1 \cdot \mathbb{T}_m(\mathbf{u}, p)(t, x) \mathrm{nd}\sigma = 0. $$

The solution $(\mathbf{u}, p)$ being linear with respect to the boundary data $\mathbf{w}$ it can be decomposed as follows:

$$ \mathbf{u}(t, x) = \dot{r}(t)\mathbf{u}_r(\mathbf{s}(t), x) + \sum_{j=1}^{N} \dot{s}_j u_d^j(\mathbf{s}(t), x); $$

$$ p(t, x) = \dot{r}(t)p_r(\mathbf{s}(t), x) + \sum_{j=1}^{N} \dot{s}_j p_d^j(\mathbf{s}(t), x); \quad (t > 0, x \in \mathcal{F}_m(\mathbf{s})), $$

where we are written $\mathbf{s} = (s_1, \dots, s_N)$ in a local chart of $\mathcal{S}$ and $\dot{s} = (\dot{s}_1, \dots, \dot{s}_N)$ in the basis $(\partial_{s_1}, \dots, \partial_{s_N})$ of the tangent space $T_{\mathbf{s}}\mathcal{S}$. It entails that the stress tensor:

$$ \mathbb{T}_m(\mathbf{u}, p) := \mu(\nabla_x \mathbf{u} + \nabla_x \mathbf{u}^T) - p \mathbb{I} \mathrm{d}, $$

can also be decomposed as:

$$ \mathbb{T}_m(\mathbf{u}, p) = \dot{r}\mathbb{T}_m(\mathbf{u}_r, p_r) + \sum_{j=1}^{N} \dot{s}_j \mathbb{T}_m(\mathbf{u}_d^j, p_d^j). $$

The elementary solutions $(\mathbf{u}_r, p_r)$ and $(\mathbf{u}_d, p_d^j)$ satisfy the Stokes system (5.8a-5.8b) with the boundary conditions:

$$ \mathbf{u}_r(t, x) = \mathbf{E}_1 \qquad t > 0, x \in \partial A_m(\mathbf{s}(t)); $$

$$ \mathbf{u}_d^j(t, x) = \mathbf{w}_d(\mathbf{s}(t), \partial_{s_j}(x)) \quad t > 0, x \in \partial A_m(\mathbf{s}(t)), j = 1, \dots, N. $$

Notice that the elementary solutions $(\mathbf{u}_r, p_r)$ and $(\mathbf{u}_d^j, p_d^j)$ ($j = 1, \dots, N$) depend on the time through the shape variable $\mathbf{s}$ only. We next introduce the scalar:

$$ M^r(\mathbf{s}) := \int_{\partial A(\mathbf{s})} \mathbf{E}_1 \cdot (\mathbb{T}_m(\mathbf{u}_r, p_r)) \mathrm{nd}\sigma, $$

and the row vector $\mathbf{N}(\mathbf{s})$ whose entries are:

$$ N_j(\mathbf{s}) = \int_{\partial A(\mathbf{s})} \mathbf{E}_1 \cdot (\mathbb{T}_m(\mathbf{u}_d^j, p_d^j)) \mathrm{nd}\sigma. $$

We can rewrite Newton's laws as:

$$ \dot{r}M^r(\mathbf{s}) + \mathbf{N}(\mathbf{s})\dot{s} = 0. $$

Upon an integration by parts, we get the equivalent definition $M^r(\mathbf{s})$:

$$ M^r(\mathbf{s}) := \mu \int_{\mathcal{F}_m(\mathbf{s})} D(\mathbf{u}_r) : D(\mathbf{u}_r) \mathrm{d}x, \qquad (5.9a) $$
where $D(\mathbf{u}_r) := (\nabla_x \mathbf{u} + \nabla_x \mathbf{u}^T)$. We deduce that $M^r(\mathbf{s})$ is always positive. The same arguments for $\mathbf{N}(\mathbf{s})$ lead to the identity:

$$N_j(\mathbf{s}) = \mu \int_{\mathcal{F}_m(\mathbf{s})} D(\mathbf{u}^r) : D(\mathbf{u}_d^j) dx. \qquad (5.9b)$$

Later on, we will also need the matrix:

$$\mathbb{M}^d(\mathbf{s}) = \left( \mu \int_{\mathcal{F}_m(\mathbf{s})} D(\mathbf{u}_d^i) : D(\mathbf{u}_d^j) dx \right)_{\substack{1 \le i \le N \\ 1 \le j \le N}}. \qquad (5.9c)$$

We eventually obtain the Euler-Lagrange equation governing the rigid displacement with respect to the shape changes:

$$\dot{\mathbf{r}} = \mathcal{L}_{\mathbf{s}}\dot{\mathbf{s}} \quad t > 0, \qquad (5.10)$$

where we have set, for every $\mathbf{s} \in \mathcal{S}$:

$$\mathcal{L}_{\mathbf{s}} = -M^r(\mathbf{s})^{-1}\mathbf{N}(\mathbf{s}).$$

Considering the expressions (5.9) and (5.10), we deduce:

**Proposition 5.2.1** The dynamics of a micro-swimmer 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 $(\mathbf{u}, p)$ be an elementary solution (as defined in the modeling above) to the Stokes equations corresponding to a viscosity $\mu > 0$, then $(\mathbf{u}, (\tilde{\mu}/\mu)p)$ is the solution corresponding to the viscosity $\tilde{\mu} > 0$. Since the Euler-Lagrange equation depends only on the Eulerian velocities $\mathbf{u}$, the proof is completed. In the sequel, we will set $\mu = 1$.

The self-propelled constraint (5.2) does not make sense any longer for low Reynolds number swimmers because $\varrho_0 = 0$. However, since we still do not want the swimmer to be able to translate itself just by self-deforming, we require the shape function to satisfy (5.2) in which we define the matrix $\mathbf{Q}^S(\mathbf{s})$ by:

$$\mathbf{Q}^S(\mathbf{s}) := \int_{\Sigma} \nabla_{\mathbf{s}}\chi(\mathbf{s}, x) \, dx, \qquad (5.11)$$

where $\Sigma = \partial B$.

### 5.2.2 High Reynolds number swimmers

Assume now that the inertia is preponderant with respect to the viscous force (it is the case when $\mathrm{Re} \ll 1$). The Navier-Stokes equations (5.7) simplify into the Euler equations:

$$\varrho_f \frac{D}{Dt} \mathbf{U}(t, X) - \nabla_X \cdot \mathbb{T}_f(\mathbf{U}, P)(t, X) = 0 \quad t > 0, X \in \mathcal{F}_f(\mathbf{s}(t)); \qquad (5.12a)$$

$$\nabla_X \cdot \mathbf{U}(t, X) = 0 \quad t > 0, X \in \mathcal{F}_f(\mathbf{s}(t)); \qquad (5.12b)$$

$$\mathbf{U}(t, X) \cdot \mathbf{n} - \mathbf{W}(t, X) \cdot \mathbf{n} = 0 \quad t > 0, X \in \partial A_f(\mathbf{s}(t)). \qquad (5.12c)$$

where the stress tensor reads:

$$\mathbb{T}_f(\mathbf{U}, P)(t, X) = -P(t, X)\mathrm{Id} \quad t > 0, X \in \mathcal{F}_f(\mathbf{s}(t)).$$

Like in the preceding Subsection, we will assume that According to Kelvin's circulation theorem, if the flow is irrotational at some moment (i.e. $\nabla \times \mathbf{U} = 0$) then, it has always
been and will always remain irrotational. Hence, we can suppose that $\nabla \times \mathbf{U} = 0$ for all times and then, according to the Helmholtz decomposition, that there exists for all time $t > 0$ a potential scalar function $\Phi(t, \cdot)$ defined in $\mathcal{F}_f(\mathbf{s})$, such that

$$ \mathbf{U}(t, X) = \nabla_X \Phi(t, X) \quad t > 0, X \in \mathcal{F}_f(\mathbf{s}(t)). $$

The divergence-free condition leads to

$$ \Delta_X \Phi(t, X) = 0 \quad t > 0, X \in \mathcal{F}_f(\mathbf{s}(t)), $$

and the boundary condition reads:

$$ \partial_{\mathbf{n}} \Phi(t, X) = \mathbf{W}(t, X) \cdot \mathbf{n} \quad t > 0, X \in \partial \mathcal{A}_f(\mathbf{s}(t)). $$

The function $\varphi(t, \cdot)$ defined by:

$$ \varphi(t, x) := \Phi(t, x - \mathbf{r}) \quad t > 0, x \in \mathcal{F}_m(\mathbf{s}(t)), $$

is harmonic in $\mathcal{F}_m(\mathbf{s}(t))$ and satisfies the boundary condition:

$$ \partial_{\mathbf{n}} \varphi(t, x) = \mathbf{w}(t, x) \cdot \mathbf{n} \quad t > 0, x \in \partial \mathcal{A}_m(\mathbf{s}(t)). $$

The potential $\varphi$ is linear in $\mathbf{w}$, so it can be decomposed into

$$ \varphi(t, x) = \dot{r}\varphi_r(t, x) + \sum_{j=1}^{N} \dot{s}_j \varphi_d(t, x) \quad t > 0, x \in \mathcal{F}_m(\mathbf{s}(t)), $$

where at every moment the elementary potentials $\varphi_r(t, \cdot)$ and $\varphi_d(t, \cdot)$ are harmonics in $\mathcal{F}_m(\mathbf{s}(t))$ and satisfy the boundary conditions:

$$ \begin{aligned} \partial_{\mathbf{n}} \varphi_r(t, x) &= \mathbf{E}_1 \cdot \mathbf{n}, \\ \partial_{\mathbf{n}} \varphi_d(t, x) &= \mathbf{w}_d(\mathbf{s}(t), \partial_{s_j}, x) \quad t > 0, x \in \partial \mathcal{A}_m(\mathbf{s}(t)). \end{aligned} $$

This process is usually referred to as Kirchhoff's law. At this point, we do not invoke Newton's laws to derive the Euler-Lagrange equation but rather use the formalism of Analytic Mechanics. Both approaches (Newton's laws of Classical Mechanics and the Least Action principle of Analytic Mechanics) are equivalent, as proved in [64], but the latter is simpler and shorter.

In the absence of buoyant force, the Lagrangian function $L$ of the body-fluid system coincides with the kinetic energy:

$$ L = m \frac{1}{2} |\dot{r}|^2 + \frac{1}{2} \int_{\mathcal{A}_m(\mathbf{s})} \varrho(\mathbf{s}, x) |\mathbf{w}_d(t, x)|^2 dx + \frac{1}{2} \int_{\mathcal{F}_m(\mathbf{s})} \varrho_f |\mathbf{u}(t, x)|^2 dx. $$

In this sum, one can identify, from the left to the right: the kinetic energy of the body connecting to the rigid motion, the kinetic energy resulting from the deformations and the kinetic energy of the fluid. We can next compute that, upon a change of variables:

$$ \int_{\mathcal{A}_m(\mathbf{s})} \varrho(\mathbf{s}, x) |\mathbf{w}_d(t, x)|^2 dx = \varrho_0 \int_B |\nabla_{\mathbf{s}} \chi(\mathbf{s}, x) \cdot \dot{\mathbf{s}}|^2 dx, $$

and

$$ \int_{\mathcal{F}_m(\mathbf{s})} \varrho_f |\mathbf{u}(t, x)|^2 dx = \varrho_f \int_{\mathcal{F}_m(\mathbf{s})} |\nabla \varphi(t, x)|^2 dx. $$
It leads us to introduce the scalar:

$$M^r(\mathbf{s}) = m + \varrho_f \int_{\mathcal{F}_m(\mathbf{s})} |\nabla \varphi_r(\mathbf{s}, x)|^2 dx, \qquad (5.13a)$$

the row vector $\mathbf{N}(\mathbf{s})$ whose entries are:

$$N_j(\mathbf{s}) = \varrho_f \int_{\mathcal{F}_m(\mathbf{s})} \nabla \varphi_r(\mathbf{s}, x) \cdot \nabla \varphi_d^j(\mathbf{s}, x) dx, \quad j = 1, \dots, N, \qquad (5.13b)$$

and the matrix $\mathbf{M}^d(\mathbf{s})$:

$$\mathbf{M}^d(\mathbf{s}) = \varrho_0 \int_B \nabla_{\mathbf{s}} \chi(\mathbf{s}, x) \otimes \nabla_{\mathbf{s}} \chi(\mathbf{s}, x) dx \\ + \varrho_f \left( \int_{\mathcal{F}_m(\mathbf{s})} \nabla_{\varphi_d^j(\mathbf{s}, x)} \cdot \nabla_{\varphi_d^j(\mathbf{s}, x)} dx \right)_{\substack{1 \le i \le N \\ 1 \le j \le N}}.$$

Observe the similarity between relations (5.9) and (5.13). We can rewrite the Lagrangian function as:

$$L(\dot{\mathbf{r}}, \mathbf{s}, \dot{\mathbf{s}}) = \frac{1}{2} M^r(\mathbf{s}) |\dot{\mathbf{r}}|^2 + \dot{\mathbf{r}} \mathbf{N}(\mathbf{s}) \dot{\mathbf{s}} + \dot{\mathbf{s}} \cdot \mathbf{M}^d(\mathbf{s}) \dot{\mathbf{s}}.$$

Invoking now the Least Action principle, we claim that the Euler-Lagrange equation is:

$$\frac{d}{dt} \frac{\partial L}{\partial \dot{\mathbf{r}}} (\dot{\mathbf{r}}, \mathbf{s}, \dot{\mathbf{s}}) - \frac{\partial L}{\partial r} (\dot{\mathbf{r}}, \mathbf{s}, \dot{\mathbf{s}}) = 0,$$

which reduces to, since $L$ does not depend on $r$:

$$\frac{d}{dt} (M^r(\mathbf{s})\dot{\mathbf{r}} + \mathbf{N}(\mathbf{s})\dot{\mathbf{s}}) = 0.$$

Assuming that the impulse $M^r(\mathbf{s})\dot{\mathbf{r}} + \mathbf{N}(\mathbf{s})\dot{\mathbf{s}}$ is zero at the initial time, we get eventually for the dynamics the exact same expression as (5.10):

$$\dot{\mathbf{r}} = \mathcal{L}_{\mathbf{s}} \dot{\mathbf{s}} \quad t > 0,$$

where, for every $\mathbf{s} \in \mathcal{S}$:

$$\mathcal{L}_{\mathbf{s}} = -M^r(\mathbf{s})^{-1} \mathbf{N}(\mathbf{s}). \qquad (5.14)$$

It is easy to verify that the dynamics does not depend on $\varrho_0$ and $\varrho_f$ independently but only on the relative density $\varrho_0/\varrho_f$, which is assumed to be equal to 1 in the sequel.

### 5.2.3 Examples of cost functionals

For low Reynolds number swimmers, a classical notion of swimming efficiency (see [56] and [8]) is defined as the inverse of the ratio between the average power expended by the swimmer during a stroke starting and ending at the shape $\mathbf{s}^\dagger$ and the power that an external force would spend to translate the system rigidly at the same average speed:

$$\text{Eff}^{-1} := \frac{\frac{1}{T} \int_{0}^{T} \left( \int_{\partial \mathcal{A}_m(\mathbf{s})} \mathbf{F}(\mathbf{s}, \dot{\mathbf{s}}, \dot{\mathbf{r}}, x) \cdot \mathbf{u}(\mathbf{s}, \dot{\mathbf{s}}, \dot{\mathbf{r}}, x) d\sigma_x \right) dt}{\bar{\mathbf{v}} \cdot \int_{\partial \mathcal{A}_m(\mathbf{s})} \mathbf{F}(\mathbf{s}^\dagger, \mathbf{0}, \bar{\mathbf{v}}, x) d\sigma_x},$$

where

$$\mathbf{F}(\mathbf{s}, \dot{\mathbf{s}}, \dot{\mathbf{r}}, x) := \left( \dot{\mathbf{r}} \mathbb{T}_m(\mathbf{u}_r, p_r)(\mathbf{s}, x) + \sum_{j=1}^{N} \dot{s}_j \mathbb{T}_m(\mathbf{u}_j^d, p_d^j)(\mathbf{s}, x) \right) \mathbf{n}$$
is the force in the normal direction exerted by the fluid at the point $x$ of the surface of the swimmer, with shape $\mathbf{s}$, shape change rate $\dot{\mathbf{s}}$ and rigid velocity $\dot{r}$. In the same way:

$$ \mathbf{u}(\mathbf{s}, \dot{\mathbf{s}}, x) := \dot{r}\mathbf{E}_1 + \sum_{j=1}^{N} \dot{s}_j \mathbf{w}_d(\mathbf{s}, \partial_{s_j}, x), $$

is the velocity of the swimmer. Eventually $\bar{\mathbf{v}}$ is the average speed:

$$ \bar{\mathbf{v}} := \left( \frac{1}{T} \int_{0}^{T} \dot{r} dt \right) \mathbf{E}_{1}. $$

With the notation (5.9), the efficiency can be rewritten as:

$$ \text{Eff}^{-1} := \frac{\frac{1}{T} \int_{0}^{T} (M^{r}(\mathbf{s})|\dot{r}|^2 + \dot{r}\mathbf{N}(\mathbf{s})\dot{\mathbf{s}} + \dot{\mathbf{s}} \cdot \mathbf{M}^{d}(\mathbf{s})\dot{\mathbf{s}}) dt}{|\bar{\mathbf{v}}|^2 M^{r}(\mathbf{s}^{\dagger})}. \quad (5.15) $$

It can easily be verified that:

$$ M^r(\mathbf{s})|\dot{r}|^2 + \dot{r}\mathbf{N}(\mathbf{s})\dot{\mathbf{s}} + \dot{\mathbf{s}} \cdot \mathbf{M}^d(\mathbf{s})\dot{\mathbf{s}} = \int_{\mathcal{F}_m(\mathbf{s})} D(\mathbf{u}, p) : D(\mathbf{u}, p) dx > 0, $$

where $(\mathbf{u}, p)$ is the solution to the Stokes system (5.8).

For high Reynolds number swimmers, we can choose the same expression (5.15) for the efficiency, in which we use the definitions (5.13). In this case, the efficiency is the inverse of the ratio between the mean energy expended by the swimmer divided by the energy required to translate rigidly the swimmer at the same average speed.

Taking into account the dynamics and replacing $\dot{r}$ by $-M^r(\mathbf{s})^{-1}\mathbf{N}(\mathbf{s})\dot{\mathbf{s}}$ in (5.15), it leads us to consider on $T_s S$ the following scalar product:

$$ \mathbf{g}_s(\dot{\mathbf{s}}_1, \dot{\mathbf{s}}_2) = \dot{\mathbf{s}}_1 \cdot \left( \mathbf{M}^d(\mathbf{s}) - \frac{\mathbf{N}(\mathbf{s}) \otimes \mathbf{N}(\mathbf{s})}{M^r(\mathbf{s})} \right) \dot{\mathbf{s}}_2, \quad (\dot{\mathbf{s}}_1, \dot{\mathbf{s}}_2 \in T_s S). \quad (5.16) $$

According to the abstract framework introduced in Section 5.1.2, the cost of an admissible shape change $\mathbf{s} : [0, T] \mapsto S$ will be:

$$ \frac{1}{2} \int_{0}^{T} \mathbf{g}_{s(t)}(\dot{\mathbf{s}}(t), \dot{\mathbf{s}}(t)) dt. \quad (5.17) $$

### 5.2.4 Regularity results

In Section 5.1.2, the manifold $S$ and the differential forms are all of them assumed to be analytic. The following Lemma ensures that, under a simple hypothesis, this regularity is ensured for swimmers in a perfect fluid and Stokesian swimmers.

We denote by $M(N_1, N_2)$ the Euclidian space of the $N_1 \times N_2$ matrices and we claim:

**Lemma 5.2.2** Assume that the map $\mathbf{s} \in S \mapsto \chi(\mathbf{s}, \cdot) \in C^1(\bar{B}, \mathbb{R}^3)$ is analytic (we refer to [90] for the definitions and the properties of analytic functions valued in Banach spaces), then for both cases (low and high Reynolds number swimmers) the maps $\mathbf{s} \in S \mapsto \mathbf{Q}^S(\mathbf{s}) \in M(3, N)$, $\mathbf{s} \in S \mapsto M^r(\mathbf{s}) \in \mathbb{R}$, $\mathbf{s} \in S \mapsto \mathbf{N}(\mathbf{s}) \in \mathbb{R}^N$ and $\mathbf{s} \in S \mapsto \mathbf{M}^d(\mathbf{s}) \in M(N, N)$ are analytic.

The proofs can be found in [28] for swimmers in a perfect fluid and in [59] for Stokesian swimmers.
<figure><img src="image_37.png" /><figcaption>Figure 5.1: Examples of manifolds $S_\mu$ for different values of $\mu$. For small values of $\mu$, $S_\mu$ turns out to be merely the surface of the ellipsoid since it is entirely included in the unit ball.</figcaption></figure>

### 5.2.5 An example of swimmer in a potential flow

All along the paper, we will illustrate our purpose by applying our approach to a concrete example. We have chosen to deal with a simplified version of a 2D swimmer in a perfect fluid introduced in [27] and improved in [65].

**Shape changes**

Recall that in 2D, at every time, $A_m(s)$ is the image of the unit disk $D$ by a diffeomorphism $\chi(s, \cdot)$ depending on the parameter $s$, and whose form (with complex notation) is:

$$ \chi(s, z) = z + s_1 \bar{z} + s_2 \bar{z}^2 + s_3 \bar{z}^3, \quad (z \in \mathbb{C}, s = (s_1, s_2, s_3) \in \mathbb{R}^3). \quad (5.18) $$

We define the following norm in $\mathbb{R}^3$:

$$ \|s\|_S = \sup_{z \in \partial D} |s_1 + 2s_2z + 3s_3z^2|, \quad (s \in \mathbb{R}^3). $$

and we claim (see [27] for details):

**Lemma 5.2.3** 1. The mapping $\chi(s, \cdot)$ is a $C^\infty$ diffeomorphism from the unit ball $D$ onto its image $A_m(s)$ if and only if $\|s\|_S < 1$.

2. The measure of the area of $A_m(s)$ is $\pi(1 - s_1^2 + 2s_2^2 + 3s_3^2)$.

Since we want both conditions (i) the mapping $\chi(s, \cdot)$ is a diffeomorphism and (ii) the area of $A_m(s, \cdot)$ is of constant (and nonzero) measure, to be fulfilled, we introduce for every $0 < \mu < 1$ the set (see Fig. 5.1):

$$ S_{\mu} = \{s \in \mathbb{R}^3 : \|s\|_{S} < 1 \text{ and } s_1^2 + 2s_2^2 + 3s_3^2 = \mu^2\}. $$

For any $0 < \mu < 1$, $S_\mu$ is a 2D analytic submanifold of $\mathbb{R}^3$. It consists in the parts the ellipsoid surface $s_1^2 + 2s_2^2 + 3s_3^2 = \mu^2$ lying inside the unit ball $\|s\|_S < 1$.

To simplify, we will consider in the following that $\mu$ is small enough. In this case, the ellipsoid $s_1^2 + 2s_2^2 + 3s_3^2 \le \mu^2$ is included in the unit ball $\|s\|_S < 1$, and hence $S_\mu$ reduces merely to the surface of the ellipsoid.

As a conclusion, once $\mu$ (and therefore the measure of the swimmer) has be chosen and fixed, the shape changes over a time interval $[0, T]$ are described by means of a function:

$$ t \in [0, T] \mapsto s(t) \in S_{\mu}. $$
<figure><img src="image_38.png" /><figcaption>Figure 5.2: Some points on the ellipsoid $S_μ$ ($μ = 0.3$) and the corresponding shapes of the swimmer.</figcaption></figure>

By direct computations, one verifies that the self-propelled constraints (5.2), ensuring that the swimmer can not modified its linear momentum by self-deforming, are automatically satisfied in this simplified case.

In Fig. 5.2, we have pictured some points of the ellipsoid and the corresponding shapes for the swimmer.

Using the conformal mapping

$$ \phi(\mathbf{s}, z) := z + \frac{s_1}{z} + \frac{s_2}{z^2} + \frac{s_3}{z^3}, \quad (z \in \mathbb{C} \setminus \bar{D}, \mathbf{s} = (s_1, s_2, s_3) \in S_\mu), $$

which maps the exterior of the unit disk onto the fluid domain $\mathcal{F}_m(\mathbf{s})$, we can compute explicitly the elementary kirchhoff's potentials $\phi_r(\mathbf{s}, \cdot)$ and $\phi_d(\mathbf{s}, \cdot)$ (again, we refer to [27] or [65] for the details). We finally get the following expressions for the mass matrices introduced in (5.13):

$$ M^r(\mathbf{s}) = 2 - 2s_1 \qquad (5.19a) $$

$$ \mathbf{N}(\mathbf{s}) = \begin{bmatrix} -3s_2 + 2s_2s_1 + 3s_2s_3 & -s_1 - 4s_3 + s_1^2 + 3s_1s_3 & -2s_2 + 3s_2s_1 \end{bmatrix} \qquad (5.19b) $$

$$ \mathbf{M}^d(\mathbf{s}) = \begin{bmatrix} 4s_2^2 - 3s_3 + \frac{9}{2}s_3^2 + 1 & s_1^2 + 6s_1s_3 + 9s_3^2 + \frac{2}{3} & 4s_2^2 - \frac{1}{2}s_1 + \frac{3}{2}s_1s_3 \\ 2s_1s_2 + 6s_2s_3 & 2s_1s_2 + 6s_2s_3 & 2s_1s_2 + 6s_2s_3 \\ 4s_2^2 - \frac{1}{2}s_1 + \frac{3}{2}s_1s_3 & 4s_2^2 + \frac{1}{2}s_1^2 + \frac{1}{2} \end{bmatrix} \qquad (5.19c) $$

The 1-form $\mathcal{L}$ defined in (5.14) as well as the scalar product $\mathbf{g}$ defined in (5.16) and the cost functional (5.17) can now be explicitly computed. Instead of writing out their (complicated) expressions, we compute rather the Ricci-curvature induced by $\mathbf{g}$ on the ellipsoid. The result is displayed on Fig 5.3.
<figure><img src="image_39.png" /><figcaption>Figure 5.3: The Ricci curvature corresponding to the Riemannian metric g defined in (5.16), on the ellipsoid.</figcaption></figure>
## 5.3 Controllability

### 5.3.1 General results

Let us consider an abstract swimmer $\mathcal{S} = (\mathcal{S}, \mathbf{g}, \mathbf{Q}^S, \mathbf{s}^\dagger, \mathcal{L})$ as described in Section 5.1.2, denote, for every $\mathbf{s} \in \mathcal{S}$:

$$ \ker \mathbf{Q}^S = \Delta_{\mathbf{s}}^S \subset T_{\mathbf{s}}\mathcal{S}. $$

Assume that the dimension of $\Delta_{\mathbf{s}}^S$ is not always zero (otherwise, it would mean that there is no shape change satisfying the self-propelled constraints (5.2)). For every $\mathbf{s} \in \mathcal{S}$, we denote by $\{\mathbf{X}_1(\mathbf{s}), \dots, \mathbf{X}_p(\mathbf{s})\}$ ($p > 0$) a spanning set of $\Delta_{\mathbf{s}}^S$ where the vector fields $\mathbf{s} \in \mathcal{S} \mapsto \mathbf{X}_j(\mathbf{s}) \in T_{\mathbf{s}}\mathcal{S}$ ($1 \le j \le p$) are assumed to be analytic. We denote $\mathcal{X} := \{\mathbf{X}_j, j = 1, \dots, p\} \subset \mathcal{X}(\mathcal{S})$ and we shall call $\mathcal{X}$ an analytic spanning family of the distribution $\Delta^S$. Notice that is general, the spanning family $\mathcal{X}$ cannot be required to be a basis of $\Delta_{\mathbf{s}}^S$ at every point $\mathbf{s}$ of $\mathcal{S}$ because the analytic vector fields $\mathbf{X}_j$ cannot be prevented from vanishing at some point of the manifold (see for instance the example in Subsection 5.3.2).

**Proposition 5.3.1** *Any absolutely continuous function with essentially bounded derivatives $\mathbf{s} : [0, T] \to \mathcal{S}$ is an admissible shape change if and only if it is solution (in the sense of Carathéodory) of a Cauchy problem*

$$ \dot{\mathbf{s}}(t) = \sum_{j=1}^{p} u_j(t) \mathbf{X}_j(\mathbf{s}(t)) \quad (t > 0), \qquad (5.20a) $$

$$ \mathbf{s}(0) = \mathbf{s}^{\dagger}, \qquad (5.20b) $$

for some $\mathbf{u} = (u_1, \dots, u_p) \in L^\infty([0, T], \mathbb{R}^p)$.

**Proof.** The proof is elementary: For every admissible shape change, the function $\mathbf{u} = (u_1, \dots, u_p) \in L^\infty([0, T], \mathbb{R}^p)$ gives the coordinates of $\dot{\mathbf{s}}$ in the spanning family $\mathcal{X}(\mathbf{s})$.

System (5.20) allows associating with every measurable function $\mathbf{u} \in L^\infty([0, T], \mathbb{R}^p)$ an admissible shape change, at least for times small enough.

We define $\mathcal{M}$ as being the analytic $(N+1)$-dimensional manifold $\mathcal{S} \times \mathbb{R}$. Then, we introduce the projectors $\pi_{\mathcal{S}}$ and $\pi_{\mathbb{R}}$ by:

$$ \pi_{\mathcal{S}}: \begin{array}{rcl} \mathcal{M} & \to & \mathcal{S} \\ (\mathbf{s}, r) & \mapsto & \pi_{\mathcal{S}}(\boldsymbol{\xi}) = \mathbf{s} \end{array} \quad \text{and} \quad \pi_{\mathbb{R}}: \begin{array}{rcl} \mathcal{M} & \to & \mathbb{R} \\ (\mathbf{s}, r) & \mapsto & \pi_{\mathbb{R}}(\boldsymbol{\xi}) = r. \end{array} $$

On $\mathcal{M}$, we define the analytic vectors fields:

$$ \mathbf{Z}_j(\boldsymbol{\xi}) := \begin{pmatrix} \mathbf{X}_j(\pi_{\mathcal{S}}\boldsymbol{\xi}) \\ \mathcal{L}_{\mathbf{s}}\mathbf{X}_j(\pi_{\mathcal{S}}\boldsymbol{\xi}) \end{pmatrix}, \quad (j = 1, \dots, p), \qquad (5.21) $$

we denote $\mathcal{Z} := \{\mathbf{Y}_j, j = 1, \dots, p\} \subset \mathcal{X}(\mathcal{M})$ and we define the distribution

$$ \Delta_{\boldsymbol{\xi}}^{\mathcal{M}} = \operatorname{span} \mathcal{Z}(\boldsymbol{\xi}), \quad \boldsymbol{\xi} \in \mathcal{M}. $$

System (5.20) and the dynamics (5.10) can now be gathered as a unique dynamical system on $\mathcal{M}$:

$$ \dot{\boldsymbol{\xi}}(t) = \sum_{j=1}^{p} u_j(t) \mathbf{Z}_j(\boldsymbol{\xi}(t)) \quad (t > 0), \qquad (5.22a) $$

$$ \boldsymbol{\xi}(0) = \boldsymbol{\xi}^{\dagger} \qquad (5.22b) $$

where $\boldsymbol{\xi}^{\dagger} = (\mathbf{s}^{\dagger}, 0)$.
**Remark 5.3.2** The choice of the initial condition $(\mathbf{s}^\dagger, 0)$ (and not $(\mathbf{s}^\dagger, r_0)$ for some $r_0 \neq 0$) is physically irrelevant, since the vector fields $\mathbf{Z}_j$ ($1 \le j \le p$), and hence the dynamics (5.22), do not depend upon the $\mathbb{R}$ component of the variable $\boldsymbol{\xi}$.

**Definition 5.3.3** For every positive time $T$, every swimmer $\mathcal{G} = (S, \mathbf{g}, \mathbf{Q}^S, \mathbf{s}^\dagger, \mathcal{L})$ and every analytic spanning family $\mathcal{X}$ of $\Delta^S$, we denote by $\mathcal{U}_{\mathcal{G}}^{\mathcal{X}}(T)$ the set of all the controls $\mathbf{u} = (u_j)_{1 \le j \le p} \in L^\infty([0, T], \mathbb{R}^p)$ for which the solution of (5.20) (and hence of (5.22)) is defined on $[0, T]$.

For any given control $\mathbf{u} \in \mathcal{U}_{\mathcal{G}}^{\mathcal{X}}(T)$, we denote

$$t \in [0, T] \mapsto \xi_{\mathcal{G}}^{\mathcal{X}}(t, \mathbf{u}) \in \mathcal{M},$$

the solution to (5.22) with control $\mathbf{u}$.

**Remark 5.3.4** According to this notation, we have:

$$\pi_{\mathbb{R}} \xi_{\mathcal{G}}^{\mathcal{X}}(T, \mathbf{u}) = \int_{0}^{T} \mathcal{L}_{\pi_S \xi_{\mathcal{G}}^{\mathcal{X}}(t, \mathbf{u})} \frac{d}{dt} \pi_S \xi_{\mathcal{G}}^{\mathcal{X}}(t, \mathbf{u}) dt.$$

One hypothesis required in order to ensure that the swimmer is controllable is that $\mathcal{X}$ is bracket generating on $S$. Observe that this condition does not depend on the particular choice of the spanning family $\mathcal{X}$ but only on the distribution $\Delta^S$ and hence on the vector valued 1-form $\mathbf{Q}^S$. It can be easily verified that if $\mathcal{X}$ and $\mathcal{X}'$ are two smooth spanning families of $\Delta^S$, then for every $\mathbf{s} \in S$:

$$\mathrm{Lie}_s \mathcal{X} = \mathrm{Lie}_s \mathcal{X}' = \mathrm{Lie}_s \Delta^S.$$

Taking into account this observation, we define:

**Hypothesis 5.3.5** The swimmer $\mathcal{G} = (S, \mathbf{g}, \mathbf{Q}^S, \mathbf{s}^\dagger, \mathcal{L})$ is such that

1. The distribution $\Delta^S$ is bracket generating on $S$, i.e.

$$\dim \mathrm{Lie}_s \Delta^S = \dim S, \quad \forall \mathbf{s} \in S;$$

2. There exists $\boldsymbol{\xi} \in \mathcal{M}$ such that $\dim \mathrm{Lie}_s \Delta^{\mathcal{M}} = \dim \mathcal{M}$.

**Lemma 5.3.6** Hypothesis 5.3.5 leads to:

$$\dim \mathrm{Lie}_s \Delta^{\mathcal{M}} = \dim \mathcal{M}, \quad \forall \boldsymbol{\xi} \in \mathcal{M}.$$

**Proof.** Let $\boldsymbol{\xi}^* = (\mathbf{s}^*, 0)$ be such that $\dim \mathrm{Lie}_{\boldsymbol{\xi}^*} \Delta^{\mathcal{M}} = \dim \mathcal{M}$. As already mentioned, the choice of 0 for the $\mathbb{R}$ component of $\boldsymbol{\xi}^*$ is irrelevant regarding the Lie algebra $\mathrm{Lie}_{\boldsymbol{\xi}^*} \Delta^{\mathcal{M}} = \mathrm{Lie}_{\boldsymbol{\xi}^*} \mathcal{Z}$ since, for every $j = 1, \dots, p$, $\mathbf{Z}_j(\boldsymbol{\xi}) = \mathbf{Z}_j(\pi_S \boldsymbol{\xi})$. Consider now any $\boldsymbol{\xi} = (\mathbf{s}, r) \in \mathcal{M}$ and denote $\mathcal{O}(\boldsymbol{\xi})$ the orbit of $\mathcal{Z}$ through $\boldsymbol{\xi}$. Since $\Delta^S$ is bracket generating on $S$, Rashevsky-Chow Theorem ensures that for any $T > 0$, there exists a control $\mathbf{u} \in L^\infty([0, T], \mathbb{R}^p)$ such that the solution to the EDO (5.20) with Cauchy data $\mathbf{s}$ is equal to $\mathbf{s}^*$ at the final time $T$. Using this control in EDO (5.22) with Cauchy data $\boldsymbol{\xi} = (\mathbf{s}, r)$, we deduce that the solution reaches a point $\tilde{\boldsymbol{\xi}}^* = (\mathbf{s}^*, r^*)$ at the time $T$ for some $r^* \in \mathbb{R}$ ($\boldsymbol{\xi}$ and $\tilde{\boldsymbol{\xi}}^*$ are both in $\mathcal{O}(\boldsymbol{\xi})$). But since $\pi_S \tilde{\boldsymbol{\xi}}^* = \pi_S \boldsymbol{\xi}^*$, we have the equality $\dim \mathrm{Lie}_{\boldsymbol{\xi}^*} \mathcal{Z} = \dim \mathrm{Lie}_{\boldsymbol{\xi}^*} \mathcal{Z} = \dim \mathcal{M}$. According to the Orbit Theorem, the dimension of the Lie algebra of $\mathcal{Z}$ is constant on $\mathcal{O}(\boldsymbol{\xi})$ and hence we have also $\dim \mathrm{Lie}_{\boldsymbol{\xi}^*} \mathcal{Z} = \dim \mathcal{M}$. The proof is now complete.

Assuming only, in the second point of Hypothesis 5.3.5, that the equality holds for one point of $\mathcal{M}$ may seem a somehow useless mathematical refinement. Quite the reverse, the explicit computation of $\mathrm{Lie}_{\boldsymbol{\xi}^*} \Delta^{\mathcal{M}}$ in concrete cases is often very involved and can still be hardly carry out for one particular $\boldsymbol{\xi}$ (see for instance §)].
**Definition 5.3.7** A swimmer satisfying Hypothesis 5.3.5 will be called controllable.

This definition is justified by the following Theorem:

**Theorem 5.3.8** Let $\mathcal{S}$ be a swimmer satisfying Hypothesis 5.3.5. Then, for every $T > 0$, every analytic spanning family $\mathcal{X}$ of $\Delta^S$, every $\xi^\dagger$ in $\mathcal{M}$ and every open, connected set $\mathcal{O} \subset \mathcal{M}$ containing $\xi^\dagger$ and $\xi^\ddagger$, there exists a control $\mathbf{u} \in L^\infty([0, T], \mathbb{R}^p)$ such that $\xi_{\mathcal{S}}^{\mathcal{X}}(T, \mathbf{u}) = \xi^\dagger$ and $\xi_{\mathcal{S}}^{\mathcal{X}}(t, \mathbf{u}) \in \mathcal{O}$ for every $t \in [0, T]$.

**Proof.** This is a straightforward consequence of the analytic Orbit theorem.

Notice that this theorem applies for the models (high and low Reynolds numbers swimmers) introduced in Section 5.2. For these models, controllability is ensured as soon as Hypothesis 5.3.5 is fulfilled.

Let us enter more into details for the example presented in Subsection 5.2.5.

## 5.3.2 Swimmer in a potential flow

In this case, as already mentioned before, we have $\mathcal{Q}^S = \mathbf{0}$ (the self-propelled constraints are always fulfilled) and hence $\Delta^S = TS_\mu$. We define the following vector fields which are, for every $\mathbf{s} \in \mathcal{S}_\mu$, an analytic spanning set of $T_\mathbf{s}\mathcal{S}_\mu$:

$$ \mathbf{X}_1(\mathbf{s}) := \begin{bmatrix} 3s_3(1-s_1) \\ 0 \\ s_1(s_1-1) \end{bmatrix}, \quad \mathbf{X}_2(\mathbf{s}) := \begin{bmatrix} 2s_2(1-s_1) \\ s_1(s_1-1) \\ 0 \end{bmatrix}, \quad \mathbf{X}_3(\mathbf{s}) := \begin{bmatrix} 0 \\ 3s_3(1-s_1) \\ 2s_2(s_1-1) \end{bmatrix}. $$

Notice that, for all $\mathbf{s} \in \mathcal{S}$, $2s_2\mathbf{X}_1(\mathbf{s}) - 3s_3\mathbf{X}_2(\mathbf{s}) - s_1\mathbf{X}_3(\mathbf{s}) = \mathbf{0}$. From this family of vectors, we build the vectors $\mathbf{Z}_j$ ($j = 1, 2, 3$) according to the definition (5.21) and the expressions (5.19). We get:

$$ \begin{align*} \mathbf{Z}_1(\boldsymbol{\xi}) &:= \begin{bmatrix} 3s_3(1-s_1) \\ 0 \\ s_1(s_1-1) \\ \frac{9}{2}s_2s_3 - 3s_1s_2s_3 - \frac{9}{2}s_2s_3^2 - s_1s_2 + \frac{3}{2}s_1^2s_2 \end{bmatrix} \\ \mathbf{Z}_2(\boldsymbol{\xi}) &:= \begin{bmatrix} 2s_2(1-s_1) \\ s_1(s_1-1) \\ 0 \\ 3s_2^2 - 2s_1s_2^2 - 3s_2s_3 - \frac{1}{2}s_1^2 - 2s_1s_3 + \frac{1}{2}s_1^3 + \frac{3}{2}s_1^2s_3 \end{bmatrix} \\ \mathbf{Z}_3(\boldsymbol{\xi}) &:= \begin{bmatrix} 0 \\ 3s_3(1-s_1) \\ 2s_2(s_1-1) \\ \frac{3}{2}s_1s_3 + 6s_3^3 - \frac{3}{2}s_1^2s_3 - \frac{9}{2}s_1s_3^2 - 2s_2^2 + 3s_1s_2^2 \end{bmatrix}. \end{align*} $$

Obviously, we have again that $2s_2\mathbf{Z}_1(\boldsymbol{\xi}) - 3s_3\mathbf{Z}_2(\boldsymbol{\xi}) - s_1\mathbf{Z}_3(\boldsymbol{\xi}) = \mathbf{0}$ for all $\boldsymbol{\xi} = (\mathbf{s}, \mathbf{r}) \in \mathcal{M}$.
By direct calculation, one can check that for all $\boldsymbol{\xi} \in \mathcal{M}$:

$$ [\mathbf{Z}_1(\boldsymbol{\xi}), \mathbf{Z}_2(\boldsymbol{\xi})] = \begin{bmatrix} 0 \\ 3s_3(2s_1-1)(s_1-1) \\ -2s_2(2s_1-1)(s_1-1) \\ -\frac{3}{2}s_1s_3 - 3s_1^2s_2^2 + \frac{3}{2}s_1^3s_3 + \frac{9}{2}s_1^2s_3^2 - \frac{21}{2}s_1s_3^2 - s_1^2 + 6s_3^2 + s_1^3 + 5s_1s_2^2 - 2s_2^2 \end{bmatrix}. $$

For $\boldsymbol{\xi}^\dagger := (\mu, 0, 0, 0) \in \mathcal{M}$, we have:

$$ [\mathbf{Z}_1(\boldsymbol{\xi}^\dagger), \mathbf{Z}_2(\boldsymbol{\xi}^\dagger), [\mathbf{Z}_1(\boldsymbol{\xi}^\dagger), \mathbf{Z}_2(\boldsymbol{\xi}^\dagger)]] = \mu(\mu - 1) \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & \frac{1}{2}\mu^2 & \mu^2 \end{bmatrix}, $$
and hence $\dim \text{Lie}_{\xi^{\dagger}}\{\mathbf{Z}_j, j = 1, 2, 3\} = 3$. According to Theorem 5.3.8, we deduce that our example of swimmer in a perfect fluid is controllable.

## 5.4 Seeking of Optimal Strokes

In this section, we address the main problems that we are interested in. A controllable swimmer $\mathcal{S}$ being given, and a cost being chosen (among those presented in (5.1)), what is the best possible stroke? By best, it is understood that the swimmer is wished to swim as far as possible with a corresponding cost as low as possible.

To be rigorously stated, the question has to be split into several closely related but not always equivalent problems:

1. What is the stroke minimizing the cost among those allowing traveling a given, fixed distance?

2. What is the stroke maximizing the travelled distance among those whose cost is not greater than a given fixed bound?

In case the cost is not important, we can also be interested in seeking the stroke maximizing the mean swimming velocity.

We shall conduct a detailed study on every one of these problems, focusing on the existence of optimal strokes and deriving their main properties.

### 5.4.1 Statement of optimal problems

To begin with, let us restrict slightly the scope of our study by introducing a new hypothesis that the swimmer has to satisfy:

**Hypothesis 5.4.1** The swimmer $\mathcal{S} = (S, g, \mathbf{Q}^S, s^\dagger, \mathcal{L})$ is such that there exists an analytic basis $\mathcal{X} = \{\mathbf{X}_j, j = 1, \dots, p\}$ of the distribution $\Delta^S$.

**Definition 5.4.2** A swimmer $\mathcal{S}$ satisfying Hypothesis 5.4.1 will be termed trivialized.

Applying a Gram-Schmidt process, we can assume that for every $\mathbf{s} \in \mathcal{S}$, the family $\{\mathbf{X}_j(\mathbf{s}), j = 1, \dots, p\}$ in Hypothesis 5.4.1 is an orthonormal basis (for the Riemannian scalar product $\mathbf{g}$ of $\mathcal{S}$) of $\Delta^S$. As already mentioned before, it is in general not possible to extract from any smooth spanning family of $\Delta^S$ a smooth basis on the whole manifold $\mathcal{S}$ (see also the computations in Subsection 5.3.2). Nevertheless, any swimmer can be locally trivialized:

**Proposition 5.4.3** Let $\mathcal{S} = (S, g, \mathbf{Q}^S, s^\dagger, \mathcal{L})$ be a swimmer. Then, there exists an open connected subset $S'$ (for the topology of $\mathcal{S}$) containing $s^\dagger$ such that $\mathcal{S}' = (S', g, \mathbf{Q}^S, s^\dagger, \mathcal{L})$ is a trivialized swimmer.

Notice in particular that any open subset of an analytic manifold is still an analytic manifold.

Let $\mathcal{S} = (S, g, \mathbf{Q}^S, s^\dagger, \mathcal{L})$ be a trivialized, controllable swimmer, $\mathcal{K}$ be a compact of $\mathcal{S}$ containing $s^\dagger$ and $\mathcal{X}$ be an orthonormal basis of $\Delta^S$. For every $\xi^\dagger$ in $\mathcal{M}$ and $T \ge 0$, we define

$$ \mathcal{U}_{\mathcal{S}}^{\chi}(\xi^{\dagger}, T) := \{ \mathbf{u} \in \mathcal{U}_{\mathcal{S}}^{\chi}(T) : \xi_{\mathcal{S}}^{\chi}(T, \mathbf{u}) = \xi^{\dagger} \}; $$

$$ \hat{\mathcal{U}}_{\mathcal{S}}^{\chi}(\xi^{\dagger}, T) := \{ \mathbf{u} \in \mathcal{U}_{\mathcal{S}}^{\chi}(\xi^{\dagger}, T) : \|\mathbf{u}(t)\|_{\mathbb{R}^p} = 1 \ \forall t \in [0, T] \}; $$

$$ \mathcal{U}_{\mathcal{S}, \mathcal{K}}^{\chi}(T) := \{ \mathbf{u} \in \mathcal{U}_{\mathcal{S}}^{\chi}(T) : \pi_{\mathcal{S}} \xi_{\mathcal{S}}^{\chi}(t, \mathbf{u}) \in \mathcal{K} \ \forall t \in [0, T] \}; $$

$$ \mathcal{U}_{\mathcal{S}, \mathcal{K}}^{\chi}(\xi^{\dagger}, T) := \{ \mathbf{u} \in \mathcal{U}_{\mathcal{S}}^{\chi}(\xi^{\dagger}, T) : \pi_{\mathcal{S}} \xi_{\mathcal{S}}^{\chi}(t, \mathbf{u}) \in \mathcal{K} \ \forall t \in [0, T] \}; $$

$$ \hat{\mathcal{U}}_{\mathcal{S}, \mathcal{K}}^{\chi}(\xi^{\dagger}, T) := \{ \mathbf{u} \in \mathcal{U}_{\mathcal{S}, \mathcal{K}}^{\chi}(\xi^{\dagger}, T) : \|\mathbf{u}(t)\|_{\mathbb{R}^p} = 1 \ \forall t \in [0, T] \}. $$
The following Lemma

**Lemma 5.4.4** If there exist an orthonormal basis $\mathcal{X}$ of $\Delta^S$ and $T > 0$ such that the set $\mathcal{U}_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(T, \xi^{\dagger})$ is empty, then it is empty for every orthonormal basis $\mathcal{X}$ of $\Delta^S$ and every $T > 0$.

**Proof.** Assume that for some $\mathcal{X}$ and $T > 0$, the set $\mathcal{U}_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(T, \xi^{\dagger})$ is nonempty and denote by $\mathbf{u}$ one of its elements. Then, for any $T' > 0$, define:

$$ \alpha = \frac{\|\mathbf{u}\|_{L^1([0,T], \mathbb{R}^p)}}{T'}; \qquad (5.23a) $$

$$ \phi(t) = \frac{1}{\alpha} \int_{0}^{t} \|\mathbf{u}(s)\|_{\mathbb{R}^p} ds, \quad t \in [0, T]; \qquad (5.23b) $$

$$ \bar{\mathbf{u}}(t) = \alpha \frac{\mathbf{u}(\phi^{-1}(t))}{\|\mathbf{u}(\phi^{-1}(t))\|_{\mathbb{R}^p}}, \quad t \in [0, T']. \qquad (5.23c) $$

It can be easily verified that $\bar{\mathbf{u}} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(T', \xi^{\dagger})$. Notice that $t \in [0, T'] \mapsto \xi_{\mathcal{G}}^{\mathcal{X}}(t, \bar{\mathbf{u}}) \in \mathcal{M}$ is nothing but a time reparameterization of the curve, also parameterized by $t \in [0, T] \mapsto \xi_{\mathcal{G}}^{\mathcal{X}}(t, \mathbf{u}) \in \mathcal{M}$. Saying that $\mathcal{U}_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(T, \xi^{\dagger})$ is nonempty means that there exist an allowable curve on $\mathcal{M}$, whose projection on $\mathcal{S}$ is contained in $\mathcal{K}$ and which links $\xi^{\dagger}$ to $\xi^{\ddagger}$. The existence of such a curve depends neither on $\mathcal{X}$ nor on $T$.

In every one of the problems stated below, we make the convention that the infimum of an empty set is equal to $+\infty$ while the supremum is equal to $-\infty$.

**Problem 5.4.5 (Minimizing the Riemannian length)** For any $\delta^{\dagger} \in \mathbb{R}$ and $T > 0$, set $\xi^{\ddagger} = (\mathbf{s}^{\dagger}, \delta^{\dagger})$ and determine:

$$ \Phi_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(\delta^{\dagger}, T) = \inf \left\{ \int_{0}^{T} \|\mathbf{u}(t)\|_{\mathbb{R}^p} dt : \mathbf{u} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(\xi^{\ddagger}, T) \right\}. \qquad (5.24) $$

Notice that, since $\mathcal{X}$ is assumed to be orthonormal, we have also:

$$ \int_{0}^{T} \|\mathbf{u}(t)\|_{\mathbb{R}^p} dt = \int_{0}^{T} \sqrt{g(\pi_{\mathcal{S}}\xi_{\mathcal{G}}^{\mathcal{X}}(t, \mathbf{u}), \pi_{\mathcal{S}}\xi_{\mathcal{G}}^{\mathcal{X}}(t, \mathbf{u})))} dt, $$

which is the length of the curve $\Gamma \subset \mathcal{K}$ parameterized by $t \in [0, T] \mapsto \pi_{\mathcal{S}}\xi_{\mathcal{G}}^{\mathcal{X}}(t, \mathbf{u}) \in \mathcal{S}$. The lengths of the curves on $\mathcal{S}$ do not depend on the parameterization, so Problem 5.4.5 is time parameterization invariant.

Modifying the cost leads to:

**Problem 5.4.6 (Minimizing the action)** For any $\delta^{\dagger} \in \mathbb{R}$ and $T > 0$, set $\xi^{\ddagger} = (\mathbf{s}^{\dagger}, \delta^{\dagger})$ and determine:

$$ \Theta_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(\delta^{\dagger}, T) = \inf \left\{ \frac{1}{2} \int_{0}^{T} \|\mathbf{u}(t)\|_{\mathbb{R}^p}^2 dt : \mathbf{u} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(\xi^{\ddagger}, T) \right\}. \qquad (5.25) $$

Remark that, unlike the cost in (5.24), the cost in (5.25) is not time parameterization independent.

Before stating results about these problems, let us introduce a last related optimal problem, studied for instance in []:

**Problem 5.4.7 (Optimizing the time)** For any $\delta^{\dagger} \in \mathbb{R}$, denote $\xi^{\ddagger} = (\mathbf{s}^{\dagger}, \delta^{\dagger})$ and determine:

$$ T_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(\delta^{\dagger}) = \inf\{T : \hat{\mathcal{U}}_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(\xi^{\ddagger}, T) \neq \emptyset\}. $$
**Problem 5.4.8 (Maximizing traveling distance with bounded Riemannian length)**

For any $l \ge 0$ and $T \ge 0$, determine:

$$ \Psi_{\mathcal{G},\mathcal{K}}^{\chi}(l, T) = \sup \left\{ \pi_{\mathbb{R}} \xi_{\mathcal{G}}^{\chi}(T, \mathbf{u}) : \mathbf{u} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^{\chi}(T), \pi_{\mathcal{S}} \xi_{\mathcal{G}}^{\chi}(T, \mathbf{u}) = \mathbf{s}^{\dagger}, \right. \\ \left. \text{and } \int_{0}^{T} \|\mathbf{u}(t)\|_{\mathbb{R}^{p}} dt \le l \right\}. \quad (5.26) $$

**Problem 5.4.9 (Maximizing traveling distance with bounded action)**

For any $l \ge 0$ and $T \ge 0$, determine:

$$ \Lambda_{\mathcal{G},\mathcal{K}}^{\chi}(l, T) = \sup \left\{ \pi_{\mathbb{R}} \xi_{\mathcal{G}}^{\chi}(T, \mathbf{u}) : \mathbf{u} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^{\chi}(T), \pi_{\mathcal{S}} \xi_{\mathcal{G}}^{\chi}(T, \mathbf{u}) = \mathbf{s}^{\dagger}, \right. \\ \left. \text{and } \frac{1}{2} \int_{0}^{T} \|\mathbf{u}(t)\|_{\mathbb{R}^{p}}^{2} dt \le l \right\}. \quad (5.27) $$

Using the control $\mathbf{u} = 0$ in the last two problems, we deduce that $\Psi_{\mathcal{G},\mathcal{K}}^{\chi}(l, T) \ge 0$ and $\Lambda_{\mathcal{G},\mathcal{K}}^{\chi}(l, T) \ge 0$.

## 5.4.2 Firsts Properties of the Optimal Strokes

In this Subsection, we will derive properties of the optimal strokes resting on the Riemannian structure of $\mathcal{S}$. Then, in the following Subsection, we will introduce an make use of the sub-Riemannian structure of $\mathcal{M}$.

To begin with, let us focus on the firsts three problems:

**Theorem 5.4.10** Let $\mathcal{G} = (\mathcal{S}, \mathcal{g}, \mathcal{Q}^{\mathcal{S}}, \mathbf{s}^{\dagger}, \mathcal{L})$ be a controllable, trivialized swimmer, and $\mathcal{K}$ be a compact of $\mathcal{S}$ containing $\mathbf{s}^{\dagger}$. Then

1. For every $\delta^{\dagger} \in \mathbb{R}$, the quantities $\Phi_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger}, T)$, $\Theta_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger}, T)$, and $T_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger})$ are either all of them infinite or all of them finite, for every $T \ge 0$ and every orthonormal basis $\mathcal{X}$ of $\Delta^{\mathcal{S}}$.

2. If $\mathbf{s}^{\dagger} \in \mathcal{K}$ then $\Phi_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger}, T)$, $\Theta_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger}, T)$, and $T_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger})$ are all of them finite, for every $T \ge 0$, every orthonormal basis $\mathcal{X}$ of $\Delta^{\mathcal{S}}$ and every $\delta^{\dagger} \in \mathbb{R}$.

3. If $\Phi_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger}, T)$, $\Theta_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger}, T)$, and $T_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger})$ are finite, then there exist minimizers or maximizers to every Problem 5.4.5, 5.4.6 and 5.4.7.

4. For every $T \ge 0$ and every $\delta^{\dagger} \in \mathbb{R}$, $\Phi_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger}, T)$, $\Theta_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger}, T)$ and $T_{\mathcal{G},\mathcal{K}}^{\chi}(\delta^{\dagger})$ do not depend on $\mathcal{X}$. So from now on, we drop $\mathcal{X}$ in the notation.

5. $\Phi_{\mathcal{G},\mathcal{K}}(\delta^{\dagger}, T)$ does not depend on $T$. So from now on, we drop $T$ in the notation.

6. The following identities hold for every $\delta^{\dagger} \in \mathbb{R}$, and every $T > 0$:

$$ \Phi_{\mathcal{G},\mathcal{K}}(\delta^{\dagger}) = T_{\mathcal{G},\mathcal{K}}(\delta^{\dagger}) \quad (5.28a) $$

$$ \Theta_{\mathcal{G},\mathcal{K}}(\delta^{\dagger}, T) = (1/2)(T_{\mathcal{G},\mathcal{K}}(\delta^{\dagger}))^{2}/T \quad (5.28b) $$

7. Any minimizer $\mathbf{u} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^{\chi}(\xi^{\dagger}, T)$ to Problem 5.4.6:

(a) is such that $\|\mathbf{u}(t)\|_{\mathbb{R}^p}$ is constant at every moment;

(b) is also a minimizer to Problem 5.4.5;
(c) is proportional to a minimizer of Problem 5.4.7.

Proof.

1. Let $\delta^\dagger$ be given. According to Lemma 5.4.4, if $\mathcal{U}_{\mathcal{G},\mathcal{K}}^\chi(T, \xi^\dagger)$ is empty for some $T > 0$, then it is empty for every $T > 0$ and therefore every quantity $\Phi_{\mathcal{G},\mathcal{K}}^\chi(\delta^\dagger, T)$, $\Theta_{\mathcal{G},\mathcal{K}}^\chi(\delta^\dagger, T)$, and $T_{\mathcal{G},\mathcal{K}}^\chi(\delta^\dagger)$ is infinite. Reciprocally, if for some $T > 0$ there exists $\mathbf{u} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^\chi(T, \xi^\dagger)$, then, for every $T > 0$, the set $\mathcal{U}_{\mathcal{G},\mathcal{K}}^\chi(T, \xi^\dagger)$ is nonempty as well and $\Phi_{\mathcal{G},\mathcal{K}}^\chi(\delta^\dagger, T)$ and $\Theta_{\mathcal{G},\mathcal{K}}^\chi(\delta^\dagger, T)$ are both finite. Moreover, from the control $\mathbf{u}$, we can build $\tilde{\mathbf{u}} \in \hat{\mathcal{U}}_{\mathcal{G},\mathcal{K}}^\chi(\|\mathbf{u}\|_{L^1([0,T],\mathbb{R}^p)}, \xi^\dagger)$ by setting $\alpha = 1$ in (5.23). We deduce that $T_{\mathcal{G},\mathcal{K}}^\chi(\delta^\dagger)$ is finite and the first assertion of the Theorem is proved.

2. Denote $\mathcal{O}_1$ the connected component of $\mathcal{K}$ containing $\mathbf{s}^\dagger$ and for every $\delta^\dagger \in \mathbb{R}$, take $\mathcal{O} = \mathcal{O}_1 \times [-|\delta^\dagger|-1, |\delta^\dagger|+1]$ in Theorem 5.3.8. The Theorem ensures that for every $T > 0$ and every orthonormal basis $\mathcal{X}$ of $\Delta^S$, the set $\mathcal{U}_{\mathcal{G},\mathcal{K}}^\chi(\xi^\dagger, T)$ is nonempty.

3. We prove now all the remaining points of the Theorem. Let $\mathcal{X}$ (an orthonormal basis of $\Delta^S$), $T > 0$ and $\delta^\dagger \in \mathbb{R}$ be given. For any control $\mathbf{u} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^\chi(\xi^\dagger, T)$, denote $\tilde{\mathbf{u}} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^\chi(\xi^\dagger, T)$ the control defined in (5.23) with $T' = T$. One can easily verify that:

$$ \int_0^T \|\mathbf{u}(s)\|_{\mathbb{R}^p} ds = \int_0^T \|\tilde{\mathbf{u}}(s)\|_{\mathbb{R}^p} ds \quad (5.29) $$

and

$$ \begin{aligned} \int_0^T \|\mathbf{u}(s)\|_{\mathbb{R}^p}^2 ds &\ge \frac{1}{T} \left( \int_0^T \|\mathbf{u}(s)\|_{\mathbb{R}^p} ds \right)^2 \quad (5.30) \\ &= \int_0^T \|\tilde{\mathbf{u}}(s)\|_{\mathbb{R}^p}^2 ds, \end{aligned} $$

with equality in (5.30) if and only if $\|\mathbf{u}(s)\|_{\mathbb{R}^p}$ is constant, i.e. $\mathbf{u} = \tilde{\mathbf{u}}$. So, replacing $\mathbf{u}$ by $\tilde{\mathbf{u}}$ does not modify the cost functional of Problem 5.4.5 and does not increase the cost functional of Problem 5.4.6. Moreover, since

$$ \int_0^T \|\tilde{\mathbf{u}}(s)\|_{\mathbb{R}^p}^2 ds = \frac{1}{T} \left( \int_0^T \|\tilde{\mathbf{u}}(s)\|_{\mathbb{R}^p} ds \right)^2, \quad (5.31) $$

if $(\mathbf{u}_n)_n$ is a minimizing sequence for either Problem 5.4.5 or Problem 5.4.6, then $(\tilde{\mathbf{u}}_n)_n$ is a minimizing sequence for both Problem 5.4.5 and Problem 5.4.6. By construction, the sequence $(\tilde{\mathbf{u}}_n)_n$ is bounded in $L^\infty([0,T], \mathbb{R}^p)$. Hence, up to a subsequence extraction, we can assume that $(\tilde{\mathbf{u}}_n)_n$ weakly converges, for instance, in $L^2([0,T], \mathbb{R}^p)$ toward $\mathbf{u}^*$. In particular, the following inequality holds:

$$ \int_0^T \|\mathbf{u}^*(s)\|_{\mathbb{R}^p}^2 ds \le \liminf_{n \to +\infty} \int_0^T \|\tilde{\mathbf{u}}_n(s)\|_{\mathbb{R}^p}^2 ds. \quad (5.32) $$

Let us verify that $\mathbf{u}^* \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^\chi(\xi^\dagger, T)$.

The functions $t \in [0, T] \mapsto \xi_{\mathcal{G}}^\chi(t, \tilde{\mathbf{u}}_n) \in \mathcal{M}$ are equi-Lipschitz continuous on $[0, T]$, because the vector fields $\mathbf{Z}_i$ are analytic on the compact $\mathcal{K}$ and hence bounded. According to Ascoli Theorem, we can assume that, up to a subsequence extraction, the sequence $(t \mapsto \xi_{\mathcal{G}}^\chi(t, \tilde{\mathbf{u}}_n))_n$ converges uniformly on $[0, T]$ toward a Lipschitz continuous
function $t \in [0, T] \mapsto \boldsymbol{\xi}^* \in \mathcal{M}$. Furthermore, the curve $t \in [0, T] \mapsto \pi_S \boldsymbol{\xi}^* \in \mathcal{S}$ is absolutely continuous, with bounded derivative and thus is an admissible shape change (with support in $\mathcal{K}$). For every $t \in [0, T]$ and every $n \in \mathbb{N}$, we have:

$$ \boldsymbol{\xi}_{\mathcal{S}}^{\chi}(t, \tilde{u}_n) = \boldsymbol{\xi}^{\dagger} + \sum_{i=1}^{p} \int_{0}^{t} \tilde{u}_{i}^{n}(s) \mathbf{Z}_{i}(\boldsymbol{\xi}_{\mathcal{S}}^{\chi}(s, \tilde{u}_{n})) \mathrm{d}s. \quad (5.33) $$

Since $\tilde{u}_i^n \to u_i^*$ in $L^2([0, T], \mathbb{R})$ and $\mathbf{Z}_i(\boldsymbol{\xi}_{\mathcal{S}}^{\chi}(\cdot, \tilde{u}_n)) \to \mathbf{Z}_i(\boldsymbol{\xi}^*)$ uniformly on $[0, T]$ (and hence also in $L^2([0, T], TM)$), passing to the limit as $n \to +\infty$ in (5.33) leads to:

$$ \boldsymbol{\xi}^*(t) = \boldsymbol{\xi}^{\dagger} + \sum_{i=1}^{p} \int_{0}^{t} u_{i}^{*}(s) \mathbf{Z}_{i}(\boldsymbol{\xi}^{*}(s)) \mathrm{d}s, \quad t \in [0, T]. $$

We have now proved that $\mathbf{u}^*$ is indeed a minimizer to Problems 5.4.6. Moreover, since equality in (5.30) holds if and only if $\|\mathbf{u}^*(t)\|_{\mathbb{R}^p}$ is constant for every $t \in [0, T]$, we infer that $\mathbf{u}^* = \tilde{\mathbf{u}}^*$. This equality leads to the following estimates:

$$ \begin{aligned} \int_0^T \|\mathbf{u}^*(s)\|_{\mathbb{R}^p} \mathrm{d}s &= \sqrt{T} \left( \int_0^T \|\mathbf{u}^*(s)\|_{\mathbb{R}^p}^2 \mathrm{d}s \right)^{1/2} \\ &\leq \liminf_{n \to +\infty} \sqrt{T} \left( \int_0^T \|\tilde{\mathbf{u}}_n(s)\|_{\mathbb{R}^p}^2 \mathrm{d}s \right)^{1/2} \\ &= \liminf_{n \to +\infty} \int_0^T \|\tilde{\mathbf{u}}_n(s)\|_{\mathbb{R}^p} \mathrm{d}s, \end{aligned} $$

and $\mathbf{u}^*$ is also a minimizer to Problem 5.4.5. Using this control in (5.31), we obtain the equality:

$$ \Theta_{\mathcal{S}, \mathcal{K}}^{\chi}(\delta^{\dagger}, T) = \frac{1}{2T} (\Phi_{\mathcal{S}, \mathcal{K}}^{\chi}(\delta^{\dagger}, T))^2. \quad (5.34) $$

Eventually, as already mentioned earlier, for every $T > 0$ and from any control $\mathbf{u} \in \mathcal{U}_{\mathcal{S}, \mathcal{K}}^{\chi}(\boldsymbol{\xi}^{\dagger}, T)$, we can build $\tilde{\mathbf{u}} \in \hat{\mathcal{U}}_{\mathcal{S}, \mathcal{K}}^{\chi}(\|\mathbf{u}\|_{L^1([0,T], \mathbb{R}^p)}, \boldsymbol{\xi}^{\dagger})$ by setting $\alpha = 1$ in (5.23). The identity (5.29) becomes:

$$ \int_0^T \|\mathbf{u}(s)\|_{\mathbb{R}^p} \mathrm{d}s = \int_0^T \|\mathbf{u}\|_{L^1([0,T], \mathbb{R}^p)} \mathrm{d}s, $$

whence we deduce that

$$ T_{\mathcal{S}, \mathcal{K}}^{\chi}(\delta^{\dagger}) = \Phi_{\mathcal{S}, \mathcal{K}}^{\chi}(\delta^{\dagger}, T). $$

This equality tells us that $\Phi_{\mathcal{S}, \mathcal{K}}^{\chi}(\delta^{\dagger}, T)$ does not depend on $T$. We conclude the proof of the theorem by observing again that $\Phi_{\mathcal{S}, \mathcal{K}}^{\chi}(\delta^{\dagger}, T)$ is the length of the curve on $\mathcal{S}$ parameterized by $t \in [0, T] \mapsto \pi_S \boldsymbol{\xi}_{\mathcal{S}, \mathcal{K}}^{\chi}(t, \mathbf{u}^*) \in \mathcal{S}$ and that this length does not depend on $\mathcal{X}$.

We address now Problems 5.4.8 and 5.4.9:

**Theorem 5.4.11** Let $\mathcal{S} = (\mathcal{S}, \mathbf{g}, \mathbf{Q}^S, \mathbf{s}^\dagger, \mathcal{L})$ be a controllable, trivialized swimmer, and $\mathcal{K}$ be a compact of $\mathcal{S}$ containing $\mathbf{s}^\dagger$. Then

1. Problems 5.4.8 and 5.4.9 admit maximizers for every $T \geq 0$, every orthonormal basis $\mathcal{X}$ of $\Delta^S$ and every $l \geq 0$.

2. Any maximizer $\mathbf{u} \in \mathcal{U}_{\mathcal{S}, \mathcal{K}}^{\chi}(\boldsymbol{\xi}^{\dagger}, T)$ to Problem 5.4.9
(a) is such that $\|\mathbf{u}(t)\|_{\mathbb{R}^p}$ is constant at every moment;

(b) is proportional to a maximizer of Problem 5.4.8.

3. $\Psi_{\mathcal{G},\mathcal{K}}^{\chi}(l, T)$ and $\Lambda_{\mathcal{G},\mathcal{K}}^{\chi}(l, T)$ do not depend on $\mathcal{X}$ so we drop it in the notation.

4. $\Psi_{\mathcal{G},\mathcal{K}}^{\chi}(l, T)$ does not depend on $T$ so we drop it in the notation.

5. The following identity holds for every $T > 0$ and $l \ge 0$:

$$ \Lambda_{\mathcal{G},\mathcal{K}}((1/2)l^2/T, T) = \Psi_{\mathcal{G},\mathcal{K}}(l). \qquad (5.35) $$

**Proof.** To prove the existence of maximizers to Problems 5.4.8 and 5.4.9, we follow the lines of the proof of Theorem 5.4.10: Let $\mathcal{X}$, $l \ge 0$ and $T \ge 0$ be given and consider first a maximizing sequence $(\mathbf{u}_n)_n$ to Problem 5.4.9. Then notice that the renormalized and reparameterized control $(\tilde{\mathbf{u}}_n)_n$ is actually not only a maximizing sequence to Problem 5.4.9 but also to Problem 5.4.8 with $l' = \sqrt{lT}$. Then, up to subsequences extractions and invoking Ascoli Theorem and the weak convergence in $L^2([0, T], \mathbb{R}^p)$ of $(\tilde{\mathbf{u}}_n)_n$, we proves the existence of a common maximizer $\mathbf{u}^* = \tilde{\mathbf{u}}^*$ to Problems 5.4.9 (with $l$) and 5.4.8 (with $l' = \sqrt{lT}$). Using the control $\mathbf{u}^*$, we also get the equality (5.35).

Once more, the time reparameterization invariance of Problem 5.4.8 leads to infer that $\Psi_{\mathcal{G},\mathcal{K}}^{\chi}$ does not depend neither on $\mathcal{X}$ nor $T$. Eventually, identity (5.35) ensures that $\Lambda_{\mathcal{G},\mathcal{K}}^{\chi}(l, T)$ does not depend on $\mathcal{X}$.

### 5.4.3 Further Properties of the Optimal Strokes

In order to prove further properties on Problems 5.4.5-5.4.9, we need to introduce the Sub-Riemannian structure on $\mathcal{M}$.

Let a trivialized swimmer $\mathcal{G} = (\mathcal{S}, \mathcal{g}, \mathcal{Q}^S, \mathbf{s}^\dagger, \mathcal{L})$ and $\mathcal{X} := \{\mathbf{X}_j, j = 1, \dots, p\}$ an orthonormal basis of $\Delta_{\mathbf{s}}^S$ be given. From the analytic vectors fields $\mathbf{X}_j$ on $\mathcal{S}$, we build the analytic vector fields $\mathbf{Z}_j$ ($j = 1, \dots, p$) on $\mathcal{M} = \mathcal{S} \times \mathbb{R}$ as described in (5.21). Then, we define

$$ \mathcal{Z} = \{\mathbf{Z}_j, \mathbf{\omega} = 1, \dots, p\}, $$

and the distribution on $\mathcal{M}$:

$$ \Delta_{\mathbf{\xi}}^{\mathcal{M}} := \operatorname{span} \mathcal{Z}(\mathbf{\xi}) \subset T_{\mathbf{\xi}}\mathcal{M}, \quad \mathbf{\xi} \in \mathcal{M}. $$

We denote by $\mathbb{Z}(\mathbf{\xi})$ the matrix whose column vectors are the $\mathbf{Z}_j(\mathbf{\xi})$ and we introduce the Euclidean bundle $\mathbf{U} := \mathcal{M} \times \mathbb{R}^p$ endowed with the Euclidean norm of $\mathbb{R}^p$ and the morphism of vector bundles

$$ f: (\mathbf{\xi}, \mathbf{u}) \in \mathbf{U} \mapsto (\mathbf{\xi}, \mathbb{Z}(\mathbf{\xi})\mathbf{u}) \in \mathcal{T}\mathcal{M}. $$

Following [4, Definition 3.1], we claim that the manifold $\mathcal{M}$ endowed with the triple $(\mathcal{M}, \mathbf{U}, f)$ is an analytic sub-Riemannian manifold. According to Definition 3.6 from the same booklet, we define the admissible curves as being the Lipschitz curves $\mathbf{\xi} : [0, T] \mapsto \mathcal{M}$ for which there exists a control function $\mathbf{u} \in L^\infty([0, T], \mathbb{R}^p)$ such that, for a.e. $t \in [0, T]$:

$$ \dot{\mathbf{\xi}} = \mathbb{Z}(\mathbf{\xi})\mathbf{u}. $$

Notice that it is exactly the dynamics (5.22) that we are dealing with. The sub-Riemannian manifold can be equipped with the so-called *Carnot-Carathédory distance* (see [4, Definition 3.13]) denoted by $d(\cdot, \cdot)$. In particular, the following identity holds for any $\delta^\dagger \in \mathbb{R}$ and $\mathbf{\xi} = (\mathbf{s}^\dagger, \delta^\dagger)$:

$$ d(\mathbf{\xi}^\dagger, \mathbf{\xi}^{\ddagger}) = \inf \left\{ \int_0^T \|\mathbf{u}(s)\|_{\mathbb{R}^p} ds, \mathbf{u} \in \mathcal{U}_{\mathcal{G}}^{\chi}(T, \mathbf{\xi}^\dagger) \right\}. $$

Be aware that actually, neither $\mathcal{X}$ nor $T$ matters in this definition.
**Theorem 5.4.12** Let $\mathcal{G} = (\mathcal{S}, \mathcal{g}, \mathcal{Q}^S, \mathbf{s}^\dagger, \mathcal{L})$ be a controllable, trivialized swimmer, and $\mathcal{K}$ be a compact of $\mathcal{S}$ such that $\mathbf{s}^\dagger \in \mathcal{K}$. Then:

1. For every $l > 0$, we have $\Psi_{\mathcal{G},\mathcal{K}}(l) > 0$. For every $l > 0$ and $T > 0$, we have $\Lambda_{\mathcal{G},\mathcal{K}}(l, T) > 0$.

2. The functions
$$l \in \mathbb{R}_+ \mapsto \Psi_{\mathcal{G},\mathcal{K}}(l) \in \mathbb{R}_+ \qquad (5.36a)$$

is increasing and right continuous.

3. For every $T > 0$, the function
$$l \in \mathbb{R}_+ \mapsto \Lambda_{\mathcal{G},\mathcal{K}}(l, T) \in \mathbb{R}_+ \qquad (5.36b)$$

is increasing and right continuous.

4. For every maximizer to Problems 5.4.8 or 5.4.9, the constraints are saturated.

5. The function
$$\delta \in \mathbb{R} \mapsto \Phi_{\mathcal{G},\mathcal{K}}(\delta) \in \mathbb{R} \qquad (5.36c)$$

is even and uniformly continuous.

6. For every $T > 0$, the function
$$\delta \in \mathbb{R} \mapsto \Theta_{\mathcal{G},\mathcal{K}}(\delta, T) \in \mathbb{R} \qquad (5.36d)$$

is even and uniformly continuous.

7. For every $l \ge 0$ and every $T \ge 0$:
$$\Phi_{\mathcal{G},\mathcal{K}}(\Psi_{\mathcal{G},\mathcal{K}}(l)) = l \qquad (5.37a)$$
$$\Theta_{\mathcal{G},\mathcal{K}}(\Lambda_{\mathcal{G},\mathcal{K}}(l, T), T) = l. \qquad (5.37b)$$

Regarding the last point of the theorem, notice that, for every $\delta^\dagger \in \mathbb{R}_+$, we have $\Psi_{\mathcal{G},\mathcal{K}}(\Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger)) \ge \delta^\dagger$ but it may happen that $\Psi_{\mathcal{G},\mathcal{K}}(\Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger)) > \delta^\dagger$ for some $\delta^\dagger$. Indeed, according to the definition of $\Psi_{\mathcal{G},\mathcal{K}}(\Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger))$, we have:

$$\Psi_{\mathcal{G},\mathcal{K}}(\Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger)) = \max \left\{ \pi_{\mathbb{R}} \xi_{\mathcal{G}}^\chi(T, \mathbf{u}) : \mathbf{u} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^\chi(T), \pi_S \xi_{\mathcal{G}}^\chi(T, \mathbf{u}) = \mathbf{s}^\dagger, \int_0^T \|\mathbf{u}(s)\|_{\mathbb{R}^p} ds \le \Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger) \right\}. \quad (5.38)$$

**Proof.**

1. For any $a \ge 0$ and $\xi \in \mathcal{M}$, denote by $B^\mathcal{M}(\xi, a)$ the sub-Riemannian ball, of radius $a$ and centered at $\xi$. According to [4, Theorem 3.8], the Carnot-Caratheodory distance induces the manifold topology on $\mathcal{M}$. We deduce that, for every $l > 0$, the set $B(\xi^\dagger, l) \cap (\mathcal{K} \times \mathbb{R})$ is open and nonempty (it contains $\mathbf{s}^\dagger$) and hence

$$((\mathbf{s}) \times \mathbb{R}) \cap B(\xi^\dagger, l) \cap (\mathcal{K} \times \mathbb{R})$$

contains an open set $\{\mathbf{s}^\dagger\} \times ] -\varepsilon, \varepsilon[$ for some $\varepsilon > 0$. We infer that, for every $l > 0$, we have $\Psi_{\mathcal{G},\mathcal{K}}(l) > \varepsilon > 0$. We use the relation (5.35) to deduce that, for every $l > 0$ and $T > 0$, we also have $\Lambda_{\mathcal{G},\mathcal{K}}(l, T) > 0$.
2. The function (5.36a) is clearly nondecreasing. Moreover, for every $l, l' \ge 0$, we have:

$$ \Psi_{\mathcal{G},\mathcal{K}}(l+l') \ge \Psi_{\mathcal{G},\mathcal{K}}(l) + \Psi_{\mathcal{G},\mathcal{K}}(l'). \quad (5.39) $$

Indeed, recall that every minimizer to Problem 5.4.8 is time parameterization invariant and consider the curves $\Gamma \subset S$ of length $l$ and $\Gamma' \subset S$ of length $l'$ corresponding to the maximizers of Problem 5.4.8 for $l$ and $l'$ respectively. Then denote $\Gamma'' = \Gamma \cup \Gamma'$. This curve is admissible, closed, of length $l+l'$ and produces a displacement no greater than $\Psi_{\mathcal{G},\mathcal{K}}(l+l')$. Inequality (5.39) together with the first point of the Theorem yield the increasing property of function (5.36a).

In order to prove the right continuity, let $l \in \mathbb{R}_+$ be given and consider a decreasing sequence $(l_n)_n$ converging to $l$. For every $l_n$ ($n \in \mathbb{N}$), denote by $\mathbf{u}_n$ the minimizer to Problem 5.4.8 such that $\|\mathbf{u}_n(t)\|_{\mathbb{R}^p}$ is constant for every $t \in [0, T]$. The sequence $(\mathbf{u}_n)_n$ is bounded in $L^2([0, T], \mathbb{R}^p)$ and the sequence $(\xi_{\mathcal{G}}^{\chi}(\cdot, \mathbf{u}_n))_n$ is bounded in $C^0([0, T], \mathcal{M})$, therefore there exists a subsequence $(l_{n_k})_k$ such that $(\Psi_{\mathcal{G},\mathcal{K}}(l_{n_k}))_k$ converges to $\limsup \Psi_{\mathcal{G},\mathcal{K}}(l_n)$ while $(\mathbf{u}_{n_k})_k$ weakly converges in $L^2([0, T], \mathbb{R}^p)$ to $\mathbf{u}^*$ and $(\xi_{\mathcal{G}}^{\chi}(\cdot, \mathbf{u}_{n_k}))_k$ uniformly converges to $\xi^*$.

On the one hand, arguing as for (5.33), we deduce that $\xi^* = \xi_{\mathcal{G}}^{\chi}(\cdot, \mathbf{u}^*)$ and then that

$$ \Psi_{\mathcal{G},\mathcal{K}}(l_{n_k}) \to \pi_{\mathbb{R}} \xi_{\mathcal{G}}^{\chi}(T, \mathbf{u}^*) \quad \text{as} \quad k \to +\infty $$

with $\|\mathbf{u}^*\|_{L^1([0,T],\mathbb{R}^p)} \le \liminf l_n = l$. Therefore we obtain that:

$$ \limsup \Psi_{\mathcal{G},\mathcal{K}}(l_n) \le \Psi_{\mathcal{G},\mathcal{K}}(l). $$

On the other hand, since $\Psi_{\mathcal{G},\mathcal{K}}$ is increasing and $l \le l_n$ for every $n$, we deduce that

$$ \Psi_{\mathcal{G},\mathcal{K}}(l) \le \Psi_{\mathcal{G},\mathcal{K}}(l_n), $$

and the conclusion arises by taking the limit inf in both sides of the inequality.

3. The proof of this point is a straightforward consequence of the preceding point and relation (5.35).

4. The same reasoning as for the second point proves that the constraints are saturated in Problems 5.4.8 or 5.4.9 for the maximizers. Indeed, if the constraints were not saturated, then it would be possible to add to the optimal curve on $S$ a small loop that would produce an extra displacement (according to the first point).

5. For $\varepsilon$ small enough, we have $B^{\mathcal{M}}(\xi^\dagger, \varepsilon) \subset \mathcal{K}$ (because the sub-Riemannian topology coincides with the manifold topology). In this case, for every $\delta^\dagger \in \pi_{\mathbb{R}}^{-1}B^{\mathcal{M}}(\xi^\dagger, \varepsilon)$, we have:

$$ \Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger) = d(\xi^\dagger, \xi^\ddagger). $$

According to [4, Theorem 3.18], the Carnot-Caratheodory distance is continuous for the manifold topology, so we deduce that the function (5.36c) is continuous in a neighborhood of 0. It is even because, for every $\delta$ and every minimizer $\mathbf{u}$ on $[0, T]$, the control $t \mapsto \mathbf{u}(T-t)$ is a minimizer with the same Riemannian length, associated to $-\delta$.

Observe now that for every $\delta^\dagger, h \in \mathbb{R}$:

$$ \Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger + h) \le \Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger) + \Phi_{\mathcal{G},\mathcal{K}}(h), $$

whence we infer that:

$$ \Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger + h) - \Phi_{\mathcal{G},\mathcal{K}}(\delta^\dagger) \le \Phi_{\mathcal{G},\mathcal{K}}(h). $$
Writing that

$$ \Phi_{\mathcal{G},\mathcal{K}}(\delta^{\dagger}) \leq \Phi_{\mathcal{G},\mathcal{K}}(\delta^{\dagger} + h) + \Phi_{\mathcal{G},\mathcal{K}}(-h), $$

and since the function is even, we finally get:

$$ |\Phi_{\mathcal{G},\mathcal{K}}(\delta^{\dagger} + h) - \Phi_{\mathcal{G},\mathcal{K}}(\delta^{\dagger})| \leq \Phi_{\mathcal{G},\mathcal{K}}(h), $$

which, with the continuity in 0, proves that the function is uniformly continuous on $\mathbb{R}$.

6. The proof of this point is a straightforward consequence of the preceding point and relation (5.28).

7. For every $l \geq 0$, we clearly have $\Phi_{\mathcal{G},\mathcal{K}}(\Psi_{\mathcal{G},\mathcal{K}}(l)) \leq l$. The inequality $\Phi_{\mathcal{G},\mathcal{K}}(\Psi_{\mathcal{G},\mathcal{K}}(l)) < l$ for some $l \geq 0$ and the existence of minimizers to Problem 5.4.5 would contradict the fact that the constraint is saturated for every maximizer of Problem 5.4.8.

The function (5.36a) has no reason to be monotone in the general case. Let us define:

**Hypothesis 5.4.13** The swimmer $\mathcal{G} = (S, g, Q^S, s^\dagger, \mathcal{L})$ is such that $Q^S = 0$ (there is no self-propelled constraints).

Every swimmer satisfying (5.4.13) is called unconstrained. For unconstrained swimmers, every absolutely continuous curve on $S$, with essentially bounded first derivative, is an admissible shape change. Under Hypothesis 5.4.13, we can state:

**Theorem 5.4.14** Let $\mathcal{G} = (S, g, Q^S, s^\dagger, \mathcal{L})$ be a controllable, trivialized, unconstrained swimmer, and $\mathcal{K}$ be a compact of $S$ such that $s^\dagger \in \mathcal{K}$. Then there exists $\varepsilon > 0$ such that the function (5.36a) is increasing on $]-\varepsilon, \varepsilon[$.

**Proof.** Denote by $B^S(s^\dagger, r)$ the Riemannian ball on $S$, where the radius $r$ is given by Lemma 5.7.1. Let $\varepsilon > 0$ be small enough such that $B^M(\xi^\dagger, \varepsilon) \subset \mathcal{K}$ (it is always possible because the sub-Riemannian topology coincides with the manifold topology) and $\pi_S B^M(\xi^\dagger, \varepsilon) \subset B^S(s^\dagger, r)$.

Assume now that there exist $0 < \delta_0 < \delta_1 < \varepsilon$ such that

$$ \Phi_{\mathcal{G},\mathcal{K}}(\delta_1) \leq \Phi_{\mathcal{G},\mathcal{K}}(\delta_0). \qquad (5.40) $$

Denote $\xi^1 = (s^\dagger, \delta_1)$ and for some $\mathcal{X} = \{\mathbf{X}_j, j = 1, \dots, p\}$ (an orthonormal basis of $\Delta^S$) and $T > 0$, denote by $\mathbf{u}^1 \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(\xi^1, T)$ a control minimizing Problem 5.4.5. Introduce as well $\gamma_1 = \pi_S \xi^1(\cdot, \mathbf{u}^1)$ and $\Gamma_1$ the curve on $S$ parameterized by $\gamma_1$. The following identity holds:

$$ \Phi_{\mathcal{G},\mathcal{K}}(\delta_1) = \int_0^T \|\mathbf{u}^1(t)\|_{\mathbb{R}^p} dt = \ell(\Gamma_1). $$

According to Lemma 5.7.1 with $x_0 = s^\dagger$ and $\gamma = \gamma_1$, there exists a continuous function $\psi: [0, 1] \times [0, T] \to S$ such that, for every $s \in [0, 1]$, $\psi(s, \cdot)$ is absolutely continuous with essentially bounded first derivative, $\psi(1, \cdot) = \gamma_1$ and $\psi(0, \cdot) = s^\dagger$. Denoting by $\mathbf{u}^s = (u_j^s)_{1 \leq j \leq p} \in \mathcal{U}_{\mathcal{G},\mathcal{K}}^{\mathcal{X}}(T)$ the control such that

$$ u_j^s(t) = g_{\psi(s,t)}(\partial_t \psi(s,t), \mathbf{X}_j(\psi(s,t))), \quad t \in [0, T], $$

and by $\Gamma_s$ the curve parameterized by $t \in [0, T] \mapsto \psi(s,t) = \pi_S \xi(t, \mathbf{u}^s)$ ($s \in [0, 1]$), we have, for every $s \in [0, 1]$:

$$ \Phi_{\mathcal{G},\mathcal{K}}(\Xi(s)) \leq \int_0^T \|\mathbf{u}^s(t)\|_{\mathbb{R}^p} dt = \ell(\Gamma_s). $$
where we have set:

$$ \Xi : s \in [0, 1] \mapsto \pi_{\mathbb{R}} \xi_{\mathcal{G}}^{\chi}(T, \mathbf{u}^s) \in \mathbb{R}. $$

The function $\Xi$ is continuous and such that $\Xi(1) = \delta_1$ and $\Xi(0) = 0$, so there exists $s^* \in ]0, 1[$ such that $\Xi(s^*) = \delta_0$. Since, according to Lemma 5.7.1, the function $s \in [0, 1] \mapsto \ell(\Gamma_s)$ is increasing, we get:

$$ \Phi_{\mathcal{G}, \mathcal{K}}(\delta_0) \le \ell(\Gamma_{s^*}) < \ell(\Gamma_1) = \Phi_{\mathcal{G}, \mathcal{K}}(\delta_1), $$

which is in contradiction with (5.40). The proof is now completed.

## 5.5 The case $N = 2$

This section is devoted to the study of optimal strokes for controllable, trivialized, unconstrained swimmers, when the dimension $N$ of the manifold $\mathcal{S}$ is equal to 2.

### 5.5.1 Optimal Strokes and Isoperimetric Inequalities

In this subsection, we wish to give a hint of how the optimal stroke problem can be interpretable as an isoperimetric problem on the manifold $\mathcal{S}$.

Recall that a stroke is a closed (oriented) admissible curve $\Gamma$. Let $\mathbf{s} : [0, T] \mapsto \mathcal{S}$ be a parameterization of a $\Gamma$. The travelled distance resulting from this stroke is:

$$ \int_{0}^{T} \mathcal{L}_{\mathbf{s}(t)} \dot{\mathbf{s}}(t) dt = \int_{\Gamma} \mathcal{L}. $$

Denoting by $\Omega$ the area enclosed by $\Gamma$, Stokes formula tells us that:

$$ \int_{\Gamma} \mathcal{L} = \int_{\Omega} d\mathcal{L}. $$

Notice that these formula are metric independent but require $\mathcal{S}$ to be orientable. The 2-form $d\mathcal{L}$ defined on the 2 dimensional manifold $\mathcal{S}$ can be seen as a signed measure on $\mathcal{S}$. On the other hand, an example of cost functional considered in this paper is just the Riemannian length of $\Gamma$.

So, roughly speaking, seeking optimal strokes consists in minimizing the Riemannian length of $\Gamma$ while maximizing the measure of the area $\Omega$ for the signed measure $d\mathcal{L}$.

In the case of the swimmer in the potential flow, we have drawn on Fig 5.4 the density function of the measure $d\mathcal{L}$. Notice in particular that, considering small strokes (and hence small closed curves), not only matter the shape of the curve but the position on the ellipsoid is also preponderant.

This approach of optimal stroke problems as isoperimetric problems would be worth some further investigation.

### 5.5.2 Pontryagin's maximum principle

Let us begin with recalling some notation. We consider a trivialized, unconstrained, controllable swimmer $\mathcal{G} = (\mathcal{S}, \mathbf{g}, \mathbf{Q}^S, \mathbf{s}^\dagger, \mathcal{L})$ (with $\mathbf{Q}^S = \mathbf{0}$ since it is *unconstrained*) and $\chi = \{\mathbf{X}_1, \mathbf{X}_2\}$ an analytic orthonormal basis of $\Delta^S = TS$. Then we define the analytic vector fields $\mathcal{Z} = \{\mathbf{Z}_1, \mathbf{Z}_2\}$ on $\mathcal{M}$, according to formula (5.21).

To pursue, we need to strengthened Hypothesis 5.3.5 as follows.

**Hypothesis 5.5.1** For every $\xi \in \mathcal{M}$, we have:

$$ \dim \operatorname{span}\{\mathbf{Z}_1(\xi), \mathbf{Z}_2(\xi), [\mathbf{Z}_1, \mathbf{Z}_2](\xi)\} = 3. \qquad (5.41) $$
<figure><img src="image_40.png" /><figcaption>Figure 5.4: Density fonction of the signed measure defined by $d\mathcal{L}$ on the ellipsoid. A stroke being a closed curve on the ellipsoid, the resulting travelled distance is obtained by measuring the area of the enclosed surface for the measure $d\mathcal{L}$. See also Fig 5.5 and 5.9.</figcaption></figure>