As already pointed out, this Hypothesis does not depend on the choice of $\mathcal{Z}$ and hence is independent of the choice of the orthonormal basis $\mathcal{X}$. Actually, (5.41) can be rewritten as:

$$ \dim \operatorname{span} \{ \Delta_{\xi}^{\mathcal{M}}, [\Delta_{\xi}^{\mathcal{M}}, \Delta_{\xi}^{\mathcal{M}}] \} = 3. $$

Recall that according to Lemma 5.3.6, Hypothesis 5.3.5 leads to:

$$ \dim \operatorname{Lie}_{\xi} \Delta^{\mathcal{M}} = \dim \mathcal{M}, \quad \forall \xi \in \mathcal{M}. $$

In Hypothesis 5.5.1, $\operatorname{Lie}_{\xi} \Delta^{\mathcal{M}}$ is required to be spanned by the Lie brackets of order no greater than 1.

In Theorem 5.4.10, it has been proved that Problems 5.4.5-5.4.7 are equivalent. Let us restate a version of this problem in the simplified 2 dimensional case we are considering in this Section:

**Problem 5.5.2** For any given $\xi_0, \xi_1$ in $\mathcal{M}$, find $T \ge 0$ and a measurable bounded function $t \in [0, T] \mapsto (u_1(t), u_2(t)) \in \mathbb{R}^2$ which minimizes the cost

$$ \frac{1}{2} \int_{0}^{T} (u_{1}(t)^{2} + u_{2}(t)^{2}) \, dt, $$

such that there exists an absolutely continuous curve $t \in [0, T] \mapsto \xi(t) \in \mathcal{M}$ satisfying

$$ \dot{\xi}(t) = u_1(t)\mathbf{Z}_1(\xi) + u_2(t)\mathbf{Z}_2(\xi) \quad \text{for almost every } t \text{ in } [0, T], \quad (5.42a) $$

$$ \xi(0) = \xi_0, \quad \xi(T) = \xi_1. \quad (5.42b) $$

and,

$$ u_1(t)^2 + u_2(t)^2 = 1 \quad \forall t \in [0, T]. \quad (5.42c) $$

In order to apply Pontryagin's maximum principle (see [2]) to this problem, we denote by $T^*\mathcal{M}$ the cotangent bundle of $\mathcal{M}$ and we introduce the Hamiltonian (see [85] Chap.7.1)

$$ H : \mathbb{R} \times \mathcal{M} \times T^*\mathcal{M} \times \mathbb{R} \times \mathbb{R}^2 \rightarrow \mathbb{R} \\ (t, \xi, \mathbf{p}, p^0, (u_1, u_2)) \mapsto H(t, \xi, \mathbf{p}, p^0, (u_1, u_2)) \quad (5.43a) $$

defined by:

$$ H(t, \xi, \mathbf{p}, p_0, \mathbf{U}) = \langle \mathbf{p}, u_1(t)\mathbf{Z}_1(\xi) + u_2(t)\mathbf{Z}_2(\xi) \rangle \\ - p_0(u_1^2(t) + u_2^2(t)) \quad (5.43b) $$

where $\langle \cdot, \cdot \rangle$ stands for the duality product $T^*\mathcal{M} \times T\mathcal{M}$. For every $\xi$ in $\mathcal{M}$, the quantity $u_1(t)\mathbf{Z}_1(\xi) + u_2(t)\mathbf{Z}_2(\xi)$ belongs to the tangent space $T_{\xi}\mathcal{M}$, $\mathbf{p}$ is in the cotangent space $T_{\xi}^{\mathcal{M}}\mathcal{M}$ and $p_0$, called the cost dual variable, is a time independent real constant.

In this setting and under Hypothesis 5.5.1, the Pontryagin's maximum principle reads (see for instance Theorem 3.28 of [4])

**Proposition 5.5.3** If $(u_1(\cdot), u_2(\cdot)) \in L^\infty([0, T], \mathbb{R}^2)$ is a solution to Problem 5.5.2 associated with curve $t \in [0, T] \mapsto \xi(t) \in \mathcal{M}$, then, there exists a non trivial

$$ \mathbf{p} : t \in [0, T] \mapsto \mathbf{p}(t) \in T^*\mathcal{M} $$

solution of the system

$$ \dot{\xi} = u_1(t)\mathbf{Z}_1(\xi) + u_2(t)\mathbf{Z}_2(\xi), \quad (5.44a) $$

$$ \dot{\mathbf{p}} = -u_1(t)\langle \mathbf{p}, D_{\xi}\mathbf{Z}_1(\xi) \rangle - u_2(t)\langle \mathbf{p}, D_{\xi}\mathbf{Z}_2(\xi) \rangle. \quad (5.44b) $$

Moreover, for all $t \in [0, T]$,

$$ u_i(t) = \langle \mathbf{p}(t), \mathbf{Z}_i(\xi(t)) \rangle \quad \text{for } i = 1, 2. $$
**Definition 5.5.4** The exponential map (from $\xi_0$), denoted by $\varepsilon_{\xi_0}$, is the mapping which associates to every $\mathbf{p}_0 \in T^*M$, the solution of System (5.44) with the initial condition $\xi(0) = \xi_0$ and $\mathbf{p}(0) = \mathbf{p}_0$. Such a trajectory is called an extremal.

We call $\gamma_{\mathbf{p}_0}$ the curve on the manifolds $M$ of the solution $\varepsilon_{\xi_0}$.

**Remark 5.5.5** The general statement of the Maximum Principle, as it can be found for instance in [2], is more intricate and involves so-called abnormal extremals. The study of such extremals is usually far from obvious, from both a theoretical and a numerical point of view. In our case, the distribution is 1-step generating, see Eq. (5.41). Hence, the Goh condition (see Chap. 10.2 of [14]) ensures that all optimal trajectories in $M$ can be lifted in $T^*M$ as normal extremals.

An obvious consequence of Proposition 5.5.3 is

**Lemma 5.5.6** For every solution of (5.44) $(\xi(\cdot), \mathbf{p}(\cdot)) : [0, T] \to T^*M$, there exists a continuous function $\theta : [0, T] \to \mathbb{R}$ such that,

$$ \langle \mathbf{p}(\cdot), \mathbf{Z}_1(\xi(\cdot)) \rangle = \cos(\theta(\cdot)), $$

$$ \langle \mathbf{p}(\cdot), \mathbf{Z}_2(\xi(\cdot)) \rangle = \sin(\theta(\cdot)). $$

### 5.5.3 Contact sub-Riemannian structure

A Riemannian manifold is endowed with a definite 2-form that induces a natural identification between the tangent space and its dual. In general, such an identification cannot be directly generalized to any sub-Riemannian structure, since the definite 2-form that defines the metric is not defined on the whole tangent space, but only on a strict subspace. In our case, from the metric $\mathbf{g}$ on $S$, we can define the definite 2-form $\mathbf{g}^M$ on $\Delta^M$ by setting:

$$ \mathbf{g}_{\xi}^{M}(\zeta_1, \zeta_2) = \mathbf{g}_{\pi_S \xi}(\pi_S \zeta_1, \pi_S \zeta_2), \quad \xi \in \mathcal{M}, \zeta_1, \zeta_2 \in \Delta_{\xi}^{M}. $$

In the particular case of so-called contact structures (see definition below), it is possible to define a natural identification between the tangent and the co-tangent spaces. We follow here the procedure exposed by Agrachev in [1], adapted to our special example (for which $\dim M = 3$).

**Definition 5.5.7** A contact structure in $M$ is a smooth distribution of 2-planes $\tilde{\Delta} : \xi \in M \mapsto \tilde{\Delta}_{\xi} \in T_{\xi}M$ that is non-integrable at every point, i.e., $\tilde{\Delta}_{\xi} + [\tilde{\Delta}_{\xi}, \tilde{\Delta}_{\xi}] = T_{\xi}M$ for every $\xi$ in $M$.

This is precisely Hypothesis 5.5.1 and therefore:

**Proposition 5.5.8** The distribution $\Delta^M$ is a contact structure.

For every $\xi$ in $M$, $\{\mathbf{Z}_1(\xi), \mathbf{Z}_2(\xi)\}$ is a basis of $\Delta_{\xi}^M$. The definite 2-form $\mathbf{g}^M$ induces a natural identification between $\Delta_{\xi}^M$ and $(\Delta_{\xi}^M)^*$. Hence the co-vectors $\mathbf{Z}_1(\xi)^*$ and $\mathbf{Z}_2(\xi)^*$ form a basis of $(\Delta_{\xi}^M)^*$. It remains to complete these bases of $\Delta_{\xi}^M$ and $(\Delta_{\xi}^M)^*$ by choosing a third vector. It can be done in a canonical way using Stokes fields (see Section 2.1 of [1]).

**Lemma 5.5.9** There exists a unique 1-form $\omega$ on $M$ such that, for every $\xi_0$ in $M$:

1. $\omega|_{\Delta_{\xi_0}^M}$ vanished,

2. $d\omega|_{\Delta_{\xi_0}^M \times \Delta_{\xi_0}^M}$ is equal to the volume form $\mathbf{g}_{\xi_0}^M$ on the oriented Euclidean space $\Delta_{\xi_0}^M$.
**Proof.** The proof of [1] is constructive. Locally, up to a restriction of a local chart, $M = \mathbb{R}^3$.

For every $\xi$, the vectors $Z_1(\xi)$ and $Z_2(\xi)$ span a 2-dimensional subspace of $T_\xi M$. Since $T_\xi^*M$ is a 3-dimensional vector space, there exists a non-trivial linear form $\tilde{\omega}_\xi$ which vanishes on $Z_1(\xi)$ and $Z_2(\xi)$, and hence on $\Delta_\xi^M$. Moreover, for any other linear form $\hat{\omega}_\xi$ that vanishes on $\Delta_\xi^M$, there exists some real number $f_\xi$ such that $\hat{\omega}_\xi = f(\xi)\tilde{\omega}_\xi$.

Expressing everything in coordinates, one can see that the coordinates of $\tilde{\omega}$ can be chosen to depend rationally on the coordinates of $Z_1$ and $Z_2$, hence we can assume that $\tilde{\omega}$ is smooth on $M$.

Any other smooth 1-form $\hat{\omega}$ that vanishes on $\Delta_\xi^M$ has the form $\hat{\omega} = f\tilde{\omega}$ with $f: M \to \mathbb{R}$ a smooth function. The problem now is to prove that there exists one, and only one, function $f$ such that $f\tilde{\omega}$ satisfies the second condition of Lemma 5.5.9.

A straightforward computation in coordinates gives, for any smooth function $f$,

$$d(f\tilde{\omega})_{|\Delta^M \times \Delta^M} = f d(\tilde{\omega})_{|\Delta^M \times \Delta^M},$$

what proves the uniqueness of the 1-form $\omega$ (if any) satisfying the conditions of Lemma 5.5.9.

As a consequence of Hypothesis 5.5.1, the 2-form $d\tilde{\omega}(Z_1, Z_2)$ does not vanish (this can be seen with a direct but tedious computation in coordinates by expressing $\det(Z_1, Z_2, [Z_1, Z_2])$ in terms of $d\tilde{\omega}(Z_1, Z_2)$). Lemma 5.5.9 follows by defining the 1-form $\omega_\xi = \frac{1}{d\tilde{\omega}_\xi(Z_1(\xi), Z_2(\xi))} \tilde{\omega}_\xi$.

From non on, we denote with $\omega$ the 1-form defined in Lemma 5.5.9. Our aim is now to define a vector field $e$ “dual” to $\omega$ in $TM$.

**Lemma 5.5.10** There exists a unique analytic vector field, called the Stokes field and denoted by $e: \xi \in M \mapsto e_\xi \in T_\xi M$ which satisfies, for every $\xi$ in $M$:

1. $\omega_\xi(e_\xi) = 1,$

2. $d\omega_\xi(e_\xi, Z_i(\xi)) = 0$, for $i = 1, 2.$

**Proof.** For every $\xi$ in $M$, since the three linear forms $\omega_\xi$, $d\omega_\xi(\cdot, Z_i(\xi))$, $i = 1, 2$, are independent, there exists an unique vector $e$ which is solution of the linear system given by the two conditions.

From now on, with a slight abuse of notations, we will denote $Z_i^* := d\omega(\cdot, Z_i)$. This emphasizes the duality between the bases $(Z_1(\xi), Z_2(\xi), e_\xi)$ and $(Z_1^*(\xi), Z_2^*(\xi), \omega_\xi)$ of $T_\xi M$ and $T_\xi^*M$.

**Remark 5.5.11** An important point in the construction of the bases $(Z_1(\xi), Z_2(\xi), e_\xi)$ and $(Z_1^*(\xi), Z_2^*(\xi), \omega_\xi)$ is that the construction depends only on $g^M$ and $\Delta^M$, up to a rotation in the 2-plane $\Delta_\xi^M$. In other words, the contact structure in $M$ is enough to induce an intrinsic isomorphism from the tangent to the cotangent space of $M$, as if $M$ were a Riemannian (not only a sub-Riemannian) manifold.

A direct consequence of the construction done in Lemmas 5.5.9 and 5.5.10 is the possibility to express uniquely any co-vector $p$ in $T_\xi^*M$ in the basis $(Z_1^*(\xi), Z_2^*(\xi), \omega_\xi)$:

$$p = \cos(\theta)Z_1^* + \sin(\theta)Z_2^* + \nu\omega.$$

As a direct consequence, any initial condition $p_0$ of the Hamiltonian system 5.44 is defined by a couple $(\theta, \nu) \in \mathbb{R}/2\pi\mathbb{R} \times \mathbb{R}$.
### 5.5.4 Local structure of small geodesics

In this Subsection, we use the adapted bases ($Z_1(\xi)$, $Z_2(\xi)$, $e_\xi$) and ($Z_1^*(\xi)$, $Z_2^*(\xi)$, $\omega_\xi$) of $T_\xi M$ and $T_{\xi^*} M$ in order to describe the geometric behaviour of the geodesics around a point $\xi_0$ of $M$.

#### Asymptotic lengths for geodesics

Fix $\xi_0 \in M$ and consider the exponential map $\epsilon_{\xi_0}$ starting from $\xi_0$.

**Definition 5.5.12** A point $\xi \neq \xi_0$ is said conjugate to $\xi_0$, if $\xi$ is a critical value for $\epsilon_{\xi_0}$. The conjugate locus is the set which contains all the points to $\xi_0$,

$$ \mathrm{Conj}(\xi_0) := \{\xi \text{ s. t. } \xi \neq \xi_0 \text{ and } \xi \text{ conjugate to } \xi_0\}. \quad (5.45) $$

**Definition 5.5.13** Fix $\xi_0 \in M$, we define the distance,

$$ C_{p_0} := \sup \{t > 0, \gamma(t) = \pi_M (\epsilon_{\xi_0}(p_0)), d(\xi_0, \gamma(t)) = t\} \quad (5.46) $$

The point $\gamma_{p_0}(C_{p_0})$ is a cut point of $\xi_0$ along $\gamma$.

The cut locus is the set which contains all the cut points of $x_0$ along all the curve $\gamma$,

$$ \mathrm{Cut}(\xi_0) := \bigcup_{\lambda} \{\xi \text{ s. t. } \xi \neq \xi_0 \text{ and } \xi \text{ cut point of } x_0 \text{ along } \gamma_{\lambda}\}. \quad (5.47) $$

For every point $\xi_0$ in $M$, the geodesics from $\xi_0$ can lose their optimal property at the set of points defined by the union of the cut locus and the conjugate locus. It is important to know when the geodesics issued from a point $\xi_0$ cut one of these sets. This question is difficult in general and there is no global answer. However, a precise description of short geodesics can be given in our particular setting (see [2]).

**Definition 5.5.14** We call constants of the structure the coefficients, denoted by the family $(c_{ij})_{\substack{i=1,2,3 \\ i=1,2}}^{j=1,2,3}$ associate with the decomposition of the two-form $dZ_i^*$ into the basis $(\omega \wedge Z_1^*, \omega \wedge Z_2^*, Z_1^* \wedge Z_2^*)$. More precisely, for $i=1,2$, $dZ_i^*$ reads,

$$ dZ_i^* = c_{01}^i \omega \wedge Z_1^* + c_{02}^i \omega \wedge Z_2^* + c_{i3} Z_1^* \wedge Z_2^* \quad (5.48) $$

As the 1-forms $Z_i^*$, notice that the constants of structure depend also on the point $\xi$ where they are computed. We will introduce the two invariants maps which describe the local geometry around a point of the cut locus and the conjugate locus.

**Definition 5.5.15** We call $\chi$ the function from $M$ to $\mathbb{R}$ defined by

$$ \chi := \sqrt{-(c_{01}^1 c_{02}^2 - c_{02}^1 c_{01}^2)}. $$

**Proposition 5.5.16** For $x_0 = (\frac{1}{16}\sqrt{3}\sqrt{2}, \frac{1}{4}, \frac{1}{2})$ the value of

$$ \chi_{x_0} := \frac{15803633355047100398207710857772250331514532656054272}{3070472626751170117064260382892590591961941813016409} $$

**Proof.** This value are calculated by using Maple.

**Proposition 5.5.17** The set $\{\xi \in M | \chi(\xi) = 0\}$ is closed with measure zero in $M$.

**Proof.** Since the mapping $\xi \in M \mapsto Z_i(x)$ is analytic, the function $\xi \mapsto \chi(\xi)$ is analytic as well. The result follows from the connectedness of $M$ and the computation of Proposition 5.5.16.

The fact that $\chi_{x_0} \neq 0$ gives an asymptotic estimate of the first cut and conjugate lengths by applying the Theorems 3.2 and 4.2 of [2]:
**Proposition 5.5.18** For $\xi$ in $\mathcal{M}$ such that $\chi_{\xi} \neq 0$, for every $\mathbf{p}_0 = (\theta, \nu) \in T_{\xi}\mathcal{M}^*$, the first conjugate length is,

$$l_1(\theta, \nu) = \frac{2\pi}{|\nu|} + O_{\nu \to \infty} \left( \frac{1}{|\nu|^3} \right), \qquad (5.49)$$

and the first cut length is,

$$l_*(\theta, \nu) = \frac{2\pi}{|\nu|} + O_{\nu \to \infty} \left( \frac{1}{|\nu|^3} \right). \qquad (5.50)$$

**Remark 5.5.19** Theorems 3.2 and 4.2 in [1] provide also the local geometric behaviour of the cut and conjugate locus. For every $\mathbf{p}_0 = (\theta, \nu)$,

$$\begin{align}
\operatorname{Conj}_{\mathbf{x}_0}(\theta, \nu) &= \mathbf{x}_0 \pm \frac{\pi}{\nu^2} \mathbf{e}_{\mathbf{x}_0} \pm \frac{2\pi \chi_{\mathbf{x}}}{\nu^3} (\cos(\theta)^3 \mathbf{Z}_1(\mathbf{x}_0) - \sin(\theta)^3 \mathbf{Z}_2(\mathbf{x}_0)) + O_{\nu \to \infty} \left(\frac{1}{\nu^4}\right) \nonumber \\
\operatorname{Cut}_{\mathbf{x}_0}(\theta, \nu) &= \mathbf{x}_0 \pm \frac{\pi}{\nu^2} \mathbf{e}_{\mathbf{x}_0} \pm \frac{2\pi \chi_{\mathbf{x}} \cos(\theta)}{\nu^3} \mathbf{Z}_1(\mathbf{x}_0) + O_{\nu \to \infty} \left(\frac{1}{\nu^4}\right). \tag{5.51}
\end{align}$$

## 5.6 Swimmer in a potential flow: Numerics

There are basically two methods to compute the optimal strokes corresponding to Problems 5.4.5-5.4.9 for the example of swimmer in a potential flow presented in Subsection (5.2.5).

The first one consists in integrating Pontryagin's maximum principle (5.44) over a time interval $[0, T]$, specifying some initial data $(\xi_0, \mathbf{p}_0)$. Since we are mostly interested in the ending point $\xi(T)$, we can implement a so-called shooting method. For instance, seeking strokes, we want the $S$ component of the ending point $\pi_S\xi(T)$ to be the same as the $S$ component of the initial point $\pi_S\xi_0$. Let $\xi_0$ be given and for every $\mathbf{p}_0$ denote by $t \in [0, T] \mapsto \xi(t, \mathbf{p}_0)$ the solution to (5.44) with Cauchy data $(\xi_0, \mathbf{p}_0)$. We seek $\mathbf{p}_0$ as a solution to the equation $F(\mathbf{p}) = 0$ where $F$ is for instance defined (in a chart) by:

$$\mathbf{p} \mapsto \|\pi_S\xi(T, \mathbf{p}) - \pi_S\xi_0\|_{\mathbb{R}^2}.$$

Notice however that not every solution to Pontryagin's maximum principle is an optimal curve. So, rather than determining optimal strokes, we integrate (5.44) a large number of times in order to draw sub-Riemannian wavefronts. Sub-Riemannian wavefronts are precisely the surfaces constituted by the ending points of all the extremal starting at a given point and having the same fixed sub-Riemannian length.

<figure><img src="image_41.png" /></figure>
<figure><img src="image_42.png" /><figcaption>Table 5.1: The sub-Riemannian wavefront of length 0.1 computed by integrating Pontryagin’s maximum principle (5.44) with initial data satisfying $\xi_0 = (\mathbf{s}_0, 0)$, $\mathbf{s}_0 = (0.3, 0, 0)$. The picture is drawn in a chart, using spherical coordinates.</figcaption></figure>
<figure><img src="image_43.png" /></figure>
<figure><img src="image_44.png" /></figure>
<figure><img src="image_45.png" /><figcaption>Table 5.2: The sub-Riemannian wavefront of length 0.2 computed by integrating Pontryagin's maximum principle (5.44) with initial data satisfying $\xi_0 = (\mathbf{s}_0, 0)$, $\mathbf{s}_0 = (0.3, 0, 0)$. The picture is drawn in a chart, using spherical coordinates.</figcaption></figure>

The second method consists in approximating any curve on $\mathcal{S}$ by means of cubic splines. In the following examples, we use a basis of 20 cubic splines (actually, two bases since, working in spherical coordinates, we need 20 cubic splines for the polar angle and others 20 for the azimuth angle). So, we dispose of 40 parameters controlling the splines and the optimal problems under consideration turn into finite dimensional optimal problems. To every set of parameters, we can associate a travelled distance and a cost. To solve our optimal problems, we use the optimal toolbox of Matlab. The main difficulty is to manage the change of chart. Indeed, starting with the classical spherical coordinates, a curve cannot pass through the south or north pole of the ellipsoid. So we have to switch the axes, in such a way that the north pole becomes a regular point in spherical coordinates.

## First example: optimizing the cost

We consider a closed curve on the ellipsoid and compute the corresponding covered distance by the swimmer. Then, we try to minimize the cost among all the closed curves for which the travelled distance is the same. Notice that it is not exactly what is stated in Problem 5.4.5, because here there is no fixed starting point. The resulting curves on the ellipsoid are pictured on Fig 5.5, while the corresponding sequences of shapes are pictured on Fig 5.6 (initial guess) and 5.7 (optimized swimmer).

## Second example: The distance-cost function

Let us draw now the graph of the function defined (5.36c). Considering a shape *at rest* for the swimmer, we compute for every $\delta \in \mathbb{R}_+$, the optimal stroke (i.e. minimizing the cost) allowing the swimmer to cover the distance $\delta$. We choose as shape *at rest*, the converging point of all the curves on Fig 5.9. On the same picture, we drawn all the curves corresponding to the optimal strokes for different values of $\delta$. A sequence of shapes corresponding to the longest curve is given in Fig 5.10. Finally, the graph of the distance-cost function is given in Fig 5.8.
<figure><img src="image_46.png" /><figcaption>Figure 5.5: The initial closed curve is the equator of the ellipsoid (the dashed line). The optimized curve (the continuous line) is supposed to have the minimum cost for the same travelled distance. The colors are the same of in Fig 5.4.</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>Figure 5.6: Sequence of 20 time equidistributed shapes for the initial guess.</figcaption></figure>
<figure><img src="image_48.png" /><figcaption>Figure 5.7: Sequence of 20 time equidistributed shapes after optimization.</figcaption></figure>
<figure><img src="image_49.png" /><figcaption>Figure 5.8: For every targeted distance (in abscissa) we compute the corresponding optimal cost (in ordinate). The curve is pseudo-periodic because, above a certain distance, the best stroke is made of two optimal smaller loops.</figcaption></figure>
<figure><img src="image_50.png" /><figcaption>Figure 5.9: Every closed curve corresponds to an optimal stroke (minimizing the cost for a given distance).</figcaption></figure>
<figure><img src="image_51.png" /><figcaption>Figure 5.10: Sequence of 20 shapes time equi distributed corresponding to the longest curve on Fig 5.9.</figcaption></figure>
## 5.7 Appendix A: Riemannian Geometry

Let $(M, g)$ be a Riemannian manifold. We denote $\ell(\Gamma)$ the length of any rectifiable curve $\Gamma \subset M$ and for every $x \in M$ and $r > 0$, we denote by $B(x, r)$ the Riemannian ball centered at $x$ and of radius $r$. The following Lemma ensures the existence of a small monotone retract at any point of $M$.

**Lemma 5.7.1** Let $x_0$ be a point of $M$. Then there exists $r > 0$ such that, for every path $\gamma: [0, T] \mapsto M$ absolutely continuous with essentially bounded first derivative, such that:

1. $\gamma(t) \in B(x_0, r)$ for every $t \in [0, T]$;

2. $\gamma(0) = \gamma(T) = x_0$;

there exists a continuous function $\psi: [0, 1] \times [0, T] \mapsto M$ satisfying:

1. For every $s \in [0, 1]$, $t \in [0, T] \mapsto \psi(s, t)$ is continuous with essentially bounded first derivative.

2. $\psi(1, \cdot) = \gamma$;

3. $\psi(0, \cdot) = x_0$;

4. The function $s \in [0, 1] \mapsto \ell(\Gamma_s)$ (where $\Gamma_s$ is the curve parameterized by $t \in [0, T] \mapsto \psi(s, t)$) is increasing.

**Proof.** Let $\text{inj}(x_0)$ be the injectivity radius at $x_0$ and denote $\mathcal{V} := B(x_0, r)$, for $0 < r < \text{inj}(x_0)$ (the constant $r$ will be fixed later on). Let $\gamma$ be a path included in $\mathcal{V}$ and satisfying the hypotheses of the lemma. Then define:

$$ \zeta: t \in [0, T] \mapsto \zeta(t) := \exp_{x_0}^{-1}(\gamma(t)) \in T_{x_0} M. $$

This function has the same regularity as $\gamma$ and $\zeta(0) = \zeta(T) = 0$. Define now, for every $(s, t) \in [0, 1] \times [0, T]$:

$$ \psi(s, t) = \exp_{x_0}(s\zeta(t)). $$

This function has the required regularity and satisfies the equalities $\psi(1, \cdot) = \gamma$ and $\psi(0, \cdot) = x_0$. So it remains only to prove that for $r$ small enough, the length of $\Gamma_s$ is increasing in $s$.

In the exponential map the metric $g$ have the following Cartan local development:

$$ g_{ij}(x) = \delta_i^j - \frac{1}{3} \sum_{k,l} R_{iklj}(0) x_k x_l + \mathcal{O}(\|x\|_E^3), \qquad (5.52) $$

where $\delta_i^j$ is the Kronecker symbol, $R_{ijkl}$ are the coefficients of the Riemann curvature tensor and $\|x\|_E$ stands for the Euclidean norm. The quantity we are interested in estimating is:

$$ \ell(\Gamma_s) = \int_0^T \|\partial_t \psi(s, t)\|_{g(\psi(s, t))} \, dt, \qquad (5.53) $$

and in the local chart, according to (5.52), we have:

$$ \|\partial_t \psi(s, t)\|_{g(\psi(s, t))}^2 = s^2 \|\dot{\zeta}(t)\|_E^2 - s^4 \left( \frac{1}{3} \sum_{i,j} \sum_{k,l} R_{iklj}(0) \zeta_k(t) \zeta_l(t) \dot{\zeta}^i(t) \dot{\zeta}^j(t) \right) + s^3 (\|\dot{\zeta}(t)\|_E^2 \mathcal{O}(\|\zeta(t)\|_E^3)). $$

So for $r$ small enough, $\zeta(t)$ is uniformly small and for every $t \in [0, T]$, the function $s \in [0, 1] \mapsto \|\partial_t \psi(s, t)\|_{g(\psi(s, t))}^2$ is increasing. We draw the same conclusion for the quantity (5.53) and proof is completed.
## 5.8 Appendix B: A brief Survey of the Orbit Theorem

In this Appendix, we aim to recall the statement of the Orbit Theorem. The material presented below is now considered as a classical part of geometric control theory.

Throughout this section, $M$ is a real analytic manifold, and $\mathcal{G}$ a set of analytic vector fields on $M$. We do not assume in general that the fields from $\mathcal{G}$ are complete.

### 5.8.1 Attainable sets

Let $f$ be an element of $\mathcal{G}$ and $q^*$ be an element of $M$. The Cauchy problem

$$ \dot{q} = f(q), \quad q(0) = q^*, \qquad (5.54) $$

admits a solution defined on the open interval $I(f, q^*)$ containing 0. For any real $t$ in $I(f, q^*)$ we denote the value of the solution of (5.54) at time $t$ by $e^{tf}(q^*)$. We denote by $I(f, q^*)^+ = I(f, q^*) \cap ]0, +\infty[$ the positive elements of $I(f, q^*)$.

For any element $q_0$ in $M$ and any positive real number $T$, we define the *attainable set at time $T$* of $\mathcal{G}$ from $q_0$ by the set $\mathcal{A}_{q_0}(T)$ of all points of $M$ that can be attained with $\mathcal{G}$ using piecewise constants controls in time $T$

$$ \mathcal{A}_{q_0}(T) = \left\{ e^{t_p f_p} \circ e^{t_{p-1} f_{p-1}} \circ \dots \circ e^{t_1 f_1}(q_0) : p \in \mathbf{N}, f_i \in \mathcal{G}, t_i \in I(f_i, e^{t_{i-1}f_{i-1}} \circ \dots \circ e^{t_1 f_1}(q_0))^+, t_1 + \dots + t_p = T \right\} $$

the times $t_i$ and the fields $f_i$ being chosen in such a way that every written quantity exists. We define also the *orbit of $\mathcal{G}$* trough $q_0$ by the set $\mathcal{O}_{q_0}$ of all points of $M$ that can be attained with $\mathcal{G}$ using piecewise constant controls, *at any positive or negative time*

$$ \mathcal{O}_{q_0}(T) = \left\{ e^{t_p f_p} \circ e^{t_{p-1} f_{p-1}} \circ \dots \circ e^{t_1 f_1}(q_0) : p \in \mathbf{N}, f_i \in \mathcal{G}, t_i \in I(f_i, e^{t_{i-1}f_{i-1}} \circ \dots \circ e^{t_1 f_1}(q_0)) \right\}. $$

Of course, if $\mathcal{G}$ is a cone, that is if $\lambda f \in \mathcal{G}$ for any positive $\lambda$ as soon as $f$ belongs to $\mathcal{G}$, the set $\mathcal{A}_{q_0}(T)$ does not depend on the positive $T$ but only on $q_0$. If $\mathcal{G}$ is assumed to be symmetric, that is if $-f$ belongs to $\mathcal{G}$ as soon as $f$ belongs to $\mathcal{G}$, then the orbit of $\mathcal{G}$ trough a point $q_0$ is the union of all attainable sets at positive time of $\mathcal{G}$ from $q_0$.

### 5.8.2 Lie algebra of vector fields

If $f_1$ and $f_2$ are two vector fields on $M$ and $q$ is a point of $M$, the *Lie bracket* $[f_1, f_2](q)$ of $f_1$ and $f_2$ at a point $q$ is the derivative at $t = 0$ of the curve $t \mapsto \gamma(\sqrt{t})$ where $\gamma$ is defined by $\gamma(t) := e^{-tf_2}e^{-tf_1}e^{tf_2}e^{tf_1}(q)$ for $t$ small enough. The Lie bracket of $f_1$ and $f_2$ at a point $q$ is an element of the tangent space $T_q M$ of $M$ at the point $q$. The Lie bracket is bilinear and skew-symmetric in $f_1$ and $f_2$, and measures the non-commutativity of the fields $f_1$ and $f_2$ (see [3, Prop 2.6]).

**Proposition 5.8.1** For any $f_1, f_2$ in $\mathcal{G}$, we have the equivalence:

$$ e^{t_1 f_1} e^{t_2 f_2} = e^{t_2 f_2} e^{t_1 f_1} \Leftrightarrow [f_1, f_2] = 0 $$

for all times $t_1$ and $t_2$ (if any) for which the expressions written in the left hand side of the above equivalence make sense.

Lie brackets of vector fields are easy to compute with the following formulas (see [3, Prop 1.3] and [3, Exercise 2.2]).
**Proposition 5.8.2** For any $f_1, f_2$ in $\mathcal{G}$, for any $q$ in $M$,

$$[f_1, f_2](q) = \frac{df_2}{dq} f_1(q) - \frac{df_1}{dq} f_2(q).$$

Further, we have the useful property:

**Proposition 5.8.3** Let $f_1$ and $f_2$ be two smooth vector fields on $M$, and let $a, b : M \to \mathbb{R}$ be two smooth functions. Then

$$[aX, bY] = ab[X, Y] + \left(\frac{db}{dq}X\right)Y - \left(\frac{da}{dq}Y\right)X.$$

From the Lie brackets, we can define the Lie algebra:

**Definition 5.8.4** The Lie algebra of $\mathcal{G}$ is the linear span of all Lie brackets, of any length, of the elements of $\mathcal{G}$

$$\text{Lie } \mathcal{G} = \text{span}\{[f_1, [\dots[f_{k-1}, f_k]\dots]], k \in \mathbb{N}, f_i \in \mathcal{G}\},$$

which is a subset of all the vector fields on $M$.

We denote by $\text{Lie}_q\mathcal{G} := \{g(q), g \in \text{Lie } \mathcal{G}\}$ the evaluation $\text{Lie}_q\mathcal{G}$ of the Lie algebra generated by $\mathcal{G}$ at a point $q$ of $M$.

### 5.8.3 The Orbit Theorem

The Orbit Theorem describes the differential structure of the orbit through a point (see for instance [3, Th 5.1] for a proof).

**Theorem 5.8.5 (Orbit Theorem)** For any $q$ and $q_0$ in $M$:

1. $\mathcal{O}(q_0)$ is a connected immersed submanifold of $M$.

2. If $q \in \mathcal{O}(q_0)$, then $T_q\mathcal{O}(q_0) = \text{Lie}_q\mathcal{G}$.

**Remark 5.8.6** The conclusion (1) of the Orbit Theorem holds true even if $M$ and $\mathcal{G}$ are only assumed to be smooth (and not analytic). The conclusion (2) is false in general when $\mathcal{G}$ is only assumed to be smooth.

The Orbit Theorem has many consequences, among them the following useful properties (see [3, Th 5.2] for a proof and further discussion).

**Theorem 5.8.7 (Rashevsky-Chow)** If $\text{Lie}_q\mathcal{G} = T_qM$ for every $q$ in $M$, then the orbit of $\mathcal{G}$ through $q$ is equal to $M$.

**Proposition 5.8.8** If $\mathcal{G}$ is a symmetric cone such that $\text{Lie}_q\mathcal{G} = T_qM$ for every $q$ in $M$, then the attainable set at any positive time of any point of $M$ is equal to $M$.
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Résumé

Cette thèse de mathématiques appliquées traite de la modélisation des déplacements de nageurs microscopiques. Nous étudions principalement les problèmes de contrôlabilité et d'optimalité associés à la mobilité d'un micro-nageur.

Dans une première partie, nous présentons un modèle de nageur simplifié, appelé le “N-link swimmer”. Ensuite, nous étudions sa contrôlabilité ainsi que l’existence de stratégies lui permettant d’atteindre un point donné le plus vite possible. Dans une deuxième partie, nous analysons les effets de la présence d’un bord sur la mobilité d’un micro-nageur. Nous montrons qu’un nageur qui est contrôlable lorsqu’il évolue dans l’espace non borné, reste “presque partout” localement contrôlable lorsqu’il nage dans un domaine délimité par un mur plat ou rugueux. Au contraire, nous prouvons qu’un nageur qui n’est pas capable d’atteindre toutes les directions lorsqu’il se déplace dans un domaine sans bord peut élargir ses directions accessibles en présence d’un mur (plat ou rugueux). Enfin, la dernière partie de la thèse fournit un cadre à l’étude de problèmes de contrôle optimal associés aux déplacements de nageurs ayant une dynamique sans dérive. Tout d’abord, nous étudions les propriétés mathématiques de plusieurs problèmes de contrôle optimal ayant des coûts fonctionnels différents (existence puis comportement). Ensuite, nous considérons les nageurs ayant deux degrés de liberté. Pour ces modèles particuliers de nageurs, nous présentons un cadre permettant d’en déduire des propriétés géométriques locales pour les solutions de certains problèmes de contrôle optimal. Tout au long de ce dernier chapitre, des simulations numériques, réalisées sur un exemple de nageur ayant une dynamique explicite, illustrent les résultats théoriques.

**Mots-clés:** Locomotion à faible nombre de Reynolds ; Interaction fluide-structure ; Contrôle optimal géométrique ; Modèles de robots microscopiques

Abstract

This thesis is devoted to the mathematical study of the swimming at low Reynolds number. The controllability and the optimal problems associated with the displacement of micro-swimmers are the main points developed in this work.

In the first part, we study the controllability and the optimal control problem in time associated with a reduced model of swimmers, called the “N-link swimmer”. In the second part, we study the boundary effect on the controllability of particular micro-swimmers made by several balls linked each others by thin jacks. Firstly, we analyze the effect of a plane wall on the mobility of these swimmers. Then, we generalize these results where the wall is rough. We demonstrate that a controllable swimmer remains controllable in a half space delimited by a wall (plane or rough) whereas the reachable set of a non controllable one is increased by the presence of a wall. The last part is devoted to provide a general framework to study optimal controllability of driftless swimmers. We focus on the study of optimal strokes i.e. periodic shape changes. More precisely, we are interested in the existence of optimal strokes, minimizing or maximizing various cost functionals, qualitative properties of the optimal strokes, regularity and monotony of the value functions.

**Keywords:** Swimming at low Reynolds number ; Fluid-structure interaction ; Geometric optimal control theory ; Natural and artificial micro-swimmers
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