ON THE CONTROL OF THE MOTION OF A BOAT

In this paper we study the motion of a boat, viewed as a rigid body S with one axis of symmetry, which is surrounded by an inviscid incompressible fluid filling R^2 -S. We take as control input the flow of the fluid through a part of the boundary of the boat. We prove that the position, orientation, and velocity of the boat are locally controllable with a bidimensional control input, even if the flow displays some vorticity.


Introduction
The control of boats or submarines has attracted the attention of the mathematical community from a long time (see e.g. [14,15,16,2].) In most of the papers devoted to that issue, the fluid is assumed to be inviscid, incompressible and irrotational, and the rigid body is supposed to have an elliptic shape. On the other hand, to simplify the model, the control is often assumed to appear in a linear way in a finite-dimensional system describing the dynamics of the rigid body, the so-called Kirchhoff laws.
A large vessel (e.g. a freighter) presents often one tunnel thruster built into the bow to make docking easier. The aim of this paper is to provide some accurate model of a boat controlled by two propellers, the one displayed in a transversal bowthruster at the bow of the ship, the other one placed at the stern of the boat (see Figure 1), and to give a rigorous analysis of the control properties of such a system. The fluid, still inviscid and incompressible, will no longer be assumed to be irrotational (i.e., the vorticity may not vanish everywhere), and the only geometric assumption for the shape of the boat will be the existence of one axial symmetry.
To be more precise, we consider a boat, represented by a rigid body with one axis of symmetry occupying a bounded, simply connected open set S(t) ⊂ R 2 of class C ∞ and which is surrounded by a homogeneous incompressible perfect fluid. We denote by Ω(t) = R 2 \ S(t) the domain occupied by the fluid, and write merely S = S(0) and Ω = Ω(0) for the domains occupied respectively by the rigid body and the fluid at t = 0. The equations for the dynamics of the system fluid + rigid body read then ∂u ∂t + (u · ∇)u + ∇p = 0, t ∈ [0, T ], x ∈ Ω(t), x ∈ Ω, (1.8) (h(0), θ(0)) = (h 0 , θ 0 ) ∈ R 2 × R, (h (0), r(0)) = (h 1 , r 0 ) ∈ R 2 × R. (1.9) In the above equations, u (resp. p) is the velocity field (resp. the pressure) of the fluid, h denotes the position of the center of mass of the solid, θ is the angle between some axis linked to the rigid body and a given fixed axis, and r denotes the angular velocity. The positive constants m and J, which denote respectively the mass and the moment of inertia of the rigid body, are defined as where ρ(·) denotes the density of the rigid body. The vector n is the outward unit vector to ∂Ω(t), so that τ = −n ⊥ = (n 2 , −n 1 ) is a unit tangent vector to ∂Ω(t). Finally, the term w(t, x), which stands for the flow through the boundary of the rigid body, is taken as control input. Its support will be strictly included in ∂Ω(t), and actually only a finite dimensional control input will be considered here (see below (1.13) for the precise form of the control term w(t, x)).
When no control is applied (i.e. w(t, x) = 0), then the existence and uniqueness of strong solutions to (1.1)-(1.9) was obtained in [18] for a ball, and in [19] for a rigid body S of arbitrary form. The result in [18] was extended to any dimension in [20] (in that paper, the issue of the persistence of regularity is also studied). We also refer to [10] for the situation when Ω(t) = Ω 0 \ S(t), with Ω 0 a bounded open set in R 3 , and for the issue of the analyticity in time. The detection of the rigid body S(t) from partial measurements of the fluid velocity has been tackled in [5] when Ω(t) = Ω 0 \ S(t) (Ω 0 ⊂ R 2 still being a bounded cavity) and in [4] when Ω(t) = R 2 \ S(t).
Here, we are interested in the control properties of (1.1)-(1.9). The controllability of Euler equations has been established in 2D (resp. in 3D) in [6] (resp. in [9]) Note, however, that there is no hope here to control both the fluid and the rigid body motion. Indeed, Ω(t) is an exterior domain, and the vorticity is transported by the flow with a finite speed propagation, so that it is not affected (at any given time) far from the boat. Therefore, we will deal with the control of the motion of the rigid body only. As the state of the rigid body is described by a vector in R 6 , namely (h, θ, h , r), it is natural to consider a finite-dimensional control input.
Note also that since the fluid is flowing through a part of the boundary of the rigid body, one more boundary condition is needed to ensure the uniqueness of the solution of (1.1)-(1.9) (see [11], [12]). In dimension two, one can impose the value of the vorticity ω(t, x) := curl v(t, x) on the inflow section of ∂Ω(t); that is, one can set ω(t, x) = ω 0 (t, x) for w(t, x) < 0 where ω 0 (t, x) is a given function.
In order to write the equations of the fluid in a fixed domain, we perform a change of coordinates. We set θ(t) = θ 0 + t 0 r(s) ds, Q(θ) = cos θ − sin θ sin θ cos θ , (1.10) and x = Q(θ(t))y + h(t), (1.11) where x (resp. y) represents the vector of coordinates of a point in a fixed frame (respectively in a frame linked to the rigid body). Note that, at any given time t, y ranges over the fixed domain Ω when x ranges over Ω(t). Next, we introduce the functions    v(t, y) := Q(θ(t)) * u(t, Q(θ(t))y + h(t)), q(t, y) := p(t, Q(θ(t))y + h(t)), l(t) := Q(θ(t)) * ḣ (t), (1.12) where˙= d/dt and * means transpose. Finally, we assume that the control takes the form w(t, x) = w(t, Q(θ(t))y + h(t)) = 1 j m w j (t)χ j (y), (1.13) where m ∈ N * stands for the number of independent inputs, and w j (t) ∈ R is the control input associated with the function χ j ∈ C ∞ (∂Ω). Often, the functions χ j have disjoint supports, i.e. χ j (y)χ k (y) = 0 ∀y ∈ ∂Ω, ∀j = k, but we shall not make this hypothesis thereafter. To ensure the conservation of the mass of the fluid, we impose the relation ∂Ω χ j (y) dσ = 0 for 1 j m. (1.14) Finally, we assume that the solid is symmetric with respect to the y 1 −axis (see Figure 2), i.e.
The paper is organized as follows.
In Section 2, we first consider potential flows. In that case, we obtain a finite-dimensional system similar to Kirchhoff laws, in which the control input w appears through both linear terms (with time derivative) and bilinear terms. To investigate the controllability of such a system, we apply the return method due to Jean-Michel Coron (we refer the reader to [7] for an exposition of that method for finite-dimensional systems and for PDEs). We consider the linearization along a certain closed-loop trajectory and obtain a local controllability result (Theorem 2.9) assuming that two conditions are fulfilled, by using a variant of Silverman-Meadows test for the controllability of a time-varying linear system. A difficulty in the previous result is that the control may be different from 0 at the final time. This inconvenient disappears for certain values of the constants in the system, leading to a (global) steady-state controllability result (Theorem 2.14).
Next, we come back to the original system (1.18)-(1.25) in Section 3. We prove that it admits a global solution for a convenient choice of the vorticity at the inflow section of ∂Ω such that the difference between the present velocity and the potential velocity can be estimated by some measurement of the vorticity at time t = 0 (Proposition 3.1).
Combining the results in Sections 2 and 3, we obtain in Section 4 the main result of the paper (Theorem 4.1), namely a local controllability result for the dynamics of the boat even if the flow is not potential. Such a result, which may be seen as a "linearization" with respect to the vorticity, involves in its proof a topological argument (Lemma 4.2).
Finally Section 5 is an appendix containing some computations that can be skipped during a first reading.
2. Potential flows 2.1. Equations of the motion in the potential case.
In this section we derive the equations describing the motion of the rigid body subject to flow controls on ∂Ω when the flow of the fluid is potential. We still denote by v 0 (·) the velocity of the fluid at t = 0, and here we assume that We also assume that the vorticity ω = curl v is null at the inflow part of ∂Ω, i.e.
The proof of the first part of Lemma 2.2 may be done along the same lines as [20,Proposition 3.1]. For the second part, it is sufficient to write Ψ(y 1 , y 2 ) = Re f (y 1 + iy 2 ) where f is some holomorphic function on Ω ⊂ R 2 ∼ C and to note that the expansion as a Laurent series of f reads f (y 1 + iy 2 ) = k 0 a k (y 1 + iy 2 ) k for some sequence (a k ) k 0 in C, since lim |y|→∞ ∇Ψ(y) = 0.
As the domain S occupied by the rigid body and the functions χ j , 1 j m, supporting the control are assumed to be smooth, we infer that the functions ∇Φ i (i = 1, 2, 3) and the functions ∇Ψ j (1 j m) are in the space H ∞ (Ω). As a consequence, we notice that for all i = 1, 2, 3 and j = 1, . . . , m, Let us now reformulate the equations for the motion of the rigid body. We define the matrix M ∈ R 3×3 by If, in addition, (y 1 , y 2 ) ∈ S ⇒ (−y 1 , y 2 ) ∈ S, (2.24) then both M and J are diagonal.
Recall thatḣ = Q(θ)l. Introduce the 3 × 3 matrix In the potential case, the dynamics of the boat can be written in the following way.
Proposition 2.4. The dynamics of (q, p) reaḋ and F (p, w) is composed of bilinear terms in p i , 1 i 3, and w j , 1 j m.
The detailed computations yielding Proposition 2.4 are given in Appendix. In particular, the equation (2.26) with the term F (p, w) is made explicit in (5.32) -(5.34).

Elementary approaches.
Let us have a look at the linearization of (2.25)-(2.26) at the origin, namelẏ The following result relates the controllability of (2.27)-(2.28) to the rank of C.
Proof. To apply Kalman rank test (see e.g. [7,22]), we compute and the result follows at once.
In particular, the controllability of the linearized system requires that the number m of control inputs satisfies m 3.
Example 2.6. When the boat has an elliptic shape, i.e.
Thus we cannot control both (q 1 , p 1 ) and w 1 . One may wonder whether it is possible to require that the fluid be at rest when the rigid body is, that is if the condition w 1 (0) = w 1 (T ) = 0 (2.37) may be imposed when p 1 (0) = p 1 (T ) = 0. The following result shows that this occurs for a very particular set of coefficients.

2.3.
Local controllability of the potential model.
Thereafter, we will still assume that (w 1 (0), w 2 (0)) = (0, 0), but we shall refrain from imposing the condition (w 1 (T ), w 2 (T )) = (0, 0). The main result in this section (see below Theorem 2.9) is derived in following a strategy inspired in part from Coron's return method. It is organized as follows: • Step 1. We construct a (non trivial) trajectory (q, p, w) such that (q(0), p(0)) = (q(T ), p(T )) = (0, 0). (2.39) To do that, proceeding as in the flatness approach by Fliess et al. [8], we consider a particular trajectory (q 1 , p 1 ) of (2.30) and next define the control input w 1 by solving the Cauchy problem (2.31) together with the initial condition w 1 (0) = 0. Notice that w 1 (T ) may be different from 0, and that w 1 is not required to be odd with respect to the time T /2. On the other hand, w 2 ≡ 0 so that q 2 = p 2 = q 3 = p 3 ≡ 0. • Step 2. We show that the linearization along this trajectory is controllable (under suitable assumptions) by combining the classical Silverman-Meadows criterion to a test for the controllability of a linear system involving a control input together with its time derivative. For notational convenience, we introduce the matrices Simple but tedious computations give We also need to introduce the matrix The following result shows that, under suitable assumptions, the local controllability holds with only two control inputs. Theorem 2.9. If both rank conditions are fulfilled, then for any T > 0 the systeṁ with state (q, p) ∈ R 6 and control w ∈ R 2 is locally controllable around the origin in time T . We can also impose that the control input w satisfies w(0) = 0. Moreover, for some , which associates to (q 0 , p 0 , q T , p T ) a control satisfying w(0) = 0 and steering the state of the system from Remark 2.10.

2.3.2.
Step 2. Controllability of the linearized system. Writing q = q + z, p = p + k, w = w + f , expanding in (2.25)-(2.26), and keeping only the first order terms in z, k, and f , we obtain the following linear system and For a time-varying linear systeṁ If φ(t, s) ∈ R n×n denotes the fundamental solution associated with A(·), i.e. the solution of the Cauchy problem We shall apply a controllability test due to Silverman-Meadows (see [7]) in a slightly extended form. For the sake of completeness, we prove it in the appendix.

Define a sequence of matrices
where φ denotes the fundamental solution defined in (2.50)-(2.51).
and denote by R the reachable set from the origin, i.e.
Note that u may take arbitrary values at t = 0, T . It may be necessary to impose that u(0) = 0 and/or that u(T ) = 0. Accordingly, we introduce the spaces Note that, by an obvious density argument, we may as well assume that u ∈ C ∞ ([0, T ], R 2 ) in the above definitions of the reachable spaces, without changing these spaces.
Then the following result holds true.
Proposition 2.13. The reachable sets from the origin for the system (2.55) are respectively . Integrating by part, and using the fact that ∂[φ( As we may assume in (2.52) that u ranges over On the other hand, taking u(t) = tu 0 (resp. u(t) = (T − t)u 0 ) where u 0 is an arbitrary constant vector, and using (2.59), we obtain that CR m ⊂ R (resp. φ(T, 0)CR m ⊂ R). This completes the proof of (2.56). (2.57) and (2.58) follow at once.
We shall establish the controllability of (2.48) by combining Propositions 2.12 and 2.13. We denote by (M i ) i 0 the sequence of matrices associated with the pair (A,

Straightforward but tedious computations give
, From Propositions 2.12 and 2.13, we know that Now to establish the controllability of the linearized system, we distinguish between two cases.

2.3.3.
Step 3. Local controllability of the nonlinear system. Let us introduce the Hilbert space endowed with its natural Hilbertian norm . We denote by B H (0, δ) the open ball in H with center 0 and radius δ, i.e.
Let us introduce the map where (q(t), p(t)) denotes the solution oḟ Note that Γ is well defined for δ > 0 small enough. Using the Sobolev embedding Theorem 1], we infer that Γ is of class C 1 on B H (0, δ) and that its tangent linear map at the origin is given by where (z(t), k(t)) solves the system is invertible, and therefore it follows from the inverse function theorem that the map Γ |V : V → R 12 is locally invertible at the origin. More precisely, there exists a number δ > 0 and an open set Ω ⊂ R 12 containing 0 such that the map Γ : B H (0, δ) ∩ V → Ω is well-defined, of class C 1 , invertible, and with an inverse map of class C 1 . Let us denote this inverse map by Γ −1 , and let us write and that for ||(q 0 , p 0 , q T , p T )|| < η, the solution (q(t), p(t)) of the systeṁ The proof of Theorem 2.9 is complete.

A global steady-state controllability result.
Theorem 2.9 is a local controllability result. A global controllability result may be obtained when γ + αβ = 0.
Theorem 2.14. If γ + αβ = 0 and then for any (q 0 , q T ) ∈ R 6 there exists a time T > 0 and a control input w ∈ H 2 (0, T ; Proof. It may be assumed without loss of generality that q 0 = (0, 0, 0). We first establish a local controllability result around the equilibrium point (q 0 , 0). This is done along the same lines as for Theorem 2.9, using again the return method with the same reference trajectory (q, p, w). (Note that w(0) = w(T ) = 0.) However, the new constraint w ∈ H 1 where the sequence of matrices (M i ) i 0 is the one associated with the pair (A, B + AC). As a continuation of the computations performed in the proof of Theorem 2.9, we obtain Sinceẇ 1 (T ) = α −1ṗ 1 (T ) and ...
The proof of Theorem 2.14 is complete.
3. Wellposedness of the system (1.18)-(1.25) with vorticity In the previous section we considered the system (1.18)-(1.25) assuming that no vorticity was present. In this section, we prove that in presence of vorticity one can still construct a regular solution of (1.18)-(1.25) having a prescribed normal velocity, as for the solution we had constructed in Section 2. Using the results of Sections 2 and 3 and a time-scaling argument, we shall derive in Section 4 a controllability result for the whole system (with vorticity).
(2) We do not assume here that w(0) = 0, as we shall consider later maximal solutions obtained by concatenation of solutions over time.

3.2.1.
Notations. Let π be a continuous linear extension operator from functions defined in Ω to functions defined in R 2 , which maps C k,α (Ω) to C k,α (R 2 ) for all k ∈ N and all α ∈ (0, 1), and the space LL(Ω) of log-Lipschitz functions on Ω to LL(R 2 ). (The construction of such an "universal" extension operator is classical, see e.g. [23].) We may also ask that π preserves the divergence-free character (acting on the stream function if necessary).

3.2.2.
Rephrasing the system. Now we rephrase a little bit the system. As before for potential flows, we show that the pressure solves some elliptic problem. Next, we replace the pressure by its expression in (3.5)-(3.6) to formulate in a new way the dynamics of the boat. The Laplacian of the pressure q is given by where we used Einstein's convention of repeated indices and the fact that div(v) = 0. Next, we compute the normal derivative of q on ∂Ω. ∂q ∂n Next, we introduce the function µ defined as the solution of the following elliptic problem: Note that for v ∈ C k+1,α (Ω) with ∇v ∈ L 2 (Ω)∩L 4 θ (Ω) and θ > 2, then f ∈ L 1 (Ω)∩L 2 θ (Ω), g ∈ C k,α (∂Ω) with the compatibility condition We see that lṙ .
Noting that ∂Ω qK i dσ = Ω ∇q · ∇Φ i dy we deduce the following form of (3.5)-(3.6) m After these preliminaries, we prove Proposition 3.1 in several steps. First, we prove the local-in-time existence of solutions (that is, up to some time T which may be less than T ), by means of Schauder's fixed point theorem. Next, we prove that such a solution can be extended up to time T by using some a priori estimates.

The operator.
We first define an operator whose fixed points will give local-in-time solutions of (3.20).

M . (3.21)
Let us now define the operator T on C: to any (l, r, ω) ∈ C, we associate T (l, r, ω) := (l,r,ω), (3.22) as follows. First, we introduce the "fluid velocity" v as the solution to the system We definel andr as follows: This completes the definition of T .

4.
It is easy to check that C is convex and closed for the uniform topology on (l, r, ω), i.e. in E = L ∞ (0, T , R 2 × R × L ∞ (Ω)). We claim that T (C) is relatively compact in E. For the (l, r) component, it is sufficient to use (3.33) and the compactness of the embedding , for 0 < α < α, and 2 < θ < θ.
On the other hand,Φ n (s, t, y) →Φ(s, t, y) uniformly on compact sets of [0, T ] 2 × R 2 , so thatω n (t, y) →ω(t, y) uniformly on compact sets of [0, T ] × R 2 . Since T (C) is relatively compact in E, we conclude thatω n →ω in L ∞ (0, T , L ∞ (Ω)). The continuity of T is proved. The conclusion of Lemma 3.3 follows then from Schauder's fixed point theorem.

Remark 3.4.
(1) If k 1, it is possible to show that the operator T contracts for small T in the norm · L ∞ (0,T ;C k−1,α (Ω)) , using a priori estimates on v in C k+1,α (Ω) and Gronwall's lemma.

3.2.5.
A priori estimates and global existence. Let us consider a solution of (3.1)-(3.8) defined on a maximal interval of existence (0, T * ) (with 0 < T * T ), fulfilling (3.13)-(3.15) for all T < T * , and satisfying as the solution constructed above that whereω(t, y) =ω 0 (Φ(0, t, y)) andΦ denotes as above the flow associated withπ(v)−l−ry ⊥ . In particular, v satisfies We will establish an a priori estimate on (l, r, ω) in a suitable space. With Lemma 3.3 and a standard procedure, this will give that T * = T .
The above computations are legitimate thanks to (3.13)-(3.15) (with T replaced by any T < T * ). Note that (3.26) is essential here to have a solution with a finite energy.
a. From div(v − l − ry ⊥ ) = 0 we infer that I 2 = 0 and that the first integral term in I 4 is nul.

It follows that
d dt E C(1 + E).
Hence, by Gronwall's lemma, E remains bounded up to time T * .
Hence the claim for |(l, r)−(l, r)| follows from Gronwall's lemma. The proof of Proposition 3.1 is complete.

We introduce the functions
which satisfy the following system ∂v ∂t In order to prove that (v, l, r) = (0, 0, 0), we establish some energy estimate for (3.56)-(3.63).
It follows from standard elliptic estimates that Therefore with Φ(s, s, y) = 0.
Combining (3.64) to (3.65), we infer that On the other hand, we have that We can estimate each part: On the other hand, Finally we have that Using (3.56) we conclude that ∇q = 0 in (0, T ) × Ω. We have proved the uniqueness of (v, q, l, r) in the class (3.13)-(3.15). Let us show now that the map w ∈ H 2 (0, T ) → (l, r) ∈ C([0, T ]) is continuous. Assume that w k → w in H 2 (0, T ), and let (v k , q k , l k , r k ) denote the solution of (3.1)-(3.8) and (3.53) associated with the initial data (v 0 , l 0 , r 0 ) and the control w k . Since we infer from the bootstrap argument in the proof of Proposition 3.1 that ∇q k is bounded in L ∞ (0, T ; L 2 (Ω)), and that (l k , r k ) is bounded in W 1,∞ (0, T ).
Extracting subsequences, we can assume that and that for some functions l, r, ω , for all 2 < θ < θ and all 0 < α < α. This yields for some fonctions v, q that We can therefore pass to the limit in (3.1)- (3.8). We also notice that if Φ k (resp. Φ) denotes the flow associated with π(v k ) − l k − r k y ⊥ (resp. with π(v) − l − ry ⊥ ), then pointwise. Thus (3.53) holds. We conclude that (v, q, l, r) is the unique solution of (3.1) -(3.8) and (3.53) associated with the data (v 0 , l 0 , r 0 ) and the control w in the class (3.13)-(3.15) (with α replaced by α ). The proof of Proposition 3.5 is achieved.

Main result
From now on, the pressure will be denoted by q. It should not be confused with the state vector q = (h 1 , h 2 , θ). We are now in a position to state and prove the main result in this paper.

Proof of Proposition 2.4.
We first express the pressure q in terms of l, r, v and their derivatives. Using (2.4), we easily obtain v · ∇v = ∇ |v| 2 2 and − ry ⊥ · ∇v + rv ⊥ = r∇(y · v ⊥ ) (5.4) Replacing q by its value in (1.22) yields Using the following identity we obtain that Therefore, from (5.6) and (5.8), we obtain Let us turn our attention to the dynamics of r. Substituting the expression of q given in (5.5) in (1.23) yields Using (5.4) and the fact that div (y ⊥ ) = 0 we obtain where we used again the identity (5.7). We conclude that Before expanding the bilinear terms in (5.9)-(5.10), we exploit the symmetries in the shape of the rigid body and in the location of the control inputs in order to write only the nonvanishing terms in the final system. Recall that we have assumed that S (hence also Ω) is symmetric with respect to the y 1 -axis, i.e.
This yields p * φ(T, t)B(t) ≡ 0. We claim the following: for all i 0, we have p * φ(T, t)M i (t) ≡ 0. The claim is proved, and we infer that p ∈ R ⊥ .