Rodrigo Lecaros
email: rlecaros@dim.uchile.cl

Lionel Rosier
email: lionel.rosier@mines-paristech.fr

CONTROL OF UNDERWATER VEHICLES IN INVISCID FLUIDS-II. FLOWS WITH VORTICITY

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Introduction

An accurate model for the motion of a boat (without rudder) equipped with tunnel thrusters was investigated in [START_REF] Glass | On the control of the motion of a boat[END_REF]. In that paper, using Coron's return method (see [4]), the authors proved that it was in general possible to control both the position and the velocity of the boat (a vector in R 6 ) by using two control inputs. The fluid was assumed to be inviscid, but not necessarily irrotational, and its motion was described by Euler equations for incompressible fluids.

In [START_REF] Lecaros | Control of underwater vehicles in inviscid fluids-I. Irrotational flows[END_REF], the authors started the study of the controllability of an underwater vehicle S (e.g. a submarine) immersed in an infinite volume of an inviscid fluid (filling R 3 \S). Assuming that the fluid was irrotational, they proved by using Coron's return method the controllability of both the position and the velocity of the vehicle (a vector in R 12 ) by using 6, or 4, or merely 3 control inputs for appropriate geometries. The aim of the present paper is to pursue this study by considering the more general case of a flow with vorticity. We will show that the local controllability of both the position and the velocity of the underwater vehicle holds in a flow with vorticity whenever it holds in a flow without vorticity. The method of proof is inspired by the one of [START_REF] Glass | On the control of the motion of a boat[END_REF]: the extension of the exact controllability to a system with a (small) vorticity is achieved by a perturbative approach relying on a topological argument. Next, the small vorticity assumption is removed by using a scaling argument. However, to prove the wellposedness of the complete system we shall use here the contraction mapping theorem instead of the Schauder fixed-point theorem as in [START_REF] Glass | On the control of the motion of a boat[END_REF]. This choice leads to a more straightforward proof.

Our fluid-structure interaction system can be described as follows. The underwater vehicle, represented by a rigid body occupying a connected compact set S(t) ⊂ R 3 , is surrounded by a homogeneous incompressible perfect fluid filling the open set Ω(t) := R 3 \ S(t) (as e.g. for a submarine immersed in an ocean). We assume that Ω(t) is C ∞ smooth and connected. Let S = S(0) and Ω(0) = R 3 \ S(0) denote the initial configuration (t = 0). Then, the dynamics of the fluid-structure system are governed by the following system of PDE's

1 ∂u ∂t + (u • ∇)u + ∇p = 0, t ∈ (0, T ), x ∈ Ω(t), (1.1)
div u = 0, t ∈ (0, T ), x ∈ Ω(t), (1.2)

u • n = (h ′ + ζ × (x -h)) • n + w(t, x), t ∈ (0, T ), x ∈ ∂Ω(t), (1.3)
lim |x|→+∞ u(t, x) = 0, t ∈ (0, T ), (1.4)

m 0 h ′′ = ∂Ω(t)
pn dσ, t ∈ (0, T ), (1.5)

d dt (QJ 0 Q * ζ) = ∂Ω(t) (x -h) × pn dσ, t ∈ (0, T ), (1.6)
Q ′ = S(ζ)Q, t ∈ (0, T ), (1.7)
u(0, x) = u 0 (x),

x ∈ Ω(0), (1.8)

(h(0), Q(0), h ′ (0), ζ(0)) = (h 0 , Q 0 , h 1 , ζ 0 ) ∈ R 3 × SO(3) × R 3 × R 3 .
(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, ζ denotes the angular velocity and Q ∈ R 3×3 the rotation matrix giving the orientation of the solid. The positive constant m 0 and the matrix J 0 , which stand for the mass and the inertia matrix of the rigid body, respectively, are defined as

m 0 = S ρ(x)dx, J 0 = S ρ(x)(|x| 2 Id -xx * )dx,
where ρ(•) represents the density of the rigid body. The vector n is the outward unit vector to ∂Ω(t), x × y is the cross product between the vectors x and y, and S(y) is the skew-adjoint matrix such that S(y)x = y × x, i.e. When f is a function depending on t, f ′ (or ḟ ) stands for the derivative of f with respect to t. For A ∈ R M×N (M, N ∈ N * ), A * denotes the transpose of the matrix A, and Id denotes the identity matrix. 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 (1.17) for the precise form of the control term w(t, x)).

S(y)

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 first in [START_REF] Ortega | Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid[END_REF] for a ball embedded in R 2 , and next in [13] for a rigid body S of arbitrary form (still in R 2 ). The case of a ball in R 3 was investigated in [14], and the case of a rigid body of arbitrary form in R 3 was studied in [17]. (See also [START_REF] Sueur | A Kato type theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body[END_REF] for the motion of a rigid body in the inviscid limit of Navier-Stokes equations and [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF] for the time regularity of the flow.) The detection of a rigid body S(t) from a partial measurement of the fluid velocity (or of the pressure) has been tackled in [START_REF] Conca | On the detection of a moving obstacle in an ideal fluid by a boundary measurement[END_REF] when Ω(t) = Ω 0 \ S(t) (Ω 0 ⊂ R 2 denoting a fixed cavity) and in [2] when Ω(t) = R 2 \ S(t).

Note also that since the fluid is flowing through a part of the boundary of the rigid body, additional boundary conditions are needed to ensure the uniqueness of the solution of (1.1)-(1.9) (see [7], [9]). In dimension three, one can specify the tangent components of the vorticity ω(t, x) := curl v(t, x) on the inflow section; that is, one can set

ω(t, x) • τ i = g 0 (t, x) • τ i for w(t, x) < 0, i = 1, 2, (1.11)
where g 0 (t, x) is a given function and τ i , i = 1, 2, are linearly independent vectors tangent to ∂Ω(t). On the other hand, since ω is divergence-free in Ω, we have that ∂Ω(t) ω(t, x) • n dσ = 0. In order to write the equations of the fluid in a fixed frame, we perform a change of coordinates. We set

x = Q(t)y + h(t), (1.12) v(t, y) = Q * (t)u(t, Q(t)y + h(t)), (1.13) q(t, y) = p(t, Q(t)y + h(t)), (1.14) l(t) = Q * (t)h ′ (t),
(1.15)

r(t) = Q * (t)ζ(t). (1.16)
Then 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 := Q * 0 (Ω(0) -h 0 ) when x ranges over Ω(t). Finally, we assume that the control takes the form

w(t, x) = w(t, Q(t)y + h(t)) = m j=1 w j (t)χ j (y), (1.17)
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). To ensure the conservation of the mass of the fluid, we impose the relation ∂Ω χ j (y)dσ = 0 for 1 ≤ j ≤ m.

(1.18)

Then the functions (v, q, l, r) satisfy the following system

∂v ∂t + ((v -l -r × y) • ∇)v + r × v + ∇q = 0, t ∈ (0, T ), y ∈ Ω, (1.19) div v = 0, t ∈ (0, T ), y ∈ Ω, (1.20) v • n = (l + r × y) • n + 1≤j≤m w j (t)χ j (y), t ∈ (0, T ), y ∈ ∂Ω, (1.21) lim |y|→+∞ v(t, y) = 0, t ∈ (0, T ), (1.22)
m 0 l = ∂Ω qn dσ -m 0 r × l, t ∈ (0, T ), (1.23)
J 0 ṙ = ∂Ω q(y × n) dσ -r × J 0 r, t ∈ (0, T ), (1.24) (l(0), r(0)) = (l 0 , r 0 ) := (Q * 0 h 1 , Q * 0 ζ 0 ), v(0, y) = v 0 (y) := Q * 0 u 0 (Q 0 y + h 0 ). (1.25)
The initial velocity field v 0 ∈ C 2,α (Ω) has to satisfy the following compatibility conditions

           curl v 0 = ω 0 in Ω, div v 0 = 0 in Ω, v 0 • n = (l 0 + r 0 × y) • n + 1≤j≤m w j (0)χ j (y) on ∂Ω, lim |y|→+∞ v 0 (y) = 0 (1.26)
where ω 0 := curl v 0 is the initial vorticity.

Once (l, r) is known, the motion of the underwater vehicle is described by the system

Q ′ (t) = Q(t)S(r(t)), (1.27) h ′ (t) = Q(t)l(t),
(1.28)

ζ(t) = Q(t)r(t). (1.29)
Using quaternions, the rotation matrix Q can be parametrized by q ∈ B 1 (0) := { q = (q 1 , q 2 , q 3 ) ∈ R 3 ; | q| := q 2 1 + q 2 2 + q 3 3 < 1} (see e.g. [START_REF] Lecaros | Control of underwater vehicles in inviscid fluids-I. Irrotational flows[END_REF]); namely, we can write Q = R( q) where R( q) :=   q 2 0 + q 2 1 -q 2 2 -q 2 3 2(q 1 q 2 -q 0 q 3 ) 2(q 1 q 3 + q 0 q 2 ) 2(q 2 q 1 + q 0 q 3 ) q 2 0 -q 2 1 + q 2 2 -q 2 3 2(q 2 q 3 -q 0 q 1 ) 2(q 3 q 1 -q 0 q 2 ) 2(q 3 q 2 + q 0 q 1 ) q 2 0 -q 2 1 -q 2 2 + q 2 3   .

with q = (q 1 , q 2 , q 3 ) ∈ B 1 (0) and q 0 := 1 -| q| 2 . Let Q 0 = R( q 0 ) with q 0 = (q 1,0 , q 2,0 , q 3,0 ). Then the dynamics of q and h are given by

           h ′ (t) = (1 -| q| 2 )l + 2 1 -| q| 2 q × l + (l • q ) q -q × l × q, q ′ (t) = 1 2 ( 1 -| q| 2 r + q × r), h(0) = h 0 , q(0) = q 0 .
(1.30) When there is no vorticity (ω ≡ 0), sufficient conditions of local exact controllability for (h, q, l, r) were derived in [START_REF] Lecaros | Control of underwater vehicles in inviscid fluids-I. Irrotational flows[END_REF]Theorem 3.10]. That result was applied to the controllability of an ellipsoidal submarine with a small number of controls: m ∈ {3, 4, 6}. The reader is referred to [START_REF] Lecaros | Control of underwater vehicles in inviscid fluids-I. Irrotational flows[END_REF] for precise statements. The method of proof of [START_REF] Lecaros | Control of underwater vehicles in inviscid fluids-I. Irrotational flows[END_REF]Theorem 3.10], inspired by the one of [5, Theorem 2.1], combined Coron's return method (see [4]), the flatness approach (for the construction of the reference trajectory) and a variant of Silverman-Meadows criteria. In the following, we shall assume that the conclusion of [START_REF] Lecaros | Control of underwater vehicles in inviscid fluids-I. Irrotational flows[END_REF]Theorem 3.10] (controllability without vorticity) holds; namely, (H) For any T > 0, there exist a number η > 0 and a map W ∈ C 1 (B R 24 (0, η), C 1 ([0, T ]; R m )) which associates with any (h 0 , q 0 , l 0 , r 0 , h T , q T , l T , r T ) ∈ B R 24 (0, η) a control w ∈ C 1 ([0, T ], R m ) with w(0) = 0 steering the state of system (1.19)-(1.25) and (1.30) without vorticity from (h 0 , q 0 , l 0 , r 0 ) at t = 0 to (h T , q T , l T , r T ) at t = T .

In (H), we used the obvious notation: B R N (0, η) := {x ∈ R N ; |x| < η}. The aim of this work is to extend this property to the more general case of fluids with vorticity. Here, we shall use the contraction mapping theorem (instead of a compactness approach as in [START_REF] Glass | On the control of the motion of a boat[END_REF]) to obtain in a direct way the existence and uniqueness of the solution of (1.19)-(1.25). The main result in this paper is the following Theorem 1.1. Assume that the assumption (H) is fulfilled, and pick any T 0 > 0. Then there exists η > 0 such that for any (h 0 , q 0 , l 0 , r 0 ) ∈ R 12 and any (h T , q T , l T , r T ) ∈ R 12 with

|(h 0 , q 0 )| < η, |(h T , q T )| < η,
and for any ω 0 ∈ C 1,α (Ω) ∩ M p 1,δ+2 ∩ M p 0,δ+3 (see below for the definition of these spaces) with

|ω 0 (y 1 ) -ω 0 (y 2 )| ≤ K [1 + min(|y 1 |, |y 2 |)] κ |y 1 -y 2 |, ∀(y 1 , y 2 ) ∈ Ω 2 , | ∂ω 0 ∂y | = O(|y| -1 ) as |y| → +∞, | ∂ω 0 ∂y (y 1 ) - ∂ω 0 ∂y (y 2 )| ≤ K 1 + min(|y 1 |, |y 2 |) |y 1 -y 2 |, ∀(y 1 , y 2 ) ∈ Ω 2 for some constants p ∈ (3, 4], δ ∈ [0, 1 -3 p ), α ∈ (0, 1 -3 p ], κ > 3 + δ + 3
p and K > 0, if v 0 denotes the solution of (1.26) with w j (0) = 0 for 1 ≤ j ≤ m, then there exist a time T ∈ (0, T 0 ] and a control input w ∈ C 1 ([0, T ]; R m ) with w(0) = 0 such that the system (1.19)-(1.25) and (1.30) admits a solution (h, q, l, r, v, q) satisfying (h, q, l, r) |t=T = (h T , q T , l T , r T ).

Remark 1.2. In our previous control result [START_REF] Lecaros | Control of underwater vehicles in inviscid fluids-I. Irrotational flows[END_REF]Theorem 3.10] for a system without vorticity, it was required that the initial/final velocities be small, but this restriction could easily be removed by using the same scaling argument as in the proof of Theorem 1.1.

The paper is organized as follows. In Section 2 we prove the existence and uniqueness of the solution of the control problem (1.19)-(1.25) (the vorticity being extended to R 3 ) by applying the contraction mapping theorem in Kikuchi's spaces. The proof of Theorem 1.1 is given in Section 3.

Wellposedness of the system with vorticity

Let us introduce some notations. For k ∈ N and α ∈ (0, 1), let C k,α (Ω) denote the classical Hölder space endowed with the norm

f C k,α (Ω) = β = (β 1 , β 2 , β 3 ) ∈ N 3 β 1 + β 2 + β 3 ≤ k ∂ β f L ∞ (Ω) + |∂ β f | 0,α , where |f | 0,α = sup |f (x) -f (y)| |x -y| α ; x ∈ Ω, y ∈ Ω, x = y .
We also need some notations borrowed from [10].

Let y = (1 + |y| 2 ) 1 2 . For s ∈ N, p ∈ [1, ∞) and λ ≥ 0, let M p
s,λ denote the completion of the space of functions in C ∞ (Ω) with compact support in Ω for the norm

u M p s,λ = β = (β 1 , β 2 , β 3 ) ∈ N 3 β 1 + β 2 + β 3 ≤ s y λ+β1+β2+β3 ∂ β u L p (Ω) .
In particular, for s = 0, u M p 0,λ = u L p pλ := ( Ω |u| p y pλ dy) 1 p . We shall mainly use the space M p 1,λ (for the vorticity) and M p 2,λ (for the velocity) endowed with the respective norms

u M p 1,λ = y λ u L p (Ω) + 1≤i≤3 y λ+1 ∂ yi u L p (Ω) , (2.1)
u M p 2,λ = y λ u L p (Ω) + 1≤i≤3 y λ+1 ∂ yi u L p (Ω) + 1≤i,j≤3 y λ+2 ∂ yj ∂ yi u L p (Ω) . (2.2)
Let π be a continuous linear extension operator from functions defined in Ω to functions defined in R 3 , which maps C k,α (Ω) to C k,α (R 3 ) for all k ∈ N and all α ∈ (0, 1). The construction of such a "universal" extension operator is classical, see e.g. [15, p. 194]. We may also ask that π preserves the divergence-free character, see e.g. [8].

We introduce some functions φ i , i = 1, 2, 3, ϕ i , i = 1, 2, 3, and ψ j , j = 1, 2, ..., m, satisfying

∆φ i = ∆ϕ i = ∆ψ j = 0 in Ω, (2.3) ∂φ i ∂n = n i , ∂ϕ i ∂n = (y × n) i , ∂ψ j ∂n = χ j on ∂Ω, (2.4) lim |y|→+∞ ∇φ i (y) = 0, lim |y|→+∞ ∇ϕ i (y) = 0, lim |y|→+∞ ∇ψ j (y) = 0. (2.5)
As the open set Ω 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), the functions ∇ϕ i (i = 1, 2, 3) and the functions ∇ψ j (1 ≤ j ≤ m) belong to H ∞ (Ω). On the other hand, it follows from [10, Proof of Lemma 2.7] that for all α = (α 1 , α 2 , α 3 ) ∈ N 3 with α 1 + α 2 + α 3 ≥ 1, we have

|∂ α φ i (y)| + |∂ α ϕ i (y)| + |∂ α ψ j (y)| ≤ C y -1-(α1+α2+α3) , i = 1, 2, 3, j = 1, 2, ..., m, y ∈ Ω. (2.6)
For notational convenience, in what follows Ω f (resp. ∂Ω f ) stands for Ω f (y)dy (resp. ∂Ω f (y)dσ(y)).

Let us introduce the matrices M, J, N ∈ R 3×3 , defined by

M i,j = Ω ∇φ i • ∇φ j = ∂Ω n i φ j = ∂Ω ∂φ i ∂n φ j , (2.7)
J i,j = Ω ∇ϕ i • ∇ϕ j = ∂Ω (y × n) i ϕ j = ∂Ω ∂ϕ i ∂n ϕ j , (2.8
)

N i,j = Ω ∇φ i • ∇ϕ j = ∂Ω n i ϕ j = ∂Ω φ i (y × n) j .
(2.9)

Next we define the matrix J ∈ R 6×6 by

J = m 0 Id 0 0 J 0 + M N N * J . (2.10)
It is easy to see that J is a (symmetric) positive definite matrix. For a potential flow (i.e. without vorticity), the dynamics of (l, r) are given by l r

′ = J -1 (Cw ′ + F (l, r, w)), (2.11)
where

F (l, r, w) = -   S(r) 0 S(l) S(r)   J l r -Cw - m p=1 w p   L M p l + R M p r + W M p w L J p l + R J p r + W J p w   , (2.12)
and

C = - C M C J , (2.13) with (C M ) i,j = Ω ∇φ i • ∇ψ j = ∂Ω n i ψ j = ∂Ω φ i χ j , (2.14) (C J ) i,j = Ω ∇ϕ i • ∇ψ j = ∂Ω (y × n) i ψ j = ∂Ω ϕ i χ j , (2.15) (L M p ) i,j = ∂Ω (∇φ j ) i χ p , (L J p ) i,j = ∂Ω (y × ∇φ j ) i χ p , (2.16) (R M p ) i,j = ∂Ω (∇ϕ j ) i χ p , (R J p ) i,j = ∂Ω (y × ∇ϕ j ) i χ p , (2.17) (W M p ) i,j = ∂Ω (∇ψ j ) i χ p , (W J p ) i,j = ∂Ω (y × ∇ψ j ) i χ p . (2.18)
We refer the reader to [START_REF] Lecaros | Control of underwater vehicles in inviscid fluids-I. Irrotational flows[END_REF] for the derivation of (2.11).

The first main result in this paper is a local existence result.

Theorem 2.1. Let p ∈ (3, 4], δ ∈ [0, 1 -3 p ), α ∈ (0, 1 -3 p ], and
T > 0. Assume given (l 0 , r 0 ) ∈ R 3 × R 3 and ω 0 ∈ C 1,α (Ω) ∩ M p 1,δ+2 ∩ M p 0,δ+3 with |ω 0 (y 1 ) -ω 0 (y 2 )| ≤ K [1 + min(|y 1 |, |y 2 |)] κ |y 1 -y 2 |, ∀(y 1 , y 2 ) ∈ Ω 2 , (2.19) | ∂ω 0 ∂y | = O(|y| -1 ) as |y| → +∞, (2.20)
| ∂ω 0 ∂y (y 1 ) - ∂ω 0 ∂y (y 2 )| ≤ K 1 + min(|y 1 |, |y 2 |) |y 1 -y 2 |, ∀(y 1 , y 2 ) ∈ Ω 2 (2.21)
for some constants κ > 3 + δ + 3 p and K > 0. Let also a control input w ∈ C 1 ([0, T ], R m ) be given. Assume that the initial velocity field v 0 ∈ C 2,α (Ω) fulfills the following compatibility conditions

           curl v 0 = ω 0 in Ω, div v 0 = 0 in Ω, v 0 • n = (l 0 + r 0 × y) • n + 1≤j≤m w j (0)χ j (y) on ∂Ω, lim |y|→+∞ v 0 (y) = 0. (2.22)
Then we can find a time T ′ ∈ (0, T ] satisfying CT ′ < 1, where

C = C( ω 0 C 1,α (Ω) + ω 0 M p 1,δ+2 , |l 0 |, |r 0 |, w C 1 ([0,T ])
) is nondecreasing in all its arguments, and a solution (v, q, l, r) of

(1.19)-(1.25) in the class v ∈ C([0, T ′ ]; C 2,α (Ω) ∩ M p 2,δ+1 ), ∇v ∈ C([0, T ′ ]; L 4 p(δ+2) (Ω)), (2.23) ∇q ∈ C([0, T ′ ]; L 2 (Ω)), (2.24) lim |y|→+∞ ∇q(t, y) = 0, ∀t ∈ [0, T ′ ], (2.25) (l, r) ∈ C 1 ([0, T ′ ]; R 3 × R 3 ). (2.26) Moreover, for w C 1 ([0,T ]) ≤ R (R > 0 being any given constant), this solution satisfies (l, r) -(l, r) L ∞ (0,T ′ ) + v -v L ∞ (0,T ′ ;C 2,α (Ω)) ≤ C ′ ω 0 C 1,α (Ω) + ω 0 M p 1,δ+2 , (2.27)
for some constant C ′ > 0, where (l, r, v) is the potential solution of (1.19)-(1.25) associated with l 0 , r 0 , {w j } 1≤j≤m , and ω 0 = 0.

Remark 2.2.

(1) Note that, using the mean-value theorem, the assumption (2.20)

implies (2.19) for κ = 1, while ω 0 ∈ M p 1,δ+2 yields |ω 0 (y)| ≤ O(|y| -δ-2 ) as |y| → +∞ by [10, Lemma 2.2].
(2) Note that the fluid-structure system considered here is more complicated than those considered in [14, 17], for we have added a control input in the boundary condition (1.21). Moreover, we require the solution to be continuous with respect to the control input in order to apply a perturbative argument at the end of the proof of Theorem 1.1. Inspired by the method developed in [START_REF] Glass | On the control of the motion of a boat[END_REF], it is quite natural to work in Kikuchi's spaces. Here, we shall prove the existence, uniqueness and continuous dependence of the solution with respect to the control input in one step, by using the contraction mapping theorem.

Theorem 2.1 will be established by using the contraction mapping principle (i.e. the Banach fixed-point theorem). We first define an operator whose fixed-points will give local-in-time solutions.

2.1. The operator. Let p ∈ (3, 4], δ ∈ [0, 1 -3 p ), α ∈ (0, 1 -3 p ],
and T > 0. We fix a control input w ∈ C 1 ([0, T ]; R m ). For N > 0 and P > 0 given, we introduce the set

F := (l, r, ω) ∈ R 3 × R 3 × C 1,α (Ω) ∩ M p 1,δ+2 ; |l -l 0 | + |r -r 0 | ≤ N, ω C 1,α (Ω) + ω M p 1,δ+2 ≤ P, div ω = 0, ∂Ω ω • n dσ = 0 .
(2.28)

Then, using Arzela-Ascoli theorem for the restrictions to closed ball centered at the origin of partial derivatives of order one, it is easy to see that F is a closed subset of the Banach space

E := R 3 × R 3 × C 0,α (Ω) ∩ M p 0,δ+2 endowed with the norm (l, r, ω) E := |l| + |r| + ω C 0,α (Ω) + ω L p p(δ+2) (Ω)
• It follows at once that for any T ′ > 0, the set

C := {(l, r, ω) ∈ C([0, T ′ ], F); (l(0), r(0), ω(0)) = (l 0 , r 0 , ω 0 )} is a closed subset of the Banach space C([0, T ′ ]; E) endowed with the norm sup t∈[0,T ′ ] (l(t), r(t), ω(t)) E , which is also complete for the equivalent norm (l, r, ω) := l L ∞ (0,T ′ ) + r L ∞ (0,T ′ ) + ω L ∞ (0,T ′ ;C 0,α (Ω)) + ω L ∞ (0,T ′ ;L p p(δ+2) (Ω)) .
(2.29) Therefore, C is complete for the distance associated with the norm • .

Here, we pick

P = e e • (C 6 π(ω 0 ) C 1,α (R 3 ) + C 7 π(ω 0 ) M p 1,δ+2 (R 3 ) ), (2.30)
where C 6 and C 7 are some universal constants arising in the computations below and that we do not intend to give explicitly, and

• C 1,α (R 3 ) and • M p 1,δ+2 (R 3 ) are defined as • C 1,α (Ω) and • M p 1,δ+2 , respectively.
Let us now define the operator τ on C: to any (l, r, ω) ∈ C, we associate τ (l, r, ω) := ( l, r, ω),

(2.31) as follows. First, we introduce the "fluid velocity"

v = η + 3 i=1 l i ∇φ i + 3 i=1 r i ∇ϕ i + 1≤j≤m w j (t)∇ψ j , (2.32)
where η is the solution to the div-curl system (see e.g. [10, Proposition 3.1])

curl η = ω, (t, y) ∈ (0, T ) × Ω, (2.33) div η = 0, (t, y) ∈ (0, T ) × Ω, (2.34) η • n = 0, (t, y) ∈ (0, T ) × ∂Ω, (2.35) lim |y|→+∞ η(t, y) = 0, t ∈ (0, T ), (2.36)
Next, we extend the velocity field and the initial vorticity by setting

v(t, •) := π[v(t, •)],
(2.37)

ω0 := π[ω 0 ]. (2.38)
The flow X associated with ṽ := v -l -r × y is defined as the solution to the Cauchy problem

     ∂ ∂s X(s; t, y) = ṽ(s, X(s; t, y)) = v(s, X(s; t, y)) -l(s) -r(s) × X(s; t, y), X(t; t, y) = y.
(2.39)

The fact that X is defined globally on [0, T ′ ] 2 × R 3 follows from the boundedness of v (see below (2.50)). We denote by G the Jacobi matrix of X, namely We infer from div(ṽ) = 0 (2.42) that det G(s; t, y) = 1.

(2.43)

We now define ω(t, y) := G -1 (0; t, y)ω 0 ( X(0; t, y)).

(2.44)

Finally, in order to define the pair ( l, r), we introduce the function µ : [0, T ′ ] × Ω → R (defined up to a function of t) which solves

               -∆µ = tr ∇v • ∇v , in (0, T ) × Ω, ∂µ ∂n = - 1≤j≤m ẇj (t)χ j (y) -(v -l -r × y) • ∇ v + r × v • n, on (0, T ) × ∂Ω, lim |y|→+∞ ∇µ(t, y) = 0 in (0, T ).
(2.45) Note that ∇µ ∈ L 2 (Ω) if ∇v ∈ L 4 (Ω) ∩ C 1 (Ω), and that, by Schauder estimates, µ ∈ C 2,α loc (Ω) if in addition v ∈ C 2,α (Ω). Then we define l and r by

l(t) r(t) := l 0 r 0 + J -1 t 0         Ω ∇µ(τ, y) • ∇φ i (y) dy i=1,2,3 Ω ∇µ(τ, y) • ∇ϕ i (y) dy i=1,2,3    - m 0 r × l r × J 0 r      dτ. (2.46)
This completes the definition of τ .

2.2. Fixed-point argument and local-in-time existence. Our first step consists in proving the following result.

Theorem 2.3. Let N > 0 and P > 0 be given. Then there exists some time T ′ > 0 such that τ is a contraction in C for the distance associated with • . Thus, τ has a unique fixed-point in C.

Proof of Theorem 2.3:

Set N := N + |l 0 | + |r 0 |.
In the sequel, the various positive constants C i will depend on the geometry, on J and on the size of the controls w i C 1 only (hence, possibly also on π, but not on T , l 0 , r 0 , ω 0 , N , etc.).

Step 1. Let (l, r, ω) ∈ C, and let v be defined by (2.32). It follows from [10, Proposition 2.11] that for any t ∈ [0, T ′ ], system (2.33)-(2.36) has a unique solution η(t) ∈ M p 2,δ+1 , and that

η(t) M p 2,δ+1 ≤ C ω(t) M p 1,δ+2 .
On the other hand, using (2.6) and the fact that 0 ≤ δ < 1 -3 p , it is easy to see that ∇φ i , ∇ϕ i , ∇ψ j ∈ M p 2,δ+1 for i = 1, 2, 3 and j = 1, 2, ..., m. It follows that

v(t) M p 2,δ+1 ≤ C( ω(t) M p 1,δ+2 + |l(t)| + |r(t)| + |w(t)|).
(2.47)

Thus v ∈ C([0, T ′ ]; M p 2,δ+1 ) with v L ∞ (0,T ′ ;M p 2,δ+1 ) ≤ C(N + P + 1). (2.48) Set N := N + P + 1 = N + |l 0 | + |r 0 | + P + 1. Now Schauder estimates combined with the embedding M p 2,δ+1 ⊂ C 1 b (Ω) (see [10, Lemma 2.2]) give that v(t) C 2,α (Ω) ≤ C ω(t) C 1,α (Ω) + v(t) C 0,α (Ω) + |l(t)| + |r(t)| + |w(t)| ≤ C ω(t) C 1,α (Ω) + v(t) M p 2,δ+1 + |l(t)| + |r(t)| + |w(t)| ≤ C ω(t) C 1,α (Ω) + ω(t) M p 1,δ+2 + |l(t)| + |r(t)| + |w(t)| . Thus v ∈ C([0, T ′ ]; C 2,α (Ω)) with v L ∞ (0,T ′ ;C 2,α (Ω)) ≤ C ω L ∞ (0,T ′ ;C 1,α (Ω)) + ω L ∞ (0,T ′ ;M p 1,δ+2 ) + + l L ∞ (0,T ′ ) + r L ∞ (0,T ′ ) + w L ∞ (0,T ′ ) .
(2.49)

Therefore, using the continuity of π, we obtain that

v L ∞ (0,T ′ ;C 2,α (R 3 )) ≤ π CN ≤ C 1 N, (2.50)
where π denotes the norm of π as an operator in L(C 2,α (Ω), C 2,α (R 3 )).

Step 2. Let us turn our attention to X and ω. Taking the scalar product of each term of the first equation in (2.39) by X results in

| X| ∂| X| ∂s = ∂ ∂s ( 1 2 | X| 2 ) = X • ∂ X ∂s = (v(s, X) -l(s)) • X ≤ |v(s, X(s))| + |l(s)| | X|.
Simplifying by | X| and using the second equation in (2.39), we obtain

| X(s; t, y)| -|y| ≤ CT ′ N. (2.51) Thus X(s, t; •) ∈ L ∞ (R 3 ) for all (s, t) ∈ [0, T ′ ] 2 .
It follows from [10, Lemma 2.2] that any function u ∈ M p 1,δ+2 satisfies |u(x)| = O(|x| -δ-1 ) as |x| → +∞, and from [10, Lemma 2.3] that M p 1,δ+2 is an algebra. Let M p 1,δ+2 (R 3 ) be defined as M p 1,δ+2 but with functions from R 3 to R. We introduce the space

V := M p 1,δ+2 (R 3 ) 3×3 ⊕ R 3×3 with norm G V := G 1 M p 1,δ+2 (R 3 ) + G 2 R 3×3 if G = G 1 + G 2 with G 1 ∈ M p 1,δ+2 (R 3 ) 3×3 and G 2 ∈ R 3×3 .
(Note that G 2 is uniquely determined by G, as it is nothing but the 3 × 3 matrix of the limits at infinity of the entries of G.) Then it is easy to see that V is a Banach space and an algebra.

Let us check that (∂v/∂y)(s, X(s; t, y)) ∈ L ∞ ((0, T ′ ) 2 , V ). From (2.51) we have that X(s; t, y) ≤ |y| + CT ′ N and proceeding as in [10, p. 587-588], we infer that for any

u ∈ M p 1,δ+2 (R 3 ) u( X(s; t, .)) M p 0,δ+2 (R 3 ) ≤ C(1 + [CT ′ N] p(δ+2) ) u M p 0,δ+2 (R 3 ) . ( 2
M p 0,δ+3 (R 3 ) ≤ C exp(CT ′ N) ∂u ∂y M p 0,δ+3 (R 3 ) .
(2.54)

We infer from (2.48) and (2.50) that ∂v ∂y L ∞ (0,T ′ ,M p 1,δ+2 (R 3 )) ≤ CN.

It follows that ∂v ∂y (s, X(s; t, y)) -S(r(s)) ∈ L ∞ ((0, T ′ ) 2 , V ) with ∂v ∂y (s, X(s; t, y)) -S(r(s)) V ≤ CN exp(CT ′ N).

(2.55)

Solving the linear Cauchy problem (2.41) in the Banach algebra V , we see that G ∈ C([0, T ′ ] 2 ; V ) and (with Gronwall's lemma) that

G L ∞ ((0,T ′ ) 2 ;V ) ≤ C 2 exp(C 2 T ′ Ne C2T ′ N ).
(2.56) By (2.43), each entry of G -1 is a cofactor of G, so that we infer that

G -1 L ∞ ((0,T ′ ) 2 ;V ) ≤ C 3 exp(C 3 T ′ Ne C3T ′ N ).
(2.57)

If f ∈ C 1 (R 3 , R 3 ) with ∂f /∂y ∈ C 0,α (R 3 , R 3 ) and g ∈ C 1,α (R 3 , R 3 ), then |g • f | 0,α ≤ C|g| 0,α ∂f ∂y α L ∞ , (2.58)
∂ ∂y (g • f ) L ∞ ≤ C ∂g ∂y L ∞ ∂f ∂y L ∞ , (2.59)
| ∂ ∂y (g • f )| 0,α ≤ C ∂g ∂y L ∞ | ∂f ∂y | 0,α + ∂f ∂y 1+α L ∞ | ∂g ∂y | 0,α . ( 2
G L ∞ ((0,T ′ ) 2 ;C 0,α (R 3 )) ≤ C exp(CT ′ Ne CT ′ N ). ( 2
G L ∞ ((0,T ′ ) 2 ;C 1,α (R 3 )) ≤ C 4 exp(C 4 T ′ Ne C4T ′ N ).
(2.62)

Using again (2.43), we obtain that

G -1 L ∞ ((0,T ′ ) 2 ;C 1,α (R 3 )) ≤ C 5 exp(C 5 T ′ Ne C5T ′ N ). (2.63)
We are in a position to derive the required estimates for ω. From (2.58)-(2.60) and (2.62), we infer that ω0 ( X(0; t, .))

L ∞ (0,T ′ ;C 1,α (R 3 )) ≤ C exp(CT ′ Ne CT ′ N ) ω0 C 1,α (R 3 ) which yields with (2.44) and (2.63) ω L ∞ (0,T ′ ;C 1,α (R 3 )) ≤ C 6 exp(C 6 T ′ Ne C6T ′ N ) ω0 C 1,α (R 3 ) .
(2.64) From (2.52)-(2.54) we obtain ω0 ( X(0; t, .))

L ∞ (0,T ′ ;M p 1,δ+2 (R 3 )) ≤ C exp(CT ′ N) ω0 M p 1,δ+2 (R 3 )
which gives with (2.57)

ω L ∞ (0,T ′ ;M p 1,δ+2 (R 3 )) ≤ C 7 exp(C 7 T ′ Ne C7T ′ N ) ω0 M p 1,δ+2 (R 3 ) .
(2.65)

If we define C 8 = max{C 6 , C 7 }, and take T ′ > 0 such that

T ′ ≤ 1 C 8 N , (2.66) then we obtain ω L ∞ (0,T ′ ;C 1,α (R 3 )) + ω L ∞ (0,T ′ ;M p 1,δ+2 (R 3 )) ≤ e e • (C 6 ω0 C 1,α (R 3 ) + C 7 ω0 M p 1,δ+2 (R 3 ) ) =: P.
(2.67)

On the other hand, if we consider any scalar function ϕ ∈ C 1 (R 3 ) with compact support, we obtain by using the change of variables y = X(t; 0, x)

R 3 ω(t, y) • ∇ϕ(y)dy = R 3 G -1 (0; t, y)ω 0 ( X(0; t, y)) • ∇ϕ(y)dy = R 3 G -1 (0; t, X(t; 0, x))ω 0 (x) • ∇ y ϕ( X(t; 0, x))dx, = R 3 G(t; 0, x)ω 0 (x) • ∇ y ϕ( X(t; 0, x))dx = R 3 ω0 (x) • ∇ x [ϕ( X(t; 0, x))]dx.
Since, by (2.51), the function ϕ( X(t; 0, •)) ∈ C 1 (R 3 ) has a compact support, we infer from div(ω 0 ) = 0 that div(ω) = 0 in R 3 .

(2.68)

Integrating over Ω, we obtain

Ω ω • ndσ = 0. (2.69)
On the other hand, ω(0) |Ω = ω 0 . Therefore, the condition about ω for ( l, r, ω) to belong to C is satisfied.

Step 3. Let us turn our attention to ( l, r). Since M p 1,δ+2 (R 3 ) ⊂ L ∞ (R 3 ) and ∇v(t) ∈ M p 1,δ+2 (R 3 ) for all t ∈ [0, T ′ ], we infer from (2.48) that for all t ∈ [0,

T ′ ] R 3 |∇v| 4 y p(δ+2) dy ≤ ∇v 4-p L ∞ (R 3 ) R 3 |∇v| p y p(δ+2) dy ≤ C ∇v 4 M p 1,δ+2 (R 3 ) < +∞. Furthermore, ∇v L ∞ (0,T ′ ,L 4 (R 3 )) ≤ C ∇v L ∞ (0,T ′ ,M p 1,δ+2 (R 3 )) ≤ C 9 N.
(2.70) Using (2.45) and (2.50), we infer that

∇µ L ∞ (0,T ′ ,L 2 (Ω)) ≤ C 10 N 2 . (2.71) From (2.46), we deduce that ( l, r) ∈ C([0, T ′ ]; R 6 ) with ( l, r) -(l 0 , r 0 ) L ∞ (0,T ′ ) ≤ C 11 T ′ N 2 .
On the other hand, ( l(0), r(0)) = (l 0 , r 0 ). Therefore, the condition about ( l, r) for τ(l, r, ω) to belong to C is satisfied provided that

T ′ ≤ N C 11 N 2 • (2.72)
Hence for T ′ satisfying (2.66) and (2.72), one has τ(C) ⊂ C.

Note also that, since ∇v(t) ∈ M p 1,δ+2 (R 3 ) for all t ∈ [0, T ], we have

y ∇v(t) ∈ M p 1,δ+1 (R 3 ) ⊂ C 0 (R 3 ) for all t ∈ [0, T ′ ].
Step 4. Now, we prove that the operator τ is a contraction in C for the distance induced by • for T ′ small enough. From now on, the constant C may depend on N, but not on T ′ or on (l k , r k , ω k ). Assume given (l k , r k , ω k ) ∈ C, k = 1, 2. Note that (l 1 , r 1 , ω 1 ) and (l 2 , r 2 , ω 2 ) correspond to the same initial data (l 0 , r 0 , ω 0 ) and the same control input ω.

Let us introduce for k = 1, 2 τ(l k , r k , ω k ) := ( lk , rk , ωk ).

Then, for k = 1, 2, ωk fulfills ωk (t, y) = A k (0; t, y)ω 0 ( Xk (0; t, y)), (2.73) where ω0 = π(w 0 ), Xk denotes the solution to

     ∂ ∂s Xk (s; t, y) = vk (s, Xk (s; t, y)) -l k (s) -r k (s) × Xk (s; t, y), Xk (t; t, y) = y, (2.74)
G k (s; t, y) := ∂ Xk ∂y (s; t, y), G(t; t, y) = Id, (2.75)
and A k := (G k ) -1 . The velocity vk = π(v k ) : R 3 → R 3 is the extension of the velocity v k : Ω → R 3 decomposed as

v k = η k + 3 i=1 l k i ∇φ i + 3 i=1 r k i ∇ϕ i + 1≤j≤m w j (t)∇ψ j ,
where η k is the solution of curl η k = ω k , (t, y) ∈ (0, T ) × Ω, (2.76) div η k = 0, (t, y) ∈ (0, T ) × Ω, (2.77)

η k • n = 0, (t, y) ∈ (0, T ) × ∂Ω, (2.78) lim |y|→+∞ η k (t, y) = 0, t ∈ (0, T ).
(2.79)

We introduce the functions

v := v 1 -v 2 , η := η 1 -η 2 , r := r 1 -r 2 , l := l 1 -l 2 , ω := ω 1 -ω 2 , and A := A 1 -A 2 .
(2.80) Thus v may be written as

v = η + 3 i=1 l i ∇φ i + 3 i=1 r i ∇ϕ i , v(0, y) = 0, (2.81)
where η is the solution to the system curl η = ω, (t, y) ∈ (0, T ) × Ω, (2.82) div η = 0, (t, y) ∈ (0, T ) × Ω, (2.83)

η • n = 0, (t, y) ∈ (0, T ) × ∂Ω, (2.84) lim |y|→+∞ η(t, y) = 0, t ∈ (0, T ).
(2.85)

Step 5. Let X := X1 -X2 . Then

        
∂ ∂s X(s; t, y) = v1 (s, X1 (s; t, y)) -v1 (s, X2 (s; t, y)) + v(s, X2 (s; t, y))

-l(s) -r 1 (s) × X(s; t, y) -r(s) × X2 (s; t, y),

X(t; t, y) = 0, (2.86)
where v := v1 -v2 = π(v).

Taking the scalar product of each term in (2.86) by X results in

| X| ∂| X| ∂s = ∂ ∂s ( 1 2 | X| 2 ) = X • ∂ X ∂s = v1 (s, X1 ) -v1 (s, X2 ) + v(s, X2 ) -l -r × X2 • X. It follows that ∂| X| ∂s ≤ C ∂v 1 ∂y L ∞ (R 3 ) | X| + v L ∞ (R 3 ) + |l| + |r| • | X2 | . Since v(s) L ∞ (R 3 ) ≤ C v(s) C 0,α (Ω) ≤ C η(s) C 0,α (Ω) + |l(s)| + |r(s)| ≤ C (l, r, ω) and | X2 | ≤ |y| + CT ′ N ≤ C y , we obtain with Gronwall's lemma that for (s, t, y) ∈ [0, T ′ ] 2 × R 3 , | X(s, t, y)| ≤ e CT ′ T ′ 0 C(1 + y ) (l, r, ω) dt ≤ CT ′ y (l, r, ω) .
(2.87)

Step 6. Let us set

A := A 1 -A 2 . (Recall that A k = (G k ) -1 for k = 1,
≤ C A ∂v 1 ∂y L ∞ (R 3 ) + |r 1 | + A 2 • ∂v 1 ∂y W 1,∞ (R 3 ) | X| + A 2 ∂v ∂y L ∞ (R 3 ) + |r| . (2.90)
From (2.50), we have that

∂v 1 ∂y (s) W 1,∞ (R 3 ) ≤ v1 (s) C 2,α (R 3 ) ≤ CN ≤ C.
Clearly,

A 2 + |r 1 | ≤ CN ≤ C.
On the other hand, it follows from Morrey's inequality that

η C 0,1-3 p (Ω) ≤ C η M p 1,δ+1 ≤ C ω M p 0,δ+2 . Let |f | 0,α,R 3 := sup{ |f (x) -f (y)| |x -y| α ; x, y ∈ R 3 , x = y}.
Then, since 0

< α ≤ 1 -3 p , we have ∂v ∂y (s) L ∞ (R 3 ) + | ∂v ∂y (s)| 0,α,R 3 ≤ C v(s) C 1,α (Ω) ≤ C η(s) C 1,α (Ω) + |l(s)| + |r(s)| ≤ C η(s) C 0,α (Ω) + ω(s) C 0,α (Ω) + |l(s)| + |r(s)| ≤ C ω(s) M p 0,δ+2 + ω(s) C 0,α (Ω) + |l(s)| + |r(s)| ≤ C (l, r, ω) .
(2.91)

We infer with (2.87) and (2.90) that ∂A ∂s (s; t, y) ≤ C A + C(T ′ y + 1) (l, r, ω) .

Since A(t; t, y) = 0, we obtain by using Gronwall's lemma that for (s, t, y) ∈ [0, T ′ ] 2 × R 3 A(s; t, y) ≤ CT ′ y (l, r, ω) .

(2.92)

Step 7. Let ω := ω1 -ω2 . We first give an estimate of ω L ∞ (Ω) . We write |ω| = A 1 (0; t, y)ω 0 ( X1 (0; t, y)) -A 2 (0; t, y)ω 0 ( X2 (0; t, y))

≤ A(0; t, y)ω 0 ( X1 (0; t, y)) + A 2 (0; t, y) ω0 ( X1 (0; t, y)) -ω0 ( X2 (0; t, y)) .

(2.93)

Since ω 0 ∈ M p 1,δ+2
ω L ∞ (0,T ′ ;L ∞ (Ω)) ≤ CT ′ (l, r, ω) .
(2.95)

Step 8. Let us now estimate the Hölder norm |ω| 0,α . Note first that it is not clear whether X ∈ C 0,α (Ω), since it could happen that X ∼ y as |y| → +∞ (and hence, X ∈ L ∞ (Ω)). Rather, we shall prove that y -1 X ∈ C 0,α (Ω).

We infer from (2.86) that ∂ ∂s ( y -1 X(s; t, y)) = y -1 v1 (s, X1 (s; t, y)) -v1 (s, X2 (s; t, y)) + v(s, X2 (s; t, y))

-l(s) -r 1 (s) × X(s; t, y) -r(s) × X2 (s; t, y) ,

= 1 0 ∂v 1 ∂y (s, X2 + σ X) y -1 Xdσ + y -1 (v(s, X2 ) -l -r 1 × X -r × X2 )
Therefore, using (2.58), we obtain

| ∂ ∂s ( X y )| 0,α,R 3 ≤ [ ∂v 1 ∂y L ∞ (R 3 ) | X y | 0,α,R 3 + | ∂v 1 ∂y | 0,α,R 3 ∂ X2 ∂y L ∞ (R 3 ) + ∂ X1 ∂y L ∞ (R 3 ) ) α X y L ∞ (R 3 ) ] +| v(s, X2 ) y | 0,α,R 3 + C(1 + | X2 y | 0,α,R 3 ) (l, r, ω) +C| X y | 0,α . (2.96) It is clear that ∂v 1 ∂y L ∞ (R 3 ) + | ∂v 1 ∂y | 0,α,R 3 + ∂ X2 ∂y L ∞ (R 3 ) + ∂ X1 ∂y L ∞ (R 3 ) ≤ C. ( 2
y ) = v2 (s, X2 ) y - l 2 y -r 2 × X2 y , X2 y (t; t, y) = y y • Since |v 2 (s, X2 )| 0,α,R 3 ≤ C|v 2 | 0,α ∂ X2 ∂y α L ∞ (R 3 ) ≤ C, v2 (s, X2 ) L ∞ (Ω) X2
≤ C, and both y -1 and y -1 y belong to W 1,∞ (R 3 ) ⊂ C 0,α (R 3 ), we obtain with Gronwall's lemma that

| X2 y | 0,α,R 3 ≤ C. (2.98)
On the other hand, we infer from (2.58) and (2.91) that

| v(s, X2 ) y | 0,α,R 3 ≤ C v(s, X 2 ) L ∞ (R 3 ) + |v(s, X2 )| 0,α,R 3 ≤ C (l, r, ω) .
It follows from (2.86), (2.96)-(2.98) and Gronwall's lemma that

| X y | 0,α ≤ CT ′ (l, r, ω) .
(2.99)

Next, we prove that a similar estimate holds for | A y | 0,α . Writing for 1 ≤ i, j ≤ 3

∂v 1 i ∂y j (s, X1 ) - ∂v 1 i ∂y j (s, X2 ) = 1 0 ∂ ∂y ( ∂v 1 i ∂y j )(s, X2 + σ X) • Xdσ,
and using (2.88), we infer that

| ∂ ∂s ( A y )| 0,α ≤ C | A y | 0,α ∂v 1 ∂y L ∞ (Ω) + |r 1 | + C A y L ∞ (Ω) | ∂v 1 ∂y | 0,α ∂ X1 ∂y α L ∞ (Ω) + C A 2 L ∞ (Ω) |v 1 | 2,α ( ∂ X2 ∂y L ∞ (Ω) + ∂ X1 ∂y L ∞ (Ω) ) α X y L ∞ (Ω) + ∂ ∂y ( ∂v 1 ∂y ) L ∞ (Ω) | X y | 0,α + C |A 2 | 0,α v1 W 2,∞ (Ω) X y L ∞ (Ω) + C | A 2 y | 0,α ∂v ∂y L ∞ (Ω) + |r| + C A 2 y L ∞ (Ω) | ∂v ∂y | 0,α ∂ X2 ∂y α L ∞ (Ω) .
Then using (2.87), (2.91), (2.92), we infer that

| ∂ ∂s ( A y )| 0,α ≤ C| A y | 0,α + C(1 + T ′ ) (l, r, ω) .
Therefore, using the fact that A(t; t, y) = 0, we obtain with Gronwall's lemma that

| A y | 0,α ≤ CT ′ (l, r, ω) .
(2.100)

We are in a position to estimate |ω| 0,α . We have |ω| 0,α ≤ A(0; t, y)ω 0 ( X1 (0; t, y))

0,α + A 2 (0; t, y) ω0 ( X1 (0; t, y)) -ω0 ( X2 (0; t, y)) 0,α ≤ A y 0,α y ω0 ( X1 ) L ∞ (Ω) + A y L ∞ (Ω) | y ω0 ( X1 )| 0,α +|A 2 | 0,α ω0 ( X1 ) -ω0 ( X2 ) L ∞ (Ω) + A 2 L ∞ (Ω) |ω 0 ( X1 ) -ω0 ( X2 )| 0,α ≤ C(1 + | y ω0 ( X1 )| 0,α )T ′ (l, r, ω) +C|ω 0 ( X1 ) -ω0 ( X2 )| 0,α
where we used (2.19), (2.87), (2.92), and (2.100). It remains to estimate |ω 0 ( X1 ) -ω0 ( X2 )| 0,α and | y ω0 ( X1 )| 0,α . For the first one, we write

|ω 0 ( X1 ) -ω0 ( X2 )| 0,α = 1 0 ∂ ω0 ∂y ( X2 + σ X) Xdσ 0,α ≤ C   sup σ∈(0,1) y ∂ ω0 ∂y ( X2 + σ X) L ∞ (Ω) X y 0,α + sup σ∈(0,1) y ∂ ω0 ∂y ( X2 + σ X) 0,α X y L ∞ (Ω) ≤ C 1 + sup σ∈(0,1) y ∂ ω0 ∂y ( X2 + σ X) 0,α T ′ (l, r, ω)
where we used (2.20), (2.21), (2.87), (2.99), and the fact that (using (2.51) for X2 )

| X2 + σ X| ≥ |y| + O(T ′ ) y ≥ 1 2
y for T ′ small enough, |y| > 1 and σ ∈ (0, 1).

We aim to prove that sup

σ ∈ (0, 1) t ∈ [0, T ′ ] y ∂ ω0 ∂y ( X2 (y) + σ X(y)) 0,α < +∞, (2.101)
where we write X(y) for X(0; t, y), etc. We have with (2.20) that sup

σ ∈ (0, 1) |y -y ′ | ≥ 1 t ∈ [0, T ′ ] y ∂ ω0 ∂y ( X2 (y) + σ X(y)) -y ′ ∂ ω0 ∂y ( X2 (y ′ ) + σ X(y ′ )) |y -y ′ | α < +∞.
On the other hand, for σ ∈ (0, 1), |y -y ′ | < 1, and t ∈ [0, T ′ ],

y ∂ ω0 ∂y ( X2 (y) + σ X(y)) -y ′ ∂ ω0 ∂y ( X2 (y ′ ) + σ X(y ′ )) |y -y ′ | α ≤ y -y ′ |y -y ′ | α ∂ ω0 ∂y ( X2 (y) + σ X(y)) + y ′ ∂ ω0 ∂y ( X2 (y) + σ X(y)) -∂ ω0 ∂y ( X2 (y ′ ) + σ X(y ′ )) |y -y ′ | α ≤ C ∂ ω0 ∂y L ∞ (Ω) + C ∂ X2 ∂y L ∞ (Ω) + ∂ X ∂y L ∞ (Ω) |y -y ′ | 1-α
where we used (2.21) and the mean value theorem for the last term. This completes the proof of (2.101). We infer that

|ω 0 ( X1 ) -ω0 ( X2 )| 0,α ≤ CT ′ (l, r, ω) .
We can prove in a very similar way that | y ω0 ( X1 )| 0,α < +∞. We conclude that

|ω(t)| 0,α ≤ CT ′ (l, r, ω) , t ∈ [0, T ′ ].
(2.102)

Step 9. Let us estimate ω L p p(δ+2) (Ω) . We write

ω p L p p(δ+2) (Ω) ≤ C Ω |Aω 0 ( X1 )| p y p(δ+2) dy + Ω |A 2 (ω 0 ( X1 ) -ω0 ( X2 ))| p y p(δ+2) dy =: C(I 1 + I 2 ),
where we have written A 1 for A 1 (0; t, y), X1 for X1 (0; t, y), etc. Then, using the fact that ω 0 ∈ M 0 p,δ+3 and (2.92), we obtain that

I 1 ≤ (CT ′ (l, r, ω) ) p Ω |ω 0 ( X1 (0; t, y)| p y p(δ+3) dy ≤ (CT ′ (l, r, ω) ) p R 3 |ω 0 (x)| p X1 (t; 0, x) p(δ+3) dx ≤ (CT ′ (l, r, ω) ) p ω0 p M p 0,δ+3 (R 3
) . Therefore, increasing the value of C if needed, we obtain

I 1 ≤ (CT ′ (l, r, ω) ) p .
For I 2 , we infer from (2.19) with κ > 3 + δ + 3 p that

I 2 ≤ C Ω |ω 0 ( X1 ) -ω0 ( X2 )| p y p(δ+2) dy ≤ C Ω | X| [1 + min(| X1 |, | X2 |)] κ p y p(δ+2) dy ≤ (CT ′ (l, r, ω) ) p R 3 y p(δ+3-κ) dy ≤ (CT ′ (l, r, ω) ) p .
We conclude that ω(t)

L p p(δ+2) (Ω) ≤ CT ′ (l, r, ω) , t ∈ [0, T ′ ]. (2.103)
Step 10. Let us proceed to the estimates of v. Since α ≤ 1 -3 p , we have M p 1,δ+1 ⊂ C 0,α (Ω), and hence

∇v C 0,α (Ω) ≤ C( ω C 0,α (Ω) + v C 0,α (Ω) + |l| + |r|) ≤ C( ω C 0,α (Ω) + v M p 1,δ+1 + |l| + |r|) ≤ C( ω C 0,α (Ω) + ω M p 0,δ+2 + |l| + |r|) ≤ CT ′ (l, r, ω) . It follows that ∇v L 4 (Ω) ≤ ∇v p 4 L p (Ω) ∇v 1-p 4 L ∞ (Ω) ≤ C v p 4 M p 1,δ+1 ∇v 1-p 4 C 0,α (Ω) ≤ C( ω C 0,α (Ω) + ω M p 0,δ+2 + |l| + |r|) ≤ CT ′ (l, r, ω) .
Step 11. We now turn our attention to l := l1 -l2 and r := r1 -r2 , where for k = 1, 2

lk (t) rk (t) := l 0 r 0 + J -1 t 0         Ω ∇µ k (τ, y) • ∇φ i (y) dy i=1,2,3 Ω ∇µ k (τ, y) • ∇ϕ i (y) dy i=1,2,3    - m 0 r k × l k r k × J 0 r k      dτ,
(2.104) and the function µ k : [0, T ′ ] × Ω → R is defined as the solution to the system

                 -∆µ k = tr ∇v k • ∇v k , in (0, T ) × Ω, ∂µ k ∂n = - 1≤j≤m ẇj (t)χ j (y) -(v k -l k -r k × y) • ∇ v k + r k × v k • n, on (0, T ) × ∂Ω, lim |y|→∞ ∇µ k (t, y) = 0 in (0, T ).
Then µ := µ 1 -µ 2 satisfies the system

                       -∆µ = tr ∇(v 1 + v 2 ) • ∇v , in (0, T ) × Ω, ∂µ ∂n = -(v -l -r × y) • ∇ v 1 + r × v 1 • n -(v 2 -l 2 -r 2 × y) • ∇ v + r 2 × v • n on (0, T ) × ∂Ω, lim |y|→∞ ∇µ(t, y) = 0 in (0, T ).
It follows that

∇µ L 2 (Ω) ≤ C ∇v 1 L 4 (Ω) + ∇v 2 L 4 (Ω) ∇v L 4 (Ω) + C( v C 0,α (Ω) + ∇v C 0,α (Ω) + |l| + |r|) ≤ CT ′ (l, r, ω) .
(2.105)

We infer from (2.104) that ( l, r) satisfies

l(t) r(t) = J -1 t 0         Ω ∇µ(τ, y) • ∇φ i (y) dy i=1,2,3 Ω ∇µ(τ, y) • ∇ϕ i (y) dy i=1,2,3    - m 0 (r × l 1 + r 2 × l) r × J 0 r 1 + r 2 × J 0 r      dτ,
and hence, with (2.105),

| l(t)| + |r(t)| ≤ CT ′ (l, r, ω) , t ∈ [0, T ′ ]. (2.106)
Gathering together (2.95), (2.102), (2.103) and (2.106), we obtain

( l, r, ω) ≤ CT ′ (l, r, ω) , t ∈ [0, T ′ ].
(2.107) Thus, for T ′ < 1/C, we have that τ (l 1 , r 1 , ω 1 )τ (l 2 , r 2 , ω 2 ) ≤ k (l 1 , r 1 , ω 1 ) -(l 2 , r 2 , ω 2 ) for some constant k ∈ (0, 1), i.e. τ is a contraction in C. The proof of Theorem 2.3 is complete.

2.3. Existence of a solution of system (1.19)-(1.25). Let us now check that the fixed-point (l, r, ω) given in Theorem 2.3 yields a solution of (1.19)- (1.25). Let v and µ be given by (2.32)-(2.36) and (2.45), respectively. Then, since (l, r, ω) ∈ C ⊂ C([0, T ′ ], F), then (2.23) holds, ∇µ ∈ C([0, T ′ ], L 2 (Ω)) and hence, with (2.46), (2.26) holds as well. Let us set

q := µ - 3 i=1 li φ i - 3 i=1 ṙi ϕ i .
(2.108) Then (2.24) holds and we have for a.e. t ∈ (0, T ′ ), q(t, .) ∈ C 2,α loc (Ω) and lim |y|→+∞ ∇q(t, y) = 0.

Proposition 2.4. Let T ′ be as in Theorem 2.3 and let (l, r, ω) denote the corresponding fixed-point of

τ in C. Then (v, q, l, r) is a solution of (1.19)-(1.25) in (0, T ′ ).
Proof of Proposition 2.4: Let

f := (v -l -r × y) • ∇ v + r × v.
(2.109)

Then we have that f (t, .) ∈ C 1,α loc (Ω) for all t ∈ [0, T ′ ]. On the other hand, since v ∈ C([0, T ′ ], M p 2,δ+1 ), we have that |v| ≤ C y -1-δ and |∇v| ≤ C y -2-δ , so that

|f (t, y)| ≤ C y -1-δ .
The divergence of f is given by

div f = div (v -l -r × y) • ∇ v + r × v = ∂ i v j ∂ j v i -∂ i l j ∂ j v i -∂ i (r × y) j ∂ j v i + div r × v = (∂ i v j ) (∂ j v i ) -(r × ∂ i y) j ∂ j v i + div r × v = tr ∇v • ∇v -∂ j (r × v i ∂ i y) j + div r × v = tr ∇v • ∇v -div r × v + div r × v = tr ∇v • ∇v = -∆µ,
where we used Einstein's convention of repeated indices and the fact that div (v) = 0. Therefore, using (2.3) and (2.108), we obtain div (f + ∇q) = 0.

(2.110)

Now we turn our attention to the curl of f . Define ṽ := v -l -r × y. Then curl ṽ = ω -2r.

(2.111)

We shall use the following identities (see e.g. [10])

curl((v • ∇)v) = (v • ∇)curl(v) -(curl(v) • ∇)v + div(v)curl(v), (2.112) curl(r × v) = div(v)r -(r • ∇)v.
(2.113)

Applying the operator curl to f and using (2.112)-(2.113), we obtain curl f = curl((ṽ • ∇)ṽ) + curl((ṽ

• ∇)(l + r × y)) + curl(r × v) = curl ((ṽ • ∇)ṽ) + curl(r × ṽ) + curl(r × v) = curl ((ṽ • ∇)ṽ) + div(ṽ)r -(r • ∇)ṽ + div(v)r -(r • ∇)v = curl ((ṽ • ∇)ṽ) -(r • ∇)ṽ -(r • ∇)v = (ṽ • ∇)curl(ṽ) -(curl(ṽ) • ∇)ṽ -(r • ∇)ṽ -(r • ∇)v = (ṽ • ∇)(ω -2r) -((ω -2r) • ∇)ṽ -(r • ∇)ṽ -(r • ∇)v = (ṽ • ∇)ω -ω • ∇ṽ.
Using (2.44), we see that ω satisfies 2.4. Proof of the estimate (2.27). The potential solution (l, r, v) of (1.19)-(1.25) associated with l 0 , r 0 , {w j } 1≤j≤m , and ω 0 = 0 is obtained in the following way. Since ω 0 = 0, π(ω 0 ) = 0 in R 3 , and hence with (2.44) the vorticity ω is null. Then we infer from (2.33)-(2.36) that η = 0 and from (2.32) that

∂ω ∂t + (ṽ • ∇)ω -(ω • ∇)ṽ = 0, t ∈ (0, T ′ ). ( 2
v = 3 i=1 l i ∇φ i + 3 i=1 r i ∇ϕ i + 1≤j≤m w j (t)∇ψ j .
(2.119)

It follows from [START_REF] Lecaros | Control of underwater vehicles in inviscid fluids-I. Irrotational flows[END_REF]Proposition 2.3] that (l, r) satisfies the ODE (2.11), whose solution is unique.

Consider a solution (l, r, ω) associated with the same l 0 , r 0 , {w j } 1≤j≤m as for (l, r, ω), but with an initial vorticity ω 0 not necessarily null. It follows from (2.28)-(2.30) that for all t ∈ [0, T ′ ] Then (2.27) follows by using Gronwall's lemma. The proof of Theorem 2.1 is complete.

ω(t) C 1,α (Ω) + ω(t) M p 1,δ+2 ≤ P = e e • (C 6 π(ω 0 ) C 1,α (R 3 ) + C 7 π(ω 0 ) M p 1,δ+2 (R 3 ) ). Now,
2.5. Uniqueness and continuity with respect to the control. The following result is concerned with the uniqueness of the solution (l, r, v, q) of (1.19)-(1.25), when the vorticity ω = curl v satisfies ω(t, y) = G -1 (0; t, y)π(ω 0 )( X(0; t, y))

(2.120) where G(s; t, y) = (∂ X/∂y)(s; t, y) and the flow X is defined on [0,

T ′ ] 2 × R 3 by     
∂ ∂s X(s; t, y) = π(v)(s, X(s; t, y)) -l(s) -r(s) × X(s; t, y), X(t; t, y) = y.

(2.121) Proposition 2.5. Let l 0 , r 0 , ω 0 , v 0 and T ′ be as in Theorem 2.1. Then the solution (v, q, l, r, ω) of (1.19)-(1.25) and (2.120)-(2.121) is unique in the class (2.23)-(2.26) (q being unique up to the addition of an arbitrary function of time). On the other hand, for any given initial data (l 0 , r 0 , ω 0 ) as above and any R > 0, the map w ∈

B := {w ∈ C 1 ([0, T ′ ], R m ); w C 1 ([0,T ]) ≤ R} → (l, r) ∈ C 0 ([0, T ′ ], R 6 ) is continuous.
Proof. Let (v, q, l, r) be a solution of ( ) with ṽ = v -l -r × y, and that it is given by (2.120) "away" from the rigid body. We assume that it is given by (2.120) everywhere, even on ∂Ω. Roughly speaking, this amounts to specifying the tangent components of the vorticity on the inflow section. Let us show that the pair (l, r) satisfies (2.46). Let µ be as in (2.108) and let f be as in (2.109). Then by (1.19) and the computations above, we have that

-∆µ = -∆q = div f = tr (∇v • ∇v), and
∂µ ∂n = ∂q ∂n + 3 i=1 li n i + 3 i=1 ṙi (y × n) i = -( ∂v ∂t + f ) • n + l • n + ṙ • (y × n) = - ∂ ∂t   [l + r × y] • n + 1≤j≤m w j (t)χ j (y)   -f • n + l • n + ṙ • (y × n) = - 1≤j≤m ẇj (t)χ j (y) -(v -l -r × y) • ∇v + r × v • n.
Thus µ solves (2.45). Integrating in (1.23)-(1.24) and using (2.108), we arrive to (2.46) with ( l, r) = (l, r).

Thus (l, r, ω) is a fixed-point of τ . As there is (for T ′ small enough) only one fixed-point of τ by the contraction mapping theorem, we infer that (l, r, ω) is unique. Then η is unique by (2.33)-(2.36), and v is unique by (2.32). Finally, ∇q is unique by (1.19) and q is unique (up to the addition of an arbitrary function of time). Let us proceed with the continuity with respect to the control. Assume given some initial data (l 0 , r 0 , ω 0 ) as above and pick any number R > 0. Let

B := {w ∈ C 1 ([0, T ], R m ); w C 1 ([0,T ]) ≤ R}.
Assume that the constants C 8 and C 11 are suitably chosen to be convenient for all w ∈ B, and pick a time T ′ > 0 convenient for all w ∈ B. Then (i) for T ′ small enough, we have for w ∈ B and (l i , r i , ω

i ) ∈ C, i = 1, 2, τ (l 1 , r 1 , ω 1 ) -τ (l 2 , r 2 , ω 2 ) ≤ k (l 1 , r 1 , ω 1 ) -(l 2 , r 2 , ω 2 ) , for some constant k ∈ (0, 1); (ii) for given (l, r, ω) ∈ C, the map w ∈ B → ( l, r, ω) ∈ C is continuous. Indeed, the map w ∈ B → v ∈ C([0, T ′ ], C 2,α (Ω) ∩ M p 2,δ+1
) is clearly continuous (using (2.32) and (2.33)-(2.36)), and hence the map w ∈ B → (l, r) ∈ C 1 ([0, T ′ ], R 6 ) is continuous (by (2.45)-(2.46)). Finally, using the assumption ω 0 ∈ M p 0,δ+3 , (2.28), (2.114), Aubin-Lions' lemma and the continuity of v, one can see (as e.g. in [START_REF] Glass | On the control of the motion of a boat[END_REF]) that the map w ∈ B → ω ∈ C([0, T ′ ], C 0,α (Ω) ∩ L p p(δ+2) (Ω)) is continuous. It follows again from the contraction mapping theorem (for a map depending on a parameter) that the map which associates with w ∈ B the fixed-point (l, r, ω) ∈ C is continuous.

Proof of main result

We are now in a position to prove the main result in this paper. Let T 0 , P, N, K and R be some given positive numbers. Then by Theorem 2.1, there exists a time T = T (T 0 , P, N, K, R) ∈ (0, T 0 ] such that system (1.19)-(1.25) has a solution (v, q, l, r) for t ∈ [0, T ], with (l, r, ω) ∈ C([0, T ], F), provided that

|(l 0 , r 0 )| ≤ 1, w C 1 ([0,T0]) ≤ R and ω 0 satisfies (2.19)-(2.21) and ω 0 C 1,α (Ω) + ω 0 M p 1,δ+2 ≤ P, div ω 0 = 0, ∂Ω ω 0 • n dσ = 0.
Let Π be a (continuous and linear) extension operator from

C 1 ([0, T ]) to C 1 ([0, T 0 ]) and pick δ := R/ Π . Then Π(w) C 1 ([0,T0]) ≤ R if w C 1 ([0,T ]) ≤ δ.
In particular, using assumption (H) for the time T , we have that Π(w) C 1 ([0,T0]) ≤ R if w = W (h 0 , q 0 , l 0 , r 0 , h T , q T , l T , r T ) for |(h 0 , q 0 , l 0 , r 0 , h T , q T , l T , r T )| < η with η > 0 small enough. Then system (1.19)-(1.25) has a solution defined for t ∈ [0, T ] corresponding to (l 0 , r 0 , ω 0 , w) as above, and also a potential solution corresponding to the same data (l 0 , r 0 , w) and to ω0 ≡ 0. Let ω 0 be as in the statement of Theorem 1.1, and write a 0 = (h 0 , q 0 ), b 0 = (l 0 , r 0 ), a T = (h T , q T ), and b T = (l T , r T ). Let a(t) := (h(t), q(t)) and b(t) = (l(t), r(t)). The proof is done in two steps. In the first step, we prove the result for ||ω 0 || C 1,α (Ω) , ||ω 0 || M p 1,δ+2 , |l 0 |, |r 0 |, |l T | and |r T | small enough, and in the second step, we remove this assumption by performing a scaling in time.

Step 1. Let the map W be as in the assumption (H) for the time T . We may pick a number η Pick any initial state (a 0 , b 0 ) = (h 0 , q 0 , l 0 , r 0 ) with |(a 0 , b 0 )| ≤ η 1 . For any given (a T , b T , v 0 ) with |(a T , b T )| ≤ η 1 , we denote by (h, q, l, r, v, q) the solution of (1.19)-(1.25) and (1.30) corresponding to the velocity v 0 and to the control w = W (a 0 , b 0 , a T , b T ), and by (h, q, l, r, v, q) the solution corresponding to (a 0 , b 0 ) together with the velocity v 0 which solves curl v 0 = 0, in Ω, div v 0 = 0, in Ω, v 0 • n = (l 0 + r 0 × y) • n, on ∂Ω,

                            
h ′ (t) = (1 -| q| 2 )l + 2 1 -| q| 2 q × l + (l • q ) q -q × l × q, q ′ (t) = 1 2 ( 1 -| q| 2 r + q × r), h ′ (t) = (1 -| q| 2 )l + 2 1 -| q| 2 q × l + (l • q ) q -q × l × q, q ′ (t) = 1 2 ( 1 -| q| 2 r + q × r), h(0) = h(0) = h 0 , q(0) = q(0) = q 0 , this gives for some constant C 2 > 0 We notice that f is continuous, by virtue of Proposition 2.5 and (1.30). Pick any ε ∈ (0, 1). From (3.1) and (3.3), we deduce that for v(t, y) := λ -1 v λ (λ -1 t, y), q(t, y) := λ -2 q λ (λ -1 t, y), w(t) := λ -1 w λ (λ -1 t), for y ∈ Ω and 0 ≤ t ≤ T λ := λT ∈ (0, T 0 ]. Then (a, b) is a trajectory for the underwater vehicle connecting (a 0 , b 0 ) at t = 0 to (a T , b T ) at t = T λ and corresponding to the initial fluid velocity v 0 .

||(h -h, q -q)|| L ∞ (0,T ) ≤ C 2 ||ω 0 || C 1,α (Ω) + ||ω 0 || M p

Acknowledgements

The first author (RL) was partially supported by Basal-CMM project, PFB 03. The second author (LR) was partially supported by the Agence Nationale de la Recherche, project ANR 15 CE23 0007 01 (Finite4SoS).

1 ∈

 1 (0, 1) such that w = W (a 0 , b 0 , a T , b T ) is defined for |(a 0 , b 0 )| ≤ η 1 and |(a T , b T )| ≤ η 1 , with ||w|| C 1 ([0,T ]) ≤ δ.

  lim |y|→∞ v 0 (y) = 0,and to the (same) control w. From (2.27) we infer that there exists some constant C 1 > 0 such that||(l -l, r -r)|| L ∞ (0,T ) ≤ C 1 ω 0 C 1,α (Ω) + ω 0 M p 0 , r 0 )| ≤ 1, ||ω 0 || C 1,α (Ω) + ||ω 0 || M p 1,δ+2 ≤ P, and ||w|| C 1 ([0,T ]) ≤ δ.(3.2)Combined to the equations

  3.2) holds. Let f : B = {x ∈ R 12 ; |x| ≤ 1} → R 12 be defined by f (x T ) = 1 η 1 (a(T ), b(T )) = 1 η 1 (h(T ), q(T ), l(T ), r(T ))where (a T , b T ) =: η 1 x T .

||ω 0

 0 || C 1,α (Ω) + ||ω 0 || M p 1,δ+2 < ν (3.4)with ν > 0 small enough, we have that|f (x T ) -x T | < ε, for |x T | ≤ 1.We need the following topological result[START_REF] Glass | On the control of the motion of a boat[END_REF] Lemma 4.1].Lemma 3.1. Let B = {x ∈ R n ; |x| < 1} and S = ∂B. Let f : B → R n be a continuous map such that for some constant ε ∈ (0, 1) |f (x) -x| ≤ ε ∀x ∈ S.(3.5)Then (1 -ε)B ⊂ f (B). (3.6)Thus, we infer from Lemma 3.1 that if (a 0 , b 0 , a T , b T ) ∈ R 24 is such that |(a 0 , b 0 )| < η 1 , |(a T , b T )| < η 2 := η 1 (1 -ε),and (3.4) is satisfied, then there exists a control w = W (a 0 , b 0 , ãT , bT ) for which the solution of (1.19)-(1.25) and (1.30) satisfies (h(T ), q(T ), l(T ), r(T)) = (a(T ), b(T )) = (a T , b T ).Step 2. To drop the assumptions|b 0 | < η 1 , |b T | < η 2 ,and (3.4) (corresponding to a given time T ∈ (0, T 0 ]), we use a scaling in time introduced in [3] for the control of Euler equations. Let (a 0 , b 0 ), (a T , b T ), and v 0 be given data with |a 0 | < η 2 and |a T | < η 2 . We set b λ 0 := λb 0 , b λ T := λb T , and v λ 0 := λv 0 . Then for λ > 0 small enough, we have that |(a 0 , b λ 0 )| < η 2 , |(a T , b λ T )| < η 2 , and ω λ 0 := curl v λ 0 satisfies ||ω λ 0 || C 1,α (Ω) + ||ω λ 0 || M p 1,δ+2 < ν. By Step 1, there exists some trajectory (a λ , b λ ) for the underwater vehicle connecting (a 0 , b λ 0 ) at t = 0 to (a T , b λ T ) at t = T , with corresponding fluid velocity v λ , pressure q λ , and control w λ . Let us set a(t) := a λ (λ -1 t), b(t) := λ -1 b λ (λ -1 t),

  where we used (2.51) to get 1 + min(| X1 |, | X2 |) ≥ C y for y ∈ Ω and t ∈ [0, T ′ ]. Thus, we have proved that for T ′ > 0 satisfying (2.66) and (2.72), we have

	, we have by [10, Lemma 2.2] that
	|ω 0 (y)| = O(|y| -δ-2 ) as |y| → +∞,	(2.94)
	so that we infer from (2.51) and (2.92) that
	A(0; t, y)ω On the other hand, by (2.19), (2.63) and (2.87), we have that
	A 2 (0; t, y) ω0 ( X1 (0; t, y)) -ω0 ( X2 (0; t, y)) ≤	C 1 + min(| X1 |, | X2 |)	| X| ≤ CT ′ (l, r, ω) ,

0 ( X1 (0; t, y)) ≤ CT ′ y (l, r, ω) |ω 0 ( X1 (0; t, y)| ≤ CT ′ (l, r, ω) .

  1.19)-(1.25) in the class (2.23)-(2.26). Then we can expand v in the form (2.32) with η as in (2.33)-(2.36). Then it is well-known that the vorticity ω = curl v satisfies the equation (2.114


Lionel Rosier
email: lionel.rosier@mines-paristech.fr.

On the Benjamin-Bona-Mahony equation with a localized damping

Keywords: AMS Subject Classification: 35Q53, 93D15 Benjamin-Bona-Mahony equation, unique continuation property, internal stabilization, boundary stabilization

We introduce several mechanisms to dissipate the energy in the Benjamin-Bona-Mahony (BBM) equation. We consider either a distributed (localized) feedback law, or a boundary feedback law. In each case, we prove the global wellposedness of the system and the convergence towards a solution of the BBM equation which is null on a band. If the Unique Continuation Property holds for the BBM equation, this implies that the origin is asymptotically stable for the damped BBM equation.

Introduction

The Benjamin-Bona-Mahony equation

v t -v xxt + v x + vv x = 0, (1.1)
was proposed in [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF] as an alternative to the Korteweg-de Vries (KdV) equation as a model for the propagation of one-dimensional, unidirectional, small amplitude long waves in nonlinear dispersive media. In the context of shallow water waves, v = v(x, t) stands for the displacement of the water surface (from rest position) at location x and time t. In the paper, we shall assume that either x ∈ R, or x ∈ (0, L) or x ∈ T = R/(2πZ) (the one-dimensional torus). The dispersive term -v xxt produces a strong smoothing effect for the time regularity, thanks to which the wellposedness theory of (1.1) is easier than for KdV (see [START_REF] Bona | Sharp well-posedness for the BBM equation[END_REF][START_REF] Roumégoux | A symplectic non-squeezing theorem for BBM equation[END_REF]). Solutions of (1.1) turn out to be analytic in time. On the other hand, the control theory is at his early stage for BBM (for the control properties of KdV, we refer the reader to the recent survey [START_REF] Rosier | Control and stabilization of the Korteweg-de Vries equation: recent progresses[END_REF]). S. Micu investigated in [START_REF] Micu | On the controllability of the linearized Benjamin-Bona-Mahony equation[END_REF] the boundary controllability of the linearized BBM equation, and noticed that the exact controllability fails due to the existence of a limit point in the spectrum of the adjoint equation. The author and B.-Y. Zhang introduced in [START_REF] Rosier | Unique continuation property and control for the Benjamin-Bona-Mahony equation[END_REF] a moving control and derived with such a control both the exact controllability and the exponential stability of the full BBM equation. For a distributed control with a fixed support, the exact controllability of the linearized BBM equation fails, so that the study of the controllability of the full BBM equation seems hard. However, it is reasonable to expect that some stability properties could be derived by incorporating some dissipation in a fixed subdomain or at the boundary. The aim of this paper is to propose several dissipation mechanisms leading to systems for which one has both the global existence of solutions and 1 a nonincreasing H 1 -norm. The conclusion that all the trajectories emanating from data close to the origin are indeed attracted by the origin is valid provided that the following conjecture is true:

Unique Continuation Property (UCP) Conjecture: There exists some number δ > 0 such that for any

v 0 ∈ H 1 (T) with v 0 H 1 (T) < δ, if the solution v = v(x, t) of v t -v xxt + v x + vv x = 0, x ∈ T, v(x, 0) = v 0 (x), x ∈ T (1.2) satisfies v(x, t) = 0 ∀(x, t) ∈ ω × (0, T ) (1.3)
for some nonempty open set ω ⊂ T and some time T > 0, then v 0 = 0 (and hence v ≡ 0).

To the best knowledge of the author, the UCP for BBM as stated in the above conjecture is still open. The main difficulty comes from the fact that the lines x = 0 are characteristic for BBM, so that the "information" does not propagate well in the x-direction. For some UCP for BBM (with additional assumptions) see [START_REF] Rosier | Unique continuation property and control for the Benjamin-Bona-Mahony equation[END_REF][START_REF] Yamamoto | One unique continuation for a linearized Benjamin-Bona-Mahony equation[END_REF]. See also [START_REF] Micu | Control and stabilization of a family of Boussinesq systems[END_REF][START_REF] Micu | Stabilization of a Boussinesq system with localized damping[END_REF] for control results for some Boussinesq systems of BBM-BBM type.

The following result is a conditional UCP in which it is assumed that the initial data is small in the L ∞ -norm and it has a nonnegative mean value. Its proof was based on the analyticity in time of the trajectories and on the use of some Lyapunov function.

Theorem 1 [START_REF] Rosier | Unique continuation property and control for the Benjamin-Bona-Mahony equation[END_REF] Let u 0 ∈ H 1 (T) be such that

T v 0 (x) dx ≥ 0, (1.4
)

and v 0 L ∞ (T) < 3. (1.5)
Assume that the solution v to (1.2) satisfies (1.3). Then v 0 = 0.

As it was noticed in [START_REF] Rosier | Unique continuation property and control for the Benjamin-Bona-Mahony equation[END_REF], the UCP for BBM cannot hold for any state in L 2 (T), for any initial data v 0 with values in {-2, 0} gives a trivial (stationary) solution of BBM. Thus, either a regularity assumption (v 0 ∈ H 1 (T)), or a bound on the norm of the initial data has to be imposed for the UCP to hold.

The paper is outlined as follows. In Section 2, we incorporate a simple localized damping in BBM equation and investigate the corresponding Cauchy problem. In Section 3, we consider another dissipation mechanism involving one derivative. The last section is concerned with the introduction of nonhomogeneous boundary conditions leading again to the dissipation of the H 1 -norm.

Stabilization of the BBM equation 2.1 Internal stabilization with a simple feedback law

We consider the BBM equation on T with a localized damping

u t -u xxt + u x + uu x + a(x)u = 0 x ∈ T, t ≥ 0, (2.1) u(x, 0) = u 0 (x) x ∈ T, (2.2)
where a is a smooth, nonnegative function on T with a(x) > 0 on a given open set ω ⊂ T.

We write (2.1)-(2.2) in its integral form

u(t) = u 0 - t 0 (1 -∂ 2 x ) -1 [a(x)u + (u + u 2 2 ) x ](τ )dτ (2.3) where (1 -∂ 2 x ) -1 f denotes for f ∈ L 2 (T) the unique solution v ∈ H 2 (T) of (1 -∂ 2 x )v = f . Define the norm • s in H s (T) as n∈Z c n e inx 2 s = n∈Z |c n | 2 (1 + |n| 2 ) s .
We have the following result.

Theorem 2 Let s ≥ 0 be given. For any u 0 ∈ H s (T), there exist T > 0 and a unique solution u of

(2.3) in C([0, T ], H s (T)). Moreover, the correspondence u 0 ∈ H s (T) → u ∈ C([0, T ], H s (T)) is Lipschitz continuous. If s = 1
, the solution exists for every T > 0, and the energy u(t) 2 1 is nonincreasing. Finally, if v 0 denotes any weak limit in H 1 (T) of a sequence u(t n ) with t n ր +∞, then the solution v of system (1.2) satisfies (1.3). In particular, if the UCP conjecture holds, then v 0 = 0, so that u(t) → 0 weakly in H 1 (T) as t → +∞, hence strongly in H s (T) for all s < 1.

Proof. We proceed as in [START_REF] Bona | Boussinesq equations and other systems for smallamplitude long waves in nonlinear dispersive media[END_REF] using the estimate

||f g|| s ≤ C||f || s+1 ||g|| s+1 ∀s ≥ -1, ∀f, g ∈ H s+1 (T). (2.4)
The estimate (2.4) follows from a similar estimate proved in [START_REF] Bona | Boussinesq equations and other systems for smallamplitude long waves in nonlinear dispersive media[END_REF] for functions defined on R, namely

|| f g|| H s (R) ≤ C|| f || H s+1 (R) ||g|| H s+1 (R) ∀s ≥ -1, ∀f, g ∈ H s+1 (R), (2.5) by letting f (x) = ϕ(x)f (x), g(x) = ϕ(x)g(x)
, where f and g are viewed as 2π-periodic functions, and

ϕ ∈ C ∞ 0 (R) denotes a cut-off function such that ϕ(x) = 1 on [0, 2π]. Indeed, we notice that ||f || H s (T) ≤ || f || H s (R) ≤ C||f || H s (T)
for some constant C > 0. Note that for any s ≥ 0

||(1 -∂ 2 x ) -1 ∂ x (f g)|| s ≤ C||f g|| s-1 ≤ C||f || s ||g|| s .
Pick any u 0 ∈ H s (T). Let us introduce the operator

(Γu)(t) := u 0 - t 0 (1 -∂ 2 x ) -1 [a(x)u + (u + u 2 2 ) x ](τ )dτ. Then sup 0≤t≤T ||(Γu -Γv)(t)|| s ≤ C 2 T 0 [||a(u -v)|| s-2 + ||u -v|| s-1 + ||u -v|| s ||u + v|| s ]dτ ≤ C 3 T (1 + 2R)||u -v|| C([0,T ],H s (T))
if we assume that u and v are in the closed ball B R of radius R centered at 0 in C([0, T ], H s (T)). We pick

T := [2C 3 (1 + 2R)] -1 so that ||Γu -Γv|| C([0,T ],H s (T)) ≤ 1 2 ||u -v|| C([0,T ],H s (T)) •
On the other hand

||Γu|| C([0,T ],H s (T)) ≤ ||u 0 || s + ||Γu -Γ0|| C([0,T ],H s (T)) ≤ ||u 0 || s + R 2 ≤ R
for the choice R = 2||u 0 || s . It follows that the map Γ contracts in B R , hence it admits a unique fixed point u in B R which solves the integral equation (2.3). Furthermore, given any ρ > 0 and any u 0 , v 0 ∈ H s (T) with ||u 0 || s ≤ ρ, ||v 0 || s ≤ ρ, one easily sees that for

T = [2C 3 (1 + 4ρ)] -1 (2.6) one has ||u -v|| C([0,T ],H s (T)) ≤ 2||u 0 -v 0 || s . (2.7)
Finally assume that s = 1. Scaling in (2.1) by u yields

1 2 ||u(T )|| 2 1 - 1 2 ||u 0 || 2 1 + T 0 T a(x)|u(x, t)| 2 dxdt = 0 (2.8) (Note that u t = -(1 -∂ 2 x ) -1 [a(x)u + (u + u 2 2 ) x ] ∈ C([0, T ], H 2 (T)
) so that each term in (2.1) belongs to C([0, T ], L 2 (T)), and the integrations by parts are valid.) It follows that the map t → ||u(t)|| 1 is nonincreasing, hence it admits a nonnegative limit l as t → ∞, and that the solution of (2.1)-(2.2) is defined for all t ≥ 0. Let T be as in (2.6) with ρ = ||u 0 || 1 . Note that ||u(t)|| 1 ≤ ||u 0 || 1 for all t ≥ 0. Let v 0 be any weak limit of {u(t)} t≥0 in H 1 (T), i.e. we have that u(t n ) → v 0 weakly in H 1 (T) for some sequence t n → +∞. Extracting a subsequence if needed, we may assume that

t n+1 -t n ≥ T for n ≥ 0. (2.9) From 1 2 ||u(t n+1 )|| 2 1 - 1 2 ||u(t n )|| 2 1 + tn+1 tn T a(x)|u(x, t)| 2 dxdt = 0, we obtain that lim n→+∞ tn+1 tn T a(x)|u(x, t)| 2 dxdt = 0. (2.10) Since u(t n ) → v 0 in H s (T) for s < 1, and ||u(t n )|| s ≤ ||u(t n )|| 1 ≤ ρ, we have from (2.7) that for all s ∈ [0, 1[ u(t n + •) → v in C([0, T ], H s (T)) as n → +∞, (2.11)
where v = v(x, t) denotes the solution of

v t -v xxt + v x + vv x + a(x)v = 0, x ∈ T, t ≥ 0, v(x, 0) = v 0 (x),
x ∈ T.

Notice that v ∈ C([0, T ]; H 1 (T)), for v 0 ∈ H 1 (T). (2.10) combined to (2.9) and (2.11) yields

T 0 T a(x)|v| 2 dxdt = 0. (2.12) Therefore v ∈ C([0, T ]; H 1 (T)) solves v t -v xxt + v x + vv x = 0, x ∈ T, t ∈ (0, T ), v(x, 0) = v 0 (x) x ∈ T, v(x, t) = 0 x ∈ ω, t ∈ (0, T ).
If the UCP conjecture is true, we have that v ≡ 0 on R × (0, T ). It follows that v 0 ≡ 0, and that as t → +∞ u(t) → 0 weakly in H 1 (T), u(t) → 0 strongly in H s (T) for any s < 1.

Internal stabilization with one derivative in the feedback law

We now pay attention to the stabilization of the BBM equation by means of a "stronger" feedback law involving one derivative. More precisely, we consider the system

u t -u xxt + u x + uu x -(a(x)u x ) x = 0, x ∈ T, t ≥ 0, (2.13) u(x, 0) = u 0 (x), x ∈ T. (2.14)
where a = a(x) denotes again any nonnegative smooth function on T with a(x) > 0 on a given open set ω ⊂ T. Scaling in (2.13) yields (at least formally)

1 2 ||u(T )|| 2 1 - 1 2 ||u 0 || 2 1 + T 0 T a(x)|u x (x, t)| 2 dxdt = 0. (2.15)
The decay of the energy is quantified by an integral term involving the square of a localized H 1 -norm in (2.15). By contrast, the integral term in (2.8) involved the square of a localized L 2 -norm. This suggests that the damping mechanism involved in (2.13) acts in a much stronger way than in (2.1). As a matter of fact, in the trivial situation when a(x) ≥ C > 0 for all x ∈ T in (2.13), it is a simple exercise to establish the exponential stability in H 1 (T) for both the linearized equation and the nonlinear BBM equation for states with zero means.

In the general case when the function a(x) is supported in a subdomain of T, we obtain the following result.

Theorem 3 Let s ≥ 0 be given. For any u 0 ∈ H s (T), there exist T > 0 and a unique solution u of (2.13)-(2.14) in C([0, T ], H s (T)). Moreover, the correspondence

u 0 ∈ H s (T) → u ∈ C([0, T ], H s (T)) is Lipschitz continuous. If s = 1, one can pick any T > 0 and u(t) -[u 0 ] H 1 (T) is nondecreasing. If the UCP conjecture is valid by replacing (1.3) by v = C on ω × (0, T ), then u(t) → [u 0 ] = (2π) -1 T u 0 (x)dx weakly in H 1 (T ) hence strongly in H s (T) for s < 1 as t → ∞.
Proof. As the proof is very similar to those of Theorem 2, we limit ourselves to pointing out the only differences. For the wellposedness, we use the estimate valid for s ≥ 0

||(1 -∂ 2 x ) -1 (au x ) x || s ≤ C||u|| s .
For s = 1, we claim that (2.15) is justified by noticing that for u ∈ H 1 (T)

-(au x ) x , u H -1 ,H 1 = au x , u x L 2 ,L 2 .
Then the wellposedness statement is established as in Theorem 2. Let us proceed to the asymptotic behavior. Pick any u 0 ∈ H 1 (T) and any v 0 ∈ H 1 (T) which is the weak limit in H 1 (T) of a sequence u(t n ) with t n → ∞ and t n+1 -t n ≥ T . Let us still denote by v the solution of (1.2). Equation (2.12) has to be replaced by

T 0 T a(x)|v x (x, t)| 2 dxdt = 0. (2.16)
To prove (2.16), we notice first that u(

t n + •) → v in C([0, T ], H s (T)) for s < 1, and that ||u(t n + •)|| L 2 (0,T ;H 1 (T))
≤ const, so that, extracting a subsequence if needed, we have that

u(t n + •) → v weakly in L 2 (0, T ; H 1 (T)).
This yields, with (2.15),

T 0 T a(x)|v x | 2 dxdt ≤ lim inf n→∞ tn+T tn T a(x)|u nx | 2 dxdt = 0. Therefore v solves v t -v xxt + v x + vv x = 0, x ∈ T, t ∈ (0, T ), (2.17) v x = 0, x ∈ ω, t ∈ (0, T ), (2.18) v ∈ C([0, T ]; H s (T)) for s < 1. (2.19)
(2.17) and (2.18

) yield v t = v x = 0 in ω × (0, T ), hence v(x, t) = C for (x, t) ∈ ω × (0, T )
for some constant C ∈ R. If the UCP Conjecture is still valid when v is constant on the band ω × (0, T ), then v ≡ C. As d[u(t)]/dt = 0, it follows that C = [u 0 ] and that as t → ∞, u(t) → [u 0 ] weakly in H 1 (T) and strongly in H s (T) for any s < 1.

Remark 1 Similar results, but with convergences towards 0, hold for the system

u t -u xxt + u x + uu x -(a(x)u x ) x = 0, u(0, t) = u(2π, t) = 0, u(x, 0) = u 0 (x)
provided that a(x) > 0 for x ∈ (0, ε) ∪ (2π -ε, 0) for some ε > 0 and the UCP Conjecture holds.

Boundary stabilization of BBM

In this section, we are concerned with the boundary stabilization of the BBM equation. We consider the following system

u t -u xxt + u x + uu x = 0, x ∈ (0, L), t ≥ 0, (2.20)
u tx (0, t) = αu(0, t) + 1 3 u 2 (0, t), (2.21)

u tx (L, t) = βu(L, t) + 1 3 u 2 (L, t), (2.22)
u(x, 0) = u 0 (x), (2.23) where α and β are some real constants chosen so that

α > 1 2 and β < 1 2 •
Scaling in (2.20) by u yields (at least formally) (2.25) u(t) → 0 strongly in H s (0, L) for all s < 1.

d dt 1 2 L 0 (|u| 2 + |u x | 2 )dx = [uu tx ] L 0 - u 2 2 + u 3 3 L 0 = (β - 1 2 )|u(L, t)| 2 + ( 1 2 -α)|u(0, t)| 2 , ( 2
(2.26)

Proof. Let us begin with the well-posedness part. Pick any u 0 ∈ H s (0, L) with s > 1 2 . Let v = u t . Then v solves the elliptic problem

(1 -∂ 2 x )v = f, x ∈ (0, L), (2.27) v x (0) = a, v x (L) = b (2.28)
with f := -u x -uu x , a := αu(0, t) + 1 3 u 2 (0, t), b := βu(L, t) + 1 3 u 2 (L, t). Note that the solution v of (2.27)-(2.28) may be written as

v = w + g where g(x) = ax + b-a 2L x 2 and w = (1 -∂ 2 x ) -1 N (f -(1 -∂ 2 x )g) solves (1 -∂ 2 x )w = f -(1 -∂ 2 x )g w x (0) = w x (L) = 0. Thus u t = v = -(1 -∂ 2 x ) -1 N (u x + uu x ) + (1 -(1 -∂ 2 x ) -1 N (1 -∂ 2 x ))g. ( 2

.29)

We seek u as a fixed point of the integral equation

u(t) = Γ(u)(t) := u 0 + t 0 -(1 -∂ 2 x ) -1 N (u x + uu x )(τ ) +(1 -(1 -∂ 2 x ) -1 N (1 -∂ 2 x )) [αu(0, τ ) + 1 3 u 2 (0, τ )]x +(2L) -1 [βu(L, τ ) + 1 3 u 2 (l, τ ) -αu(0, τ ) - 1 3 u 2 (0, τ )]x 2 dτ.(2.30) Note that D((1 -∂ 2 x ) s 2 N ) = H s (0, L) for -1/2 < s < 3/2.
Let R > 0 be given and let B R denote the closed ball in C([0, T ], H s (0, L)) of center 0 and radius R. For 1/2 < s < 5/2 and u ∈ B R , v ∈ B R , we have that (2.24). Thus ||u(t)|| H 1 (0,L) is nonincreasing, and T may be taken as large as desired. Let us turn our attention to the asymptotic behavior. Let (t n ) n≥0 be any sequence with t n → ∞ as n → ∞. Extracting a subsequence if needed, we can assume that t n+1 -t n ≥ T for each n and that, for some v 0 ∈ H 1 (0, L), u(t n ) → v 0 weakly in H 1 (0, L) (hence strongly in H s (0, L) for s < 1) as n → ∞. The continuity of the flow map (which follows at once from the fact that the map Γ is a contraction) yields

||Γ(u)(t) -Γ(v)(t)|| H s (0,L) ≤ C t 0 (1 + R)||u(τ ) -v(τ )|| H s (0,L) dτ ≤ CT (1 + R)||u -v|| C([0,T ],H s (0,L)) and ||Γ(u)(t)|| H s (0,L) ≤ ||u 0 || H s (0,L) + CT (1 + R)||u|| C([0,T ],H s (0,L)) • Pick R = 2||u 0 || H s (0,L) and T = (2C(1 + R)) -1 . Then Γ is a contraction in B R ,
u(t n + •) → v(•)
in C([0, T ], H s (0, L)) for s < with v(x, t) = 0 for x ∈ R \ (0, L), t ∈ (0, T ).

Since v ∈ C([0, T ], H s (R)) with 1/2 < s < 1, we infer from the UCP Conjecture (if true) that v ≡ 0, hence v 0 = 0.

  .24) hence ||u(t)|| H 1 is nonincreasing if α and β fulfill (2.3). The wellposedness of (2.20)-(2.23) and the asymptotic behavior are described in the following result.

	Theorem 4 Let s ∈ (1/2, 5/2) and u 0 ∈ H s (0, L). Then there exist a time T > 0 and a
	unique solution u ∈ C([0, T ], H s (0, L)) of (2.20)-(2.23). Furthermore, if s = 1, then T may
	be taken arbitrarily large, and if the UCP Conjecture holds, as t → ∞
	u(t) → 0 weakly in H 1 (0, L)

  hence it admits a unique fixed point which solves (2.30), or (2.29) and (2.23). Assume now that s = 1. It follows from (2.29) that u t ∈ C([0, T ], H 2 (0, L)), hence we can scale by u in (2.20) to derive

  1, where v denotes the solution of (2.20)-(2.23) issued from v 0 at t = 0. Integrating (2.24) on [t n , t n+1 ] yields Extending v(x, t) by 0 for x ∈ R \ (0, L) and t ∈ (0, T ), we infer from (2.20), (2.21), (2.22) (for v) and (2.31) that v t -v txx + v x + vv x = 0 for x ∈ R, t ∈ (0, T )

	Letting n → ∞ yields
		(	1 2	-β)	0	T	|v(L, t)| 2 dt + (α -	1 2	)	0	T	|v(0, t)| 2 dt = 0.	(2.31)
	1 2	||u(t n+1 )|| 2 H 1 (0,L) -	1 2	||u(t n )|| 2 H 1 (0,L) +(	1 2	-β)	tn+1 tn	|u(L, t)| 2 dt+(α-	1 2	)	tn+1 tn	|u(0, t)| 2 dt.


Saber Dakhli
email: saber.dakhli@gmail.com

Hatem Rmili
email: hmrmili@kau.edu.sa

Jean-Marie Floc'h
email: jean-marie.floch@insa-rennes.fr

Muntasir Sheikh

Abdallah Dobaie
email: adobaie@kau.edu.sa

Kourosh Mahdjoubi
email: kouroch.mahdjoubi@univ-rennes1.fr

Fethi Choubani
email: fethi.choubani@supcom.rnu.tn

Richard W Ziolkowski
email: ziolkowski@ece.arizona.edu

Keywords: Capacitively loaded loop (CLL), metamaterial-inspired structures, multiband behavior, printed dipole antennas, radiation patterns. I

A family of multiband metamaterial-inspired dipole antennas are reported. These antennas are based on a dipole integrated in a planar configuration with capacitively loaded loops (CLLs) as their near-field resonant parasitic (NFRP) elements. In the realized structures, the number of operating frequencies is determined by the size of the CLLs and their proximity to the printed dipole.. Simulation results explain the design features and demonstrate that the structures present a multiband behavior. Three corresponding prototypes were fabricated and tested. The measured impedance mismatch and radiation pattern results are presented and are shown to be in good agreement with their simulated values.

INTRODUCTION

Metamaterials (MTMs) are characterized as exhibiting exotic physics behaviors and, as a consequence, they have been attractive for use in many electromagnetic engineering applications [START_REF] Engheta | Metamaterials: Physics and Engineering Explorations[END_REF][START_REF] Eleftheriades | Negative-Refraction Metamaterials Fundamental Principles and Applications[END_REF][START_REF] Caloz | Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications[END_REF]. In particular, their material properties can be tailored to a given application. Several prototypes have been developed for radiating and scattering applications operating from UHF to optical frequencies. In contrast to bulk metamaterials, metamaterial-inspired structures are those that are based on one or a few metamaterial unit cells. They have provided extremely interesting flexibility in the engineering design process for manipulating a variety of electromagnetic, acoustic and thermal wave phenomena.

A number of metamaterial-inspired electrically small antenna (ESA) designs, emphasizing high efficiency, high directivity and large bandwidth have been realized. The concept of metamaterial-based antennas was proposed in the early 2000's [START_REF] Ziolkowski | Application of double negative metamaterial to increase the power radiated by electrically small antennas[END_REF], [START_REF] Ziolkowski | Metamaterial-based efficient electrically small antennas[END_REF]. The paradigm of metamaterial-inspired electrically small antennas was introduced in [START_REF] Erentok | Metamaterial-inspired efficient electrically-small antennas[END_REF]. A variety of examples of these types of antennas have been reported [START_REF] Lee | A compact and low-profile tunable loop antenna integrated with inductors[END_REF]- [START_REF] Zhu | A compact tri-band monopole antenna with single-cell metamaterial loading[END_REF]. Reviews describing the progress of recent applications of metamaterials to antennas have appeared recently [START_REF] Ziolkowski | Metamaterial-inspired engineering of antennas[END_REF], [START_REF] Dong | Metamaterial-based antennas[END_REF].Additional studies have reported configurations that are low profile and operate at low frequencies [START_REF] Ziolkowski | Design and Experimental Verification of Z Antennas at UHF Frequencies[END_REF], have multi-functionalities [START_REF] Jin | Multi-functional, electrically small, planar near-field resonant parasitic antennas[END_REF], and have high directivities [START_REF] Jin | High directivity, electrically small, lowprofile, near-field resonant parasitic antennas[END_REF]. Recent considerations have emphasized the inclusion of active non-Foster (NF) elements in these near-field resonant parasitic (NFRP) antenna designs to achieve several interesting electrically small antenna properties simultaneously, including broad impedance and directivity bandwidths [START_REF] Zhu | Active metamaterial-inspired broad bandwidth, efficient, electrically small antennas[END_REF]- [START_REF] Tang | Augmenting a modified Egyptian axe dipole antenna with non-Foster elements to enlarge its directivity bandwidth[END_REF]. These NF designs were integrated together to achieve an efficient, high directivity, large front-to-back ratio, and wide impedance and directivity bandwidth, electrically small antenna [START_REF] Ziolkowski | An efficient, broad bandwidth, high directivity, electrically small antenna[END_REF]. Active unit cells have been of great research interest for low loss and broadbandwidth metamaterials from low to very high frequencies [START_REF] Hrabar | Negative capacitor paves the way to ultra-broadband metamaterials[END_REF]- [START_REF] Xiao | Loss-free and active optical negative-index metamaterials[END_REF].

In this paper, we report printed planar multiband metamaterial-inspired antennas that are based on the compact augmentation of a driven dipole antenna with two capacitively loaded loop (CLL) elements. Single [START_REF] Turkmen | Metamaterial inspired, electrically small, GSM antenna with steerable radiation patterns and high radiation efficiency[END_REF]- [START_REF] Dakhli | Family of low-profile and directive metamaterial-inspired antennas[END_REF] and multiple [START_REF] Jin | Metamaterial-inspired, electrically small, Huygens sources[END_REF], [START_REF] Turkmen | Single, dual and triple-band metamaterial-inspired electrically small planar magnetic dipole antennas[END_REF] CLLs have been used previously as NFRP elements to successfully control the radiation patterns of printed monopole antennas at one and several frequencies. The CLLs are simpler metamaterial unit cells than their splitring resonator (SRR) counterparts that have been used in similar applications, e.g., [28], [START_REF] Barbuto | Design of a multifunctional SRR-loaded printed monopole antenna[END_REF].. This work represents an extension of the results presented in [START_REF] Dakhli | Compact and Multi-Band Metamaterial-Inspired Dipole Antenna[END_REF]; it includes a more complete analysis of the design features of the antenna and improved experimental results. In particular, it will be shown that the addition of the CLLs leads to the desired multifrequency behavior and depends simply on their size and their proximity to the printed dipole.

II. BASIC STRUCTURE

A printed dipole antenna was selected as the driven element and the two CLLs were selected as the NFRP elements because they could be very simply integrated with it. The resulting NFRP antenna was fabricated with a Rogers Duroid TM 5880 substrate with thickness 0.8 mm, relative permittivity 2.2, relative permeability r = 1.0 and loss tangent equal to 0.0009. The structure was simulated by using the ANSYS/ANSOFT high frequency structure simulator (HFSS).

A. Printed dipole antenna design

As shown in Fig. 1, the printed dipole is constructed in a three layer fashion. One dipole arm and the positive microstrip feed-line connected to it are printed on the top side of the substrate. The second dipole arm and the negative microstrip feed-line connected to it are printed on the bottom side. A ground strip near the feed point of the structure on the bottom side is combined with the negative feed-line. The two arms have a total length L, which was selected to be L = 50 mm. The distance D between the arms and the ground plane influences the antenna impedance matching. The ground strip length was chosen to be 66 mm, slightly larger than L, in order for it to act as a reflector, concentrating the radiated power into the front, broadside direction (θ = 0°). The width of the ground strip (6 mm) and the width of its feed-point portion (15 mm) were selected to accommodate the SMA connector and for ease of fabrication. With an optimized distance D= 34 mm, between the dipole arms and the ground strip, the antenna operates at 2.45 GHz in the ISM band without any additional matching network. The details of the design are given in Fig. 1.

B. Simulation results

The simulated performance characteristics of this printed dipole radiator are summarized in Fig. 2. The input impedance mismatch values as a function of the excitation frequency are given in Fig. 2(a). It shows that the resonance frequency is 2.45GHz with a 10dB fractional impedance bandwidth approximately equal to 19%. Fig. 2(b) presents the 3D-radiation pattern of the antenna at its resonance frequency, 2.45GHz. It clearly shows a dipole-like pattern with its null along the X-axis and with a maximum towards the positive Y-direction. The maximum gain is 2.32dB. This value is slightly larger than expected from a small dipole because of the presence of the ground-strip reflector, which slightly reduces the back radiation from the radiator while enhancing its forward value. The distribution of the surface current vectors on the copper of the antenna at 2.45 GHz is illustrated in Fig. 2(c). While the current is large along the Y axis on the positive feedline, the associated radiated field is cancelled by the field created by the opposite flowing current on the negative feedline. The currents that create the dipole pattern shown in Fig. 2(b) are dominated by the contributions from the arms and the ground strip.

III. ANTENNAS WITH TWO CLL NFRP ELEMENTS

In order to obtain a multi-band behavior, we have added two printed CLL elements to the structure, one on each side of the substrate to be coupled to the associated dipole arm and feedline as depicted in Fig. 3. As proved in previous works [START_REF] Turkmen | Metamaterial inspired, electrically small, GSM antenna with steerable radiation patterns and high radiation efficiency[END_REF], [START_REF] Dakhli | Analysis of a compactand superdirectivemtamaterial-inspired monopole antenna[END_REF], the presence of a CLL element close to a driven dipole antenna generates a second resonance at a frequency lower than the driven element. This effect reduces the effective overall size of the antenna, which improves its miniaturization [START_REF] Turkmen | Metamaterial inspired, electrically small, GSM antenna with steerable radiation patterns and high radiation efficiency[END_REF], [START_REF] Dakhli | Analysis of a compactand superdirectivemtamaterial-inspired monopole antenna[END_REF]. In our case, two CLL NFRP elements have been added close to the segments of the dipole antenna where the highest surface currents are localized. This significantly increases the coupling between the CLL elements and the driven dipole. It must be noted that configuration A2 was obtained from configuration A1 after an extensive parametric study which took into account the main design parameters and optimized the structure to achieve the lowest operating frequencies with good impedance matching. It was found that the main parameters affecting these two main properties are the dimensions of the CLLs and the coupling distance between them and the feed line.

A. Optimized structures

B. Simulation results

1-Impedance matching

The simulated |S11| values of these two mutli-band antennas as a function of the source frequency are plotted in Fig. 4.As shown in Fig. 4 (a), antenna A1 is characterized by a single low resonance frequency at 1.61GHz, in addition to the dipole resonance at 2.17GHz. This single, low frequency resonance result is due to the fact that both CLLs are identical and symmetrically located with respect to the driven dipole and, therefore, resonate at the same frequency. On the other hand, in the case of antenna A2, where the two loops are sized differently and are asymmetrically positioned, Fig. 4(b) shows that two different low frequency resonances are obtained, at 1.52 and 1.62 GHz, in addition to the dipole resonance at 2.20GHz. These low frequency resonances are related to the size of each CLL element and to its level of coupling to the dipole, as well as its coupling to the other CLL element. The coupling is controlled by the distances the CLL elements are from the driven dipole and each other. In

2-Surface currents

In order to analyze the origin of these resonances, we have studied the distribution of the surface currents on the copper elements for both structures. Fig. 5 illustrates the simulated surface current distributions for antennas A1 and A2 at their resonance frequencies. From Fig. 5-a, it is verified that the lowest frequency resonance at 1.61 GHz corresponds to the simultaneous resonance of both CLL elements and that the highest one corresponds to the dipole resonance at 2.17GHz. Similarly, in Fig. 5-b, the lowest frequencies at 1.52 GHz and 1.62 GHz correspond to the larger (left) and the same-sized (right) CLL elements, respectively. The resonance at 2.20 GHz clearly shows the dipole behavior. Notice that when each of the structures is resonant, the others are not. Again, this indicates that each of the resonance frequencies could be tuned without impacting the other ones.

3-Radiation patterns

The simulated gain patterns of antennas A1 and A2at their resonance frequencies are presented in Fig. 6. These patterns closely correspond to the behavior expected from the current distributions shown in Fig. 5. Fig. 6(a) shows that antenna A1 acts as a pair of magnetic dipoles at the low frequency 1.61 GHz that are oriented along the z-axis, i.e., they result from the loop modes excited in the CLL elements as illustrated in Fig. 5(a). The pattern has its maximum gain in the (x-y) plane. It also shows that antenna A1 acts as an electric dipole placed along x-axis at the higher resonance frequency, 2.17GHz, where the dipole element is resonant but the CLL elements are not. This pattern clearly corresponds to the reference dipole case shown in Fig. 2(b), which shows the modification of the dipole pattern by the presence of the ground strip. The corresponding simulated gain patterns for antenna A2 at its resonance frequencies are given in Fig. 6(b). The antenna clearly acts as two independent, resonant magnetic dipoles at the lower resonance frequencies, 1.52 and 1.62GHz. At 1.52GHz, the maximum gain is along the x-axis; at 1.62 GHz it is along the y-axis. At the higher resonance frequency, 2.20 GHz, it again acts as an electric dipole modified by the presence of the ground strip.

C. Experimental Validation

Prototypes of the two antennas, A1 and A2, were fabricated. Photos of them are shown in Fig. 7. They were characterized experimentally by measuring their impedance mismatch over the frequency band 1-3 GHz and their gain patterns at their resonant frequencies. The impedance mismatch, i.e., the |S11| values, as a function of the source frequency was measured using a network analyzer Agilent N5230A. The gain patterns were obtained in a "SATIMO Stargate32" anechoic chamber.

1-Impedance matching

2-Radiation patterns

Fig. 9 presents the 3D measured radiation patterns of the fabricated prototypes of the two antennas at their resonance frequencies. We can observe different shapes of the radiation patterns due to the different excited parts of the antenna A2. In fact, at the lowest frequency, 1.53GHz, the first CLL element (on the right in Fig. 3) is strongly excited and resonant; the antenna acts as a dipole oriented along the yaxis. At the frequency, 1.61GHz, the second CLL element (on the left in Fig. 3) is resonant. At this frequency, we note that the pattern is more directive along the +z-axis. In this case, the structure behaves like a Huygens source in that the CLL element acts as a magnetic dipole in combination with the electric dipole. At 2.18 GHz, the electric dipole behavior is clearly dominant once again. In the case of the antenna A1 and at the lowest resonance frequency 1.58 GHz, the two CLL elements are simultaneously highly excited and visually act as a dipole perpendicular to the principal dipole. For the resonance occurring at 2.18GHz, the two CLLs are not resonating and their effects are disabled. The radiation pattern of the antenna is highly affected by the resonance of the principal dipole. IV.

ANTENNA WITH TWO NFRP CLL ELEMENTS AND A DIRECTOR ELEMENT

In order to generate an additional resonance frequency, a parasitic element is introduced. This geometry, included the additional director element, is shown in Fig. 10. This structure was based on the A2 antenna design optimized to generate a new higher frequency resonance.

A. Antenna design

The geometry and the parameters design of the antenna are shown in Figure 11. The dimensions of the dipole and the two CLLs are the same as the antenna A2.

B. Results and discussions 1-Impedance matching

Fig. 12 represents simulated and measured impedance mismatch for the proposed antenna. Again, the agreement between the measured and the predicted results is good. We can remark from Fig. 12 that we have generated especially an additional resonant frequency higher than the monopole's one. Simulations performed on the structure show multi resonant frequencies at 1.52 GHz, 1.61 GHz, 2.38 GHz and 3.65 GHz, while measurement gives resonances at 1.54 GHz, 1.63GHz, 2.25 GHz and 3.69 GHz.

2-Surface currents

To investigate the origin of these resonances, we have studied the distribution of surface currents on the structure at two selected resonant frequencies. Fig. 13 represents the surface currents simulated at 2.38 GHz and 3.65 GHz. At the frequency 2.38 GHz, only the dipole is active, the two resonators and the parasitic element are disabled, whereas the frequency 3.65 GHz corresponds mainly to the parasitic element excitation.

3-Radiation patterns

Fig. 14 presents 3D simulated radiation patterns of the proposed antenna at its resonance frequencies. We can observe different shapes of radiation patterns due to the different excited parts of the antenna. In fact, at the lowest frequency 1.54 GHz, the first resonator element (on the right in Fig. 3) is strongly excited and acts as a dipole with a zero on y-axis. The frequency 1.63 GHz, corresponds to the resonance of the second resonator (on the left in Fig. 3). At this frequency, we note a directive beam toward +y-axis. In this case, the structure behaves like a Huygens source, where the CLL element acts as a magnetic dipole which is combined to the electric dipole.

The parasitic element and the two CLLs are disabled at the frequency 2.38 GHz. The resulting radiation pattern corresponds to the dipole one. Finally, the frequency 3.65 GHz, corresponds to the resonance of the parasitic element alone. The measured values of the maximum gain obtained for the proposed antenna are 0.87 dBi, 0.48 dB, 3.46 dB and 4.47 dB at frequencies 1.54 GHz, 1.63 GHz, 2.25 GHz and 3.69 GHz, respectively

V. CONCLUSION

In this paper, a class of multi-band metamaterial-inspired printed dipole antennas is reported. The structures consist of a principal dipole antenna associated with two CLLs parasitic elements in order to obtain supplementary resonances. Simulated and experimental results are in good agreement. These results show that placing CLLs and a parasitic element close to the printed dipole creates resonant frequencies depending on their size and proximity to the feeding line. In addition to the multifrequency behavior, the proposed antennas present different shape of radiation patterns due to the possibility to able or disable some parts of the structure as the dipole, the CLLs elements and the parasitic element. These properties make the proposed antennas well suitable for some emerging multiband wireless applications.

Figure 1 :

 1 Geometry of the basic structure (dimensions in millimeters).

Figure 2 :

 2 Simulation results of the basic printed dipole structure. (a) Input impedance mismatch values, |S11| (dB), versus the excitation frequency show the resonance frequency is 2.45 GHz;(b) 3D-gain pattern and (c) surface current distribution at this resonance frequency.

  We have studied two configurations: the first structure shown in Fig 3(a) denoted A1 consists of two identical, symmetrically located (with respect to the microstrip line) CLL elements. The second antenna, denoted A2, is shown in Fig. 3(b). It consists of two CLL elements having slightly different sizes and placements with respect to the microstrip line. The configurations were optimized to achieve good impedance matching to a 50  source.

Figure 3 :

 3 Figure 3: Geometry and dimensions (in millimeters) of the multiband antenna structures. (a) Antenna A1, and (b) antenna A2.

Fig. 4 (

 4 b) the larger sized CLL element gives the 1.51 GHz value and the different distance to the dipole of the samesized CLL element slightly shifts the corresponding resonance frequency from 1.61 to 1.62 GHz. The asymmetric coupling of the two CLL elements, particularly the close proximity of the same-sized one to the dipole element, lowers the dipole resonance. By varying their sizes and the coupling distances, the various resonance frequencies can be tuned to desired values.

Figure 4 :

 4 Simulated input impedance mismatch, |S11,| as a function of the source frequency. (a) Antenna A1, and (b) antenna A2.

Figure 5 :

 5 Figure 5: Simulated surface current distributions for the two proposed antennas at their resonance frequencies. (a) Antenna A1, and (b) antenna A2.

Figure 6 :

 6 Figure 6: Simulated 3D gain patterns for the two proposed antennas at their resonance frequencies. (a) Antenna A1, and (b) antenna A2.

Figure 7 :

 7 Figure 7 :Photos of the realized prototypes: antenna A1: (a) top view, (b)bottom view; andantenna A2: (c) top view, (d) bottom view.

Fig. 8

 8 Fig. 8 compares both the simulated and measured |S11| values for antennas A1 and A2. As predicted, when the two CLLs are identical and symmetric, only two resonance frequencies are obtained as illustrated in Fig.8 (a). On the other hand, when the CLLs are asymmetrically located and have different sizes, three resonance frequencies are obtained as confirmed in Fig. 8(b).Fig. 8 confirms the multi-band behavior of the proposed structures. Resonances occur at frequencies 1.55 GHz and 2.18 GHz for antenna A1; and at 1.48 GHz, 1.59 GHz and 2.18 GHz for antenna A2. The agreement between the simulated and measured results is very reasonable given the fabrication tolerances.

Figure 8 :

 8 Figure 8: Comparison of the measured and simulated input impedance mismatch, |S11|, as a function of the source frequency.(a) Antenna A1, and (b) antenna A2.

F1

Figure 9 :

 9 Figure 9: Measured 3D gain patterns for the two proposed antennas at their resonance frequencies. (a) Antenna A1, and (b) antenna A2.

Figure 10 :

 10 Figure 10: Geometry and dimensions (in millimeters) of the multiband antenna structure including a director element.The frequency F4 corresponds to the length λeff/2 of the parasitic element.The prototype is fabricated and characterized. Fig.11presents photos of the realized prototype.

Figure 11 :

 11 Figure 11: Photos of the realized prototype: (a) Top View; (b) Bottom View.

Figure 12 :

 12 Figure 12: Measured and Simulated input impedance mismatch, |S11| as a function of the source frequency

F3 = 2

 2 .38 GHz F4 = 3.65 GHz

Figure 13 :

 13 Figure 13:Simulated surface current for the proposed antenna at the resonance frequencies.

F1=Figure 14 :F1Figure 15 :

 1415 Figure 14: Simulated 3D-radiation patterns for the single NFRP element reconfigurable antenna.

ACKNOWLEDGMENT

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. . The authors, therefore, acknowledge technical and financial support of KAU.

REFFERENCES