Arnaud Münch
email: arnaud.munch@math.univ-bpclermont.fr.

Pablo Pedregal
email: pablo.pedregal@uclm.es.

Pablo Pedregal About

ABOUT LEAST-SQUARES TYPE APPROACH TO ADDRESS DIRECT AND CONTROLLABILITY PROBLEMS

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Introduction

Let Ω ⊂ R N , N = 2 or N = 3 be a bounded connected open set whose boundary ∂Ω is Lipschitz. Let ω ⊂ Ω be a (small) nonempty open subset, and assume that T > 0. We use the notation Q T = Ω × (0, T ), q T = ω × (0, T ), Σ T = ∂Ω × (0, T ) and we denote by n = n(x) the outward unit normal to Ω at any point x ∈ ∂Ω. Bold letters and symbols denote vector-valued functions and spaces; for instance L 2 (Ω) is the Hilbert space of the functions v = (v 1 , . . . , v N ) with v i ∈ L 2 (Ω) for all i.

This work is related to the null controllability problem for the non-stationary Stokes system (1.1) y t -ν∆y + ∇π = f 1 ω , ∇ • y = 0 in Q T y = 0 on Σ T , y(•, 0) = y 0 in Ω which describes a viscous incompressible fluid flow in the bounded domain Ω. We use as a control function the density of external forces f = f (x, t) concentrated in the arbitrary subdomain ω during the time interval (0, T ); y is the vector field of the fluid velocity, and π is the scalar pressure. The real ν denotes the constant viscosity of the fluid. The symbol 1 ω stands for the characteristic function of ω. We introduce the following spaces (1.2)

H = {ϕ ∈ L 2 (Ω) : ∇ • ϕ = 0 in Ω, ϕ • n = 0 on ∂Ω}, V = {ϕ ∈ H 1 0 (Ω) : ∇ • ϕ = 0 in Ω}, U = ψ ∈ L 2 (Ω) : Ω ψ(x) dx = 0 .
Then, for any y 0 ∈ H, T > 0, and f ∈ L 2 (q T ), there exists exactly one solution (y, π) of (1.1) with the following regularity :

y ∈ C 0 ([0, T ]; H) ∩ L 2 (0, T ; V) , π ∈ L 2 (0, T ; U )
(see [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF]). The null controllability problem for (1.1) at time T is the following:

For any y 0 ∈ H, find f ∈ L 2 (q T ) such that the corresponding solution to (1.1) satisfies

(1.3) y(•, T ) = 0 in Ω.
The controllability properties of evolution PDEs have attracted a lot of works in the last decades: some relevant references are [3,[START_REF] Fursikov | Controllability of Evolution Equations[END_REF][START_REF] Lasiecka | Control theory for partial differential equations: continuous and approximation theories. I. Abstract parabolic systems[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF][START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions[END_REF]. In particular, the Stokes -and more generally the Navier-Stokes -system has received a lot of attention: we mention the references [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF][START_REF] Yu | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF].

The following result is proved in [START_REF] Fursikov | On approximate controllability of the Stokes system[END_REF] (see also [START_REF] Fabre | Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems[END_REF][START_REF] Fursikov | Controllability of Evolution Equations[END_REF][START_REF] Yu | Remarks on exact controllability for the Navier-Stokes equations[END_REF]) by the way of Carleman estimates.

Theorem 1.1 (Fursikov-Imanuvilov). -The linear system (1.1) is null-controllable at any time T > 0.

On the other hand, the (numerical) approximation of controls either distributed or located on the boundary for the Stokes system has received much less attention, due to the underlying ill-posedness of the approximation. In practice, such approximation is usually addressed in the framework of an optimal control reformulation. Precisely, one seeks to minimize the quadratic functional J(f ) := 1 2 f 2 L 2 (qT ) over the non-empty set C(y 0 , T ) = {(y, f ) : f ∈ L 2 (q T ), y solves (1.1) and satisfies (1.3)}.

Following [START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems: a numerical approach Encyclopedia of Mathematics and its Applications[END_REF], duality arguments allow to replace this constrained minimization by the unconstrained minimization of its conjugate function J ⋆ defined as

J ⋆ (ϕ T ) = 1 2 qT |ϕ| 2 dx dt + Ω y 0 • ϕ(•, 0) dx over ϕ T ∈ H
, where (ϕ, σ) solves the adjoint backward Stokes system associated with (1.1) :

(1.4)

-ϕ t -ν∆ϕ + ∇σ = 0, ∇ • ϕ = 0 in Q T ϕ = 0 on Σ T , ϕ(•, T ) = ϕ T in Ω
H is the Hilbert space defined as the completion of any smooth space functions included in H for the norm ϕ L 2 (qT ) . The control of minimal square-integrable norm is then given by f = φ 1 ω where φ is associated with the unique minimizer φT in H of J ⋆ (see [3,[START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems: a numerical approach Encyclopedia of Mathematics and its Applications[END_REF]). The difficulty, when one wants to approximate such control in any finite dimensional space, that is when one likes to minimize numerically J ⋆ , is that the space H is huge, in particular, contains ∪ s∈N H -s (Ω), and even elements that may not be distributions. Numerical experiments do suggest that the minimizer φT is very singular (we refer to [START_REF] Fernández-Cara | Numerical null controllability of the 1D heat equation: primal algorithms[END_REF][START_REF] Fernández-Cara | Numerical null controllability of the 1D heat equation: Duality and Carleman weights[END_REF][START_REF] Münch | Numerical approximation of null controls for the heat equation: illposedness and remedies[END_REF] for a detailed analysis in the close case of the heat equation). This phenomenon is independent of the choice of J, but is related to the use of dual variables. Actually, the equality (1.3) can be viewed as an equality in a very small space (due to the strong regularization effect of the heat kernel). Accordingly, the associated multiplier ϕ T belong to a large dual space, much larger than L 2 (Ω), that is hard to represent (numerically) in any finite dimensional space.

An alternative way of looking at these problems and avoiding the introduction of dual variables has been proposed in [START_REF]A variational perspective on controllability[END_REF]. It is based on the following simple strategy. Instead of working all the time with solutions of the underlying state equation, and looking for one that may comply with the final desired state, one considers a suitable class of functions complying with required initial, boundary, final conditions and appropriate regularity, and seeks one of those that is a solution of the state equation. This is in practice accomplished by setting up an error functional defined for all feasible functions, and measuring how far those are from being a solution of the underlying state equation.

One advantage of this variational approach is that the way to get closer to a solution of the problem is by minimizing a functional that cannot get stuck on local minima because the only critical points of the error turn out to be global minimizers with zero error. Therefore a general strategy for approximation consists in using a descent algorithm for this error functional. This approach which has the flavor of a least-squares type method has been employed successfully in our null controllability context for the linear heat equation in [START_REF] Münch | Numerical null controllability of the heat equation through a least squares and variational approach[END_REF] and for an hyperbolic system in [START_REF] Münch | A variational approach to approximate controls for systems with essential spectrum: application to membranal arch[END_REF].

We apply this approach to the Stokes system. In Section 2, we describe the ingredients of the variational approach for the system (1.1) and reduce the search of one controlled trajectory for the Stokes system to the minimization of the quadratic functional E defined by (2.2) over the affine space A defined by (2.1). In Section 3, by a general-purpose lemma (Lemma 3.2), using the very specific structure of the functional E, we prove that we may construct minimizing sequences for the error functional E that do converge strongly to an extremal point for E (see Proposition 3.1). Section 4, we adapt the argument for the direct problem of the steady Navier-Stokes system.

Results of this work were partially announced in the note [START_REF] Münch | A least-squares formulation for the approximation of null controls for the Stokes system[END_REF].

A least-squares reformulation

Following [START_REF] Münch | Numerical null controllability of the heat equation through a least squares and variational approach[END_REF][START_REF]A variational perspective on controllability[END_REF], we define the non-empty space (2.1)

A = (y, π, f ); y ∈ L 2 (0, T, H 1 0 (Ω)), y t ∈ L 2 (0, T ; H -1 (Ω)), y(•, 0) = y 0 , y(•, T ) = 0, π ∈ L 2 (0, T ; U ), f ∈ L 2 (q T ) .
These hypotheses on y imply that it belongs to C([0, T ], L 2 (Ω)) and give a meaning to the equalities y(•, 0) = y 0 , y(•, T ) = 0 in L 2 (Ω). Note also that A is defined in agreement with the regularity of any solution (y, π) of the Stokes system with a source term f ∈ L 2 (Q T ). Then, we define the functional

E : A → R + by (2.2) E(y, π, f ) = 1 2 QT (|v t | 2 + |∇v| 2 + |∇ • y| 2 ) dx dt
where the corrector v is the unique solution in H 1 (Q T ) of the (elliptic) boundary value problem

(2.3) -v tt -∆v + (y t -ν∆y + ∇π -f 1 ω ) = 0, in Q T , v = 0 on Σ T , v t = 0 on Ω × {0, T }.
For any (y, π, f ) ∈ A, the term y t -ν∆y + ∇π -f 1 ω belongs to L 2 (0, T ; H -1 (Ω)) so that the functional E is well-defined in A. The approach developed here is based on the following result.

Proposition 2.1. -(y, π) is a controlled solution of the Stokes system (1.1) by the control function f 1 ω ∈ L 2 (q T ) if and only if (y, π, f ) is a solution of the extremal problem :

(2.4) inf

(y,π,f )∈A E(y, π, f ).
Proof-From the controllability of the Stokes system given by Theorem 1.1, the extremal problem (2.4) is well-posed in the sense that the infimum, equal to zero, is reached by any controlled solution of the Stokes system. Note that, without additional assumptions, the minimizer is not unique. Conversely, we check that any minimizer for E is a solution of the (controlled) Stokes system: let (Y, Π, F) ∈ A 0 be arbitrary where (2.5)

A 0 = (y, π, f ); y ∈ L 2 (0, T, H 1 0 (Ω)), y t ∈ L 2 (0, T ; H -1 (Ω)), y(•, 0) = y(•, T ) = 0, π ∈ L 2 (0, T ; U ), f ∈ L 2 (q T ) .
The first variation of E at the point (y, π, f ) in the admissible direction (Y, Π, F) defined by

(2.6) E ′ (y, π, f ), (Y, Π, F) = lim η→0 E((y, π, f ) + η(Y, Π, F)) -E(y, π, f ) η ,
exists, and is given by

(2.7) E ′ (y, π, f ), (Y, Π, F) = QT v t • V t + ∇v • ∇V + (∇ • y)(∇ • Y) dx dt where the corrector V ∈ H 1 (Q T ), associated with (Y, Π, F), is the unique solution of (2.8) -V tt -∆V + (Y t -ν∆Y + ∇Π -F 1 ω ) = 0 in Q T , V = 0 on Σ T , V t = 0 on Ω × {0, T }.
Multiplying the main equation of this system by v (recall that v is the corrector associated with the minimizer (y, π, f )), integrating by parts, and using the boundary conditions on v and V, we get

(2.9)

E ′ (y, π, f ), (Y, Π, F) = - QT (-Y • v t + ν∇Y • ∇v -Π ∇ • v -F • v 1 ω ) dx dt + QT (∇ • y)(∇ • Y) dx dt, ∀(Y, Π, F) ∈ A 0 ,
where we have used that

- T 0 Y t , v H -1 (Ω),H 1 (Ω) dt = QT Y • v t dx dt - Ω [Y • v] T 0 dx = QT Y • v t dx dt,
and that

T 0 ∇Π, v H -1 (Ω),H 1 (Ω) dt = - QT Π ∇ • v dx dt. Therefore if (y, π, f ) minimizes E, the equality E ′ (y, π, f ), (Y, Π, F) = 0 for all (Y, Π, F) ∈ A 0 implies that the corrector v = v(y, π, f ) solution of (2.3) satisfies the conditions (2.10) v t + ν∆v -∇(∇ • y) = 0, ∇ • v = 0, in Q T , v = 0, in q T .
But from the unique continuation property for the Stokes system (see [START_REF] Fabre | Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems[END_REF][START_REF] Fabre | Prolongement unique des solutions de l'équation de Stokes[END_REF]), it turns out that v = 0 in Q T and that ∇ • y is a constant in Q T . Eventually, the relation (2.9) is then reduced to

E ′ (y, π, f ), (Y, Π, F) = (∇ • y) QT ∇ • Y dx dt = 0, ∀(Y, Π, F) ∈ A 0
and then implies that this constant is zero. Consequently, if (y, π, f ) ∈ A is a minimizer for E, then ∇•y = 0 in Q T , and the corrector v is zero in Q T , so that E(y, π, f ) = 0. Therefore, (y, π, f ) solves (1.1), and since (y, π, f ) ∈ A, the state y is controlled at the time T by the function f which acts as a control distributed in ω.

Remark 2.2. -The proof of Proposition 2.1 only utilizes optimality of (y, π, f ) and not its minimality. Therefore in the statement of the proposition, we could have written instead : (y, π) is a controlled solution of the Stokes system (1.1) by the control function f 1 ω ∈ L 2 (q T ) if and only if (y, π, f ) is a stationary point for the functional E(y, π, f ) over (y, π, f ) ∈ A. This is relevant from the perspective of the numerical simulation for it guarantees that the numerical procedure based on a descent strategy cannot get stuck in local minima.

Remark 2.3. -For any (y, π, f ) ∈ A, the cost E can be formulate as follows

E(y, π, f ) = 1 2 y t -ν∆y + ∇π -f 1 ω 2 H -1 (QT ) + 1 2 ∇ • y 2 L 2 (QT )
. This justifies the least-squares terminology. The use of least-squares type approaches to solve linear and nonlinear problem is not new and we refer to [1, 13, 2] for many applications. The use of least-squares type approaches in the controllability context comes from [START_REF]A variational perspective on controllability[END_REF].

Remark 2.4. -The quasi-incompressibility case is obtained in the same way. It suffices to add ǫπ (for any ǫ > 0) to the divergence term in the functional E. This is also in practice a classical trick to fix the constant of the pressure π (see Section 4).

Remark 2.5. -The approach allows to consider compact support control jointly in time and space. It suffices to replace the function 1 ω in (1.1) by any compact support function in time and space, say 1 qT , where qT denotes a non-empty subset of Q T . Since Theorem 1.1 holds for any controllability time T and any subset ω of Ω, the controllability of (1.1) remains true as soon as qT contains any non-empty cylindrical domain of the form ω 1 × (t 1 , t 2 ) ⊂ Ω × (0, T ).

Remark 2.6. -A fortiori, the approach is well-adapted to address the direct problem (which consists, f being fixed, in solving the boundary value problem (1.1): it suffices to remove from A and A 0 the condition (1.3), and fix the forcing term f (see [START_REF] Bochev | Least-squares finite element methods[END_REF] using a similar least-squares type point of view). In that case, the second line of (2.10) is replaced by v(•, T ) = 0, which implies with the first line, that v and ∇ • y = 0 both vanish in Q T .

It is worth to notice that this approach allows to treat at the same time the null controllability constraint and the incompressibility one. In this sense, the pair (π, f ) can be regarded as a control function for the set of constraints

(2.11) y(•, T ) = 0 on Ω, ∇ • y = 0 in Q T .
These two conditions are compatibles: there is no competition between them. In the uncontrolled situation, from the uniqueness, the pressure π is unique as soon as the source term (here f 1 ω ) is fixed. On the other hand, in our controllability context, the pair (π, f ) is not unique: the pressure π depends on f 1 ω and vice versa. Therefore, the optimization with respect to both variables at the same time makes sense. From this point of view, we may reformulate the problem as a general controllability problem for the heat equation:

y t -ν∆y = V := f 1 ω -∇π in Q T ,
V being a control function such that (2.11) holds. The control function V acts on the whole domain, but on the other hand, should take the specific form V := f 1 ω -∇π.

Again, this perspective is different with the classical one, which consists in finding a control v ∈ L 2 (q T ), such that y(•, T ) = 0 in Ω where (y, π) solves (1.1). This can done by duality, penalization technique, etc. Conversely, one may also consider iteratively first the null controllability constraint, that is, for any π fixed in L 2 (0, T ; U ), find a control f π 1 ω such that (1.3) holds, and then find the pressure π such that ∇ • y π = 0 holds in Q T . Using again a least-squares type approach (for the heat equation, as developed in [START_REF] Münch | Numerical null controllability of the heat equation through a least squares and variational approach[END_REF]), the first step reduces to solve, for any π ∈ L 2 (0, T, U ) fixed, the problem inf

(yπ,fπ)∈A1 Ẽ(y, f ) := 1 2 v 2 H 1 (QT )
where v = v(y π , π, f π ) solves (2.3) and A 1 is given by

A 1 = (y, f ); y ∈ L 2 (0, T, H 1 0 (Ω)), y t ∈ L 2 (0, T ; H -1 (Ω)), y(•, 0) = y 0 , y(•, T ) = 0, f ∈ L 2 (q T ) .
The second step consists in updating the pressure according to a descent direction for the function

G : L 2 (0, T, U ) → R defined by G(π) := 1/2 ∇ • y π 2 L 2 (QT )
. We get that the first variation of G at π in the direction π ∈ L 2 (0, T ; U ) is given by < G ′ (π), π >= QT ∇π • p dx dt where p solves

-p t -ν∆p = ∇(∇ • y π ) in Q T , p(•, T ) = 0 in Ω, p = 0 on Σ T .
Again, this direct problem may be solved within the variational approach developed in this work (see Remark 2.6).

3. Convergence of some minimizing sequences for E Proposition 2.1 reduces the approximation of a null control for (1.1) to a minimization of the functional E over A. As a preliminary step, since A is not a vectorial space, we remark that any element of A can be written as the sum of one element of A, say s A , plus any element of the vectorial space A 0 . Thus, we consider for any s A := (y A , π A , f A ) ∈ A the following problem: ) can be decomposed as the sum s A +s A0 , for some s A0 in A 0 . We endow the vectorial space A 0 with its natural norm • A0 such that :

(3.2) y, π, f 2 A0 := QT (|y| 2 + |∇y| 2 ) dx dt + T 0 y t (•, t) 2 H -1 (Ω) dt + QT (|f | 2 + |π| 2 ) dx dt,
recalling that y t H -1 (Ω) = g H 1 0 (Ω) where g ∈ H 1 0 (Ω) solves -∆g = y t in Ω. We denote , A0 the corresponding scalar product. (A 0 , • A0 ) is an Hilbert space.

The relation (2.9) allows to define a minimizing sequence in A 0 for E sA . It turns out that minimizing sequences for E sA which belong to a precise subset of A 0 remain bounded uniformly. This very valuable property is not a priori guaranteed from the definition of E sA . The boundedness of E sA implies only the boundedness of the corrector v for the H 1 (Q T )norm and the boundedness of the divergence ∇ • y of the velocity field for the L 2 (Q T )-norm. Actually, this property is due to the fact that the functional E sA is invariant in the subset of A 0 which satisfies the state equations of (1.1).

In order to construct a minimizing sequence bounded in A 0 for E sA , we introduce the linear continuous operator T which maps a triplet (y, π, f ) ⊂ A into the corresponding vector (v, ∇ • y) ∈ H 1 (Q T ) × L 2 (Q T ), with the corrector v as defined by (2.3). Then we define the space A = Ker T ∩ A 0 composed of the elements (y, π, f ) satisfying the Stokes system and such that y vanishes on the boundary ∂Q T . Note that A is not the trivial space : it suffices to consider the difference of two distinct null controlled solutions of (1.1). Finally, we note A ⊥ = (Ker T ∩ A 0 ) ⊥ the orthogonal complement of A in A 0 and P A ⊥ : A 0 → A ⊥ the (orthogonal) projection on A ⊥ .

We then define the following minimizing sequence (y k , π k , f k ) k≥0 ∈ A ⊥ as follows:

(3.3)    (y 0 , π 0 , f 0 ) given in A ⊥ , (y k+1 , π k+1 , f k+1 ) = (y k , π k , f k ) -η k P A ⊥ (y k , π k , f k ), k ≥ 0
where (y k , π k , f k ) ∈ A 0 is defined as the unique solution of the formulation

(3.4) (y k , π k , f k ), (Y, Π, F) A0 = E ′ s0 (y k , π k , f k ), (Y, Π, F) , ∀(Y, Π, F) ∈ A 0 .
η k denotes a positive descent step. In particular, (3.4)

implies that π k = -∇ • v k ∈ L 2 (Q T ) and f k = -v k 1 ω ∈ L 2 (q T ) (actually in H 1 (q T )).
One main issue of our variational approach is to establish the convergence of the minimizing sequence defined by (3.3). We have the following result.

Proposition 3.1. -For any s A ∈ A and any {y 0 , π 0 , f 0 } ∈ A ⊥ , the sequence s A + {(y k , π k , f k )} k≥0 ∈ A converges strongly to a solution of the extremal problem (2.4).

This proposition is the consequence of the following abstract result which can be adapted to many different situations where this variational perspective can be of help. Lemma 3.2. -Suppose T : X → Y is a linear, continuous operator between Hilbert spaces, and H ⊂ X, a closed subspace, u 0 ∈ X. Put

E : u 0 + H → R + , E(u) = 1 2 Tu 2 , A = Ker T ∩ H.
1. E : u 0 +A ⊥ → R is quadratic, non-negative, and strictly convex, where A ⊥ is the orthogonal complement of A in H. 2. If we regard E as a functional defined on H, E(u 0 + •), and identify H with its dual, then the derivative E ′ (u 0 + •) always belongs to A ⊥ . In particular, a typical steepest descent procedure for E(u 0 + •) will always stay in the manifold u 0 + A ⊥ . 3. If, in addition, min u∈H E(u 0 + u) = 0, then the steepest descent scheme will always produce sequences converging (strongly in X) to a unique (in u 0 + A ⊥ ) minimizer u 0 + u with zero error.

Proof of Lemma 3.2-Suppose there are u i ∈ A ⊥ , i = 1, 2, such that

E u 0 + 1 2 u 1 + 1 2 u 2 = 1 2 E(u 0 + u 1 ) + 1 2 E(u 0 + u 2 ).
Due to the strict convexity of the norm in a Hilbert space, we deduce that this equality can only occur if Tu 1 = Tu 2 . So therefore u 1 -u 2 ∈ A ∩ A ⊥ = {0}, and u 1 = u 2 . For the second part, note that for arbitrary U ∈ A, TU = 0, and so

E(u 0 + u + U ) = 1 2 Tu 0 + Tu + TU 2 = 1 2 Tu 0 + Tu 2 = E(u 0 + u).
Therefore the derivative E ′ (u 0 + u), the steepest descent direction for E at u 0 + u, has to be orthogonal to all such U ∈ A.

Finally, assume E(u 0 + u) = 0. It is clear that this minimizer is unique in u 0 + A ⊥ (recall the strict convexity in (i)). This, in particular, implies that for arbitrary u ∈ A ⊥ , (3.5)

E ′ (u 0 + u), u -u ≤ 0,
because this inner product is the derivative of the section t → E(u 0 + tu + (1 -t)u) at t = 0, and this section must be a positive parabola with the minimum point at t = 1. If we consider the gradient flow

u ′ (t) = -E ′ (u 0 + u(t)), t ∈ [0, +∞),
then, because of (3.5),

d dt 1 2 u(t) -u 2 = u(t) -u, u ′ (t) = u(t) -u, -E ′ (u 0 + u(t)) ≤ 0.
This implies that sequences produced through a steepest descent method will be minimizing for E, uniformly bounded in X (because u(t)u is a non-increasing function of t), and due to the strict convexity of E restricted to u 0 + A ⊥ , they will have to converge towards the unique minimizer u 0 + u.

Remark 3.3. -Despite the strong convergence in this statement, it may not be true that the error is coercive, even restricted to u 0 + A ⊥ , so that strong convergence could be very slow.

Because of this same reason, it may be impossible to establish rates of convergence for these minimizing sequences.

The element u 0 determines the non-homogeneous data set of each problem: source term, boundary conditions, initial and/or final condition, etc. The subspace H is the subset of the ambient Hilbert space X for which the data set vanishes. T is the operator defining the corrector, so that Ker T is the subspace of all solutions of the underlying equation or system. The subspace A is the subspace of all solutions of the problem with vanishing data set. In some situations A will be trivial, but in some others will not be so. The important property is (iii) in the statement guaranteeing that we indeed have strong convergence in X of iterates. The main requirement for this to hold is to know, a priori, that the error attains its minimum value zero somewhere, which in the situation treated here is guaranteed by Theorem 1.1.

Proof of Proposition 3.1-The result is obtained by applying Lemma 3.2 as follows. If we put B = {y ∈ L 2 (0, T, H 1 0 (Ω)) :

y t ∈ L 2 (0, T ; H -1 (Ω))}, X is taken to be B × L 2 (0, T ; U ) × L 2 (q T ).
H is taken to be A 0 as given in (2.5) and u 0 = s A ∈ A ⊂ X. The operator T maps a triplet (y, π, f

) ∈ A ⊂ X into (v, ∇ • y) ∈ Y := H 1 (Q T ) × L 2 (Q T ) as explained earlier. Remark 3.4.
-The construction of the minimizing sequence only requires the resolution of standard well-posed elliptic problems over Q T , well-adapted to general situations (time dependent support, mesh adaptation, etc). On the other hand, it is important to highlight that the L 2 (q T ) control function f obtained from the minimizing procedure does not a priori minimize any specific norm (for instance the L 2 -norm).

Without the projection on (Ker T ∩ A 0 ) ⊥ in (3.3), the sequence (y k , π k , f k ) remains a minimizing sequence for E sA : actually, the values of the cost E sA along the sequence (y k , π k , f k ) are equal with or without the projection. This is due to the fact that the component of the descent direction (y k , π k , f k ) on (Ker T ∩ A 0 ) does not affect the value of the cost : on the other hand, without the projection, the minimizing sequence may not be bounded uniformly in A 0 , in particular the control function f may not be bounded in L 2 (q T ).

The subset A ⊥ is not explicit, so that in practice the projection P A ⊥ (y k , π k , f k ) may be defined by

P A ⊥ (y k , π k , f k ) = (y k , π k , f k ) -p,
where p solves the extremal problem :

(3.6) min p∈A p -(y k , π k , f k ) A0 .
Recalling that A is by definition the set of triplets (y, π, f ) satisfying y t -ν∆y + ∇π -f 1 ω = 0, ∇ • y = 0 in Q T such that y vanishes on ∂Q T , this extremal problem is nothing else than a controllability problem for the Stokes system, similar to the one considered in this work. Therefore, we shall bypass this projection and shall introduce instead a stopping criteria for the descent method measuring how far from A ⊥ the descent direction is. We refer to [START_REF] Münch | A least-squares formulation for the approximation of controls for the Stokes system[END_REF] for numerical experiments.

Direct problem for the steady Navier-Stokes system

The least-squares approach allows to address non-linear problem. We consider here simply the steady Navier-Stokes and address the direct problem: find (y, π) solution of (4.1)

-ν∆y + (y • ∇)y + ∇π = f , ∇ • y = 0 in Ω y = 0 on ∂Ω.

We recall the following result (see [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF]) :

Theorem 4.1. -For any f ∈ H -1 (Ω), there exists at least one (y, π) ∈ H 1 0 (Ω)×L 2 0 (Ω) solution of (4.1). Moreover, if ν -2 f H -1 (Ω) is small enough, then the couple (y, π) is unique.

In order to solve this boundary value problem, we use a least-squares type approach. We consider the space A = H 1 0 (Ω) × L 2 (Ω) and then we define the functional

E : A → R + by (4.2) E(y, π) := 1 2 Ω (|∇v| 2 + |∇ • y| 2 ) dx
where the corrector v is the unique solution in H 1 0 (Q T ) of the (elliptic) boundary value problem

(4.3) -∆v + (-ν∆y + div(y ⊗ y) + ∇π -f ) = 0, in Ω, v = 0 on ∂Ω.
We then consider the following extremal problem We recall the following equality From Theorem 4.1, the infimum is equal to zero and is reached by an element solution of (4.1). Conversely, we would like to state that the only critical point for E correspond to solution of (4.1).

In view of [27, Corollary 1.8], let us first prove the following result.

Proposition 4.3. -E is an error functional, that is E : A → R is a C 1 -functional over the Hilbert space A, E is non-negative and lim E(y, π) = 0 as E ′ (y, π) → 0. Proof. -We write that (4.6) E ′ (y, π) • (Y, Π) = Ω ∇v • ∇V + (∇ • y)(∇ • Y)dx where V ∈ H 1 0 (Ω) solves (4.7) -∆V + (-ν∆Y + div(y ⊗ Y) + div(Y ⊗ y) + ∇Π) = 0, in Ω, V = 0 on ∂Ω.
Multiplying the state equation by v and using (4.2), we get (4.8)

E ′ (y, π) • (Y, Π) = Ω ν∆v • Y + (∇v + (∇v) T )y) • Y + (∇ • v)Πdx + Ω (∇ • y)(∇ • Y)dx
or equivalently (4.9)

E ′ (y, π) • (Y, Π) = Ω -ν∇v • ∇Y + (y ⊗ Y + Y ⊗ y) : ∇v + (∇ • v)Πdx + Ω (∇ • y)(∇ • Y)dx.
We then check that we can take Y = v, i.e. v = v(y, π) ∈ H 1 0 (Ω) uniquely given by the solution of (4.7) remains bounded with respect to (y, π) ∈ H 1 0 × L 2 (Ω). By definition, the corrector v solves the variational formulation:

(4.10) Ω ((∇v + ν∇y -y ⊗ y) : ∇w -π∇ • w -f • w) = 0, ∀w ∈ H 1 0 (Ω)
Taking w = v, we get that (4.11)

Ω |∇v| 2 dx = Ω (y ⊗ y : ∇v -ν∇y : ∇v + π∇ • v + f • v)dx so that, in view of the Poincaré inequality, (4.12) v H 1 0 (Ω) ≤ C( y ⊗ y L 2 (Ω) + y H 1 0 (Ω) + π L 2 (Ω) + f H -1 (Ω)
). leading to the result. Then, taking Y = v in (4.9), we get (4.13)

E ′ (y, π) • (v, Π) = Ω -ν|∇v| 2 -(y ⊗ v + v ⊗ y) : ∇v + (∇ • v)Πdx + Ω (∇ • y)(∇ • v)dx = Ω -ν|∇v| 2 -(v ⊗ v) : ∇y + 1 2 (∇ • y)|v| 2 dx + Ω (∇ • v)(∇ • y + y • v + Π)dx
Similarly, in view of (4.12), Π s = -(∇ • y + y • v) ∈ L 2 (Ω) remains bounded with respect to (y, π) and we write

(4.14) E ′ (y, π) • (v, Π s ) = Ω -ν|∇v| 2 -(v ⊗ v) : ∇y + 1 2 (∇ • y)|v| 2 dx
We then use the following result (consequence of the well-posedness of the Oseen equation)

Lemma 4.4. -For any y ∈ H 1 0 (Ω), F ∈ L 2 (Ω), there exists (Y, Π) ∈ H 1 0 (Ω) × L 2 (Ω) with ∇ • Y = 0 such that (4.15) Ω (ν∇Y -(Y ⊗ y + y ⊗ Y)) : ∇w -Π∇ • w -F • w = 0, ∀w ∈ H 1 0 (Ω) such that Y, Π H 1 0 (Ω)×L 2 (Ω) ≤ C( y H 1 0 (Ω) + F L 2 (Ω)
) for some C > 0. Using this lemma for F = v and w = v (v is the corrector associated to the pair (y, π)), we obtain that (Y, Π) ∈ H 1 0 (Ω) × L 2 (Ω) satisfies ∇ • Y = 0 and (4. 16)

Ω (ν∇Y -(Y ⊗ y + y ⊗ Y)) : ∇v -Π∇ • v -v • v = 0, ∀w ∈ H 1 0 (Ω)
With this pair (Y, Π) bounded with respect to v and to y, and so with respect to (y, π), we have from (4.9), (remind that ∇ • Y = 0) (4.17 The property E ′ (y, π) • (Y, Π) → 0 then implies that v L 2 (Ω) → 0. Then, from (4.14), the property E ′ (y, π) • (v, Π s ) → 0 then implies from the equality (4.16) that ∇v L 2 (Ω) → 0. Then, 4.9 implies that Ω ∇ • y∇ • Ydx → 0 for all Y ∈ H 1 0 (Ω) so that ∇ • y L 2 (Ω) → 0.

We then recall the following result.

Lemma 4.5. -[27, Corollary 1.8] Let H be a Hilbert space. Suppose J : H → R is an error functional for which there is a unique u 0 ∈ H with J(u 0 ) = 0. If J(u j ) → 0, then u j → u 0 strongly in H, and J complies with the Palais-Smale condition.

In view of Proposition 4.3 and of the previous Lemma, we have the following result. As a very interesting consequence, Proposition 4.6 reduces the approximation of the vanishing elements of E to the construction of one arbitrary minimizing sequence, say (y k , π k ) k>0 , for E.

The unsteady case as well as the controllability case are more involved: we refer to [START_REF] Münch | A least-squares formulation for the approximation of null controls for the Navier-Stokes system[END_REF] based on [START_REF] Fernández-Cara | Numerical null controllability of semi-linear 1D heat equations : fixed points, least squares and Newton methods[END_REF][START_REF] Yu | Remarks on exact controllability for the Navier-Stokes equations[END_REF][START_REF]A variational approach for the Navier-Stokes system[END_REF].

E

  sA (y, π, f ), E sA (y, π, f ) := E(s A + (y, π, f )). Problems (2.4) and (3.1) are equivalent. Any solution of Problem (3.1) is a solution of the initial problem (2.4). Conversely, any solution of Problem (2.4

Lemma 4 .

 4 2. --For all y, z, div(y ⊗ z) = y∇ • z + (∇y)z. -For all y, z, p ∈ H 1 0 (Ω), (4.5) Ω div(y ⊗ z) + div(z ⊗ y) • p = -Ω (y ⊗ z + z ⊗ y) : ∇p = -Ω (∇p + (∇p) T )y • z

  ) E ′ (y, π) • (Y, Π) = Ω -ν∇v • ∇Y + (y ⊗ Y + Y ⊗ y) : ∇v + (∇ • v)Πdx

Proposition 4 . 6 .

 46 -If ν -1 f H -1 (Ω) is small enough, then any minimizing sequence (y k , π k ) k>0 for E converges strongly toward (y, π), unique solution of the boundary value problem (4.1).Proof. -It suffices to apply the previous lemma for J = E and H = A. The uniqueness here follows from Theorem 4.1.Remark 4.7. -We write again that(4.18) E ′ (y, π) • (Y, Π) = Ω ∇v • ∇V + (∇ • y)(∇ • Y)dxwhere V ∈ H 1 0 (Ω) solves the well-posed formulation(4.19) -∆V + (-ν∆Y + div(y ⊗ Y) + div(Y ⊗ y) + ∇Π) = 0, in Ω, V = 0 on ∂Ω.Multiplying the state equation by v and using (4.2), we get (4.20)E ′ (y, π) • (Y, Π) = Ω ν∆v • Y + (∇v + (∇v) T )y) • Y + (∇ • v)Πdx + Ω (∇ • y)(∇ • Y)dxso that, at the optimality, the corrector funcion v solves the following linear boundary problem :(4.21) ν∆v + (∇v + (∇v) T )y -∇(∇ • y) = 0, ∇ • v = 0 in Ω v = 0 on ∂Ω Proposition 4.3 implies that if E ′ (y, π) = 0, then E(y, π) = 0. Therefore, the formulation (4.21) implies that v = 0 and ∇ • y = 0. This may be seen as a unique continuation property for Navier-Stokes equation. Remark 4.8. -The previous results hold true if we replace the cost E by

2 +

 2 |∇ • y + επ| 2 ) dxfor any ε > 0, where v solves (4.3).

Research supported in part by MTM2007-62945 of the MCyT (Spain), and PCI08-0084-0424 of the JCCM (Castilla-La Mancha).


Bruno Mery
email: bruno.mery@me.com

The Relative Complexity of Constraints in Co-Predicative Utterances

We explore the particular variations of semantic felicity in co-predicative utterances the constraints on the combination of facets of polysemous words, and the possible adaptations that can be proposed to existing formal frameworks that support lexical semantics. As the linguistic data is incomplete and disputed, we also include a proposal for a linguistic survey aimed at clearing up many outstanding issues. 1

1 The Generative Approach to Lexical Semantics and Polysemy

The Issue of Polysemy in Compositional Semantics

The inarguable facts that words in human languages can be employed with many different meanings, according to the context they are used in, is seemingly at odds with the principle of compositional semantics inherited from Montague, which presupposes that singular meaningful term can be provided for every lexical item. This can be resolved by Word Sense Disambiguation techniques (associating the most probable meaning to each word before composition, using data refined from corpora). Another approach is to hold polysemy to be an important feature of language, and to re-design the compositional semantics around it. In this approach, the lexicon becomes much more complex than a map of words to meanings, as each lexical entry can be combined with others in order to select the appropriate sense for a word used in a specific context: this is the so-called Generative approach to lexical semantics developped in [START_REF] Pustejovsky | The Generative Lexicon[END_REF] and many subsequent works. We will discuss a few basic features of this approach in Section 1.2.

In the present paper, we will study the specific issue of the felicity of co-predicative utterances. Those constructs are detailed in Section 2.1, the issues of felicity in Section 2.3, and we will focus on the characterisation of constraints on felicity in Section 3, and the necessary adaptations of formal semantics in Section 4.

1.2 GL Qualia, Dots, Processes and Uses [START_REF] Pustejovsky | The Generative Lexicon[END_REF] and subsequent works such as [START_REF] Asher | Lexical Meaning in Context: a Web of Words[END_REF] have detailed the implications on Montagovian semantics of relational polysemy, distinguishing (among many other phenomena) four kinds of meanings that are distinct, but directly related (by opposition to accidental polysemy, which consists of different words that happen to be homonyms).

The qualia are derived from the "modes of explanation" of Aristotle; the idea is that a noun for something can refer to :

1. The thing itself (formal quale, as in long sword), 2. A salient part of the thing (constitutive quale, as in dull sword → dull edge), and, for artificial items, 3. Their creation process (agentive quale, as in master's sword → sword made by a master blacksmith), 4. Their use, for objects that have one (telic quale, as in fine sword → sword fine for fighting).

Pustejovsky argues that such uses are extremely common, and that a competent speaker of a language has access to a qualia, thus justifying that such information should be part of the lexicon itself.

Complex objects, called dot-objects (sometimes written •-objects) are words that denote a singular concept that can be envisioned on two or more different facets. Canonical examples include books (with physical and informational facets), meals (with an event and a food facet), newspapers (with an organisational and a physical aspect), etc. There are many examples which seem to stem from compounding essential and existential information. As the number and type of facets is not determined, contrary to qualia, and as those facets enjoy differentiated individuation conditions, they have proved to be one of the most serious difficulties for formal models of lexical semantics.

In addition, some implicit processes can shift the meaning of a word referring to something to different states: grinding can turn materials into artefacts, animals or plants into food (delicious salmon), packaging can provide implicit containers for masses (there is water and wine on the table) -see [START_REF] Mery | Computing the Semantics of Plurals and Massive Entities using Many-Sorted Types[END_REF] for details. Deverbal nouns are notoriously polysemous between process and result (the construction is at the end of the street) -see [START_REF] Jacquey | Ambiguïtés lexicales et traitement automatique des langues : modélisation de la polysémie logique et application aux déverbaux d'action ambigus en franc ¸ais[END_REF], [START_REF] Real-Coelho | A generative montagovian lexicon for polysemous deverbal nouns[END_REF].

Finally, additional facets can be bestowed upon a lexical item via explicit constructions, which are easier to account for. They include as-phrases (I do not have issues with Mr. X as a lawyer; however, as a candidate. . . ) and for-phrases (this knife is sharp enough for shaving).

The Many Facets of Co-Predicative Sentences

Co-Predications

Co-predicative utterances, phrases or sentences, or co-predications for short, consist in the explicit reference to two (or more) different facets of the same lexical items at the same time. The most classical example is heavy and interesting book, but there are many others. The difficulty of co-predications is that the two predicates apply to different types (here, to physical and informational objects respectively). Lexical semantics frameworks that do not consider this and simply coerce the type of the argument to the one required by the predicate fail to provide a suitable typing for book in such sentences.

Mixing Polysemy Facets

In the original formulation by Pustejovsky, co-predications were mostly studied between facets of dot-objects. However, they can be equally valid between various sources:

two or more qualia:

• good, expensive wine (telic+formal);

• fast blue car (constitutive+formal). . . ; two or more facets of a dot-object:

• red closed door (physical+aperture);

• liberal, picture-less tabloid (organisation+physical). . . ; a facet of a dot-object and a qualia of another facet:

• I have an inspired article in my briefcase (agentive quale of the informational facet + physical facet); process, result, and other facets:

• the translation, printed right after its lengthy completion, is considered bold yet naive (physical and agentive quale of the informative facet of the result + process); an arbitrary number of references to different facets in a coherent discourse:

• The book is 5lbs, and has 400 pages. It has been set in Times 12, with large margins. Its leather cover is aged but sound. Its writing took five years, and the completion of the hundred in-quarto printings four months. [. . . ] It certainly is an interesting read.

Felicitous and Infelicitous Co-Predications

There are thus many possible combinations of facets that can be predicated on simultaneously. However, some co-predications are infelicitous. We have elaborated previously on the case of grinding : * The salmon was lightning-fast and delicious, many examples of process/result alternations: * The construction took three months and stands tall, as well as capital-governments alternations: * Washington is an old American city and has denounced Teheran. 2 3 Characterising Constraints on Felicity

Relaxable and Fixed Constraints

The most evident constraints are the following: facets that can cognitively co-exist for the same object are not constrained in their co-predications, while facets that are only present after a transformative process (grinding, packaging, resultatives) are exclusive to all others.

In addition, a proper name of a polysemous entity (such as a city or place) can be used as a paraphrase for a specific group of people (such as the name of the capital being a proxy for the national government), to the exclusion of all other possible senses. We can thus distinguish between two rough classes of facets: flexible and rigid.

However, the rigidity constraint can be relaxed in some cases. A syntactic split of the components of the co-predications can render some infelicitous phrases acceptable : the salmon, which was very fast, is delicious, or the construction, that took three months to complete, stands tall. Beyond the syntactical separation, the use of different tenses or explicit contrast using yet or but can also relax the constraints on the use of facets that correspond to different, exclusive states of the same entity.

Capital/government (and related) alternation constraints are much more difficult to relax, considering: Washington is an old American city. It is located on the banks of the Potomac. * It has denounced Teheran..

That distinction has given us the three classes of constraints that we have been using until now, flexible facets, semi-rigid facets (the constraints can be relaxed via syntactic means) and rigid facets (the constraints are fixed).

City Names: a Short Case Study

However, this does not cover the whole complexity of constraints on the felicity of co-predications. See [START_REF] Real-Coelho | A generative montagovian lexicon for polysemous deverbal nouns[END_REF] for a case study on deverbal co-predications; we will discuss here the specific case of city names.

City names are highly polysemous, with many possible facets that can be separated in two groups. The first is composed of facets that pertain to the city as a whole: architecture, location, size, atmosphere, climate, population, lifestyle. . . that we will call characteristics, and the second, of facets we will call essentials, that use the name of the city to refer to a specific group of people, such as city/metropolis/regional/national government and sport clubs. In our first approximation, characteristics are flexible and essentials are rigid.

However, the following constructions can allow some of the essentials to copredicate:

lexical proximity -if two facets share the same lexical field, as in Barcelona dominates Europe in football as well as in handball; object identity -if the predication is made on a single direct object, as in Barcelona dominates Europe in football as in architecture; zeugma -if zeugma are considered acceptable, as in Barcelona dominates Europe in football as they do Madrid in politics;

discourse flow -logical relations in the narration can provide felicitous readings, as in Detroit, by lake Michigan, has filed for bankruptcy; the city has become desolate and lifeless. -The above can be daisy-chained in order to co-predicate over seemingly incompatible facets in a long discourse: Bordeaux, struggling with traffic issues due to its growing population and its geographical position on both sides of the Garonne, wishes to build a new bridge in order to ease commuting between those.

Evaluating the Idiosyncrasies of Felicity in Co-Predications

We want to stress that felicity is an highly subjective notion, that varies from speaker to speaker. Additionally, it appears that some of the constraints are idiosyncratic; anecdotical evidence, including separate personal communications from Asher, Lecomte and Luo indicate that some common co-predications are felicitous in English but not in Chinese. While this should not stop formal frameworks to develop mechanisms that integrate constraints on co-predications, establishing a robust catalogue of the actual usages is important.

We would like to propose a set of templates for sentences that can be adapted to valid local city data, with co-predications we consider felicitous, infelicitous, and forced by the various mechanisms evoked above, as well as straightforward predications on a single facet as controls. This survey should be conducted by native speakers of several languages in different linguistic groups (minimally Romance, Germanic and East-Asiatic), and we would like it to be part of collaborative international projects that are being constructed as a result of the recent interest in lexical semantics.

A Proposal for Linear Types and Terms

Constraints on co-predications can be added to current lexical frameworks as an external mechanism3 . However, it is also possible to allow for every possible combination with a redesign of the formalism. Our proposal incorporates a λ -calculus whose terms are typed with formulae of the second-order linear intuitionistic logic.

Linear Λ TY n is adapted from our framework in the following way:

-Simple predications are implemented by linear application (heavy is H ϕ t ).

-Basic morphisms (type transformations) also (selecting the physical facet is done via Λ α. f α ϕ ). -Lexical transformations (giving access to specific facets of each term) are implemented as pairs, the first component being an accessor to the facet (a type transformation, as above) and the second assessing the compatibility of the facet with other transformations.

The present abstract is intended as a discussion material, detailing a specific issue in order to foster collaborations around the subject. Having been prepared by the author in his spare time, it is, by necessity, incomplete. The work presented here is closely related to Λ TY n , a formal framework for lexical semantics developed since

with Christian Bassac, Richard Moot, Christian Retoré and many other researchers at LaBRI and LIRMM. As a team, we would like to thank everyone that has joined in TYTLES and engaged in discussions and research around type-theoretical lexical semantics.

Those examples are subject to personal interpretation and linguistic idiosyncrasies that we will discuss in Section

3.3.

It is sufficient to keep track of the flexible/rigid characteristic of terms, relaxing the constraints when appropriate using a set of rules that detect syntactic or discursive features, and to stop the composition when incompatible co-predications are detected; this is the approach we have proposed so far, see[START_REF] Retoré | The Montagovian Generative Lexicon Lambda Ty n : a Type Theoretical Framework for Natural Language Semantics[END_REF] for a recent synthesis in our Λ TY n framework.

-Additional terms, that modify this compatibility, can be provided by syntax, discourse and pragmatics in order to relax the constraints when appropriate.

A detailed account of this proposal is given in [START_REF] Mery | Lexical Semantics with Linear Types[END_REF].

Summary

While the work presented here is still in progress, we are convinced that the issue of constraints on co-predication is of importance in the establishment of precise natural language semantics. We hope to gather enough linguistic data in order to characterise the extent of the phenomena, and have the necessary formal tools to treat it accurately. Our goal remains to integrate issues of lexical semantics and polysemy directly in the flow of analysis of natural language, from syntax to semantics and logical representation.