Abderrahmane Habbal
email: habbal@unice.fr

Moez Kallel
email: moez.kallel@enit.utm.tn

Marwa Ouni
email: marwa1ouni@gmail.com

Nash strategies for the inverse inclusion Cauchy-Stokes problem

Keywords: Data completion, Cauchy-Stokes problem, shape identification, Nash games Mathematics Subject Classifications (2010): 49J20, 65K10, 65N06, 90C30

may come

Nash strategies for the inverse inclusion Cauchy-Stokes problem

Introduction

Fluid dynamics are central in many industrial, biological and biomedical processes. The good functioning of the involved systems could be dramatically damaged in the presence of undesired small obstacles (impurities) or inclusions (cavitation). For example, polymer material degradation is related to the formation of inclusions during polymer extrusion [START_REF] Son | Cavitation of polyethylene during extrusion processing instabilities[END_REF] ; as well, the mechanism of joint cracking is related to cavity formation [START_REF] Gregory N Kawchuk | Real-time visualization of joint cavitation[END_REF].

A large spectrum of the processes above can be considered as Stokes flows, though they should be taken unsteady and anisotropic to render satisfactorily the complex phenomenon of the formation of cavities [START_REF] Stieger | Hydrodynamic cavitation in stokes flow of anisotropic fluids[END_REF]. The shape and location of the inclusions is generally out of reach for direct observation, hence the need for effective nondestructive monitoring solutions, known as geometric inverse problems when mathematics and algorithms are involved. Popular mathematical models build on the assumption that some specific measurements are available over the whole boundary of the structure under investigation, dealing with partial differential equations of boundary value -BVP-type. However, it should be noticed that from a technological point of view, when industrial devices are involved, the assumption above is in general impossible to fulfill, either because it is too expensive, or simply because part of the boundary is not accessible to probing, think of a heart valve [START_REF] Lo | Cavitation phenomena in mechanical heart valves: Studied by using a physical impinging rod system[END_REF]. Such restrictions lead to develop complex protocoles like for the detection of flaws in metal melts in foundry industry [START_REF]On the Detection, Behaviour and Control of Inclusions in Liquid Metals[END_REF]. Industrial solutions use in general protocoles where emission and reception of the probing signals are set on the same location of the boundary. From a mathematical point of view, we have access to over specified boundary data (e.g. temperature and thermal flux) on the probing location, and no data elsewhere. Thus, we deal with partial differential equations, having access to over specified boundary data, and missing data to recover as well as unknown inclusions to detect. We are then in the framework of geometric inverse problems for the so called Cauchy-Stokes system. We shall restrict ourselves to the case of steady and Newtonian Stokes flows.

                     ν∆u -∇p = 0 in Ω \ ω * , divu = 0 in Ω \ ω * , σ(u, p)n = 0 on ∂ω * , u = f on Γ c , σ(u, p)n = Φ on Γ c , (1)
where n is the unit outward normal vector on the boundary, and σ(u, p) the fluid stress tensor defined as follows:

σ(u, p) = -pI d + 2νD(u)

with D(u) = 1/2(∇u + ∇u T ) being the linear strain tensor and I d the identity matrix.

For the sake of simplicity, from now on, the viscosity ν of the fluid is set to ν = 1.

Additionally to the geometric identification problem (i.e. detect the inclusions ω * ) one has to complete the boundary data, that is to recover the missing traces of the velocity u and of the normal stress σ(u, p).n over Γ i the inaccessible part of the boundary. Remark that the difference between obstacles and inclusions amounts to which boundary condition is used : homogeneous Dirichlet one for the obstacles and homogeneous Neumann condition for the inclusions (considered as free surfaces).

Even when restricted to elliptic equations, mostly Laplace and Stokes systems, there exists a prolific literature dedicated to each of these two problems separately, and because of their well known ill-posedness (in the sense of Hadamard) [START_REF] Hadamard | The Cauchy Problem and the Linear Hyperbolic Partial Differential Equations[END_REF], most of the literature addresses as well (if not exclusively) the ensuing stability and other computational issues. For the Cauchy problem, far from being exhaustive, an excerpt of popular approaches are the least-square penalty techniques used in [START_REF] Falk | Logarithmic convexity for discrete harmonic functions and the approximation of the cauchy problem for poisson's equation[END_REF] and in the earlier paper [START_REF] Colli | On the inverse potential problem of electrocardiology[END_REF], Tikhonov regularization methods [START_REF] Cimetiere | Solution of the cauchy problem using iterated tikhonov regularization[END_REF], quasi reversibility methods [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the cauchy problem for laplace's equation[END_REF], alternating iterative methods [11] [12] and control type methods andrieux2006solving [START_REF] Ben Abda | A control type method for solving the cauchy-stokes problem[END_REF]. Recently, an approach based on game theory, using decentralized strategies, was proposed in [START_REF] Habbal | Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems[END_REF].

Let us mention that many of the papers dedicated to data completion or to obstacle detection and based on control or optimization approaches, minimize a so-called Kohn-Vogelius type functional, an energy error function introduced in the framework of parameter identification in [START_REF] Robert | Relaxation of a variational method for impedance computed tomography[END_REF].

Regarding the obstacle identification problem, a more challenging geometric inverse problem, and again with a very partial on the existing literature, the authors in [START_REF] Alvarez | Identification of immersed obstacles via boundary measurements[END_REF] address the obstacle detection problem for unsteady Stokes and Navier-Stokes flows, quasi reversibility coupled to a level set approach is used in [START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF] for the Laplace equation, shape optimization [START_REF] Caubet | Detecting an obstacle immersed in a fluid by shape optimization methods[END_REF] and topological gradient [START_REF] Caubet | On the detection of several obstacles in 2d stokes flow : topological sensitivity and combination with shape derivatives[END_REF] are used for the Stokes system, and in [START_REF] Ballerini | Stable determination of an immersed body in a stationary stokes fluid[END_REF] stability issues are addressed for the inverse obstacle problem in a Stokes flow.

In contrast, rather a few papers address the joint geometric and data completion inverse problems, at least regarding its computational aspects. Close to our present work, the inverse obstacle problem for the Cauchy-Laplace equation is studied in [START_REF] Caubet | On the data completion problem and the inverse obstacle problem with partial cauchy data for laplace's equation[END_REF] where a control-type approach is used and applied to a Kohn-Vogelius functional. In [START_REF] Bourgeois | The exterior approach to solve the inverse obstacle problem for the stokes system[END_REF] the authors use quasi reversibility coupled to a level set approach to solve the inverse obstacle problem for the Cauchy-Stokes equations. A formulation based on nonlinear integral equations arising from the reciprocity gap [START_REF] Andrieux | The reciprocity gap: a general concept for flaws identification problems[END_REF] principle is used in [START_REF] Cjs Alves | Integral equations for an inverse boundary value problem for the two-dimensional stokes equations[END_REF].

Presently, we consider the inverse inclusion problem for the Cauchy-Stokes system. In order to solve the joint completion/detection problem, we reformulate it as a three players Nash game, following the ideas introduced earlier in [START_REF] Habbal | Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems[END_REF] to solve the Cauchy-Laplace (completion) problem. minimizing the gap over Γ c between second player's and the prescribed Dirichlet data f . The fading and regularizing difference between the solutions to these two BVPs is shared by the two players. The third player has no own BVP, but has access to the two previous ones, and uses as control variable the shape of the inclusion(s). The third player's criteria to minimize is a Kohn-Vogelius type functional. The three players play a static Nash game with complete information, whose relevant solution concept is the so-called Nash equilibrium (NE).

We shall present and prove some theoretical results for the Cauchy-Stokes problem, precisely that a Nash equilibrium exists and is unique, and coincides with the missing data as soon as the Cauchy problem has a solution (that is, when the over specified data are compatible). Then, we propose a new algorithm dedicated to the joint computation of the missing data and the obstacle shapes. In this algorithm, a Nash subgame is played by the completion first and second players in order to precondition the Cauchy problem and tackle its ill-posedness. A level set approach is used for the latter geometric identification problem. We lead a sensitivity analysis, and present several numerical experiments that corroborate the efficiency of our approach and its nice stability with respect to noisy data.

The paper is organized as follows. In Section 2, we extend our previous [START_REF] Habbal | Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems[END_REF] Nash game approach to the data completion for the steady Stokes flows. In view of the formulation of the geometric inverse problem, we first recall in Section 3 a now classical identifiability proof [START_REF] Alvarez | Identification of immersed obstacles via boundary measurements[END_REF] usually established for obstacles, so with homogeneous Dirichlet boundary condition, with a minor adaption to fit the case of inclusions, whose boundary conditions are of homogeneous Neumann type. Then, we formulate in Section 4 the Nash game approach to tackle the joint completion and geometric identification problem. We detail our algorithm, and some numerical aspects of the level set method, used to capture the inclusion boundary. Section 5 is devoted to the presentation of three numerical 2D test cases which assess the ability of our algorithm to jointly recover the missing boundary data and the location and shape of the inclusions as well. We finally draw some concluding remarks in Section 6.

Data completion for the Stokes problem

We consider in the present section the case where possible obstacles or inclusions are known, which amounts to simply not consider them, focusing solely on the data completion problem. In the following, we apply the Nash game formulation of [START_REF] Habbal | Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems[END_REF] to the Stokes problem. Results and proofs in the cited reference extend easily to the present case.

With the notations introduced previously, let be f ∈ (H

1 2 (Γ c )) d and Φ ∈ (H -1 2 (Γ c )) d
given Cauchy data. The pressure field p being determined up to a constant, it s convenient to introduce the following space :

L 2 0 (Ω) = {q ∈ L 2 (Ω) ; Ω q = 0}.
The Cauchy-Stokes problem is stated as follows : find u ∈ (H 1 (Ω)) d and p ∈ L 2 0 (Ω) such that

               ∆u -∇p = 0 in Ω, divu = 0 in Ω, u = f over Γ c , σ(u, p)n = Φ over Γ c ,
The data completion problem, which is simply a reformulation of the Cauchy-Stokes one, amounts to find

τ * ∈ (H 1 2 (Γ i )) d and η * ∈ (H -1 2 (Γ i )) d such that u = τ * and σ(u, p)n = η * over Γ i .
For any given η ∈ (H -1 2 (Γ i )) d and τ ∈ (H

1 2 (Γ i )) d , we define the states (u 1 (η), p 1 (η)) ∈ (H 1 (Ω)) d × L 2 0 (Ω) and (u 2 (τ ), p 2 (τ )) ∈ (H 1 (Ω)) d × L 2 0 (Ω)
as the unique weak solutions of the following Stokes boundary value problems

(SP 1
) and (SP 2 ) :

(SP 1 )                ∆u 1 -∇p 1 = 0 in Ω, divu 1 = 0 in Ω, u 1 = f on Γ c , σ(u 1 , p 1 )n = η on Γ i , (SP 2 )                ∆u 2 -∇p 2 = 0 in Ω, divu 2 = 0 in Ω, u 2 = τ on Γ i , σ(u 2 , p 2 )n = Φ on Γ c .
We then define the following cost functionals :

J 1 (η, τ ) = 1 2 ||σ(u 1 (η), p 1 (η))n -Φ|| 2 (H -1 2 (Γ c )) d + 1 2 ||u 1 (η) -u 2 (τ )|| 2 (H 1 2 (Γ i)) d (2)
J 2 (η, τ ) = 1 2 ||u 2 (τ ) -f || 2 (H 1 2 (Γ c )) d + 1 2 ||u 1 (η) -u 2 (τ )|| 2 (H 1 2 (Γ i)) d ) (3)
We are now in a position to formulate the two-player Nash game. The first player is defined by its strategy η ∈ (H -1 2 (Γ i )) d and cost J 1 , while the second one has control on τ ∈ (H 1 2 (Γ i )) d and aims at minimizing the cost J 2 . The two players play a static Nash game with complete information. The most popular solution concept for such games is the one of a Nash equilibrium (NE) given by the

Definition 1 A strategy pair (η N , τ N ) ∈ (H -1 2 (Γ i )) d × (H 1 2 (Γ i ))
d is a Nash equilibrium if the following holds:

   J 1 (η N , τ N ) ≤ J 1 (η, τ N ), ∀η ∈ (H -1 2 (Γ i )) d , J 2 (η N , τ N ) ≤ J 2 (η N , τ ), ∀τ ∈ (H 1 2 (Γ i )) d . (4)
The recourse to a game formulation and to a NE solution finds its justification in the following result :

Proposition 1 Consider the Nash game defined above, with costs given by ( 2) and ( 3).

(i) There always exists a unique Nash equilibrium

(η N , τ N ) ∈ (H -1 2 (Γ i )) d × (H 1 2 (Γ i )) d , which is also the minimum of the potential L(η, τ ) = 1 2 ||σ(u 1 (η), p 1 (η))n -Φ|| 2 (H -1 2 (Γ c )) d + 1 2 ||u 2 (τ ) -f || 2 (H 1 2 (Γ c )) d + 1 2 ||u 1 (η) -τ || 2 (H 1 2 (Γ i)) d .
(ii) If the Cauchy problem has a solution (u, p), then (u

1 (η N ), p 1 (η N )) = (u 2 (τ N ), p 2 (τ N )) = (u, p) and
(η N , τ N ) are the missing data, i.e. η N = σ(u, p)n |Γ i and τ N = u |Γ i .

Proof of (i). We first prove the uniqueness of a NE. It is easy to check that the potential L is strictly convex by computing its second order differential with respect to (η, τ ), see [START_REF] Ben Abda | A control type method for solving the cauchy-stokes problem[END_REF]. Thus, L has at most a one minimum. Moreover, if it exists, the minimum of L is a Nash equilibrium, and conversely. Indeed, let be (η 0 , τ 0 ) the minimum of L, then, we have

     L(η 0 , τ 0 ) ≤ L(η, τ 0 ), ∀η ∈ (H -1 2 (Γ i )) d , L(η 0 , τ 0 ) ≤ L(η 0 , τ ), ∀τ ∈ (H 1 2 (Γ i )) d .
Thanks to the specific structure of L, this is equivalent to write

     J 1 (η 0 , τ 0 ) ≤ J 1 (η, τ 0 ), ∀η ∈ (H -1 2 (Γ i )) d , J 2 (η 0 , τ 0 ) ≤ J 2 (η 0 , τ ), ∀τ ∈ (H 1 2 (Γ i )) d . That is, (η 0 , τ 0 ) is a Nash equilibrium. Conversely, if (η 0 , τ 0 ) is a Nash equilibrium then,      J 1 (η 0 , τ 0 ) ≤ J 1 (η, τ 0 ), ∀η ∈ (H -1 2 (Γ i )) d , J 2 (η 0 , τ 0 ) ≤ J 2 (η 0 , τ ), ∀τ ∈ (H 1 2 (Γ i )) d . Adding the term 1 2 ||u 2 (τ ) -f || 2 (H 1 2 (Γ c )) d in the first inequality and the term 1 2 ||σ(u 1 , p 1 )n -Φ|| 2 (H -1 2 (Γ c )) d in the second one, we get,      L(η 0 , τ 0 ) ≤ L(η, τ 0 ), ∀η ∈ (H -1 2 (Γ i )) d , L(η 0 , τ 0 ) ≤ L(η 0 , τ ), ∀τ ∈ (H 1 2 (Γ i )) d .
By the optimality conditions, we have

       ∂L ∂η (η 0 , τ 0 ) = 0, ∂L ∂τ (η 0 , τ 0 ) = 0,
thus, (η 0 , τ 0 ) is the minimum of L, the uniqueness of which implies that of the Nash equilibrium.

The proof of existence follows the same lines as in [START_REF] Habbal | Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems[END_REF], the main ingredient being the uniform ellipticity of the convex partial maps η -→ J 1 (η, τ ) and of τ -→ J 2 (η, τ ) which allows for a direct application of the Nash Theorem, see ibidem references to the Nash games and theorem.

Proof of (ii). If we assume that the Cauchy-Stokes problem has a solution (u, p), which is then unique by the Holmgren theorem, then let us define the following

η C = σ(u, p)n |Γ i and τ C = u |Γ i . It is then straightforward to check that the solutions (u 1 (η C ), p 1 (η C )) to (SP 1 ) and (u 2 (τ C ), p 2 (τ C )) to (SP 2 )
coincide with the Cauchy solution (u, p), thanks to the uniqueness of the solution of the boundary value Stokes problem. Thus,

L(η C , τ C ) = 0 so that (η C , τ C ) is a minimum of L ≥ 0. Thanks to the uniqueness result above, (η N , τ N ) = (η C , τ C ).
For the computation of the NE for the Cauchy-Stokes problem, we used a popular algorithm [START_REF] Attouch | Alternating proximal algorithms for weakly coupled convex minimization problems. applications to dynamical games and pde's[END_REF] which amounts basically to solve iteratively the following coupled problem, using gradient descent methods,

(η N , τ N ) = argmin η J 1 (η, τ N ), (η N , τ N ) = argmin τ J 2 (η N , τ ).
We describe in Algorithm 1 below the main steps of the method, with a version where the Cauchy data of the Dirichlet type f are possibly perturbed by a noise with some magnitude σ, yielding for the Cauchy problem a noisy Dirichlet data f σ : Algorithm 1: Computation of a Cauchy-Stokes Nash equilibrium Given : ε > 0 a convergence tolerance, Kmax a computational budget, σ a noise level and ρ(σ) a -tunedfunction which depends on the noise. Choose an initial guess

S (0) = (η (0) , τ (0) ) ∈ ((H -1 2 (Γ c )) d × (H 1 2 (Γ c )) d . Set k = 1. • Step 1: Compute η (k) solution of min η J 1 (η, τ (k-1) )
and determine

η (k) = tη (k-1) + (1 -t)η (k) with 0 ≤ t < 1 . • Step 2: Compute τ (k) solution of min τ J 2 (η (k-1) , τ ) and determine τ (k) = tτ (k-1) + (1 -t)τ (k) with 0 ≤ t < 1 . • Step 3: Compute s k = ||u (k) 2 -f σ || L 2 (Γc)
, where (u

(k) 2 , p (k)
2 ) is the solution of the following direct problem

               ∆u (k) 2 -∇p (k) 2 = 0 in Ω, divu (k) 2 = 0 in Ω, u (k) 2 = τ (k) on Γ i , σ(u (k) 2 , p (k)
2

)n = Φ on Γ c . While s k ≥ ρ(σ)ε and k < Kmax set k = k + 1, return back to step 1.
The gradient descent methods used to solve steps 1 and 2 in the algorithm above do require the computation of the gradients of the costs J 1 and J 2 , with respect to their respective strategies. The fast computation of the latter is classical, and led by means of an adjoint state method, as shown by the Proposition 2 below, see [START_REF] Ben Abda | A control type method for solving the cauchy-stokes problem[END_REF] for details.

We shall use the following classical notation :

H 1 Γ (Ω) = {ϕ ∈ (H 1 (Ω)) d /ϕ |Γ = 0}
whenever Γ is a non empty subset of the boundary of Ω.

Proposition 2 We have the following two partial derivatives:

(AP 1 )                                ∂J 1 ∂η ψ = - Γ i ψλ 1 dΓ i , ∀ψ ∈ (H -1 2 (Γ i )) d , with (λ 1 , κ 1 ) ∈ H 1 Γ c (Ω) × L 2 0 (Ω) solution of the adjoint problem:                  Γ c (σ(u 1 , p 1 )n -Φ)((∇γ + ∇γ T )n)dΓ c + Γ i (u 1 -τ )γdΓ i + Ω (∇γ + ∇γ T ) : ∇λ 1 dΩ - Ω κ 1 divγdΩ = 0, ∀γ ∈ H 1 Γ c (Ω). -Γ c (σ(u 1 , p 1 )n -Φ)δndΓ c -Ω δdivλ 1 dΩ = 0, ∀δ ∈ L 2 0 (Ω), (AP 2 )                              ∂J 2 ∂τ µ = Γ i (σ(λ 2 , κ 2 )n -(u 1 (η) -u 2 (τ )))µdΓ i , ∀µ ∈ (H 1 2 (Γ i )) d , with (λ 2 , κ 2 ) ∈ H 1 Γ i (Ω) × L 2 0 (Ω) solution of the adjoint problem:                ∆λ 2 -∇κ 2 = 0 in Ω, divλ 2 = 0 in Ω, λ 2 = 0 on Γ i , σ(λ 2 , κ 2 )n = f -u 2 (τ ) on Γ c ,
where, by a classical convention, ∇u :

∇v = T r(∇u∇v T ) = i,j ∂u i ∂x j ∂v i ∂x j .
The existence and uniqueness of the solutions to the problems (AP 1 ) and (AP 2 ), namely the adjoint states

(λ 1 , κ 1 ) ∈ H 1 Γ c (Ω) × L 2 0 (Ω) and (λ 2 , κ 2 ) ∈ (H 1 (Ω)) d × L 2 0 (Ω) is straightforward, thanks to the regularity assumption on the Cauchy data (f, Φ) ∈ (H 1 2 (Γ c )) d ×(H -1 2 (Γ c
)) d and to regularity results on the solutions to the Stokes problems (SP 1 ) and (SP 2 ), see e.g. [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems[END_REF] (or [START_REF] Bastay | An alternating method for the stationary stokes system[END_REF] Theorem 5.2).

Later on, Algorithm 1 described above will be embedded into an overall algorithm with the specific task of processing the data recovery problem. We shall then use the partial derivatives given by Proposition 2. The overall algorithm stems from the Nash game played by the data recovery problem against the inclusion inverse problem. Next section is then devoted to a mandatory preamble for geometric inverse problems, that is the identifiability question.

An identifiability result for the inverse inclusion Cauchy-Stokes problem

In the present section, we adapt an identifiability result in [START_REF] Alvarez | Identification of immersed obstacles via boundary measurements[END_REF], established for the case of obstacles, that is with a homogeneous Dirichlet condition, to the case of inclusions defined by Neumann (or free surface) boundary conditions.

The set of admissible inclusions is definded by :

D ad = {ω ⊂⊂ Ω is open, Lipschitz and Ω \ ω is connected}.
We follow grosso modo the same proof technique of [START_REF] Alvarez | Identification of immersed obstacles via boundary measurements[END_REF], noticing that, differently from the obstacle (Dirichlet) case, inclusions are not identifiable in case of over specified data f of affine free divergence form. Consequently, even if the over specified fluid stress Φ is identically zero, it is enough for the identifiability to hold, that the velocity data f be non affine. Assume there exists a pair of compatible data (f, Φ) ∈ (H

1 2 (Γ c )) d × (H -1 2 (Γ c
)) d for the Cauchy-Stokes problem, such that either Φ ≡ 0 or f is not the trace of an affine divergence free function.

Consider two admissible open sets ω 1 and ω 2 in D ad . For i = 1, 2, let be (u i , p i ) the solutions to the following Cauchy-Stokes inclusion problem :

                     ∆u i -∇p i = 0 in Ω \ ω i , divu i = 0 in Ω \ ω i , σ(u i , p i )n = 0 on ∂ω i , u i = f on Γ c , σ(u i , p i )n = Φ on Γ c . (5)
Then

ω 1 = ω 2 .
Proof. Denote by ω = ω 1 ∪ ω 2 and define, over the set Ω \ ω, v = u 1 -u 2 and q = p 1 -p 2 , where (u 1 , p 1 ) and (u 2 , p 2 ) are the solutions to the system (5).

One sees that

(v, q) satisfies                ∆v -∇q = 0 in Ω \ ω, divv = 0 in Ω \ ω, v = 0 on Γ c , σ(v, q)n = 0 on Γ c .
Thus, thanks to the Holmgren uniqueness theorem for the Stokes system, we have v = 0 and q = 0 in Ω \ ω and then of Ω. We know from system (5) that :

u 1 = u 2 in Ω \ ω.
∆u 2 -∇p 2 = 0 in ω 1 \ ω 2 .
We multiply the equation above by u 2 and take the integral over ω 1 \ ω 2 . Observe then that, thanks to

divu 2 = 0 in ω 1 \ ω 2 , one has ∆u 2 = 2div(D(u 2 )) where we recall that D(u 2 ) = 1/2(∇u 2 + ∇u T 2 ).
By use of the Green formula, we obtain

ω1\ω2 D(u 2 ) : ∇u 2 dx - ω1\ω2 p 2 divu 2 dx = ∂(ω1\ω2) (-p 2 I d + D(u 2 ))nu 2 ds.
which can be rewritten as follows :

1 2 ω1\ω2 |D(u 2 )| 2 dx = ∂(ω1\ω2) σ(u 2 , p 2 )nu 2 ds.
Now, since σ(v, q) vanishes in Ω\ω, and thanks to the continuity of the involved -normal-traces, one has σ(v, q)n = 0 on ∂ω. From other part, one has σ(u 1 , p 1 )n = 0 on ∂ω 1 thanks to equations [START_REF]On the Detection, Behaviour and Control of Inclusions in Liquid Metals[END_REF]. We then have σ(u 2 , p 2 )n = 0 on ∂ω 1 \ ∂(ω 1 ∩ ω 2 ). Now, since we know that σ(u 2 , p 2 )n = 0 on ∂ω 2 thanks to equations (5), we obtain

∂(ω1\ω2) σ(u 2 , p 2 )nu 2 ds = 0, that is, 1 2 ω1\ω2 |D(u 2 )| 2 dx = 0.
Since ||D(u 2 )|| 2 L 2 (ω1\ω2) = 0, the components of the matrix of D(u 2 ) are a.e. zero. Consequently, the velocity field u 2 has an affine form in ω 1 \ω 2 , shortly given by u 2 (x) = Ax+b where A is a constant matrix with null diagonal.

We know from above and from equations ( 5) that (u 2 , p 2 ) satisfies the following system,

         ∆u 2 -∇p 2 = 0 in ω 1 \ ω 2 , divu 2 = 0 in ω 1 \ ω 2 , σ(u 2 , p 2 )n = 0 on ∂(ω 1 \ ω 2 ).
Thus, by application (to u 2 (x) -(Ax + b) which fulfills the system above) of the unique continuation theorem for the steady Stokes equation established in [START_REF] Fabre | Unique continuation property of solutions of the stokes equation[END_REF], we conclude that u 2 (x) = Ax + b and p 2 = 0 in the whole domain Ω \ ω 2 .

Finally, reasoning with the traces on the boundary of the domain ∂Ω, we observe that u 2 (x) = Ax +b and p 2 = 0

in Ω \ ω 2 yields σ(u 2 , p 2 )n = Φ = 0 and u 2 (x) = f (x) = Ax + b over Γ c , which, by assumption, is impossible. We conclude that ω 1 \ ω 2 = ∅, and so ω 1 = ω 2 .

The identifiability result suggests that there is no need for a third party state equation, the two state equations (SP 1 ) and (SP 2 ) formulated with inclusions and dedicated to the completion problem should suffice. Only a third player's cost functional should be defined, playing with inclusions as strategies. Hence we enforced the data completion steps, by letting the first and second players lead a Nash subgame during the overall iterations, see next section. Numerical experiments show that this choice turned out to be efficient.

Coupled data completion and geometry identification for the Stokes problem

The aim of the present section is to introduce an algorithm dedicated to recover the missing boundary data while solving the inverse inclusion problem for steady Stokes flows. We extend the two-player Nash game set for the completion problem to a three-player Nash game, the third player being in charge of the inverse inclusion problem.

We recall that the inverse inclusion problem amounts to find ω * ∈ D ad such that the fluid velocity u and the pressure p are solution to the following Cauchy-Stokes problem:

                     ∆u -∇p = 0 in Ω \ ω * , divu = 0 in Ω \ ω * , σ(u, p)n = 0 on ∂ω * , u = f on Γ c , σ(u, p)n = Φ on Γ c , (6)
Thanks to the identifiability result stated in section 3, a single pair of -compatible-measurements (f, Φ) is enough to recover the inclusion(s) as well as the missing data. Next, we shall set up a three-player Nash game following the same philosophy than in section 2 dedicated for the sole completion.

For η ∈ (H -1 2 (Γ i )) d , τ ∈ (H 1 2 (Γ i )) d
and ω ∈ D ad , let us define the following three cost functionals:

J 1 (η, τ ; ω) = 1 2 ||σ(u ω 1 (η), p ω 1 (η))n -Φ|| 2 (H -1 2 (Γ c )) d + 1 2 ||u ω 1 (η) -u ω 2 (τ )|| 2 (H 1 2 (Γ i)) d , (7)
J 2 (η, τ ; ω) = 1 2 ||u ω 2 (τ ) -f || 2 (H 1 2 (Γ c )) d + 1 2 ||u ω 1 (η) -u ω 2 (τ )|| 2 (H 1 2 (Γ i)) d , (8)
J 3 (η, τ ; ω) = ||σ(u ω 1 (η), p ω 1 (η) -σ(u ω 2 (τ ), p ω 2 (τ ))|| 2 L 2 (Ω\ω) d + µ|∂ω|, (9)
where (u ω 1 (η), p ω 1 (η)) and (u ω 2 (τ ), p ω 2 (τ )) are the solutions of the respective BVP (P 1 ) and (P 2 ) :

(P 1 )                      ∆u ω 1 -∇p ω 1 = 0 in Ω \ ω, divu ω 1 = 0 in Ω \ ω, σ(u ω 1 , p ω 1 )n = 0 on ∂ω, u ω 1 = f on Γ c , σ(u ω 1 , p ω 1 )n = η on Γ i , (P 2 )                      ∆u ω 2 -∇p ω 2 = 0 in Ω \ ω, divu ω 2 = 0 in Ω \ ω, σ(u ω 2 , p ω 2 )n = 0 on ∂ω, σ(u ω 2 , p ω 2 )n = Φ on Γ c , u ω 2 = τ on Γ i .
In a few words, there are three players: Player (1) controls the strategy variable η ∈ (H -1 2 (Γ i )) d and aims at minimizing the cost J 1 and Player (2) controls the strategy variable τ ∈ (H Notice that the state variables (u ω 1 (η), p ω 1 (η)) and (u ω 2 (τ ), p ω 2 (τ )) belong to the space (H 1 (Ω\ω)

) d × L 2 (Ω\ω),
which obviously depends on ω, a variable intended to be a control. In order to circumvent this tricky dependence, we recourse to a level-set formulation, before stating the actual three-player Nash game effectively implemented.

A level-set formulation

The level-set approach is a very convenient tool in shape identification, see [START_REF] Malladi | Shape modeling with front propagation: A level set approach[END_REF] for a general introduction, or [START_REF] Santosa | A level-set approach for inverse problems involving obstacles[END_REF] where the approach is applied to detect obstacles in a Stokes flow. The boundary of the shape to be identified is postulated to be a zero level-set of a smooth enough (say Lipschitz) function φ : Ω -→ R. In other words, when φ varies in some -admissible-functional space, admissible open subsets ω ∈ Ω are those defined by the following

         φ(x) < 0 in ω, φ(x) > 0 in Ω \ ω, φ(x) = 0 on ∂ω.
The open set Ω \ ω is then given in terms of the level-set function as follows:

Ω \ ω = {x ∈ Ω such that H(φ(x)) = 1}, (10)
where H(.) is the Heaviside function. The perimeter of ω can then be formally given by

|∂ω| = Ω |∇H(φ)|dx = Ω δ(φ)|∇φ|dx.
where δ is the Dirac distribution.

For regularity reasons, and for ensuring the well-posedness of the modified Stokes system as well, it is usual to use smoothed versions of the Heaviside and Dirac distributions. Given two small enough parameters ε > 0 and β > 0, we used smoothed versions denoted respectively by H ε,β (.) and δ ε,β (.), expressed as follows, for s ∈ R,

H ε,β (s) =          1 if s > ε, 1 2 (1 + 2 π arctan( s ε )) if |s| ≤ ε, β if s < -ε, δ ε,β (s) =    1 π ( ε s 2 + ε 2 ) if |s| ≤ ε, β if |s| > ε.
Let us now define the -control free-Sobolev state spaces : Given g ∈ (H

1 2 (Γ c )) d , ψ ∈ (H 1 2 (Γ i )) d V g = {v ∈ H 1 (Ω) d /divv = 0 and v |Γc = g} and W ψ = {v ∈ H 1 (Ω) d /divv = 0 and v |Γi = ψ}.
Problems (P 1 ) and (P 2 ) are then rephrased in terms of the level-set, yielding the modified weak form :

(P 1 ε,β )      Find (u φ 1 , p φ 1 ) ∈ V f × L 2 0 (Ω) such that Ω (σ(u φ 1 , p φ 1 ) : ∇v 1 )H ε,β (φ)dΩ = Γ i ηv 1 dΓ, ∀v 1 ∈ H 1 Γ c (Ω), (11)
(P 2 ε,β )      Find (u φ 2 , p φ 2 ) ∈ W τ × L 2 0 (Ω) such that Ω (σ(u φ 2 , p φ 2 ) : ∇v 2 )H ε,β (φ)dΩ = Γ c Φv 2 dΓ, ∀v 2 ∈ H 1 Γ i (Ω), (12)
It is not the scope of the present paper to discuss the dependence of the modified Stokes problems with respect to (ε, β), which is known to behave consistently [START_REF] Duan | Shape-topology optimization of stokes flow via variational level set method[END_REF] [32], so we still refer to problems (P 1 ε,β ) and (P 1 ε,β ) as (P 1 ) and (P 2 ), and we omit to underline the dependence of the state variables w.r.t. (ε, β) as well.

Level-set sensitivity and optimality condition

The player (3) in charge of the inverse inclusion problem has now control on the level-set function φ instead of the open subset ω ∈ D ad . The new form of the third player's cost functional is now as follows :

J 3 (η, τ ; φ) = Ω |σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 )| 2 H ε,β (φ)dx + µ Ω δ ε,β (φ)|∇φ|dx
where (u φ 1 , p φ 1 ) and (u φ 2 , p φ 2 ) solve respectively problems [START_REF] Kozlov | An iterative method for solving the cauchy problems for elliptic equations[END_REF] and [START_REF] Johansson | Reconstruction of a stationary flow from incomplete boundary data using iterative methods[END_REF]. We choose as convenient space for the levelset variables the Sobolev space S = H 1 (Ω) though it is not optimal (in the sense that it may introduce too much regularity requirement, hampering the capture of non H 1 inclusions).

In order to perform the partial optimization of J 3 (η, τ ; φ) w.r.t. φ for (η, τ ) given by players ( 1) and ( 2), one needs to compute the derivative of J 3 w.r.t. φ. We have the following :

Proposition 3 If φ ∈ S satisfies the boundary condition ∂φ ∂n = 0 over ∂Ω then the partial derivative of J 3 (η, τ ; φ) with respect to φ, in any direction ψ ∈ S, is given by

( ∂J 3 ∂φ (η, τ ; φ), ψ) = Ω δ ε,β (φ) |σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 )| 2 -µdiv( ∇φ |∇φ| ) + σ(u φ 1 , p φ 1 ) : ∇λ 1 + σ(u φ 2 , p φ 2 ) : ∇λ 2 ψdx, where (λ 1 , π 1 ) ∈ H 1 Γ c (Ω) × L 2 0 (Ω) and (λ 2 , π 2 ) ∈ H 1 Γ i (Ω) × L 2 0
(Ω) are respective solutions of the adjoints problems,

                   -2 Ω (σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 )) : (∇h 1 + ∇h T 1 )H ε,β (φ) - Ω (divh 1 )π 1 H ε,β (φ) + Ω ((∇h 1 + ∇h T 1 ) : ∇λ 1 )H ε,β (φ) = 0, ∀h 1 ∈ H 1 Γ c (Ω), 2 Ω (σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 ))(k 1 I)H ε,β (φ) - Ω k 1 (divλ 1 )H ε,β (φ) = 0, ∀k 1 ∈ L 2 0 (Ω), (13)
                   2 Ω (σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 )) : (∇h 2 + ∇h T 2 )H ε,β (φ) - Ω (divh 2 )π 2 H ε,β (φ) + Ω ((∇h 2 + ∇h T 2 ) : ∇λ 2 )H ε,β (φ) = 0, ∀h 2 ∈ H 1 Γ i (Ω), - 2
Ω (σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 ))(k 2 I)H ε,β (φ) - Ω k 2 (divλ 2 )H ε,β (φ) = 0, ∀k 2 ∈ L 2 0 (Ω), (14)
and where (u φ 1 , p φ 1 ) and (u φ 2 , p φ 2 ) are the solutions to respectively [START_REF] Kozlov | An iterative method for solving the cauchy problems for elliptic equations[END_REF] and [START_REF] Johansson | Reconstruction of a stationary flow from incomplete boundary data using iterative methods[END_REF].

The proof of Proposition 3 above is given in Appendix.

The necessary optimality condition for the minimization problem min φ∈S J 3 (η, τ ; φ) is then formulated as the following Euler-Lagrange equation :

     δ ε,β (φ)[|σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 )| 2 -µdiv( ∇φ |∇φ| ) + σ(u φ 1 , p φ 1 ) : ∇λ 1 + σ(u φ 2 , p φ 2 ) : ∇λ 2 ] = 0, in Ω, ∂φ ∂n = 0, on ∂Ω. ( 15
)
The strongly nonlinear equation above -with implicit terms-is solved iteratively as the stationary state of the following evolution equation

           ∂φ ∂t = -δ ε,β (φ)[|σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 )| 2 -µdiv( ∇φ |∇φ| ) + σ(u φ 1 , p φ 1 ) : ∇λ 1 + σ(u φ 2 , p φ 2 ) : ∇λ 2 ] = 0 in R + × Ω, ∂φ ∂n = 0 on R + × ∂Ω, φ(0, x) = φ 0 (x) in Ω, (16)
where φ 0 ∈ S is a given initial condition.

The variational formulation associated to the problem ( 16) above reads

Ω ∂φ ∂t ψdx = - Ω δ ε,β (φ) |σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 )| 2 + σ(u φ 1 , p φ 1 ) : ∇λ 1 + σ(u φ 2 , p φ 2 ) : ∇λ 2 ψdx + Ω µδ ε,β (φ)div( ∇φ |∇φ| )ψdx, ∀ψ ∈ H 1 (Ω).
But, since one has

div(δ ε,β (φ) ∇φ |∇φ| )) = δ ε,β (φ)∇φ ∇φ |∇φ| + δ ε,β (φ)div( ∇φ |∇φ| ),
we get,

Ω ∂φ ∂t ψdx = - Ω δ ε,β (φ) |σ(u φ 2 , p φ 2 ) -σ(u φ 1 , p φ 1 )| 2 + σ(u φ 1 , p φ 1 ) : ∇λ 1 + σ(u φ 2 , p φ 2 ) : ∇λ 2 ψdx - Ω µδ ε,β (φ) |∇φ| ∇φ∇ψdx -µ Ω δ ε,β (φ)|∇φ|ψdx, ∀ψ ∈ H 1 (Ω).
The problem above is solved numerically by means of a semi-implicit Euler scheme with ∂φ ∂t approximated by

φ n+1 -φ n δt
, where φ n (.) = φ(t n , .) and t n = nδt, with δt > 0 a given time step.

We obtain the following iterative scheme:

   Given φ n , Find φ n+1 ∈ H 1 (Ω) such that: a(φ n+1 , ψ) = l(ψ), ∀ψ ∈ H 1 (Ω), (17)
where

             a(φ n+1 , ψ) = Ω φ n+1 ψdx + δt Ω µδ ε,β (φ n ) |∇φ n | ∇φ n+1 ∇ψdx, l(ψ) = -δt Ω δ ε,β (φ n ) |σ(u n 2 , p n 2 ) -σ(u n 1 , p n 1 )| 2 + σ(u n 1 , p n 1 ) : ∇λ n 1 + σ(u n 2 , p n 2 ) : ∇λ n 2 ψdx -µδt Ω δ ε,β (φ n )|∇φ n |ψdx + Ω φ n ψdx with (u n 1 , p n 1 ) = (u 1 (η, φ n ), p 1 (η, φ n )), (u n 2 , p n 2 ) = (u 2 (τ, φ n ), p 2 (τ, φ n ))
are solutions to [START_REF] Kozlov | An iterative method for solving the cauchy problems for elliptic equations[END_REF] and ( 12) for a given φ n , and λ n {i=1,2} = λ {i=1,2} (φ n ) are the adjoint state solutions of the problems ( 13) and [START_REF] Habbal | Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems[END_REF].

In order to prevent the level set iterates from being too flat or too steep, we trigger from time to time a regularization pass that reinitializes the level set to a signed distance (see e.g. [START_REF] Allaire | Structural optimization using sensitivity analysis and a level-set method[END_REF]). This step is mandatory in order to keep the iterated level sets smooth enough, but is also necessary to have a non vanishing |∇φ n | that ensures the ellipticity of a(., .) in [START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF]. The update is performed by solving the following equation :

           ∂ψ ∂t + sign(φ n )(|∇ψ| -1) = 0 in R + × Ω, ∂ψ ∂n = 0 on R + × ∂Ω, ψ(0, x) = φ n (x) in Ω, (18)
In practice, equation ( 18) above is solved for a few time steps (typically 5 or 6) then one reassigns the last computed ψ to φ n (x).

The six variational problems ( 11)-( 14) ( 17) and ( 18) are solved by means of ad hoc Finite Element methods (see Section 5 below).

The three-player Nash algorithm

We are now ready to state the three-player identification/completion Nash game. As aforementioned in section 4, players ( 1) and ( 2) aim at solving the Cauchy problem, while player ( 3) is aimed at minimizing a Kohn-Vogelius type energy, intended to capture the shape of the inclusion. The game is of Nash type, which means that it is static with complete information [START_REF] Habbal | Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems[END_REF] and hence its solution is a Nash equilibrium (NE), see Definition 4.

Given a triplet (η, τ, φ)

∈ (H -1 2 (Γ i )) d × (H 1 2 (Γ i )) d × S, let (u φ 1 (η), p φ 1 (η)
) be the solution to the approximate Stokes problem [START_REF] Kozlov | An iterative method for solving the cauchy problems for elliptic equations[END_REF] and (u φ 2 (τ ), p φ 2 (τ )) the solution to the approximate Stokes problem [START_REF] Johansson | Reconstruction of a stationary flow from incomplete boundary data using iterative methods[END_REF], then the three players and their respective costs are defined as follows:

• Player (1) has control on the Neumann strategies η ∈ (H -1 2 (Γ i )) d , and its cost functional is given by

J 1 (η, τ, φ) = 1 2 ||σ(u φ 1 (η), p φ 1 (η))n -Φ|| 2 (H -1 2 (Γ c )) d + 1 2 ||u φ 1 (η) -u φ 2 (τ )|| 2 (H 1 2 (Γ i)) d (19)
• Player (2) has control on the Dirichlet strategies τ ∈ (H

1 2 (Γ i )) d ,
and its cost functional is given by

J 2 (η, τ, φ) = 1 2 ||u φ 2 (τ ) -f || 2 (H 1 2 (Γ c )) d + 1 2 ||u φ 1 (η) -u φ 2 (τ )|| 2 (H 1 2 (Γ i)) d ) (20)
• Player (3) has control on the inclusion level-set strategies φ ∈ S, and its cost functional is given by

J 3 (η, τ ; φ) = Ω |σ(u φ 2 (τ ), p φ 2 (τ )) -σ(u φ 1 (η), p φ 1 (η))| 2 H ε,β (φ)dx + µ Ω δ ε,β (φ)|∇φ|dx (21)
In Algorithm 2 below, we describe the main steps in computing the Nash equilibrium. This algorithm is unusual in the sense that, first, it introduces a completion-oriented Nash subgame, solved incompletely (Kmax is small, around ten iterations), and second, it processes the third player's minimization step by iterating on the necessary optimality condition. Classical algorithms compute Nash equilibria with Kmax=1. It is easy to check (by writing down the stationarity equations) that when the two algorithms converge, they lead to the same limit point, which is a Nash equilibrium of the three-player's game defined above.

As we shall see in section 5 below, Algorithm 2 outperforms the classical one.

Algorithm 2: Computation of the coupled inclusion-completion Nash equilibrium Given : convergence tolerances ε N > 0, ε S > 0, Kmax a computational budget per Nash iteration, Nmax a maximum Nash iterations, σ a noise level and ρ(σ) a -tuned-function which depends on the noise. Set n = 0, choose an initial level-set φ (0) ∈ S.

• Step I: (a completion Nash subgame) Set k = 1.

Choose an initial guess

S (k-1) = (η (k-1) , τ (k-1) ) ∈ ((H -1 2 (Γ c )) d × (H 1 2 (Γ c )) d . • Step 1: Compute η (k) solution of min η J 1 (η, τ (k-1) , φ (n) )
and set

η (k) = αη (k-1) + (1 -α)η (k) with 0 ≤ α < 1 . • Step 2: Compute τ (k) solution of min τ J 2 (η (k-1) , τ, φ (n) )
and set

τ (k) = ατ (k-1) + (1 -α)τ (k) with 0 ≤ α < 1 . • Step 3: While S (k) -S (k-1) > ε S and k < Kmax, set k = k + 1, return back to step 1. • Step II: Compute r k = ||u (k) 2 -f σ || L 2 (Γ c ) , where (u (k) 2 , p (k)
2 ) is the solution of the problem ( 12) with the level-set φ = φ (n) and with the Dirichlet condition u

(k) 2 = τ (k) over Γ i . •
Step III: While r k ≥ ρ(σ)ε and n < Nmax update the level-set : compute φ (n+1) solution to the variational problem [START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF] and set n = n + 1, go back to step I.

Numerical experiments

In this section, we provide and discuss the numerical results of experiments led for three test cases, named A, B and C. These 3 test-cases share the following common settings :

The domain:

Ω = -1 2 , 1 2 × -1 2 , 1 2
The boundaries:

Γ i = { 1 2 } × -1 2 , 1 2 ; Γ c = ∂Ω \ Γ i
Normal stress: Φ(x, y) = -2(y 2 -1/4; 0) prescribed over ∂Ω Initial strategies: for Step I, we always used S (0) = (η 0) , τ (0) ) = (0, 0) and took α = 0.10

Parameters: for Step III : we took δt = 0.02 for solving equation [START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF].

The test-cases differ in the shape and/or number of connexe components of the inclusions.

Given a known shape and location of the inclusion ω * ∈ D ad , we solve the following Stokes problem :

               ∆u -∇p = 0 in Ω \ ω * , divu = 0 in Ω \ ω * , σ(u, p)n = 0 on ∂ω * , σ(u, p)n = Φ on ∂Ω,
where the (phantom) exact solution (u, p) is used to build the remaining Cauchy data f = u |Γ c , and the exact missing data u |Γ i and σ(u, p)n |Γ i . The two latter data together with the known inclusion shape ω * are used to compute the following relative errors :

err D = ||τ N -u |Γi || L 2 (Γi) ||u |Γi || L 2 (Γi) , err N = ||η N -σ(u, p)n |Γi || L 2 (Γi) ||σ(u, p)n |Γi || L 2 (Γi) , (22)
err O = mes(ω * ∪ ω N ) -mes(ω * ∩ ω N ) mes(ω * ) .
where (η N , τ N , φ N ) is the approximate Nash equilibrium output from Algorithm 2, and ω N = 1 { φ N < 0}.

These metrics are used to assess the efficiency of our approach. The stability w.r.t. noise was stressed by solving the joint inverse inclusion/completion problem with noisy perturbations of the Dirichlet data f σ = f + σN with N being a Gaussian white noise.

Two different initial level-sets were used : φ

1 has as zero level-set the disk B(c 0 , r 0 ) where c 0 = (0, 0) and r 0 = 0.30, and φ

2 is a periodic function with as zero level-set 30 fairly uniformly distributed small holes filling the whole domain Ω, see Figures 3(a) and 4(a).

The solvers for Stokes, equation ( 17) and adjoint systems, the sensitivity routines, and the minimization algorithms as well, were implemented using the Finite Element package FreeFem++ [START_REF] Hecht | New development in freefem++[END_REF].

Test-case A.

The exact inclusion is a disk ω * = B(c, r) centered at c and with a radius r where c = (0, 0) and r = 0.10.

The FreeFem++ implementation of Algorithm 2 was ran for two different initial contours, leading to very close results, both of them in good accordance with the exact solutions (inclusion and missing data). It can be however

For the case of noisy Dirichlet data f σ given over Γ c , it can be seen from the profiles presented in Figure 5 that the boundary data recovery is remarkably stable with respect to the noise magnitude, and even more striking is the stability of the detected inclusion.

The relative errors defined by formulas [START_REF] Bourgeois | The exterior approach to solve the inverse obstacle problem for the stokes system[END_REF] are summarized in Table 1 for the test-case A. Test-case B.

The exact inclusion ω * has a peanut-like shape, with a boundary parameterized as follows :

∂ω =      x 0 y 0   + r(θ)   a sin θ b cos θ   ; θ ∈ [0, 2π)    ,
where r(θ) = sin 2 θ + 0.25 cos 2 θ, (x 0 , y 0 ) = (0, 0) and (a, b) = (0.15, 0.18).

In this test-case, the shape of the inclusion is nonconvex. We observe from Figure 6(b) that while the computed zero level-set is in good accordance with the exact one, it is however unable to accurately capture the nonconvex features of the real inclusion. This is not very surprising in view of the different smoothed approximations used for the inclusion Stokes problems as well as for the level-set equation. The data completion results are however very satisfactory, as shown by the velocity and normal stress profiles in Figures 6(c)-(e). There is also a remarkable stability with respect to noisy data of both the inclusion detected and the recovered boundary data, see Figure 7.

Test-case C.

The inclusion to be detected is the union of two separate disks ω * 1 = B(c 1 , r 1 ) centered at c 1 = (0.2, 0.2) and with a radius r 1 = 0.10 and ω * 2 = B(c 2 , r 2 ) centered at c 2 = (-0.2, -0.2) and with a radius r 2 = 0.12.

This third and last test-case was set up to assess the ability of our algorithm to identify inclusions with several components. One observes from Figure 8(b) that the locations and the shapes of the two components of the inclusion are well detected, as well as the recovered data Figure 8(c)-(f). The recovery of missing boundary data is stable with respect to noisy Dirichlet measurements while there is a barely slight shift in the location of the detected approximation of ω * 1 for the noise levels 3% and 5%, as shown Figure 9.

The relative errors presented in Algorithm 2 vs Classical.

In a classical algorithm [START_REF] Li | Distributed algorithms for the computation of noncooperative equilibria[END_REF] dedicated to the computation of a Nash equilibrium, there would be no preconditioning step as step I in Algorithm 2, or in other words, Kmax=1. We have compared these two approaches, for two noise free test-cases. We used the same number of total calls (400) to the Stokes Finite Element solvers. We see from

Conclusion

We addressed in the present paper the delicate problem of detecting unknown cavities immersed in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a Stokes regime, the cavities are inclusions and the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse inclusion Cauchy-Stokes problem is ill-posed for both the cavities and missing data reconstructions, and designing stable and efficient algorithms, which is the main goal of our work, is not straightforward.

The ill-posedness is tackled by decentralization : we reformulate it as a three players Nash game, following the ideas introduced earlier in [START_REF] Habbal | Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems[END_REF] to solve the Cauchy-Laplace (completion) problem. Thanks to a simple yet strong identifiability result for the Cauchy-Stokes system, it is enough to set up two Stokes BVP, then use them as state equations. The Nash game is then set between 3 players, the two first targeting the data completion while the third one targets the inclusion detection. The latter problem is formulated using a level-set approach, and we provided the third player with the level-set function as strategy, while its cost functional is of Kohn-Vogelius type.

The class of algorithms we propose are summarized in Algorithm 2, the involved computational apparatus being rather classical : use of descent algorithms for the different minimizations, use of adjoint state method to compute the sensitivities, and use of Finite Element methods to solve the state and adjoint state equations, as well as to update the level-sets. We used Freefem++ to implement these routines.

We led 2D numerical experiments for three different test-cases. For noise free, as well as for noisy -Cauchy data-Dirichlet measurements, we obtained satisfactory results, exhibiting very stable behaviour with respect to the noise level (1%, 3%, 5%). The obtained results favor our 3-player Nash game approach to solve parameter or shape identification for Cauchy problems. Finally, our approach rises difficult theoretical questions that we did not address here, such as the existence, uniqueness and convergence issues for the level-set solution to the implicit optimality condition [START_REF] Robert | Relaxation of a variational method for impedance computed tomography[END_REF] and, related to the game-theoretic approach, the existence and convergence issues for the 3-player Nash equilibrium. In the same way, we find that

where (λ 2 , π 2 ) solves the adjoint problem

                  

Figure 1 :

 1 Figure 1: An example of the geometric configuration of the problem : the whole domain including cavities is denoted by Ω. It contains an inclusion ω * . The boundary of Ω is composed of Γ c , an accessible part where over-specified data are available, and an inaccessible part Γ i where the data are missing.

Theorem 1

 1 Let be Ω ⊂ R 2 an open bounded Lipschitz domain and Γ c a non-empty open subset of the boundary ∂Ω.

Figure 2 :

 2 Figure 2: Different situations

1 2 (

 2 Γ i )) d and aims at minimizing the cost J 2 . These two players may be interpreted exactly the same way than in the completion game stated section 2: they are given Dirichlet (resp. Neumann) data and try to minimize the gap with the Neumann (resp. Dirichlet) remaining condition. The player (3) controls the strategy variable ω ∈ D ad and aims at minimizing the Kohn-Vogelius type functional J 3 .

1 ,

 1 observed from Figure 3(c)(e) and Figure 4(c)(e) that the initial contour φ computed first component of the fluid velocity and normal stress are more accurate with the initial contour φ so we shall later on present only those results obtained with φ

Figure 3 :

 3 Figure 3: Test case A. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over Γ c . (a) initial contour is φ(0) 1 (b) exact inclusion shape -green line-and computed one -blue dashed-(c) exact -line-and computed -dashed line-first component of the velocity over Γ i (d) exact -line-and computed -dashed linesecond component of the velocity over Γ i (e) exact -line-and computed -dashed line-first component of the normal stress over Γ i (f) exact -line-and computed -dashed line-second component of the normal stress over Γ i .

Figure 4 :

 4 Figure 4: Test case A. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over Γ c .(a) initial contour is φ

  ) exact inclusion shape -green line-and computed one -blue dashed-(c) exact -lineand computed -dashed line-first component of the velocity over Γ i (d) exact -line-and computed -dashed line-second component of the velocity over Γ i (e) exact -line-and computed -dashed line-first component of the normal stress over Γ i (f) exact -line-and computed -dashed line-second component of the normal stress over Γ i .

Figure 5 :

 5 Figure 5: Test case A. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over Γ c with noise levels σ = {1%, 3%, 5%}.(a) initial contour is φ

  ) exact inclusion shape -green line-and computed ones for different noise levels (c) exact and computed first components of the velocity over Γ i (d) exact and computed second components of the velocity over Γ i (e) exact and computed first components of the normal stress over Γ i (f) exact and computed second components of the normal stress over Γ i .

Figure 6 :

 6 Figure 6: Test case B. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over Γ c . (a) initial contour is φ

  ) exact inclusion shape -green line-and computed one -blue dashed-(c) exact -line-and computed -dashed line-first component of the velocity over Γ i (d) exact -line-and computed -dashed linesecond component of the velocity over Γ i (e) exact -line-and computed -dashed line-first component of the normal stress over Γ i (f) exact -line-and computed -dashed line-second component of the normal stress over Γ i .

Figure 7 :

 7 Figure 7: Test case B. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over Γ c with levels σ = {1%, 3%, 5%}.(a) initial contour is φ

  ) exact inclusion shape -green line-and computed ones for different noise levels (c) exact and computed first components of the velocity over Γ i (d) exact and computed second components of the velocity over Γ i (e) exact and computed first components of the normal stress over Γ i (f) exact and computed second components of the normal stress over Γ i .

Figure 8 :

 8 Figure 8: Test case C. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over Γ c . (a) initial contour is φ

  ) exact inclusion shape -green line-and computed one -blue dashed-(c) exact -line-and computed -dashed line-first component of the velocity over Γ i (d) exact -line-and computed -dashed linesecond component of the velocity over Γ i (e) exact -line-and computed -dashed line-first component of the normal stress over Γ i (f) exact -line-and computed -dashed line-second component of the normal stress over Γ i .

Figure 9 :

 9 Figure 9: Test case C. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over Γ c with levels σ = {1%, 3%, 5%}.(a) initial contour is φ

2 Ω 2 Ω

 22 ) exact inclusion shape -green line-and computed ones for different noise levels (c) exact and computed first components of the velocity over Γ i (d) exact and computed second components of the velocity over Γ i (e) exact and computed first components of the normal stress over Γ i (f) exact and computed second components of the normal stress over Γ i .where (λ 1 , π 1 ) solves the adjoint state problem(σ(u 2 , p 2 ) -σ(u 1 , p 1 )) : (∇h 1 + ∇h T 1 )H ε,β (φ) -Ω π 1 divh 1 H ε,β (φ) + Ω ((∇h 1 + ∇h T 1 ) : ∇λ 1 )H ε,β (φ) = 0, ∀h 1 ∈ H 1 Γ c (Ω), (σ(u 2 , p 2 ) -σ(u 1 , p 1 ))(k 1 I)H ε,β (φ) -Ω k 1 divλ 1 H ε,β (φ) = 0, ∀k 1 ∈ L 2 0 (Ω).

2 Ω(σ(u 2 , p 2 )

 222 -σ(u 1 , p 1 ))σ(u 2 , p 2 )H ε,β (φ) = Ω (σ(u 2 , p 2 ) : ∇λ 2 )δ ε,β (φ)ψ,

2 Ω 2 Ω

 22 (σ(u 2 , p 2 ) -σ(u 1 , p 1 )) : (∇h 2 + ∇hT 2 )H ε,β (φ) -Ω π 2 divh 2 H ε,β (φ) + Ω ((∇h 2 + ∇h T 2 ) : ∇λ 2 )H ε,β (φ) = 0, ∀h 2 ∈ H 1 Γ i (Ω), -(σ(u 2 , p 2 ) -σ(u 1 , p 1 ))(k 2 I)H ε,β (φ) -Ω k 2 divλ 2 H ε,β (φ) = 0, ∀k 2 ∈ L 2 0 (Ω).Using (24) and (25), we obtain∂J 3 ∂φ (η, τ ; φ).ψ = Ω |σ(u 2 , p 2 ) -σ(u 1 , p 1 )| 2 δ ε,β (φ)ψ + Ω (σ(u 1 , p 1 ) : ∇λ 1 )δ ε,β (φ)ψ + Ω (σ(u 2 , p 2 ) : ∇λ 2 )δ ε,β (φ)ψ +µ Ω δ ε,β (φ)|∇φ|ψ + µ Ω δ ε,β (φ) ∇φ∇ψ |∇φ| .On the one hand, we have Ω div(δ ε,β (φ) φ satisfies the boundary condition δ ε,β (φ) |∇φ| ∂φ ∂n = 0, on ∂Ω, one can conclude that ∂J 3 ∂φ (η, τ ; φ).ψ = Ω |σ(u 2 , p 2 ) -σ(u 1 , p 1 )| 2 δ ε,β (φ)ψ + Ω (σ(u 1 , p 1 ) : ∇λ 1 )δ ε,β (φ)ψ + Ω (σ(u 2 , p 2 ) : ∇λ 2 )δ ε,β (φ)ψ -µ Ω δ ε,β (φ)div( ∇φ |∇φ| )ψ.

Table 1 :

 1 Test-case A. L 2 relative errors on missing data on Γ i (on Dirichlet and Neumann data), and the error between the reconstructed and the real shape of the inclusion for various noise levels.

	Noise level σ = 0%	σ = 1%	σ = 3%	σ = 5%
	err D	0.010	0.015	0.039	0.063
	err N	0.031	0.033	0.051	0.07
	err O	0.032	0.043	0.066	0.117

  Table 2 corroborate the stability of the detected contours and missing data with respect to noisy Dirichlet measurements.

	Noise level σ = 0%	σ = 1%	σ = 3%	σ = 5%
	err D	0.042	0.044	0.046	0.08
	err N	0.095	0.1	0.13	0.16
	err O	0.099	0.11	0.13	0.15

Table 2 :

 2 Test-case C. L 2 -errors on missing data over Γ i (on Dirichlet and Neumann data), and the error between the reconstructed and the real shape for various noise levels.

Table 3 that

 3 , for both test-cases, Algorithm 2 outperforms the classical one. The preconditioning Nash subgame in Step I, which is dedicated to enforce the data completion part does indeed enforce the identifiability property as well, since from the result established in Proposition 3, it is enough for a candidate velocity u(ω), for some inclusion ω, to be a Cauchy solution for the pair of boundary measurements (f, Φ), to ensure that ω = ω * , the real inclusion.

	Case A Classical algorithm Algorithm 2	Case C Classical algorithm Algorithm 2
	err D	0.058	0.033	err D	0.067	0.058
	err N	0.106	0.032	err N	0.208	0.122
	err O	0.358	0.140	err O	0.566	0.167

Table 3 :

 3

Relative errors on the reconstructed missing data and inclusion shape for the Stokes problem (with noise free measurements), compared for a classical Nash algorithm and Algorithm 2 : (left) test-case A (right) test-case C.

Appendix : proof of Proposition 3

For fixed (η, τ ) ∈ (H -1 2 (Γ i )) d × (H 1 2 (Γ i )) d , let us define the Lagrangian L by :

where the control φ ∈ S, the state variables

A formal derivation of the functional J 3 (η, τ ; φ) with respect to φ in some direction ψ ∈ H 1 (Ω) yields

+µ

where we have used the notations (u We know that (u 1 , p 1 ) solves the variational equation

Then, (u 1 , p 1 ) fulfills the following weak formulation

Now, we derive the Lagrangian L with respect to u 1 and with respect to p 1 , we get

If the pair (h 1 , k 1 ) is replaced by (u 1 , p 1 ) in ( 23) and because of divu 1 = 0 implies divu 1 = 0, using the weak formulation for the couple (u 1 , p 1 ), we get -2 Ω (σ(u 2 , p 2 ) -σ(u 1 , p 1 ))σ(u 1 , p 1 )H ε,β (φ) = Ω (σ(u 1 , p 1 ) : ∇λ 1 )δ ε,β (φ)ψ, [START_REF] Cjs Alves | Integral equations for an inverse boundary value problem for the two-dimensional stokes equations[END_REF]


Angélique Villière

Sarah Le Roy

Catherine Fillonneau

Carole Prost

Innoscent

InnOscent chromatographic system: An innovative device to revisit aroma analysis and recombination perspectives

InnOscent chromatographic system:

An innovative device to revisit aroma analysis and recombination perspectives ONIRIS -College of Veterinary Medicine and Food Science, Flavor group GEPEA -UMR CNRS 6144, Nantes, France

Angélique VILLIÈRE, Sarah LE ROY, Catherine FILLONNEAU, Carole PROST GC-MS-O is very helpful to identify odorants in aromas. However, there is a gap between this individual characterization and the effective contribution of compounds to aromas, notably due various interactions. Thus, many aroma investigations turn to reconstituted models and recombination experiments with chemicals, a valuable but time-consuming approach.

InnOscent device was designed to realize both conventional GC-MS-O analysis and recombination-omission experiments. These capabilities are illustrated hereafter through a wine aroma analysis and contribution of odorants to global aroma was evaluated.

angelique.villiere@oniris-nantes.fr  InnOscent device permits to achieve both conventional GC-O analysis and recombination or omission experiments, enabling a comprehensive analysis of aromas.

 It offers a valuable alternative to current approaches notably those using chemicals and overcome limitations related to identification, quantitation, and commercial availability of compounds. Hidden reference

The similarity of the hidden reference vs reference, is scored at 8,2/10 instead of 10, due to natural reluctance of judges to use ends scale.

  Compounds perceived by at least 4 judges N ≥ 4 Compounds perceived by at least 6 judges N ≥ 6 Total extract Compounds perceived by at least 3 judges N ≥ 3 Compounds perceived by at least 8 judges N ≥ 8 Dual facilities are achieved by a wise combination of splitter and Deans switch microfluidics transfer modules.GC-MS-dual Olfactometric port analysisRecovering selective fractions with GC-MS-FID control mixture of the only major odorants does not recreate the odor of the original wine. Freeing from chemicals, InnOscent makes it possible to evaluate the contribution of any compound to an overall aroma.


Déborah González

Ramón Arrué

Edison Matamala-Cea

Rodrigo Arancibia

Paul Hamon

Olivier Cador

Thierry Roisnel

Jean-René Hamon
email: jean-rene.hamon@univ-rennes1.fr

Néstor Novoa
email: nenovoa@udec.cl

Homoleptic Co(II), Ni(II), Cu(II), and Zn(II) complexes based on 8hydroxylquinoline Schiff base derivative: a combined synthetic, spectral, structural, and magnetic study †

Keywords: ONN ligands, Schiff bases, Homoleptic Complexes, X-ray diffraction, Hirshfeld Surface Analysis

A series of four new homoleptic Co(II), Ni(II), Cu(II) and Zn(II) complexes (1-4) of the type M(HL)2, where HL -stands for the deprotonated form of the tridentate O,N,Ndonor Schiff base ligand obtained by condensation between 8-hidroxy-2quinolinecarboxaldehyde and 2-aminophenol, have been prepared in yields ranging from 70 to 85%. The five compounds were characterized using elemental analysis, FT-IR, ESI-MS, UV-Vis, and 1 H & 13 C NMR spectroscopies (for the diamagnetic species H2L and Zn(II) complex 4). The molecular structures of H2L and its Ni(II) and Zn(II) complexes 2 and 4, respectively, were further confirmed by single crystal X-ray diffraction methods. X-ray crystal structure determination revealed that in complex 2, the six-coordinate Ni(II) metal ion adopts a N4O2 distorted octahedral geometry with the two tridentate HL -ligands coordinated in a perfect orthogonal fashion. In complex 4, however, the Zn(II) metal center is pentacoordinated and lies in a distorted N3O2 trigonal bipyramidal environment, one HL - ligand acting as an O,N-bidentate ligand. In both cases, a free hydroxyl unit remains available for supramolecular interactions. Hirshfeld surface (HS) analysis along with 2D fingerprint plots were employed to consider the intermolecular forces, including hydrogen bonds and π-π stacking interactions, and their quantification in the crystal lattice. SQUID magnetometry experiments showed high spin Co(II) (S = 3/2), Ni(II) (S = 1) and Cu(II) (S = ½) isolated spins at room temperature with no interaction between magnetic centers on cooling.

Introduction

Schiff bases are ubiquitous ligands facilely synthesized and coordinated to almost all the metal ions, thus creating a very fertile and dynamic research field in coordination chemistry. [START_REF] Hernandez-Molina | Acyclic and Macrocyclic Schiff Base Ligands[END_REF][START_REF] Gavey | [END_REF][3][4][5][6] They are readily formed upon condensation reactions of a primary amine with a carbonyl group, and are characterized by an imine (>C=N-) or azomethine (-HC=N-) function. Due to their highly modular synthesis allowing the control of the nature of donor atoms, chelating ability, as well as their electronic and steric properties, Schiff base ligands can readily form stable complexes with most of the transition metals in various oxidation states and multiple forms of coordination towards the metallic ions. [7,8] Consequently, Schiff base complexes have found applications in different branches of science such as catalysis, [9][10][11][12] magnetism, [13][14][15] sensing, [16,17] optical, [18,19] and energy sciences, [20] as well as medicinal chemistry. [21,22] Among the Schiff bases, N2O2 tetradentate based salen-type ligands where H2-salen (N,N-bis(salicylidene)ethylenediamine) is the quintessential representative of this class of symmetrical di-imines compounds, [23] continue to be the most popular and the most studied ones. [17,24] They are generally prepared in high yields by condensation of any primary diamine with two equivalents of salicylaldehyde derivatives. [START_REF] Hernandez-Molina | Acyclic and Macrocyclic Schiff Base Ligands[END_REF]25] Mono-condensed Schiff bases have also been obtained either by direct, [26,27] or metal template, [28,29] reactions between carbonyl units and appropriate amino alcohol, diamine or aminothiol substrates, yielding a large variety of tridentate ligands featuring ONX-donor sets (X = O, N, and S). [7,[30][31][32][33] Metal complexes prepared with these ligands show diverse structure features, topology, nuclearity as well as supramolecular architectures, depending on the experimental conditions (metal-toligand ratios, supporting anions, solvents, etc). [3,7,[34][35][36] At this stage, it is worthy to emphasize that ONN-tridentate derivatives bearing a pendant free amino group are useful building blocks for the synthesis of unsymmetrically-substituted tetradentate and compartmental Schiff base ligands. [7,37] In the tridentate ONX-type family, albeit much less well developed, quinoline salicylaldimine based ligands [38,39] have been investigated for their versatility towards various transition metal with interesting structural, biological, optical, magnetic and catalytic properties. [40][41][42][43][44][45][46][47][48][49][50][51] The tridentate quinoline Schiff bases act as planar monoanionic ligands incorporating the pyridine moiety and exhibiting an extended p-system. The fused aromatic rings provide are rather rigid skeleton that enforces a meridional configuration at the metal center. The most studied ones result from condensation of aminoquinoline with Accepted Manuscript salicylaldehyde producing variously substituted ((quinolinyl-imino)methyl)phenol derivatives. [38,39] 8-hydroxylquinoline based Schiff bases are less common. 5-amino substituted 8-hydroxylquinoline derivatives were recently used by Cui and co-workers to prepare a series of tetranuclear lanthanide complexes, some of them featuring magneto-caloric effect and slow magnetic relaxation (SMM) behavior. [52,53] In 1972, Hata and Uno reported the synthesis of N-2-(8-hydroxylquinolin-2-ylazomethine)phenol (H2L, Scheme 1) obtained by Schiff condensation of 8-hydroxylquinoline-2-carbaldehyde with 2-aminophenol. [54] Interestingly, this ONN tridentate quinoline Schiff base ligand possesses a pendant hydroxyl group which could, as observed in the salicylidene-(2-hydroxylethyl)imine series, generate weak noncovalent interactions leading to the formation of extended supramolecular structures, [33,55] or engage into coordination with a neighbouring metal center to form onedimensional coordination polymeric chains. [56] In the present work, four new homoleptic metal(II) complexes [M(HL)2] (M = Co, 1; Ni, 2; Cu, 3; Zn, 4) derived from HL ligand, the monodeprotonated form of the diprotic H2L Schiff base precursor, have been synthesized and characterized. The structures of the pro-ligand H2L and of the Ni(II) and Zn(II) complexes have been confirmed by single crystal X-ray crystallographic analysis. Solid-state supramolecular interactions of the three compounds have also been described; Hirshfeld surface (HS) analysis along with 2D fingerprint plots were used to quantify intermolecular forces. Furthermore, magnetic properties of the paramagnetic complexes 1-3 were investigated.

Results and discussion

Synthesis and some properties of ligand precursor H2L and metal(II) complexes 1-4.

The known diprotic Schiff-base ligand N-(8-Hydroxyl-2-quinolylmethylene)-o-aminophenol (H2L) only characterized by its melting point and elemental analysis, [54] was synthesized following a slightly modified procedure. H2L was, indeed, formed upon reacting equimolar amount of 8-hidroxyquinoline-2-carbaldehyde and 2-aminophenol in refluxing toluene for 1 h (Scheme 1). It was isolated as a yellowish powder in 97% crude yield upon cooling the reaction mixture to room temp. Recrystallization from dichloromethane afforded a yellow microcrystalline solid (see Exp. Sect.).

Accepted Manuscript

Scheme 1. Synthesis of the diprotic ligand H2L and its metal(II) complexes 1-4.

The neutral complexes 1-4 were straightforwardly obtained by reacting the diprotic Schiff-base precursor H2L with the appropriate hydrated metal(II) acetate salt (2:1 molar ratio), in refluxing methanol for 1.5 h (Scheme 1). The four metal(II) complexes were obtained in moderate to good yield (60-84%) and isolated as microcrystalline dark solids. The Schiff-base proligand H2L as well as its four respective coordination compounds 1-4 are air and thermally stable, and moisture insensitive on storage under ordinary condition. They exhibit good solubility in common polar organic solvents but are not soluble in diethyl ether and hydrocarbon solvents. Complexes 1-4 having two uncoordinated phenol groups are soluble in buffered water (pH = 9.0, 25 °C). Under more basic conditions (pH = 14, NaOH), they start to decompose after a few minutes. Satisfactory elemental analysis, FT-IR, 1 H and 13 C NMR (only for H2L and 4, see Supporting Information), UV-Vis spectroscopy, and single-crystal X-ray diffraction study (for H2L, 2 and 4) helped to confirm their purity, composition and identity. In addition, complexes 1-4 were characterized by their respective molecular ion peak [M] + or as the sodium aggregate [M+Na] + in their HRMS-ESI (positive mode) mass spectra (see Exp. Sect.).

Spectroscopic characterization

The solid-state FT-IR spectra of complexes 1-4 exhibited similar absorption band patterns suggesting the analogy of their molecular structures. They reveal the disappearance of the quinolinic O-H stretching vibrations observed in the spectra of the ligand precursor H2L at 3413 cm -1 , whereas bands in the range 3367 -3371 cm -1 attributed to the o-phenylene O-H

Accepted Manuscript

stretching vibrations of the free ligand H2L remained. In addition, a bathochromic shift of 27-31 cm -1 of the C=N stretching vibration of the azomethine group found at 1616 cm -1 in the free H2L pro-ligand is observed upon complex formation. Such observations indicate that the chelating Schiff base ligand is bonded to the metal(II) ion in its mono-anionic form and in a tridentate fashion through the O,N,N -donor set (O-Phenolic, N-Pyridine and N-imine). Thus, complexes 1-4 adopt a M(HL)2-type structure where each HL ligand bears a free o-phenylene hydroxyl group.

The electronic absorption spectra in the UV-Vis region of the free H2L and its corresponding complexes 1-4 were recorded in DMSO solution at room temperature. The experimental spectra of H2L, 2 and 4 are shown in Figure 1, while the deconvoluted ones are displayed with those of 1 and 3 in Figures S1 andS2 (Supporting Information), respectively.

The deconvoluted spectral data are gathered in Table 1. The high energy absorption bands at 272 nm (for H2L) and in the 289-304 nm range (for 1-4) may be assigned as intraligand p-p* charge transfer (ILCT) transition, and relatively lower energy absorption peaks at 365 nm (for H2L) and in the 349-355 nm range (for 1-4) may be ascribed to the ligand centered n-p* transition. The observed hypsochromic shift associated with the decrease in intensity of the absorption bands (Table 1) may be attributed to the coordination of the HL -ligand heteroatoms to the metal center. Complexes 1-4 displayed absorptions around 380 nm and shoulder in the 445-498 nm range, which is consistent with ligand to metal charge transfer (LMCT) transitions. Similar results were found for other bis-(tridentate Schiff base) late firstraw transition metal complexes, [33,45,57,58] suggesting that the octahedral geometry around the metal(II) core is preserved in DMSO solution. It was further confirmed with X-ray crystal structure analysis, at least in the case of the Ni(II) derivative 2 (see below). In addition, a weak absorption band due to d-d transition appeared at 602 nm for the Cu(II) complex 3. [35,59] Accepted Manuscript [a] Spectra recorded in DMSO at room temp.

Accepted Manuscript

Description of the X-ray crystal structures

Single crystals suitable for X-ray structure investigation were obtained for the ligand precursor H2L and its Ni(II) and Zn(II) complexes 2 and 4, respectively, by either slow evaporation or vapor diffusion techniques (see Exp. Sect). The molecular structure of H2L is displayed in Figure 2 with selected bond distances and angles given in the caption. Those of 2 and 4 are depicted in Figures 3 and4, respectively. Selected bond lengths and angles of the first metal(II) coordination sphere of 2 and 4 are given in Table 2, while those of the HL - ligands of the two complexes are provided in Table S1 (Supporting Information). The ligand precursor H2L crystallizes in the non-centrosymmetric space group P21, whereas complexes 2 and 4 crystallize in the centrosymmetric space groups P-1 and P21/n, respectively, with in all the three cases one molecule per asymmetric unit. The single-crystal X-ray diffraction studies confirm the formation of the ligand precursor H2L through a Schiff base condensation between 8-hidroxylquinoline-2-carbaldehyde and 2-aminophenol, and that both 2 and 4 are neutral species in which the two tridentated anionic monodeprotonated HL -ligand adopts a meridional coordination mode. Such a "mer" coordination type presumably favors the flatness of the ligand with an extended p-delocalization that would probably be preferred over a more constraint facial arrangement. The other aspects of the structures are discussed below in three sections for the sake of simplicity.

Accepted Manuscript

Ligand precursor H2L. The molecular structure reveals an almost planar Schiff base molecule with a dihedral angle of 10.8° between the planes of the pendant phenol and quinolyl rings (Figure 2). Proligand H2L Adopts an E-conformation, [27] with respect to the C=N double bond. The azomethine C=N bond distance (1.295 (7), Å) is similar to those previously measured for structurally related Ar-C(H)=N-Ar Schiff base compounds. [60][61][62] The The distortion from ideal octahedral coordination geometry at the metal center is markedly noticeable with mean deviations from ideal angles (90° and 180°) of 7.525° and

metric

Accepted Manuscript

19.85°, and with maximum deviations of 14.33 ° and 25.41°, respectively (Table 2). Those significant deviations from the idealized values could be due to the rigidity of the coordinated planar ligands, as previously observed for distorted octahedral Ni(II) complexes featuring a N4O2 donor set. [45,46,57,63] The Ni(II)-N bond distances vary from 1.9657 (15) 2). Such distorted geometry has already been encountered for pentacoordinated Zn(II) species. [64] The Zn-N bond distances of the basal plane (2.005(4) and 2.039(5) Å) are much shorter than the axial one (2.391(5) Å). This is also true for the Zn-O bond lengths (Table 2). Moreover, a short contact (2.722(4) Å) exists between the metal ion and the azomethine nitrogen N(3) of the bidentate ligand (Figure 4). Interestingly, although in the crystal structure of 4 the two ligands are not equivalent, in solution the molecular structure is symmetric as established by [START_REF] Hernandez-Molina | Acyclic and Macrocyclic Schiff Base Ligands[END_REF]

Accepted Manuscript

Solid-state supramolecular interactions

The molecules of H2L are connected through a series of intermolecular hydrogen bonds (Table 3, and Figure S3 (Supporting Information)). The projection of the packing shows an extended parallel-displaced disposition, possibly attributable to the p-stacking interactions between the aromatic systems and the nitrogen atoms (Table 3, and Figure S4 (Supporting Information)) due to an intermolecular close contact involving the two rings (Cg2-Cg3) of the quinolic fragment of each molecule. In addition, this arrangement can be explained by the hydrogen bonds formed beside parallel-displaced molecular-chains (see Figure S3 (Supporting Information)). Table 3. Close contact interactions in H2L and its complexes 2 and 4. [a] 0.84 2.169 2.962 145.3 C(9)-H(9) ---O(1) [a] 0.95 2.440 3.386 174.3 C(3)-H(3) ---O(2) [a] 0.88 2.492 3.439 174.4 C( 14)-H( 14) ---O(2) [a] 0.95 2.492 3.339 174.4 C(4)-H(4) ---O(2) [a] 0.95 2.776 3.370 121.4 C(5)-H(5) ---O(2) [a] 0.95 2.632 3.305 128.3 Complex 2 O(1)-H(1) ---O(4) [a] 0.83 1.790 2.618 175.76 C(6)-H(6) ---O(2) [a] 0.95 2.654 3.319 127.48 C(9)-H(9) ---O(2) [a] 0.95 2.568 3.445 153.72 C(4)-H(4) ---O(1) [a] 0.95 2.643 3.439 141.72 C(3)-H(3) ---N(3) [b] 0 3.946 1.00 Complex 2 Cg3-Cg3 [a] 3.541 0.00 Complex 4 Cg2-Cg3 [a] 3.845 3.18 Cg1-Cg2 [a] 3.748 18.36 On the other hand, within the crystal packing of nickel(II) derivative 2, intermolecular O-H … O hydrogen bonds (Table 3) generate a virtual infinite chain-type structure (Figure S5,

Hydrogen bond calculated by X-ray diffraction Donor-H ---Acceptor D-H(Å) H ---A(Å) D ---A(Å) D-H ---A(°)

Ligand H2L O(1)-H(1) ---O(1)

Accepted Manuscript

Supporting Information). These interactions occur between the hydroxyl hydrogen of each phenolic with the phenolic group of the neighbouring complex. In this regard, the packing of 2 shows the presence of an intramolecular H-bond between C(3)-H (phenolic ring) and the coordinated atom N(1). In addition, the analysis of the intermolecular p-p stacking interactions between Cg3-Cg3 rings (Figure S6, Supporting Information) determines the layout of the molecular network in the crystal lattice (Table 3).

Finally, the analysis of the molecular and crystal structure of the zinc(II) complex 4, reveals the presence of two hydrogen bonds specifically centred in the hydroxyl fragments containing in each molecule (Table 3, and Figure S7 (Supporting Information)). In particular, the presence of an intramolecular H-bond of different nature with respect to the one found for 2. In this case, the analysis reveals a hydrogen bond between the OH fragment of the phenolring and the oxygen coordinated to the metal-core (second HL -ligand of the same complex).

In addition, we detected the presence of two types of p-p stacking interactions between the quinolic and phenolic rings (Figure S8, Supporting Information). These short contact interactions can be responsible of the finale orientation in the crystal lattice, and of the good stability of the resulting single-crystal.

Hirshfeld Surface Analysis (HSA)

The supramolecular interactions observed by X-ray diffraction are, surely, determined by indirect approximations. In this regard, performing HSA is a good way to complement the presence of several intermolecular contacts. This analysis is defined as the electron density boundary surfaces between the molecules in a crystal lattice. [65] During the mapping, the surfaces are kept transparent for the visualization of various supramolecular interactions responsible for the packing stabilization. The HS of H2L mapped in Figure 5 (top) shows the most important supramolecular interaction centred in the -OH unit (red spots), which highlights the close contact through which the intermolecular interaction in the crystal lattice takes place. The colour intensity can be used for visualizing the strength of the interactions. In this regard, the most important interaction is that exhibited by O-H•••O bond between two molecules, over the weaker interactions C-H•••O previously described in Table 3. Information). This important supramolecular interaction provides high thermal and moisture

Accepted Manuscript

Accepted Manuscript

stability (see Exp. Sect.) to the crystal packing, and allows us to project this type of complexes for diverse applications in solid phase. The crucial role of these H-bond interactions can be contrasted by the analysis of the fingerprint plots (Figure 6).

Accepted Manuscript

Magnetic properties

The magnetization, M, of compounds 1, 2 and 3 have been measured 2 and 300 K at two different external magnetic fields: 2 kOe below 20 K and 10 kOe above 20 K. The thermal variation of the magnetic susceptibility is plotted on Figure 7 by means of the cMT product (cM being the molar magnetic susceptibility calculated from the ratio M/H, H the magnetic field in Oe and T the temperature in Kelvin). At room temperature, the cMT values for 2 and 3 are equal to 1.27 and 0.39 cm 3 K mol -1 , respectively. These values are in agreement with Ni(II) (S = 1) and Cu(II) (S = ½) isolated spins with gNi = 2.25 and gCu = 2.04.

On cooling cMT remains constant for 3 which indicates that there is no interaction between magnetic centers. For 2, cMT remains quasi constant down to 25 K. On cooling further cMT decreases continuously down to 0.6 cm 3 K mol -1 which matches the expected value in the presence of negative Zero Field Splitting (axial local anisotropy). [START_REF] Kahn | Molecular Magnetism[END_REF] It also indicates that, like in 3, there is no interaction between magnetic centers. For 1, the room temperature value (2.62 cm 3 K mol -1 ) is far above the expected value for a spin free S = 3/2 with g = 2 (1.875 cm 3 K mol -1 ). However, owing to the presence of spin orbit coupling important deviations are observed for high spin Co(II). [START_REF] Kahn | Molecular Magnetism[END_REF] Also, on cooling, cMT decreases continuously because of the combined effect of the spin orbit coupling and the distortion of the coordination polyhedron, which might be analysed, although inadequate, in a framework of a spin Hamiltonian. [START_REF] Rigamonti | [END_REF] This is, however, beyond the scope of this work. interaction between magnetic centers on cooling. The synthetic versatility and simplicity of the approach, allowing a facile scale-up of the procedure, depicted here suggest application in the chemical design of selective molecular scaffolds for multi-centered binding and reactivity.

Experimental Section

General considerations: Reactions were performed under nitrogen atmosphere using standard Schlenk techniques. The solvents were dried and distilled according to standard procedures. [START_REF] Armarego | Purification of Laboratory Chemicals[END_REF] 8-hidroxy-2-quinolinecarboxaldehyde, 2-aminophenol, nickel(II) acetate tetrahydrate, copper(II) acetate hydrate, cobalt(II) acetate tetrahydrate and zinc(II) acetate dihydrate were purchased from commercial sources and used without further purification.

Infrared spectra were recorded on a Nicolet Magna 550, FT-IR spectrophotometer, as KBr disks in the 4000 to 400 cm -1 range. [START_REF] Hernandez-Molina | Acyclic and Macrocyclic Schiff Base Ligands[END_REF]

Accepted Manuscript

magnetizations for powdered samples have been measured with a SQUID magnetometer design MPMS-XL5) operating between 2 and 300 K at a constant field of 2 kOe below 20 K and 10 kOe above, and diamagnetic corrections were applied by using Pascal's constants. [START_REF] Pascal | [END_REF] Melting points were determined in evacuated capillaries on a Kofler Bristoline melting point apparatus and were not corrected. [54] A three-necked round bottom flask equipped with a Dean-Stark apparatus was loaded with a magnetic stir bar, 8-hidroxy-2quinolinecarboxaldehyde (0.500 g, 2.89 mmol), 2-aminophenol (0.315 g, 2.89 mmol) and 15 mL of toluene. The reaction mixture was refluxed for 1 h. Then, the reaction was cooled at room temp. giving a microcrystalline yellow solid. The compound was collected by filtration, washed with 3 x 5 mL portion of cold MeOH, and dried under vacuum. Slow evaporation (7 days) of a dichloromethane solution yielded yellow single crystals of H2L (0.415 g, 1.57 mmol, 83%). A suitable crystal from this crop was used for X-ray structure determination.

Synthesis of the diprotic Schiff base precursor H2L:

Synthesis of complex Co(HL)2 (1):

A Schlenk tube was charged with a magnetic stir bar, H2L (0.15 g, 0.57 mmol), cobalt(II) acetate tetrahydrate (0.70 g, 0.28 mmol) and 10 mL of MeOH. After 5 min of stirring, the reaction mixture was refluxed for 1.5 h. The reaction was cooled to room temp. giving a dark red precipitate. The solid material was filtered off, washed

Accepted Manuscript

Synthesis of complex Ni(HL)2 (2):

The synthesis of this complex was carried out a procedure similar to that described for complex 1, using, in this case, H2L (0.100 g, 0.38 mmol), nickel(II) acetate tetrahydrate (0.047 g, 0.19 mmol) and 10 mL of MeOH.

Recrystallization from vapor diffusion of diethyl ether into a methanol solution of the compound for 7 days provided red microcrystals of 2 (0.041 g, 67%). A suitable crystal from this crop was used for X-ray structure determination. M.p.

Synthesis of complex Cu(HL)2 (3):

The synthesis of this dark orange complex was carried out following a procedure similar to that described for complex 1, using, in this case, H2L (0.15 g, 0.57 mmol), copper(II) acetate monohydrate (0.058 g, 0.29 mmol) and 10 mL of

Synthesis of complex Zn(HL)2 (4):

The synthesis of this complex was carried out following a procedure similar to that described for complex 1, using, in this case, H2L (0.15 g, 0.57 mmol), zinc(II) acetate dihydrate (0.064 g, 0.29 mmol) and 10 mL of MeOH.

Recrystallization from vapor diffusion of diethyl ether into a chloroform/dichloromethane solution mixture of the compound for 7 days afforded red microcrystals of 4 (0.075 g, 79%).

A suitable crystal from this crop was used for X-ray structure determination. M.p.

Accepted Manuscript

X-ray crystal structure determination: A well-shaped crystal of each compound H2L, 2 and was coated in Paratone-N oil, mounted on a Kaptan loop and transferred to the cold gas stream of the cooling device. Intensity data were collected at 150(2) K on a D8 VENTURE Bruker-AXS diffractometer equipped with a multilayer monochromated Mo-Ka radiation (l = 0.71073 Å) and a CMOS Photon100 detector. The three structures were solved by dualspace algorithm using the SHELXT program, [70] and then refined with full-matrix least-square method based on F 2 (SHELXL-2014). [71] All non-hydrogen atoms were refined with anisotropic atomic displacement parameters. Except oxygen linked hydrogen atoms that were introduced in the structural model through Fourier difference maps analysis, all H atoms were finally included in their calculated positions. In addition, the contribution of the disordered solvents to the calculated structure factors of 2 and 4 were estimated following the BYPASS algorithm, [72] implemented as the SQUEEZE option in PLATON. [73] Then, a new data set, free of solvent contribution, was used in the final refinement. Crystal data and details of data collection and refinement for H2L, 2 and 4 are summarized in Table 4. ORTEP views were drawn using OLEX2 software. [74] CCDC 1856459-1856461 contain the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif Accepted Manuscript

Accepted Manuscript

Hirshfeld Surface (HS) determination: The analyses of the surfaces were mapped using CrystalExplorer, [75] starting from the crystallographic information file (CIF) of H2L, 2 and 4.

The normalized contact distance (dnorm), defined in terms of de, di and the vdW raddi of the atoms, was calculated using the Eq. 1, where the distance from the HS to the nearest nucleus internal (di) or external (de), with respect to the surface involved, and vdW is the van der Waals raddi of atoms, according to the literature. [65] The dimensional (3D) HS maps are generated by dnorm, appears as a red spots over surfaces given a close-contact interactions, responsible of molecular packing in the crystal lattice.

Also, 3D dnorm surfaces can be plotted in 2D fingerprint representations, to quantitatively summarize, the nature and the type of all intermolecular interactions present in each crystal packing. HS for H2L and its complexes 2 and 4, mapped over dnorm (range -0.76 to 2.38Å), are illustrated in Figure 5.

Figure 1 .

 1 Figure 1. Experimental UV-Vis spectra of H2L (top), 2 (middle) and 4 (bottom), recorded in DMSO at room temp.

  parameters observed in the aromatic cycles reveal a strong delocalization throughout the p-system, with average C-C bond lengths of 1.382 Å, 1.395 Å, and 1.398 Å, for the ophenylene, fused C6 and pyridine rings, respectively. In the 8-hydroxylquinolyl fragment, intramolecular hydrogen bonding between the hydroxyl hydrogen and the nitrogen of the pyridinic ring is observed with an O(2)•••N(2) separation of 2.698(7) Å.

Figure 2 .Complex 2 .

 22 Figure 2. Molecular structure of ligand precursor H2L with partial atom labelling scheme. Hydrogen atoms have been omitted for clarity. Thermal ellipsoids are drawn at 40% probability. Selected bond lengths [Å] and angles [°]: O(1)-C(1) 1.351(7), C(1)-C(2) 1.402(8), N(1)-C(2) 1.397(9), N(1)-C(7) 1.295(7), C(7)-C(8) 1.461(9), N(2)-C(8) 1.339(7), N(2)-C(16) 1.354(8), C(15)-C(16) 1.417(8), O(2)-C(15) 1.363(8); O(1)-C(1)-C(2) 120.60(6), N(1)-C(2)-C(1) 114.10(5), C(2)-N(1)-C(7) 120.90(5), N(1)-C(7)-C(8) 121.10(5), N(2)-C(8)-C(7) 114.30(5), N(2)-C(16)-C(15) 116.80(5), C(8)-N(2)-C(16) 117.50(5), O(2)-C(15)-C(16) 119.40(6).

Figure 3 .

 3 Figure 3. Molecular structures of nickel(II) complex 2, showing a partial atom-numbering scheme. Hydrogen atoms are omitted for clarity. Thermal ellipsoids are drawn at 40% probability.

Complex 4 .

 4 The molecular structure of 4 (Figure4), shows that the complex consists of two mono-anionic HL -ligands coordinating to the zinc(II) ion, one in a meridionally tridentate fashion through the O-Phenolic, N-quinolyl and N-imine atoms, and the other one in a bidentate mode through the O,N-donor atoms set of the hydroxyquinolyl fragment, to afford a five-coordinate geometry. Zn(II) is therefore sitting in a N3O2 coordination environment from the two Schiff base ligands that define a distorted trigonal bipyramidal geometry about the metal ion. Thus, the two O-Zn-N chelate bite angles are different: O(2)-Zn(1)-N(2) = 79.15(17)° is similar to the corresponding angle measured in the Ni(II) complex 2 (see Accepted Manuscript above), whereas O(4)-Zn(1)-N(4) of 82.80(2)° is slightly greater. This could arise from the bidentate coordination of the second HL -ligand by releasing of steric strain, highlighting the flexibility of the HL -ligand. In the above mentioned description, The axial sites are occupied by the N(1) and O(2) atoms, while the trigonal plane is defined by the N(2), N(4), and O(4) atoms, with the Zn atom located 0.024(4) Å out of this plane in the direction of O(2) atom. Distortion of the coordination sphere around the Zn(II) metal center from the ideal trigonal bipyramidal geometry, is manifested in: (i) the N(1)-Zn(1)-O(2) angle of 151.28(16)° which reflects the restricted bite angle of the ligand; (ii) the three equatorial angles ranging from 82.80(2)° to 143.00(2)°; and (iii) the axial-Zn(1)-equatorial angles comprised between 73.97(18)° and 113.97(17)° (Table

  H and 13 C NMR spectra showing a single set of signals for the two equivalent ligands (see above and Exp. Sect.). A fast equilibrium on the NMR timescale involving the decoordination/coordination of the two iminic nitrogen atoms does presumably occur in solution.

Figure 4 .

 4 Figure 4. Molecular structures of the Zinc(II) compound 4, showing a partial atom numbering scheme and the intramolecular H-bond O(1)H•••O(4). Hydrogen atoms are omitted for clarity with exception of hydroxilic H atoms. Thermal ellipsoids are drawn at 20% probability.

  [a] neighboring molecule; [b] intramolecular H-bond; where: Cg1 = C1-C6; Cg2 = C7(N8)-C9; Cg3 = C11-C16.

Figure 5 .

 5 Figure 5. Views of the dnorm Hirshfeld surface plotted for H2L (top), 2 (middle), and 4 (bottom). H-bonded are described inTable 3 and text.

Figure 6 . 2 ( 3 .

 623 Figure 6. Decomposed fingerprint plots related with HS areas and contributions (including reciprocal contacts) to the total HS for O-H•••O hydrogen bonds present in the crystal packing of H2L (top), 2 (middle), and 4 (bottom).

Figure 7 .Conclusion

 7 Figure 7. temperature dependences of cMT for the three paramagnetic complexes 1, 2 and 3.

  H and 13 C NMR spectra were recorded on a Bruker Avance 400 Digital Instrument. All NMR spectra are reported in parts per million (ppm, d) relative to tetramethylsilane (Me4Si), with the residual solvent proton and carbon resonances used as internal standards. The following abbreviations are used to describe peak patterns: br, broad; s, singlet; t, triplet and m, multiplet. High resolution electrospray ionization mass spectra (ESI-MS) were conducted on a Bruker MAXI 4G mass spectrometer in positive mode, at the Centre Régional de Mesures Physiques de l'Ouest (CRMPO, Université de Rennes 1, France). Elemental analyses were conducted on a Thermo-Finnigan Flash EA 1112 CHNS/O analyzer by the Microanalytical Service of the CRMPO. The temperature dependences of the

  M.p. 180-183 °C. IR (KBr): ῡ = 3413(s) and 3363(s) (O-H), 3062(w) (C-H aryl), 1616(w), 1576(m), 1508(s) (C … N) and (C … C), 1465(s) (O-H), 1284(w) (C-O), 752(s) (C-H) cm -1 . 1 H NMR (400 MHz, CD3OD; 315 K): δ = 10.21 (br s, 1 H, OH), 9.00 (s, 1 H, N=CH), 8.42 (t, 1 H, C3H2N), 8.36 (t, 1 H, C3H2N), 8.03 (br s, 1 H, OH), 7.52 (t, 1 H, C6H4), 7.42 (m, 1 H, C6H3), 7.40 (m, 1 H, C6H3), 7.18 (m, 2 H, C6H4), 6.96 (m, 2 H, C6H4 + C6H3) ppm. 13 C{ 1 H} NMR (101 MHz, (CD3)2SO, 298 K): δ = 155.9 (N=CH), 155.0 (Cquat, C3H2N-CH=), 153.9 ((Cquat, C6H3-OH), 153.3 (Cquat, C6H4-OH), 139.8 (Cquat, C6H4-N), 138.1(Cquat, C6H3), 137.6 (CH, C3H2N), 131.0 (Cquat, C6H3), 129.7 (CH, C6H4), 129.6 (CH, C6H4), 120.8 (CH, C6H4), 120.7 (2 CH, C6H4 + C6H3), 119.8 (CH, C6H3) ppm. C16H12N2O2 calcd. C 72.72, H 4.58, N 10.60; found C 72.56, H 4.44, N 10.27.

with 3 x 3

 3 mL portion of cold MeOH, diethyl ether (2 x 3 mL portion), and dried under vacuum for 2 h, to afford 1 (0.76 g, 84%). M.p. 170-172 °C. IR (KBr): ῡ = 3399 (br,m) (O-H), 3050(w) (C-H aryl), 1587(m) and 1498(w) (C … N) and (C … C), 1448(s) (O-H), 1277(w) (C-O), 754(s) (C-H) cm -1 . ESI-MS [m/z], calcd. C32H22N4O4NaCo: 608.08707, found: 608.0866 (0 ppm) [M+Na] + . C32H22N4CoO4•2H2O calcd. C 61.71, H 4.24, N 9.01; found C 61.98, H 3.81, N, 8.81.

  168-170 °C. IR (KBr): ῡ = 3371(m) (O-H), 3049(w) (C-H aryl), 1585(m) and 1498(w) (C … N) and v(C … C), 1448(s) (O-H), 1269(w) (C-O), 750(s) (C-H) cm -1 . ESI-MS (m/z), calcd for C32H23N4O4 58 Ni: 585.10672, found: 585.1069 (0 ppm) [M+H] + . C32H22N4NiO4•H2O calcd. C 63.71, H 4.01, N 9.29; found C 63.69, H 3.95, N 9.18.

  MeOH. (0.056 g, 60%). M.p. 168-170 °C. IR (KBr): ῡ = 3367(w) (O-H), 3028(w) (C-H aryl), 1589(w) and 1552(m) (C … N) and (C … C), 1450(s) (O-H), 1238(m) (C-O), 744(s) (C-H) cm -1 . ESI-MS [m/z], calcd. C32H23N4O4 63 Cu: 590.10153, found: 590.1012 (0 ppm) [M+H] + . C32H22N4CuO4 calcd. C 65.13, H 3.76, N 9.49; found C 64.93, H 3.70, N 9.53.

  168-170 °C. IR (KBr): ῡ = 3375(w) (O-H), 3055(w) (C-H aryl), 1589(m) and 1494(m) (C … N) and (C … C), 1448(s) (O-H), 1278(m) (C-O), 750(s) (C-H) cm -1 . 1 H NMR (400 MHz, (CD3)2SO, 298 K): δ = 8.70 (s, 2 H, OH), 8.57 (m, 2 H, C3H2N), 7.65 (m, 4 H, C3H2N + C6H3), 6.62 (m, 2 H, C6H4), 6.48 (s, 2 H, CH=N), 6.26 (m, 2 H, C6H4), 6.20 (m, 2 H, C6H4), 6.00 (m, 4 H, C6H4 + C6H3), 5.96 (m, 2 H, C6H3) ppm. 13 C{ 1 H] NMR (100 MHz, (CD3)2SO, 298 K): δ = 162.6 (C-O-Zn), 158.1 (CH=N), 152.3 (C-OH), 149.1 (CN, N-C6H4-2-OH), 139.5 (Cquat, C6H3), 138.5 (CH, C6H3), 136.7 (Cquat, C3H2N), 130.3 (Cquat, C6H3), 131.1, 128.7 (CH, N-C6H4-2-OH), 119.7 (CH, C3H2N), 119.2 (CH, C6H3), 118.6 (CH, C(H)=N), 116.3 (CH, C6H3), 112.5, 110.0 (CH, N-C6H4-2-OH) ppm. ESI-MS [m/z], calcd. C32H22N4O4Na 64 Zn: 613.08302, found: 613.0830 (1 ppm) [M+Na] + . C32H22N4O4Zn•0.75H2O calcd. C 63.47, H 3.92, N 9.26; found C 63.58, H 3.70, N 8.75.

  of fit / F 2

Table 1 .

 1 Electronic spectral data for compounds

H2L and 1-4 [a]

	Compound	Band	l/nm (log e)
	H2L	A	272 (4.44)
		B	365 (4.13)
		A	296 (4.36)
	1	B C	349 (3.96) 378 (3.97)
		D	463 (3.72)
		A	304 (4.30)
	2	B	355 (3.96)
		C	383 (3.81)
		D	498 (3.46)
		A	289 (4.35)
		B	351 (3.80)
	3	C	383 (4.02)
		D	472 (3.56)
		E	602 (2.87)
		A	296 (4.34)
	4	B C	350 (3.85) 380 (3.88)
		D	445 (3.62)

Table 2 .

 2 Selected bond distances[Å] and angles[°] for the first metal(II) coordination sphere of compounds 2 and 4.

	to 2.2393(15) Å

stacking interactions determine by X-ray diffraction

		.95	2.636	3.478	147.93
		Complex 4
	O(3)-H(3) ---O(2) a	0.84	1.769	2.600	170.16
	O(1)-H(1) ---O(4) b	0.84	1.745	2.558	162.24
	p-p Centroid-Centroid	Distance(Å)	angle(º)
		Ligand H2L
	Cg2-Cg3 [a]

Table 4 .

 4 Crystallographic data, details of data collection and structure refinement parameters for compounds H2L, 2 and 4.

		H2L	2	4
	Empirical Formula	C16H12N2O2	C32H22N4NiO4	C32H22N4O4Zn
	Formula mass, g mol -1	264.28	585.24	591.90
	Collection T, K	150(2)	150(2)	150(2)
	Crystal size (mm)	0.58 x 0.09 x 0.04	0.31 x 0.16 x 0.08	0.08 x 0.05 x 0.015
	Crystal color	Yellow	Black	Light brown
	crystal system	Monoclinic	Triclinic	Monoclinic
	space group	P21	P-1	P21/n
	a (Å)	11.930(5)	9.8744(10)	11.611(4)
	b (Å)	4.4633(16)	11.4788(13)	15.595(5)
	c (Å)	12.352(6)	14.6534(17)	16.306(5)
	α (°)	90	111.161(4)	90
	b (°)	106.526(17)	101.541(4)	107.960(10)
	γ (°)	90	97.567(4)	90
	V (Å 3 )	630.5(5)	1479.3(3)	2808.6(16)
	Z	2	2	4
	Dcalcd (g cm -3 )	1.392	1.314	1.400
	F(000)	276	604	1216
	abs coeff (mm -1 )	0.094	0.698	0.919
	q range (°)	3.395 to 27.380	2.924 to 27.482	2.924 to 27.483
	range h,k,l	-15/15, -5/5, -15/15 -12/11, -14/14, -19/18 -14/15, -20/18, -21/21
	No. total refl.	4548	31658	28478
	No. unique refl.	2723	6764	6414
	Comp. qmax (%)	98.5	99.9	99.7
	Max/min transmission	0.996/0.840	0.946/0.838	0.986/0.829
	Data/Restraints/Parameters	2723/2/187	6764/3/376	6414/0/372
	Final R
	[I>2s(I)]

Acknowledgements

The authors thank Drs. F. Lambert and P. Jehan (CRMPO, Rennes) for helpful assistance with HR-MS measurements. Financial support from the Fondo Nacional de Desarrollo Científico y Tecnológico [FONDECYT (Chile), grant No. 11160658 (IR: N.N.)] and the Vicerrectoría de Investigación y Desarrollo, Universidad de Concepción (Chile), grant VRID Iniciación No. 216.021.034-10-1.0IN (IR: N.N.), the Centre National de la Recherche Scientifique (CNRS), the Université de Rennes 1, and the Chilean-French International Associated Laboratory for "Multifunctional Molecules and Materials'' (LIAM3-CNRS N°1207) is gratefully acknowledged.