Takéo Takahashi
email: takeo.takahashi@inria.fr

Luz De Teresa
email: ldeteresa@im.unam.mx.

Ying Wu-Zhang

Controllability results for cascade systems of m coupled N -dimensional Stokes and Navier-Stokes systems by N -1 scalar controls

Keywords: Null controllability, Navier-Stokes systems, Carleman estimates 2010 Mathematics Subject Classification. 76D05, 35Q30, 93B05, 93B07, 93C10

main results 4.1 Final state observability . . . . . . . . . .

Introduction

Controllability issues related to a single parabolic equation or to a single Stokes or Navier-Stokes system have been intensively studied in the last fifty years giving rise to interesting techniques, new challenges and open problems. See some seminal results [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF][START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Lebeau | Contrôle exacte de l'équation de la chaleur[END_REF] for the heat equation and [START_REF] Coron | Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary[END_REF][START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF][START_REF] Imanuvilov | On exact controllability for the Navier-Stokes equations[END_REF] for the Navier-Stokes system. The literature is vast and it is difficult to mention all the intensive studies about this subject. However, it is only in the last fifteen years that the challenging issue of controlling coupled parabolic systems has attracted the interest of the control community. This kind of systems appears mathematically in optimal control theory as a characterization of the optimal control (with one equation coupled to its adjoint) but also appears, for example, in the study of chemical reactions (see e.g. [START_REF] Érdi | Mathematical models of chemical reactions[END_REF], [START_REF] Conforto | Some asymptotic limits of reaction-diffusion systems appearing in reversible chemistry[END_REF]), and in a wide variety of mathematical biology and physical situations (see e.g. [START_REF] Iida | Vanishing, moving and immovable interfaces in fast reaction limits[END_REF] ). In the case of scalar (heat) coupled equations an important number of challenging problems has been solved (see [START_REF] Ammar-Khodja | Recent results on the controllability of linear coupled parabolic problems: a survey[END_REF] for a survey of results until 2011) and sometimes the results have been surprising [START_REF] Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF][START_REF] Khodja | New phenomena for the null controllability of parabolic systems: minimal time and geometrical dependence[END_REF][START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF]. In the case of coupled Stokes or Navier-Stokes systems, to our knowledge, only some cases of two coupled systems have been treated [START_REF] Carreño | Insensitizing controls with two vanishing components for the threedimensional Boussinesq system[END_REF][START_REF] Carreño | Insensitizing controls with one vanishing component for the Navier-Stokes system[END_REF][START_REF] Guerrero | Controllability of systems of Stokes equations with one control force: existence of insensitizing controls[END_REF][START_REF] Montoya | Robust Stackelberg controllability for the Navier-Stokes equations[END_REF]. Here our aim is to generalize results for a m scalar cascade system [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF] to a m N -dimensional Stokes or Navier-Stokes cascade system but including an extra deal: to eliminate one component on the N -dimensional control.

Let us be more specific: we consider a bounded domain Ω of R N (N = 2, 3) whose boundary ∂Ω is regular enough. Let T > 0 and let ω ⊂ Ω be a (arbitrary small) nonempty open subset which will usually be referred as the control domain. We will use the notation Q = Ω × (0, T ) and Σ = ∂Ω × (0, T ).

In this article, we are interested in the null controllability of a coupled system of m Stokes or Navier-Stokes systems, with m 2:

           ∂ t y (i) -∆y (i) + ∇p (i) + ε y (i) • ∇ y (i) = m j=1 A i,j y (j) + D i v1 ω in Q, (1 i m) ∇ • y (i) = 0 in Q, (1 i m) y (i) = 0 on Σ, (1 i m) y (i) (•, 0) = y (i) 0 in Ω, (1 i m)
with ε = 0 for Stokes systems and ε = 1 for Navier-Stokes systems, where A i,j ∈ M N (R) and where D i ∈ M N,r (R) for some r ∈ N * . We have denoted by 1 ω the characteristic function of ω. Let us notice that we take in this work the viscosities of the fluids constant and equal to 1 to simplify.

We can write the above systems in a more compact way as

         ∂ t y -∆y + ∇p = Ay + Dv1 ω in Q, ∇ • y = 0 in Q, y = 0 on Σ, y| t=0 = y 0 in Ω, (1.1)
or

         ∂ t y -∆y + (y • ∇)y + ∇p = Ay + Dv1 ω in Q, ∇ • y = 0 in Q, y = 0 on Σ, y| t=0 = y 0 in Ω, (1.2)
where we set y = y (1) , . . . , y (m) , p = p (1) , . . . , p (m) , y 0 = y , ∆y = ∆y (1) , . . . , ∆y (m) , ∇p = ∇p (1) , . . . , ∇p (m) , (y • ∇) y = y (1) • ∇ y (1) , . . . , y (m) • ∇ y (m) , ∇ • y = ∇ • y (1) , . . . , ∇ • y (m) and

Ay = m j=1
A 1,j y (j) , . . . ,

m j=1
A m,j y (j) Dv = D 1 v, . . . , D m v .

In this work, we will focus in the particular case where the partitioned matrix A has the form

A =        A 1,1 A 1,2 A 1,3 . . . A 1,m A 2,1 A 2,2 A 2,3 . . . A 2,m 0 A 3,2 A 3,3 . . . A 3,m . . . . . . . . . . . . . . . 0 0 . . . A m,m-1 A m,m        (1.3)
with all the blocks under the diagonal non zero and such that all its blocks are scalar matrices. More precisely, our hypotheses on A are

A i,j = a i,j I N , a i,i-1 = 0 (2 i m), a i,j = 0 if i j + 2. (1.4)
We also control (1.1) or (1.2) by acting only on one of the Stokes or of the Navier-Stokes systems, for instance the first one, and with N -1 scalar controls on this system. Thus, without loss of generality, we assume

r = N -1, D j = 0 (j 2), D 1 = 1 0 (if N = 2) or D 1 =   1 0 0 1 0 0   (if N = 3). (1.5)
The above choice of the matrix A corresponds to a particular coupling considered in the context of the nullcontrollability of systems of m linear heat equations, see [START_REF] González-Burgos | Controllability results for cascade systems of m coupled parabolic PDEs by one control force[END_REF] and our aim is to extend this result in the case of coupled of Stokes or Navier-Stokes systems.

In order to state our main results, we recall some standard functional spaces associated with the Stokes system:

H = y ∈ L 2 (Ω) N : ∇ • y = 0 in Ω, y • n = 0 on ∂Ω (1.6) and V = y ∈ H 1 0 (Ω) N : ∇ • y = 0 in Ω . (1.7)
Our main result is the following theorem.

Theorem 1.1. Assume (1.4) and (1.5). Then, for any T > 0 and for any y 0 ∈ H m , there exists a control v ∈ L 2 (0, T ; L 2 (ω) N -1 ) such that the corresponding solution y = y (1) , ..., y (m) to (1.1) satisfies y(•, T ) = 0 in Ω.

Remark 1.2. As a consequence, we deduce that we can control the system (1.1) of N × m scalar equations with N -1 scalar controls.

From Theorem 1.1 and a general method to deal with the controllability of nonlinear parabolic systems, we deduce the local null controllability of the system (1.2): Theorem 1.3. Assume (1.4) and (1.5). Then, for any T > 0, there exists δ > 0 such that, for any

y 0 ∈ V m satisfying y 0 V m δ, there exists v ∈ L 2 (0, T ; L 2 (ω) N -1 ) such that the corresponding solution y to (1.2) satisfies y(•, T ) = 0 in Ω.
In order to prove Theorem 1.1, we introduce the adjoint system:

                     -∂ t ϕ (i) -∆ϕ (i) + ∇π (i) = i+1 j=1 A j,i ϕ (j) in Q, (1 i m -1) -∂ t ϕ (m) -∆ϕ (m) + ∇π (m) = m j=1 A j,m ϕ (j) in Q, ∇ • ϕ (i) = 0 in Q, (1 i m) ϕ (i) = 0 on Σ, (1 i m) ϕ (i) (•, T ) = ϕ (i) T in Ω, (1 i m) (1.8) with ϕ (i)
T ∈ H (1 i m) and we denote by ϕ (i) j , j = 1, . . . , N the coordinates of ϕ (i) . Note that by setting ϕ (m+1) ≡ 0, a m+1,i = 0 (1 i m), we can write the first above equations as

-∂ t ϕ (i) -∆ϕ (i) + ∇π (i) = i+1 j=1 a j,i ϕ (j) in Q, (1 i m).
(1.9)

Following a standard duality argument (see, for instance, [34, Theorem 11.2.1, p.357]), Theorem 1.1 will be obtained as a consequence of the following observability inequality:

Ω m i=1 ϕ (i) (x, 0) 2 dx C(T ) ω×(0,T ) N -1 j=1 ϕ (1) j 2 dxdt. (1.10)
for some C depending on T , Ω and ω.

The proof of (1.10) is based on (global) Carleman inequalities introduced in [START_REF] Fursikov | Controllability of evolution equations[END_REF] for the controllability of parabolic equations. Such inequalities have been used by many authors to deal with Stokes of Navier-Stokes systems (for instance, [START_REF] Imanuvilov | Remarks on exact controllability for the Navier-Stokes equations[END_REF] or [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF]). The case of controls with some vanishing components was considered in [START_REF] Fernández-Cara | Some controllability results for the Ndimensional Navier-Stokes and Boussinesq systems with N -1 scalar controls[END_REF], [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF] and [START_REF] Carreño | Insensitizing controls with one vanishing component for the Navier-Stokes system[END_REF]. We follow here the method introduced in [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF]. As a first step, one can get rid of the pressure by applying a differential operator on (1.9) (or on components of (1.9)) such as curl or ∆. This leads to a system of coupled heat equations but without prescribed boundary conditions. Using results such as [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF] or [START_REF] Imanuvilov | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF], one can obtain a Carleman estimate with boundary terms that can be absorbed by some standard arguments. Let us note that to recover the observability on the components that are not observed, one has to use the divergence-free condition on ϕ (i) and the Dirichlet boundary conditions. Unhappily in this process, one loses part of the weights on these components. The last part of the proof of (1.10) consists in estimating the local terms associated with ϕ (i) j , i > 1 and this is done by using (1.4). An important work related to this subject is [START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF] where the authors obtain the local null controllability of the Navier-Stokes system in dimension 3 with a control having two vanishing components. In that case, the method is based on a different linearization and on a different approach based on results of Gromov.

This paper is organized as follows. In Section 2, we introduce the weights for our Carleman estimates and we recall some results, in particular some Carleman estimates for other systems. Section 3 corresponds to the statement and to the proof of the Carleman estimate for (1.8). Finally, in Section 4, we use this estimate to prove Theorem 1.1 and Theorem 1.3.

Preliminaries

In the whole article, we use the notation C for a generic positive constant that depends on Ω, ω. We can assume to simplify that T ∈ (0, 1) and that will allow us to avoid some dependence on T for some constants. Finally we only write the proof in the case N = 2, the case N = 3 can be done similarly.

To write our Carleman inequalities, we introduce standard weights and functions. First, we consider a nonempty domain ω 0 such that ω 0 ⊂ ω. Then, using [START_REF] Fursikov | Controllability of evolution equations[END_REF] (see also [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Theorem 9.4.3,p.299]), there exists

η 0 ∈ C 2 (Ω) satisfying η 0 > 0 in Ω, η 0 = 0 on ∂Ω, max Ω η 0 = 1, ∇η 0 = 0 in Ω \ ω 0 . (2.1)
Then, we define the following functions:

α(x, t) = exp {λ(2 + 2)} -exp{λ(2 + η 0 (x))} t (T -t) , ξ(x, t) = exp{λ(2 + η 0 (x))} t (T -t) , (2.2)
α * (t) = max x∈Ω α(x, t) = exp {λ(2 + 2)} -exp{2λ } t (T -t) , ξ * (t) = min x∈Ω ξ(x, t) = exp{2λ } t (T -t) , (2.3)
where 14, λ > 1. In literature > 3 is enough. However, to prove our result we need to use 14 (see Proof of Proposition 3.2).

Note that we have the following useful relations: there exists C > 0 depending on Ω such that

|∂ t α| + |∂ t ξ| CT ξ 1+1/ , (2.4)
(α * ) + (ξ * ) CT (ξ * ) 1+1/ , (α * ) + (ξ * ) CT 2 (ξ * ) 1+2/ , (2.5)
ξ * C T 2 , (2.6
)

|∇α| = |∇ξ| Cλξ, |∆α| = |∆ξ| Cλ 2 ξ. (2.7)
Weights of the kind (2.2) were first considered in [START_REF] Fursikov | Controllability of evolution equations[END_REF]. In its present form, these weights have already been used in [START_REF] Guerrero | Null controllability of some systems of two parabolic equations with one control force[END_REF] in order to obtain Carleman estimates for the controllability of strongly coupled parabolic equations and later in [START_REF] Guerrero | Controllability of systems of Stokes equations with one control force: existence of insensitizing controls[END_REF] for the existence of insensitizing controls for Stokes systems. Now, we recall some standard results. The first one is a Carleman estimate for the gradient operator, it is stated and proved in [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF]: Lemma 2.1. Let r ∈ R. There exists C > 0 depending only on r, Ω and ω 0 such that, for every T > 0, s C and every u ∈ L 2 (0, T ;

H 1 (Ω)), Q e -2sα s r+2 ξ r+2 |u| 2 dxdt C    Q e -2sα s r ξ r |∇u| 2 dxdt + ω0×(0,T ) e -2sα s r+2 ξ r+2 |u| 2 dxdt.    .
The second result is a Carleman estimate for the heat equation with non-homogeneous Dirichlet boundary conditions. It is proved in [START_REF] Imanuvilov | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF].

Lemma 2.2. There exists a constant

C > 0 such that for any λ C, s C, f 0 , f 1 , ..., f N ∈ L 2 (Q) and u ∈ L 2 0, T ; H 1 (Ω) ∩ H 1 0, T ; H -1 (Ω) satisfying ∂ t u -∆u = f 0 + N j=1 ∂ j f j in Q, we have Q e -2sα s -1 ξ -1 |∇u| 2 + sξ |u| 2 dxdt C ω0×(0,T ) e -2sα sξ |u| 2 dxdt + e -sα * s -1/4 (ξ * ) -1/4+1/ u 2 L 2 (Σ) + e -sα * s -1/4 (ξ * ) -1/4 u 2 H 1 4 , 1
2 (Σ) + Q e -2sα s -2 ξ -2 |f 0 | 2 dxdt + N j=1 Q e -2sα |f j | 2 dxdt   . (2.8) Recall that u H 1 4 , 1
2 (Σ) = u 2 H 1/4 (0,T ;L 2 (∂Ω)) + u 2 L 2 (0,T ;H 1/2 (∂Ω)) 1/2
.

The next result concerns the regularity of the solutions of the Stokes system

         ∂ t u -∆u + ∇p = f in Q, ∇ • u = 0 in Q, u = 0 on Σ, u(•, 0) = 0 in Ω.
(2.9)

This result can be found in [START_REF] Temam | Behaviour at time t = 0 of the solutions of semilinear evolution equations[END_REF].

Lemma 2.3. Assume T > 0. For any f ∈ L 2 (Q) N , there exists a unique solution u ∈ L 2 (0, T ; H 2 (Ω) N ) ∩ H 1 (0, T ; H), p ∈ L 2 (0, T ; H 1 (Ω)/R)
to the Stokes system (2.9) and there exists a constant C > 0 depending only on Ω such that

u 2 L 2 (0,T ;H 2 (Ω) N ) + u 2 H 1 (0,T ;L 2 (Ω) N ) C f 2 L 2 (0,T ;L 2 (Ω) N ) .
(2.10)

Assume moreover that f ∈ L 2 (0, T ; V ) then u ∈ L 2 (0, T ; H 3 (Ω) N ) ∩ H 1 (0, T ; V )
and there exists a constant C > 0 depending only on Ω such that

u 2 L 2 (0,T ;H 3 (Ω) N ) + u 2 H 1 (0,T ;V ) C f 2 L 2 (0,T ;V ) . (2.11) Assume moreover that f ∈ L 2 0, T ; H 3 (Ω) N ∩ H 1 (0, T ; V ) , f (•, 0) = 0 in Ω. Then, u ∈ L 2 (0, T ; H 5 (Ω) N ) ∩ H 1 (0, T ; H 3 (Ω) N ) ∩ H 2 (0, T ; V ) and there exists a constant C > 0 depending only on Ω such that u 2 L 2 (0,T ;H 5 (Ω) N ) + u 2 H 1 (0,T ;H 3 (Ω) N ) + u 2 H 2 (0,T ;V ) C f 2 L 2 (0,T ;H 3 (Ω) N ) + ∂ t f 2 L 2 (0,T ;V ) .
3 The Carleman estimate for the adjoint system

As before, we denote by C various positive constants which depends only on Ω and ω (they depend also in general on the choice of η 0 and ω 0 but one can consider that η 0 as well as ω 0 depend Ω and ω). Without any lack of generality, we treat the case of dimension N = 2 and j = 2. The same proof can be performed in the general case.

Our aim is to estimate the following quantity associated with the solutions of the system (1.8):

I 1 (s, ϕ (i) ) := Q e -2sα s -1 ξ -1 ∇ 3 ∆ϕ (i) 1 2 + sξ ∇ 2 ∆ϕ (i) 1 2 + s 3 ξ 3 ∇∆ϕ (i) 1 2 + s 5 ξ 5 ∆ϕ (i) 1 2 dxdt + Q e -2sα * (sξ * ) 5 ϕ (i) 2 dxdt. (3.1)
Thus, our main result states as follows:

Theorem 3.1. There exists C > 0 depending on the geometry such that for any s C and for any ϕ T ∈ H m , the solution (ϕ (1) , ..., ϕ (m) ) of (1.8) satisfies

m i=1 I 1 (s, ϕ (i) ) C ω×(0,T ) e -2sα s 2 m+3 -6 ξ 2 m+3 -6 ϕ (1) 1 2 dxdt. (3.2)
In order to prove the above proposition, we first start by estimating each I 1 (s, ϕ (i) ) (i = 1, . . . , m) independently.

Proposition 3.2. Let ω ⊂ Ω be a nonempty open set such that ω 0 ω. Then, there exists a constant C such that for any s C and for any ϕ T ∈ H m , the solution ϕ of (1.8) satisfies

I 1 (s, ϕ (i) ) C ω×(0,T ) e -2sα s 5 ξ 5 ∆ϕ (i) 1 2 dxdt + m j=1 Q e -2sα * s 9/2 (ξ * ) 5 ϕ (j) 2 + e -2sα s -2 ξ -2 ∇ 2 ∆ϕ (j) 1 2 dxdt   (1 i m). (3.3)
Proof of Proposition 3.2. First taking the divergence of (1.9), we remark that

∆π (i) = 0 in Q (1 i m).
Thus, using a standard method for the Stokes system (see, for instance [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF]), we apply the operator ∇ 2 ∆ to the first components of (1.9), and we deduce

-∂ t ∇ 2 ∆ϕ (i) 1 -∆∇ 2 ∆ϕ (i) 1 = i+1 j=1 a j,i ∇ 2 ∆ϕ (j) 1 in Q, (1 i m). (3.4)
Applying Lemma 2.2 to the above equations, we deduce that for λ λ 1 and for s s 1 ,

Q e -2sα s -1 ξ -1 ∇ 3 ∆ϕ (i) 1 2 + sξ ∇ 2 ∆ϕ (i) 1 2 dxdt C ω0×(0,T ) e -2sα sξ ∇ 2 ∆ϕ (i) 1 2 dxdt + e -sα * s -1/4 (ξ * ) -1/4+1/ ∇ 2 ∆ϕ (i) 1 2 L 2 (Σ) 4 + e -sα * (sξ * ) -1/4 ∇ 2 ∆ϕ (i) 1 2 H 1 4 , 1
2 (Σ) 4 + i+1 j=1 Q e -2sα s -2 ξ -2 ∇ 2 ∆ϕ (j) 1 2 dxdt   . (3.5)
The rest of the proof is divided into several steps:

• In Step 1, we complete the left-hand side of (3.5) with weighted integrals of ϕ (i) in Q, and adding some local terms in the right-hand side.

• In Step 2, we obtain an upper bound of the boundary terms.

• Finally, in Step 3, we estimate the local terms that do not appear in (3.3).

Step 1. We apply Lemma 2.1 with u = ∇∆ϕ (i)

1 and r = 1: for any s C and λ C,

Q e -2sα s 3 ξ 3 ∇∆ϕ (i) 1 2 dxdt C    Q e -2sα sξ ∇ 2 ∆ϕ (i) 1 2 dxdt + ω0×(0,T ) e -2sα s 3 ξ 3 ∇∆ϕ (i) 1 2 dxdt    . (3.6)
Then, we apply Lemma 2.1 with u = ∆ϕ

(i) 1 2 dxdt C    Q e -2sα s 3 ξ 3 ∇∆ϕ (i) 1 2 dxdt + ω0×(0,T ) e -2sα s 5 ξ 5 ∆ϕ (i) 1 2 dxdt    . (3.7)
Now, using the divergence condition of ϕ (i) , we have

∂ 2 ϕ (i) 2 = ∂ 1 ϕ (i) 1 ∇ϕ (i) 1
.

Then using the Poincaré inequality and the ellipticity of the Laplace operator with Dirichlet boundary conditions, we deduce the existence of a constant C depending on Ω such that

Ω ϕ (i) 2 dx C Ω ∆ϕ (i) 1 2 dx.
Combining the above relation with (3.5), (3.6) and (3.7), we deduce that I 1 (s, ϕ (i) ) defined by (3.1) satisfies

I 1 (s, ϕ (i) ) C ω0×(0,T ) e -2sα sξ ∇ 2 ∆ϕ (i) 1 2 + s 3 ξ 3 ∇∆ϕ (i) 1 2 + s 5 ξ 5 ∆ϕ (i) 1 2 dxdt + e -sα * s -1/4 (ξ * ) -1/4+1/ ∇ 2 ∆ϕ (i) 1 2 L 2 (Σ) 4 + e -sα * (sξ * ) -1/4 ∇ 2 ∆ϕ (i) 1 2 H 1 4 , 1
2 (Σ) 4 + i+1 j=1 Q e -2sα s -2 ξ -2 ∇ 2 ∆ϕ (j) 1 2 dxdt   . (3.8)
Step 2. In this step, we get rid of the boundary terms in the right-hand side of (3.8). To estimate the first term, we notice that since 4, (ξ * ) -1/4+1/ is bounded in (0, T ) (see (2.6)). Thus

e -sα * s -1/4 (ξ * ) -1/4+1/ ∇ 2 ∆ϕ (i) 1 2 L 2 (Σ) 4 Cs -1/2 e -sα * ∇ 2 ∆ϕ (i) 1 2 L 2 (Σ) 4 Cs -1/2 e -sα * ∇ 2 ∆ϕ (i) 1 2 L 2 (Q) 4 + s 1/2 e -sα * (ξ * ) 1/2 ∇ 2 ∆ϕ (i) 1 L 2 (Q) 4 s -1/2 e -sα * (ξ * ) -1/2 ∇ 3 ∆ϕ (i) 1 L 2 (Q) 8 Cs -1/2 Q e -2sα * sξ ∇ 2 ∆ϕ (i) 1 2 + s -1 ξ -1 ∇ 3 ∆ϕ (i) 1
2
dxdt Cs -1/2 I 1 (s, ϕ (i) ). (3.9)

For the second boundary term, we will use a trace inequality (see, for instance, [27, Theorem 2.1, p.9]) and by an interpolation argument (see, for instance, [26, Corollary 9.2, p.46]), we have

(ξ * ) -1/4 e -sα * ∇ 2 ∆ϕ (i) 1 2 H 1 4 , 1 2 (Σ) 4 C (ξ * ) -1/4 e -sα * ∇ 2 ∆ϕ (i) 1 2 L 2 (0,T ;H 1 (Ω) 4 )∩H 1/2 (0,T ;L 2 (Ω) 4 ) C (ξ * ) -1/4 e -sα * ∇ 2 ∆ϕ (i) 1 2 L 2 (0,T ;H 1 (Ω) 4 )∩H 1 (0,T ;H -1 (Ω) 4 )
. (3.10)

Our goal is to estimate the last term. First we consider the function

θ 1 (t) := s 3/2 (ξ * ) 3/2-1/ e -sα * . From (2.5)-(2.6), for s C, |θ 1 | Cs 5/2 (ξ * ) 5/2 e -sα * . (3.11) Then, from (1.8), (θ 1 ϕ (i) , θ 1 π (i) ) is the solution of              -∂ t θ 1 ϕ (i) -∆ θ 1 ϕ (i) + ∇ θ 1 π (i) = i+1 j=1 a j,i θ 1 ϕ (j) -θ 1 ϕ (i) in Q, ∇ • θ 1 ϕ (i) = 0 in Q, θ 1 ϕ (i) = 0 on Σ, θ 1 ϕ (i) | t=T = 0 in Ω.
(3.12)

Applying Lemma 2.3 to the above system and using (3.11), we obtain

θ 1 ϕ (i) 2 L 2 (0,T ;H 2 (Ω) 2 )∩H 1 (0,T ;L 2 (Ω) 2 ) C   i+1 j=1 s 3/2 (ξ * ) 3/2 e -sα * ϕ (j) 2 L 2 (Q) 2 + s 5/2 (ξ * ) 5/2 e -sα * ϕ (i) 2 L 2 (Q) 2   .
Now, combining the above estimate with an interpolation inequality, we deduce

s 2 (ξ * ) 2-1/(2 ) e -sα * ϕ (i) 2 L 2 (0,T ;H 1 (Ω) 2 ) C θ 1 ϕ (i) L 2 (0,T ;H 2 (Ω) 2 ) s 5/2 (ξ * ) 5/2 e -sα * ϕ (i) L 2 (Q) 2 C   i+1 j=1 s 3/2 (ξ * ) 3/2 e -sα * ϕ (j) 2 L 2 (Q) 2 + s 5/2 (ξ * ) 5/2 e -sα * ϕ (i) 2 L 2 (Q) 2   . (3.13)
Second, we introduce

θ 2 = s(ξ * ) 1-3/(2 ) e -sα * . From (2.5)-(2.6), for s C, |θ 2 | Cs 2 (ξ * ) 2-1/(2 ) e -sα * . (3.14) Then, from (1.8), (θ 2 ϕ (i) , θ 2 π (i) ) is the solution of              -∂ t θ 2 ϕ (i) -∆ θ 2 ϕ (i) + ∇ θ 2 π (i) = i+1 j=1 a j,i θ 2 ϕ (j) -θ 2 ϕ (i) in Q, ∇ • θ 2 ϕ (i) = 0 in Q, θ 2 ϕ (i) = 0 on Σ, θ 2 ϕ (i) | t=T = 0 in Ω. (3.15)
Using that the right-hand side of the first equation is divergence free and with null trace on ∂Ω, we can apply Lemma 2.3 to the above system and we deduce

θ 2 ϕ (i) 2 L 2 (0,T ;H 3 (Ω) 2 )∩H 1 (0,T ;V ) C i+1 j=1 θ 2 ϕ (j) 2 L 2 (0,T ;V ) + θ 2 ϕ (i) 2 L 2 (0,T ;V )
.

Using θ 2 Cs 2 (ξ * ) 2-1/(2 ) e -sα * and combining the above result with (3.14) and (3.13)

θ 2 ϕ (i) 2 L 2 (0,T ;H 3 (Ω) 2 )∩H 1 (0,T ;V ) C m j=1 s 5/2 (ξ * ) 5/2 e -sα * ϕ (j) 2 L 2 (Q) 2 . ( 3.16)
Finally, we introduce

θ 3 = (ξ * ) -5/(2 ) e -sα * . From (2.5)-(2.6), for s C, |θ 3 | Cθ 2 , |θ 3 | Cs 2 (ξ * ) 2-1/(2 ) e -sα * . (3.17)
Then, from (1.8), (θ 3 ϕ (i) , θ 3 π (i) ) is the solution of

             -∂ t θ 3 ϕ (i) -∆ θ 3 ϕ (i) + ∇ θ 3 π (i) = i+1 j=1 A j,i θ 3 ϕ (j) -θ 3 ϕ (i) in Q, ∇ • θ 3 ϕ (i) = 0 in Q, θ 3 ϕ (i) = 0 on Σ, θ 3 ϕ (i) | t=T = 0 in Ω. (3.18)
We can again apply Lemma 2.3 to the above estimate to deduce

θ 3 ϕ (i) 2 L 2 (0,T ;H 5 (Ω) 2 )∩H 1 (0,T ;H 3 (Ω) 2 )∩H 2 (0,T ;V ) C   i+1 j=1 θ 3 ϕ (j) 2 L 2 (0,T ;H 3 (Ω) 2 ) + ∂ t θ 3 ϕ (j) 2 L 2 (0,T ;H 1 (Ω) 2 ) + θ 3 ϕ (i) 2 L 2 (0,T ;H 3 (Ω) 2 ) + ∂ t θ 3 ϕ (i) 2 L 2 (0,T ;H 1 (Ω) 2 )
.

θ 3 ϕ (i) 2 L 2 (0,T ;H 5 (Ω) 2 )∩H 1 (0,T ;H 3 (Ω) 2 )∩H 2 (0,T ;V ) C   m j=1 s 5/2 (ξ * ) 5/2 e -sα * ϕ (j) 2 L 2 (Q) 2   . (3.20)
Using the trace inequality (3.10) and that 14, we have

(ξ * ) -1/4 e -sα * ∇ 2 ∆ϕ (i) 1 2 H 1 4 , 1 2 (Σ) 4 C (ξ * ) -1/4 e -sα * ∇ 2 ∆ϕ (i) 1 2 L 2 (0,T ;H 1 (Ω) 4 )∩H 1 (0,T ;H -1 (Ω) 4 ) C (ξ * ) -1/4 e -sα * ϕ (i) 1 2 L 2 (0,T ;H 5 (Ω))∩H 1 (0,T ;H 3 (Ω)) C θ 3 ϕ (i) 2 L 2 (0,T ;H 5 (Ω) 2 )∩H 1 (0,T ;H 3 (Ω) 2 ) . (3.21)
Combining (3.20) and (3.21), we deduce that

e -sα * (sξ * ) -1/4 ∇ 2 ∆ϕ (i) 1 2 H 1 4 , 1 2 (Σ) 4 Cs -1/2 m j=1 s 5/2 (ξ * ) 5/2 e -sα * ϕ (j) 2 L 2 (Q) 2
.

Putting together (3.8), the above relation and (3.9), we deduce at this step the following inequality:

I 1 (s, ϕ (i) ) C ω0×(0,T ) e -2sα sξ ∇ 2 ∆ϕ (i) 1 2 + s 3 ξ 3 ∇∆ϕ (i) 1 2 + s 5 ξ 5 ∆ϕ (i) 1 2 dxdt + m j=1 Q e -2sα * s 9/2 (ξ * ) 5 ϕ (j) 2 + e -2sα s -2 ξ -2 ∇ 2 ∆ϕ (j) 1 2 dxdt   . (3.22)
Step 3. To estimate the local terms, we proceed in a standard way: we consider ω 1 an open subset satisfying ω 0 ω 1 ω and η 1 ∈ C 2 c (ω 1 ), η 1 ≡ 1 in ω 0 , η 1 0. Then, an integration by parts gives

ω0×(0,T ) e -2sα sξ ∂ 2 ∂x k ∂x q ∆ϕ (i) 1 2 dxdt ω1×(0,T ) η 1 e -2sα sξ ∂ 2 ∂x k ∂x q ∆ϕ (i) 1 2 dxdt = - ω1×(0,T ) ∂ ∂x k η 1 e -2sα sξ ∂ 2 ∂x k ∂x q ∆ϕ (i) 1 ∂ ∂x q ∆ϕ (i) 1 dxdt - ω1×(0,T ) η 1 e -2sα sξ ∂ 3 ∂x 2 k ∂x q ∆ϕ (i) 1 ∂ ∂x q ∆ϕ (i) 1 dxdt. 10
Using (2.7) and Young's inequality, we deduce from the above relation that there exists C > 0 such that for all ε > 0,

ω0×(0,T ) e -2sα sξ ∇ 2 ∆ϕ (i) 1 2 dxdt ε Q e -2sα s -1 ξ -1 ∇ 3 ∆ϕ (i) 1 2 + sξ ∇ 2 ∆ϕ (i) 1 2 dxdt + C ε ω1×(0,T ) e -2sα s 3 ξ 3 ∇∆ϕ (i) 1 2 dxdt. (3.23)
Now we estimate, in an analogous way, the local term associated with ∇∆ϕ (i)

1 : we consider

η 2 ∈ C 2 c ( ω), η 2 ≡ 1 in ω 1 , η 2 0.
Then, integrating by parts, we obtain

ω1×(0,T ) e -2sα s 3 ξ 3 ∂ ∂x k ∆ϕ (i) 1 2 dxdt ω×(0,T ) η 2 e -2sα s 3 ξ 3 ∂ ∂x k ∆ϕ (i) 1 2 dxdt = - ω×(0,T ) ∂ ∂x k η 2 e -2sα s 3 ξ 3 ∂ ∂x k ∆ϕ (i) 1 ∆ϕ (i) 1 dxdt - ω×(0,T ) η 2 e -2sα s 3 ξ 3 ∂ 2 ∂x 2 k ∆ϕ (i) 1 ∆ϕ (i) 1 dxdt.
Using (2.7) and the Young's inequality, we deduce from the above relation that there exists C > 0 such that for all ε > 0, ω1×(0,T ) e -2sα s 3 ξ 3 ∇∆ϕ

(i) 1 2 dxdt ε Q e -2sα s 3 ξ 3 ∇∆ϕ (i) 1 2 + sξ ∇ 2 ∆ϕ (i) 1 2 dxdt + C ε ω×(0,T ) e -2sα s 5 ξ 5 ∆ϕ (i) 1 2 dxdt. (3.24)
The above estimate, together with (3.22) and (3.23) implies (3.3). This concludes the proof of Proposition 3.2.

We are now in a position to prove Theorem 3.1.

Proof of Theorem 3.1. Summing (3.3) for i = 1, . . . , m, and taking s C for a constant C large enough, we deduce that

m i=1 I 1 (s, ϕ (i) ) C m i=1 ω×(0,T ) e -2sα s 5 ξ 5 ∆ϕ (i) 1 2 dxdt. (3.25)
In order to get rid of the local terms in the right-hand side (except the term corresponding to i = 1), we introduce a sequence of open sets O i , (0 i m) such that

ω =: O 0 O 1 ... O i ... O m ω
and functions

ζ i ∈ C 2 c (O i ) such that ζ i ≡ 1 in O i-1 , ζ i 0 (1 i m)
. Then, we consider the equation m-1 of (1.9), we apply the Laplace operator on the first component of this equation and we multiply it by ζ 1 :

ζ 1 a m,m-1 ∆ϕ (m) 1 = -ζ 1 ∂ t ∆ϕ (m-1) 1 -ζ 1 ∆ 2 ϕ (m-1) 1 -ζ 1 m-1 j=1 a j,m-1 ∆ϕ (j) 1 .
(3.26)

Then, using the above equation combined with (1.4) and integrating by parts, we deduce ω×(0,T ) e -2sα s 5 ξ 5 ∆ϕ

(m) 1 2 dxdt O1×(0,T ) ζ 1 e -2sα s 5 ξ 5 ∆ϕ (m) 1 2 dxdt = -1 a m,m-1 O1×(0,T ) ζ 1 e -2sα s 5 ξ 5 ∆ϕ (m) 1   ∂ t ∆ϕ (m-1) 1 + ∆ 2 ϕ (m-1) 1 + m-1 j=1 a j,m-1 ∆ϕ (j) 1   dxdt. (3.27)
Integrating by parts and applying ∆ to the first component of the equation (1.9) with i = m, we obtain

O1×(0,T ) ζ 1 e -2sα s 5 ξ 5 ∆ϕ (m) 1 ∂ t ∆ϕ (m-1) 1 dxdt = - O1×(0,T ) ζ 1 ∂ t e -2sα s 5 ξ 5 ∆ϕ (m) 1 ∆ϕ (m-1) 1 dxdt + O1×(0,T ) ζ 1 e -2sα s 5 ξ 5 ∆ 2 ϕ (m) 1 ∆ϕ (m-1) 1 dxdt + m j=1 O1×(0,T )
ζ 1 e -2sα s 5 ξ 5 a j,m ∆ϕ

(j) 1 ∆ϕ (m-1) 1 dxdt. (3.28)
By integrating by parts, we also find

O1×(0,T ) ζ 1 e -2sα s 5 ξ 5 ∆ϕ (m) 1 ∆ 2 ϕ (m-1) 1 dxdt = O1×(0,T ) ∆ ζ 1 e -2sα s 5 ξ 5 ∆ϕ (m) 1 ∆ϕ (m-1) 1 dxdt + O1×(0,T ) ζ 1 e -2sα s 5 ξ 5 ∆ 2 ϕ (m) 1 ∆ϕ (m-1) 1 dxdt + O1×(0,T ) 2∇ ζ 1 e -2sα s 5 ξ 5 • ∇∆ϕ (m) 1 ∆ϕ (m-1) 1 dxdt. (3.29)
Let us now estimate the right-hand side of (3.28) and (3.29) and the last term in (3.27). Using (2.4), we have ∂ t e -2sα s 5 ξ 5 Ce -2sα s 6 ξ 6+1/ .

Combining this inequality and Young's inequality, we deduce the existence of C such that for any s C and for any ε > 0,

O1×(0,T ) ζ 1 ∂ t e -2sα s 5 ξ 5 ∆ϕ (m) 1 ∆ϕ (m-1) 1 dxdt ε Q e -2sα s 5 ξ 5 ∆ϕ (m) 1 2 dxdt + C ε O1×(0,T ) e -2sα s 7 ξ 7+2/ ∆ϕ (m-1) 1 2 dxdt. (3.30) Using (2.7), we have ∆(ζ 1 e -2sα s 5 ξ 5 ) Ce -2sα s 7 ξ 7 .
Combining this inequality and Young's inequality, we deduce the existence of C such that for any s C and for any ε > 0,

O1×(0,T ) ∆ ζ 1 e -2sα s 5 ξ 5 ∆ϕ (m) 1 ∆ϕ (m-1) 1 dxdt ε Q e -2sα s 5 ξ 5 ∆ϕ (m) 1 2 dxdt + C ε O1×(0,T ) e -2sα s 9 ξ 9 ∆ϕ (m-1) 1 2 dxdt. (3.31) ∇(ζ 1 e -2sα s 5 ξ 5 ) Ce -2sα s 6 ξ 6 .
Combining this inequality and Young's inequality, we deduce the existence of C such that for any s C and for any ε > 0,

O1×(0,T ) 2∇ ζ 1 e -2sα s 5 ξ 5 • ∇∆ϕ (m) 1 ∆ϕ (m-1) 1 dxdt ε Q e -2sα s 3 ξ 3 ∇∆ϕ (m) 1 2 dxdt + C ε O1×(0,T ) e -2sα s 9 ξ 9 ∆ϕ (m-1) 1 2 dxdt. (3.32)
Finally, the other terms can be estimated by using Young's inequality:

O1×(0,T ) ζ 1 e -2sα s 5 ξ 5 ∆ 2 ϕ (m) ∆ϕ (m-1) 1 dxdt ε Q e -2sα sξ ∆ 2 ϕ (m) 1 2 dxdt + C ε O1×(0,T ) e -2sα s 9 ξ 9 ∆ϕ (m-1) 1 2 dxdt, (3.33)
m j=1 O1×(0,T ) ζ 1 e -2sα s 5 ξ 5 a j,m ∆ϕ (j) ∆ϕ (m-1) 1 dxdt ε m j=1 Q e -2sα s 5 ξ 5 ∆ϕ (j) 1 2 dxdt + C ε O1×(0,T ) e -2sα s 5 ξ 5 ∆ϕ (m-1) 1 2 dxdt, (3.34)
-1 a m,m-1

O1×(0,T ) ζ 1 e -2sα s 5 ξ 5 ∆ϕ (m) 1 m-1 j=1 a j,m-1 ∆ϕ (j) 1 dxdt ε Q e -2sα s 5 ξ 5 ∆ϕ (m) 1 2 dxdt + C ε m-1 j=1 O1×(0,T ) e -2sα
I 1 (s, ϕ (j) ) + C ε O1×(0,T ) e -2sα s 9 ξ 9 ∆ϕ (m-1) 1 2 dxdt + C ε m-2 j=1 O1×(0,T ) e -2sα s 5 ξ 5 ∆ϕ (j) 1 2 dxdt. (3.36)
We can repeat the same analysis for ∆ϕ (m-1) 1

and we obtain the existence of a constant C such that for any s C and for any ε > 0,

O1×(0,T ) e -2sα s 9 ξ 9 ∆ϕ (m-1) 1 2 dxdt ε m j=1 I 1 (s, ϕ (j) ) + C ε O2×(0,T ) e -2sα s 17 ξ 17 ∆ϕ (m-2) 1 2 dxdt. + C ε m-3 j=1 O2×(0,T ) e -2sα s 13 ξ 13 ∆ϕ (j) 1 2 dxdt. (3.37)
Proceeding by induction, we can estimate all the local terms and we deduce from (3.25) that

m i=1 I 1 (s, ϕ (i) ) C Om×(0,T ) e -2sα s 2 (m+1) +1 ξ 2 (m+1) +1 ∆ϕ (1) 1 2 dxdt. (3.38)
Finally, we estimate the above local term in terms of ϕ

1 . In order to do this, we consider ω an open subset satisfying O m ω ω and

ζ ∈ C 2 c ( ω) such that ζ ≡ 1 in O m , ζ 0.
Then by integrating by parts, we obtain

Om×(0,T ) e -2sα s 2 (m+1) +1 ξ 2 (m+1) +1 ∆ϕ (1) 1 2 dxdt ω×(0,T ) ζe -2sα s 2 (m+1) +1 ξ 2 (m+1) +1 ∆ϕ (1) 1 2 dxdt = - ω×(0,T ) ∇ ζe -2sα s 2 (m+1) +1 ξ 2 (m+1) +1 ∆ϕ (1) 1 • ∇ϕ (1) 1 dxdt - ω×(0,T ) ζe -2sα s 2 (m+1) +1 ξ 2 (m+1) +1 ∆∇ϕ (1) 1 • ∇ϕ (1) 1 dxdt. (3.39)
Using (2.7), we have

∇( ζe -2sα s 2 (m+1) +1 ξ 2 (m+1) +1 ) Ce -2sα s 2 (m+1) +2 ξ 2 (m+1) +2 .
Combining this inequality and Young's inequality, we deduce the existence of C such that for any s C and for any ε > 0,

Om×(0,T ) e -2sα s 2 (m+1) +1 ξ 2 (m+1) +1 ∆ϕ (1) 1 2 dxdt ε Q e -2sα s 3 ξ 3 ∇∆ϕ (1) 1 2
+ s 5 ξ 5 ∆ϕ (1) 1 2 dxdt
+ C ε ω×(0,T ) e -2sα s 2 (m+2) -1 ξ 2 (m+2) -1 ∇ϕ (1) 1 2 dxdt. (3.40)
Then, we consider ζ ∈ C 2 c (ω) such that ζ ≡ 1 in ω, ζ 0 and we integrate by parts:

ω×(0,T ) e -2sα s 2 (m+2) -1 ξ 2 (m+2) -1 ∇ϕ (1) 1 2 dxdt ω×(0,T ) ζe -2sα s 2 (m+2) -1 ξ 2 (m+2) -1 ∇ϕ (1) 1 2 dxdt = - ω×(0,T ) ∇ ζe -2sα s 2 (m+2) -1 ξ 2 (m+2) -1 • ∇ϕ (1) 1 ϕ (1) 1 dxdt - ω×(0,T ) ζe -2sα s 2 (m+2) -1 ξ 2 (m+2) -1 ∆ϕ (1) 1 ϕ (1) 1 dxdt. ∇(ζe -2sα s 2 (m+2) -1 ξ 2 (m+2) -1 ) Ce -2sα s 2 (m+2) ξ 2 (m+2) .
Combining this inequality and Young's inequality, we deduce the existence of C such that for any s C and for any ε > 0, ω×(0,T ) e -2sα s 2 (m+2) -1 ξ 2 (m+2) -1 ∇ϕ

(1) 1 2 dxdt ε   Q e -2sα s 6 ξ 6 |∇ϕ (1) 1 | 2 dxdt + Q e -2sα s 5 ξ 5 |∆ϕ (1) 1 | 2 dxdt   + C ε ω×(0,T ) e -2sα s 2 (m+3) -6 ξ 2 (m+3) -6 ϕ (1) 1 2 dxdt. (3.41)
Using Lemma 4 in [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF] (see [START_REF] Imanuvilov | Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems[END_REF] for the proof) and the definition (3.1) of I 1 , we deduce that

I 1 (s, ϕ (i) ) + ω×(0,T ) e -2sα s 8 ξ 8 ϕ (i) 1 2 dxdt Q e -2sα s 5 ξ 5 ∆ϕ (i) 1 2 dxdt + ω×(0,T ) e -2sα s 8 ξ 8 ϕ (i) 1 2 dxdt C   Q e -2sα s 8 ξ 8 ϕ (i) 1 2 dxdt + Q e -2sα s 6 ξ 6 ∇ϕ (i) 1 2 dxdt   .
This estimate, combined with (3.38), (3.40) and (3.41) yields the conclusion of Theorem 3.1.

4 Proof of the main results

Final state observability

In this section, we use Theorem 3.1 in order to prove the final state observability of the adjoint system (1.8). Proof. First, we consider an energy estimate of the adjoint system (1.8). Multiplying each equation (1.9) by ϕ (i) and integrating by parts, we deduce

- 1 2 d dt Ω m i=1 ϕ (i) 2 dx + Ω m i=1 ∇ϕ (i) 2 dx = Ω m i,j=1 A j,i ϕ (i) • ϕ (j) dx. (4.2)
Thus, using the Grönwall lemma, there exists C > 0 such that and consequently, for some constant

t → e Ct Ω m i=1 ϕ (i) (x,
C 3 > 0, e -2sα * (sξ * ) 5 e -C 3 T 2 . (4.5)
Similarly, from (2.2), there exist two constants c 1 , c 2 > 0 such that for (x, t)

∈ Ω × [0, T ], α(x, t) c 1 t (T -t) , ξ(x, t) c 2 t (T -t)
and consequently, for some constant c 3 > 0, e -2sα (sξ)

2 m+3 -6 e -2s c 1 t (T -t) s c 2 t (T -t)
ϕ (i) (x, 0) 2 dx C T e C(T + 1 T 2 ) ω×(0,T ) ϕ (1) 1 2 dxdt.
This implies that (4.1).

Proof of Theorem 1.1

We recall that H and V are defined by (1.6) and (1.7). We also set

H := H m , V := V m . (4.7)
We define P 0 : L 2 (Ω) N → H the Leray projector and the projection P defined as: 1) , . . . , y (m) → P 0 y (1) , . . . , P 0 y (m) .

P : L 2 (Ω) N m → H, y = y (
We also consider the unbounded operator in H defined by

D(A) := H 2 (Ω) N ∩ V m , Ay := P∆y + Ay. (4.8)
Finally, we define the control operator B ∈ L(U, H) by

U := L 2 (ω) N -1 , Bv = B v 1 , . . . , v N -1 := P 0 v 1 , . . . , v N -1 , 0 1 ω , 0, . . . , 0 . (4.9)
With the above definition, we can write (1.1) as

y = Ay + Bv, y(0) = y 0 . (4.10)
As it is well-known (see, for instance, [34, p.357]), system (4.10) is null-controllable in time T > 0 if and only if there exists K(T ) > 0 such that

e T A * ϕ 0 2 H K(T ) 2 T 0 B * e tA * ϕ 0 2 H dt (ϕ 0 ∈ H). (4.11)
One can check that

D(A * ) = H 2 (Ω) N ∩ V m , A * y = P∆y + A * y,
where A * is the transpose of the matrix A and that

B * ϕ = ϕ (1) 1 |ω , . . . , ϕ (1)
N -1 |ω .

Thus, we deduce Theorem 1.1 from Lemma 4.1.

Proof of Theorem 1.3

In order to prove Theorem 1.3, we recall a method introduced in [START_REF] Liu | Single input controllability of a simplified fluid-structure interaction model[END_REF][START_REF] Takahashi | Boundary local null-controllability of the Kuramoto-Sivashinsky equation[END_REF] to deal with the controllability of nonlinear parabolic systems. We consider H and U two Hilbert spaces, A : D(A) → H the infinitesimal generator of an analytic semigroup e tA t 0 and B ∈ L(U, H) a control operator. We also assume the final state observability for all T > 0

e T A * ϕ 0 2 H K(T ) 2 T 0 B * e tA * ϕ 0 2 H dt (ϕ 0 ∈ H), (4.12)
with K : (0, ∞) → [0, ∞) continuous and non increasing. Let us consider T > 0 and suppose there exist ρ 0 , ρ 1 , ρ ∈ C 0 ([0, T ], R + ), non increasing, positive in [0, T ) such that ρ 0 (T ) = ρ 1 (T ) = ρ(T ) = 0 and such that, for some constant q > 1,

ρ 0 (t) := ρ 1 (q 2 (t -T ) + T )K((q -1)(T -t)) t ∈ T 1 - 1 q 2 , T , (4.13)
ρ 0 Cρ, ρ 1 Cρ, |ρ |ρ 0 Cρ 2 (t ∈ [0, T ]) . (4.14)
for some constant C > 0. We denote by L 2 ρ1 (0, T ; H) the space

L 2 ρ1 (0, T ; H) := f ∈ L 2 (0, T ; H) ; f ρ 1 ∈ L 2 (0, T ; H)
and we define similarly L 2 ρ0 (0, T ; U). Then we can consider the control problem y = Ay + Bv + f, y(0) = y 0 .

(4.15)

We have the following result (see [START_REF] Liu | Single input controllability of a simplified fluid-structure interaction model[END_REF]):

Theorem 4.2. With the above assumptions, there exists a bounded operator

E T ∈ L D (-A) 1/2 × L 2 ρ1 (0, T ; H), L 2 ρ0 (0, T ; U)
such that for any y 0 ∈ D (-A)

1/2
and for any f ∈ L 2 ρ1 (0, T ; H), the solution y of (4.15)

with u = E T (y 0 , f ) satisfies y ρ ∈ L 2 (0, T ; D(A)) ∩ C 0 [0, T ]; D (-A) 1/2 ∩ H 1 (0, T ; H).
Moreover there exists a constant C such that

y ρ L 2 (0,T ;D(A))∩C 0 ([0,T ];D((-A) 1/2 ))∩H 1 (0,T ;H) C y 0 D((-A) 1/2 ) + f L 2 ρ 1 (0,T ;H) .
Remark 4.3. Note that in [START_REF] Liu | Single input controllability of a simplified fluid-structure interaction model[END_REF], A is assumed to be self-adjoint negative but the result can be extended to the case where A is the generator of an analytic semigroup. Indeed, the hypothesis used in the proof is the maximal regularity of (4.15) for v = 0.

Remark 4.4. Since ρ(T ) = 0, the above result implies in particular that y(T ) = 0, that is the null-controllability of (4.15).

In the previous section, we have defined for our problem the spaces H, U, A and B, see (4.7), (4.8) and (4.9). To show that A is the generator of an analytic semigroup, we first note that P∆ is self-adjoint negative (see, for instance, [31, Theorem 2.1.1, p.128]) and thus is the generator of an analytic semigroup. Then using a perturbation argument (see, for instance, [30, Theorem 2.1, p.80]), we deduce that A is the generator of an analytic semigroup in H.

Finally, applying Lemma 4.1, we deduce that (4.12) holds with

K(T ) = C K e C K T 2 ,
for some constant C K > 0.

Let us consider q ∈ 1, 2 with C 0 , C 1 , C some positive constants such that

C 0 := C 1 q 4 - C K (q -1) 2 > C 1 2 , C 1 2 < C < C 0 < C 1 .
Then we can check that (4.13) and (4.14) hold and we have moreover that

ρ 2 ρ 1 . (4.16)
Consequently, we deduce from Theorem 4.2 a controllability result on the system 1) -∆y (1) + ∇p (1) = m j=1 A 1,j y (j) + ve 1 1 ω + f (1) in Q, ∂ t y (i) -∆y (i) + ∇p (i) = m j=i-1

                   ∂ t y (
A i,j y (j) + f (i) in Q, (2 i m) ∇ • y (i) = 0 in Q, (1 i m) y (i) = 0 on Σ, (1 i m) y (i) (•, 0) = y such that for any y 0 ∈ V and for any f = f (1) , . . . , f (m) ∈ L 2 ρ1 (0, T ; L 2 (Ω) N m ), the solution y of (4.17 We are now in a position to prove Theorem 1.3:

Proof of Theorem 1.3. First we notice that y is solution of (1.2) if it is a solution of (4.17) with f = -(y (1) • ∇)y (1) , ..., -(y (m) • ∇)y (m) .

Thus, we consider the mapping N T : f ∈ B R → (-(y (1) • ∇)y (1) , ..., -(y (m) • ∇)y (m) ), where

B R := f ∈ L 2 ρ1 (0, T ; L 2 (Ω) N m ) ; f ρ 1 L 2 (0,T ;[L 2 (Ω) N ] m ) R
where R > 0 is such that y 0 V R.

We are going to show that for R small enough (and thus y 0 V small enough), N T (B R ) ⊂ B R and that (N T ) |B R is a strict contraction. Using the Banach fixed point theorem we deduce the existence of a fixed point of N T . The corresponding solution y of (4.17) is a solution of (1.2) and from (4.18), we deduce that y(•, T ) = 0. It thus remains to prove that for R small enough, N T (B R ) ⊂ B R and that (N T ) |B R is a strict contraction. In order to do this, we first note that, using (4.16), Sobolev's embeddings and Hölder's inequalities, we have

N T (f ) ρ 1 L 2 (0,T ;[L 2 (Ω) N ] m ) C y 0 V + f L 2 ρ 1 (0,T ;[L 2 (Ω) N ] m ) 2 4CR 2 R,
for R small enough. For such R, we have N T (B R ) ⊂ B R . Now, let us consider f , f ∈ B R and let us write f = f -f . We consider the solution y (resp. y) the solution of (4.17) associated with the control v = E T (y 0 , f ) (resp. v = E T (y 0 , f )). Then, y := y -y is the solution of (4.17

1

  and r = 3: for any s C and λ C, Q e -2sα s 5 ξ 5 ∆ϕ

s 5 ξ 5 ∆ϕ

 5 (3.27) with (3.30), (3.31), (3.32), (3.33), (3.34), and (3.35) yields the existence of a constant C such that for any s C and for any ε > 0, ω×(0,T ) e -2sα s 5 ξ 5 ∆ϕ

Lemma 4 . 1 .

 41 Assume T ∈ (0, 1) and ω is non empty open set of Ω. Then, there exists C > 0 and 14 such that for any ϕ 0 ∈ H m , the solution ϕ of (1.8) satisfies

  3), (4.4), (4.5) and (4.6), we deduce that for some constant C, Ω m i=1

1 4 . 1 (

 141 and let us setρ 0 (t) := C K e -C 0 (T -t) 2 , ρ 1 (t) := e -C T -t) 2 , ρ(t) := e -C (T -t) 2

  Ω. (1 i m)

( 4 . 17 )

 417 More precisely, there existsE T ∈ L V × L 2 ρ1 (0, T ; L 2 (Ω) N m ), L 2 ρ0 (0, T ; L 2 (ω))

2 ρ 1 (

 21 ) with the control v = E T (y 0 , f ) satisfiesy ρ ∈ L 2 (0, T ; H 2 (Ω) N m ) ∩ C 0 [0, T ]; H 1 (Ω) N m ∩ H 1 (0, T ; H).(4.18)Moreover we have the following estimatey ρ L 2 (0,T ;[H 2 (Ω) N ] m )∩C 0 ([0,T ];[H 1 (Ω) N ] m )∩H 1 (0,T ;H) C y 0 V + f L 0,T ;[L 2 (Ω) N ] m ) .(4.19)

2 L 2 L 2 v ρ 2 Lρ 2 L 2

 222222 ∞ (0,T ;L6 (Ω) N ) ∇w ρ (0,T ;L 6 (Ω) N ) C ∞ (0,T ;H 1 (Ω) N ) w (0,T ;H 2 (Ω) N ). (4.20)Using this relation and (4.19), we deduce that

1 (

 1 ) associated with the control v := E T (0, f ) and thusy ρ L 2 (0,T ;[H 2 (Ω) N ] m )∩C 0 ([0,T ];[H 1 (Ω) N ] m )∩H 1 (0,T ;H) C f L 2 ρ 0,T ;[L 2 (Ω) N ] m ) .

  t)

	is nondecreasing. In particular, for some constant C > 0,
	Ω	m i=1	ϕ (i) (x, 0)	2	dx	2 T	e CT	T /4 Ω 3T /4	i=1 m	ϕ (i) (x, t)	2	dx dt.	(4.3)
	On the other hand, from (3.2) and (3.1), we deduce that
	e -2sα *	(sξ * ) 5	m	ϕ (i) 2	dxdt C	e -2sα (sξ) 2 m+3 -6 ϕ (1) 1	2	dxdt.	(4.4)
	Q				i=1						ω×(0,T )
	Using that for t ∈ [T /4, 3T /4],
								3T 2 16	t(T -t)	T 2 4	,
	we deduce, from (2.3), the existence of two constants C 1 , C 2 > 0 such that for t ∈ [T /4, 3T /4],
							α * (t)	C 1 T 2 , ξ * (t)	C 2 T 2
													2
														dx

Using this and (4.20), we obtain

Thus for R small enough, (N T ) |B R is a strict contraction and this ends the proof of Theorem 1.3


Anastasia Bolotnikova

Pejman Rasti

Andres Traumann

Iiris Lüsi

Morteza Daneshmand

Fatemeh Noroozi

Kadri Samuel

Suman Sarkar

Gholamreza Anbarjafari

Iiris Lusi

Block based image compression technique using rank reduction and wavelet difference reduction

Keywords: Lossy Image Compression, Wavelet Difference Reduction, Block Processing, Singular Value Decomposition. 1

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

With the growing demand for multimedia applications especially high definition images, efficient storage and transmission of images have been issues of great concern [START_REF] Fujii | Digital Cinema and Super-High-Definition Content Distribution on Optical High-Speed Networks[END_REF]- [START_REF] Tsan-Jieh | High Definition Image Pre-Processing System for Multi-Stripe Satellites' Image Sensors[END_REF]. Image processing deals with the reduction of the amount of bits used to represent an image.

There are two categories of image compression techniques namely lossless and lossy image compression techniques [START_REF] Groach | DCSPIHT: Image Compression Algorithm[END_REF], [START_REF] Jayaraman | Digital Image Processing[END_REF]. In lossless image compression, the original image can be perfectly recovered from the compressed image while in lossy compression the original image cannot be perfectly recovered from the compressed image because some information is lost as a result of compression. Lossless compression is used in applications with high requirements such as medical imaging. Lossy compression techniques are very popular because they offer higher compression ratio. The objective of image compression is to achieve as much compression as possible with little loss of information [START_REF] Boliek | Beyond compression: a survey of functionality derived from still image coding[END_REF], [START_REF] Gonzalez | Digital Image Processing[END_REF].

In this research work, a new lossy compression technique which employs Singular value decomposition (SVD) and wavelet difference reduction (WDR) is presented. SVD is a lossy image compression technique which can be regarded as a quantization process where it reduces the physcovisual redundancies of the image [START_REF] Yang | Combined Techniques Of Singular Value Decomposition and Vector Quantization For Image Coding[END_REF], [START_REF] Greenberg | Differential equations & Linear algebra[END_REF]. Wavelet transforms are used frequently in many image processing applications [START_REF] Anbarjafari | Image Super Resolution Based on Interpolation of Wavelet Domain High Frequency Subbands and the Spatial Domain Input Image[END_REF]- [START_REF] Ozcinar | Video Resolution Enhancement By Using Complex Wavelet Transform[END_REF]. One of these applications is in image compression. WDR is one of the state-of-the-art techniques in image compression which uses wavelet transform. It is a lossy image compression technique which achieves compression by first taking the wavelet transform of the input image and then applying the difference reduction method on the transform values [START_REF] Tian | A lossy image codec based on index coding[END_REF]- [START_REF] Raja | Image Compression Using WDR and ASWDR Techniques With Different Wavelet Codecs[END_REF].

In the proposed compression technique, the input image is firstly divided into the equal sized blocks and each block is quantized using the SVD technique and then WDR compression is carried out on the output image from the SVD compression for further compression. The proposed technique has been tested on several well-known images such as, Lena, Peppers, Boat, and Airfield. The results of this technique have been compared with those of JPEG2000 and WDR techniques. The quantitative experimental results based on PSNR show that the proposed technique overcomes the aforementioned techniques. The SVD and WDR image compression techniques are discussed in the next section.

2.

RELATED WORK

Singular Value Decomposition

SVD has been used in many image processing application [START_REF] Lu | Image Denoising via Improved Sparse Coding[END_REF][START_REF] Demirel | Image equalization based on singular value decomposition[END_REF][START_REF] Chin | Incremental Kernel PCA for Efficient Non-linear Feature Extraction[END_REF]. Due to its nature, SVD has been always an important mathematical tool for image compression [START_REF] Rufai | Lossy image compression using singular value decomposition and wavelet difference reduction[END_REF][START_REF] Chung | A novel SVD-and VQ-based image hiding scheme[END_REF][START_REF] Dongdong | Image Compression Transmission Algorithm Based on the Singular Value Decomposition Applied in the Wireless Multimedia Sensor Networks[END_REF]. An image is represented by a matrix in digital form. SVD is a technique used to decompose a given matrix A, into three matrices U, Σ, and V. U and V are orthogonal while Σ is a diagonal matrix containing the singular values A [START_REF] Kalman | A Singularly Valuable Decomposition[END_REF][START_REF] Nivedita | Performance Analysis of SVD and SPIHT Algorithm for Image Compression Application[END_REF][START_REF] Knochaert | SVD compression, unitary transforms, and computational complexity[END_REF]. Σ can be approximated by ignoring small singular values. This approximation will result into a lower rank singular matrix. In order to reconstruct the input matrix, U and V matrices also need to be updated. Eqn. [START_REF] Fujii | Digital Cinema and Super-High-Definition Content Distribution on Optical High-Speed Networks[END_REF] shows the reconstruction of the input matrix with approximated Σ.

( ) () ( )
where ,

T pq mn m p nq mm m p nn nq mmp nn q AU V UU U V V V × ×× × ×× × × ×- ×- =Σ ⎡⎤ ⎡ ⎤ == ⎣⎦ ⎣ ⎦ (1)
The SVD compression is a lossy compression technique because singular values of the image are ignored in the procedure cannot be recovered, which results in losses [START_REF] Rufai | Lossy image compression using singular value decomposition and wavelet difference reduction[END_REF], [START_REF] Knochaert | SVD compression, unitary transforms, and computational complexity[END_REF].

Wavelet Difference Reduction

Wavelet transformations have been employed in many of applications of computer vision in last couple of decays [START_REF] Antonini | Image coding using wavelet transform[END_REF][START_REF] Plonka | Optimally sparse image representation by the easy path wavelet transform[END_REF][START_REF] Zhao | The Image Compression Technology Based on Wavelet Transform[END_REF]. The WDR is a compression technique based on the difference reduction method. The wavelet transform of the input image is first taken; bit plane encoding is then applied to the transform values. The bit plane encoding procedure starts with the initialization stage, where a threshold T o is chosen such that To is greater than all the transform values and at least one of the transform values has a magnitude of T o /2. The next stage is the initialization stage where the threshold T = T k-1 is updated to T = T k , where T k = T k-1/2 . New significant transform values (w(i)) which are satisfying T ≤ | w(i) | ≤ 2T are then identified at the significant pass stage. The transform values of these significant transform values are then encoded using the difference reduction method. At the significant pass stage, already quantized values (w Q ) which satisfy |w Q | ≥ 2T are then refined in order to reduce error [START_REF] Yuan | Novel embedded image coding algorithms based on wavelet difference reduction[END_REF][START_REF] Walker | A Low-power, Low-memory System for Wavelet-based Image Compression[END_REF][START_REF] Walker | Wavelet-based image compression[END_REF].

THE PROPOSED TECHNIQUE

The proposed image compression technique is a lossy compression technique. It begins by dividing the input image into equal sized blocks. In this work block sizes 16×16, 32×32, 64x64, and 128×128 are investigated for input images of size 256×256. SVD compression is then carried out on each block. As it was mentioned in the previous section, the SVD based compression is lossy due to the nature of the process. However the qualitative loss is not psychovisually noticeable up to some point. After SVD compression, WDR compression is carried out to further compress the image. The compression ratio is obtained by multiplication of the compression ratio obtained by using SVD with that of WDR. The compression ratio of WDR in the proposed block based compression technique is calculated by dividing the total number of bits required for representing the original image by the total number of bits stream which obtained by WDR. Decompression is carried out by taking the inverse WDR of the bit streams in order to reconstruct each block and then all blocks are combined together to form the decompressed image.

The block diagram of the proposed image compression technique is shown in

EXPERIMENTAL RESULTS AND DISCUSSION

The proposed lossy image compression, as it was mentioned in the Introduction, was tested on Airfield, Boats, Lena and Peppers images. All the input images are 256x256, 8-bit grey scale. Table 1 and 2 are showing the quantitative comparison between the proposed techniques and JPEG2000 and WDR by use of PSNR for compression ratio of 60:1 and 80:1 respectively. As the PSNR values shows in Table 1 and 2 the performance of the proposed technique overcomes the JPEG2000 and WDR based image compression techniques. Figure 2 is showing a portion of the magnified Lena image being compressed with compression ratio of 60:1 by using JPEG200, WDR, and the proposed technique with block size of 64x64.

5.

CONCLUSION

This paper was proposing a new block based lossy image compression technique which uses rank reduction and WDR technique. The input image is divided into blocks of equal sizes. Then each block was quantized by reducing its rank using SVD and then compressed by applying WDR technique. Reconstruction was carried out by decompressing each blocks bit streams and then merging all of them to obtain the decompressed image. The proposed image compression technique was compared with WDR technique and JPEG2000. The excremental results were showing that the proposed image compression technique outperforms the WDR and JPEG2000 techniques.

Figure 1 .

 1 The experimental qualitative and quantitative results are represented and discussed in the next section.

Figure 1 :

 1 Figure 1: The block diagram of the proposed blocked based lossy image compression technique.

Figure 2 :

 2 Figure 2: Zoomed (a) original Lena image, and compressed images by using (b) JPEG2000, (c) WDR, and (d) the proposed image compression technique using 64x64 block sizes at 60:1 compression ratio.

Figure 3 Figure 3 :

 33 Figure 3 is illustrating PSNR versus the compression ratio for the aforementioned images comparing the JPEG2000 and WDR techniques and the proposed block based lossy image compression technique with 128x128 block sizes. The graphs are showing the superiority of the proposed image compression over the other state-of-the-art techniques.

Table 1 :

 1 PSNR values in dB for 60:1 compression

	Image/Technique	Proposed Technique JPEG2000 WDR
	Airfield	42.23	23.76	23.71
	Boats	43.45	28.26	28.14
	Lena	44.75	31.18	31.71
	Peppers	44.50	30.96	30.29

Table 2 :

 2 PSNR values in dB for 80:1 compression

	Image/ Technique	Proposed Technique	JPEG2000 WDR
	Airfield	41.45	22.64	22.71
	Boats	43.00	26.76	26.82
	Lena	44.24	29.62	29.71
	Peppers	44.22	29.54	28.93


Christer Loob

Pejman Rasti

Iiris Lüsi

Julio Jacques Junior

Xavier Baró

Sergio Escalera

Tomasz Sapinski

Dorota Kaminska

Gholamreza Anbarjafari

Julio C S Jacques Junior

Tomasz Sapiński

Dominant and Complementary Multi-Emotional Facial Expression Recognition Using C-Support Vector Classification

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Facial expressions are one of the most effective ways for humans to communicate due to the fact that they contain critical and necessary information regarding human affective states [START_REF] Adolphs | Cognitive neuroscience of human social behaviour[END_REF]- [START_REF] Nagaraju | Evaluation of lbp-based facial emotions recognition techniques to make consistent decisions[END_REF]. Automatic recognition of facial expressions is a complex issue because of significant variations in the physiognomy of faces with respect to person's identity, environment illumination and head pose [START_REF] Gu | Facial expression recognition using radial encoding of local gabor features and classifier synthesis[END_REF], [START_REF] Rahulamathavan | Facial expression recognition in the encrypted domain based on local fisher discriminant analysis[END_REF].

Automatic facial expression recognition is widely used in many applications such as assistive robotics, human emotion analysis, and human-machine interfaces [START_REF] Dai | An associate memory model of facial expressions and its application in facial expression recognition of patients on bed[END_REF], [START_REF] Anbarjafari | Expression recognition by using facial and vocal expressions[END_REF]. As a result, recognition of facial expressions has become an important research topic in human-computer interaction, as well as in the fields of machine learning, image processing, human cognition, and pattern recognition.

The general approach to facial expression recognition consists of 5 stages [START_REF] Kumari | Facial expression recognition: A survey[END_REF]: preprocessing, face acquisition, image segmentation, feature extraction and classification. Preprocessing consists of image enhancement techniques such noise removal or illumination enhancement [START_REF] Anbarjafari | Face recognition using color local binary pattern from mutually independent color channels[END_REF]. The face acquisition involves utilisation of some face detection algorithm, such as the Viola-Jones [START_REF] Viola | Robust real-time face detection[END_REF] algorithm. The facial This work is supported Estonian Research Council Grant (PUT638), the Estonian Centre of Excellence in IT (EXCITE) funded by the European Regional Development Fund, the Spanish Projects TIN2013-43478-P and TIN2016-74946-P (MINECO/FEDER, UE), CERCA Programme / Generalitat de Catalunya and the European Network on Integrating Vision and Language (iV&L Net) ICT COST Action IC1307. components of regions of interest such as mouth, eyes, ears, cheeks, nose, forehead, and eyebrow are detected during the third step. At the fourth stage a feature extraction algorithm such as local binary patterns (LBP) [START_REF] Ojala | A comparative study of texture measures with classification based on featured distributions[END_REF], Gabor filters [START_REF] Lyons | Automatic classification of single facial images[END_REF], linear discriminant analysis (LDA) [START_REF] Belhumeur | Eigenfaces vs. fisherfaces: Recognition using class specific linear projection[END_REF], principal component analysis (PCA) [START_REF] Turk | Face recognition using eigenfaces[END_REF], local gradient code (LGC) [START_REF] Tong | Facial expression recognition algorithm using lgc based on horizontal and diagonal prior principle[END_REF], and independent component analysis (ICA) [START_REF] Bartlett | Face recognition by independent component analysis[END_REF] is used in order to extract the feature points from the regions of interest. Finally, a classification method is used to find the recognition rate. Some of the most popular classification methods are support vector machines (SVM) [START_REF] Hsu | A practical guide to support vector classification[END_REF] and artificial neural networks (ANN) [START_REF] Altman | An introduction to kernel and nearest-neighbor nonparametric regression[END_REF].

An automated Facial Action Coding System (FACS) system is presented by Jihun Hamm et al. [START_REF] Hamm | Automated facial action coding system for dynamic analysis of facial expressions in neuropsychiatric disorders[END_REF]. In their work, dynamic changes of facial actions in neuropsychiatric patients' videos were analyzed by measuring the movements of facial action units. As mentioned by the authors, they approach has some advanges in comparison with other methods. It analyses dynamical expression changes through videos (instead of still images), it measures individual (and combined) facial muscle movements through Action Units (AUs), instead of a few prototypical expressions, and finally, it performs automatically without requiring interventions from an operator. In summary, geometric and texture features are automatically extracted from video and tracked, and then temporal profiles of facial muscle movements are produced.

Luo et al. [START_REF] Luo | Facial expression recognition based on fusion feature of pca and lbp with svm[END_REF] presented a facial expression recognition algorithm by combining PCA and LBP features in a SVM classifier. In their work, local features are extracted from the mouth region, which gives support the global features.

In this work, we present a new facial expression recognition model that uses illumination enhancement [START_REF] Anbarjafari | Image illumination enhancement with an objective noreference measure of illumination assessment based on gaussian distribution mapping[END_REF] and Voila-Jones face detection algorithms, as well as automatic facial feature extraction techniques and SVM for feature extraction and classification.

The remainder of this paper is organised as follows. A detailed overview of the proposed method is presented in Section II. Section III contains the outcome of experimental results. Finally, Section IV concludes the paper.

II. THE PROPOSED METHOD

Although most research works are focused on classical emotions which are anger, contempt, disgust, fear, happiness, sadness, and surprise as well as an eighth emotion referred to as neutral [START_REF] Ekman | Unmasking the face: A guide to recognizing emotions from facial clues[END_REF]- [START_REF] Hipp | Primary vision and facial emotion recognition in early parkinson's disease[END_REF], in this work we are proposing a step forward and introducing further facial expression recognition, by dividing the facial expressions into two main categories, namely, dominant expressions and complementary expressions. Dominant expressions include the seven main expressions. The complementary expression are those minor expressions that people express while they are in one of the seven classical states. Based on the aforementioned basic emotions, one can have 30+ different expressions which are combinations of the seven basic expressions. These new expressions are referred to as surprisingly angry, disgustingly sad, happily surprised, and so on. The list of the possible emotions are given in Table I. By introducing such detailed facial expression recognition technique, a human-computer interaction system will be able to understand the emotional state of the user more accurately and make a more appropriate decision. The dominant facial expression is recognised by using all features and detected landmarks on facial images, whereas the complementary expressions have been recognised only by using lower facial landmarks.

The general steps of the proposed facial multi-emotional recognition system is shown in Fig. 1. The details of each part is explained below as well.

A. Feature extraction

In the proposed method we extract 68 intransigent feature points including 6 points for both eyes, 5 points for both eyebrows, 9 points for the nose, 20 points for the mouth region. The remaining 17 extracted points represent the boundaries of the face. For feature extraction, we adopted the FERA 2015 baseline system described in [START_REF] Valstar | Fera 2015 -second facial expression recognition and analysis challenge[END_REF] and code by Baltrusaitis [START_REF] Baltrusaitis | 3d constrained local model for rigid and non-rigid facial tracking[END_REF]. The face is segmented so that the eyes, eyebrows, nose, 6 of the upper boundary points are considered as upper face points and rest as lower face points.

It should be mentioned that number of extracted feature points are varied in different methods according to their purpose or their limitation such as 34 points in [START_REF] Zhang | Feature-based facial expression recognition: Sensitivity analysis and experiments with a multilayer perceptron[END_REF] or 24 facial points in [START_REF] Afzal | Perception of emotional expressions in different representations using facial feature points[END_REF]. Fig. 2 shows some sample facial images with their extracted features marked. In the proposed method faces on the images in the dataset are first detected using Viola & Jones face detector [START_REF] Viola | Rapid object detection using a boosted cascade of simple features[END_REF]. The images are then normalized in terms of size, translation, rotation and the background is removed. Only prelabelled image sets are used and all images excluding neutral and full emotion samples are removed from further processing.

The feature extraction process begins by grayscaling the images. Face localisation and segmentation is accomplished using the Viola & Jones face detector [START_REF] Viola | Rapid object detection using a boosted cascade of simple features[END_REF]. The image is split into a 4x4 grid to maintain some local information. Appearance features are extracted using Local Gabor Binary Patterns (LGBP) due to its robustness to complex variations in expression and illumination [START_REF] Zhang | Local gabor binary pattern histogram sequence (lgbphs): a novel non-statistical model for face representation and recognition[END_REF]. The image is first convolved with a set of Gabor filters after which LBP feature extraction occurs over the set of Gabor magnitude response images [START_REF] Valstar | Fera 2015 -second facial expression recognition and analysis challenge[END_REF], [START_REF] Zhang | Local gabor binary pattern histogram sequence (lgbphs): a novel non-statistical model for face representation and recognition[END_REF].

Euclidean distances between corresponding points on neutral and full emotion images are calculated. The resulting data is normalised and sent to the support vector machine for dominant emotion classification. The data is then divided into two data sets describing the lower-and the upper half of the face. The resulting distances are normalised and distances below 0.2 will be eliminated. The data is then sent to a support vector machine for complementary emotion classification.

B. C-support vector classification

We approach the problem by using the leave-one-out method by training on all the data except for the current test sample. The sample is classified by SVM a maximum of 7 times for the 7 basic emotions or until a successful classification of any emotion is obtained.

For this type of SVM training involves the minimisation of the error function:

1 2 w T w +C l ∑ i=1 ξ i (1)
Subject to the constraints:

y i (w T φ (x i ) + b) ≥ 1 -ξ i , (2)
ξ i ≥ 0, i = 1, ..., l, ( 3
)
Where w is the width of the hyperplane margin, C is the penalty parameter, the index i labels the l training cases, ξ represents the independent variables and y ∈ ±1 represents the class labels. The kernel function φ is used to transfrom data(x i ) from the input to the feature space. The kernel function [START_REF] Hsu | A practical guide to support vector classification[END_REF] is also called

K(x i , x j ) ≡ φ (x i ) T φ (x j ).
The kernel function used for the proposed method is Radial Basis Function (RBF) kernel in which case:

K(x i , x j ) = exp(-γ||x i -x j || 2 ), γ > 0 ( 4
)
Where γ represents the influence a single training sample has and ||x ix j || 2 is the squared Euclidean distance between the two feature vectors.

The kernel nonlinearly maps samples into a higher dimensional space so it can handle the case when the relation between class labels and attributes is nonlinear. Furthermore, the RBF kernel uses less hyperparameters, which influence the complexity of the model selection, than for example a polynomial kernel would have. The RBF kernel also has fewer numerical difficulties in contrast to polynomial kernels. Find the recognition rate by using SVM.

C. Dominant and complimentary expression

It is clear that different regions of the face convey different types of affective information and some parts of the face convey some emotions better than the others. While there is some evidence that the eye region is more informative for human subjects in recognising angry, fearful and sad faces, disgust and happiness appears to be mainly conveyed by the mouth. Surprise may be similarly recognised from both regions. Human subjects recognise happiness from the bottom half of the face as accurately and even faster than from the whole face [START_REF] Galvo | Facial expression recognition in peripheral versus central vision: role of the eyes and the mouth[END_REF], [START_REF] Calder | Configural information in facial expression perception[END_REF].

Affective information is not distributed equally across the face and different emotions utilise different parts of the face. Blais [START_REF] Blais | The eyes are not the window to basic emotions[END_REF] found that the mouth area appears to be the most informative part of the face when recognising emotions. The mouth region is the biggest region of the face in area and has the ability to move the most. Hence it is the most informational part of the human face [START_REF] Blais | The eyes are not the window to basic emotions[END_REF].

Due to the intricacies of the human face the seven basic emotions are not nearly enough to cover the whole emotional range of a person. Emotions can often come in combinations, for example, a person can be both happy and surprised at the same time.

In an attempt to alleviate this problem we have utilised a complementary emotion in addition to the dominant one. In the proposed method, the complementary emotion is extracted from the either the lower or upper half of the face while the information about the dominant emotion is extracted from the whole face. If the dominant emotion is classified as one of the emotions that is mainly influenced by the upper part of the face (anger, fear, sadness), then we search for the complementary emotion in the lower half of the face and vice versa.

Such method allows to classify an expression as a combination of two emotions and provides us with 30+ variations, if we classify for seven basic emotions. If the dominant emotion is classified as 'happy' and the complementary emotion as 'surprise', we would call the resulting emotion combination 'Surprisingly happy'. The experimental results, reported in the Section III, show that not all 49 multiemotional expression exist. However the results state that the proposed method can always recognises 30+ emotions.

III. THE EXPERIMENTAL RESULTS AND DISCUSSIONS

In this work Cohn-Kanade AU-Coded Facial Expression (CK+) database [START_REF] Ambadar | All smiles are not created equal: Morphology and timing of smiles perceived as amused, polite, and embarrassed/nervous[END_REF], [START_REF] Kanade | Comprehensive database for facial expression analysis[END_REF] and Japanese Female Facial Expression (JAFFE) image database [START_REF] Dailey | Evidence and a computational explanation of cultural differences in facial expression recognition[END_REF], [START_REF] Lyons | Coding facial expressions with gabor wavelets[END_REF] have been used in order to conduct the experimental results. The first database contains 327 samples of the 8 facial expressions (seven basic facial expressions an a neutral expression) of 100 distinctive people. The second database contains 213 images of 7 facial expressions (six basic facial expressions and one neutral) posed by 10 Japanese female models.

The results of applying the proposed method on all of the faces in CK+ database are shown in Table II. It has the recognition rate of 100% for angry, happy and surprised emotion. Minimum recognition rates belong to contempt and sadness emotions.

Table III illustrates the recognition rates of complimentary emotions for all available dominant emotions in the CK+ database. The results are a combination of what proposed in [START_REF] Blais | The eyes are not the window to basic emotions[END_REF] and [START_REF] Calder | Configural information in facial expression perception[END_REF]. Results marked in bold are considered to be The emotion recognition rates that is shown in Table IV and Table V are less than those for CK+ database. In this database, most recognised images belong to the happy emotion category.

In order to have quantitative results on the proposed dominant and complementary emotion recognition, sets of images with the recognised labels, as shown in Figs 3,4, 5 and 6, from CK+ and JAFFE databases were shown to randomly chosen 64 people. They were asked to vote whether the recognised dominant and complementary emotion have been recognised correctly or not. The voting results showed that 73.88% of the people agreed on the tagged dominant and complementary emotions by the proposed method. The results of the voting is shown in Fig. 7. As it is illustrated in Fig. 7, except for angrily fearful in both CK+ and JAFFE databases the majority of the voters agreed on the correct tagging by the proposed method. the concept of a complementary emotion. In this research, the focus was on automatic recognition of a complementary emotion in addition to the dominant emotion. This research has accomplished recognising complementary emotions from samples labelled for only dominant emotions illustrating that the emotional range of a person is much greater than seven basic emotions. The developed methodology has been tested on existing datasets but to our knowledge there is currently no database with labels for complementary emotions to compare the results with. We hope to pave way for new research by the introduction of the complementary emotion concept. Creation of a database with labels for both dominant and complementary emotions would greatly promote research in this area. Inclusion of psychologists in future research would provide deeper insight into human emotions.

Fig. 1 .

 1 Fig. 1. Flowchart of the proposed method.

Fig. 2 .

 2 Fig. 2. Extracted feature points.

Algorithm 1

 1 PROPOSED METHOD SCHEMA Input: Facial images from database Output: Emotional Recognition Illumination enhancement for each image of database. localization of enhanced images. Normalization of localized images. Face detection by using Viola-Jones and remove background. feature extraction by using LGBP. Find the euclidean distance between neutral and full emotion. Divided data to upper and lower face.

Fig. 3 .

 3 Fig. 3. Emotion pairs with the dominant emotions being angry, fear and sadness respectively for each row, complementary emotions are recognised from the lower part of the face for the CK+ database. The first row illustrates from left to right emotion pairs angry, fearfully angry and surprisingly angry. The second row illustrates from left to right emotion pairs angrily feared, fear, happily feared and sadly feared. The third row illustrates from left to right emotion pairs angrily sad, fearfully sad and sad. Due to copyright issues all example emotion pairs from CK+ database are not displayable.

Fig. 4 .

 4 Fig. 4. Emotion pairs with the dominant emotions being angry, fear and sadness respectively for each row, complementary emotions are recognised from the lower part of the face for the JAFFE database. The first row illustrates from left to right emotion pairs angry, disgustingly angry, fearfully angry, sadly angry and surprisingly angry. The second row from left to right illustrates emotion pairs fearfully angry, disgustingly feared, fear, sadly feared and surprisingly feared. The third row illustrates from left to right emotion pairs angrily sad, happily sad, sad and surprisingly sad.

Fig. 5 .

 5 Fig. 5. Emotion pairs with the dominant emotions being disgust, happy and surprise respectively for each row, complementary emotions are recognised from the upper part of the face for the CK+ database. The first row illustrates from left to right emotion pairs angrily disgusted and disgust. The second row illustrates from left to right emotion pairs fearfully happy, happy and surprising happy. The third row illustrates from left to right emotion pairs fearfully surprised and sadly surprised. Due to copyright issues all example emotion pairs from CK+ are not displayable.

Fig. 6 .

 6 Fig. 6. Emotion pairs with the dominant emotions being disgust, happy and surprise respectively for each row, complementary emotions are recognised from the upper part of the face for the JAFFE database. The first row illustrates from left to right emotion pairs disgust, fearfully disgusted, sadly disgusted and surprisingly disgusted. The second row illustrates from left to right emotion pairs fearfully happy, happy, sadly happy and surprisingly happy. The third row illustrates from left to right emotion pairs angrily surprised, happily surprised, sadly surprised and surprise.

Fig. 7 .

 7 Fig. 7. The average result of the voting of 64 randomly chosen people on whether the facial emotion image is tagged correctly with the recognised emotion. The blue bar shows the percentage of the people who agreed with the recognised emotion and the red bar shows the percentage of the people who disagreed with the recognised emotion.

TABLE I DOMINANT

 I

			-COMPLEMENTARY EMOTION COMBINATIONS
		Angry	Contempt	Disgust	Fear	Happy	Sadness	Surprise
	Angry	angry	contemptly angry	disgustingly angry	fearfully angry	happily angry	sadly angry	surprisingly angry
	Contempt	angrily contempt	contempt	disgustingly contempt	fearfully contempt	happily contempt	sadly contempt	surprisingly contempt
	Disgust	angrily disgusted	contemptly disgusted	disgust	fearfully disgusted	happily disgusted	sadly disgusted	surprisingly disgusted
	Fear	angrily fearful	contemptly fearful	disgustingly fearful	fearful	happily fearful	sadly fearful	surprisingly fearful
	Happy	angrily happy	contemptly happy	disgustingly happy	fearfully happy	happy	sadly happy	surprisingly happy
	Sadness	angrily sad	contemptly sad	disgustingly sad	fearfully sad	happily sad	sad	surprisingly sad
	Surprise	angrily surprised	contemptly surprised	disgustingly surprised	fearfully surprised	happily surprised	sadly surprised	surprised

TABLE II THE

 II DOMINANT EMOTION FOR THE CK+ DATABASE

		Recognition
		Rate (%)
	Angry	100
	Contempt	89
	Disgust	98
	Fear	96
	Happy	100
	Sadness	89
	Surprise	100
	pure emotions as the dominant and complementary emotions
	overlap.

TABLE III ANGRY

 III , FEAR, SADNESS GET COMPLEMENTARY EMOTION FROM LOWER FACE AND CONTEMPT, DISGUST, HAPPY, SURPRISE FROM UPPER FACE FOR THE CK+ DATABASE.

		Angry	Contempt	Disgust	Fear	Happy	Sadness	Surprise
	Angry	34	3	1	2	1	2	2
	Contempt	3	8	1	1	0	3	2
	Disgust	7	2	49	1	0	0	0
	Fear	3	1	1	14	2	1	3
	Happy	3	2	4	5	51	2	2
	Sadness	4	1	2	3	0	15	3
	Surprise	5	2	3	3	2	3	65

TABLE IV THE

 IV DOMINANT EMOTION FOR THE JAFFE DATABASE.

		Recognition
		Rate (%)
	Angry	67
	Disgust	55
	Fear	84
	Happy	87
	Sadness	45
	Surprise	73
	IV. CONCLUSION
	In this paper, a new methodology was developed for recog-
	nising more than the seven classical emotions by introducing

TABLE V ANGRY

 V , FEAR, SADNESS GET COMPLEMENTARY EMOTION FROM LOWER FACE AND DISGUST, HAPPY, SURPRISE FROM UPPER FACE FOR THE JAFFE DATABASE.

		Angry	Disgust	Fear	Happy	Sadness	Surprise
	Angry	19	3	1	0	2	1
	Disgust	0	18	3	0	1	0
	Fear	2	3	15	1	1	1
	Happy	0	0	1	25	2	2
	Sadness	4	0	0	1	16	2
	Surprise	1	0	0	2	2	20