L I Galtchouk
email: leonid.galtchouk@math.unistra.fr

S M Pergamenshchikov
email: serge.pergamenchtchikov@univ-rouen.fr

Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions. *

Keywords: drift coefficient estimation, discrete data, efficient procedure, ergodic diffusion process, minimax, nonparametric sequential estimation AMS (1991) Subject Classification : primary 62G07; secondary 62G20

A truncated sequential procedure is constructed for estimating the drift coefficient at a given state point based on discrete data of ergodic diffusion process. A nonasymptotic upper bound is obtained for a pointwise absolute error risk. The optimal convergence rate and a sharp constant in the bounds are found for the asymptotic pointwise minimax risk. As a consequence, the efficiency is obtained of the proposed sequential procedure.

Introduction

In this paper we consider the following diffusion model: dy t = S(y t ) dt + σ(y t )dW t , 0 ≤ t ≤ T , (1.1) where (W t ) t≥0 is a scalar standard Wiener process, S(•) and σ(•) are unknown functions. This model appears in a number of applied problems of stochastic control, filtering, spectral analysis, identification of dynamic system, financial mathematics and others (see [START_REF] Arato | Linear stochastic systems with constant coefficients[END_REF], [START_REF] Bensoussan | Optimal control of partially observed systems[END_REF], [START_REF] Karatzas | Methods of Mathematical Finance[END_REF], [START_REF] Yu | Statistical Inferences for Ergodic Diffusion Processes[END_REF], [START_REF] Sh | Statistics of a random process[END_REF] and others for details).

The problem is to estimate the function S(x) at a point x 0 based on the discrete time observations

(y t j ) 1≤j≤N , t j = jδ , (1.2)
where N = [T /δ] and the frequency δ = δ T ∈ (0, 1) is a function of T that will be specified later.

The estimation problem of the function S was studied in a number of papers in the case of complete observations, that is when a continuous trajectory (y t ) 0≤t≤T was observed. In the parametric case this problem was considered apparently for the first time in the paper [START_REF] Arato | On parameter estimation of a complex stationary gaussian process[END_REF] for diffusion model of the axis of the equator precession. In that paper a non-asymptotic distribution of the maximum likelihood estimator was found for a special Ornstein-Uhlenbeck process.

It should be noted that investigating non asymptotic properties of parametric estimators in the models like to (1.1) comes to the analysis of non linear functionals of observations. At the most cases this analysis is unproductive in non asymptotic setting. In order to overcome the technique difficulties the sequential analysis methods were used in [START_REF] Sh | Statistics of a random process[END_REF] and [START_REF] Novikov | Sequential estimation of the parameters of the diffusion processes[END_REF] for estimating a scalar parameter. In [START_REF] Konev | Sequential estimation of the parameters of random processes with continuous time[END_REF] these methods were extended to estimating a multi-dimensional parameter as well. Moreover, in [START_REF] Konev | On truncated sequential estimation of the parameters of diffusion processes[END_REF] truncated sequential procedures were developed that economizes the observation time.

In [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF] and [START_REF] Galtchouk | Non parametric sequential minimax estimation of the drift coefficient in diffusion processes[END_REF] a sequential approach was proposed for the pointwise nonparametric estimation in the ergodic models (1.1). Later in [START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF] the efficiency was studied of the proposed sequential procedures .

A sufficiently complete survey one can find in [START_REF] Yu | Statistical Inferences for Ergodic Diffusion Processes[END_REF] on the nonparametric estimation in the ergodic model (1.1) when non sequential approaches are used.

In the cited papers estimation problems were studied based on complete observations (y t ) 0≤t≤T . In practice, usually one has at disposal discrete time observations even for continuous time models.

A natural question arises about proprieties and the behavior of estimates based on discrete time observations for such models. These problems were studied for several models. We cite some of them.

The LAN property was studied in the papers [START_REF] Gobet | Lan property for ergodic diffusions with discrete observations[END_REF], [START_REF] Kessler | Estimation of an ergodic diffusion from discrete observations[END_REF] and [START_REF] Yoshida | Estimation for diffusion processes from discrete observations[END_REF] for parametric ergodic diffusion models observed at discrete times. Parametric estimation was investigated in [START_REF] Jacod | Parametric inference for discretely observed non-ergodic diffusions[END_REF] for a non ergodic diffusion. Nonparametric estimation setting for models of kind (1.1) was considered firstly for estimating the unknown diffusion coefficient σ(•) based on discrete time observations on the interval [0, 1] (see, for example, [START_REF] Florens-Zmirou | On estimating the diffusion coefficient from discret observations[END_REF], [START_REF] Genon-Catalot | Non-parametric estimation of the diffusion coefficient by wavelets methods[END_REF], [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF], [START_REF] Jacod | Non-parametric kernel estimation of the coefficient of a diffusion[END_REF] and the references therein). Later, in [START_REF] Gobet | Nonparametric estimation of scalar diffusions based on low frequency data[END_REF] kernel estimates of drift and diffusion coefficients were studied for non ergodic process (1.1) taking the values into the interval [0, 1].

So far as concerning the estimation in ergodic case, it should be noted that a sequential procedure was proposed in [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF] for nonparametric estimating the drift coefficient of the process (1.1) in the integral metric. Some upper and lower asymptotic bounds were found for the L p -risks. Later, in the paper [START_REF] Comte | Nonparametric estimation for a discretely observed integrated diffusion model[END_REF] a nonasymptotic oracle inequality was proved for the drift coefficient estimation problem in a special empiric quadratic risk based on discrete time observations. This paper deals with the drift coefficient efficient nonparametric estimating at a given state point based on discrete time observations (1.2) in the absolute error risk. We find the optimal minimax convergence rate and we study the lower bound normalized by this convergence rate.

Our approach is based on the sequential analysis and it was developed in the papers [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF], [START_REF] Galtchouk | Non parametric sequential minimax estimation of the drift coefficient in diffusion processes[END_REF], and [START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF] for the nonparametric estimation.

Let us remind that in the case of complete observations (that is, when a whole trajectory is observed) the sequential estimate efficiency was proved by making use of uniform concentration inequalities (see [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes[END_REF]), besides the weak Hölder space functions S were used.

As it turns out later in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF], the efficient kernel estimate in the above given sense provides to construct a selection model adaptive procedure that appears efficient in the quadratic L 2 -metric.

Therefore, in order to realize this program in the case of discrete time observations, one needs to obtain the relative concentration inequalities, that is done in [START_REF] Galtchouk | Uniform concentration inequality for discrete-time ergodic diffusion processes[END_REF]. It should be noted that to obtain nonasymptotic concentration inequalities we make use of nonasymptotic bounds uniform over functions S and σ for the convergence rate in the ergodic theorem for the process (1.1). The latest result is proved in [START_REF] Galtchouk | Geometric ergodicity for families of homogenuous markov chains[END_REF] and it based on a new approach using Lyapounov's functions and the coupling method.

Further in this paper, by making use of the concentration inequalities we find the explicit constant in the upper bound for weak Hölder's risk normalized by the optimal convergence rate and we prove that this upper bound is best over all possible estimates. It means the procedure is efficient.

The paper is organized as follows. In Section 2 we describe the functional classes. In Section 3 the sequential procedure is constructed. In Section 4 we obtain a nonasymptotic upper bound for the absolute error pointwise risk of the sequential procedure. In Section 5 we show that the proposed procedure is asymptotically efficient for the pointwise risk. All proofs are given in Section 6. In the Appendix we give all necessary technical results.

Functional class

We consider the pointwise estimation problem for the function S(•) at a fixed point x 0 ∈ R for the model (1.1) with unknown diffusion coefficient σ. It is clear that to obtain a good estimate for the function S(•) at the point x 0 it is necessary to impose some conditions on the function ϑ = (S, σ) which provide that the observed process (y t ) 0≤t≤T returns to any vicinity of the point x 0 infinitely many times.

In this Section we describe the weak Hölder functional class which guarantees the ergodicity property for this model. Firstly, for some x * ≥ |x 0 | + 1, M > 0 and L > 1 we denote by Σ L,M the class of functions S from C 1 (R) such that sup

|x|≤x * |S(x)| + | Ṡ(x)| ≤ M and -L ≤ inf |x|≥x * Ṡ(x) ≤ sup |x|≥x * Ṡ(x) ≤ -L -1 .
Moreover, for some fixed parameters 0 < σ min ≤ σ max we denote by V the class of the functions σ from C 2 (R) such that

0 < σ min ≤ inf x∈R min (|σ(x)| , | σ(x)| , |σ(x)|) ≤ sup x∈R max (|σ(x)| , | σ(x)| , |σ(x)|) ≤ σ max < ∞ . (2.1)
In this paper we make use of the week Hölder functions introduced in [START_REF] Galtchouk | Asymptotically efficient estimates for non parametric regression models[END_REF].

Definition 2.1. We say that a function S satisfies the weak Hölder condition at the point x 0 ∈ R with the parameters h,ǫ > 0 and exponent β = 1 + α, α ∈ (0, 1), if S ∈ C 1 (R) and its derivative satisfies the following inequality

1 -1 z 1 0 ( Ṡ(x 0 + uzh) -Ṡ(x 0 )) du dz ≤ ǫ h α . (2.2)
We will denote the set of all such functions by H w x 0 (ǫ, β, h).

Note that the inequality (2.2) implies that sup

S∈H w x 0 (ǫ,β,h) Ω x 0 ,h (S) ≤ ǫ h β , (2.3)
where

Ω x 0 ,h (S) = 1 -1 (S(x 0 + hz) -S(x 0 )) dz .
Let us denote by H w x 0 ,M (ǫ, β, h) the set of all functions D from H w x 0 (ǫ, β, h)

such that sup x∈R (|D(x)| + | Ḋ(x)|) ≤ M/2 and D(x) = 0 for |x| ≥ x * . Let S 0 be a function from Σ L,M/2 such that lim h→0 h -β Ω x 0 ,h (S 0 ) = 0 . (2.4)
We denote

U M (x 0 , β) = S 0 + H w x 0 ,M (ǫ, β, h) , (2.5)
where h = T -1/(2β+1) and

ε = ε T = 1 (ln T ) 1+γ (2.6)
for some 0 < γ < 1. Obviously that U M (x 0 , β) ⊂ Σ L,M . Now we set

Θ β = U M (x 0 , β) × V . (2.7)
It should be noted that, for any ϑ ∈ Θ β , there exists the invariant density which is defined as

q ϑ (x) = R σ -2 (z) e S(z) dz -1 σ -2 (x) e S(x) , (2.8)
where S(x) = 2

x 0 σ -2 (v)S(v)dv (see,e.g., [START_REF] Gihman | Stochastic differential equations[END_REF], Ch.4, 18, Th2). It is easy to see that this density is uniformly bounded in the class (2.7) , i.e.

q * = sup x∈R sup ϑ∈Θ β q ϑ (x) < +∞ (2.9)
and bounded away from zero on the interval [x 0 -1, x 0 + 1], i.e.

q * = inf x 0 -1≤x≤x 0 +1 inf ϑ∈Θ β q ϑ (x) > 0 . (2.10) For any R → R function f from L 1 (R) we set m ϑ (f ) = R f (x) q ϑ (x) dx . (2.11)
Assume that the frequency δ in the observations (1.2) is of the following asymptotic form (as T → ∞)

δ = δ T = O ε T T , (2.12)
where the function ε T is introduced in (2.6). Now, for any estimate (i.e. any (y t ) 0≤t≤T measurable function) ST (x 0 ) of S(x 0 ), we define the pointwise risk as follows

R ϑ ( S T ) = E ϑ | S T (x 0 ) -S(x 0 )| .
(2.13)

Sequential procedure

In order to construct an efficient pointwise estimate of S we begin with estimating the ergodic density q = q ϑ at the point x 0 from first N 0 observations. We choose

N 0 = N γ 0 and 2/3 < γ 0 < 1 . (3.1)
We will make use of the following kernel estimate

q T (x 0 ) = 1 2(N 0 -1)ς N 0 -1 j=0 Q y t j -x 0 ς , (3.2)
where

Q(y) = 1 (|y|≤1) and ς = ς T is a function of T such that ς T = o(T -γ 0 /2 ) as T → ∞ .
For T ≥ 3 we set

q T (x 0 ) =      (υ T ) 1/2 , if q T (x 0 ) < (υ T ) 1/2 ; q T (x 0 ) , if (υ T ) 1/2 ≤ q T (x 0 ) ≤ (υ T ) -1/2 ; (υ T ) -1/2 , if q T (x 0 ) > (υ T ) -1/2 , (3.3)
where

υ T = 1 (ln T ) a 0 and a 0 = √ γ + 1 -1 10 .
The properties of the estimates q T (x 0 ) and q T (x 0 ) are studied in the Appendix. Let us define the following stopping time

̟ = ̟ T = inf    j ≥ N 0 : j i=N 0 φ i ≥ H T    , (3.4)
where

φ i = χ h,x 0 (y t i-1 )1 {i≤N } + 1 {i>N } , χ h,x 0 (y) = Q ((y -x 0 )/h) and h is a positive bandwidth. We put ̟ = ∞ if the set {•} is empty. Obviously, that in our case ̟ < ∞ a.s. since i≥N 0 φ i = +∞ a.
s. Now we have to choose the threshold H T . Note that in order to construct an efficient estimate one should use all, i.e. N, observations. Therefore, the threshold H T should provide the asymptotic relations ̟ T ≈ N and

N i=N 0 φ i = N i=N 0 χ h,x 0 (y t i-1 ) as T → ∞ .
In order to obtain these relations, note that due to the ergodic theorem

N i=N 0 χ h,x 0 (y t i-1 ) ≈ 2h(N -N 0 )q T (x 0 ) .
Hence, replacing in the right-hand side term the ergodic density with its corrected estimate yields the following definition of the threshold

H = H T = h(N -N 0 )(2 q T (x 0 ) -υ T ) . (3.5)
Note that in [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF] it has been shown that the such form of the threshold H T provides the optimal convergence rate. It is clear that

̟ ≤ N + H T < N + h(N -N 0 )/ √ υ T ,
i.e. the stopping time ̟ is bounded. Now on the set Γ T = {̟ ≤ N} we define the correction coefficient α = α T as

α T = H T -̟-1 j=N 0 χ h,x 0 (y t j-1 ) χ h,x 0 (y t ̟-1 ) , i.e. on the set Γ T ̟-1 j=N 0 χ h,x 0 (y t j-1 ) + αχ h,x 0 (y t ̟-1 ) = H T .
Moreover, on the Γ c T we set α T = 1. Using this definition we introduce the weight sequence

α j = 1 {j<̟} + α1 {j=̟} . (3.6)
One can check directly that, for any j ≥ 1, the coefficients α j are F t j-1 measurable, where F t = σ (y u , 0 ≤ u ≤ t). Now we define the sequential estimate for S(x 0 ) as

S * h,T (x 0 ) = 1 δH T   N j=N 0 α j χ h,x 0 (y t j-1 ) ∆y t j   1 Γ T .
(3.7)

In the next section we study the non asymptotic properties of this procedure.

Remark 3.1. Note that the correction coefficient of type (3.6) was used firstly in the paper [START_REF] Borisov | Sequential estimation of parameters of discrete processes[END_REF] in order to construct an unbiased estimate of a scalar parameter in autoregressive processes AR [START_REF] Arato | Linear stochastic systems with constant coefficients[END_REF]. Here we make use of the same idea but for a nonparametric procedure.

Non asymptotic estimation

As we will see later in studying the estimate (3.7), the approximation term plays the crucial role. In our case, this term is of the following form

Υ 1,T = 1 δH T N j=N 0 α j χ h,x 0 (y t j-1 ) ̺ j , (4.1)
where

̺ j = t j t j-1
(S(y u ) -S(y t j-1 ))du. One can show the following result.

Proposition 4.1. For any T ≥ 3,

sup ϑ∈Σ L,M ×[0,σ max ] E ϑ Υ 2 1,T ≤ L 2 L 1 δ , (4.2)
where L = max(L, M) and

L 1 = 2 σ 2 max + 2δ(M 2 + L 3 D * + L 2 x 2 * ) . Moreover, we set Υ 2,T = 1 δH T N j=N 0 α j χ h,x 0 (y t j-1 ) ̺ * j , (4.3)
where

̺ * j = t j t j-1 (σ(y u ) -σ(y t j-1 ))dW u .
Proposition 4.2. For any T ≥ 3 for which 0 < δ ≤ 1, one has

sup ϑ∈Σ L,M ×[0,σ max ] E ϑ Υ 2,T 2 ≤ σ 2 max L 1 h(N -N 0 ) √ υ T . (4.4)
Proofs of Propositions 4.1-4.2 are given in the Appendix. Now we introduce the approximation term, i.e.

B T = 1 H T N j=N 0 α j f h (y t j-1 ) (4.5)
with f h (y) = χ h,x 0 (y)(S(y) -S(x 0 )). Taking into account this formula, we can represent the error of estimate (3.7) on the set Γ T as

S * h,T (x 0 ) -S(x 0 ) = Υ 1,T + B T + M T , (4.6)
where

M T = 1 δH T   N j=N 0 α j χ h,x 0 (y t j-1 )η j   with η j = t j t j-1
σ(y u ) dW u . Obviously, for any function S from Σ L,M , the term B T can be bounded as

|B T | ≤ h max |x-x 0 |≤h | Ṡ(x)| ≤ M h .
sup ϑ∈Σ L,M ×[0,σ max ] E ϑ M 2 T ≤ σ 2 max δh(N -N 0 ) √ υ T . (4.7)
Hence, we obtain the following upper bound.

Theorem 4.4. For any h > 0 and T ≥ 3, one has

sup ϑ∈Σ L,M ×[σ min , σ max ] E ϑ |S * h,T (x 0 ) -S(x 0 )| ≤ U * (δ, h, T ) + M Π * T , (4.8)
where

U * (δ, h, T ) = L δL 1 + M h + σ max δ h (N -N 0 )υ 1/4 T and Π * T = sup ϑ∈Σ L,M ×[σ min , σ max ] P ϑ Γ c T .
Let us study now the last term in (4.8).

Proposition 4.5. Assume that the parameter δ is of the following form (2.12) and h ≥ T -1/2 . Then, for any a > 0, lim

T →∞ T a Π * T = 0 . (4.9)
Proof of this proposition is given in the Appendix.

Asymptotic efficiency

First of all we study a lower bound for the risk (2.13). To this end we set

ς * ϑ (x 0 ) = 2q ϑ (x 0 ) σ(x 0 ) . (5.1)
This parameter provides a sharp asymptotic lower bound for the pointwise risk normalized by the minimax rate ϕ T = T β/(2β+1) .

Theorem 5.1. The risk defined in (2.13) admits the following lower bound

lim T →∞ ϕ T inf S T sup ϑ∈Θ β ς * ϑ (x 0 ) R ϑ ( S T ) ≥ E|ξ| , (5.2)
where infimum is taken over all possible estimate S T , ξ is a (0, 1) gaussian random variable.

Theorem 5.2. The kernel estimate S * h,T defined in (3.7) with h = T -1/(2β+1) satisfies the following asymptotic inequality

lim T →∞ ϕ T sup ϑ∈Θ β ς * ϑ (x 0 ) R ϑ (S * h,T ) ≤ E|ξ| .
where ξ is a (0, 1) gaussian random variable.

Notice that the Theorem 5.1 and Theorem 5.2 imply the following efficiency property Theorem 5.3. The sequential procedure (3.7) with h = T -1/(2β+1) is asymptotically efficient in the following sense

lim T →∞ ϕ T sup ϑ∈Θ β ς * ϑ (x 0 ) R ϑ (S * h,T ) = lim T →∞ ϕ T inf S T sup ϑ∈Θ β ς * ϑ (x 0 ) R ϑ ( S T ) ,
where infimum is taken over all possible estimate S T .

Remark 5.1. Note that the constant (5.1) provides the sharp lower bound for the minimax pointwise risk. The calculation of this constant is possible by making use of the weak Hölder class. This functional class was introduced in [START_REF] Galtchouk | Asymptotically efficient estimates for non parametric regression models[END_REF] for regression models. For the first time, the constant (5.1) was obtained in the paper [START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF] at the pointwise estimation problem of the drift based on continuous time observations of the process (1.1) with the unit diffusion.

Later this constant was used in the paper [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] to obtain the Pinsker constant for a quadratic risk at the adaptive estimation problem of the drift in the model (1.1) based on continuous time observations. Remark 5.2. Note also that in this paper the efficient procedure is constructed when the regularity is known of the function to be estimated. In the case of unknown regularity we shall use an approach based on the model selection similarly to that in the paper [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] which deals with continuous time observations. The announced result will be published in the next paper which is in the work.

6 Proofs

Lower bound

In this section the Theorem 5.1 will be proved. Let us introduce the model (1.1) with σ = 1, i.e. dy t = S(y t ) dt + dW t . (6.1)

Now we define the risk corresponding to this model as follows

R * S ( S T ) = E S | S T (x 0 ) -S(x 0 )| , (6.2)
where E S denotes the expectation with respect to the distribution P S of the process (6.1) in the space of continuous functions

C[0, T ]. It is clear that sup ϑ∈Θ β ς * ϑ (x 0 ) R ϑ ( S T ) ≥ sup S∈U M (x 0 ,β) 2q S (x 0 ) R * S ( S T ) , (6.3)
where q S is the invariant density for the process (6.1) which equals to q ϑ with σ = 1. Let now g be a continuously differentiable probability density on the interval [-1, 1]. Then, for any u ∈ R and 0 < ν < 1/4, we set

S u,ν (x) = S 0 (x) + u ϕ T V ν x -x 0 h ,
where h = T -1/(2β+1) and

V ν (x) = 1 ν ∞ -∞ 1 (|u|≤1-2ν) + 21 (1-2ν≤|u|≤1-ν) g u -x ν du .
It is easy to see directly that, for any 0 < ν < 1/4, V ν (0) = 1 and

1 -1 V ν (x) dx = 2 .
Therefore, denoting D(x) = S u,ν (x) -S 0 (x), we obtain

1 -1 (D(x 0 + hz) -D(x 0 ))dz = 0 .
Moreover, note that

| Ḋ(x)| = |u| ϕ -1 T h -1 | Vν x -x 0 h | ≤ |u| T -α/(2β+1) ν -2 ġ * ,
where ġ * = sup x | ġ(x)|. Therefore, for any fixed u ∈ R and 0 < ν < 1/4, the functions S u,ν belong to the class U M (x 0 , β) for sufficiently large T . It implies that, for any b > 0 and for sufficiently large T , we can estimate from below the right-hand term in the inequality (6.3) as sup

S∈U M (x 0 ,β) 2q S (x 0 ) R * S ( S T ) ≥ sup |u|≤b 2q S u (x 0 ) R * S u ( S T ) .
Taking into account here that lim

T →∞ sup |u|≤b q S u (x 0 ) -q S 0 (x 0 ) = 0 ,
we obtain the inequality (5.2) by making use of the Theorem 4.1 from [START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF].

Thus we obtain the Theorem 5.1.

Upper bound

We begin with stating the following result for the term (4.5).

Proposition 6.1. The function B T defined in (4.5) satisfies the following asymptotic property lim sup

T →∞ ϕ T sup ϑ∈Θ β E ϑ |B T | = 0 . (6.4)
The result is proved in the Appendix. Now we prove Theorem 5.2. To this end we set

φ(u) = +∞ j=N 0 φ i 1 {t i-1 <u≤t i } ,
where the random variables (φ i ) i≥1 are defined in (3.4). Using this function we introduce the stopping time

τ = τ T = inf t ≥ T 0 : t T 0 φ(u) du ≥ δH T , where T 0 = t N 0 = δN 0 . As usually, we put τ = ∞ if the set {•} is empty. Obviously that τ ≤ T + δH ≤ T + δh(N -N 0 )/ √ υ T .
Due to the equality

∞ T 0 φ(u) du = ∞, we obtain immediately that the ran- dom variable ξ T = 1 δH T τ T 0 φ(u) dW u (6.5)
is gaussian N (0, 1) (see, for example [START_REF] Sh | Statistics of a random process[END_REF], Ch.17). Now, using this property, we can rewrite the deviation (4.6) on set Γ T as

S * h,T (x 0 ) -S(x 0 ) = B * T + M (1)
T + σ(x 0 )M

(2)

T + σ(x 0 ) δH T ξ T , (6.6)
where

B * T = Υ 1,T + Υ 2,T + B T , M (1)
T = 1 δH T N j=N 0 α j χ h,x 0 (y t j-1 ) (σ(y t j-1 ) -σ(x 0 ))∆W t j and M (2) T = 1 δH T   ̟ j=N 0 α j φ j ∆W t j - τ T 0 φ(u) dW u   .
First we note that the definition of the sequence ( α j ) j≥1 in (3.6) implies N j=N 0 α j χ h,x 0 (y t j-1 ) ≤ H T a.s. (6.7)

Therefore, through the condition (2.1)

E ϑ M (1) T 2 = E ϑ   1 δH 2 T N j=N 0 α 2 j χ h,x 0 (y t j-1 ) σ(y t j-1 ) -σ(x 0 ) 2   ≤ E ϑ hσ 2 max δH T .
Taking into account here that, for T ≥ 3,

H T ≥ h(N -N 0 )(2 √ υ T -υ T ) ≥ h(N -N 0 ) √ υ T , (6.8)
we obtain lim

T →∞ ϕ T sup ϑ∈Θ β E ϑ |M (1)
T | = 0 .

Now we study the term M

(2)

T . To this end note that t ̟-1 < τ ≤ t ̟ . Therefore, we can represent this term as

M (2) T = 1 δH T α φ ̟ ∆W t ̟ -φ ̟ W τ -W t ̟ -1 .
Moreover, taking into account that the stopping times ̟ and τ are bounded, one gets

E ∆W t ̟ 2 = δ and E W τ -W t ̟-1 2 = E τ -t ̟-1 ≤ δ .
Therefore, from here and (6.8), we get

E ϑ M (2) T 2 ≤ 2E ϑ 1 δH 2 T ≤ 2 δ h 2 υ T (N -N 0 ) 2 and lim T →∞ ϕ T sup ϑ∈Θ β E ϑ |M (2)
T | = 0 .

To put an end to the proof of this theorem we have to show that lim

T →∞ sup ϑ∈Θ β E ϑ K T |ξ T | = 0 , (6.9)
where

K T = 1 2 q T (x 0 ) -υ T - 1 2q ϑ (x 0 )
.

It is easy to see that, for any T > 0, the random variable ξ T is conditionally gaussian with respect to F T 0 with the parameters (0, 1). Therefore,

E ϑ K T |ξ T | = 2 π E ϑ K T .
Taking into account here Lemma 8.5 we come to the equality (6.9). Hence Theorem 5.2.

Conclusions

In the paper we studied the estimation problem of the function S when its smoothness is known. In the case of unknown smoothness, in order to construct an adaptive estimate based on discrete time observations (1.2) in the model (1.1) we shall use the approach developed in [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF] for continuous time observations. The approach make use of Lepskii's procedure and sequential estimating. Note that Lepskii's procedure works here just thanks to sequential estimating since, for the sequential estimate of the function S, the stochastic term in the deviation (6.6) is a gaussian random variable. This provides correct estimating the tail distribution of a kernel estimate and adapting for the pointwise risk. Moreover, for adaptive estimating in the case of quadratic risk, we shall apply the selection model developed in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] to sequential kernel estimates (3.7). Note once more, that gaussianity of the stochastic term in (6.6) is a cornerstone result for obtaining a sharp oracle inequality. It permits to find Pinsker's constant like to [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] and then to study the proposed procedure efficiency. These both programs will be realized in the next paper.

8 Appendix

Geometric ergodicity

First of all we recall that in [START_REF] Galtchouk | Geometric ergodicity for families of homogenuous markov chains[END_REF] we have proved the following result.

Theorem 8.1. For any ǫ > 0, there exist constants R = R(ǫ) > 0 and κ = κ(ǫ) > 0 such that

sup u≥0 e κu sup g * ≤1 sup x∈R sup ϑ∈Σ L,M ×V |E ϑ,x g(y u ) -m ϑ (g)| (1 + x 2 ) ǫ ≤ R , where E ϑ,x (•) = E ϑ (•|y 0 = x), g * = sup x |g(x)|.

Concentration inequalities

For any R → R function f belonging to L 1 (R) we set

D n (f ) = n k=1 f (y t k ) -m ϑ (f ) . (A.1)
Now we assume that the frequency δ in the observations (1.2) is of the following form

δ = δ T = 1 (T + 1)l T , (A.2)
where the function l T is such that, lim Then, for any a > 0, lim

T →∞ l T T 1/2 = 0 and lim T →∞ l T ln T = +∞ , (A.
T →∞ T a sup h≥T -1/2 sup ϑ∈Θ β P ϑ |D N (χ h,x 0 )| ≥ κ T T = 0 . (A.5)

Proof of Proposition 4.1

First, we note that by the Bunyakovskii -Cauchy -Schwarz inequality

E ϑ ̺ 2 j (S)|F t j-1 ≤ δ L 2 t j t j-1 E ϑ (y u -y t j-1 ) 2 |F t j-1 du , where L = max(L, M). Note now that, for t j-1 ≤ u ≤ t j , E ϑ ((y u -y t j-1 ) 2 |F t j-1 ) ≤ 2δ u t j-1 E ϑ (S 2 (y v )|F t j-1 ) dv + σ 2 max ≤ 2δ 2 u t j-1 (M 2 + L 2 E ϑ (y 2 v |F t j-1 ) )dv + σ 2 max .
Due to Proposition 8.6, we can estimate the last conditional expectation as

sup ϑ∈Σ L,M ×[0,σ max ] sup t j-1 ≤u≤t j E ϑ y 2 u |F t j-1 ≤ D * L + y 2 t j-1
.

Therefore, taking into account that χ h,x 0 (y t j-1 )y 2

t j-1 ≤ x 2 * , we obtain sup t j-1 ≤u≤t j sup ϑ∈Σ L,M ×[0,σ max ] χ h,x 0 (y t j-1 ) E ϑ (y u -y t j-1 ) 2 |F t j-1 ≤ L 1 δ . (A.6) Therefore, sup j≥1 sup ϑ∈Σ L,M ×[0,σ max ] χ h,x 0 (y t j-1 ) E ϑ ̺ 2 j (S)|F t j-1 ≤ L 2 L 1 δ 3 .
Making use of the inequality (6.7) yields the following upper bound, through the Bunyakovskii -Cauchy -Schwarz inequality,

E ϑ Υ 2 T ≤ E ϑ 1 δ 2 H T N j=N 0 α j χ h,x 0 (y t j-1 )̺ 2 j = E ϑ 1 δ 2 H T N j=N 0 α j χ h,x 0 (y t j-1 )E ϑ ̺ 2 j |F t j-1 ≤ L 2 L 1 δ .
Hence Proposition 4.1.

Proof of Proposition 4.2

Note that by the condition (2.13)

E ϑ (̺ * j ) 2 |F t j-1 = t j t j-1 E ϑ σ(y u ) -σ(y t j-1 )) 2 |F t j-1 du ≤ σ 2 max t j t j-1 E ϑ (y u -y t j-1 ) 2 |F t j-1 du .
Therefore, using the inequality (A.6) we obtain sup

j≥1 sup ϑ∈Σ L,M ×[0,σ max ] χ h,x 0 (y t j-1 ) E ϑ (̺ * j ) 2 |F t j-1 ≤ σ 2 max L 1 δ 2 .
From here and (6.7) and, taking into account that 0 < α j ≤ 1, we obtain

E ϑ Υ 2 2,T = E ϑ 1 δ 2 H 2 T N j=N 0 α 2 j χ h,x 0 (y t j-1 )E ϑ (̺ * j ) 2 |F t j-1 ≤ σ 2 max L 1 E ϑ 1 H T .
Now the inequality (6.8) yields (4.4). Hence Proposition 4.2.

Proof of Proposition 4.3

Indeed, taking into account the inequalities (6.7) and (6.8), we obtain that, for any T ≥ 3,

E ϑ M 2 T = E ϑ 1 δ 2 H 2 T N j=N 0 χ h,x 0 (y t j-1 ) α 2 j E ϑ η 2 j |F t j-1 ≤ E ϑ 1 δ 2 H 2 T N j=N 0 α j χ h,x 0 (y t j-1 ) t j t j-1 σ 2 (y u ) du ≤ σ 2 max δh(N -N 0 ) √ υ T .
Hence Proposition 4.3.

Proof of Proposition 6.1

We start with setting

r T = (2 q T -υ T )h m ϑ (χ h,x 0 ) and N 1 = N 0 + r T (N -N 0 ) . Note that N 1 -N 0 ≤ q * √ υ T -1 N := N * 1 ,
for sufficiently large T . Moreover, we set

G T = 1 H T N 1 j=N 0 f h (y t j-1 ) and G T = G T -B T .
Using (2.11) we can represent the term G T as

G T = N 1 -N 0 H T m ϑ (f h ) + 1 H T N 1 j=N 0 f h (y t j-1 ) := G 1 (T ) + G 2 (T ) ,
where f h (y) = f h (y)m ϑ (f h ). Taking into account that m ϑ (χ h,x 0 ) ≥ 2hq * , we obtain

|G 1 (T )| = r T (N -N 0 )h H T |m * ϑ (h)| ≤ 1 2q * |m * ϑ (h)| , m * ϑ (h) = m ϑ (f h ) h .
Let us represent the last term as

m * ϑ (h) = q ϑ (x 0 )Ω x 0 ,h (S) + m ϑ (h) ,
where m ϑ (h) = 1 -1 (S(x 0 + hz) -S(x 0 )) (q ϑ (x 0 + hz)q ϑ (x 0 )) dz. Furtherer, by the definition (2.5), one has

Ω x 0 ,h (S) = Ω x 0 ,h (S 0 ) + Ω x 0 ,h (D) ,
for some function D from H w x 0 (ǫ, β, h). Therefore, the properties (2.3)-(2.4) and (2.6) yield lim

h→0 ϕ T sup S∈U M (x 0 ,β) Ω x 0 ,h (S) = 0 .
Obviously, that lim sup

h→0 h -2 sup ϑ∈Θ β | m ϑ (h)| < ∞ .
Hence, lim sup

T →∞ ϕ T sup ϑ∈Θ β E ϑ |G 1 (T )| = 0 .
Now we note that,

E ϑ G 2 2 (T ) = E ϑ 1 H 2 T   N 1 -1 j=N 0 Ψ j + f 2 h (y t N 1 -1 )   ,
where

Ψ j = f 2 h (y t j-1 ) + 2 f h (y t j-1 ) N 1 l=j+1 E ϑ f h (y t l-1
)|F t j-1 and F t = σ{y s , 0 ≤ s ≤ t}. Taking into account that (y t ) t≥0 is a homogeneous Markov process and that | f h (y)| ≤ 2Mh, we estimate from above the last conditional expectation, through the Theorem 8.1 (for ǫ = 1/2), as

E ϑ f h (y t l-1 )|F t j-1 = E ϑ,y t j-1 f h (y t l-j ) ≤ 2MhR 1 + y 2 t j-1 1/2 e -κt l-j ≤ 2MhR 1 + |y t j-1 | e -κδ(l-j) .
Therefore,

|Ψ j | ≤ 4M 2 h 2 1 + 2R(1 + |y t j-1 |)
e κδ -1 .

From here, bounding e κδ -1 by κδ, we get

E ϑ G 2 2 (T ) ≤ 8M 2 h 2 E ϑ 1 H 2 T N 1 j=N 0 1 + R κδ (1 + |y t j-1 |) ≤ 8M 2 h 2 1 + R κδ E ϑ (N 1 -N 0 ) H 2 T + 8M 2 h 2 R κδ E ϑ 1 H 2 T N 1 j=N 0 E ϑ (y 2 t j-1 |F t N 0 -1 ) 1/2
.

By making use of Proposition 8.6 one obtains

E ϑ G 2 2 (T ) ≤ 8M 2 h 2 1 + R κδ E ϑ N 1 -N 0 H 2 T 1 + D * L + |y t N 0 -1 | .
Now from (6.8) it follows that

N 1 -N 0 H 2 T = 1 H T m ϑ (χ h,x 0 ) ≤ 1 2h 2 √ υ T (N -N 0 )q * .
Thus,

sup ϑ∈Θ β E ϑ G 2 2 (T ) ≤ G * δ √ υ T (N -N 0 )
,

where

G * = 4M 2 (κ + R) 1 + 2 D * L + |y 0 | /(κq * ).
From this equality we obtain immediately lim

T →∞ ϕ T sup ϑ∈Θ β E ϑ |G 2 (T )| = 0 .
Let us estimate the term G T . Taking into account the lower bound (6.8) we get

| G T | ≤ Mh H T N 1 j=N 0 χ h,x 0 (y t j-1 ) -H T + 2Mh H T ≤ M √ υ T (N -N 0 ) N 1 j=N 0 χ h (y t j-1 ) + 2M √ υ T (N -N 0 )
, where χ h (y) = χ h,x 0 (y)m ϑ (χ h,x 0 ). By making use of the Theorem 8.1 with ǫ = 1/2 one gets

j≥N 0 E ϑ,y 0 | χ h (y t j-1 )| ≤ Re -κδ(N 0 -1) 1 -e -κδ 1 + D * L + |y 0 | .
This inequality implies directly lim

T →∞ ϕ T sup ϑ∈Θ β E ϑ | G T | = 0 .
Hence Proposition 6.1.

8.7

Properties of the estimate (3.3) Lemma 8.3. Assume that the parameter δ is of the form (2.12). Then, for any a > 0, lim

T →∞ T a sup ϑ∈Θ β P ϑ (| q T (x 0 ) -q ϑ (x 0 )| > υ T ) = 0 . Proof. Denoting ψ ς (y) = (1/ς)Q((y -x 0 )/ς) one has q T (x 0 ) -q ϑ (x 0 ) = 1 2 1 -1 (q ϑ (x 0 + ςz) -q ϑ (x 0 )) dz + 1 2(N 0 -1) D N 0 -1 (ψ ς ) . Therefore P ϑ (| q T (x 0 ) -q ϑ (x 0 )| > υ T ) ≤ P ϑ (| 1 -1 (q ϑ (x 0 + ςz) -q ϑ (x 0 )) dz| > υ T ) + P ϑ ( 1 (N 0 -1) D N 0 -1 (ψ ς ) > υ T ) .
The first term on the right-hand side equals to zero for sufficiently large T since

| 1 -1 (q ϑ (x 0 + ςz) -q ϑ (x 0 )) dz| ≤ ς 2 q * < υ T ,
for sufficiently large T , where q * = sup x sup ϑ | qϑ (x)| < ∞. Applying Theorem 8.2 to the second term on the right-hand side of the same inequality yields the Lemma 8.3.

Lemma 8.4. Assume that the parameter δ is of the form (2.12). Then, for any a > 0, lim

T →∞ T a sup ϑ∈Θ β P ϑ (| q T (x 0 ) -q ϑ (x 0 )| > υ T ) = 0 .
Proof. Note, that for sufficiently large T (for that ln T ≥ max(q * 2 , 1/q 2 * )),

| q T (x 0 ) -q ϑ (x 0 )| ≤ | q T (x 0 ) -q ϑ (x 0 )| .
The Lemma follows immediately from Lemma 8.3.

Lemma 8.5. Assume that the parameter δ is of the form (2.12). Then, lim sup

T →∞ 1 υ 1/2 T sup ϑ∈Θ β E ϑ 1 q T (x 0 ) -υ T - 1 q ϑ (x 0 ) ≤ 4 q * < ∞ . (A.7)
Proof. Indeed, for sufficiently large T for which

υ T ≤ min 1/(q * ) 2 , 1/4 ,
we obtain

E ϑ 1 q T (x 0 ) -υ T - 1 q ϑ (x 0 ) ≤ 2υ 1/2 T q * + 2 q * υ 1/2 T E ϑ | q T (x 0 ) -q ϑ (x 0 )| ≤ 4υ 1/2 T q * + 2 q * υ T P ϑ (| q T (x 0 ) -q ϑ (x 0 )| > υ T ) .
Now Lemma 8.4 implies the equality (A.7). Hence Lemma 8.5.

Moment inequality for the process (1.1)

We state the moment bound from [START_REF] Galtchouk | Geometric ergodicity for families of homogenuous markov chains[END_REF].

Proposition 8.6. Let (y t ) t≥0 be a solution of the equation (1.1). Then, for any z ∈ R and m ≥ 1,

sup u≥0 sup ϑ∈Σ L,M ×[0,σ max ] E ϑ,z (y u ) 2m ≤ (2m -1)!! (D * L + z 2 ) m , (A.8) where E ϑ,z (•) = E ϑ (•|y 0 = z) and D * = (M + Lx * + 2x * ) 2 (L + M) + σ 2 max .
Proof. To obtain this inequality we make use of the method proposed in ( [START_REF] Yu | Two-Scale Stochastic Systems[END_REF], p.20) for linear stochastic equations. First of all, note that thanks to Theorem 4.7 from [START_REF] Sh | Statistics of a random process[END_REF], for any T > 0, there exists some ǫ > 0 such that, for each ϑ ∈ Θ β and z ∈ R, |q ϑ (x 0 + hz)q ϑ (x 0 )| dz ≤ υ T /4 .

Therefore, for such T , P ϑ (̟ > N) ≤ P ϑ (| q T (x 0 )q ϑ (x 0 )| > υ T /8) By applying Lemma 8.3 and the inequality (A.5) we obtain (A.12).

+ P ϑ |D N 0 ,N -1 (χ h,x 0 )| > Nhυ T /2 .

Now we estimate the last term as

Proposition 4 . 3 .

 43 For any T ≥ 3, one has

Theorem 8 . 2 .

 82 [START_REF] Bensoussan | Optimal control of partially observed systems[END_REF] in particular, the function l T = (ln T ) 1+γ from (2.6) is of this kind. Moreover, let κ = κ T be a positive function satisfying the following properties lim ([START_REF] Galtchouk | Uniform concentration inequality for discrete-time ergodic diffusion processes[END_REF]) Assume that the frequency δ satisfies (A.2)-(A.3).

sup 0≤t≤T E 2 ≤ 2 (e

 0≤t≤T22 ϑ,z e ǫy 2 t < ∞ .(A.9)Let us denote D ϑ (y) = 2yS(y) + σ 2 (y) + κy 2 and κ = L -1 . Taking into account that 0 < κ < 1 and x * ≥ 1 we obtain that, for |y| ≤ x * ,|D ϑ (y)| ≤ x 2 * (2M + 1) + σ 2 max .Let now |y| ≥ x * . Denoting by y * the projection of y onto the interval [-x * , x * ] we obtain that 2yS(y) = 2yS(y * ) + 2y * (S(y) -S(y * )) + 2(yy * ) (S(y) -S(y * ))≤ 2|y|M + 2Lx * |yy * | -2κ|yy * | M + Lx * + 2x * )|y| -2κy 2 .Therefore, supϑ∈Σ L,M ×[0,σ max ] sup y∈R D ϑ (y) ≤ D * .By the Ito formula we obtaindy 2m u = -mκy 2m u dt + my 2(m-1) u D ϑ (y u ) + 2(m -1)σ 2 (y u ) dt + 2my 2m-1 u σ(y u )dW t .Moreover, the property (A.9) yields that, for any m ≥ 1, -mκ(t-s) y 2m-1 s σ(y s )dW s = 0 .Therefore,E ϑ y 2m t ≤ z 2m + m(2m -1)D * t 0 e -mκ(t-s) E ϑ y 2(m-1)s ds. Now the induction implies directly the bound (A.8). Hence Proposition 8.6. Proposition 8.7. Let (y t ) t≥0 be a solution of the equation (1.1). Then, for any z ∈ R and m ≥ 1, and for any stopping time τ taking values in [0, T ], one has sup ϑ∈Σ L,M ×[0,σ max ] E ϑ,z (y τ ) 2m ≤ B * (m, z) T (A.10) and sup ϑ∈Σ L,M ×[0,σ max ] E ϑ,z sup 0≤u≤T (y u ) 2m ≤ B * 1 (m, z) T , (A.11) where B * (m, z) = (2m -1)!! (D * L + z 2 ) m (D * + 2(m -1)σ 2 max ) and B * 1 (m, z) = 1 + mB * (m + 1, z).The proof of this proposition follows immediately from Proposition 1.1.5 in[START_REF] Yu | Two-Scale Stochastic Systems[END_REF].

5 1 - 1 q, sup ϑ∈Θ β 1 - 1

 51111 It is clear, that to show(3.7) it suffices to check that, for any a > 0, limT →∞ T a sup ϑ∈Σ L,M ×[σ min , σ max ] P ϑ (Γ c T ) = 0 . (A.12)Indeed, by the definition of ̟P ϑ (Γ c T ) = P ϑ x 0 (y t j-1 ) < H T   = P ϑ D N 0 ,N -1 (χ h,x 0 ) < (2 q Tυ Tm * ϑ (χ h,x 0 )) (N -N 0 )h , where D k,n (f ) = D n (f ) -D k (f ) and m * ϑ (χ h,x 0 ) = m ϑ (χ h,x 0 ) h = ϑ (x 0 + hz) dz .Taking into account the definition of υ T in (3.3) we obtain that, for sufficiently large T

P 4 +

 4 ϑ |D N 0 ,N -1 (χ h,x 0 )| > Nhυ T /2 ≤ P ϑ |D N -1 (χ h,x 0 )| > Nhυ T /P ϑ |D N 0 (χ h,x 0 )| > Nhυ T /4 .

The second author is partially supported by the RFBR-Grant 09-01-00172-a.


Ali Chamas

Al Ghouwayel
email: ali.ghouwayel@liu.edu.lb

Haj-Ali Amin
email: amin.hajali@liu.edu.lb

Zouhair El-Bazzal
email: zouhair.bazzal@liu.edu.lb

Yves Louët
email: yves.louet@supelec.fr

Towards a Triple Mode Common Operator FFT for SoftWare Radio Systems

A scenario to design a Triple Mode FFT is addressed. Based on a Dual Mode FFT structure, we present a methodology to reach a triple mode FFT operator (TMFFT) able to operate over three different fields: complex number domain C, Galois Fields GF (Ft) and GF (2 m ). We propose a reconfigurable Triple mode Multiplier that constitutes the core of the Butterflybased FFT. A scalable and flexible unit for the polynomial reduction needed in the GF (2 m ) multiplication is also proposed. An FPGA implementation of the proposed multiplier is given and the measures show a gain of 18 % in terms of performance-to-cost ratio compared to a "Velcro" approach where two self-contained operators are implemented separately.

I. INTRODUCTION

In recent years there has been an enormous proliferation of standards in broadcast television, radio and in mobile communications. Current examples include digital television (DVB, ISDB), digital radio (DAB), wireless LAN(Hiperlan, 802.11a, 802.11b, ..., 802.16m, ...), 2.5/3G, 4G and future mobile communications. These standards form the basis for an ever-growing number of sophisticated consumer electronic devices, each with the potential to sell in very high volumes.

In typical designs, these complex standards are implemented using dedicated architectures, which are optimized to reduce cost to the absolute minimum. Products developed using dedicated architectures are often difficult to upgrade in order to support changes to the standards or to add new features.

At the beginning of the 90's, a concept called SoftWare Radio (SWR) has emerged from demonstrations in military research to become a cornerstone of the third generation strategy for affordable, ubiquitous and global communications [START_REF] Mitola | Software Radio Architecture[END_REF]. This SWR technology is a way to design a sufficiently programmable and reconfigurable architecture able to support many different transmission standards on a common platform. The reconfigurability of a SWR system can offer a range of benefits at different levels. A radio system implemented on a reconfigurable architecture can be upgraded to fix bugs or to add functionalities, and it can also support new standards as it is assumed that there is sufficient flexibility in the architecture.

The communication chains of different standards, intended to be implemented on a common platform, have some common signal processing operations such as channel coding, modulation, equalization, etc. In order to exploit to a great advantage the commonalities among these communication tasks for different standards, one need firstly to identify these commonalities and secondly find the optimal way to implement a generic hardware platform with programmable modules. In this sense, a technique called parametrization has been introduced [START_REF] Jondral | A Software Defined Radio Structure for 2nd and 3rd Genaration Mobile Communications Standards[END_REF]. The key idea is to get an optimal sharing between hardware and software resources and find a best way to reuse some hardware and software modules without affecting the system's performances.

This paper addresses the problem of parametrization technique under the CO approach where it is in line with the optimal design of a SWR system intended to support several communication standards. In [START_REF] Ghouwayel | A Reconfigurable Architecture for the FFT Operator in a Software Radio Context[END_REF], we have investigated the frequency processing of cyclic codes particularly RS codes with the aim to insert the classical FFT operator, initially used for complex Fourier transform, in the encoding and decoding processes of RS codes. By examining the characteristics of the Fourier transform FFT-GF2 used to process the encoding and some decoding processes of classical RS codes defined over GF (2 m ), we found that the adequacy of its structure to the structure of FFT defined over C (FFT-C) is challenged by the transform length. That is, the most efficient algorithms applied to a transform of length 2 m in the case of FFT-C computations cannot be applied to compute FFT-GF2 since its transform length is of the form 2 m -1. For this, we thought to seek out a transform satisfying the FFT-C criteria while keeping in mind to find a way to include the FFT-GF2, used in the RS coding defined over GF (2 m ), in the intended common and reconfigurable FFT structure.

To be able to exploit the whole FFT-C structure we explored finite field transforms having a highly composite length. The candidate transforms were the Fermat transforms defined over GF (F t ) [3] [START_REF] Agarwal | Fast Convolution Using Fermat Number Transforms with Applications to Digital Filtering[END_REF]. These transforms are used to deal with a specific class of RS codes defined over GF (F t ). These codes were studied in 1976 where the authors have emphasized the importance of their codeword length allowing the use of efficient algorithms to compute their associated Fermat transforms [START_REF] Justesen | On the Complexity of Decoding Reed-Solomon Codes[END_REF] [START_REF] Reed | The Fast Decoding of Reed-Solomon Codes Using Fermat Transforms[END_REF]. These codes were also recommended for the use in spacecraft communications [7] where the RS(256,224) over GF (257) was studied to be used together with a convolutional code.

As a first contribution, we have investigated the redesign of the FFT-C in such a way to be able to provide two functionalities: complex Fourier transform and Fermat transform. Conceptually, the design of such a dual mode operator implies the design of arithmetical operators capable to operate over the two domains: C and GF (F t ). We have proposed reconfigurable architectures for the multiplier, adder and subtracter [START_REF] Ghouwayel | A Reconfigurable Butterfly Architecture for Fourier and Fermat Transforms[END_REF] [10]. These operators implemented on the complex butterfly (available in the FFT-C) led to a reconfigurable butterfly that will constitute the core of the dual mode operator. Based on the FFT-C structural strategy, we have designed the architecture of the DMFFT operator [START_REF] Ghouwayel | FPGA implementation of a reconfigurable FFT for multi-standard systems in software radio context[END_REF].

To evaluate the complexity and speed performances of this operator, we have considered its implementation on ALTERA's FPGA devices. Compared to a Velcro FFT/FNT operator, the DMFFT presented an important gain in terms of ALUTs and memory saving. We have shown that for a transform length N = 64 implemented with different wordlengths (9 ≤ n c ≤ 16), the DMFFT presents a memory saving between 20 and 30%, a gain in ALUTs and performanceto-cost ratio gain that go from 9.2 % up to 26 % and from 9.7 % up to 37.4 % respectively. This deviation of gain in ALUTs and in performance-to-cost ratio is directly related to the wordlength, so as n c increases, these gains decrease. For N = 256, the DMFFT presents the same memory saving and the other gains evolve in the same manner of that of DMFFT-64 but with lower values. This is due to the fact that the complexity of the DMFFT architecture is mainly dominated by the FFT-C architecture complexity which increases with the increase of the wordlength and the transform length. The high n c values are necessary to get a good FFT-C computations precision while in the Velcro FFT/FNT, a 9-bit wordlength is sufficient to implement the FNT. Then, for a transform length N ≤ 256, the use of the DMFFT is largely efficient and allows to process RS codes which have codeword length N ≤ 256 and whose principals are N =64, 128 and 256.

In [START_REF] Al Ghouwayel | On the FPGA Implementation of the Fourier Transform over Finite Fields GF (2 m )[END_REF], we have proposed a hardware design and efficient implementation of the FFT over GF (2 m ). The proposed design is based on the cyclotomic decomposition of polynomials representing the input data. A very high throughput rate is attained, that makes the FFT to be a Common Operator while it is capable to produce its functionality at least twice faster than the existing solutions.

Nevertheless, RS codes implemented in the various actual standards (ADSL, ATSC, DVB, GSM,... ) are codes defined over GF (2 m ). Then, the inclusion of the finite field Fourier transform defined over GF (2 m ) as an extra functionality to be provided by our designed CO is a necessity to be on actual standardization's trend.

In this paper we propose a scenario that aims to combine the FFT-GF2 with the DMFFT by exploiting the same logical cells at the gate level to obtain a reconfigurable Triple Mode FFT (TMFFT) operator. We propose a generic combined multiplier able to perform a standard binary multiplication and GF (2 m ) multiplications for m=6, 7 and 8. We also show that this multiplier can be easily implemented into the reconfigurable multiplier proposed in [START_REF] Ghouwayel | Complexity Evaluation of a Re-Configurable Butterfly with FPGA for Software Radio Systems[END_REF] which leads to a triple mode multiplier able to operate over three different fields: C, GF (F t ) and GF (2 m ). Having this triple mode multiplier and taking into account that the addition in GF (2 m ) can be realized by XOR gates, the necessary arithmetical tools to implement the FFT-GF2 are available in the DMFFT operator.

II. TOWARDS A COMBINED TRIPLE MODE FFT OPERATOR (TMFFT)

We propose a scenario that aims to combine the FFT-GF2 structure with the DMFFT structure on the same die area to obtain a combined and reconfigurable operator TMFFT. This combination can be achieved if the following two steps can be realized.

1) Providing the GF (2 m ) operations, mainly the multiplications, with the binary multipliers implemented in the DMFFT operator. 2) Incorporation of the FFT-GF2 physical structure into the DMFFT structure. We begin the discussion with the first step where the approach for a combined multiplier exploits the fact that the partial product generations of both GF (2 m ) multiplier and standard binary multiplier can be performed using the same array and interconnections between cells.

A. Basic binary multiplication

Let us consider the basic principle of the standard binary arithmetic multiplier. In 1964, Wallace [START_REF] Wallace | A Suggestion For A Fast Multiplier[END_REF] introduced a notion of a carry-save tree constructed from one-bit full adders as a way of reducing the number of partial product bits in a fast and efficient way. Later, Dadda [START_REF] Dadda | Some Schemes for Parallel Multipliers[END_REF] refined Wallace's method by defining a cell placement strategy that minimizes the number of full-adders and half adders, at the cost of a larger carry propagate adder. For our multiplier design, we will consider the Wallace's method since it is structurally more regular.

Wallace'method is based on parallel counters and the multiplication of two binary numbers is performed in the following sequence.

• Form all partial products in parallel with an array of AND gates. As for the GF (2 m ) multiplication, it is viewed as a polynomial multiplication modulo f (x), where f (x) is the irreducible polynomial characterizing the field GF (2 m ).

Let us consider the multiplication of two GF (2 m ) numbers A and B. A and B can be represented by means of the vector basis {α m-1 , ..., α 1 , α 0 }, where α is a primitive element of the field, as

A = a m-1 α m-1 + ... + a 1 α 1 + a 0 α m-1 , B = b m-1 α m-1 + ... + b 1 α 1 + b 0 α m-1 ,
where a i , b i are coefficients in GF [START_REF] Jondral | A Software Defined Radio Structure for 2nd and 3rd Genaration Mobile Communications Standards[END_REF]. The polynomial multiplication C(x) = A(x)B(x) mod f (x) can be calculated by firstly building the partial products

P i (x) = A(x)b i , f or i = 0, ..., m -1.
Secondly, since the polynomial's coefficients belong to GF (2), all partial products P i (x) have to be added modulo 2. A partial result C p (x) can be obtained with

C p (x) = ∑ P i(x) mod 2 C p (x) = c 0 + c 1 x + ... + c 2m-2 x 2m-2 ,
which is obviously not an element of the field GF (2 m ) and should be reduced modulo f (x).

A GF (2 m ) multiplier then performs two basic operations. The product of two elements and the modulo f (x) correction. The first operation can be performed by ANDing the corresponding a i and b i , for i = 0, ..., m -1, and subsequently adding the partial products modulo 2. These partial products must be arranged in rows, with each row shifted i positions to the left as shown in Fig. 2. This step is similar to the standard binary product with the only difference that the sum of partial products in this case is done modulo 2. Thus, an opportunity to exploit cells and the interconnection structure of partial product generator of a typical binary multiplier unit is possible. The modulo correction should be performed separately. Early designs of GF (2 m ) multipliers used a serial approach. Although serial GF (2 m ) multipliers have low hardware requirements, they are very slow. Consequently, several parallel designs have been proposed in the literature [START_REF] Mastrovito | VLSI Designs for Multiplications over Finite Fields GF (2 m )[END_REF] [16] [17] [START_REF][END_REF]. In [START_REF] Drescher | VLSI Architecture For Non-Sequential Inversion Over GF (2 m ) Using the Euclidean Algorithm[END_REF] [START_REF] Garcia | A Combined 16-bit Binary and Dual Galois Field Multiplier[END_REF], the idea to combine GF (2 m ) with a standard binary multiplier on DSP processor was investigated. In [START_REF] Drescher | VLSI Architecture For Non-Sequential Inversion Over GF (2 m ) Using the Euclidean Algorithm[END_REF], the proposed design is based on a Wallace tree multiplier which has been modified to perform either conventional binary or GF (2 m ) multiplication. Their polynomial reduction (or modulo correction) introduces a linear delay. In [START_REF] Garcia | A Combined 16-bit Binary and Dual Galois Field Multiplier[END_REF], a new wiring scheme, to avoid adding carries of partial product reduction, is proposed and a parallel polynomial reduction is used. The authors designed a multiplier capable of performing either 16-bit two's complement or unsigned multiplication, or two independent 8-bit GF (2 8 ) multiplications. In the following, we propose a general approach allowing the implementation of any GF (2 m ) multiplier, for 6 ≤ m ≤ 8, in the reconfigurable multiplier proposed in [START_REF] Ghouwayel | Complexity Evaluation of a Re-Configurable Butterfly with FPGA for Software Radio Systems[END_REF]. We start here from the standard binary structure of the multiplier shown in Fig. 3 The partial product reduction should be redesigned in such a way to avoid the carries propagation if the GF (2 m ) multiplication is to be performed. This can be realized by reconfiguring the wire connects of the "W3" cells. The new wiring scheme is illustrated in Fig. 4. This wiring scheme can be easily realized in a reconfigurable way on FPGA devices using a reprogrammable LUTs. Arriving at

W 3 2 0 2 0 2 0 2 0 2 1 W 2 2 0 2 0 2 0 2 1
Fig. 4. The (3,2) and (2,2) counters in GF (2 m ) multiplication the last step (the polynomial reduction), it can be performed in parallel by using the method considered in [START_REF] Gao | Custom VLSI Design of Efficient Low Latency and Low Power Finite Field Multiplier for Reed-Solomon Codec[END_REF]. Let P (α) be the extended result before the polynomial reduction. P (α) can be expressed as

P (α) = 2m-2 ∑ i=m p i α i + m-1 ∑ i=0 p i α i , ( 1
)
where α i for m ≤ i ≤ 2m -2 can be substituted by

A i (α) = m-1 ∑ j=0 a i,j α j .
The A i (α) are the canonical representations in the field's base of the field elements α i that appear in the first summation in Equation 1. Let us consider an example of multiplication of two elements A and B in GF (2 4 ). Let f (x) = x 4 + x + 1 be the primitive polynomial and

A = α 10 = 0.α 3 + α 2 + α + 1 = (0111), B = α 12 = α 3 + α 2 + α + 1 = (1111).
The multiplication operation is described in Fig. 5. The classical method of multiplication shown in the upper left side of Fig. 5 is expressed by hardware circuit which is based on Wallace tree. This tree, originally designed to perform the standard binary multiplication, is modified in such a way to avoid the carries propagation between the cells constituting the tree. In this approach we have considered the disconnection of the carry's outputs and the corresponding neighboring cell inputs that are set to zero. The disconnected wires are represented by black dots. This wiring scheme can be realized by means of reprogrammable LUT1s that connect the carry outputs of each A i B i unit to the corresponding entries of the W 3 cells. A LUT1 maintain the carry propagation (by connecting the input to the output) in the case of standard binary multiplication and set its output to zero in the case of GF (2 m ) multiplication.

1 1 0 1 : . 0 . 0 . 0 ) ( 0 1 2 3 4 5 6 P 0 1 1 0 0 . 0 2 3 5 0 0 0 0 0 1 1 0 1 0 1 1 ) 1 1 0 1 ( 1 .
. .

1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 XOR W 3 4 3 W 2 2 1 W 3 3 2 W 3 5 4 W 2 6 5 W 3 4 3 W 2 5 4 W 2 6 5 W 3 6 7 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 0 1 0 1 0 1

Partial product reduction

Partial product generation Fig. 5 shows an example of GF (2 4 ) multiplier whose whole architecture, except the polynomial reduction, is implemented on the standard binary multiplier. This implementation can be extended for any GF (2 m ) multiplier provided that the size of the original binary multiplier can support the size of the GF (2 m ) multiplier to be implemented.

In the next section, we propose a reconfigurable architecture of a combined multiplier that can support either a binary multiplication or GF (2 m ) multiplication for m=6, 7 and 8.

III. PROPOSED TRIPLE MODE MULTIPLIER

In this section we consider the combination of GF (2 m ) multiplier, for m = 6, 7, 8, with the standard binary multiplier. Fig. 6 shows a block diagram of the combined triple mode multiplier 8 × 8 bits. In this design, we restrict the sizes of GF multipliers according to the code lengths of RS codes used in the practical applications. The architecture we propose is based on the Wallace's tree for the partial product reduction. A tree of size 8 × 8 is sufficient to perform GF (2 8 ) multiplication. However, the tree size is fixed according to the desired precision for the complex Fourier transform. As discussed in [START_REF] Ghouwayel | FPGA implementation of a reconfigurable FFT for multi-standard systems in software radio context[END_REF], a 13 bit-wordlength represents a good complexity-precision tradeoff. Thus, for GF (2 m ) multiplication, the Wallace's tree can process any two words for m ≤ 13 but a further attention should be taken into account for the realization of the polynomial reduction. For this, in the following we restrict the values of m to the more practical ones. The multiplier receives a control signal S to pilot the switching from the standard binary addition to modulo 2 addition and to manage the configuration of the interconnection between the neighboring cells according to the model previously shown in Fig. 5. The multiplier performs the partial product reduction regardless of the size of the operands. If the size is smaller than 13, the operands can be concatenated by "0". The rest of the architecture is the carry-propagate adder (for the standard binary multiplication) and the polynomial reduction unit (for the GF (2 m ) multiplication).

The part which will be influenced by the variable length of the GF multiplier is the polynomial reduction unit since the reduction is based on the binary representation of the powers of α which varies with the order of the Galois field. For this, we propose a reconfigurable polynomial reduction unit as shown in Fig. 7.

This polynomial reduction unit receives 15 bits from the partial product reduction block and a parameter m which defines the size of GF (2 m ) multiplication to be performed. According to the chosen GF , the corresponding powers of α are selected from the ROM blocks. By ANDing these powers ). These LUT1s are configured to provide their outputs according to Table I. If m < 8, the most significant bits of P i s are pre-initialized to zero and the rest of bits will contain the useful result. The latency of the polynomial reduction unit is equivalent to the time needed to perform one 8-bit AND and four 8-bit XOR.

The combined multiplier shown in Fig. 6 can be easily integrated within the reconfigurable multiplier designed in [START_REF] Ghouwayel | Complexity Evaluation of a Re-Configurable Butterfly with FPGA for Software Radio Systems[END_REF] leading to triple mode multiplier (Fig. 8) able to perform three different multiplications: (i) conventional binary multiplication (ii) multiplication over GF (F t ) and (iii) multiplication over GF (2 m ). We note that n c and n represent the complex and Galois field wordlengths respectively.

With this multiplier and by taking into account that a modulo 2 addition can be realized by means of XOR gates, the arithmetical resources required to design the TMFFT are designed.

To achieve the TMFFT scenario, further studies, considered as future, will focus on the design of a triple mode butterfly, appropriate dataflow schedule and a new wiring scheme allowing the integration of the structure of the GF (2 m )-FFT within the DMFFT operator. In order to evaluate the complexity of the proposed multiplier circuit, we consider its implementation on FPGA (Field Programmable Gate Arrays) devices. The complexity C of the FPGA-based circuit is determined by the number of logic blocks called Adaptive LUTs (i.e. ALUTs). The performance is measured in terms of the execution time T expressed in nanoseconds (ns). The advantage the common operator approach presents is emphasized when compared to an obvious approach called V elcro approach. This latest considers the separately implementation of two self-contained operators: Dual Mode Multiplier and GF (2 m ) multiplier.

The comparison of the two approaches can be drawn by evaluating an overall metric that takes into account both the hardware complexity and the execution time. This metric, called performance-to-cost ratio and denoted by η, is expressed as 1T C * 10 6 [START_REF] William | Shanzhen Xing Fixed-Point Multiplier Evaluation and Design with FPGA[END_REF]. The higher the metric η, the better the complexity-performance tradeoff is.

The choice of the target device is based on the considered devices in [START_REF] Ghouwayel | Complexity Evaluation of a Re-Configurable Butterfly with FPGA for Software Radio Systems[END_REF] [START_REF] Ghouwayel | FPGA implementation of a reconfigurable FFT for multi-standard systems in software radio context[END_REF]. We have written a synthesizable VHDL code 1 of the dual, Velcro and triple mode multipliers. Table II summarizes the complexity and performances measures for different number of bits n that represents the multiplier size. As shown, the integration of the GF (2 m ) multiplier causes an excess of only 6% of the total complexity. The metrics η V and η C denotes the performance-to-cost ratio of the Velcro and Common Operator approach respectively. The last row of Table II indicates the performance-to-cost ratio gain which is around 18 % in favor of the common operator approach. These measures show a good result which enforces the effectiveness of our adopted approach of reconfigurability and common operators.

V. CONCLUSION

We have presented a hardware design and efficient implementation of a triple mode multiplier able to operate over three different fields. A parameterizable polynomial reduction unit is proposed allowing the execution of the GF (2 m ) multiplication for various sizes of Galois field. The proposed multiplier is implemented on FPGA devices and the obtained measures show an excess of only 6% in terms of ALUTs compared to the complexity of the dual mode multiplier proposed in [START_REF] Ghouwayel | Complexity Evaluation of a Re-Configurable Butterfly with FPGA for Software Radio Systems[END_REF]. In terms of performance-to-cost ratio, the common operator approach represented by the triple mode multiplier outperforms the Velcro approach by a gain of around 18 %. An approach to design a TMFFT operator is described which will be further developed in near future works. The intended TMFFT operator can be used in two different contexts: modulation/demodulation OFDM and Reed Solomon encoding and decoding over GF (F t ) and GF (2 m ).

Fig. 1 .

 1 Fig. 1. The W3 and W2 architectures

Fig. 2 .

 2 Fig. 2. Partial product matrix

Fig. 3 .

 3 Fig. 3. Steps for N by N multiplication

Fig. 5 .

 5 Fig. 5. Multiplication of two elements in GF (2 4 )

Fig. 6 .

 6 Fig. 6. Block diagram of the combined multiplier

6 Fig. 7 .

 67 Fig. 7. Parallel polynomial reduction for m = 6, 7 and 8.

Fig. 8 .

 8 Fig. 8. Triple mode multiplier

All experiments described in this paper were performed on a PC (Intel(R) Core(TM)2 Duo CPU,

2.1 GHz, 3 GB of memory) running Windows Vista. The VHDL code was synthesized using Quartus II version 6 and implemented on STRATIX II, EP2S15F484C3 Device with the option "Standard Fit" as the level of the Fitter's effort.


Giuseppe Geymonat

Christian Licht

Thibaut Weller

Plates made of piezoelectric materials: when are they really piezoelectric?

This paper aims at presenting in a synthetic way mathematical results that have been rigorously and recently derived by the authors. These results deal with the simplified but accurate modeling of linearly piezoelectric thin plates. It is shown how mathematical tools lead to two different situations linked to either sensors or actuators devices. Moreover we enlighten the fact that for some piezoelectric crystal classes, the coupling between the electrical and the mechanical effects disappear in the plate. Finally, we furnish a detailled example of such models in the case of a plate constitued by a 222 symmetry class material.

Introduction

Piezoelectric materials are widely used in the design of smart structures (see [START_REF] Chopra | Review of State of Art of Smart Structures and Integrated Systems[END_REF] for a widespread presentation of such structures). For example (see [START_REF] Yang | The Mechanics of Piezoelectric Structures[END_REF] and [START_REF] Yang | Analysis of Piezoelectric Devices[END_REF]) thin piezoelectric plates devices can be either bonded or embedded in these structures to determine strains and displacements (sensing effect) or to provide localized strains through which the deformation of the structure can be controlled (actuation effect). It is thus of major technological interest to provide efficient modelings of such structures. In [START_REF] Weller | Analyse asymptotique de plaques minces linéairement piézoélectriques[END_REF], [START_REF] Weller | Réponse dynamique asymptotique de plaques minces linéairement piézoélectriques dans l'approximation quasi-électrostatique[END_REF], [START_REF] Weller | Etudes des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF] and [START_REF] Weller | Asymptotic modeling of thin piezoelectric plates[END_REF], taking advantage of the particular shapes of such devices, the authors have derived new models by a rigorous study of the asymptotic behavior of a three dimensional body whose thickness, considered as a parameter, tends to zero. The important result is that two different models appear at the limit, enlightening the different behavior of piezoelectric sensors and actuators devices. In particular, it is shown how the kind of electrical loading is connected to the obtained limit model.

Starting from a general three-dimensional piezoelectric problem denoted by P3D, we outline the method that lead to simplified but accurate plate models denoted by P2D because they are two-dimensional in essence. The ground of the method is to view the thickness of the plate as a small normalizing parameter whose aim is to tend to zero. Then, in particular, it can be shown that the two limit generalized kinematics do not have the same number of variables. Moreover, we investigate the influence of crystalline symmetries on the properties of our models and show that some crystal classes lead to a striking structural switch-off of the piezoelectric effect: even if the material is piezoelectric, it is not anymore the case for the thin plate. More precisely, this switch-off does not depend only on the crystal class of the piezoelectric material that constitutes the plate but also on the electrical boundary conditions, i.e. crystal symmetries do not have the same influence on sensors and actuators. It is shown that if the plate is used as a sensor, the decoupling occurs for the classes 2, 222, 2mm, 4, 4, 422, 4mm, 42m, 6, 622, 6mm, 23 and 43m, while, if the plate is used as an actuator, the decoupling takes place with the classes m, 32, 422, 6, 622 and 6m2.

We therefore observe that the decoupling occurs for sensors and actuators for only two classes: 422 and 622. These results are detailled in Section 4.

Finally, we give an example of our results in the case of a 222 symmetry class material. The distinction between the two purely mechanical behaviors is also made explicit by showing that the difference lies in a non negligible inplane shear coefficient.

Theoretical considerations

It is useful to recall the basic equations governing the electroelastic behavior of piezoelectric continua. It is the starting point of many problems that can either be of mathematical or numerical nature.

Basic piezoelectric equations

Latin lower indexes run from 1 to 3 and the lower index ′ , i ′ stands for the derivation with respect to the i th coordinate. Moreover, the convention of summation over repeated indexes is used.

The equilibrium of a piezoelectric body whose reference configuration is a 3D domain Ω with boundary S leads to: σij,j + fi = 0, Di,i -q = 0, rot E = 0.

(

Obviously, the electric field E being irrotational, it derives from an electric scalar potential ϕ. In the equations above, fi, q are the mechanical body force components and the electric body charge, while σij and Di stand for the Cauchy stress tensor and the electric displacement vector components. These latter components are related to those of small strain tensor eij(u) = 1 2 (ui,j + uj,i) (u denotes the displacement vector field) and of electric vector field Ei = -ϕ,i by the constitutive equations (see [8] for example):

σij = a ijkh e kh (u) -b ijk ϕ ,k , Dp = bpqr eqr(u) + cpq ϕ,q. (2)
In this expression, a ijkh , b ijk and cpq denote the elastic, piezoelectric and dielectric material constants, respectively.

------------Remark 1. It is possible to define an operator M : R 12 -→ R 12 governing the constitutive law (2) by writing that

1 σ D = M e(u) ∇ϕ . (3)
The couple σ D is called the generalized stress, while e(u) ∇ϕ is the generalized strain.

Most of the time, the mapping ( 2) is seen in a 9 × 9 matrix-form representation. In this direction, we introduce

M = aIJ -b Kl b lK cmn , (4)
with cmn = cmn and where indexes l, m and n take their values in {1, 2, 3} while I, J and K satisfy the Voigt contraction convention, taking their values in {1, 2, 3, 4, 5, 6}.

We recall that the Voigt contraction convention is a mapping which associates to a couple of indexes (i, j) a sole index I such that

(i, j) (1, 1) (2, 2) (3, 3) (2, 3) (3, 1) (1, 2) I 1 2 3 4 5 6
Thus, the elastic tensor a can be seen as a 6 × 6 real matrix which is written in another font by a. In the same way, the piezoelectric tensor b takes the form of a 6 × 3 real matrix denoted by b. However, due to the scalar product, it is necessary to adjust the physical constants:

aIJ = a ijkh if 1 ≤ I, J ≤ 3, aIJ = √ 2 a ijkh if 1 ≤ I, J ≤ 3 and 4 ≤ J, I ≤ 6, aIJ = 2 a ijkh if 4 ≤ I, J ≤ 6, b Ik = b ijk if 1 ≤ I ≤ 3, b Ik = √ 2 b ijk if 4 ≤ I ≤ 6.
The relation [START_REF] Yang | Analysis of Piezoelectric Devices[END_REF], which governs the linearly piezoelectric constitutive law, then takes the form

1 We recall that ∇ϕ =   ϕ,1 ϕ,2 ϕ,3   .               σ11 σ22 σ33 √ 2 σ23 √ 2 σ31 √ 2 σ12 D1 D2 D3               =              
a11 a12 a13 a14 a15 a16 -b11 -b12 -b13 a12 a22 a23 a24 a25 a26 -b21 -b22 -b23 a13 a23 a33 a34 a35 a36 -b31 -b32 -b33 a14 a24 a34 a44 a45 a46 -b41 -b42 -b41 a15 a25 a35 a45 a55 a56 -b51 -b52 -b53 a16 a26 a36 a46 a56 a66 -b61 -b62 -b63 b11 b21 b31 b41 b51 b61 c11 c12 c13 b12 b22 b32 b42 b52 b62 c12 c22 c23 b13 b23 b33 b43 b53 b63 c13 c23 c33

              •               e11(u) e22(u) e33(u) √ 2 e23(u) √ 2 e31(u) √ 2 e12(u) ϕ,1 ϕ,2 ϕ,3               . (5)
We therefore note that the generalized three dimensional kinematics of a piezoelectric solid is described by nine variables (six mechanical and three electrical).

- ----------The piezoelectric body can be submitted to either essential or natural mechanical and electrical boundary conditions, or a combination of them, on S:

ui = Ui , σij nj = Fi, ϕ = V , Di ni = Q. (6)
We do not precise the associated partitions of S, the reader is refered to [START_REF] Weller | Analyse asymptotique de plaques minces linéairement piézoélectriques[END_REF] for the details. Here Ui, Fi, V , Q and ni denote the specified mechanical displacement and surface force components, the electrical potential, the flux through S of the electric induction and the outward unit normal vector components, respectively. The local three-dimensional electroelastic problem P3D(Ω) consists of finding the electromechanical state s = (u, ϕ) satisfying equations (1), ( 2) and (6).

Variational piezoelectric equations

It is necessary to rewrite P3D(Ω) in another form in order to gather useful informations on the electromechanical state s = (u, ϕ). Multiplying by sufficiently smooth2 kinematically admissible virtual displacements vi and electrical potential ψ, equation (1) becomes equivalent to

Ω (σij,j + fi) vi dΩ + Ω (Di,i -q) ψ dΩ = 0. (7)
Integrating by part this expression leads to

- Ω σij,j vi,j dΩ + S σij njvi dS + Ω fi vi dΩ - Ω Di ψ,i dΩ + S Di ni ψ dΩ - Ω q ψ dΩ = 0. (8)
The symmetry of the stress tensor (σij = σji), the definition of the small strain tensor and the boundary conditions ( 6) thus give

- Ω σij,j eij(v) dΩ + S Fi vi dS + Ω fi vi dΩ - Ω Di ψ,i dΩ + S Q ψ dS - Ω q ψ dΩ = 0. ( 9
)
For any kinematically admissible virtual electromechanical state r = (v, ψ), we introduce the linear form

L(r) = S Fi vi dS + Ω fi vi dΩ + S Q ψ dS - Ω q ψ dΩ. (10)
We notice that

- Ω σij,j eij(v) dΩ - Ω Di ψ,i dΩ = - Ω M e(u) ∇ϕ • e(v) ∇ψ dΩ, (11)
and, for brevity, define the bilinear form m associated with the electroelastic po-

tential 1 2 (σ • e + D • E) m(s, r) = m((u, ϕ), (v, ψ)) = Ω M e(u) ∇ϕ • e(v) ∇ψ dΩ. (12)
It is then possible to reformulate the problem of determining the electromechanical state at equilibrium:

P3D(Ω)

Find s = (u, ϕ) sufficiently smooth such that m(s, r) = L(r), for all virtual electromechanical state r = (v, ψ).

This expression of the electromechanical problem is at the starting point of either finite element formulations or mathematical questions in (linear) piezoelectricity (see [START_REF] Benjeddou | Advances in piezoelectric finite element modeling of adaptive structural elements: a survey[END_REF] and [START_REF] Robaldo | A unified formulation for finite element analysis of piezoelectric adaptive plates[END_REF] for example).

The problem of piezoelectric plates

In pure elasticity, the problem of finding simplified but accurate plate models is quite old. Recently however, [START_REF] Ciarlet | Mathematical Elasticity[END_REF] has presented a rigorous mathematical justification of the Kirchhoff-Love model in the homogeneous and isotropic case. This method has been successfully generalized to (possibly heterogeneous) linearly piezoelectric plates (see [START_REF] Weller | Analyse asymptotique de plaques minces linéairement piézoélectriques[END_REF], [START_REF] Weller | Réponse dynamique asymptotique de plaques minces linéairement piézoélectriques dans l'approximation quasi-électrostatique[END_REF], [START_REF] Weller | Etudes des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF] and [START_REF] Weller | Asymptotic modeling of thin piezoelectric plates[END_REF]), both in the static and transient siuations. At this point, it has been made possible to reconciliate some existing divergent models (see [START_REF] Bisegna | A consistent theory of thin piezoelectric plates[END_REF], [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, Part 1 Models without homogenization[END_REF], [START_REF] Maugin | An asymptotic theory of thin piezoelectric plates[END_REF], [START_REF] Sène | Modélisation asymptotique de plaques : Contrôlabilité exacte frontière[END_REF]) or to precise the mathematical framework of some existing results (see [START_REF] Rahmoune | New thin piezoelectric plate models[END_REF]).

In the plate models derivation, a crucial role is played by the thickness direction (also called the outplane direction). For commodity, this direction corresponds to the third coordinate axis. The role played by the thickness is crucial because it is very small compared to the other dimensions of the plate :

Ω = ω × (-ε/2, +ε/2),
where ω is a bounded domain of R 2 with smooth boundary and where ε denotes the thickness of the plate. That leads to the idea of considering ε as a small parameter and of connecting this parameter to the data of our problem, i.e. the electromechanical coefficients, loading and state. In a sense, by this way, plates models can be interpreted as a peculiar electromechanical state resulting of a given class of electromechanical loading imposed to a thin flat piezoelectric plate.

From the mathematical point of view, the method consists in studying what does happen to the unique solution of P3D(Ω) when Ω is the reference configuration of a flat piezoelectric body whose thickness goes to zero (this is the reason why this method belongs to the field of asymptotic analysis). The striking fact is that two models, i.e. two different kinds of behavior, appear at the limit. These two models are intimately connected to the type of electric loading subjected to the plate, thus giving a rigorous theoretical background to the study of sensors and actuators plate-like devices. In order to emphasize the fact that the models we get are arising through a dimension reduction process, they will be denoted by P2D(Ω). More precisely, by different averagings through the thickness, it is possible to show that our limit models can be fully described by taking into account only the inplane coordinates.

In the sequel, we consider the following four electromechanical boundary conditions on the set Γ ± constituted by the lower and the upper faces of the flat thin plate occupying Ω:

(BC)sensor : D • n = Q on Γ ± , (BC)actuator : V = V ± 0 on Γ ± .
Here, we focus on the presentation of the obtained models. For the mathematical arguments underlying the whole analysis of this problem, we refer the reader to [START_REF] Weller | Analyse asymptotique de plaques minces linéairement piézoélectriques[END_REF], [START_REF] Weller | Etudes des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF] and [START_REF] Weller | Asymptotic modeling of thin piezoelectric plates[END_REF].

The sensor model

Three kinds of information are needed to fully describe an electromechanical model. These are the generalized kinematics (or generalized strain), the inner loading (or generalized stress) and the constitutive equations (which link them).

The generalized kinematics

The generalized kinematics involves the tensor of small strains and the electrical potential gradient. In [START_REF] Weller | Etudes des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF] and [START_REF] Weller | Asymptotic modeling of thin piezoelectric plates[END_REF], we have shown that these two mathematical objects appear in reduced forms in the limit models.

The displacements field We obtain a Kirchhoff-Love displacements field, which in particular means that the model cannot render shear effects (see [START_REF] Fernandes | Accurate modelling of piezoelectric plates: single-layered plate[END_REF], [START_REF] Nicotra | Piezoelectric plates with changing thickness[END_REF] [2] and [START_REF] Yang | Analysis of Piezoelectric Devices[END_REF] in which shear or thickness effects are rendered with different methods). More precisely, a Kirchhoff-Love displacement v satisfies:

e(v) =   e11(v) e12(v) 0 e12(v) e22(v) 0 0 0 0   . (13)
It is possible to show that Kirchhoff-Love displacements can be decomposed into a membrane and a flexural parts (see [START_REF] Ciarlet | Mathematical Elasticity[END_REF], [START_REF] Weller | Etudes des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF] and [START_REF] Weller | Asymptotic modeling of thin piezoelectric plates[END_REF]). This result is a very classical one in pure elasticity. However, it is important to note that because of the symmetry of the small strain tensor, the number of purely mechanical variables comes down from six to three.

The electrical field The asymptotic analysis of the three-dimensional problem shows a crucial difference between the actuator and the sensor cases: in the actuator case, the electric potential intervenes only (at the first order) through the outplane direction while, in the sensor case, the electric potential does not depend (at the first order) on the outplane direction. Focusing here on the sensor case, we are in the situation for which the outplane direction plays no role, i.e. the electrical potential does not depend on x3 so that the limit model only takes into account two variables (the inplane electrical ones).

We can therefore conclude that the limit generalized kinematics is described by only five variables instead of nine in the full 3D situation. It is represented by the e1(u, ϕ) vector:

e1(u, ϕ) =       e11(u) e22(u) √ 2 e12(u) ϕ,1 ϕ,2       . (14)
The generalized stress The generalized stress involves the stress tensor and the electrical displacement vector. As a result of the asymptotic analysis one finds that this mathematical object reduces to its inplane components, so that it takes the reduced form:

σ1 =       σ11 σ22 √ 2 σ12 D1 D2       . ( 15
)
The sensor constitutive law We now have to identify the mathematical object Msensor which link the generalized stress to the generalized strain that live on the plate. The algebraic arguments that lead to the exact formula of the limit constitutive law are presented and justified in [START_REF] Weller | Etudes des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF] and [START_REF] Weller | Asymptotic modeling of thin piezoelectric plates[END_REF]. In fact, the limit constitutive equations emerge from a recombination of the electromechanical components of M (see [START_REF] Weller | Réponse dynamique asymptotique de plaques minces linéairement piézoélectriques dans l'approximation quasi-électrostatique[END_REF]). This recombination is imposed by the structure of e1(u, ϕ) and σ1 described above. In the sensor case, the recombination lead to rewrite (5) as

              σ11 σ22 √ 2 σ12 D1 D2 σ33 √ 2 σ23 √ 2 σ31 D3               = M ′ 1 •               e11(u) e22(u) √ 2 e12(u) ϕ,1 ϕ,2 e33(u) √ 2 e23(u) √ 2 e31(u) ϕ,3               . ( 16
)
with

M ′ 1 =              
a11 a12 a16 -b11 -b12 a13 a14 a15 -b13 a12 a22 a26 -b21 -b22 a23 a24 a25 -b23 a16 a26 a66 -b61 -b62 a36 a46 a56 -b63 b11 b21 b61 c11 c12 b31 b41 b51 c13 b12 b22 b62 c12 c22 b32 b42 b52 c23 a13 a23 a36 -b31 -b32 a33 a34 a35 -b33 a14 a24 a46 -b41 -b42 a34 a44 a45 -b43 a15 a25 a56 -b51 -b52 a35 a45 a55 -b53 b13 b23 b63 c13 c23 b33 b43 b53 c33

              . Now, let M 00 1 =      
a11 a12 a16 -b11 -b12 a12 a22 a26 -b21 -b22 a16 a26 a66 -b61 -b62 b11 b21 b61 c11 c12 b12 b22 b62 c12 c22

      , M 0- 1 =       a13 a14 a15 -b13 a23 a24 a25 -b23 a36 a46 a56 -b63 b31 b41 b51 c13 b32 b42 b52 c23       , M -0 1 =     a13 a23 a36 -b31 -b32 a14 a24 a46 -b41 -b42 a15 a25 a56 -b51 -b52 b13 b23 b63 c13 c23     , M -- 1 =     a33 a34 a35 -b33 a34 a44 a45 -b43 a35 a45 a55 -b53 b33 b43 b53 c33     . (17)
Because the asymptotic analysis of P3D(Ω) associated with the boundary conditions (BC)sensor shows that the vector

    σ33 √ 2 σ23 √ 2 σ31 D3   
 can be neglected, the sensor constitutive equations appears as the Schur complement (or the condensation) of the block M --

1 of M ′ 1 : Msensor = M1 = M 00 1 -M 0- 1 (M -- 1 ) -1 M -0 1 . (18)
Introducing the mechanical (m) and electrical (e) components of the generalized stress and strain, we associate to Msensor the sub-operators M1 mm , M1 me , M1 em and M1 ee :

Msensor = M1 mm M1 me M1 em M1 ee . ( 19
)
It is shown in [START_REF] Weller | Etudes des symétries et modèles de plaques en piézoélectricité linéarisée[END_REF] that M and Msensor share the same inner structure, that is:

M1 mm = M T 1mm , M1 me = -M T 1em , M1 ee = M T 1ee . ( 20
)
Variational formulation of the sensor model Similarly to [START_REF] Bisegna | A consistent theory of thin piezoelectric plates[END_REF], we define

m1(s, r) = m1((u, ϕ), (v, ψ)) = Ω Msensor e1(u, ϕ) • e1(v, ψ) dΩ. ( 21
)
Our proposed model which allows us to determine the electromechanical state of a plate-like sensor at equilibrium then reads as:

P sensor 2D (Ω)
Find s = (u, ϕ) sufficiently smooth such that m1(s, r) = L(r), for all virtual electromechanical state r = (v, ψ).

As an asymptotic result, the thinner the plate (compared to its other dimensions), the more accurate the model is. In this direction, the degree of accuracy of this result is precised in [START_REF] Weller | Analyse asymptotique de plaques minces linéairement piézoélectriques[END_REF]. Mathematically speaking, it is of importance to precise that the function space on which live the (limit) admissible electromechanical state is not the same that in the three-dimensional case. This is the reason why it is often spoken of "singular perturbations" problems.

Practically speaking, this case corresponds to a device which is able to measure (directly or indirectly) the flux of the electric induction, so that the linear form L is perfectly determined. A numerical treatment of P sensor

2D

(Ω) then gives the piezoelectric state in the plate. That is why we can call this model sensor.

The actuator model

As it has been specified earlier, the difference between sensor and actuator models lies in the informations that the electrical potential can take into account. Here, in the actuator case, these informations are collected only upon the outplane direction, while in the sensor case these informations were collected upon the two inplane directions. Of course, the purely mechanical informations do not change, but the fact that only ϕ,3 appears in the actuator model radically changes the generalized kinematics and stress together with the constitutive law.

The generalized kinematics and stress

As it has just been pointed out, displacements field is always of Kirchhoff-Love type (see [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, Part 1 Models without homogenization[END_REF]). As to the electrical potential, it can be shown that only E3 appears. The generalized kinematics is therefore described by four variables at the limit. It is represented by the vector:

e2(u, ϕ) =     e11(u) e22(u) √ 2 e12(u) ϕ,3     . ( 22
)
Similarly, the generalized stress takes the form:

σ2 =     σ11 σ22 √ 2 σ12 D3     . (23)
The actuator constitutive law The method to find the constitutive relations is similar to the one presented in the sensor case. However, the difficulty lies in an adequate electromechanical coefficients recombination. We precise this point here.

First of all, we rewrite (5) as

              σ11 σ22 √ 2 σ12 D3 σ33 √ 2 σ23 √ 2 σ31 D1 D2               = M ′ 2 •               e11(u) e22(u) √ 2 e12(u) ∂3ϕ e33(u) √ 2 e23(u) √ 2 e31(u) ∂1ϕ ∂2ϕ               , (24)
so that:

M ′ 2 =              
a11 a12 a16 -b13 a13 a14 a15 -b11 -b12 a12 a22 a26 -b23 a23 a24 a25 -b21 -b22 a16 a26 a66 -b63 a36 a46 a56 -b61 -b62 b13 b23 b63 c33 b33 b43 b53 c13 c23 a13 a23 a36 -b33 a33 a34 a35 -b31 -b32 a14 a24 a46 -b43 a34 a44 a45 -b41 -b42 a15 a25 a56 -b53 a35 a45 a55 -b51 -b52 b11 b21 b61 c13 b31 b41 b51 c11 c12 b12 b22 b62 c23 b32 b42 b52 c12 c22

             
. and define

M 00 2 =     a11 a12 a16 -b13 a12 a22 a26 -b23 a16 a26 a66 -b63 b13 b23 b63 c33     , M 0- 2 =     a13 a14 a15 a23 a24 a25 a36 a46 a56 b33 b43 b53     , M -0 2 =   a13 a23 a36 -b33 a14 a24 a46 -b43 a15 a25 a56 -b53   , M -- 2 =   a33 a34 a35 a34 a44 a45 a35 a45 a55   . (25)
Here, the asymptotic analysis of P3D(Ω) associated with the boundary conditions (BC)actuator shows that the two vectors

  σ33 √ 2 σ23 √ 2 σ31
  and ∂1φ ∂2φ can be neglected, so that the actuator constitutive equations reads as

Mactuator = M2 = M 00 2 -M 0- 2 (M -- 2 ) -1 M -0 2 . ( 26
)
This operator shares the same structure and symmetry properties as those exhibited in ( 19)-(20).

Similarly to the sensor case, in order to get the variational formulation of the plate-like actuator problem, we define

m2(s, r) = m2((u, ϕ), (v, ψ)) = Ω M2 e2(u, ϕ) • e2(v, ψ) dΩ, (27)
and the problem of determining the electromechanical state of a plate-like actuator at equilibrium then takes the form:

P actuator

2D

(Ω) Find s = (u, ϕ) sufficiently smooth such that m2(s, r) = L(r), for all virtual electromechanical state r = (v, ψ), which is also a singularly perturbed problem.

Remark 2. To be more precise, in the expression of the model P actuator

2D

(Ω), the terms 'sufficiently smooth' mean that s has to satisfy (BC)2 while r has to satisfy (BC)2 with V ± 0 = 0, see [START_REF] Weller | Analyse asymptotique de plaques minces linéairement piézoélectriques[END_REF] for the technical details.

This case corresponds to a device subjected to given electric potential at its boundary. A numerical treatment of P actuator

2D

(Ω) supplies the piezoelectric state in the plate. Therefore, the mechanical state can be controlled through electric loading. That is why we call this model actuator.

Influence of crystalline symmetries

It is interesting to give some properties of the operator Mp (p = 1, 2), which supplies the constitutive equations of the piezoelectric thin plates. As we saw, the fundamental coupling property of M remains true for Mp:

Mp me = -( Mp em ) T , (28)
where m and e respectively denote the mechanical and electrical components of the generalized kinematics and stress.

Considering the influence of crystalline symmetries on the three-dimensional constitutive law (see [8] for example), we can deduce, in the case of a polarization normal to the plate, that3 : -M2 mm involves mechanical terms only, -M1 mm = M2 mm for the crystalline classes m, 32, 422, 6, 622 and 6m2, -M1 mm involves electrical terms except for these previous classes, -when p = 1, there is an electromechanical decoupling ( Mp me = 0) for the classes 2, 222, 2mm, 4, 4, 422, 4mm, 42m, 6, 622, 6mm, 23 and 43m, when p = 2, this decoupling occurs with the classes m, 32, 422, 6, 622 and 6m2, nevertheless the operators Mp mm and Mp ee involve a mixture of elastic, piezoelectric and dielectric coefficients. In these cases, the plate can be considered as no more piezoelectric. We are then in a situation of a structural switch-off of the piezoelectric effect.

We then enlighten situations for which piezoelectric materials lead to nonpiezoelectric structures. For recent results concerning the reverse situation, that is the possibility of conceiving piezoelectric composites without using piezoelectric materials, the reader is refered to [START_REF] Sharma | On the possibility of piezoelectric nanocomposites without using piezoelectric materials[END_REF].

Application and example: 222 crystalline class

Let's consider a thin piezoelectric plate constituted by a material whose crystalline symmetry class is 222. Then (5) takes the form:

              σ11 σ22 σ33 √ 2 σ23 √ 2 σ31 √ 2 σ12 D1 D2 D3               =              
a11 a12 a13 0 0 0 0 0 0 a12 a22 a23 0 0 0 0 0 0 a13 a23 a33 0 0 0 0 0 0 0 0 0 a44 0 0 -b41 0 0 0 0 0 0 a55 0 0 -b52 0 0 0 0 0 0 a66 0

0 -b63 0 0 0 b41 0 0 c11 0 0 0 0 0 0 b52 0 0 c22 0 0 0 0 0 0 b63 0 0 c33               •               e11(u) e22(u) e33(u) √ 2 e23(u) √ 2 e31(u) √ 2 e12(u) ϕ,1 ϕ,2 ϕ,3               . (29)
Therefore, ( 18) and (26) respectively lead to

      σ11 σ22 √ 2 σ12 D1 D2       =          a11 - a 2
a 55          •       e11(u) e22(u) √ 2 e12(u) ϕ,1 ϕ,2       (30)
in the sensor case and to

    σ11 σ22 √ 2 σ12 D3     =      a11 - a 2 13 a 33 a12 -a 13 a 23 a 33 0 0 a12 -a 13 a 23 a 33 a22 - a 2 23 a 33 0 0 0 0 a66 -b63 0 0 b63 c33      •     e11(u) e22(u) √ 2 e12(u) ϕ,3     (31)
in the actuator case.

As outlined in the preceeding section, the relation (30) shows that σ and D respectively depend solely on e(u) and ∇ϕ when the plate acts as a sensor, so that it can be considered as no more piezoelectric. However, when the same plate acts as an actuator, the piezoelectric coupling does not vanish as it can be seen in ( 31). Moreover, we observe that the difference between both Mp mm lies in the inplane shear coefficient: if p = 1, this coefficient is equal to a66 + b 2 63 c 33 while it is equal to a66 if p = 2. Since the order of magnitude of the permittivity c33 is most of the time very low compared to the piezoelectric constants (see [8] p. 146), the term b 2 63 c 33 cannot be neglected. Therefore, from the purely mechanical point of view, it appears a significative difference between the two models.

Conclusion

Through classical tools of variational analysis, two of the authors derived in [START_REF] Weller | Analyse asymptotique de plaques minces linéairement piézoélectriques[END_REF] the mathematical modeling of linearly piezoelectric plates. In this paper, the properties of the two models that arise from this derivation are detailed. These two models are related to the kind of electrical boundary conditions imposed to the plate (i.e. whether the piezoelectric plate is electroded or not). When the upper and lower faces are not covered with electrodes, the plate acts as a sensor whose model uses a richer generalized kinematics4 than the one of the actuator obtained when the upper and lower faces are electroded. The procedure that leads to the constitutive laws of our models is presented and detailled.

As our results come from the field of theoretical mathematical modeling, our paper does not directly deals with a numerical treatment of the studied problem. However, the theorems proven in [START_REF] Weller | Analyse asymptotique de plaques minces linéairement piézoélectriques[END_REF] and presented here can be straightforwardly used in the field of numerical studies through finite element method. It has to be emphasized that standard 3D mechanical problems are often prohibitive from the numerical point of view. It is one of the many reasons for which the of asymptotic analysis of dimension reduction problems is so important: in the case presented here, our limit models are bidimensional. Therefore, the numerical treatment is made much more easier. Of course, the accuracy of our models is clearly related to the order of magnitude of the thickness of the plate : the thinner the plate, the more accurate the modeling. We also emphasize on the point that our modeling is carried out for every crystal classes that are piezoelectrically compatible. The phenomenon of structural switch-off is theoretically enlightened for the first time.

To improve the modeling of smart structures, it is however necessary to look more precisely at a rigorous modeling of multiparameterized multiphysical structures, in order to take into account the thickness of the electrodes, but also the fact that piezoelectric devices are often glued as patches on host structures. This kind of multiphysical structures has to be very carefully studied. The difficulty lies on the fact that many different orders of magnitude of geometrical and mechanical natures appear in such problem : the thicknesses of the plate, of the electrodes and of the bonded joint, but also the orders of magnitude of the stiffness of the glue, of the electrodes and, eventually, of the dielectric constants of the plate.

In the sequel, we do not precise the exact mathematical background of such a formulation.

In the following, the letter m in Sans Serif font stands for 'mechanical' while the same letter m in italic stands for 'mirror', as it is usually understood in crystallography.

In the sense that the generalized kinematics has more components in this situation. Note, however, that the two electric components ϕ,1 and ϕ,2 do not depend on x3 while, for the actuator, the electric potential may depend on x1, x2 and x3 even if the sole (vertical) electric component appears in the generalized kinematics.