Nils Altm

Lars Gr

A Comparative Stability Analysis of Neumann and Dirichlet Boundary MPC for the Heat Equation ⋆

Keywords: predictive control, stability, partial differential equations, boundary control

We continue our study on stability properties of model predictive control without terminal constraints applied to the heat equation. Here, we focus on boundary control and in particular on the differences between Neumann and Dirichlet boundary conditions. We first illustrate the differences by means of numerical simulations and then explain our findings by a theoretical analysis.

INTRODUCTION

In this paper we continue our study of stability properties for model predictive control (MPC) applied to the one dimensional heat equation. We particularly investigate the case of Neumann boundary control and compare this case to our earlier findings for Dirichlet boundary control and distributed control. MPC, also called receding horizon control (RHC), is a well known method in which a feedback controller is synthesized from the iterative solution of finite horizon optimal control problems. The application of MPC to parabolic PDEs for boundary control can be found, e.g., in [START_REF] Dubljevic | Boundary Predictive Control of Parabolic PDEs[END_REF] and [START_REF] Ito | Receding horizon optimal control for infinite dimensional systems[END_REF]. In [START_REF] Dubljevic | Boundary Predictive Control of Parabolic PDEs[END_REF], the authors use a modal decomposition technique in order to obtain low dimensional problems and focus on state and control constraints. In [START_REF] Ito | Receding horizon optimal control for infinite dimensional systems[END_REF], a receding horizon method for infinite dimensional systems is presented. These references have in common that the stability of the closed loop solution is guaranteed by adding terminal constraints and control Lyapunov functionals as terminal costs to the finite horizon optimal control problems. However, as the construction of terminal constraints and costs is a challenging task, here we consider MPC without such constraints and costs, which is also often preferred in practical applications, cf. [START_REF] Qin | A survey of industrial model predictive control technology[END_REF].

One important parameter in unconstrained MPC schemes is the length of the optimization horizon N . It is well known that under suitable conditions stability holds for sufficiently large N , see Grüne and Pannek (2011, Chapter 6 and the references therein). In this paper we investigate the dependence of the minimal stabilizing N on the system structure and parameters. Since the computational effort in each MPC step grows with N , this information is of particular interest. We investigate this problem for the linear heat equation, extending [START_REF] Altmüller | Distributed and boundary model predictive control for the heat equation[END_REF] by including the case of Neumann boundary control and comparing this case with our earlier results on Dirichlet boundary control and distributed control.

Our paper is organized as follows. In Section 2 we recall stability results for MPC schemes without terminal constraints.

⋆ Lars Grüne was supported by the European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN, Grant agreement nr. 264735-SADCO The PDE control system under consideration is introduced in Section 3 in which we also present numerical simulation results motivating our study. The parameters needed for the stability analysis are derived in Section 4 in which we first summarize useful earlier results from [START_REF] Altmüller | Distributed and boundary model predictive control for the heat equation[END_REF] and then present our new results on Neumann boundary control. Finally, in Section 5 we use these results in order to compare Neumann with Dirichlet boundary and distributed control and analytically explain the numerical findings. Section 6 concludes the paper.

PRELIMINARIES

In this section we summarize results on MPC and its stability. We consider nonlinear discrete time control systems given by z(n + 1) = f (z(n), u(n))

(1) with state space Z and set of control values U , which are both possibly infinite dimensional normed spaces. Continuous time PDE models can be converted into this discrete form by sampling, for details see Section 3. The solution trajectory for given initial state z 0 and control sequence (u(n)) n∈N0 ⊆ U is denoted by z u (•) = z u (•; z 0 ) and the space of control sequences u : N 0 → U by U. Our goal is to compute an optimal asymptotically stabilizing feedback law for (1) by MPC. To this end we define the finite horizon cost functional

J N (z 0 , u) := N -1 n=0 ℓ(z u (n; z 0 ), u(n)) (2)
with optimization horizon N ∈ N ≥2 and the optimal value function

V N (z 0 ) := inf u∈U J N (z 0 , u). (3)
Given an initial value z 0 , the MPC feedback µ N : Z → U and the corresponding solution z µ N of the closed loop system

z µ N (n + 1; z 0 ) = f (z µ N (n; z 0 ), µ(z µ N (n; z 0 ))) (4
) are then obtained in the following iterative way:

1. Fix some N ∈ N, set z µ N (0; z 0 ) := z 0 and n := 0 2. Minimize (2) with initial value z µ N (n; z 0 ) in order to obtain the optimal control sequence u ⋆ (0), u ⋆ (1), . . . , u ⋆ (N -1) and set µ N (z µ N (n; z 0 )) := u ⋆ (0) 3. Compute z µ N (n + 1; z 0 ) by ( 4), set n := n + 1 and go to 2.

In order to achieve stability of the closed loop system additional terminal constraints or terminal costs are often introduced in (2), cf. [START_REF] Rawlings | Model Predictive Control: Theory and Design[END_REF] or Grüne and Pannek (2011, Chapter 5) and the references therein. In contrast to this approach, here we use MPC without additional constraints or costs. We remark that state and control constraints coming from modelling considerations can be included by choosing appropriate subsets Z ⊆ Z and U ⊆ U but since this topic is not in the focus of this paper we will not go into details concerning feasibility issues.

Next we state a stability result from [START_REF] Grüne | Analysis of unconstrained nonlinear MPC schemes with time-varying control horizon[END_REF] which requires the following assumption. Assumption 1. The system (1) is called exponentially controllable with respect to the stage costs ℓ if there exist an overshoot bound C ≥ 1 and a decay rate σ ∈ (0, 1) such that for each z ∈ Z there exists u z ∈ U satisfying

ℓ(z uz (n; z), u z (n)) ≤ Cσ n min u∈U ℓ(z, u) =: Cσ n ℓ ⋆ (z) (5) for all n ∈ N 0 .
Based on this assumption, the following stability theorem is proven in [START_REF] Grüne | Analysis of unconstrained nonlinear MPC schemes with time-varying control horizon[END_REF]. Theorem 2. (Stability Theorem). Let the Assumption 1 hold with overshoot constant C ≥ 1 and decay rate σ ∈ (0, 1). Let the prediction horizon N be such that the stability condition

α N := 1 - (γ N -1) N i=2 (γ i -1) N i=2 γ i - N i=2 (γ i -1) > 0 (6)
holds with γ i := C i-1 n=0 σ n . Assume, moreover, that K ∞functions η, η exist satisfying η( z -z ⋆ ) ≤ ℓ ⋆ (z) and η( zz ⋆ ) ≥ V N (z) for all z ∈ Z. Then the closed loop system (4) is asymptotically stable and thus in particular each trajectory converges to the equilibrium z ⋆ as n → ∞.

Our goal is now to use this theorem in order to determine the minimal horizon N which is needed in order to ensure (6). While quantitative estimates of horizons N for which stability holds derived from Theorem 2 tend to be conservative (see, e.g., [START_REF] Altmüller | Receding horizon optimal control for the wave equation[END_REF]), it was observed in [START_REF] Altmüller | Distributed and boundary model predictive control for the heat equation[END_REF] that Assumption 1 and Condition (6) allow for a very precise explanation of the qualitative behavior of such minimal horizons N in dependence on system parameters and system structure. This leads to the following road map for our analysis:

• Find (not necessarily optimal) controls u z such that the exponential controllability condition is fulfilled. • Calculate C and σ for this particular control.

• Use this information to explain the qualitative behavior of the minimal horizon N guaranteeing (6).

In order to give an intuition for Condition (6), in Figure 1 we visualize the minimal N satisfying (6) depending on C and σ: the areas in Figure 1 visualize the pairs (C, σ) corresponding to the minimal stabilizing horizon N = . . . indicated within the respective area. The key information from this visualization for our subsequent analysis is the different impact of the constants: For fixed σ it is always possible to obtain stability with the shortest possible horizon N = 2 by reducing C. In contrast to this, for fixed C it is in general impossible to arbitrarily reduce N by reducing σ. Therefore, the overshoot constant C plays an important role in the stability analysis.

HEAT EQUATION

In this section we introduce the equation which we want to stabilize using MPC. We consider a one dimensional linear diffusion-reaction equation, henceforth briefly denoted as heat equation, with Dirichlet and Neumann boundary control. We assume that the control can only act on the right boundary, i.e., we consider

y t (x, t) = y xx (x, t) + µy(x, t) on Ω × (0, ∞) (7) y(x, 0) = y 0 (x). on Ω y(0, t) = 0 on (0, ∞) with either Dirichlet control y(1, t) = v(t) on (0, ∞) (8) or Neumann control y x (1, t) = v(t) on (0, ∞).
(9) On the left side we impose a homogenous Dirichlet condition, the domain is given by Ω = (0, 1) and y 0 ∈ H 1 0 (Ω) denotes the initial condition. The uncontrolled equation (v(t) ≡ 0) is known to be unstable if the reaction parameter µ becomes larger than the least eigenvalue of (-∂ xx ), cf. [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF]. Thus, the equation is unstable for µ ≥ π 2 in the Dirichlet controlled case and for µ ≥ π 2 /4 for the Neumann controlled PDE. results of the optimal control problem for both Dirichlet and Neumann boundary control can be found in [START_REF] Lasiecka | Control Theory for Partial Differential Equations: Continuous and Approximation Theories[END_REF].

Although we focus on boundary control in this paper, for our analysis it will turn out to be useful to also consider the heat equation with distributed control, i.e., the equation in which the control v(x, t) acts on the whole domain Ω. Imposing homogeneous Dirichlet boundary conditions this equation reads

y t (x, t) = y xx (x, t) + µy(x, t) + v(x, t) on Ω × (0, ∞) y(x, 0) = y 0 (x) on Ω y(0, t) = y(1, t) = 0 on (0, ∞) (10)
The uncontrolled PDE is again unstable for µ ≥ π 2 . The existence and regularity theory of the distributed optimal control problem can be found in [START_REF] Lions | Optimal Control of Systems Governed by Partial Differential Equations[END_REF].

In order to use the theory presented in Section 2 we have to rewrite the continuous time PDE as a discrete time system (1). Let ϕ(x, t; v, y 0 ) be the solution of (7, 8), (7, 9) or ( 10) with control v and initial function y 0 . Then we define the sampled data system with sampling period T > 0 as z(n + 1) := ϕ(x, T ; u(n), z(n)) (11) with z(0) = y 0 . In the boundary control case we define the discrete time control as u(n

) := v(• + nT )| [0,T ) with u(n) ∈ L ∞ ([0, T ), R); in the distributed case we use u(n) ∈ L ∞ ([0, T ), L 2 (Ω)) defined as u(n) := v(•, • + nT )| Ω×[0,T ) .
In the sequel, this relation between v and u will be implicitly used when verifying (5). The discrete time n corresponds to the continuous time nT which implies z

(n) = y(•, nT ) ∈ H 1 (Ω) =: Z.
It is known that the choice of the running cost plays an important role for stability of the MPC closed loop. Results of the influence of different stage cost on the horizon can be found in [START_REF] Altmüller | Distributed and boundary model predictive control for the heat equation[END_REF]. Since in this paper we do not focus on the stage cost but rather on the different types of boundary control, we exclusively use the stage cost

ℓ(z(n), u(n)) = 1 2 y(•, nT ) 2 L 2 (Ω) + λ 2 |v(nT )| 2 , (12)
i.e., we penalize the state in the L 2 (Ω)-norm and the squared absolute value of the control where λ > 0 denotes a so called regularization or Tikhonov parameter. The analogous choice for the distributed problem reads

ℓ(z(n), u(n)) = 1 2 y(•, nT ) 2 L 2 (Ω) + λ 2 u(•, nT ) 2 L 2 (Ω) .
(13) In order to motivate our subsequent study we first numerically evaluate the stability properties for Dirichlet and Neumann boundary control. The following simulations illustrate certain phenomena which we will later explain in our theoretical analysis. To this end, using the sampling time T = 0.01 and the initial function y 0 (x) = 1 5 sin(πx), by performing numerical simulations of the MPC closed loop we have determined the minimal horizon N N for which asymptotic stability could be observed. In these simulations, the optimization problem within each MPC step was solved by a Newton-CG method, cf. [START_REF] Borzì | Computational optimization of systems governed by partial differential equations[END_REF]. The arising PDEs were discretized by a finite difference scheme. The results are shown in Table 1. Note that with our numbering N N = 2 is the smallest possible horizon. For comparison, Table 2 shows the corresponding values N D for Dirichlet boundary control and N dis for distributed control, see also [START_REF] Altmüller | Distributed and boundary model predictive control for the heat equation[END_REF]. Note that in both tables we have only considered those values of µ for which the respective uncontrolled equation is unstable.

DERIVATION OF C AND σ

Following our road map from Section 2 we will now derive estimates for the constants C and σ in Assumption 1. We start by recalling results from Altmüller and Grüne (2012) for the distributed control case, as these will turn out to be useful for the analysis of the boundary control case we discuss afterwards.

Distributed Control

In order to perform the first step of the road map presented in Section 2 we have to find a control u z for any initial value z such that the exponential controllability condition (5) is satisfied. A key issue in this method is that we do not need the optimality of this control. It is known that the linear feedback law v(x, t) := -Ky(x, t) (14) stabilizes (10) for a sufficient large constant K ∈ R. This property is used in the following theorem in order to establish (5). Theorem 3. The heat equation (10) with control ( 14), K > µπ 2 , and stage costs (13) fulfills the exponential controllability condition (5). The corresponding constants are given by σ = e -2T (π 2 -µ+K) ∈ (0, 1) and C = 1 + λK 2 ∈ R.

Proof. We sketch the main steps of the corresponding proof of Altmüller and Grüne (2012, Theorem 4.1). The particular control ( 14) reduces the equation ( 10) to y t (x, t) = y xx (x, t) + (µ -K)y(x, t).

(15) In order to obtain an estimation for σ we use classical Lyapunov methods presented, e.g., in [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF]. For this purpose we define

V (t) := 1 2 y(•, t) 2 L 2 (Ω) = 1 2 1 0 y(x, t) 2 dx.
Differentiating V (t) with respect to time and using Wirtinger's inequality result in V (t) ≤ 2(-π 2 + µ -K)V (t). This inequality yields V (t) ≤ e -2t(π 2 -µ+K) V (0) and thus

ℓ ⋆ (z(n)) ≤ 1 2 e -2nT (π 2 -µ+K) y(•, 0) 2 L 2 (Ω) = σ n ℓ ⋆ (z 0 ) with σ := e -2T (π 2 -µ+K) . The overshoot constant C is determined via ℓ(z(n), u(n)) = 1 2 y(•, nT ) 2 L 2 (Ω) + λ 2 v(•, nT ) 2 L 2 (Ω) = 1 2 y(•, nT ) 2 L 2 (Ω) + λ 2 K 2 y(•, nT ) 2 L 2 (Ω) = Cℓ ⋆ (z(n)) with C := 1 + λK 2 .
Combining this with the inequality for ℓ ⋆ then yields (5).

Boundary Control

In this subsection we derive the exponential controllability constants for the boundary controlled PDE. Like in the distributed case we intend to find an exponentially stabilizing feedback law for (7). One approach to obtain such a feedback is the method of backstepping, cf. [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF] for details. The idea behind this technique is to transform (7) into a so called "target system" which is exponentially stable. The transformation is done with a Volterra kernel w(x, t) = y(x, t) -x 0 k(x, z)y(z, t) dz.

(16) A suitable target system is given by w t (x, t) = w xx (x, t) + (µ -K)w(x, t) (17) w(x, 0) = w 0 , w(0, t) = 0 and w(1, t) = 0 in the case of Dirichlet control or w x (1, t) = 0 for the Neumann control. One advantage of this backstepping approach is that the resulting Volterra kernel is given as an explicit formula. For our system (7) and the target system (17) the transformation and the inverse transformation are given by w(x, t) = y(x, t)

+ x 0 k 1 (x, z)y(z, t) dz = y(x, t) + x 0 Kz I 1 K(x 2 -z 2 ) K(x 2 -z 2 ) y(z, t) dz (18) y(x, t) = w(x, t) + x 0 k 2 (x, z)w(z, t) dz = w(x, t) - x 0 Kz J 1 K(x 2 -z 2 ) K(x 2 -z 2 ) w(z, t) dz, (19)
where I 1 (x) denotes the first modified Bessel function and J 1 (x) the first classical Bessel function. The feedback control law coming from the Volterra transformation is given by

u(t) = - 1 0 Kz I 1 K(1 -z 2 ) K(1 -z 2 ) y(z, t) dz (20)
for the Dirichlet control and by

u(t) = - K 2 y(1, t) - 1 0 Kz I 2 K(1 -z 2 )
1 -z 2 y(z, t) dz (21) for the Neumann controlled system. A derivation that this transformation converts the system ( 7) into ( 17) can be found in [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF]. For a proof concerning the kernel properties see [START_REF] Liu | Boundary feedback stabilization of an unstable heat equation[END_REF]. In the subsequent Theorem 5 we use this control in order to establish (5). For its proof we need the following lemma whose proof is straightforward and thus omitted.

Lemma 4. Let u(x) ∈ L 2 (0, 1) and for i = 1, 2 define u(x) = v(x) + x 0 k i (x, z)v(z) dz. Then the estimate u(•) 2 L 2 (Ω) ≤ (1 + L i ) 2 v(•) 2 L 2 (Ω) (22)
holds with the constant L i :=

1 0 x 0 k i (x, z) 2 dz dx 1/2
.

Dirichlet Boundary Control Now we specialize our results to the case of Dirichlet boundary control, for which we can prove the following theorem, cf. also [START_REF] Altmüller | Distributed and boundary model predictive control for the heat equation[END_REF].

Theorem 5. The Dirichlet boundary controlled heat equation ( 7) with control (20), K > µ -π 2 , and stage costs (12) satisfies the exponential controllability condition (5). The corresponding constants are given by σ = e -2T (π 2 -µ+K) ∈ (0, 1) and

C = (1 + λK 2 η(K))ξ(K) ∈ R with η(K) := 1 0 x I1( √ K(1-x 2 )) √ K(1-x 2 ) 2 dx and ξ(K) := (1 + L 1 (K)) 2 (1 + L 2 (K)) 2 .
Proof. Again we explain the main steps of the proof of Altmüller and Grüne (2012, Theorem 5.2). First we note that the target system is exactly the reduced system (15) we used in the proof of Theorem 3. Thus, we can use the information about the decay rate. By combining Theorem 3 and Lemma 4 we obtain

y(•, t) 2 L 2 (Ω) ≤ (1 + L 2 ) 2 w(•, t) 2 L 2 (Ω) ≤ (1 + L 1 ) 2 (1 + L 2 ) 2 e -2t(π 2 -µ+K) y(•, 0) 2 L 2 (Ω)
and thus µ+K) . Furthermore, we get

ℓ ⋆ (z(n)) ≤ ξ(K)σ n ℓ ⋆ (z 0 ) (23) with ξ(K) := (1 + L 1 (K)) 2 (1 + L 2 (K)) 2 and σ := e -2T (π 2 -
ℓ(z(n), u(n)) = 1 2 y(•, nT ) 2 L 2 (Ω) + λ 2 v(nT ) 2 ≤ 1 2 y(•, nT ) 2 L 2 (Ω) + λ 2 K 2 y(•, nT ) 2 L 2 (Ω) 1 0 x I 1 ( K(1 -x 2 )) K(1 -x 2 ) 2 dx = Cℓ ⋆ (z(n)) (24)
with C := (1 + λK 2 η(K)) and

η(K) := 1 0 x I 1 ( K(1 -x 2 )) K(1 -x 2 ) 2 dx.
Combining ( 23) and ( 24) then yields the assertion.

Neumann Boundary Control Finally we now consider the heat equation with Neumann boundary control. In contrast to before, here we perform a different analysis for small and large values of the reaction parameter µ. First we will derive the exponential constants for sufficiently small values of µ. Afterwards, we consider the case of arbitrary values for µ.

In order to construct a control we use again the method of backstepping. The stage cost is given by ( 12).

It is known that the feedback

y x (1, t) = -Ky(1, t) (25
) stabilizes the problem for small values of the reaction parameter µ. In order to derive the corresponding constants C and σ, we need the following generalization of the Friedrichs inequality. Theorem 6. For u ∈ H 1 (Ω) the following inequality holds

Ω n j=1 |D j u| 2 dx + K ∂Ω |u| 2 dS ≥ C F (K) Ω |u| 2 dx (26)
where the optimal Friedrichs constant C F (K) is given by the least eigenvalue of

-∆u = µu in Ω (27) ∂ ν u + Ku = 0 in ∂Ω. Moreover, C F (K) is increasing and lim K→∞ C F (K) = C D
where C D is the least eigenvalue of the Dirichlet eigenproblem -∆u = µu in Ω, u = 0 in ∂Ω.

Proof. For a generalization of the first result see [START_REF] Auchmuty | Optimal coercivity inequalities in W 1,p[END_REF], for the second result see [START_REF] Auchmuty | Dependence of Friedrichs' constant on boundary integrals[END_REF]. Theorem 7. For µ < C D = π 2 the Neumann boundary controlled heat equation (7, 9) with control (25), K > 0 with C F (K) > µ, and stage costs (12) satisfies the exponential controllability condition (5). The corresponding constants are given by σ = e -2T (C F (K)-µ) ∈ (0, 1), C = 1 + λK 2 M ∈ R for some M ∈ R.

Proof. In order to use similar Lyapunov methods as in the proof of Theorem 3 we define

V (t) := 1 2 y(•, t) 2 L 2 (Ω) = 1 2 Ω y(x, t) 2 dx
and differentiate V (t) with respect to time

V (t)= Ω y(x, t)y t (x, t) dx = Ω y(x, t)(y xx (x, t) + µy(x, t))dx = y(1, t)y x (1, t) - Ω y x (x, t) 2 dx + µ Ω y(x, t) 2 dx = -Ky(1, t) 2 - Ω y x (x, t) 2 dx + µ Ω y(x, t) 2 dx ≤ (-C F (K) + µ) Ω y(x, t) 2 dx = -2(C F (K) -µ)V (t)
and thus V (t) ≤ e -2t(C F (K)-µ) V (0).

(28) Because of Theorem 6 and µ < C D = π 2 for the domain Ω = (0, 1) under consideration, there exists K ∈ R such that C F (K) > µ. This yields

ℓ ⋆ (z(n)) = 1 2 y(•, nT ) 2 L 2 (Ω) ≤ 1 2 e -2nT (C F (K)-µ) y(•, 0) 2 L 2 (Ω) = σ n ℓ ⋆ (z 0 )
with σ = e -2T (C F (K)-µ) ∈ (0, 1). With similar arguments one gets 1 0 y x (x, t) dx ≤ M e -γt 1 0 y x (x, 0) dx and with Agmon's inequality (cf. [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF]) the boundedness of y(x, t) for t ≥ 0. Thus, there exists

M ∈ R with y(1, t) 2 ≤ M 1 0 y(x, t) 2 dx.
The overshoot constant can be determined by

ℓ(z(n), u(n)) = 1 2 y(•, nT ) 2 L 2 (Ω) + λ 2 |u(nT )| 2 = 1 2 y(•, nT ) 2 L 2 (Ω) + λ 2 K 2 |y(1, nT )| 2 ≤ Cℓ * (z(n)) with C := (1 + λK 2 M ).
The drawback of the feedback control ( 25) is that it works only for µ < π 2 . In order to construct an exponential stabilizing control for arbitrary µ, we use again the backstepping method. Theorem 8. The Neumann boundary controlled heat equation with control ( 21), where K is chosen such that K > µ -π 2 /4, and stage costs (12) satisfies the exponential controllability condition (5). The corresponding constants are given by σ = e -2T (π 2 /4-µ+K) ∈ (0, 1) and

C = (1 + λK 2 ( √ M /2 + η 2 (K)) 2 )ξ(K) ∈ R with η 2 (K) :=   1 0 x I 2 ( K(1 -x 2 )) (1 -x 2 ) 2 dx   1 2 and ξ(K) := (1 + L 1 (K)) 2 (1 + L 2 (K)) 2 .
Proof. First, we note that the Volterra kernel for the Neumann control is exactly the same as in the Dirichlet case (only the feedback control and the eigenvalue change). Thus, we obtain with the same calculation as in Theorem 5 4-µ+K) . Moreover, we get with similar arguments as in Lemma 4

ℓ ⋆ (z(n)) = 1 2 y(•, nT ) 2 L 2 (Ω) ≤ 1 2 ξ(K)σ n y(•, 0) 2 L 2 (Ω) = ξ(K)σ n ℓ ⋆ (z 0 ) (29) with ξ(K) := (1 + L 1 (K)) 2 (1 + L 2 (K)) 2 and σ := e -2T (π 2 /
ℓ(z(n), u(n)) ≤ Cℓ ⋆ (z(n)) (30)
with 29) and (30) now yields the assertion.

C := (1 + λK 2 ( √ M /2 + η 2 (K)) 2 ). Combining (
Note that Theorem 8 is valid for larger range of reaction parameters µ than Theorem 7, however, at the expense that the derived constants attain larger values.

EXPLANATION OF NUMERICAL OBSERVATIONS

We can now use Theorem 7 and 8 to explain the observations from Table 1. From the spectral properties we know that the uncontrolled equation is unstable for µ ≥ π 2 /4. Since C D = π 2 holds in Theorem 7, we can apply this theorem for π 2 /4 ≤ µ < π 2 . Consequently, for these values of µ we obtain the overshoot constant C = 1 + λK 2 M which we can bring arbitrarily close to one by reducing the value of the regularization parameter λ. Therefore, Figure 1 reveals that for µ < π 2 we can always achieve stability with the smallest possible horizon N = 2. This explains our numerical findings from Table 1 for µ < π 2 . The similarity of the derived overshoot constant with that from the distributed control case derived in Theorem 3, i.e., C = 1 + λK 2 , moreover explains the qualitatively similarity of the two cases for µ < π 2 .

For µ ≥ π 2 we have to resort to Theorem 8 which yields the larger overshoot constant (1 + λK 2 ( √ M /2 + η 2 (K)) 2 )ξ(K). This constant is similar to that derived in Theorem 5 for Dirichlet boundary control (C = (1 + λK 2 η(K))ξ(K)), which explains why the dependence of N N on λ in Table 1 is qualitatively similar to N D in Table 2. Particularly, we obtain C → ξ(K) for λ → 0. Since ξ(K) is monotonically increasing with ξ(K) > 1 for K ∈ R + we cannot obtain C → 1 as λ → 0 and therefore we cannot expect stability for N = 2 even for small values of λ, cf. Figure 1.

An interesting question is whether the bound µ = π 2 which limits the applicability of the "better" Theorem 7 in our theoretical stability analysis is actually a tight bound. In order to get evidence for this fact, we have performed further numerical simulations of the MPC closed loop. Since our analysis depends on both C and σ but we only investigated the behavior of C, we need to keep the influence of σ small in our numerical simulations. One way to achieve this is be reducing the sampling time T , since for T → 0 we obtain σ → 1 and thus we can observe the pure influence of C. Figure 3 displays the largest value of µ for which stability is obtained with N = 2 depending on the sampling time T (solid blue line) while the theoretical bound π 2 is indicated by the dashed line. It is clearly visible that the curve tends to the theoretical bound for small values of T . For even smaller values of T , the simulations did not yield reasonable information because the numerical errors became predominant. Still, the simulations strongly indicate that the bound µ = π 2 separating the two different stability behaviors is indeed tight.

CONCLUSIONS

We have shown that the control structure and the stage cost significantly influence the length of the minimal stabilizing horizon in MPC without stabilizing terminal constraints. Our controllability based analysis is able to explain several -at the first glance puzzling -numerically observed phenomena for the heat equation. While it is not clear whether a similar analysis is feasible for more complex PDE models, we nevertheless think that our analysis provides insight for the choice of stage costs guaranteeing good performance of MPC with short optimization horizons, which can also be valuable in practical applications.

Fig. 1 .

 1 Fig. 1. Stability areas for different optimization horizons N in dependence of C and σ

Fig. 2 .

 2 Fig. 2. Solution of the heat equation (10) with µ = 15 and initial condition y(x, 0) = 1 5 sin(πx).

Fig. 3 .

 3 Fig. 3. Largest reaction parameter µ for which stability with N = 2 is possible in dependence on the sampling time T

Table 2 .

 2 Minimal stabilizing horizon N dis for distributed and N D for Dirichlet boundary control with varying λ and µ determined from numerical simulations of the MPC closed loop

		λ = 0.01	λ = 0.001	λ = 0.0001
	µ	N N	N N		N N
	4	2		2		2
	5	3		2		2
	7	4		2		2
	9	6		2		2
	11	7		3		3
	13	8		5		4
	15	9		6		5
	Table 1. Minimal stabilizing horizon N N for Neu-
	mann boundary control with varying λ and µ de-
	termined from numerical simulations of the MPC
				closed loop
		λ = 0.01	λ = 0.001	λ = 0.0001
	µ	N dis	N D	N dis	N D	N dis	N D
	10	2	2	2	2	2	2
	11	3	2	2	2	2	2
	12	4	3	2	3	2	2
	13	4	3	2	3	2	3
	14	5	4	2	4	2	3
	15	6	5	2	5	2	4

One observes that in all cases a smaller value of the regularization parameter λ leads to a smaller horizon. However, for Neumann boundary control only for values of µ below µ ≈ 10 it is possible to obtain stability for the shortest possible horizon N N = 2 by reducing λ, while for larger values of µ the decrease stops at a value N N ≥ 3 even for very small values of λ. Comparing this behavior with that of the Dirichlet boundary control and the distributed control from Table

2

, one sees that for values of µ below µ ≈ 10 the dependence of N N on λ is qualitatively similar to distributed control while for larger values of µ it is similar to Dirichlet boundary control. It is the purpose of the remainder of the paper to theoretically explain this numerically observed behavior.


Dominique Berger

Salah-Eddine Khzami

Jourdan Didier

Sandie Bernard

Graça De Carvalho

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Keywords: Didactique des sciences, éducation à la santé, éducation à la sexualité, conceptions, valeurs, enseignants, primaire, secondaire

représentations sociales et conceptions d'enseignants et futurs enseignants, incidences sur les pratiques et la formation ?

INTRODUCTION

This study is included in the Biohead-Citizen research project « Biology, Health and Environmental Education for better Citizenship » (FP6, CIT2-CT2004-506015, [START_REF] Carvalho | Biology, Health and Environmental Education for better Citizenship[END_REF]). One of its two axes aims to analyze in service teachers or pre-service teachers' conceptions. We will present the results coming from 12 countries on the reproductive health education /sex education [RHE/SE]. Our objective is firstly to identify the teachers' conception variation, in each country and between countries. Are these conceptions linked with knowledge, with believes, philosophical or/and political issues? What is the role of local and social practices in each country? How are the interactions between "knowledge", "values" and "social practices" [START_REF] Clément | Science et idéologie : exemples en didactique et épistémologie de la biologie[END_REF])? In fine, what are the in service teachers or pre-service teachers' conceptions on RHE/SE and on the impact of RHE/SE would be promote better citizenship. We aim to build training programmes on teachers' knowledge of and attitude towards reproductive health education /sexuality education [RHE/SE].

SCIENTIFIC BACKGROUND

Sex education, sometimes called sexuality education or sex and relationships education, is the process of acquiring information and forming attitudes and beliefs about sex, sexual identity, relationships and intimacy. It is also about developing young people's skills so that they make informed choices about their behavior, and feel confident and competent about acting on these choices. It is widely accepted that young people have a right to sex education, partly because it is a means by which they are helped to protect themselves against abuse, exploitation, unintended pregnancies, sexually transmitted diseases and HIV/AIDS. Human sexuality is not enclosing in the life transmission and includes four important dimensions:

-Anatomy and biology with sex physiology, procreation and survival mankind -Social dimension with cultural influence, social norms and rules and theirs political, juridical and religious incidences -Psychological dimension with gender issue, the personality's construction and self esteem.

-Affective and relational dimension with feelings (love, desire), points of vue and emotions… Sexuality presents a multidimensional aspect and concern deeply humans in the interweaving between sexuality, social influences, cognitive and affective development. So sexuality can be reduce in a dichotomy biology/psychology and involve in same time psycho-affective and biological maturation and social learning. Sexuality education seeks both to reduce the risks of potentially negative outcomes from sexual behavior like unwanted or unplanned pregnancies and infection with sexually transmitted diseases, and to enhance the quality of relationships. It is also about developing young people's ability to make decisions over their entire lifetime. Sex education that works, by which we mean that it is effective is sexuality education that contributes to this overall aim.

Although most interventions on Sex education concern teenagers (see ref in Kirby, 2002) or young adults (Ergene and al., 2005), the WHO texts insist on the necessity for implementing early sexuality education, particularly in primary schools [START_REF] Who | Preventing HIV/AIDS/STI and related discrimination : an important responsability of health promoting schools[END_REF]2004a). Many authors underline the importance of the educational dimension of this: "the action must be integrated into a global life-education strategy; the way it is spoken of must be adapted to the situation, the parents' involvement, as they are the first resource for the children, must be as great as possible, and the educators must come from different fields" (AFLS, 1993).

It is necessary, in order to go beyond a prescriptive approach, and adopt a decidedly educational perspective, to implement a learner-centred approach (Develay, 1993), taking into account the children's peculiarities, expectations, needs, and also their own representations (Fischer, 2001). In this way, taking into account pupils' representations is an important part of the learning process. The interest of taking them into account in an HIV/AIDS or sex education programme for children under twelve has already been underlined (Fassler and al., 1989;Ferron and al., 1989;Thomas, 1991;Sly and al., 1992;[START_REF] Schaalma | Determinants of consistent condom use by adolescents: the impact of experience of sexual intercourse[END_REF]Shonfeld and al., 1993;Kelly, 1995;[START_REF] Who | Preventing HIV/AIDS/STI and related discrimination : an important responsability of health promoting schools[END_REF]2004a ;2004b).

The model on which these propositions was based assumes that learners build knowledge from their own lives. They learn through their mental representations, which depend on each person's social and biological experience, and individual disposition. Learning is a highly active mental process which works in an integrative mode in the conflict between what the learner has in his mind and what he can find and understand through his conceptions about his environment. When a learner makes a new model, all his mental models must be rebuilt in an interaction between the pre-existent representation and outside information (Giordan, 2000). Sex education requires the teacher to take into account the pupils' representations and to help them build other more relevant

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ones. Moreover, the child's environment must be taken into account in the programme as children's representations are not only based on what they learn at school, but also on all the other aspects of a child's life (Downie, Tannahill and Tannahill, 1996).

Representations are to be taken as coherent models used by learners and teachers to reason when faced with a problematic situation (Jodelet, 1991;Farr, 1997). The building of these is rather complex and depends on values and beliefs shared by a social group and which give rise to a common outlook manifest during social interactions. As these representations are linked to the subject's emotional responses, the cultural and social group he belongs to, and constitute a decisive element in his relationship with the world, they are resistant to change. Representations are closely linked to behavior (Abric, 1997) and they could be considered as being a good target, as it is well known that knowledge is easy to change, but behaviors are much more challenging. Thus if a study aiming at impacting these representations is to be relevant, it has not only to target the transmission of knowledge, but also to take into account the social and cultural aspects of the children's and teacher's daily environment (Doise and Mugny, 1997).

Most researches are quasi-experimental designs. This method is of great evidentiary value but, due to its peculiarities, sex education also needs other designs (Victoria and al. 2004). In our context using the controlled randomized trial as a method for assessing the effects of the social representations is really difficult. The impact of representations means that using a control group is delusive, and that randomization is tricky (Tones and Tilford, 2001). This is mainly because causal chains in public health interventions are very complex. Although such an approach has its limits, several authors conclude that they are in favour of new research on teachers' representation being set up, aiming at determining exactly what content and what tools would be most suitable (Darroch J. and Silverman J. 1989 ;Heymans, 1993 ). Associating multidisciplinary partners in the drawing up and the implementation of the researches' programmes makes it possible to better understand the interactions between teachers' representations and social issues (believes, knowledge and practices) (Martinand, 2003;Clément, 2005). If these interactions were not considered, they would be confounding factors.

METHODOLOGY

Between the 19 countries of the Biohead-Citizen project, 12 give us data on RHE/SE. Four are outside Europe and widely (Tunisia, Morocco, Senegal) or in large part (Lebanon) muslin. Height countries are Europeans members (Portugal, France, Italy, Finland, Cyprus, Estonia, Hungry and Romania) and resent different characteristics: north and south, east and west Europe, catholic, protestant or orthodox culture.

In each country, we have data coming from pre (PRE) and in (IN) service teachers in primary school (P) and in secondary schools in biology (B) and national language (L) in 6 balanced data groups. The corpus include responses from 5189 questionnaires: Cyprus (CY, 322), Estonia (ES, 183), Finland (FI, 306), France (FR, 732), Hungry (HU, 334), Italy (IT, 559), Lebanon (LB, 722), Morocco (MA, 330), Portugal (PT, 351), Romania (RO, 273), Senegal (SN, 324), Tunisia (TN, 753).

Questionnaire and coding

The questionnaire was elaborated in the Biohead-Citizen Project in a common work with all the research teams and translated in national language (validate by retro-translation). It was pre-tested before implementation in each country.

Our work concerns only Sexuality education (cf. annex) and 35 questions (A2, A9, A21, A 30, A31, A36, A41, A 46, A57 to A60, A65, A85 to A90, B3, B5, B11, B13, B17 to B19, B24, B30 to B32, and B37 to B41). Responses are coding with a number scale. Questions A2, A9, A21, A 30, A31, A36, A41, B1 to B3, B5, B6, B9, B12, B13, B15 to B19, B21 to B27 are coding from 1 to 4 from « agree to totally disagree".

Few questions are on values (e.g. A41: "Homosexual couples should have the same rights as heterosexual couples"), on biological knowledge (e.g. "B17: After ovulation, the follicle changes into corpus luteum which produces high levels of progesterone and estrogens"). More are on teaching practices, values and knowledge (e.g. B19: Psychological and social aspects of sex education should be taught primarily by biology teachers). Questions A57 to A59 and A65 are about abortion. Responses are coding from 1 "In this case, abortion is morally acceptable" to 4 "In this case, abortion is morally unacceptable" and in A 65 "Abortion is acceptable" , from 1 "never" to 5 " at any moment".

Question A 60: "There are several behaviours that can help to decrease the spreading of AIDS world-wide. In your view, what is the behaviour you find most relevant to be considered in school sexual education? ", is coded from 1 "To have sex only within a stable relationship (not have several sexual partners)" to 4 "To have safer sex, for instance by using a condom in sexual intercourse".

The questions A85 to A90 and B37 to B41 are on the age teachers think the following topics should be first introduced at school by teachers and/or external specialists. Responses are coded from 1 "less than 6 years old" to 5 "never in school".

Responses' Analysis

Our hypothesis is that teachers' personal values influence their conceptions on HRE/SE. These conceptions may be linked to the school disciplinary (biology or national language), to the teaching level (primary or secondary level), to the training (in service or pre service). But the variables linked to country's context, religion or gender, are determining. What may most differentiate the conceptions of individuals with varying religions, specialities or any other feature? In fact, how to highlight the link between conceptions related to scientific knowledge or educative system in one hand and the teacher's personal, social and cultural background, namely their values on the other hand? Multivariate analysis allows representing the more structuring components of individuals. These components state better individual's variance. Statistical multivariate analysis has become a standard to investigate complex data featuring the behaviour of many individuals, according to many variables (Lebart et al., 1995). Here variables are questions in a questionnaire for which we gathered answers. To analyze the HRE/ SE and teaching practices data, we use principal component analysis (PCA, Lebart et al., 1995). Variables are coding as numbers. We have done a discriminating analysis (Between group analysis, [START_REF] Dolédec | Rythmes saisonniers et composantes stationnelles en milieu aquatique I-Description d'un plan d'observations complet par projection de variables[END_REF] in complement of PCA to show differences between groups' conceptions (country context, religion, faith and gender). We use instrumental variables in responses' structures as disciplinary (P, B, L), training (In or Pre service) and countries.

RESULTS AND ANALYSIS

In this results' presentation, we have chosen to don't show calculation, just graphics which are more explicit. We present the first results and just few variables. We don't propose an exhaustive study, just a skimming over to explore the new research field founded by Biohead-Citizen Project.

Global approach

Figure 1: Principal component analysis (PCA), global correlation circle

The schema shows us three distinct groups of variables. The group of questions (A) contributes strongly to axe 1 and illustrate that teachers think it is not possible to teach the social components of sexuality education to young pupils. Moreover they don't agree to teach in school before 15 years old contents as homosexuality, pedophilia, pleasure organs, abortion, incest and sex violence, orgasm and pleasure, eroticism and pornography (A85 to A90 and B37 to B41). Netherless, pupils found all these topics in their daily life with their peers, in the streets or in the media (press, radio, TV…).

We found a group (B) composed by questions about possibility of abortion (A57 to 59); the respondents who don't agree with abortion think also that women have not the same rights then men (A2), that it's not important to have same number women deputies than men (A30) and that women, biologically, have not same intelligence than men (A21). For them, Homosexual couples must not have the same rights then other couples… And they think also, it's the hazard who determines the sex of a baby (B30).

The group (C) of questions is defining the same oblique axis as the group (B), with just opposite positions: it's possible to try to choose baby sex by specific diet or medically assisted techniques (B31, B32) and who agree

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with the idea that abortion would be acceptable at any time (A65); a man can be as sensitive and emotional as women (A46) and there is no biological reason which would justify inequality between men and women (A6, A36, A46). Finally the question related to safer sex (as a behavior most relevant to be considered in school sexuality education) is included in this group (C) of questions.

Training issue

The analysis (figure 3) shows us a difference between training. A PCA has been done to differentiate the six "groups of teaching" controlled in our sampling design (pre or in service, primary or secondary level, biology or letters. In fact Biology teachers (in service or pre service) have conceptions more based on knowledge and on the conviction that they can teach the whole dimensions of sexuality education (Q A31, B5, B18 B17, B22, B3 B24, B30). They think that they have to teach biological aspects of sexuality (human reproduction and ISD) and say that they are able o speak with their pupils about emotion, feelings, pleasure but they have the same difficulties as the others with safer sex (A60) and homosexual rights (A41). In relation with the others groups, we found a clear difference between in service and pre service teachers in particular for primary teachers. The in service primary teachers are in difficulty in front of the content of sexuality education especially with social and affective issues like homosexuality, abortion, contraception, pleasure organs, paedophilia, safer sex (Questions : A 85 to A90, A 60) for example their conception is based on a false knowledge about the homosexuality origin (Question: B11).

The pre service primary teachers are in opposition with in service primary teachers and agree with safer sex, abortion, homosexual rights… and prefer teach these contents early in the curriculum.

There is an effect of age in the teachers' conceptions. Inside the three thematic groups (primary, secondary biology and secondary letters), older teachers (in service; mean age = 40 years old) are always on the upper part of the axis 2 and the younger teachers (pre service; mean age = 23 years old) are in the lower position (figure 3). Morocco and Senegal at the other end France, Finland, Portugal, Estonia and Italy, in the middle, Hungary, Romania and Cyprus. Crossing this graphic with the figure 1, we can see that there is a correlation between countries and conceptions.

Country variable

The charts are linked by axes' superposition. It seems that European countries have different point of view and conceptions than south countries. News entrants in Europe are in middle position. According to figure 1, we can see that south countries have conceptions founded on inequality between men and women and about their rights and same thing about homosexual rights. They have conservative conceptions about abortion, safer sex and sexuality education. They do not agree to teach sexuality topics in primary school or in first years in secondary school before 15 years. In opposition around France (Fr), we can see the correlation with open view about abortion, equality between men and women and homosexual rights.

But we have to nuance this analysis. The opposition is also in teaching practices about topics like homosexuality, pedophilia, pleasure organs, contraception, abortion, sexual diseases (A85 to A90). France, Finland, Estonia, Portugal and Italy agree more to teach theses contents. The charts are explicit on these issues. We see on the left chart the distribution of theses questions on the correlation circle. Their scores are superior to 3 or 4. It seems that teachers think that the following topics should be first introduced at school by teachers not before 15 years old. The vectors show us where theses conceptions are strongly implanted. The right chart indicates the countries concerned.

The questions about social aspect of sexuality (B37 to B41; pregnancy and birth, sexual intercourses, incest and sexual violence, orgasm and sexual pleasure, eroticism and pornography) have scores near 5. The teachers clearly think that it's not their job to teach these contents. The scores are powered by south countries.

On the right charts we can see two poles. One is constituted by south countries (Tunisia, Lebanon, Morocco, and beneath Senegal) and the other one by european countries (France, Estonia, Finland, Portugal, Italy, and beneath Cyprus). New entrant's countries are in the middle (Romania, Hungary). In the right chart we can see also the large distribution of the answers reveal the conceptions' diversities. The four categories (AGN, agnostic, atheist, without religion; MUS, Moslem [Sunnite, Chiite, Druze, Alevi]; ELS, other religions or believes [Jewish, Hinduist, Animist…]; CHR, [catholic, protestant, orthodox]). We have two poles on an axe with in one hand Moslem and in the other hand Agnostic. In the middle part, we found others religions and Christians. With a similar process; we have crossed religion and teachers' conceptions (cf. fig1).

The religious impact

There is a strong correlation between religion

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and teachers' conceptions. Agnostics and Atheists agree with the idea that abortion would be acceptable at any time (A65). They think also that a man can be as sensitive and emotional as women (A46) and there is no biological reason to inequality between men and women (A6, A36, A46).

For them, safer sex is the behavior they find most relevant to be considered in school sexuality education. They agree with rights' equality between men and women and homosexual couples. At the opposite we find Moslem. This statement needs to be qualified. So, we crossed this data with God's believe levels.

4.4 Crossing the data with God's believe levels. In question P12, The responses are coded from 1 "most important" to 5 "I don't believe in God". To do that, we use a PCA which illustrate that the level of faith is an important variable more than religion's membership. The variable "faith" contributes heavily to axe 1 as also shown by this graph. There is a great difference between those who have a high level in faith ( 1) and all the others (2, 3, 4 and 5). We note a small scale progression between 2 to 5 along axe 1.

All the data are clearly oriented in two blocks. Those who believe deeply in God are positioned on conservative point of view about men and women equality about women rights and homosexual rights. They don't agree with abortion and they think that it is not to school to teach social content in sexuality education.

In the opposite, more the teachers don't believe in god more they agree with social contents of SE and think that men and women are equal, more they agree with women rights an homosexual rights. They think also it's possible to teach social contents as pedophilia, homosexuality, incest and violence, sexual intercourses… pregnancy and birth and they agree abortion at any moment.

It is also interesting to cross these data (faith level) with religion. In fact the results show us that faith' level is the most important factor which impact on teachers' conceptions. Christians with high faith's level have same kind of responses to the questionnaires than Moslem with same level.

Crossing the data with religious practices Figure 8: interclass analysis, religious practices

This schema shows us the evaluation of religious practices' level. The responses are coded from 1 "I practice a religion" to 5 "I don' practice a religion". The distribution near axe 1 indicate that the religious practices' factor is also relevant

In coherence with the faith level we can observe two groups. The first composed by those who declare practicing a religion (1 and 2) and the other characterized by low practices or no practice. These groups are also linked with the teachers' conceptions. We found the same orientations about teaching practices in social components of sexuality education abortion and men and women equality and rights. Teacher's who have a religious practice don't agree with abortion, homosexual couple's rights and safer sex, they don't agree also with the social contents of sexuality education (A 85-90 and B37-41). To the teaching practices, they think that psychological and social aspects of sexuality education should be taught by health professionals.

DISCUSSION

The implementation of early educational actions on health risk behaviors (e.g. nutrition, drugs abuse and sexually transmitted infections) can answer more effectively to the strong social demand, which has also been taken constantly by the political sector for many years in order to answer to major societal needs (WHO, 2001). Since the efficacy of the preventive policies is largely dependent of the citizens' adhesion to them, Health Education and Sexuality education turned out to be a rather important social issue. Although being a school mission, health authorities have reinforced the idea that Sexuality Education should be carried out in partnership with other social organizations involved in Health Education, in particular local organisms of the Health sector (WHO, 2006).

To study the effects of the context components and to identify the elements that interact with the sexuality learning domain has been a rather difficult task due to the lack of adequate instruments of research. It is in this field of educational determinants of the person's structuration (especially his/her self-esteem) and his/her ability to communicate, that educational actions must be taken, contributing in particular to the development of the psychosocial competencies. This involves moving beyond practices that rely mainly on classroom-based sexuality education models, to a more comprehensive, integrated approach that focuses both on child-youth attitudes and behaviors, and their environment. Sexuality is regarded as an important determinant of the personal development and wellbeing.

These first results must be considered with prudence. Deep analysis must be done to highlight precisely the variances between groups and their links with social representations and individuals' conceptions.

Forever, we have found that individuals' conceptions have a great incidence on teachers' social representations and on their practices. Recent publications (St Léger, 1998;Lister-Sharp et al. 1999;INSERM, 2001;WHO 2006) have shown that certain pedagogical activities have some positive effect, others no effect at all and yet other ones have a negative effect. This negative impact can be due not only to the quantitative increase of pupils' risk behaviors but also to the intensification of pupils' uneasy feeling or of their perception of a tension between their social/family life experience and the school prescriptions. So the teacher's conceptions are really important especially in prevention to promote health and sexual health. Sexuality Education is determined directly by both the teaching pedagogical practices and the social context [START_REF] References Allensworth | The comprehensive school health program: exploring an expanded concept[END_REF][START_REF] Gold | The science base for comprehensive health education[END_REF] especially in skills developments.

To be effective, sexuality education needs to include opportunities for young people to develop skills, only having information is not sufficient. The kinds of skills young people develop as part of sexuality education are linked to more general life-skills. Being able to communicate, listen, negotiate, ask for and identify sources of help and advice, are useful life-skills and can be applied in terms of sexual relationships. Effective sexuality education develops young people's skills in negotiation, decision-making, assertion and listening. Other important skills include being able to recognize pressures from other people and to resist them, deal with and challenge prejudice, seek help from adults -including parents, carers and professionals -through the families, community and health services. Sexuality education also helps young people to be able to differentiate between accurate and inaccurate information, discuss a range of moral and social issues and perspectives on sex and sexuality, including different cultural attitudes and sensitive issues like sexuality, abortion and contraception.

The first prevention level is to be in capacities to access to information (WHO). Sexual health education involves a combination of educational experiences that allows individuals to do the following:

• to acquire knowledge that is relevant to their specific sexual health issues; • to develop the motivation and personal insight that they will need to act on the knowledge; • to acquire the skills necessary to enhance sexual health and avoid negative sexual health outcomes; • and to help create an environment that is conducive to sexual health.

Research consistently shows that positive sexual health outcomes are most likely to occur when sexual health education integrates knowledge, motivation and skill-building opportunities and occurs in an environment conducive to sexual health.

School-based sexuality education is an important and effective way of enhancing young people's knowledge, attitudes and behavior (Kirby, 2001(Kirby, , 1992)). There is widespread agreement that formal education should include sexuality education and what works has been well-researched. Evidence researches suggest that effective school programmes will include the following elements:

• A focus on scientific information about human reproduction, providing accurate information about contraception and birth control; Working on psychosocial abilities, self esteem, dealing with peer and other social pressures on young people; providing opportunities to practice communication, negotiation and assertion skills;

• Uses a variety of approaches to teaching and learning that involve and engage young people and help them to personalize the information; • Uses approaches to teaching and learning which are appropriate to young people's age, experience and cultural background;

Formal programmes with these elements have been shown to increase young people's levels of knowledge about sex and sexuality, decrease risk when they do have sex. All the elements are important and inter-related, and sexuality education needs to be supported by links to sexual health services, otherwise it is not going to be so effective. It also takes into account the messages about sexual values and behavior young people get from other sources, like friends and the media. It is also responsive to the needs of the young people themselves -whether they are girls or boys, on their own or in a single sex or mixed sex group, and what they know already, their age and experiences.

Therefore the great challenge we have done with this study, is to identify better not only the nature of the teaching practices but also the teachers' conceptions, especially in sexuality education and their links with the practices. When teacher's have a high level in believes in God, they don't agree to men and women equality, homosexual rights, abortion and safer sex. They don't agree to teach social component of sexuality education before 15 years old. Or, the early access to information is the best ways to prevent sexual violence, sexual transmitted diseases, and promote sexual health and more largely health education. Providing effective sexuality education can seem daunting because it means tackling potentially sensitive issues. However, because sex education comprises many individual activities, which take place across a wide range of settings and periods of time, there are lots of opportunities to contribute. School-based education programmes are particularly good at providing information and opportunities for skills development and attitude clarification in more formal ways, through lessons within a curriculum. So it's very important to Public Health, especially to prevent AIDS and sexual transmitted diseases and sexual violence to well know the obstacles to sexuality education's implementation.

For a long time, mainly because human societies were particularly concerned with hygiene, it was considered appropriate for schools, with their uncontested basis of universal knowledge, to convey rules for good health. Nowadays, health education and promotion is not just about encouraging children and young people to eat well and to exercise; it encompasses a much broader holistic approach (15). In both cases, school should not only be the automatic transmitter of public health policy. The educational system has to act as mediator. It cannot ignore public health issues or social demands but it has to confront such claims with its own values and missions. However, for a teacher, who has many priorities of schooling, including building literacy and numeracy skills; scientific and artistic competencies; societal, historical and cultural dimensions, and who have in fact to provide the means for all to succeed, it is not easy to have a clear view of his or her own contribution to health promotion including sex education. Defining the teacher's role is all the more important since responsibility for children's health lies primarily with the parents and that health promotion and sex education is by no means neutral. Instead, it lies at the intersection between the private and public domains, related to behavioural issues that are determined culturally and to the most intimate of personal decisions.

In the field, it is not easy to identify the school's mission in an environment marked by the power of the models transmitted by the media and the political use of religious points of vue. The position of teaching staff is, therefore, difficult to maintain. The first aim of teacher training in sex education is then to help them have a clear view of their mission and its ethical limits. Before giving them methodological tools, teacher training aims at helping them build their professional identity (16).

Sex education is based on the faith in the ability of people to take charge of their own destiny. Trust in teachers and a contribution to the development of their capacity to undertake health promotion and sex education activities can be a base for international co-operation and sex education development.

APPENDIX

Questions about HRSE A2.

In a modern society, men and women should have equal rights. A9.

Women are less intelligent than men are because their brains are smaller than men's brains are. A21. Biologically, women can be as intelligent as men. A30. It is important that there are as many women as men in parliaments. A31. When a couple has already had two girls, the chances that their third child be a boy are higher. A36. Men might be more able to think logically than women, because men might have different brain bilateral symmetry. A37. Religion and politics should be separated. A41. Homosexual couples should have the same rights as heterosexual couples. A46. Biologically, men cannot be as sensitive and emotional as women.

Here are stories of couples, or women, who are considering the necessity of having an abortion. If you were in these situations, would you too consider the possibility of abortion? (Tick only ONE of the four boxes for EACH situation): A57. A couple already has one child, and the mother is at risk of dying from a complication during her pregnancy. A58. A young couple in severe economic difficulty. A59. A woman who has been informed of the high probability of giving birth to a severely handicapped child. A60. There are several behaviours that can help to decrease the spreading of AIDS world-wide. In your view, what is the behaviour you find most relevant to be considered in school sexual education? (Tick only ONE of the four boxes)

To have sex only within a stable relationship (not have several sexual partners)

To have safer sex, for instance by using a condom in sexual intercourse A65. Abortion is acceptable (tick only ONE answer): Never Up to 2 weeks after conception (i.e. before implantation in uterus) Up to 12 weeks after conception (the legal period in countries where abortion is authorized). Up to 6 months (before the foetus can survive outside the uterus) At any moment

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Figure 2 :

 2 Figure 2: teachers' statute variable

Figure 3 :

 3 Figure 3: interclass analysis to differentiate countries When we link the data (fig 1) with countries variables, the schema indicate clearly the incidence of a cultural effect. We have on the first axe the distribution of the countries between two poles. At one end we have Lebanon, Tunisia,

Figure 4 :

 4 Figure 4: questions A85 to A90 and B37 to B41

Figure

  Figure 5: religion distribution

Figure 6 :

 6 Figure 6: interclass groups analysis, faith's distribution

Figure 7 :

 7 Figure 7: relative contribution of main factors (faith's level)

Figure 9 :

 9 Figure 9: interclass analysis, variables' distribution This chart linked with the previous one, indicates that issues about abortion (A65) and safer sex (A60) and homosexual couple's rights (A41) and about the biological aspect of sensitivity and emotion (A46) are the most relevant.

  • A basis in theories which explain what influences people's sexual choices and behavior and gender issue • A clear, and continuously reinforced message about sexual behavior and risk reduction;

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Acknowledgements

This work had the financial support of the European project FP6 Biohead-Citizen CIT2- CT-2004-506015.

Sexually transmitted diseases should be taught primarily by biology teachers. B5.

Sexually transmitted diseases should be taught primarily by health professionals (doctor, nurse). B11. There are genetic factors in parents that predispose their children to become homosexual. B13. When women stop taking the contraceptive pill, menstruation occurs, due to the absence of progesterone and estrogens hormones. B17. After ovulation, the follicle changes into corpus luteum which produces high levels of progesterone and estrogens. B18. Teachers avoid teaching sex education because these topics are private. B19. Psychological and social aspects of sex education should be taught primarily by biology teachers. B24. Psychological and social aspects of sex education should be taught primarily by health professionals (doctor, nurse).

Admitting that you don't have any children and that you wish to have only one child, would you do the following?