André P Calmon
email: acalmon@dt.fee.unicamp.br.

Thomas Vallée
email: thomas.vallee@univ-nantes.fr.

João B R Do Val

Lemna

MONETARY POLICY AS A SOURCE OF UNCERTAINTY

Keywords: Uncertainty, Monetary Policy. JEL: C61, E42, E52, E58

This paper proposes a model in which control variations induce an increase in the uncertainty of the system. The aim of our paper is to provide a stochastic theoretical model that can be used to explain under which uncertainty conditions monetary policy rules should be less or more aggressive, or, simply, applied or not.

Introduction

This paper examines a model of systems for which the control action has an ensuing impact on the uncertainty that pervades the system. For systems that have origin in physical, chemical, biological or engineering phenomena, the inherent system uncertainty is not dependent upon the controller action, whereas in some macroeconomic systems considerable changes in the policy can create an impact on the confidence of the various economic agents and aggravate the overall system uncertainty. If the action is interpreted as a change of course, for example a new insight from the central bank into the health of the economy, the economic agents may have uncertain reactions to the new view. Thus, any change of course induced by a change in a policy is bound to create further uncertainty, and policy-makers should act with extra caution. There is no parallel to this reinforcing effect in the study of physical phenomena, and to our knowledge, this behavior is not dealt with in the field of control theory either.

Recently there has been a great deal of research on monetary policy making under uncertainty. This research can be explained by the need to reach beyond the classical linear-quadratic framework to model accurately the policy-maker's problem. During the sixties and the seventies, with the development of modern optimal control theory, monetary models were constructed under the assumption that the parameters of the model would remain fixed over time, i.e., they are not affected by political decisions. Hence, the main idea was that it is possible to deal with uncertainty using, for example, the Bellman's certainty equivalence. In a classic paper, [START_REF] Brainard | Uncertainty and the effectiveness of policy[END_REF] already criticized this point of view. He argued that the certainty equivalence can be employed only in the case of additive uncertainty (e.g. shock uncertainty). Brainard highlighted the fact that, in the case of multiplicative uncertainty (e.g. model uncertainty), uncertainty about the parameters of a model would lead to less aggressive policies than those used when uncertainty is ignored. This result is known as the "Brainard conservatism principle". Another conclusion from this work is that we need robustness. This conclusion is not a new one. One can find a similar conclusion in [START_REF] Rosenbrock | Good, Bad, or Optimal ?[END_REF]. For Rosenbrock and McMorran a distinction exists between Good, Bad and Optimal decision rules. It may happen that optimal rules are not Good rules since the optimality characteristic comes from an exact specification of the optimization problem (including the constraints). Hence, any deviations of the constraints, for example, may lead to poor performances of the ex-ante optimal rules. As a consequence, and as argued by [START_REF] Onatski | Modeling Model Uncertainty[END_REF], model uncertainties has to be explicitly described.

Based on the use of Bayesian decision theory and robust optimal control theory, there have recently been many research papers challenging the need for robustness (see for example [START_REF] Edge | Welfaremaximizing monetary policy under parameter uncertainty[END_REF] or [START_REF] Onatski | Modeling Model Uncertainty[END_REF]). Among these approaches, Markov switching models have also provided an interesting tool. The use of Markov switching models in the optimal monetary policy literature is explained by their ability to model both exogenous and endogenous regime changes. [START_REF] Blake | Optimal Monetary Policy in Markov-Switching Models with Rational Expectations Agents[END_REF], for example, provide two algorithms to compute the solution of a model with regime shifts. They show that these methods can be applied to decision-making processes in order to incorporate the case of different beliefs between the policy-maker and the private sector. In a recent paper, [START_REF] Svensson | Monetary Policy with Model Uncertainty: Distribution Forecast Targeting[END_REF] combine Bayesian learning theory and Markov jump-linear-quadratic (MJLQ) systems to handle the problem of a policy-maker who does not know the structure of the economy.

While interesting, these papers are far from having reached any consensus on the theoretical aggressiveness of the monetary policy rules. As noticed by [START_REF] Issing | Monetary Policy in a World of Uncertainty[END_REF], uncertainty about the persistence of the inflation process can lead policy-makers to adjust interest rates more vigorously, since this can reduce uncertainty about the future development of inflation. Similarly, when such uncertainty arises from imperfect credibility, policy-makers may be well-advised to act more decisively. Furthermore, empirical monetary policies, such as official interest rate changes, tend to be adjusted relatively infrequently and in small steps. Usually, theoretical policies do not exhibit such a smooth behavior.

As noticed by [START_REF] Cagliarini | Monetary Policy-making in the Presence of Knightian Uncertainty[END_REF], interest rates tend to move in a sequence of steps in a given direction, or remain constant for some time, rather than experiencing the frequent reversals that commonly arise from optimal policy simulations. In other words, models of optimal monetary policy behaviour tend to generate much more volatile paths of interest rates, for which policy reversals are frequent. [START_REF] Cagliarini | Monetary Policy-making in the Presence of Knightian Uncertainty[END_REF] develop a model based on Bewley preferences with Knightian uncertainty which has the potential to explain the inertia in the level of the interest rate observed in actual interest rate paths. This Knightian uncertainty is directly related to the fact that policy-makers may be uncertain not only about the parameters, but also about the general specification of the model being used. For [START_REF] Cagliarini | Monetary Policy-making in the Presence of Knightian Uncertainty[END_REF], such a Knightan/risk approach is preferable to some robust approaches since monetary policy-makers usually communicate in terms of the balance of risks rather than in terms of avoiding worst-case scenarios.

Our paper aims to take in account the step behaviour of the monetary policy-makers through a different approach. We assume that the policymakers will choose an interest rate that decreases the amount of uncertainty they are likely to face in the subsequent period. This can be used to explain the collective decision of the FED, ECB and BOE to decrease all the federal funds rates in the subprime crisis. Rather than restoring trust in the financial markets, this decision was followed by a worsening of this situation.

The effect described in the previous paragraph is well known. When dealing with monetary policy rules, or more generally with economic systems, the actions of the policy-maker are observed by the distinct economic agents, who react according to their interests. As a consequence, a change of course represented by a change in a policy is bound to create further uncertainty, and policy-makers should act with extra caution. This has been noted in a seminal paper by [START_REF] Lucas | Econometric Policy Evaluation: A Critique[END_REF] for whom political decisions have an impact on the economy and induce changes in the value of parameters. Within this context, the aim of our paper is to provide a stochastic theoretical model that can be used to explain under which uncertainty conditions a policy rule should be less or more aggressive, or, simply applied or not. The paper is structured as follows. Section 2 first surveys the literature on monetary policy and uncertainty. Then we introduce a model that endogenizes uncertainty. Section 3 applies this model to a general macroeconomic monetary policy model. We show that, over 1964-2008, the FED may have been more aggressive than predicted in our model. Then we conclude.

Monetary Policy and Uncertainty

Since the seminal work of [START_REF] Brainard | Uncertainty and the effectiveness of policy[END_REF] the link between uncertainty and aggressiveness of monetary policy has been considerably studied. We briefly highlight some of the main results in the following section. We then present the CVIU approach developed by [START_REF] Calmon | Control Variation as a Source of Uncertainty: Single Input Case[END_REF] that endogenizes uncertainty.

Dynamic Uncertainty

The System

Consider a discrete-time system described by the state equation:

x k+1 = A k x k + b k u k + ω k , (1)
where

A ∈ R n×n , b ∈ R n×1 , x k ∈ R n , u k ∈ R and ω k ∈ R n are respectively,
the state, the input and a the noise (a stochastic process). Suppose that in a given time window 0 ≤ k ≤ N , the system performance is evaluated by means of the cost functional,

J(x 0 , µ) = E N -1 k=0 C k (x k , u k ) + C N (x N ) , (2)
where E[•] stands for the expected value of the corresponding random variable, C k is non-negative for each 0 ≤ k < N -1 and convex in both arguments. Also, the terminal cost C N is non-negative and convex. The following quadratic cost function is a possible candidate

J(x 0 , µ) = E N -1 k=0 (x T k Qx k + u k P T x + u T k Ru k ) + x T N F x N
We assume that at each time instant k the policy maker can determine the input control, u k , having perfect state information, and µ = {u 0 , u 1 , . . . , u N -1 } stands for an admissible policy.

Results on Uncertainty

Following [START_REF] Onatski | Robust Monetary Policy Under Model Uncertainty in a Small Model of the U.S. Economy[END_REF], different sources of uncertainty exist that affect the formulation of monetary policy :

1. Shock uncertainty: Future events, shocks and disturbances.

Parameter/Model uncertainty :

• The actual workings of the economy;

• Market reactions to central bank policy;

• Market expectations of central bank policy; 3. Econometric estimation error: Limitations of the date.

If we refer to our dynamic discrete system, these sources become

• Additive uncertainty → Shock: uncertainty on ω k ;

• Parameter uncertainty → Multiple uncertainty (e.g. policy effectiveness): uncertainty about the vector b, → Persistence uncertainty: uncertainty about the matrix A.

→ Objective uncertainty: uncertainty about the weights of the loss function, Q,R and P .

• Econometric uncertainty: for example uncertainty about the final date N .

Among some of the results (see [START_REF] Sahuc | Règles de politique monétaire en présence d'incertitude : une synthèse[END_REF] for a survey), we know that additive uncertainty does not impact the monetary policy because of the certainty equivalence principle [START_REF] Brainard | Uncertainty and the effectiveness of policy[END_REF]). The same author showed that multiplicative uncertainty leads to caution unless a negative correlation between additive and multiple uncertainties exists. But, if persistence uncertainty is more important than the one about policy effectiveness (multiplicative uncertainty), then optimal policy may be aggressive (see [START_REF] Craine | Optimal monetary policy with uncertainty[END_REF] or [START_REF] Mercado | Parameter Uncertainty and Policy Intensity: Some Extensions and Suggestions for Further Work[END_REF]Kendrick (2000,2006)). As [START_REF] Söderström | Monetary policy with uncertain parameters[END_REF] says when there is uncertainty about the persistence of inflation, it is optimal for the central bank to respond more aggressively to shocks than under certainty equivalence, since, this way, the central bank reduces uncertainty about future changes in inflation. Finally, if the cost of target variability (e.g. matrix Q of the loss function) is directly related to certain structural parameters of the model (as B), the classic (Brainard) attenuation of monetary policy can be overturned (see, for example, [START_REF] Levin | Parameter Uncertainty and the Central Bank's Objective Function[END_REF]).

One of the main results is that there is no consensus about the link between uncertainty and the optimal degree of aggressiveness of monetary policy. Furthermore, these results also depend on the "optimization strategy" adopted.1 .

Historical Federal Funds and Endogenous Uncertainty

As previously discussed, the sources of uncertainty are many and their consequences on the degree of aggressiveness of monetary policy are varied. When looking at empirical data, (see figure 1), one can easily conclude that the ECB is less aggressive in its conduct of monetary policy than the FED. • The ECB is facing more uncertainty than the FED, and, their conservatism is compatible with a negative correlation between uncertainty and the aggressiveness of their policy.

• The uncertainty was higher in USA, and this uncertainty is positively correlated with the aggressiveness of their policy.

If we go back to a constant parameter case of our general state equation (1):

x k+1 = Ax k + bu k + ω k , (3)
The existence of uncertainty about some parameters of A and B will lead to

x k+1 = (A + Ã)x k + (b + b)u k + ω k , (4)
which can be reformulated as

x k+1 = Ax k + bu k + ω k + Ãx k + bu k , (5)
If x k is a known vector at time k, Ãx k is a shift in the initial additive uncertainty ω k , while bu k becomes an endogeneous uncertainty due to the unknown exact impact of u k on the state. So it is likely that any change in monetary policy can impact the overall level of uncertainty.

One framework used to analyze such an interdependence is the one of signalling game (see for example [START_REF] Canzoneri | Monetary policy games and the role of private information[END_REF] and [START_REF] Walsh | Announcements, Inflation Targeting and Central Bank Incentives[END_REF]) where any change in monetary policy will be interpreted by private agents as a positive or a negative signal. In the case of a loss of credibility (bad signal), the monetary policy may increase GDP and inflation variability, and thus increase the level of uncertainty. Although interesting, this approach generally assumes that the central bank knows the way its signal will impact private agents' expectations. Let assume, that the monetary authority knows only that any particular change of its monetary policy, such as an increase of the federal funds rate, may increase the level of uncertainty on the system and, as a consequence, the future effectiveness of its policy. Following Brainard's conservatism principle, a risk neutral policy-maker may found optimal to keep unchanged the federal funds rate.

We know that the ECB, like the FED, defines its optimal policy by looking at both the inflation and the output gap dynamics. As a consequence, many monetary policy models, such as the well known Rudebush-Svensson model [START_REF] Rudebusch | Policy Rules for Inflation Targeting[END_REF]), integrate these two dynamics. So, two sources of uncertainty are taken into account. Uncertainty concerning future inflation dynamics, and uncertainty on other economic dynamics like the output gap changes. Of course, uncertainty faced by the policy-makers can influence these both dynamics with the same strength or may impact them differently. In a similar way, the policy used can feedback on both uncertainties or only one.

In the next section, we will introduce the principle of a system for which control variations increase state uncertainty (CVIU). With such a model, one can endogenize the level of uncertainty of the different dynamics of the system.

The CVIU Approach

Traditionally, in stochastic control problems, the imbedded system uncertainty is modeled by means of additive or multiplicative disturbances [START_REF] Bertsekas | Dynamic Programming & Optimal Control[END_REF]) and of parameter uncertainty, as in the Markov Jump Linear Systems [START_REF] Costa | Discrete-Time Markov Jump Linear Systems[END_REF]). However, these models may not suffice to describe uncertainty in many situations. Consider, again, the problem faced by a National Central Bank (NCB) when defining monetary policy [START_REF] Onatski | Robust Monetary Policy Under Model Uncertainty in a Small Model of the U.S. Economy[END_REF], [START_REF] Svensson | Monetary Policy with Model Uncertainty: Distribution Forecast Targeting[END_REF]). As it increases or decreases the interest rate, the NCB has an uncertainty about the expectations of the economic agents. Significant course change in the monetary policy may induce unexpected and undesired consequences such as an increase in inflation or a reduction in GDP. On the other hand, if the variation of the interest rate is too small, the NCB objectives may not be accomplished. This is an example of a system for which any change of policy leads to an increase in system uncertainty.

The monetary policy problem is an example of a system where control variations increase state uncertainty (CVIU). From a control theory perspective, CVIU systems can be used to control systems with complex, underdetermined dynamics, for which the behavior near a given point and for a given control policy can be fairly well described by a linear model. However, large variations of the control action can drift the system to regions where the linear approximation error is too large. In this case, the approximation error corresponds to the uncertainty generated by policy variations.

We consider that the magnitude of the control action acts as a source of system uncertainty in such a way that the noise sequence ω k is modulated by the absolute value of the control |u k |, as follows:

ω k = (σ k + σ k |u k |) ε k ,
where σ k > 0, σ k > 0, and ε k is an i.i.d. random vector with a normalized covariance matrix cov(ε k ) = I n×n .

We aim at the dynamic programming method, and in a preliminary step we are interested in characterizing the function V : R n × R → R and V * : R n → R, defined as

V (x, u) = C(x, u) + E [F (x 1 )] , (6)
and

V * (x) = inf u∈R m V (x, u), (7)
where C : R n × R → R and F : R n → R are both convex, non-negative, Lipschitz functions. The random vector x 1 is determined by (1) with x 0 = x and u 0 = u. Note that the system is time homogeneous in the sense that if x k = x and u k = u, one evaluates equivalently the expected value in (6) of x k+1 .

The following lemma is important for the characterization of V * .

Lemma 1. The functions V (x, u) and V * (x) given by ( 6) and ( 7) , respectively, are convex.

Proof. See [START_REF] Calmon | Control Variation as a Source of Uncertainty: Single Input Case[END_REF].

Even though V * is defined as a piecewise function, it will be differentiable if V is strictly convex and differentiable. Moreover, based on V , we can characterize the generalized gradient of V * . These facts are stated in the following lemma.

Lemma 2. With V as defined as in (6) and V * as in (7), and u * (x) defined as

u * (x) = arg min u∈R V (x, u). (8) then 1. ∂V * (x) = co{∂ x V (x, u) : u ∈ u * (x)};
2. V * (x) will be differentiable if V is a strictly convex differentiable function.

Proof. See [START_REF] Calmon | Control Variation as a Source of Uncertainty: Single Input Case[END_REF].

With the fact that the value function is convex, we can determine the sign of u * based solely on the value of the state x (see figure 2). Assume that a function (x, u) → f (x, v) is differentiable. Provided that f is also convex in u, one can obtain the sign of the minimum in u by analyzing

∇ u f | u=0 for each x. If ∇ u f | u=0 > 0 (< 0)
, then the function is increasing (decreasing) at the origin and, consequently, the minimum will be in the negative (positive) half-plane. Of course, if ∇ u f | u=0 = 0 the optimal solution is u * = 0. Note that this analysis can not be applied to V in (6) since, even though V is convex, it will not necessarily be differentiable at u = 0 (in fact, this will never be the case). The following Lemma presents a result concerned with this issue and shows that there will be a region in the state space where u * = 0 if the cost function is Lipschitz. Furthermore, for the case where C and F are differentiable, we will show that a region where u * = 0 will always exist.

u * 0 u ∂ u V (x, u) (a) x ∈ R1 u * 0 u ∂ u V (x, u) (b) x ∈ R2 0 u ∂ u V (x, u) (c) x ∈ R3 Figure 2: Behaviour of ∂ u V (x, u) for different x. In (c), u * = 0.
Lemma 3. For the function V described in (6) and u * given by (8) we have

       u * (x) > 0, if x ∈ R 1 (V ), u * (x) < 0, if x ∈ R 2 (V ), u * (x) = 0, if x ∈ R 3 (V ). ( 9
)
where

R 1 (V ) = x : x ∈ R n , lim u i ↓0 ∇V (x, u i ) < 0 , (10)
R 2 (V ) = x : x ∈ R n , lim u i ↑0 ∇V (x, u i ) > 0 , (11)
R 3 (V ) = R 1 ∪ R 2 . ( 12
)
Proof. See [START_REF] Calmon | Control Variation as a Source of Uncertainty: Single Input Case[END_REF].

We can now provide the following theorem.

Theorem 1. Suppose that for each 0 ≤ k < N we have that x → C k (x, u), u → C k (x, u) and x → C N (x) are convex functions and that (2) is limited for all x 0 ∈ R n and µ. For the system in (1) and evaluated by means of the cost function ( 2), the optimal policy can be obtained recursively as follows.

1. Define

J * N (x) = C N (x), x ∈ R n and set k = N -1; 2. Define J k (x, u) for each x ∈ R n , as J k (x, u) = C k (x, u) + E[J * k+1 (A k x + b k u + (σ k + σ k |u|)ε k )],
3. For each x ∈ R n , determine if the optimal action u * k will be positive, negative or zero with

       u * k (x) > 0, if x ∈ R 1 (J k ), u * k (x) < 0, if x ∈ R 2 (J k ), u * k (x) = 0, if x ∈ R 3 (J k ). ( 13
) If u * k ∈ R 1 (J k ) or u * k ∈ R 2 (J k ), determine u * k such that J k (x, u * k ) ≤ J k (x, u) for each u ∈ R. This is equivalent to requiring that 0 ∈ ∂ u C k (x, u * k ) + ∂ u E[J * k+1 (A k x + b k u * k + (σ k + σ k |u * k |)ε k )]
4. Define the function J * k (x) by

J * k (x) = J k (x, u * k (x)) = C(x, u * k (x)) + E[J * k+1 (x k+1 )] with x k+1 = A k x + b k u * k (x) + (σ k + σ k |u * k (x)|)ε k . If k = 0, stop. If else, return to step 2.
The optimal policy u k for each 0 ≤ k < N is thus obtained, in such a way that

J * k (x) = J k (x, u * k (x)) ≤ J k (x, u), holds for each (x, u) and, in particular, J * (x) = J 0 (x, u * 0 (x)) ≤ J 0 (x, u), ∀(x, u).
Proof. See Calmon, Vallée, and do Val ( 2009)

-2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 v * 0 x 0 (1) x 0 (2)
Figure 3: Optimal input u * as a function of the state value for the system described in the example.

A two Dimensional Example

In order to represent graphically the region in the state-space where the optimal policy is to maintain the same control as before, i.e., v * = 0, consider a two dimensional system described by

x k+1 = Ax k + Bv k + (1 + |v k |)ε k , ε k ∼ N (0, I n×n ) where A = 2 1 1 -2 ; b = 1 2 .
Suppose that the system is evaluated through the quadratic cost function:

J(x 0 , π) = 4 k=0 (x T k x k + 2v k P T x + 0.5v 2 k ) + x T 5 x 5
with P = 0 1 T . Figure 3 shows v * 0 as a function of the initial state x 0 . The region where v * 0 = 0 can be clearly seen in the center of the graph. In the other regions, the optimal input will be a linear function of the initial state.

3 The CVIU Approach in a Monetary Model

The Model

Let consider a backward-looking model of the US economy given by:

π k+1 = n 1 j=0 α j π k-j + δy k + ǫ π,k , (14)
y k+1 = n 2 j=0 β j π k-j + n 3 j=0 λ j y k-j + n 4 j=0 γ j i k-j + ǫ y,k , (15)
where π k is the quarterly annualized inflation, y k is the output gap and i k is the quarterly average federal funds rate in percentage points given at an annual rate.

The two shocks, ǫ π,k and ǫ y,k , are responsible for state uncertainty. This model is derived from the model proposed by [START_REF] Rudebusch | Policy Rules for Inflation Targeting[END_REF] (see also [START_REF] Svensson | Monetary Policy with Model Uncertainty: Distribution Forecast Targeting[END_REF]). The first equation is a version of the Phillips curve, which relates inflation to a lagged output gap term and to three lags of inflation. The second equation represents an aggregatedemand relation that relates the output gap to its own lag, to inflation over the previous three quarters and to two lags in the interest rate.

It is interesting to observe that large variations in the interest rate lead to a high state uncertainty. Consequently, this system fits into our CVIU approach with ǫ y,k defined as a function of the interest rate variations. Denoting the variations in the interest rate from instant k to instant k + 1 as v k , we have:

i k = i k-1 + v k
Using a model similar to the one presented in Section 2.3, we obtain:

ǫ k,π = σ π + σ π |v k | ξ k,π (16) ǫ k,y = σ y + σ y |v k | ξ k,y (17)
where ξ k,π ∼ N (0, 1) and ξ k,y ∼ N (0, 1) .

To determine the optimal decision policy, we must first describe the system in a suitable form. First, by rewriting (15), we obtain the new system:

π k+1 = n 1 j=0 α j π k-j + δy k + ǫ π,k (18)
y k+1 = n 2 j=0 β j π k-j + n 3 j=0 λ j y k-j + γ 0 (v k + i k-1 ) + n 4 j=1 γ j i k-j + ǫ y,k (19)
This system can be put in a state-space form:

x k+1 = Ax k + Bv k + E k (v k ), (20)
where

n max = max{n 1 , n 2 } x k ≡ [π k . . . π k-nmax y k . . . y k-n 3 i k-1 . . . i k-n 4 ] T , E k (v k ) ≡ ǫ π,k+1 0 1×nmax ǫ y,k+1 0 1×(n 3 +n 4 ) T .
Defining:

a := α 0 . . . α n 1 T , b := β 0 . . . β n 2 T , c := λ 0 . . . λ n 3 T , d := γ 0 + γ 1 γ 2 . . . γ n 4
T , the transition matrix A will be given by:

A ≡    A 11 A 12 0 nmax+1×n 4 A 21 A 22 A 23 0 n 4 ×nmax+1 0 (n 4 )×(n 3 +1) A 33   
where

A 11 =                  a T [I n 1 ×n 1 0 n 1 ×1 ] , if n max = n 1 a T 0 1×n 2 -n 1 [I n 2 ×n 2 0 n 2 ×1 ] , if n max = n 2 and A 21 =                      b T 0 1×n 1 -n 2 0 n 3 ×n 1 +1   , if n max = n 1 b T 0 n 3 ×n 2 +1 , if n max = n 2
The other terms of A are:

A 12 = δ 0 1×n 3 0 n 1 ×n 3 +1 , A 22 = c T I n 3 ×n 3 0 n 3 ×1 , A 23 = d T 0 n 3 ×n 4 , A 33 =   1 0 1×n 4 -1 I n 4 -1×n 4 -1 0 n 4 -1×1  
The matrix B is given by

B ≡ 0 1×nmax+1 γ 0 0 1×n 3 1 0 1×(n 4 -1)

T

In this example, the policy-maker's goals are to reduce inflation and output gap analyzing a period of N quarters. In the monetary literature (see [START_REF] Dennis | How Important Is Precommitment for Monetary Policy?[END_REF] for example), two different loss functions are used. The first one does not include the interest rate smoothing, focusing mainly on the two policy objectives of the Federal Reserve, while the second one takes the rate smoothing parameter into account. Since it is widely believed that central banks smooth interest rates, that is, they aim at limiting interest rate volatility, following [START_REF] Rudebusch | Policy Rules for Inflation Targeting[END_REF] and [START_REF] Svensson | Monetary Policy with Model Uncertainty: Distribution Forecast Targeting[END_REF], we consider the following criterion2 to minimize at each current period k:

L = π 2 k + y 2 k + 1 2 (i k -i k-1 ) 2 (21)
The cost function can be rewritten as a function of v, leading to:

J(x) = E x ′ N Qx N + N -1 k=0 x ′ k Qx k + 2P x k v k + rv 2 k (22)
Using ( 20) and ( 22) we can apply Theorem 1 directly to obtain the optimal input policy.

Empirical State Space Form

In order to simulate the model, we suppose that the model's parameters are as shown in Table 1, with n 1 = 2, n 2 = 2, n 3 = 0 and n 4 = 1. This estimation is taken from [START_REF] Pardo | Optimal versus realized policy rules in a regime-switching framework[END_REF]. They used quarterly data for the US economy, from the first quarter of 1960 to the fourth quarter of 2008. The interest rate (i k ), is four-quarter average federal funds rate from the Board of Governors. Inflation (π k ) is the GDP chaintype price index in percent at an annual rate, i.e. 400(ln p kln p k-1 ). The output gap (y k ) is built as 100(q k -q * k )/q * k , where q k is the actual real GDP and q * k is the potential GDP. The data used are available from BEA and CBO. All the variables were de-meaned prior to estimation. 1, the system we are considering is:

x k+1 = Ax k + Bv k + (σ + σ|v k |)ξ k , ξ k ∼ N (0, Σ) (23) x k = π k π k-1 π k-2 y k i k-1 T , v k = i k -i k-1 , A =        0
σ k = σ π 0 0 σ y 0 T , σ k = σ π 0 0 σ y 0 T
Finally, the system is evaluated through the cost function ( 22) with

Q =       
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

      
; P = 0 0 0 0 0 0 T ; r = 0.5.

will consider two numerical approaches. In the first one, we will look at the impact of σ starting from a given state value x 0 . We will compare the results with standard Linear Quadratic Regulator (LQR) simulations. Then, in a second approach, we will compare our numerical solutions with some real data.

Remark: for all the simulations we add a constraint on the non negative value of i(k), ∀k. That is, if at some time k the measure of change

v(k) is such that i(k) = v(k) + i(k -1) < 0 then we set v(k) = -i(k -1).
That is, the maximum decrease of the interest rate is fixed to the step between the previous positive interest rate and 0.

LQR versus CVIU

We start the simulation with the historical values for the year 2008 (last quarter), that is x 0 = 0.4869 3.8615 1.1101 -4.280 1.94 T . We will look at the impact of adding uncertainty, that is additive and multiplicative shocks. As stated earlier, when the uncertainty generated by input variation (represented by σ) rises, the region in the state space where v * k = 0 will become larger. Conversely, if σ = 0 the problem becomes the traditional LQR problem and the optimal input variation will be a linear function of the state. In order to concentrate on the impact of rising uncertainty, we assume that the shocks impact identically both economic equations (inflationary and GDP), in that

σ k = σ 0 0 σ 0 T , σ k = σ 0 0 σ 0 T
where σ and σ are some constant scalar.

We simulate realizations of the system for σ = 0.3 and σ = 0, with σ = 0.8. The results are shown in Figure 4. In the case where interest rate variations increase system uncertainty (σ = 0.3) there exists many time instants where v * k = 0, leading to a smoother behaviour of the interest rate (Figures 4a-4b). At the opposite, when σ = 0 the interest rate varies more frequently. If we set σ = 0.1, as shown in Figures 4c-4d, then reaction is more frequent, and the descent towards 0 is faster. Although the CVIU monetary policy involves a small steps behaviour, this policy can be stronger than the LQR one, as shown by Figure 4c, if we are far away from R 3 (J k ).

Comparison with Real Data

In order to study the impact of changing the historical starting date, Figure 5 shows the change in the interest rate based on one specific run (5a) and based on an average evolution on 200 runs (5b). We compare these changes with historical data when the simulations start at two different initial dates: t 0 ∈ {2006.1, 2008.4}3 . As one can check, the descent in the last quarter of 2008 is faster than the one in the first quarter of 2006, which is consistent with the historical data. The difference with the historical rate of change can be explained either by a different level of uncertainty than the one used in the simulations (ceteris paribus an increase of σ will generate conservatism), or by a different gap on objective and current state value (ceteris paribus an increase of this gap will increase the aggressiveness of the monetary policy).

Moreover, in a time horizon from 1962 to 2008, we compare at each period of time k what would have been the next period CVIU monetary policy and the LQR one. We compare these values with the historical ones. We made two CVIU's run defined by σ = 0.8 and σ ∈ {0.3, 0.8}. The same vector of gaussian noises ǫ were used for each simulation. The results are shown in Figure 6. Observe first that, as expected, the LQR policy does worst than the CVIU one for all σ. Second, the CVIU interest rates are closer to the real ones when uncertainty is high. This result means that the Federal Reserve Bank monetary policy was most likely conducted in an highly uncertain world, mainly in the 70's, up to the middle of the 80's, where the inflation and interest rates were high and increasing. Third, although the CVIU policy is close to the real ones in the low inflation regime (mainly from the middle of the 80's to the 00's), this is not true in the 70's. In order to improve these results one must add non constant and asymmetric standard deviations σ, and/or modify the cost weights (Q).

3 -2 -1 0 1 2 -5 -4 -3 -2 -1 0 1 2 i ii v * k k (d)

A Sensitivity Analysis

From our previous results, one could ask whether or not the FED was running its monetary policy under high or low level of uncertainty. Although our theoretical model is not directly formulated as an econometric model, we try to answer this question based on the following steps. First, we ran a Monte Carlo analysis (10000 runs), where at each run we randomly draw σ π , σ y , σ π , σ y ∼ N [0, 1]. Then we compare4 the results of our theoretical model with real data taken from 1962 (first quarter) to 2008 (fourth quarter). That is, we test the values of the uncertainty parameters, σ i and σ i , that allow the CVIU solution to be the closest of the real ones. We add the following constraint to our previous simulations: ǫ k = ǫ. We assume that only one similar shock exists that affects both equations in a similar way. This last assumption does not affect the general results.

The results of the Monte Carlo simulations are plotted using boxplots (see Figures on Appendix A) were we underline the mean trend over different quintile of the uncertainty parameters. Furthermore, we carried out some regressions in order to analyze more precisely the impacts of the different sources of uncertainty (σ i , σ i ) on the discrepancy between real data and our theoretical estimations (see Appendix B). From Tables 2-4 and Figures 8-10 the following remarks can be made.

• Additive uncertainty appears to be less important than multiplicative uncertainty.

• We observe that smaller values in the interest rate errors are linked to highest values of both additive and multiplicative uncertainty parameters. This confirms the result of Figure 6b. This may primarily be interpreted as the fact that the interest rates policy is conducted in a world of uncertainty.

• One should also notice that the reduction in the variance of the errors when the value of the multiplicative uncertainty parameter increases is mainly due to the property of our theoretical model that says to do less (and sometimes nothing).

• The inflation rate has a better fit when uncertainty is high on GDP rather on inflation. Increasing the multiplicative inflation uncertainty increases the errors. A possible interpretation is that the activism of the FED was stronger than it should be according to the CVIU solutions. This is a possible interpretation we have already noted in Section 2.2.

• Similarly, the errors on GDP gap are smaller when uncertainties lay on inflation rather on GDP.

As shown by Figures 11-14 in Appendix C, these results are not modified by a change in the loss function parameters Q and r.

Sensitivity Analysis à la Diebold-Mariano

Our previous results are based on mean square errors minimization. We analyze how these results are modified when using a more sophisticated method. That is, we rank in a vector the different simulations using an ordinal procedure. Then, we analyze a possible correlation between the value of a given parameter and the order of the simulation in this rank vector. We use the test of predictive accuracy proposed by [START_REF] Diebold | Comparing predictive accuracy[END_REF] in order to construct the rank vector.

Assume that two different values for σ's vector involves two different models. The interest rate forecast errors from the two models are

ξ 1,t+1|t = i t+1 -i 1,t+1|t (24)
ξ 2,t+1|t = i t+1 -i 2,t+1|t (25)
These one-step forecast are computed for t = t 0 , .., T , where t 0 = 1961.1 and T = 2008.4, for a total of T 0 = 188 forecasts. The accuracy of each forecast is measured by a particular loss function. We restrict ourself to the absolute error loss:

L(ξ i,t+h|t ) = |ξ i,t+h|t | (26)
As in [START_REF] Diebold | Comparing predictive accuracy[END_REF], one may test the null hypothesis of equality performance

H 0 : E[L(ξ 1,t+h|t )] = E[L(ξ 2,t+h|t )] ( 27
)
against the alternative

H 0 : E[L(ξ 1,t+h|t )] = E[L(ξ 2,t+h|t )] (28)
Using

d t = L(ξ 1,t+h|t ) -L(ξ 2,t+h|t ) (29)
one can rewrite the null hypothesis as

H 0 : E[d(t)] = 0 (30)
The Diebold-Mariano test statistic is

S = d V (d) T (31)
where

d = 1 T T t=1 d t (32) V (d) = γ 0 + 2 T j=1 γ j ( 33
)
with γ j = cov(d t , d t-j ).

Rather than testing the null hypothesis using the result of S values in a normal law, we compare the S values between two models. When a model fits better than another one, one should rank this model in a better place. So, we construct a rank vector based on the results of the Diebold-Mariano test. The Pseudo-Code of the algorithm we used is as follows, where a total number of runs N was done:

• BEGIN

• Initializations:

-1) Creation of a rank vector (0 1×N ).

-2) The loop counter is set to k = 1.

-3) A vector of run's number is created V ecN umb = [1, 2, .., N ].

• Loop:

-1) Initialization: the best run is set to i = 1.

-2) While i ≤ N -k do * a) Test the run V ecN umb(i) against other runs V ecN umb(j), with j > i and j ≤ N -k + 1. * b) If T est(V ecN umb(i)) < T est(V ecN umb(j)), the best run is set to j. * c) Go back to a) with i ← j.

-3) Set the loop counter to k ← k + 1.

-4) Set the score N -k + 1 at the position V ecN umb(i) in the rank vector.

-Reduce the order of V ecN umb: V ecN umb ← V ecN umb -V ecnumb(i).

-6) Go Back to 1).

• END First notice that the higher the number in the rank vector, the higher the accuracy of the model. Figure 7 shows the link between the values in the rank vector and the different values of σ's vector. This underlines the accuracy of the model with a high level of uncertainty since highest uncertainties involve highest rank order. The main positive impact of increasing uncertainty is due to multiple uncertainty on GDP. That is, the FED policy is compatible with our CVIU approach if, and only if, the uncertainty facing the FED is higher for GDP than for inflation. A natural counterintuitive corollary of this result is that the FED thinks it is more able to manage inflation than GDP.

Conclusion

In this paper, we developed a theoretical framework and the optimal control strategy for CVIU systems associated with a convex cost functional. The convexity of the cost-to-go functions was asserted, making it simpler to obtain the optimal policy using dynamic programming. Since the state equation is not differentiable, an algorithm for determining the optimal policy was described using generalized gradients. The optimal strategy yields that the state-space will be divided into three disjoint regions, representing the regions where the optimal control policy is to increase, decrease, or maintain the previous input. Furthermore, for the case where the Cost Functional is differentiable, it was asserted that a region where no variation is optimal will always exist. This characteristic of the optimal policy is intuitively sound, since in many real-world problems, in face of the uncertainty generated by changing the control policy, the best strategy is to maintain the same policy as before. The presented CVIU model and analysis can be directly applied to numerous practical scenarios, ranging from monetary policy problems to medicine and biology and, in general, to problems for which a complete dynamic model is too complex to be feasible.

From our numerical analysis, we obtain two main results. First, and on the last two decades, monetary policy was mainly conducted on the basis of an uncertain (multiple) world. Second, the activism of the FED was stronger that it should be following our CVIU approach unless we admit that uncertainty mainly occurs in GDP dynamics rather than in inflation one. Our result is similar to the one of [START_REF] Tillmann | Parameter Uncertainty and Non-Linear Monetary Policy Rules[END_REF] who shows that the optimal monetary policy under parameter uncertainty can motivate a nonlinear interest rate rule which involves that the policy response to inflation becomes stronger, the higher the inflation rate and the larger the output gap.

Finally, a comparison between our theoretical CVIU approach and the real data lead to two general remarks. First, we known from regime switching modelization (see [START_REF] Pardo | Optimal versus realized policy rules in a regime-switching framework[END_REF]) that there have been at least two historical regimes of inflation: a high level one (mainly in the 70's) and a low one (mainly the 90's and 00's). One should question how our results are modified if evaluated in such a switching regime framework. Second, and obviously, a link exists between our CVIU approach and signalling monetary game theory [START_REF] Canzoneri | Monetary policy games and the role of private information[END_REF], [START_REF] Walsh | Announcements, Inflation Targeting and Central Bank Incentives[END_REF]). A characteristic of these "political business cycle models" is that the central bank knows the signaling effects of its monetary policy on private agents' expectations. By losing credibility, monetary policy can increase GDP and inflation variability, and thus increase the level of uncertainty. Proposing a link between regime switching modelization (e.g. jump Markov model) and monetary signalling game is the focus of our current research. [0.0 [0.2 [0.4 [0.6 [0.8 [0.0 [0.2 [0.4 [0.6 [0.8 [0.0 [0.2 [0.4 [0.6 [ [0.0 [0.2 [0.4 [0.6 [0.8 [0.0 [0.2 [0.4 [0.6 [0.8 [0.0 [0.2 [0.4 [0.6 [0. .2 [0.4 [0.6 [0.8 [0.0 [0.2 [0.4 [0.6 [0.8 [0.0 [0.2 [0.4 [0.6 [ Given : Qy y [0.0 [0.2 [0.4 [0.6 [0.8 [0.0 [0.2 [0.4 [0.6 [0.8 [0.0 [0.2 [0.4 [0.6 [

A Figures of the Sensitivity

Figure 1 :

 1 Figure 1: Evolution of the FED and ECB funds rate

  σ and σ being diagonal matrices where:

Figure 4 :

 4 Figure 4: Comparison between realizations of the system considering the following scenarios: (i) σ = 0.8 and σ = 0.3 (a,b) or σ = 0.1 (c,d); (ii) σ = 0.8 and σ = 0 (a,b,c,d), which is the standard LQR case.

Figure 5 :

 5 Figure 5: Comparison between realizations of the system for n = 50 considering t 0 ∈ {2006.1, 2008.4}: one isolated run (left), average run (right).

Figure 6 :

 6 Figure 6: Comparison between theoretical interest rate values and historical ones considering the following scenarios: σ = 0.8 and (a) σ = 0.3; (b) σ = 0.8 and, (c) σ = 0. Grey bars highlight lower historical inflation regime.

Figure 7 :

 7 Figure 7: Relationship between Diebold-Mariano test value and the σ's vector values.

Figure 8 :

 8 Figure 8: Impact of uncertainty on the sum of the total square difference to the real inflation rate.

Figure 9 :

 9 Figure 9: Impact of uncertainty on the sum of the total square difference to the real interest rate.

Figure 11 :

 11 Figure 11: Sensitivity Analysis: Impact of r and σ π on Σ i .

Figure 12 :

 12 Figure 12: Sensitivity Analysis: Impact of r and σ y on Σ i .

Figure 13 :

 13 Figure 13: Sensitivity Analysis: Impact of Q y,y and σ π on Σ i .

Table 1 :

 1 Estimates of the US quarterly modelUsing the parameters of Table

	Parameters Estimation Parameters Estimation
	α 0	0.554	β 0	0.015
	α 1	0.145	β 1	-0.033
	α 2	0.256	β 2	-0.02
	δ	0.115	γ 0	0.133
	λ	0.902	γ 1	-0.197

Table 4 :

 4 B.3 GDP Gap Errors:Σ y = N t=1 (y t|t-1y t ) 2 .Impact of the values of the uncertainty parameters on Σ y C Impact of Q and rWe ran 20000 simulations with Q π,π = 1. At each run we randomly drewσ π , σ y , σ π , σ y ∼ [0, 1], Q π,π = 1, Q y,y ∈ [0.1,10], and r ∈ [0, 1.5].

	Σ y	Estimate Std. Error t-value	P r(> |t|)	Adjusted R-squared
	Constant case					0.967
	(Intercept)	256.174	0.588	436	< 2e -16 * * *
	σ π	-61.808	0.567	-109	< 2e -16 * * *
	σ y	154.532	0.562	275	< 2e -16 * * *
	σ π	-111.043	0.570	-195	< 2e -16 * * *
	σ y	233.825	0.571	410	< 2e -16 * * *

  Figure 14: Sensitivity Analysis: Impact of Q y,y and σ y on Σ i .

				Given : Qy y
	0	2	4	6	8	10
						0.8
	Intetrest rate errors					0 10000 20000 30000
			Multiplicative GDP Uncertainty σy

It is well known now that using an optimal robust control approach will generally increase the aggressiveness.

We will show in section

3.5 that relaxing the interest rate smoothing goal by setting r = 0 will not change our general results.

We used an empirical state space estimated up to the first quarter of 2006 for the run started at 2006.1: α0 = 0.608, α1 = 0.108, α2 = 0.236 , β0 = 0.038 , β1 = -0.075, β2 = 0.000, δ = 0.106, γ0 = 0.121,γ1 = -0.195 and λ = 0.896.

The comparison is made based on the measure of the total square of error of the one-period ahead forecast: N t=1 (ξt) 2 , with ξt = x t|t-1 -xt

This work was supported by the State of São Paulo Research Foundation (FAPESP)

A.3 Output Gap

B Some Regression Statistics

All the regressions have been carried out using the "lm" procedure of the R software.

Signif


Clément Prunier
email: clement.prunier@u-bourgogne.fr

Jérôme Rousseau
email: jerome.rousseau@u-bourgogne.fr

Pauline Butaud
email: pauline.butaud@univ-fcomte.fr

Thomas Jeannin

Vincent Placet
email: vincent.placet@univ-fcomte.fr

Etude de la résistance au cisaillement et du vieillissement de contreplaqués collés par résine thermoplastique et renforcés par des fibres naturelles Shear strength and aging study of plywood bonded with thermoplastics and reinforced with natural fibers

Keywords: Contreplaqué, thermoplastiques, lin, adhésion, vieillissement Plywood, thermoplastic, flax, adhesion, aging

Dans l'industrie du bois, le remplacement des colles traditionnelles utilisées pour les contreplaquées par des adhésifs thermoplastiques recyclés ou biosourcés est une alternative intéressante. Ce travail étudie l'hybridation de contreplaqués utilisant des résines thermoplastiques polypropylène recyclé greffé anhydride maléique (rPPAM) ou acide polylactique (PLA) comme adhésifs et des renforts en fibres de lin. Les performances de ces contreplaqués sont caractérisées à travers leur résistance au cisaillement interlaminaire. Les résultats obtenus montrent que ces adhésifs présentent des caractéristiques satisfaisantes en comparaison à celles des colles traditionnelles. L'effet de l'ajout de fibres de lin à l'interface entre placages varie en fonction du type d'adhésif, de l'essence de bois et de l'orientation des fibres, et n'est pas toujours positive quant à la résistance au cisaillement. Enfin, le vieillissement des assemblages selon deux méthodes différentes est étudié. Les contreplaqués utilisant du PP se distinguent par une bonne résistance au vieillissement, contrairement au PLA. Dans tous les cas de figures, le vieillissement entraîne une baisse des performances lorsque des fibres sont présentes à l'interface.

Introduction

La maîtrise de l'empreinte environnementale des matériaux utilisés dans l'industrie est devenue un enjeu extrêmement important. Le bois représente une solution intéressante dans ce contexte. Lorsqu'il est utilisé sous forme de contreplaqués, il nécessite l'emploi d'adhésifs tels que les colles à base d'urée formaldéhyde (UF), mélamine urée formaldéhyde (MUF) et phénol urée formaldéhyde (PUF). Il s'agit de composés pétrosourcés qui posent un problème d'impact environnemental tant lors de leur production que lors de la gestion de leur fin de vie, en plus de certains risques sanitaires [1]. Leur remplacement par des adhésifs à plus faible impact environnemental constitue donc un sujet d'étude de premier ordre. Parmi les candidats à ce remplacement, on compte les résines biosourcées basées sur les tannins et la lignine [START_REF] Mansouri | Lignin-based polycondensation resins for wood adhesives[END_REF], mais aussi différentes résines thermoplastiques comme le polyéthylène basse densité (PEBD), le polyamide (PA), le polyester (PE) [START_REF] Bekhta | Properties of Thermoplastic-Bonded Plywood: Effects of the Wood Species and Types of the Thermoplastic Films[END_REF], le polyéthylène haute densité (PEHD), le polyuréthane (PU) [START_REF] Gaugler | A new methodology for rapidly assessing interfacial bonding within fibre-reinforced thermoplastic composites[END_REF], le polypropylène recyclé (rPP) [START_REF] Kajaks | Recycled Thermoplastic Polymer Hot Melts Utilization for Birch Wood Veneer Bonding[END_REF] parfois greffé d'anhydride maléique (AM) pour améliorer l'adhésion [START_REF] Song | Utilization of Polypropylene Film as an Adhesive to Prepare Formaldehydefree, Weather-resistant Plywood-like Composites: Process Optimization, Performance Evaluation, and Interface Modification[END_REF], ainsi que l'acide polylactique (PLA) [START_REF] Luedtke | Understanding the development of interfacial bonding within PLA/wood-based thermoplastic sandwich composites[END_REF][START_REF] Grigsby | Bonding Wood Veneer with Biobased Poly(Lactic Acid) Thermoplastic Polyesters: Potential Applications for Consolidated Wood Veneer and Overlay Products[END_REF]. L'étude présentée ici se concentre sur l'utilisation de ces deux dernières résines, soient le rPPAM et le PLA.

L'évaluation de la résistance à l'humidité et à l'immersion est cruciale lors de l'introduction de nouveaux adhésifs pour l'assemblage de contreplaqués, car elle permet d'avoir une idée rapide de leur réaction au vieillissement et de les qualifier pour différents domaines d'application. Les assemblages réalisés avec du PLA montrent de bons résultats pour des essais d'immersion en eau froide, mais ne sont cependant pas utilisables dans des milieux humides à partir de 60 °C du fait de la dégradation par hydrolyse des groupements esters de la résine [START_REF] Banjo | Moisture-induced changes in the mechanical behavior of 3D printed polymers[END_REF]. En ce qui concerne le PP, l'étude de Kajaks et al. [START_REF] Kajaks | Recycled Thermoplastic Polymer Hot Melts Utilization for Birch Wood Veneer Bonding[END_REF] montre une dégradation des contreplaqués assemblés avec cette résine lors de l'immersion en eau froide, avec une baisse de plus de 50% des propriétés de résistance au cisaillement interlaminaire après 24 h d'immersion. Cependant, dans une autre étude, le caractère hydrophobe du PP rend les assemblages résistant à des vieillissements accélérés par immersion en eau chaude et bouillante et leur permet d'atteindre certains standards d'adhésifs conventionnels [START_REF] Song | Utilization of Polypropylene Film as an Adhesive to Prepare Formaldehydefree, Weather-resistant Plywood-like Composites: Process Optimization, Performance Evaluation, and Interface Modification[END_REF].

Par ailleurs, plusieurs études ont montré un réel intérêt à renforcer des produits bois comme des contreplaqués par des matériaux composites fibrés, dans le but d'améliorer leurs propriétés mécaniques et de diminuer leur variabilité [START_REF] Kajaks | Recycled Thermoplastic Polymer Hot Melts Utilization for Birch Wood Veneer Bonding[END_REF][START_REF] Valdes | Reinforcement of maritime pine cross-laminated timber panels by means of natural flax fibers[END_REF]. Il est important de maitriser l'effet de cet ajout de renfort sur le comportement adhésif des interfaces, car cela va influer sur le transfert de charge entre les différents éléments. Jorda et al. [START_REF] Kajaks | Recycled Thermoplastic Polymer Hot Melts Utilization for Birch Wood Veneer Bonding[END_REF] ont étudié l'effet du renforcement par ajout de fibres de lin à l'interface de contreplaqués collés par différents adhésifs, et ont observé que l'effet de cet ajout sur la résistance au cisaillement interlaminaire dépendait de l'adhésif utilisé : une amélioration a été observée lors d'utilisation d'epoxy ou de colle urée-formaldéhyde (UF), alors que la résistance diminuait pour des colles mélamines-urée-formaldéhyde (MUF) ou polyuréthane (PUR).

Fabrication des contreplaqués

Les placages sont assemblés suivant la séquence d'empilement [0/90/0]s (s pour symétrique), 0° est le sens du fil du bois. Les contreplaqués sont fabriqués par thermocompression en utilisant une presse Fontijne® LabPro 600 à 180 °C, une température permettant une bonne fluidité du rPPAM et du PLA, tout en restant en dessous des températures de dégradation des fibres de lin [START_REF] Van De Velde | Thermal and Mechanical Properties of Flax Fibres as Potential Composite Reinforcement[END_REF] et du bois [START_REF] Avat | Contribution à l'étude des traitements thermiques du bois jusqu'à 300°C: Transformations chimiques et caractérisations physico-chimiques[END_REF]. Le processus complet de fabrication comprend plusieurs étapes.

-(a) Géométrie et composition des éprouvettes des essais de résistance en cisaillement ; (b) : Visualisation par corrélation d'images du champ de déformation en cisaillement dans les éprouvettes testées en cisaillement interlaminaire. (c) Montage d'essai de cisaillement interlaminaire

Lors de l'essai, la force et le déplacement de la traverse sont enregistrés. La force à rupture est ensuite divisée par la surface de la zone de recouvrement pour obtenir la valeur de contrainte en cisaillement interlaminaire, notée ILSS (pour InterLaminar Shear Strength). Une étude statistique utilisant le test de Student est réalisée afin de déterminer si les variations entre les résultats sont statistiquement significatives. La valeur significative de p (valeur de probabilité du test de Student) est fixée à 0,05.

Résultats

Résistance au cisaillement des contreplaqués

ε LT L'effet de l'ajout de fibres naturelles de lin à l'interface entre deux placages sur la résistance au cisaillement interlaminaire est discuté dans cette section. La Fig. 2 présente une comparaison des résistances au cisaillement interlaminaire pour des contreplaqués avec et sans renforts en fibres de lin.

Les éprouvettes de hêtre collées au rPPAM et au PLA ont des valeurs de résistance au cisaillement interlaminaire de respectivement 7 MPa et 7,7 MPa, ce qui est en deçà des valeurs pour des colles traditionnelles qui se trouvent aux alentours des 9 MPa [START_REF] Gavrilović-Grmuša | Influence of the degree of condensation of urea-formaldehyde adhesives on the tangential penetration into beech and fir and on the shear strength of the adhesive joints[END_REF]. Les éprouvettes de douglas collées au rPPAM et au PLA ont des valeurs de résistance au cisaillement interlaminaire de 4,2 MPa et 2,5 MPa, ce qui est similaire pour le rPPAM et inférieur pour le PLA aux résultats obtenus avec des colles traditionnelles, qui se trouvent autour des 4,5 MPa.

Dans le cas des éprouvettes hêtre collées au rPPAM, l'ajout de fibres naturelles à 0° vient améliorer les performances de l'interface, la valeur de résistance au cisaillement interlaminaire passe de 6,96 MPa à 8,83 MPa. Cette amélioration est confirmée par l'étude statistique, avec p < 0,007. L'ajout de fibres permet d'augmenter l'épaisseur du joint et de localiser la rupture dans l'interface, alors qu'elle apparaît dans le substrat pour les éprouvettes sans fibres. Lorsque les fibres sont disposées à 90° dans l'interface, la variation de valeur de résistance au cisaillement de l'interface n'est pas statistiquement significative (p > 0,3), mais la rupture reste localisée dans le joint.

Les collages réalisés avec du PLA ont une rupture toujours localisée dans le joint de colle, même en l'absence de fibres. L'ajout de fibres à 0° a un effet négatif statistiquement significatif (p < 0,0005) sur la valeur de résistance au cisaillement interlaminaire, avec une diminution de 7,66 MPa à 6,76 MPa. Cette diminution est également observée lorsque l'orientation des fibres est à 90° (p < 0,00004), la résistance au cisaillement interlaminaire passant de 7,66 MPa à 6,57 MPa.

Les éprouvettes de douglas ont une rupture qui est toujours localisée dans le substrat. Les éprouvettes de douglas collées au rPPAM ont bénéficié d'une amélioration statistiquement significative (p < 0,04) de leur résistance au cisaillement interlaminaire lors de l'ajout de fibres à 0°. On passe alors de 4,17 MPa à 5,69 MPa. Aucune variation statistiquement significative n'est observée lorsque les fibres sont ajoutées à 90°. Pour les éprouvettes collées au PLA, une amélioration statistiquement significative (p < 0,007) de la valeur de résistance au cisaillement interlaminaire est observée lorsque les fibres sont ajoutées à 0°, avec des valeurs passant de 2,52 MPa à 3,77 MPa. Aucune variation statistiquement significative n'est observée lorsque les fibres sont ajoutées à 90° (p > 0,06).

Effet du vieillissement accéléré

Les résultats après vieillissement accéléré (à 90% HR et 40 °C pendant 40 jours, noté Ag1) sont présentés en Fig. 3.

Les contreplaqués collés au rPPAM sans fibres montrent une augmentation statistiquement significative (p < 0,0009) de la résistance en cisaillement interlaminaire suite au vieillissement, par rapport aux éprouvettes non vieillies, en passant de 6,96 MPa à 9,43 MPa (+35%). Lorsque les éprouvettes contiennent des fibres, une diminution significative de la résistance (p < 0,0004) est observée, avec une baisse de 8,83 MPa à 6,63 MPa (-25%). Comme dans le cas des éprouvettes non vieillies, l'ajout de fibres dans l'interface permet de localiser la rupture dans le joint, alors qu'elle a lieu dans le substrat lorsque les fibres sont absentes.

Les éprouvettes collées au PLA montrent une baisse significative (p < 0,00004) de leur résistance au cisaillement interlaminaire après vieillissement avec une diminution de 7,66 MPa à 6,48 MPa (-15%). Une baisse significative (p < 0,0004) est également observée pour les éprouvettes avec des fibres, avec une diminution de 6,76 MPa à 4,82 MPa (-38%). Les éprouvettes avec ou sans fibres présentent une rupture localisée dans l'adhésif.

Concernant le douglas, tous les essais se caractérisent par une rupture localisée dans le substrat. Les éprouvettes collées au rPPAM ne montrent pas de variation statistiquement significative de résistance au cisaillement interlaminaire lors du vieillissement, qu'il y ait présence ou non de fibres à l'interface, (respectivement p > 0,9 et p > 0,3).

Les éprouvettes collées au PLA ne montrent pas de variation statistiquement significative (p > 0,05) lors du vieillissement. Lorsque les éprouvettes contiennent des fibres, une diminution statistiquement significative (p < 0,004) est observée, avec une diminution de 3,77 MPa à 2,82 MPa (-25%). Les résultats de résistance au cisaillement interlaminaire des éprouvettes qui ont subi un vieillissement accéléré par immersion en eau bouillante (Ag2) sont présentés en Fig. Les éprouvettes douglas -rPPAM vieillies ne montrent pas de différence significative (p > 0,7) par rapport aux éprouvettes vieillies. Lorsque des fibres sont présentes à l'interface, une diminution statistiquement significative (p < 0,002) est observée, avec une diminution de 5,69 MPa à 2,94 MPa (-48%). Les éprouvettes avec ou sans fibres présentent une rupture localisée dans l'adhésif. La suite de l'étude sera consacrée à l'interprétation de ces résultats. Il est nécessaire d'analyser les phénomènes physico chimiques et mécaniques à l'échelle microscopique intervenant dans les différents constituants et leurs interfaces pour pouvoir proposer des mécanismes de dégradation et de rupture expliquant les différences de comportement observées.

Conclusion

Ce travail s'est intéressé à la résistance au cisaillement interlaminaire de contreplaqués de hêtre ou de douglas, collés avec des thermoplastiques rPPAM ou PLA, et renforcés ou non par des fibres naturelles. L'ajout de fibres de lin à 0° à l'interface a permis d'améliorer les valeurs de résistance au cisaillement interlaminaire pour les éprouvettes hêtre et douglas collées au rPPAM. Les éprouvettes collées au PLA ont montré une amélioration des performances de l'interface uniquement pour le douglas.

La deuxième partie de l'étude, étudiant le vieillissement de ces matériaux hybrides selon deux types de vieillissement, montre une excellente résistance à l'humidité du rPPAM, mais une dégradation dans le cas du PLA. L'ajout de fibres s'est montré défavorable à la résistance au cisaillement interlaminaire des contreplaqués. L'étude montre que le remplacement des adhésifs pétrosourcés des contreplaqués par des résines thermoplastiques recyclées ou biosourcées est une solution cohérente du point de vue des performances d'adhésion entre les placages. Le renforcement par fibres naturelles en condition d'humidité faible et stable possède un aspect bénéfique, mais semble être à éviter pour des utilisations dans des conditions à fort taux d'humidité.

Ce travail a été soutenu par le programme « Investissements d'Avenir », projet ISITE-BFC (contrat ANR-15-IDEX-0003) dans le cadre du projet WooFHi.

Bibliographie

[1] « Formaldehyde ». https://www.cancer.org/cancer/cancer-causes/formaldehyde.html (consulté le 18 février 2022).

2. 4

 4 Fig. 1 -(a) Géométrie et composition des éprouvettes des essais de résistance en cisaillement ; (b) : Visualisation par corrélation d'images du champ de déformation en cisaillement dans les éprouvettes testées en cisaillement interlaminaire. (c) Montage d'essai de cisaillement interlaminaire

Fig. 2 -

 2 Fig. 2 -Valeurs de résistance au cisaillement interlaminaire pour des contreplaqués avec/sans renforts, écarts types et analyse statistique de Student (a→h)

Fig. 3 -

 3 Fig. 3 -Valeurs de résistance au cisaillement après Ag1 pour des éprouvettes de hêtre (a) et de douglas

Fig. 4 -

 4 Fig. 4 -Valeurs de résistance au cisaillement après Ag2 pour des éprouvettes de hêtre (a) et de douglas (b)

  Les contreplaqués utilisés dans cette étude sont constitués de deux essences de bois : le hêtre et le douglas. Le bois se présente sous forme de placages, de fines couches de bois obtenues par un procédé de déroulage, réalisé au LaBoMaP de Cluny. Les placages ont une épaisseur de 2 mm pour le hêtre et 3 mm pour le douglas, pour une masse volumique de 710 kg/m 3 pour le hêtre et 500 kg/m 3 pour le douglas. Les placages sont stockés en intérieur à température ambiante, et sont découpés en plaques de 240 x 240 mm 2 au moment de la fabrication.Deux adhésifs thermoplastiques sont étudiés. Le premier est un polypropylène recyclé (rPP) provenant de déchets agricoles, fourni par la société Adivalor ®. Il est mélangé à 5 % en masse de polypropylène greffé d'anhydride maléique (PPAM) Polybond® 3200, qui contient 1% d'anhydride maléique (AM). Cet ajout a pour effet de favoriser l'adhésion du polypropylène, de nature hydrophobe, avec le bois et les fibres naturelles en permettant une estérification entre l'AM et les groupements alcools du bois et des fibres. Le matériau a été transformé en film de 70 μm d'épaisseur par extrusion calandrage. Son "Melting Flow Index" (MFI) est de 6 g/10 min (230 °C -2,16 kg). Le deuxième adhésif retenu est une résine PLA fournie sous forme de pellets par la société NaturePlast sous la référence « PLE 005 ». Il a également été transformé en film de 100 μm d'épaisseur. Il possède un MFI de 10 g/10min (190°C, 2,16 kg).Les renforts sont composés de fibres unidirectionnelles de lin (FlaxTape®), se présentant sous forme de nappes d'une densité surfacique de 110 g/m 2 , et fournies par la société Lineo®. Elles sont stockées à 23 °C et 50% d'humidité relative pendant au moins 48 h avant fabrication.

		les
	combinant à des adhésifs constitués de résines recyclées ou biosourcées. L'objectif poursuivi est
	d'améliorer et d'homogénéiser les propriétés structurelles globales du matériau bois, mais aussi dans
	un premier temps d'obtenir localement une meilleure résistance interlaminaire des interfaces entre
	placages. On s'intéresse de plus aux effets du vieillissement sur ces renforcements, en les soumettant
	d'une part à un conditionnement de 40 jours dans des conditions sévères de température et d'humidité,
	et d'autre part à un vieillissement accéléré par immersion de courte durée dans l'eau bouillante. Ces
	méthodes de vieillissement correspondent aux normes relatives aux contreplaqués permettant de
	catégoriser leurs différentes classes d'application [11]. Une première partie présentera les matériaux
	utilisés, puis le processus de fabrication des contreplaqués et de préparation des éprouvettes, et la
	méthode d'essai utilisée. Dans un deuxième temps, les résultats obtenus seront présentés afin
	d'évaluer l'effet de la résine et de l'apport ou non de fibres, et l'effet du vieillissement selon les
	différentes configurations. Enfin, la dernière section conclut cet article et présente les perspectives
	associées à ces travaux.
	2.	Matériel et méthodes
		2.1 Matériaux

Le projet dans lequel s'inscrit cette étude s'intéresse aux effets de ce renforcement par fibres, en

3 Réalisation des éprouvettes Les

  plaques sont découpées d'après la norme EN302 en 7 éprouvettes de 220 x 25 mm². Elles sont ensuite entaillées sur chaque face de façon à obtenir des rainures de 3 mm de largeur espacées de 25 mm. Une représentation schématique de l'éprouvette est donnée en Fig 1a.La surface comprise entre les deux rainures est appelée surface de recouvrement, et mesure donc 25 x 25 mm². La sollicitation de traction imposée aux éprouvettes entraîne une contrainte de cisaillement dans cette surface de recouvrement, ce qui permet de qualifier sa résistance et ainsi d'évaluer les performances d'adhésion des différentes configurations d'interface.7 éprouvettes de chaque groupe de constituants présentés en Tab.1 ont subi un vieillissement accéléré, réalisé selon deux méthodes différentes, plus ou moins sévères. La première méthode, notée « Ag1 » (pour Aging 1), s'inspire des travaux de Scida et al.[START_REF] Scida | Influence of hygrothermal ageing on the damage mechanisms of flax-fibre reinforced epoxy composite[END_REF], qui étudient le vieillissement hygrothermique de composites lin/epoxy. Dans cette méthode, les éprouvettes sont vieillies dans une enceinte climatique Q-SUN XE-3® avec un taux d'humidité de 90%HR et une température de 40 °C pendant 40 jours. Ce vieillissement est appliqué aux éprouvettes de hêtre et de douglas, utilisant des adhésif rPPAM ou PLA, avec ou sans présence de renfort en lin. La deuxième méthode, notée « Ag2 » est basée sur la norme EN 314 concernant les essais d'adhésion pour contreplaqués. Elle consiste en une immersion de 6 h de l'échantillon dans l'eau bouillante, avant un séchage en enceinte conditionnée. Son caractère très exigeant permet de tester les limites d'adhésifs connus pour leur caractère hydrophobe, comme le PP. Cette méthode a été utilisée pour tester des éprouvettes de hêtre et de douglas collés avec du rPPAM ou du PLA, et en présence ou non de renforts de lin.

	Essence	Vieillissement	Adhésif	Renfort de lin
	Douglas Hêtre	40 °C 90% HR 40j (Ag1) Immersion eau bouillante -6h (Ag 2)	rPPAM PLA	Avec/Sans
		Tab. 1. Récapitulatif des paramètres de l'étude de vieillissement

La presse se ferme et la température des plateaux monte à 180 °C à une vitesse de 15 °C/min. La pression est ensuite appliquée pendant une durée correspondante au temps nécessaire pour atteindre 180 °C au coeur du contreplaqué, cette température étant ensuite maintenue pendant une durée variant de 5 à 15 minutes. Les plateaux sont ensuite refroidis à une vitesse de 20 °C/min. Après retour à la température ambiante, la presse est ouverte et le contreplaqué est stocké à 50 % HR et 23 °C pendant au moins 72 h avant essai.

Les contreplaqués hybridés sont obtenus par ajout d'un pli de lin unidirectionnel FlaxTape ® entre les deux plis centraux, à 0° ou 90° par rapport au sens du fil du bois. Le processus de fabrication reste strictement identique.

2.

Remerciements