Jean-Sébastien Pentecôte

Marilyne Huchet-Bourdon

Shock asymmetries and distance to the Euro Area

Keywords: JEL Classification: E3, E42, C02, C32 shock asymmetries, distance, Euro area enlargement

the Euro Area has enlarged for the third time to Cyprus and Malta. As Slovakia is now planned to join in 2009, these waves of new entries revive the debate around greater asymmetries which may threaten the stability of the whole monetary union. This paper extends Bayoumi and Eichengreen's (1992) centre-periphery approach. We show how a suitable decomposition of the correlations between supply and demand disturbances enables to get two new indices to give a more intuitive assessment of the distance to the Euro area and the origin of shock asymmetries. Using monthly data over 1995-2008 on 21 countries, asymmetries are measured by correlations among the structural shocks from a VAR process. We then translate these correlations estimates into two synthetic indices. One can be interpreted as the relative distance of the candidate country to a fully symmetric currency area. The other reveals the relative magnitude of shock asymmetries. Our very first results show that most of the countries under study are closer to the seminal Euro area rather than to Germany. New comers remain at the periphery of the Euro area with pronounced shock asymmetries than either the founder members or the three Opt-Outs.

Introduction

Since January 2008, the Euro Area has enlarged for the third time to Cyprus and Malta. As Slovakia is now planned to join in 2009, these waves of new entries revive the debate around greater asymmetries which may threaten the stability of the whole monetary union. While little attention has been paid to the adhesion of the Drachma, the perspective of a widening to the currencies of the Central and Eastern European Countries has revived the debate around the participation in a monetary union. Many and fast accessions could jeopardize the stability of the enlarged union as well as the definition and the exercise of stabilization policies. It would be the case if the eligible countries add to the heterogeneity of the whole system opening the way to new asymmetries or reinforcing the existing ones. Indeed, it is common now to make the following assessment: if national economies in the Euro area diverge considerably, the common monetary policy will not be optimal for all countries concerned.

A large literature is interested in the asymmetries either of business cycles or shocks of countries in the EMU (see Huchet-Bourdon and Pentecôte (2008) for a survey).

The current financial turmoil has also revived the debate about the instability associated to either explicit or implicit Euro targeting as exchange rate policies followed by many EU countries.

Reconsidering their attitude towards the Euro requires that States must be able to assess their own eligibility for a soon entry into the single currency area. The current Euro members may also worry about the capacity of the union itself to withstand new entries in the near future.

Recent works have stressed the predominant role of shock asymmetries in the dynamics and in the welfare cost of forming a monetary union [START_REF] Jondeau | Testing Heterogeneity Within The Euro Area[END_REF]. As also questioned by [START_REF] Plasmans | Macroeconomic stabilization policies in the EMU: spillovers, asymmetries and institutions[END_REF], the enlargement of a monetary always induces welfare losses when the accession country is hit by asymmetric price shocks. Any transfer mechanism is then unable to compensate losers for the implied negative spillovers.

The pessimistic view is reinforced by the fact that EMU insiders would suffer, on average, from next enlargements. Further, either full or partial fiscal coalitions jointly with the ECB's monetary policy appear to be instable in the absence of a well-design institutional setting. In this view, our paper extends Bayoumi and Eichengreen's (1992) core-periphery approach. Their work has been widely used to assess empirically the eligibility of a given country to join a currency union. Their method relies on the identification of the so-called "structural" supply and demand shocks using Vector Auto-Regressive (VAR) models before estimating correlations between common and idiosyncratic disturbances or between the dynamic responses of representative aggregates (economic activity and prices) to them.

The aim of this paper is to show how a suitable decomposition of the correlations between supply and demand disturbances enables to get two new indices to give a more intuitive assessment of the distance to the Euro area and the origin of shock asymmetries: one index in terms of distance of the candidate country to a fully symmetric currency area, and the other in terms of the relative magnitude of shock asymmetries.

The paper is organized as follows: The next section provides technical details on the measure of the two indexes. Section 2 describes data and presents the results. The final section concludes.

Size and side of shock asymmetries: two new indexes

Shock correlations and distance from a currency area

By construction, correlation coefficients between either supply (ρ s ) or demand shocks (ρ d ) take their values in the [-1,1] interval. Graphically, these correlations lie somewhere in the square box delimited by the dashed line as depicted on Figure 1 below.

Fig. 1: Vector representation of shock correlations and asymmetries indices

Point S at the upper right corner corresponds to the full symmetric case between any candidate member and the reference area or country. At the opposite, the lower left point A reflects complete asymmetric shocks, in terms of demand as well as in term of supply. Following Bayoumi and Eichengreen's approach, the closer the candidate country to point S is, the lower is the cost of joining the European currency union. From this perspective, it would be interesting to translate these correlations into a more intuitive measure of distance to EMU.

As illustrated on the above Figure 1, this leads us to build two new indices which derive from basic trigonometric calculus according to a vector representation in the correlations space of aggregate supply and demand disturbances (ρ s , ρ d ). These two complementary indicators indeed receive an intuitive appealing when one wishes to gauge the eligibility of a State to a given monetary union.

The first one can be interpreted as a measure of the relative distance of the candidate country (C 1 on fig. 1) from the "fully symmetric" case (S). This index derives from the ratio of the norms of two

H 1 A -1 1 S ρ ρ ρ ρ d ρ ρ ρ ρ s C 4 C 5 C 6 C 7 -1 C 3 C 8 O C 2 C 1
vectors with the "complete asymmetric" case (A) as their common starting point (see figure 1): the first one relates (A) to the accession country (C 1 ), while the second one links the "worst" to the "best" situations which can be observed in terms of stochastic asymmetries. More precisely, we define the distance index I D :

AS AC I D 1 1 - = .
(1)

By construction, the distance index I D varies in the [0,1] range such that the closer to one the index is, the more is the candidate country subject to common (symmetric) shocks. In this sense the candidate country would be more synchronized with the currency area if I D increases.

What matters for the moment is only the relative magnitude of asymmetries irrespective of the nature of the shocks. To illustrate this point let us consider another accession country whose asymmetries are reflected by point C 2 such that:

1 2 C d C s ρ ρ = and 1 2 C s C d ρ ρ = . 1
C and 2 C are thus equidistant to the first "secant" and the vector 2 1 C C is orthogonal to AS . In this circumstance both countries will be characterized by the same "distance" index, namely:

1 2 C D C D I I = since 1 2 AC AC = .
The cost of asymmetries in the wake of an adhesion to a monetary union may differ according to the origin of shocks. It is often acknowledged that supply shocks have a permanent effect on output whereas demand shocks are assumed to be neutral on real macro-variables in the long run (see Fidrmuc and Korhonen, 2006, for a survey).

But this traditional view has recently been challenged. Given the debate on this issue it is interesting to build a second index to reveal the relative intensity of shock asymmetries.

The side of shock asymmetries

For this purpose, let us define the angulus

( ) 1 , 1 OC OH ang C = θ together with the complex number 1 1 1 C d C s C i z ρ ρ + =
. We know that:

( )

1 1 arg C C z = θ
. It appears from the above figure 1 that the argument of the complex number associated to the vector 1 OC will take the magnitude as well as the sign of each correlation coefficient into account.

Since point S lies on the first "secant", we can build the asymmetry index I A as: While the candidate States C 1 and C 2 were found to be equally eligible to the monetary union on the basis of the distance index, this is no longer true if we focus now on the extent to which there is asymmetry among shock asymmetries. To decide whether an entry into a currency union is suitable or not it also matters having an idea about the induced costs given the nature of shock idiosyncrasies. One may furthermore wonder about the weights to be assigned to each of these sources of costs in some representative social loss function. Unfortunately, little is known about how to link these stochastic asymmetries to the cost of adhesion to a currency area (see although [START_REF] Hughes-Hallett | On the enlargement of currency unions: incentives to join and incentives to reform[END_REF] for a tentative assessment).

As shown on figure 1, the two new indices based on shock correlations can be used to identify countries belonging to the core or to the periphery of a given currency area (which is referred to here by point S). However, as pointed out by [START_REF] Artis | Reflections on the optimal currency area (OCA) criteria in the light of EMU[END_REF], there is no well-established theoretical foundation about the related trigger distance levels in terms of stochastic asymmetries. Assuming equal weights of the welfare effects induced by a lack of synchronization of macroeconomic shocks on either the demand or the supply side would lead to the depicted circular zoning. It depends on distance which separates the candidate country C i to the perfect symmetric monetary union (S).

Instead, unequal weights would produce rather different (elliptic) zones.

To sum up the above discussion, table 1 below shows how the correlations of structural shocks vary with the values taken by our distance and asymmetric indices.

Results

Data

The data include price and output time series. As a proxy for output we consider the industrial production index in volume. Both the consumer price index (CPI) and the harmonised CPI are also used according to the countries. Data are taken from the Eurostat database on a monthly basis over the period 1995:01-2008:04. Twenty-one countries are included in the sample: the eleven founders, countries which joined the EMU after 1999 (Greece in 2001, Slovenia in 2007, Cyprus in 20081 ), the "Outs" (United Kingdom, Sweden and Denmark), future EU countries candidates to the EMU (Estonia, Lithuania, Latvia, and Slovakia).

A related issue is how to specify the reference area: several studies discuss this point (see Huchet-Bourdon and Pentecôte (2008) for a survey). We can indeed take one country as reference; in this case, the choice often bears on Germany. But we can also make our comparisons with the Euro area as a whole. This raises further questions since the Euro did not exist before 1999, and also because the number of countries participating to the EMU has grown over/through time. This requires a weighted average of several European economies. Both cases are first adopted (German and EMU-11) but in a second step only the aggregated series are kept2 .

We also need aggregated series corresponding to the EMU with 11 countries. Which aggregated data can we consider for the Euro area still remains a great question.

The answer is not obvious for several reasons. First, we need long historical data series but the Euro area did not exist before 1999. We have consequently to build the aggregated series with the national data for the period pre-1999 at least. Second, the Euro area has been enlarged several times so which countries do we consider in the aggregated series. Do we consider aggregated series corresponding to 11, 12, 13 and 15 countries between 1999 and 2008? Third we have to determine the weights of the different countries of the Euro area. The common approach is to aggregate across these countries using a constant pre-specified set of weights3 .

In this study Eurostat data are used for the following reasons. First, Eurostat data are available over our sample: from 1995 to 2008. Second, it takes the various enlargements into account. Third, HCPI series are built by Eurostat. Fourth, we need to make these aggregations for two series, industrial production and prices.

Average data for the EMU-11 about IPI, CPI and HCPI are collected from Eurostat Database. The HCPI series for the EMU with 11 countries is only available until 2000 (before the first enlargement). Hence, we rebuilt the series by initially transforming by retropolation Euro-12 in Euro-11 over the period before 2001, and then by extrapolating the principle of calculation of the weights to supplement EMU-11 after 2001.

Estimated results

The first step has been to estimate structural VARs in order to measure correlations among the structural shocks. These estimations are not reported here for mainly two reasons. Firstly, this is not the main object of this paper4 , and secondly because of the size of the sample (21 countries, and two kinds of shocks, supply and demand).

On the other hand these estimations enable to then compute two synthetic indices. One corresponds to the relative distance of the candidate country to a fully currency area, and the other measures the relative magnitude of shock asymmetries. Both indices are computed for each country of the sample at each date between 1995 and 2008.

To ease the interpretation of so many results, we looked for judicious graphic representations.

We first represent each index over the period 1998-2008 for each country relative to Germany and the Euro area with the 11 founder countries. Graphs are represented in Annex. Our first objective here is to see whether the choice of the reference zone is important when we make our comparisons.

According to the figures, it seems that the shape of the evolution of both indices is not so many different if computations are made either with Germany as the reference zone or with the EMU-11.

Nevertheless, this first look do not must occult the fact that there are substantially differences in the size of the relative strength of asymmetries: the value is generally higher and positive if we are interested in comparisons with the EMU-11 whereas it is rather negative when we take Germany as the reference zone.

However, the figures show how it is difficult to study the impacts of such measures for all countries and on the whole period.

So, in a second step, we represent on the same graph both indices, measured either compared to Germany or to the Euro area with 11 countries, for three key dates: 1998:5 which corresponds to the announcement of the eleven founders countries of the EMU, 2004:5 because of the huge enlargement of the EU, and 2008:4 that is the last available data whatever the country.

A black triangle symbolises the measures relative to the EMU-11, and a grey one means that Germany is the target country for computations. -1,9 -1,7 -1,5 -1,3 -1,1 -0,9 -0,7 -0,5 -0,3 -0,1 0,1 0,3 0,5 0,7 0,9 1,1 1,3 1,5

Distance

-1,9 -1,7 -1,5 -1,3 -1,1 -0,9 -0,7 -0,5 -0,3 -0,1 0,1 0,3 0,5 0,7 0,9 1,1 1,3 1,5 -1,9 -1,7 -1,5 -1,3 -1,1 -0,9 -0,7 -0,5 -0,3 -0,1 0,1 0,3 0,5 0,7 0,9 1,1 1,3 1,5

Distance

Distance

Asymme try

Note : AT: Austria, BE: Belgium, CY : Cyprus, DK: Denmark, DE: Germany, ET: Estonia, FI: Finland, FR: France, GR: Greece, IR: Ireland, IT: Italy, LT: Lithuania, LU: Luxembourg, LV: Latvia, NL: Netherlands, PT: Portugal, SK: Slovakia, SP: Spain, SV: Slovenia, SW: Sweden, UK: United Kingdom.

When comparisons are made with the aggregated EMU with 11 countries, figure 2 reveals that both indices are higher than those corresponding to the comparisons with Germany, whatever the chosen date. Besides, both indices reflect a decrease in the asymmetries through the period under study.

The last part of the graph shows that new comers (now in the EMU or still applicant countries) like Cyprus, Greece, Slovakia, and Latvia, remain at the periphery of the Euro area with more pronounced shock asymmetries than either the founder members or the three "Opt-Outs".

Third, it should be very interesting to investigate more precisely our results according to the origin of the asymmetries (supply or demand side) as well as the level of distance of the country to the Euro area (either the band is changing or the values are those of May 1998). We make this exercise for two reference zones again: Germany or EMU-11.

Conclusion

Using monthly data over 1995-2008 on 21 countries, asymmetries are measured by correlations among the structural shocks from a VAR process. We then translate these correlations estimates into two synthetic indices. One can be interpreted as the relative distance of the candidate country to a fully symmetric currency area. The other reveals the relative magnitude of shock asymmetries.

These indices enable to know on which side are dominated shock asymmetries given the country's distance to the Euro area.

Our very first results show that most of the countries under study are closer to the seminal Euro area rather than to Germany. New comers remain at the periphery of the Euro area with pronounced shock asymmetries than either the founder members or the three Opt-Outs. Our results demonstrate also a general movement of the founder members to get closer to the Euro area through time.

Finally, we find that three founders have moved towards the core but still lie at the (first) periphery:

Portugal and, more surprisingly, the Luxemburg and the Netherlands. Since the latter were often viewed as members of the former D-Mark currency union, this may signal that Germany is no longer the sole centre of gravity of the EMU.

We have to complete this work by looking for the link between the asymmetries and the cost of adhesion to a currency area: the idea is to consider a loss function (see Hughes-Hallett and Jensen ( 2004)). It would enable us to answer to the following questions: What does imply the enlargement process on the cost of (next) entry and on the cost of membership for its current members? What is the overall cost of the growing EMU? Is there a boundary to enlargement?

.

  experiences greater asymmetry from the demand side than from the supply side. In this case we verify that either cos(θ C ) > cos(θ S ) or sin(θ C ) < sin(θ S ). This is precisely the case of the accession state (C 1 ) on figure1: it records relatively more asymmetries in terms of demand disturbances (thereby relatively less on the supply The same conclusion could be drawn for candidates C 6 , C 7 , and C 8 which lie in the ]-1,0[, ]-2,-1[, and ]-3,-2[ intervals respectively. Unlike C 1 , the three others exhibit negative correlations between demand shocks. Furthermore, C 8 is characterized by even stronger supply shock discrepancies. the accession country are more asymmetric or less synchronized relative to the currency area than demand shocks are. This means that we verify either cos( 1 C θ ) < cos(θ S ) or sin( 1 C θ ) > sin(θ S ). Such situations are depicted by countries like C 2 , C 3 , C 4 , and C 5 on the above graphic: the asymmetry side index takes then its values in the corresponding ] 1,2[, ]2,3[,]3,4[, and ]-4,-3[ range. between supply shocks is equivalent to its demand counterpart. Point C would then lie somewhere along the [AS] segment.

Figure

  Figure 2: Countries' eligibility to EMU from size and side of shock asymmetry (1 comparisons with Euro-11, with Germany) Classification, 1998:5

  row (red): May 1998; Second row (blue): April 2008

Table 2a : Asymmetries and distance (Euro-11 as the reference zone) AT BE GR IT LT NL PT SW SV UK AT BE SP LU SW UK FR FR IT

 2a

	] 1 , 2 ]

Table 2b : Asymmetries and distance (Euro-11 as the reference zone)

 2b The country classification according to their distance (by columns) and their relative asymmetry (by rows) indices is reported on tables 2a and 2b. Columns refer to the distance to each country from the Euro area given the lower (d l ) and upper (d u ) distance bounds, computed as the sample mean minus or plus one standard deviation. We adopt two approaches: limits fluctuate with the current distribution of the distance indices in table 2a, while these bounds are kept fixed at their 1998 levels in table2b. The latter case enables to show how the classification between countries would have changed if the initial distribution of distances was maintained.By construction, the trigger values for the relative asymmetry index remain the same. From our results, asymmetries are always in the[-2,2] range. This implies that the estimated correlations of supply shocks are always positive (or nil), while those on demand side may change sign.Major differences from table 2a to table 2b are stressed in italic case because they correspond to a switch of one country from one group to another. A striking feature is the general movement of the founder members to get closer to the Euro area through time. If we had maintained the distance bounds of 1998, six other economies (Austria, Belgium, Spain, Germany, Finland and Ireland), belong now to what can be defined as the core, in addition to France and (more recently) Italy. The remaining three founders have moved towards the core but still lie at the (first) periphery: Portugal and, more surprisingly, the Luxemburg and the Netherlands. Since the latter were often viewed as members of the former D-Mark currency union, this may signal that Germany is no longer the sole centre of gravity of the EMU.

			Distance to the Euro Area (u l and u u in May 1998)
			Far	Medium	Close
			[ 0 , 0.547 [	[ 0.547 , 0.728 [	[ 0.728 , 1 ]
	Asymmetries mainly from the Demand side	] -2 , 0 ]	SP ET IR LU CY SK	DE DK FI GR	-DE FI IR
	Asymmetries mainly from the Demand side	[ 0 , 1 [	SV ET LV	AT BE GR IT LT NL PT SW SV UK DK LT LU NL PT SW SV UK	FR AT BE SP FR IT
	Asymmetries mainly from the Supply side	] 1 , 2 ]	DK -

Malta is not included in our sample due to the unavailability of monthly data on the studied period.

[START_REF] Mink | Measuring synchronicity and comovement of business cycles with an application to the Euro Area[END_REF] select the cycle that lies the closest to all individual countries' cycles in the region in terms of synchronicity and co-movement.

[START_REF] Anderson | Constructing historical Euro Area data[END_REF] discuss the various approaches that have been used for constructing Euro area data.

See Huchet-Bourdon and Pentecôte (2008) for details on correlations asymmetries.


Eduardo Cerpa
email: eduardo.cerpa@usm.cl.

Jean-Michel Coron
email: coron@ann.jussieu.fr.

Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition

Keywords: Korteweg-de Vries equation, stabilization by feedback, backstepping AMS subject classifications. 35Q53, 93D15 I

This paper deals with the stabilization problem for the Korteweg-de Vries equation posed on a bounded interval. The control acts on the left Dirichlet boundary condition. At the right end-point, Dirichlet and Neumann homogeneous boundary conditions are considered. The proposed feedback law forces the exponential decay of the system under a smallness condition on the initial data. Moreover, the decay rate can be tuned to be as large as desired. The feedback control law is designed by using the backstepping method.

posed on a bounded domain [0, L] can be seen as a nonlinear control system where the inputs are the boundary data. From the nature of this equation, one boundary condition at the left end-point and two boundary conditions at the right end-point have to be imposed. The most studied case considers boundary conditions on u(t, 0), u(t, L), and u x (t, L).

Surprisingly, the control properties of this system are very different depending on where the controls are located. If we act on the left Dirichlet boundary condition and homogeneous data is considered at the right, then the system behaves like a heat equation and only null-controllability can be proven [START_REF] Rosier | Control of the surface of a fluid by a wavemaker[END_REF], [START_REF] Glass | Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit[END_REF]. On the other hand, if we act on the two right data and homogeneous boundary condition is considered at the left, then the system behaves like a wave equation with an infinite speed of propagation, in the sense that exact controllability holds for any time of control [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF]. Another fascinating phenomena occurs when we put only one control input at the right end-point and keep homogeneous the other two boundary conditions: there exist some spatial domains (intervals of some given lengths) for which the corresponding linearized KdV equation is not any more controllable [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF], [START_REF] Glass | Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition[END_REF]. In despite of that, in these critical cases the nonlinearity gives the exact controllability of the nonlinear KdV equation [START_REF] Coron | Exact boundary controllability of a nonlinear KdV equation with critical lengths[END_REF], [START_REF] Cerpa | Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain[END_REF], [START_REF] Cerpa | Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain[END_REF]. However, for some critical intervals the applied method requires a minimal time of control, which is not known to be really necessary.

Due to the existence of these critical intervals for the linear system, we do not know if the energy of the KdV equation is decreasing when the three boundary conditions considered are homogeneous (free control case). For instance, if L = 2π, the time-independent function given by (1 -cos(x)) satisfies u t (t, x) + u x (t, x) + u xxx (t, x) = 0, u(t, 0) = u(t, 2π) = u x (t, 2π) = 0.

(

) 3
Therefore we find out a stationary solution of the linear KdV equation with homogeneous boundary conditions and consequently a solution with non-decreasing energy. This implies that DRAFT for the free control case with L = 2π, the linearized system associated to the KdV equation ( 1)

is not exponentially stable.

With a feedback control law acting at the left hand side such phenomenon does not appear and the method we propose in this case allows to address the problem of rapid exponential stabilization: given a desired decay rate, we find a feedback law exponentially stabilizing the system at that rate.

Based on the hyperbolic nature of the KdV equation controlled from the right, a method introduced in [10], [START_REF] Urquiza | Rapid exponential feedback stabilization with unbounded control operators[END_REF] is used in [START_REF] Cerpa | Rapid exponential stabilization for a linear Korteweg-de Vries equation[END_REF] to get the rapid stabilization of the linear KdV equation from the Neumann boundary condition on the right. This gramian-based approach comes from the finite dimensional theory [START_REF] Lukes | Stabilizability and optimal control[END_REF], [START_REF] Kleinman | An easy way to stabilize a linear control system[END_REF] and was first introduced for PDE in the context of internal control [START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF]. This method requires the controllability of the linear system and therefore a non critical interval has to be considered.

In this paper, we applied the backstepping method to design the feedback control law. The backstepping method is very known as a ODE control method (see [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF] and [START_REF] Coron | Control and nonlinearity[END_REF]Section 12.5]).

The first extensions to PDE have appeared in [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF] and [START_REF] Liu | Backstepping boundary control of Burgers' equation with actuator dynamics[END_REF]. Later on, Krstic and his collaborators introduced a modification of the method by means of an integral transformation of the PDE.

This invertible transformation map the original PDE into an asymptotically stable one. In this context, the first continuous backstepping designs were proposed for the heat equation [START_REF] Liu | Boundary feedback stabilization of an unstable heat equation[END_REF], [START_REF] Smyshlyaev | Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations[END_REF].

The applications to wave equation appeared later in [START_REF] Krstic | Output-feedback stabilization of an unstable wave equation[END_REF], [START_REF] Smyshlyaev | Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary[END_REF], [START_REF] Smyshlyaev | Boundary stabilization of a 1-D wave equation with in-domain antidamping[END_REF]. An excellent starting point to get inside this method is the book [START_REF] Krstic | Boundary control of PDEs. A course on backstepping designs[END_REF] by Krstic and Smyshlyaev.

This paper is organized as follows. In Section II, we formulate the problem and state the main result. Section III contains the backstepping design of the feedback control law. Section IV is devoted to prove the exponential decay of the L 2 -norm of the solutions for the linearized system around the origin. In Section V, we prove that this result still holds for the nonlinear KdV control system when the initial condition is small enough. Some extensions to the non-constant coefficient case and different boundary conditions are considered in Section VI. Finally, some final remarks are given in Section VII.

Remark 1: The stabilization of the KdV equation by using an internal feedback control law DRAFT was addressed in [START_REF] Menzala | Stabilization of the Korteweg-de Vries equation with localized damping[END_REF], [START_REF] Pazoto | Unique continuation and decay for the Korteweg-de Vries equation with localized damping[END_REF] (see also [START_REF] Rosier | Global Stabilization of the generalized Korteweg-de Vries Equation Posed on a finite Domain[END_REF], [START_REF] Linares | On the exponential decay of the critical generalized Korteweg-de Vries with localized damping[END_REF] for KdV equations with other nonlinearities).

They have proved the following semi-global stabilization. Let L > 0, R > 0 and a = a(x) a damping term satisfying a(x) ≥ a 0 > 0, forall x ∈ ω where ω is nonempty open subset of (0, L). Then, there exist C = C(R) > 0 and µ = µ(R) > 0 such that

u(t, x) L 2 (0,L) ≤ Ce -µt u(0, •) L 2 (0,L) , ∀t ≥ 0 (4)
for any solution of

   u t (t, x) + u x (t, x) + u xxx (t, x) + a(x)u(t, x) + u(t, x)u x (t, x) = 0, u(t, 0) = u(t, L) = u x (t, L) = 0, (5)
with u(0, •) L 2 (0,L) ≤ R. This result is different from ours in the sense that the exponential decay rate can not be imposed as large as desired. A key role in their design is played by the damping term a = a(x), which prevents the existence of critical domains and allows to work with a dissipative system for any L. Other internal feedback control laws (static or time-varying ones) for the KdV equation with periodic boundary conditions can be found in [START_REF] Russell | Exact controllability and stabilizability of the Korteweg-de Vries equation[END_REF], [START_REF] Laurent | Control and stabilization of the Korteweg-de Vries equation on a periodic domain[END_REF].

II. PROBLEM STATEMENT AND MAIN RESULT

Given L > 0, we consider the following nonlinear control system on the interval [0, L]

   u t (t, x) + u x (t, x) + u xxx (t, x) + u(t, x)u x (t, x) = 0, u(t, 0) = κ(t), u(t, L) = 0, u x (t, L) = 0. (6)
For any positive λ, we address the problem of building some feedback control law κ(t) = κ(u(t, •)) such that the origin is exponentially stable for the corresponding closed-loop system [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF] and the exponential decay rate is λ.

By using the backstepping method, we are able to find such a control law. The design is based on the linearized system around the origin. The linear closed-loop system is exponentially stable and the same result is obtained for the nonlinear KdV equation by adding a smallness condition on the initial data.

Our main theorem is the following.

DRAFT Theorem 2: For any λ > 0, there exist a feedback control law κ = κ(u(t, •)), r > 0 and

D > 0 such that u(t, •) L 2 (0,L) ≤ De -λt u(0, •) L 2 (0,L) , ∀t ≥ 0, (7)
for any solution of (6) satisfiying u(0, •) L 2 (0,L) ≤ r.

The feedback law κ is explicitly defined as follows

κ(t) = L 0 k(0, y)u(t, y)dy, (8)
where the function k = k(x, y) is characterized in Section III as the solution of a given partial differential equation depending on λ.

Remark 3: As we shall see in Section VI, Theorem 2 also holds if system ( 6) is replaced by

   u t (t, x) + u x (t, x) + u xxx (t, x) + u(t, x)u x (t, x) = 0, u(t, 0) = κ(t), u x (t, L) = 0, u xx (t, L) = 0. (9)
or

   u t (t, x) + γ(x)u x (t, x) + u xxx (t, x) + u(t, x)u x (t, x) + δ(x)u(t, x) = 0, u(t, 0) = κ(t), u(t, L) = 0, u x (t, L) = 0, (10)
where γ, δ are given functions in C 1 ([0, L]) and C([0, L]), respectively.

III. CONTROL DESIGN

The backstepping method applied here is based on the linear part of the equation. In this way, we consider the control system linearized around the origin

   u t (t, x) + u x (t, x) + u xxx (t, x) = 0, u(t, 0) = κ(t), u(t, L) = 0, u x (t, L) = 0. (11)
Given a positive parameter λ, we look for a transformation Π :

L 2 (0, L) → L 2 (0, L) defined DRAFT by w(x) = Π(u(x)) := u(x) - L x k(x, y)u(y)dy, (12)
such that the trajectory u = u(t, x), solution of [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF] with

κ(t) = L 0 k(0, y)u(t, y)dy, (13)
is map into the trajectory w = w(t, x), solution of the linear system

   w t (t, x) + w x (t, x) + w xxx (t, x) + λw(t, x) = 0, w(t, 0) = 0, w(t, L) = 0, w x (t, L) = 0. (14)
For system [START_REF] Laurent | Control and stabilization of the Korteweg-de Vries equation on a periodic domain[END_REF], called the target system, we have for any t ≥ 0

d dt L 0 |w(t, x)| 2 dx = -|w x (t, 0)| 2 -2λ L 0 |w(t, x)| 2 dx ≤ -2λ L 0 |w(t, x)| 2 dx (15)
and therefore we easily obtain for w = w(t, x) the exponential decay at rate λ

w(t, •) L 2 (0,L) ≤ e -λt w(0, •) L 2 (0,L) , ∀t ≥ 0. (16)
In Section IV, we prove that, thanks to the invertibility of the map Π, the exponential decay ( 16) also holds for system [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF].

Naturally, we can wonder if this decay rate is sharp. Let us notice that the eigenvalues of system ( 14) are the eigenvalues of

   w t (t, x) + w x (t, x) + w xxx (t, x) = 0, w(t, 0) = 0, w(t, L) = 0, w x (t, L) = 0, (17)
shifted to the left λ units. Thus, we are lead to study the eigenvalues σ of

   -φ (x) -φ (x) = σφ(x), φ(0) = 0, φ(L) = 0, φ (L) = 0. (18)
Surprisingly, the location of the eigenvalues of ( 18) depends on the length of the interval.

DRAFT From [START_REF] Rosier | Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain[END_REF], we know that there exist some eigenvalues located on the imaginary axis if and only

if L ∈ {2π k 2 +k + 2 3 (k, ) ∈ N 2 }
, which is called the set of critical lengths for this problem.

In Figure 1, the first five eigenvalues of system ( 17 In conclusion, the decay in ( 16) is optimal for some values of L. Let us focus in the key step, which is finding the kernel k = k(x, y) such that w(t, x) = Π(u(t, x)) satisfies [START_REF] Laurent | Control and stabilization of the Korteweg-de Vries equation on a periodic domain[END_REF]. For that, we perform the following computations

w t (t, x) = u t (t, x) - L x u t (t, y)k(x, y)dy = u t (t, x) + L x (u y (t, y) + u yyy (t, y)) k(x, y)dy = u t (t, x) - L x u(t, y) (k y (x, y) + k yyy (x, y)) dy -k(x, x)(u(t, x) + u xx (t, x)) + k y (x, x)u x (t, x) -k yy (x, x)u(t, x) + k(x, L)u(t, L) + k(x, L)u xx (t, L) -k y (x, L)u x (t, L) + k yy (x, L)u(t, L), (19)
w x (t, x) = u x (t, x) + k(x, x)u(t, x) - L x k x (x, y)u(t, y)dy, (20)
DRAFT

w xx (t, x) = u xx (t, x) + u(t, x) d dx k(x, x) + k(x, x)u x (t, x) + k x (x, x)u(t, x) - L x k xx (x, y)u(t, y)dy, (21)
and

w xxx (t, x) = u xxx (t, x) + u(t, x) d 2 dx 2 k(x, x) + 2u x (t, x) d dx k(x, x) + k(x, x)u xx (t, x) + u(t, x) d dx k x (x, x) + k x (x, x)u x (t, x) + k xx (x, x)u(t, x) - L x k xxx (x, y)u(t, y)dy. (22)
Thus, given λ ∈ R and using [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF], we have

w t (t, x) + w x (t, x) + w xxx (t, x) + λw(t, x) = - L x u(t, y) k xxx (x, y) + k x (x, y) + k yyy (x, y) + k y (x, y) + λk(x, y) dy + k(x, L)u xx (t, L) + u x (t, x) k y (x, x) + k x (x, x) + 2 d dx k(x, x) + u(t, x) λ + k xx (x, x) -k yy (x, x) + d dx k x (x, x) + d 2 dx 2 k(x, x) . ( 23
)
After the above computations and since d dx k(x, x) = k x (x, y) + k y (x, y), we obtain that one has [START_REF] Laurent | Control and stabilization of the Korteweg-de Vries equation on a periodic domain[END_REF] for every u solution of [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF] with [START_REF] Krstic | Boundary control of PDEs. A course on backstepping designs[END_REF] if the kernel k = k(x, y) defined in the triangle

T := {(x, y) x ∈ [0, L], y ∈ [x, L]} satisfies                  k xxx (x, y) + k yyy (x, y) + k x (x, y) + k y (x, y) = -λk(x, y), in T , k(x, L) = 0, in [0, L], k(x, x) = 0, in [0, L], k x (x, x) = λ 3 (L -x), in [0, L]. (24)
Let us make the following change of variable

t = y -x, s = x + y, (25)
DRAFT and define G(s, t) := k(x, y). We have k(x, y) = G(x + y, y -x) and therefore

k x = G s -G t , k y = G s + G t , (26)
k xx = G ss -2G st + G tt , k yy = G ss + 2G st + G tt , (27)
k xxx = G sss -3G sst + 3G stt -G ttt , (28)
k yyy = G sss + 3G sst + 3G stt + G ttt . (29)
Now, the function G = G(s, t), defined in

T 0 := {(s, t) t ∈ [0, L], s ∈ [t, 2L -t]}, satisfies                  6G tts (s, t) + 2G sss (s, t) + 2G s (s, t) = -λG(s, t), in T 0 , G(s, 2L -s) = 0, in [L, 2L], G(s, 0) = 0, in [0, 2L], G t (s, 0) = λ 6 (s -2L), in [0, 2L]. (30)
Let us transform this system into an integral one. We write the equation in variables (η, ξ), integrate ξ in (0, τ ) and use that 6G ts (η, 0) = λ. Next, we integrate τ in (0, t) and use that G s (η, 0) = 0. Finally, we integrate η in (s, 2L -t) and use that G(2L -t, t) = 0. Thus, we can write the following integral form for G = G(s, t)

G(s, t) = - λt 6 (2L -t -s) + 1 6 2L-t s t 0 τ 0 2G sss (η, ξ) + 2G s (η, ξ) + λG(η, ξ) dξdτ dη. (31)
To prove that such a function G = G(s, t) exists, we use the method of successive approximations. We take as an initial guess

G 1 (s, t) = - λt 6 (2L -t -s) (32)
and define the recursive formula as follows,

G n+1 (s, t) = 1 6 2L-t s t 0 τ 0 2G n sss (η, ξ) + 2G n s (η, ξ) + λG n (η, ξ) dξdτ dη. (33)
DRAFT Performing some computations, we get for instance

G 2 (s, t) = 1 108 t 3 λ -λ 2 L + λ 2 t 4 2L -t -s + t 3 λ 2 4 (2L -t) 2 -s 2 , (34)
and more generally the following formula

G k (s, t) = k i=1 a i k t 2k-1 + b i k t 2k (2L -t) i -s i , (35)
where the coefficients satisfy b k k = 0 and more importantly, there exist positive constants M, B such that, for any k ≥ 1 and any (s, t) ∈ T 0

G k (s, t) ≤ M B k (2k)! (t 2k-1 + t 2k ). (36)
This implies that the series ∞ n=1 G n (s, t) is uniformly convergent in T 0 . Therefore the series defines a continuous function G :

T 0 → R G(s, t) = ∞ n=1 G n (s, t) (37)
and we get a solution of our integral equation. Indeed, we can write

G = G 1 + ∞ n=1 G n+1 = G 1 + 1 6 ∞ n=1 2L-t s t 0 τ 0 2G n sss (η, ξ) + 2G n s (η, ξ) + λG n (η, ξ) dξdτ dη = G 1 + 1 6 2L-t s t 0 τ 0 2 ∞ n=1 G n sss (η, ξ) + 2 ∞ n=1 G n s (η, ξ) + λ ∞ n=1 G n (η, ξ) dξdτ dη = G 1 + 1 6 2L-t s t 0 τ 0 2G sss (η, ξ) + 2G s (η, ξ) + λG(η, ξ) dξdτ dη. ( 38
)
where we have used that the corresponding series n≥1 G n s and n≥1 G n sss are also uniformly convergent.

Once we have found the function G = G(s, t), we get the existence of the kernel k = k(x, y).

It is easy to see that the map Π : L 2 (0, L) → L 2 (0, L), defined by [START_REF] Krstic | Output-feedback stabilization of an unstable wave equation[END_REF], is continuous and consequently we have the existence of a positive constant D κ such that

Π(f ) L 2 (0,L) ≤ D κ f L 2 (0,L) , ∀f ∈ L 2 (0, L). (39)
In Figure 2, we plot the gain kernel k(0, y) (see ( 12

IV. STABILITY OF THE LINEAR SYSTEM

We know that the target system ( 14) is exponentially stable. In order to get the same conclusion for the linear system [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF], the method we are applying uses the inverse transformation Π -1 . For that, we introduce a kernel function (x, y) which satisfies

                 xxx (x, y) + yyy (x, y) + x (x, y) + y (x, y) = λ (x, y), in T , (x, L) = 0, in [0, L], (x, x) = 0, in [0, L], x (x, x) = λ 3 (L -x), in [0, L], (40)
The existence and uniqueness of such a kernel = (x, y) are proven in the same way than for the kernel k = k(x, y) in Section III. Once we have defined = (x, y), it is easy to see that DRAFT the transformation Π -1 is characterized by

u(x) = Π -1 (w(x)) := w(x) + L x (x, y)w(y)dy. ( 41
)
Let us see that k(x, y) and (x, y) are related by the formula

(x, y) -k(x, y) = y x k(x, η) (η, y)dη, (42)
which in fact proves that Π -1 maps a trajectory of ( 14) into a trajectory of ( 11) with control κ(t) defined by [START_REF] Krstic | Boundary control of PDEs. A course on backstepping designs[END_REF]. Indeed, by plugging (41) into ( 12)and using Fubini's theorem we get, for

any w ∈ L 2 (0, L), that for any x ∈ [0, L] L x ( (x, y) -k(x, y))w(y)dy = L x y x k(x, η) (η, y)w(y)dηdy, (43)
which proves (42) for any (x, y) ∈ T .

The map Π -1 : L 2 (0, L) → L 2 (0, L) is continuous and therefore we get the existence of a positive constant D such that

Π -1 (f ) L 2 (0,L) ≤ D f L 2 (0,L) , ∀f ∈ L 2 (0, L). ( 44
)
Let us prove that system (11)-( 13) is exponentially stable. In fact, given u 0 ∈ L 2 (0, L), we define

w 0 (x) = Π(u 0 (x)) := u 0 (x) - L x k(x, y)u 0 (y)dy. ( 45
)
The solution of ( 14) with initial condition w(0, x) = w 0 (x) satisfies ( 15), i.e.

w(t, •) L 2 (0,L) ≤ e -λt w 0 (•) L 2 (0,L) , ∀t ≥ 0. ( 46
)
Moreover, the solution of ( 11) is given by u(t, x) = Π -1 (w(t, x)). Thus, from (39),(44) and DRAFT (46) we have for any t ≥ 0

u(t, •) L 2 (0,L) ≤ D w(t, •) L 2 (0,L) ≤ D e -λt w 0 (•) L 2 (0,L) ≤ D D k e -λt u 0 (•) L 2 (0,L) , (47)
which proves the exponential decay at rate λ for system [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF] with feedback law (13).

V. STABILITY OF THE NONLINEAR SYSTEM

Let u = u(t, x) be a solution of the nonlinear equation ( 6) with the control κ given by [START_REF] Krstic | Boundary control of PDEs. A course on backstepping designs[END_REF].

Then, w = Π(u(t, x)) satisfies

w t (t, x) + w x (t, x) + w xxx (t, x) + λw(t, x) = -w(t, x) + L x (x, y)w(t, y)dy w x (t, x) + L x x (x, y)w(t, y)dy (48)
with homogeneous boundary conditions

w(t, 0) = 0, w(t, L) = 0, w x (t, L) = 0. (49)
We multiply (48) by w and integrate in (0, L) to obtain

d dt L 0 |w(t, x)| 2 dx = -|w x (t, 0)| 2 -2λ L 0 |w(t, x)| 2 dx -2 L 0 w(t, x)F (t, x)dx (50)
where the term F = F (t, x) is given by

F (t, x) = w(t, x) L x x (x, y)w(t, y)dy + w x (t, x) L x (x, y)w(t, y)dy + L x (x, y)w(t, y)dy L x x (x, y)w(t, y)dy . (51)
We can prove that there exists a positive constant

C = C( C 1 (T ) ) such that 2 L 0 w(t, x)F (t, x)dx ≤ C L 0 |w(t, x)| 2 3/2 (52)
DRAFT and therefore, if there exists t 0 ≥ 0 such that

w(t 0 , •) L 2 (0,L) ≤ λ C , (53)
then we obtain

d dt L 0 |w(t, x)| 2 dx ≤ -λ L 0 |w(t, x)| 2 dx, ∀t ≥ t 0 . (54)
Thus, we get

w(t, •) L 2 (0,L) ≤ e -λ 2 t w(0, •) L 2 (0,L) , ∀t ≥ 0, (55)
provided that

w(0, •) L 2 (0,L) ≤ λ C . (56)
As we did for the linear system, by using the continuity of the transformations Π and Π -1 (see ( 39) and ( 44)) and (55), we obtain the exponential decay of the nonlinear equation [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF]. From (56), we have to add a smallness condition on the initial data of system [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF]. This concludes the proof of Theorem 2.

VI. SOME EXTENSIONS

As mentioned in Remark 3, this method can be applied to stabilize other related KdV systems.

In this Section, we focus in the linear control design because the nonlinear part of the argument is the same as in Section V. More precisely, we show the equations that define the kernel functions corresponding to each case.

A. Different boundary conditions

In order to stabilize system

   u t (t, x) + u x (t, x) + u xxx (t, x) = 0, u(t, 0) = κ(t), u x (t, L) = 0, u xx (t, L) = 0, (57)
DRAFT we have to consider a kernel k = k(x, y) satisfying on the triangle

T := {(x, y) x ∈ [0, L], y ∈ [x, L]} the equation                  k xxx (x, y) + k yyy (x, y) + k x (x, y) + k y (x, y) = -λk(x, y), in T , k yy (x, L) = 0, in [0, L], k(x, x) = 0, in [0, L], k x (x, x) = λ 3 (L -x), in [0, L]. (58)
The transformation

w(x) = Π(u(x)) := u(x) - L x k(x, y)u(y)dy, (59)
map the solution u = u(t, x) into the trajectory w = w(t, x), solution of the target system

   w t (t, x) + w x (t, x) + w xxx (t, x) + λw(t, x) = 0, w(t, 0) = 0, w x (t, L) = 0, w xx (t, L) = 0, (60)
which is exponentially asymptotically stable for λ > 0, with a decay at least equals to λ. By making the change of variable

t = y -x, s = x + y, (61)
and defining G(s, t) := k(x, y), we get that G is solution of

G(s, t) = - λ 6 (2L -s) -2 t 0 G s (s, τ )dτ -2 t 0 τ 0 G ss (s, ξ)dξdτ + 1 6 t 0 τ 0 2L-ξ s -4G sss (η, ξ) + 2G s (η, ξ) + λG(η, ξ) -12G sst (η, ξ) dξdτ dη (62)
that can be studied by applying again the method of successive approximations..

DRAFT

B. Non-constant coefficient case

Given two functions γ ∈ C 1 ([0, L]) and δ ∈ C([0, L]), we consider the non-constant coefficient linear KdV equation

   u t (t, x) + γ(x)u x (t, x) + u xxx (t, x) + δ(x)u = 0, u(t, 0) = κ(t), u(t, L) = 0, u x (t, L) = 0, (63)
and the target system

   w t (t, x) + γ(x)w x (t, x) + w xxx (t, x) + (λ + δ(x))w(t, x) = 0, w(t, 0) = 0, w(t, L) = 0, w x (t, L) = 0, (64)
which is exponentially asymptotically stable for λ large enough. In fact,

λ ≥ max x∈[0,L] |γ (x) -2δ(x)|
is a sufficient condition to ensure an exponential decay rate equals to λ/2 for solutions of (64).

In this case, the kernel k = k(x, y) to be considered in order to define the corresponding transformation (59) is the solution of

                 k xxx (x, y) + k yyy (x, y) + γ(x)k x (x, y) + γ(y)k y (x, y) = α(x, y)k(x, y), in T , k(x, L) = 0, in [0, L], k(x, x) = 0, in [0, L], k x (x, x) = λ 3 (L -x), in [0, L]. ( 65
)
where α is defined by

α(x, y) = -(λ + γ (y) + δ(x) + δ(y)). (66)
DRAFT By using the change of variable (61), system (65) can be lead to the equation

G(s, t) = - λt 6 (2L -t -s) + 1 6 2L-t s t 0 τ 0 2G sss (η, ξ) + a 1 (η, ξ)G s (η, ξ) + a 2 (η, ξ)G t (η, ξ) + α η -ξ 2 , η + ξ 2 G(η, ξ) dξdτ dη, (67)
where

a 1 (η, ξ) = γ η + ξ 2 + γ η -ξ 2 and a 2 (η, ξ) = γ η + ξ 2 -γ η -ξ 2 . ( 68
)
As before, this equation can be studied by applying the method of successive approximations.

C. A non-zero equilibrium state

We consider u = u(t, x) solution of ( 6). If instead of linearizing this system around zero, we do that around a non-zero equilibrium state z = z(x) solution of

   z (x) + z (x) + z(x)z (x) = 0, z(0) = U 0 , z(L) = 0, z (L) = 0, (69)
with U 0 = 0, we have to consider the control system

   v t (t, x) + (1 + z(x))v x (t, x) + v xxx (t, x) + z x (x)v(t, x) = 0, v(t, 0) = h(t), v(t, L) = 0, v x (t, L) = 0, (70)
where v = v(t, x) and h = h(t) are the first-order approximation of the state (u(t, x) -z(x)) and the control (κ(t) -U 0 ), respectively. Thus, by using the control design in Subsection VI-B for non-constant coefficients, we can locally stabilizes system (6) around a non-zero equilibrium.

VII. CONCLUSIONS

We have applied the backstepping method to build some boundary feedback laws, which locally stabilize the Korteweg-de Vries equation posed on a finite interval. Our control acts on the Dirichlet boundary condition at the left hand side of the interval where the system evolves.

DRAFT

The closed-loop system is proven to be locally exponentially stable with a decay rate that can be chosen to be as large as we want. This approach allows to consider the KdV equation [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF] with other boundary conditions and low order terms as u x or u with non-constant coefficients depending on the space variable.

The situation where we act on the right end-point is different. If we consider homogeneous Dirichlet condition on the left and one or two control inputs at the right hand side of the interval, then we are not able to prove the backstepping method works with the transformation [START_REF] Krstic | Output-feedback stabilization of an unstable wave equation[END_REF]. Indeed, when imposing w t + w x + w xxx + λw = 0 (71) on the target system, we get the following expression at x = L to vanish k(x, L)u xx (t, L) + k(x, L)u(t, L) + k yy (x, L)u(t, L) -k y (x, L)u x (t, L).

As we do not have to our disposal u xx (t, L), the first term above arises the condition k(x, L) = 0.

Even if we do not care about the two last terms in (72), in order to keep w(t, 0) = u(t, 0) = 0, we have to impose k(0, y) = 0 for any y ∈ (0, L). With these four boundary restrictions (the other two are on k(x, x)), the third order kernel equation satisfied by k = k(x, y) becomes overdetermined. Therefore, it is not clear if such a function k = k(x, y) exists.

A natural idea to deal with controls at x = L is to use the transformation w(t, x) = u(t, x) -x 0 k(x, y)u(t, y)dy,

instead of [START_REF] Krstic | Output-feedback stabilization of an unstable wave equation[END_REF]. However, it is not clear if that approach works. In fact, if we do that, we have to deal now with the extra condition k y (x, 0) = 0 for any x ∈ (0, L). This is due to the fact that when imposing w t + w x + w xxx + λw = 0 on the target system, we get the extra term u x (t, 0)k y (x, 0) to be cancelled. As previously, this fourth restriction gives an overdetermined kernel equation for k = k(x, y). Moreover, the existence of critical lengths when only one control is considered at the right end-point suggests that either the existence of the kernel or the DRAFT invertibility of the corresponding map Π should fail for some spatial domains.

Two related problems that remain still open are the boundary global or semi-global stabilization and the output feedback control problem. The boundary global stabilization is hard because it is needed a really nonlinear design as in [START_REF] Laurent | Control and stabilization of the Korteweg-de Vries equation on a periodic domain[END_REF] for the KdV equation with periodic boundary conditions. Concerning the output feedback control problem, we believe it could be solved by applying the backstepping approach in order to built some observers as done in [START_REF] Smyshlyaev | Backstepping observers for a class of parabolic PDEs[END_REF], [START_REF] Krstic | Output-feedback stabilization of an unstable wave equation[END_REF] for the heat and the wave equations respectively.

  ) are plot for different values of L. In case (a), L = 1 (non-critical) and the first eigenvalue σ 1 is approximately -72. The system behaves like a dissipative one. In (b), L = 2π (critical) and we have σ 1 = 0. The system has one conservative component given by the eigenfunction φ(x) = 1 -cos(x). In (c), L = 2π 7/3 and the first two eigenvalues are imaginary numbers σ 1 = 0.2i and σ 2 = -0.2i. This examples show the different behaviors system[START_REF] Laurent | Control and stabilization of the Korteweg-de Vries equation on a periodic domain[END_REF] can have and the important role played by the parameter λ in our design.

Figure 1 .

 1 Figure 1. The first five eigenvalues of system (17) for (a) L = 1, (b) L = 2π, and (c) L = 2π 7/3, respectively.

  )) as a function of y ∈ [0, L] for different lengths (a) L = 1 (non-critical), (b) L = 2π (critical) and (c) L = 2π 7/3 (critical). The kernel functions are defined with λ = 1. This illustrate the fact that case (a) is easier to stabilizes than case (b), which is easier to stabilizes than case (c). This is due to the location of the corresponding open-loop eigenvalues as shown in Figure 1.

Figure 2 .

 2 Figure 2. Gain kernel k(0, y) corresponding to λ = 1 for (a) L = 1, (b) L = 2π, and (c) L = 2π 7/3, respectively.

Acknowledgements. This work has been done while the first author was visiting the Laboratoire Jacques-Louis Lions at Université Pierre et Marie Curie in Paris. He thanks this institution for its hospitality.

This author was partially supported by the ERC advanced grant FP7-266907 CPDENL of the 7th Research Framework Programme (FP7), by Fondecyt grant #11090161, by CMM Basal grant and by MathAmsud project CIP-PDE. J.-M. Coron is with the Institut For this work, this author was supported by the ERC advanced grant FP7-266907 CPDENL of the 7th Research Framework Programme (FP7). DRAFT


S Lepoutre

A Gauguet

G Trénec

M Büchner

J Vigué

He-McKellar-Wilkens topological phase in atom interferometry

Keywords: numbers: 03.65.Vf, 03.75.Dg, 39.20.+q

We report the first experimental test of the topological phase predicted by He and McKellar and by Wilkens in 1993: this phase, which appears when an electric dipole propagates in a magnetic field, is connected to the Aharonov-Casher effect by electric-magnetic duality. The He-McKellar-Wilkens phase is quite small, at most 27 mrad in our experiment, and this experiment requires the high phase sensitivity of our atom interferometer with spatially separated arms as well as symmetry reversals such as the direction of the electric and magnetic fields. The measured value of the He-McKellar-Wilkens phase differs by 31% from its theoretical value, a difference possibly due to some as yet uncontrolled systematic errors.

In quantum mechanics, propagation can be modified without any force, the first example being the Aharonov-Bohm effect [1] discovered in 1959: a magnetic field shifts the fringes of an electron interferometer, even if the field vanishes on the electron paths. This was the discovery of topological phases [START_REF] Berry | Topological phases belong to the group of geometric phases introduced by[END_REF], which differ considerably from ordinary dynamic phases because they are independent of the particle velocity and non-reciprocal, i.e. they change sign when propagation is reversed. In 1984, Aharonov and Casher [START_REF] Aharonov | [END_REF] discovered another topological phase, which appears in a matter wave interferometer operated with a particle carrying a magnetic dipole, the interferometer arms encircling a line of electric charges. In 1993, He and McKellar [4] applied electricmagnetic duality [5] to the Aharonov-Casher phase, thus exhibiting a topological phase when the particle carries an electric dipole and the interferometer arms encircle a line of magnetic monopoles: this phase appeared as speculative but a possible experiment was rapidly proposed by Wilkens [6]. Whereas the Aharonov-Bohm and Aharonov-Casher (AC) effects were rapidly tested by experiments [7][8][9][10][11][12], no experimental test of the He-McKellar-Wilkens (HMW) phase has been available so far. Here, we report the first experimental attempt to detect the HMW phase, with results in reasonable agreement with theory. The HMW phase is the last member of the family of topological phases when free particles propagate in electromagnetic fields [13,14]: one might expect similar phases for higher order electromagnetic multipoles but the calculated values for quadrupoles [15] are so small that their detection is presently out of reach.

Although Wilkens experiment proposal [6] is 18 years old, no experimental proof of the HMW phase has been reported. A test was discussed in 1996 by Wei et al. [16] and by Schmiedmayer et al. [START_REF] Schmiedmayer | Atom interferometry[END_REF] and an experiment with a superfluid helium interferometer was proposed in 2009 by Sato and Packard [START_REF] Sato | [END_REF], without any published result yet. Let us compare the detection of the AC and HMW phases, in order to understand why this test is difficult. All accurate tests of the AC phase [10][11][12] have used a Ramsey interferometer [19], in which the particle propagates in a superposition of two spin states: this type of interferometer is ideal for the detection of a spindependent phase and it provides an excellent cancelation of systematic errors. The use of a Ramsey interferometer for the HMW phase would require the production of a quantum superposition of states with opposite electric dipole moments, which is feasible if states of opposite parity are quasi-degenerate [14], a situation which does not exist with ground state atoms. Consequently, the HMW phase must be measured by alternating field configurations and by studying differences of measured phases: this procedure makes an experiment more subject to systematic errors than Ramsey interferometry. In addition, in order to create a non-zero HMW phase shift, the two interferometer arms must propagate in different electric or magnetic fields, which is possible only if the two arms are spatially separated. Following the pioneering work of Pritchard and co-workers [START_REF] Schmiedmayer | Atom interferometry[END_REF]20], very few separated-arm atom interferometers have been built.

Although not associated with a classical force, the AC and the HMW phases can be explained by the interaction of a dipole with a motional field [14]. The HMW phase is due to the interaction of an electric dipole d with the motional electric field E mot = γv×B where v is the atom velocity, B is the magnetic field and γ = 1/ 1 -v 2 /c 2 ≃ 1 is the relativistic factor. U = -d • E mot is the interaction energy, which induces the HMW phase shift:

ϕ HMW = U dt/h (1)
where the closed loop follows the interferometer paths. Because of parity, atoms or molecules do not have a permanent dipole and an electric field E is needed to induce a dipole d = 4πε 0 αE, where α is the electric polarizability. Finally, the HMW phase shift is given by:

ϕ HMW = (d × B) • vdt/h (2)
Because vdt is the infinitesimal path length, ϕ HMW is independent of the modulus v of the atom velocity but it changes sign with the direction of propagation, and it is maximum when E, B and v are orthogonal. The interaction region is placed where the distance between interferometer arms is largest, close to 100 µm. Panel b: the interaction region producing the electric and magnetic fields (not to scale). The interferometer arms (dotted blue lines) are separated by a septum, which is the common electrode of two plane capacitors producing opposite electric fields (high voltage electrodes in red; grounded electrodes in black). Two rectangular coils (brown rectangle) produce the magnetic field.

To detect the HMW phase, we use a Mach-Zehnder atom interferometer [21] shown in fig. 1a. A supersonic lithium beam (mean velocity near 1000 m/s with a distribution half-width close to 100 m/s) is strongly collimated and then diffracted by three laser standing waves in the Bragg regime: the laser frequency is chosen such that only the dominant lithium isotope 7 Li contributes to the signal [21,22]. The diffraction events play the roles of beam-splitters and mirrors for the atomic wave. This interferometer produces two output beams, labeled C and D on fig. 1a, with complementary fringe signals. Beam C is selected by a slit and the atoms of this beam are ionized by a Langmuir-Taylor "hot-wire" detector [23]. The resulting ions are counted, thus providing the interferometer signal I given by:

I = I 0 [1 + V cos(ϕ p + ϕ d )] ( 3
)
I 0 is the mean intensity and V is the fringe visibility. The phase ϕ p is due to perturbations. The phase ϕ d , due to atom diffraction, depends on the positions of the laser standing wave mirrors M i . To record interference fringes (see fig. 2), we move mirror M 3 with a piezoelectric device and its displacement, measured by a Michelson interferometer, gives very accurately the variations of ϕ d . Typical values of the mean intensity and of the fringe visibility are I 0 ≈ 50000 atoms/s and V ≈ 70%, leading to a phase sensitivity near 20 -30 mrad/ √ Hz. The interaction region is schematically represented in fig. 1b (more details in [5]): homogeneous electric fields are produced by two plane capacitors sharing a thin "septum" electrode [24], which is inserted between the two interferometer arms without modifying their propagation. With a 1.10 mm electrode spacing, the electric field is E(kV/m)≈ 0.9V where the applied voltage V (Volt) can be positive or negative, in order to test field reversal. The double capacitor is placed inside a pair of coils producing a fairly homogeneous magnetic field: the use of coils rather than permanent magnets limits the field value near 14 mT but gives rise to a much better control and permits field reversal. For a current I circulating in the coils, the field B at the center is given by B/I ≈ 0.56 mT/A. This configuration with opposite electric fields on the two arms, proposed by Wei et al. [16], differs from the idea of He, McKellar and Wilkens [4,6] in which the same electric dipole propagates in opposite magnetic fields but we think that the phase shift predicted by equation 2 is truly a HMW phase shift [5]. We have replaced the charged wire of Wei et al. [16] by plane capacitors to improve the field homogeneity and to minimize the dispersion of the polarizability phase (a large dispersion reduces the fringe visibility). With this interaction region and lithium atoms [25], the HMW phase shift predicted by equation ( 2) is ϕ HMW (V, I)(rad) = -1.28 × 10 -6 V I, corresponding to a maximum value |ϕ HMW | max = 27 mrad for |V | = 800 V and |I| = 25 A.

Let ϕ E+B (V, I) be the measured phase shift when the electric and magnetic fields are both applied. ϕ E+B includes the HMW phase shift but also the dynamic phase shifts [26] induced by each field acting separately. The electric field induces a Stark (or polarizability) phase shift [24] ϕ S = -2πε 0 α E 2 dt/h. We tune the ratio of the voltages applied to the capacitors to ensure ϕ S < ∼ 100 mrad, a very small value compared to the phase shifts induced on each arm which can exceed 300 rad.

The magnetic field induces a Zeeman energy shift U Z (F, m F ), function of the F, m F hyperfine sub-level, and a phase shift [22,27] given by ϕ Z (F, m F ) = U Z (F, m F , B)dt/h. If a magnetic field B = 14 mT was applied to one arm only, the Zeeman phase shift would be extremely large, ϕ Z (F = 2, m F = ±2) ≈ ±10 5 rad. This phase shift would be perfectly canceled if the magnetic field had exactly the same value on the two arms but a weak field gradient exists near the septum, sufficient to induce a Zeeman phase shift ϕ Z (F = 2, m F = ±2) ≈ ±11 rad when B = 14 mT. The detected signal is an average over the ground state sub-levels with almost equal contributions [5,22]. These sub-levels form 4 pairs with opposite Zeeman phase shifts, so that the measured Zeeman phase shift ϕ B is very weak but the large dispersion of ϕ Z with F, m F reduces the fringe visibility. To reduce the Zeeman phase shifts, another coil producing an opposite magnetic field gradient was introduced outside the interaction region of fig. 1b, and this compensation is excellent when hyperfine uncoupling is negligible [5].

In order to extract the HMW phase, we perform measurements with the voltage only to get the polarizability phase ϕ E (V ) and with the current only to get the Zeeman phase ϕ B (I), and we then get ϕ EB (V, I):

ϕ EB (V, I) = ϕ E+B (V, I) -ϕ E (V ) -ϕ B (I) (4)
The diffraction phase ϕ d is very sensitive to the x-positions of the standing wave mirrors M i , with δϕ d /δx i ≈ 20 rad/µm for M 1 , M 3 or 40 rad/µm for M 2 . This sensitivity induces phase drifts, near 2 rad/hour, due to small displacements δx i of thermal origin. To minimize the effect of these drifts, we alternate voltagecurrent configurations over a 20 second-long fringe scan, during which the phase drift is linear and it only mod-ifies slightly the scan slope. We have used either a 4field configuration with the following (V, I) values (0, 0), (V, 0), (V, I), (0, I) or a 6-field configuration including E-reversal, by adding (-V, 0), (-V, I). Fits extract the characteristics of the individual fringe patterns (see fig. 2). If ϕ(V, I) is the fringe phase of the (V, I) configuration, the phases needed to evaluate equation 4 are given by the following differences, ϕ E (V ) = ϕ(V, 0) -ϕ(0, 0), ϕ B (I) = ϕ(0, I) -ϕ(0, 0) and ϕ E+B (V, I) = ϕ(V, I)ϕ(0, 0). The statistical uncertainty on ϕ EB (V, I), near 30 mrad for a 20 second-long scan, is reduced near 3 mrad by averaging about 100 scans. The phase shift thus deduced is still influenced by various stray phase shifts due to geometrical defects of the interaction region [5]. These systematic effects have been studied for a large set of (V, I) data points and an approximate analytical model was developed in order to evaluate their contributions to ϕ EB . As the stray phase shifts increase rapidly with I, we discuss here only the data collected with |I| ≤ 12 A, but the experiments carried with larger |I|-values have been useful to understand experimental defects. The dominant part of the stray phase shifts is an even function of I and we separate it from ϕ HMW which is odd with V and I by combining measurements with opposite I-values in ϕ f inal = [ϕ EB (V, I) -ϕ EB (V, -I)]/2. Similarly as for the Zeeman phase shifts, cancelations between hyperfine sublevels limit the contribution of the Aharonov-Casher effect to very small values in ϕ EB ; this contribution, always smaller than 3 mrad, was evaluated thanks to our model and subtracted from the data plotted in fig. 3. Our results agree with the expected linear dependence ϕ HMW ∝ V I, but the slope ϕ f inal (V, I)/(V I) = (-1.68 ± 0.07) × 10 -6 rad/(V.A) differs by 31% from the expected value, presumably due to a lack of accuracy of our analytical model [5].

The magnetic field direction changes over the interferometer, thus inducing a Berry's phase [START_REF] Berry | Topological phases belong to the group of geometric phases introduced by[END_REF]. The measured Berry's phase is the difference between the two arms and it is expected to be very small because of the magnetic field homogeneity. Moreover, it is canceled by our procedure because it has the same value in ϕ E+B (V, I) and in ϕ B (I). The more complex Berry's phase involving the electric and magnetic fields discussed in [28] also appears to be negligible [5].

In conclusion, we have performed the first experimental test of the He-McKellar-Wilkens topological phase by atom interferometry: this experiment has taken advantage of the high phase sensitivity of our interferometer and of its arm separation. Our measurement is not limited by the interferometer sensitivity but by systematic effects due to several small experimental defects. The slope of the HMW phase ϕ f inal (V, I) as a function of the V I product differs from the theoretical value and this difference is probably due to a lack of perfect understanding of the corrections due to field gradients and other experimental defects. These defects are small but they could be further reduced by a better design of the interaction region. Moreover, the stray phase shifts due to these defects are enhanced by the fact that the signal is an average over 8 hyperfine-Zeeman sublevels and optical pumping in a single F, m F sub-level should greatly reduce these systematic effects and improve the accuracy of the measurements. The HMW phase is expected to independent of the atom velocity v and we can test this property by tuning lithium velocity by changing the supersonic beam carrier gas [5]. This test is feasible and very interesting but we must first reduce the error bar in order to distinguish this behavior from the 1/v-dependence of dynamic phases. This experiment continues the development of new tools in quantum manipulation of atoms. For instance, the HMW effect could be used to build a coherent atom diode based on the non-reciprocal feature of topological phases. While existing atom diodes [29,30] use optical pumping and do not conserve the coherence of the atom wave, the HMW effect is coherent. In an interferometer as shown in fig. 1, with a HMW phase ϕ HMW = π/2 for left to right propagation and a diffraction phase ϕ d = -π/2, a wave packet would be fully transmitted from entrance A to exit C but a wave packet entering by C would be transmitted to B and not to A. Although very intriguing, such a behavior does not violate any fundamental law and cannot be used to build a Maxwell demon [START_REF] Maxwell | Theory of Heat[END_REF].

FIG. 1 :

 1 FIG. 1: (Color online) Schematic top views of the setup. Panel a: our atom interferometer, with two entrances A and B and two exits C and D (C is detected). An atomic beam (dotted blue lines) entering by A is diffracted by three quasiresonant laser standing waves produced by the mirrors Mi.The interaction region is placed where the distance between interferometer arms is largest, close to 100 µm. Panel b: the interaction region producing the electric and magnetic fields (not to scale). The interferometer arms (dotted blue lines) are separated by a septum, which is the common electrode of two plane capacitors producing opposite electric fields (high voltage electrodes in red; grounded electrodes in black). Two rectangular coils (brown rectangle) produce the magnetic field.

  FIG. 2: (Color online) Panel a: the atom interferometer signal is plotted as a function of M3 x-position, using the 4-field configuration procedure with E = 680 kV/m and B = 7.8 mT. In panels b-e, the signals corresponding to each field configuration, are plotted with their best fits. The mean detected flux I0 ≈ 47 × 10 3 atoms/s is constant. We use the no-field case as reference for visibility (V0 = (57.2 ± 0.6) %) and phase. In panel c, V (0, I) /V0 = (50.0 ± 1.2)% and ϕB (I) = -183 ± 25 mrad. In panel d, V (V, I) /V0 = (48.7 ± 0.2)% and ϕE+B (V, I) = -139 ± 24 mrad. In panel e, V (V, 0) /V0 = (102.4 ± 1.6)% and ϕE (V ) = 118 ± 18 mrad. ¿From these measurements, we get ϕEB (V, I) = (-74 ± 35) mrad. The average over about 100 similar scans produce phase measurements with a few mrad error bar.

FIG. 3 :

 3 FIG. 3: He-McKellar-Wilkens phase shift as a function of the voltage-current product: the phase ϕ f inal (V, I) is plotted as a function of the V I product. The data points (squares), represented with their statistical error bars, are in good agreement with a linear behavior ϕ f inal (V, I) ∝ V I, as shown by the best fit to the data (dotted line). The fit slope (-1.68 ± 0.07)10 -6 rad/V.A is 31% larger than the expected slope of the HMW phase (full line).

We thank the laboratory technical staff for their help and A. Cronin for fruitful discussions. The development of our atom interferometer owes much to A. Miffre and to M. Jacquey who also made the first plans of the HMW experiment. We thank CNRS INP, ANR (grants ANR-05-BLAN-0094 and ANR-11-BS04-016-01 HIPATI) and Région Midi-Pyrénées for support.