Laura Pometcu

Chloé Méjean

Ratiba Benzerga
email: r.benzerga@univ-rennes1.fr

Ala Sharaiha

Philippe Pouliguen

Claire Le Paven

On the choice of the dielectric characterization method for foam composite absorber material

Keywords: Dielectric properties, composites, organic compounds, Carbon fibers, Epoxy foams

Highlights

 Dielectric characterization of carbon fibers loaded epoxy composites has been done.

 Free space, waveguide and coaxial probe technics are used for the characterization.

 Free space is the adapted technic for the heterogeneous composite characterization.

 Simulation and measurement of the reflectivity of absorber porotype have been done.

INTRODUCTION

Microwave absorber materials are used extensively in anechoic chambers (AC) to emulate free-space conditions and to suppress unintended radiations of the in-situ electronic equipment [START_REF] Drake | Extension and verification of absorbing material effectiveness on reducing electromagnetic emissions[END_REF]. The choice of an absorber material for an AC use is based on its absorption performance, as well as its density and also its cost since a significant amount of materials is necessary to cover the entire surface of the AC. Different associations «polymer» + «carbon filler» are currently studied as absorbers, for example, resins loaded with carbon particles [START_REF] Chin | Development of the composite RAS (radar absorbing structure) for the X-band frequency range[END_REF], resins loaded with carbon nanotubes [START_REF] Saib | Carbon nanotube composites for broadband microwave absorbing materials[END_REF], polymer foams loaded with carbon nanotubes [START_REF] Yang | Novel Carbon Nanotube-Polystyrene Foam Composites for Electromagnetic Interference Shielding[END_REF] or resins loaded with carbon fibers [START_REF] Chu | Effects of diameter and hollow structure on the microwave absorption properties of short carbon fibers[END_REF]. The most used absorber material in AC is the polyurethane (PU) foam loaded with fine carbon particles [START_REF] Siepel | Microwave pyramidal absorber APM[END_REF]. Here, we propose to study the dielectric characteristics of an original composite which associates long carbon fibers with epoxy foam.

Several studies have been carried out on carbon fibers mixed with epoxy resin [START_REF] Zhao | Microwave absorbing properties of activated carbon-fiber felt screens (vertical-arranged carbon fibers)/epoxy resin composites[END_REF][START_REF] Paligová | Electromagnetic shielding of epoxy resin composites containing carbon fibers coated with polyaniline base[END_REF], but, up to now, the density of the resin has limited its use as an absorber in AC. Then, the advantage of the epoxy foam compared to bulk resin is obviously the low density of the produced material, appropriate for its application in AC. In addition, another advantage of the epoxy foam, compared to the PU foam, lies in its high mechanical rigidity [9] which allows a machining under complex form to optimize the absorption performances [START_REF] Pometcu | Straight wedge absorber geometry optimization for normal and oblique incidence[END_REF][START_REF] Dewitt | Electromagnetic scattering by pyramidal and wedge absorber[END_REF].

The characterization of absorbing materials usually shows a decrease of the electromagnetic signal in certain frequency ranges depending on the composition as well as the overall geometry of the measured material. Moreover, available dielectric characterization techniques often present constraints, such as the size of samples needed for the characterization. For example, free space techniques require materials with large dimensions, contrary to coplanar techniques which need very thin samples. On the other hand, the waveguide technique requires samples with precise dimensions that will decrease as the measurement frequency increases.

Another constraint is the frequency range of the measurement, sometimes limited, such as in the case of a resonant cavity for which only one frequency is explored, or in the case of a waveguide which works in a very short frequency range. The broadband techniques often need very large samples (several tens of cm 3 ) that cannot be obtained for all type of materials.

The purpose of the present article is to characterize by three different techniques an absorbing composite made of the association of epoxy foam and carbon fibers. The objective is to find the most suitable technique for this type of "heterogeneous" (i.e. composite) material.

Free space, X-band waveguide and dielectric probe measurement methods were used. The article is composed as follows: the first part briefly explains the elaboration of the composite samples, the second part details each of the three dielectric characterization techniques, the third part presents and discusses the results of the different measurements and the final part introduces the achievement and the measurement of a prototype, having a commercial geometry, in order to validate the dielectric properties obtained by one or more of the characterization methods.

ELABORATION OF CARBON FIBERS LOADED EPOXY FOAMS

The carbon fibers used in this study are 3 mm length and 7 μm diameter (provided by Apply Carbon SA). For the composite elaboration, we have used epoxy foam made from commercial epoxy resin and hardener from Sicomin [9]. The elaboration process of our material is described as follows. The chosen weight percentage of carbon fibers (here, between 0 wt. % and 1 wt.%) is added to the epoxy resin and mixed for a few minutes before the hardener agent is added. The latter induces the foaming process and the polymerization reaction of the resin (6 hours step). At last, thermal treatment is achieved during 6 hours at 60°C in order to complete the polymerization reaction and to fix the mechanical properties of the foam composite.

Finally, the samples are cut to the desired geometry required for the measurements. Here, three techniques, with samples having different sizes, have been used to determine the dielectric properties of the composites.

BASIC CONCEPTS OF THE CHARACTERIZATION METHODS

FREE SPACE TECHNIQUE

The free space method [START_REF] Brancaccio | A Free-space Method for Microwave Characterization of Materials in Aerospace Application[END_REF] is a nondestructive technique that can be used in a large frequency band. A schematic of the microwave measurement setup is given in Fig. 1. Two reflection coefficients (S11) of the material are measured, respectively, with and without a metallic ground plane behind the sample. The complex permittivity of the characterized material is extracted from the two measured S11 coefficients using the method of Fenner and al [START_REF] Fenner | A Comprehensive Analysis of Free-space and Guided-wave Techniques for Extracting the Permittivity and Permeability of Materials Using Reflection-only Measurements[END_REF].

For this technique, the dimensions of samples are 15 x 15 x 6.5 cm 3 . Two ultrawideband (UWB) antennas (ETS-Lindgren's Model 3115 Double-ridged Waveguide Horn) of size 24.4 cm x 15.9 cm x 27.9 cm (length x width x depth) are used. The antennas are placed in front of the characterized sample (in order to have a normal incidence of the electromagnetic wave) and are coupled to a Vector Network Analyzer. The distance between sample and antennas is 113 cm. The latter is calculated considering far field (Fraunhofer) conditions (Eq.1-

3) [START_REF] Yadava | Antenna and wave propagation[END_REF] at the lowest measured frequency.

𝑅 = 2 * 𝐷 2 𝜆 (𝐸𝑞. 1) 𝑅 ≫ 𝐷 (𝐸𝑞. 2) 𝑅 ≫ 𝜆 (𝐸𝑞. 3)
where R is the distance between sample and antennas, D is the antenna aperture diameter and 𝜆 the higher measured wavelength.

X-BAND WAVEGUIDE

The second technique used for the characterization of the epoxy composite material is a waveguide. This technique is very popular and widely used [15 -17] but it is narrowband and it requires a sample with a specific size depending on the frequency band of measurement (large samples for low frequencies and, inversely, very small samples for high frequencies).

Moreover, samples have to be cut to very precise dimensions in order to perfectly enter the slot and so to prevent the formation of any air voids between the sample and the walls of the waveguide. In this study, we have performed measurements with an X-band waveguide working between 8.2 GHz and 12.4 GHz (Fig. 2.). For this frequency range, the samples are smaller than those used for free space technique and must have a (theoretical) size of 22.86 x 10.16 x 9.7 mm 3 .

The method used to extract the complex permittivity of the material is the Nicolson-Ross-

Weir technique [START_REF] Nicolson | Measurement of the Intrisic Properties of Materials by Timedomain Techniques[END_REF][START_REF] Weir | Automatic Measurement of Complex Dielectric Constant and Permeability at Microwave Frequencies[END_REF] which uses the measured reflection (S11) and transmission (S21) coefficients.

OPEN ENDED COAXIAL PROBE

The third technique uses a dielectric coaxial probe 85070E from Keysight (Fig. 3.a). It is connected to an 8510C Keysight vector network analyzer. For this characterization, the sample surface state is very important. Indeed, the surface in contact with the dielectric probe should be as flat as possible to avoid any air gap which would induce an underestimation of the complex permittivity. Samples have a surface of at least 2 x 2 cm 2 , greater than the surface of the probe (Fig. 3.b). With the dielectric coaxial probe, the complex permittivity of the characterized sample ( = ε' -jε") is directly obtained; the dielectric loss tan is calculated as the ratio of the imaginary part (ε") upon the real part (ε') of the complex permittivity.

To minimize the measurement uncertainty, 8 measurements are performed for each sample, and the mean value is calculated. Standard deviations of 5% and 20% have been estimated from the 8 measurements, for permittivity and dielectric loss, respectively.

DIELECTRIC CHARACTERIZATION RESULTS

FREE SPACE CHARACTERIZATION

Photos of the composite samples used for the free space measurement are shown in Supplementary data 1. Different carbon fiber load percentages (between 0 wt% and 1 wt% in weight) were used. It should be noted that the grey coloration of samples is becoming darker as the percentage of carbon fibers increases.

Remind that, for this technique, two measurements of the reflection coefficient have to be done to extract the complex permittivity: the sample alone and the sample backed by a metallic plate. At least, 4 measurements have been done on each sample in order to check the reproducibility of the measurements. All the reflection coefficients are very close. For the measurement without metallic plate (Fig. 4.a) and for the unloaded sample, S11

values are below -20 dB on the entire studied frequency range. This signifies a low reflection for this sample at the interface air/sample. As the load rate of carbon fibers in the composite increases, the reflection coefficient increases; for example, the value of S11 parameter is around -8 dB at 10 GHz for the 1 wt.% carbon fibers loaded composite. Concurrently, the permittivity of the material increases (Fig. 4.c); for example, the permittivity of the material loaded with 1 wt.% is 2.03 at 10 GHz (unloaded: ' = 1.20). In fact, the increase of the reflection coefficient without metallic plate is due to an increasing mismatch of the relative impedances at the interface air/material as the permittivity of the composite increases under the effect of the carbon load. If the impedance characteristic of the composite is lower than 377 ohms (relative impedance of free space), the reflection at the interface air/material is high and the material is no longer a good absorber.

For the measurement with metallic plate (Fig. These results are very different from the ones obtained in free space. This is probably due to the dimensions of the samples used for the waveguide which are very small compared to the ones used for the free space measurements. In fact, the materials studied here have heterogeneous structure and composition by their nature of composites and foams. The porous nature of the epoxy foam (with millimetric porosities) and the addition of carbon fibers 3 mm length, make the composite heterogeneous at a millimetric scale, which is the scale of the waveguide measurement in the X-band. As the carbon fibers load increases, this heterogeneity increases as shown by the different sizes of porosities in Fig. 6. For this reason, it appears very difficult to characterize this type of material using the waveguide technique because the small characterized sample is not representative of the entire composite material. Permittivity, at 10 GHz, of the 1 wt.% loaded foam is ' = 1.76, smaller than those obtained by free space measurement (' = 2.03) and waveguide measurement (' = 2.99). Losses are also significantly lower than previously, for example, the measured value here is tan = 0.24 at 10 GHz for the 1 wt.% carbon load composite, while it was 1.39 and 1.06, respectively, for the free space and waveguide measurements. These discrepancies can be explained by the existence of an air gap (due to imperfect planar surface of sample) as well as millimetric dimension of porosities, both under-estimating ' and tan. Losses determined in free space are significantly higher than those measured by waveguide and coaxial probe. This can be explained by the fact that only a small volume is analyzed by the latter two techniques as emphasized by Fig. 8.b.c, too low to be representative of the entire foam sample with porosities and fibers in the millimetric scale (Fig. 8.d). In addition, as the surface of the foam is irregular, this inexorably leads to a certain roughness of the samples, which is known to greatly influence the characterization by the waveguide and coaxial probe methods [START_REF] Baker-Jarvis | NlST Technical Note 1536[END_REF][START_REF] Agarwal | Interaction of electromagnetic waves at rough dielectric surfaces[END_REF]. These two techniques are also very sensitive to the presence of any air gaps as pointed out by [START_REF] Baker-Jarvis | NlST Technical Note 1536[END_REF][START_REF] Chen | Microwave Electronics: Measurement and Materials Characterization[END_REF][START_REF] Ness | Air-Gap Effect in Rectangular Waveguide Containing a Lossy H-Plane Dielectric Slab[END_REF]. All these features will finally lead to an under-estimation of the measured dielectric properties [START_REF] Baker-Jarvis | NlST Technical Note 1536[END_REF][START_REF] Chen | Microwave Electronics: Measurement and Materials Characterization[END_REF][START_REF] Agilent | Basics of Measuring the Dielectric Properties of Materials[END_REF]. In the case of the coaxial probe, this is accentuated by the fact that the measurement is punctual, and therefore much localized. In fact, this technique is more suitable for liquid samples [START_REF] Agilent | Basics of Measuring the Dielectric Properties of Materials[END_REF][START_REF] Venkatesh | An overview of dielectric properties measuring techniques[END_REF], but it is often used for the characterization of solid samples for its simplicity of use [START_REF] Marsland | Dielectric measurements with an open-ended coaxial probe[END_REF][START_REF] Filali | Design and Calibration of a Large Open-Ended Coaxial Probe for the Measurement of the Dielectric Properties of Concrete[END_REF][START_REF] Atwater | Complex permittivities and dielectric relaxation of granular activated carbons at microwave frequencies between 0.2 and 26 GHz[END_REF].

OPEN ENDED COAXIAL PROBE

To summarize, the free space technique seems to be the most likely to access the intrinsic characteristics of loaded composite foams than the coaxial probe or waveguide techniques.

Moreover, in our case, the frequency range of measurement of the waveguide technique is very narrow, compared to the frequency range of targeted application of our absorbers. The achievement of a prototype absorber (i.e., at the actual scale) and its measurement in anechoic chamber will come to confirm it.

ELABORATION AND CHARACTERIZATION OF A CARBON FIBERS LOADED EPOXY FOAM PROTOTYPE

In order to validate the dielectric properties obtained with the different techniques, the simulation of the reflection coefficient of a pyramidal absorber was conducted using the dielectric characteristics obtained from each of the three techniques. These simulations, achieved with the CST Microwave Studio software, were compared to the measurement of a prototype in an anechoic chamber.

The chosen absorber geometry for a prototype is the one of the APM12 commercial absorber from SIEPEL [START_REF] Siepel | Microwave pyramidal absorber APM[END_REF]. It consists of a combination of square-based pyramids, of 9 cm high and 3.8 x 3.8 cm² base deposited on a 2.5 cm thick base (Fig. 9.a). The photo of the achieved prototype, which contains 64 pyramids, is presented in Fig. 9.b. For this prototype, the 0.5 wt.% loaded foam was chosen because it presents the best compromise to access a high absorption and prevent from too much reflection: high losses and a low permittivity, as well as a good homogeneity. Previous work [START_REF] Méjean | Electromagnetic absorber composite made of carbon fibers loaded epoxy foam for anechoic chamber application[END_REF] has emphasized the high absorption performance of this composition.

Fig. 10 presents the simulations of the reflection coefficient (S11) achieved with the dielectric characteristics obtained from the three techniques for the 0.5 wt.% carbon loaded foam composite. The simulation from coaxial probe predicts a low reflection (S11 < -40 dB) for frequencies above 9 GHz, whereas a rapid increase of the reflectivity is observed below. For example, the reflection coefficient reaches -10 dB at 5.5 GHz. Here, it must be remembered that the material must have a maximum threshold of reflectivity near -10 dB to be considered as a good absorber [START_REF]ANSI C63.5[END_REF]. The simulation obtained with the waveguide shows a reflection coefficient lower than -45 dB between 8. The crucial point is that the simulation results obtained with the dielectric characteristics extracted from the free space measurements are very close to the measurement of the prototype, especially for low frequencies. This confirms our deductions on the relevance of the measurement of heterogeneous materials, such as porous or composite materials, by using a free space technique. It is adapted because it needs a large sample, so representative of the heterogeneous structure of loaded foam composite. Waveguide and coaxial probe techniques are less appropriate, especially at low frequencies for the coaxial technique. Nevertheless, due to its high sample size, the free space technique may not be suitable for any material, especially in routine tests.

CONCLUSION

This paper presents the dielectric characterization by three techniques of a new material for electromagnetic absorption in anechoic chambers. The composite, made of carbon fiber loaded epoxy foam, presents a heterogeneous composition and structure due to the presence of porosities and long fibers. It was characterized by free space, waveguide and coaxial probe techniques in order to extract its dielectric properties: permittivity and dielectric loss.

Measurements show very different dielectric characteristics and led us to conclude that free space is the more adapted technique for the characterization of a heterogeneous structure such as foam composites. A 0.5 wt.% carbon fiber loaded material, which presents the best compromise of low permittivity, high loss and best homogeneity, was used to achieve an absorber prototype. The simulation of its reflection coefficient performed with the dielectric characteristics extracted with the free space technique is very close to the measured one. This

List of

Fig. 4 .

 4 Fig. 4.a and 4.b present the evolution of the reflection parameter (S11) as function of

  Fig. 4.b). This definitively validates the method of extraction of the complex permittivity.

Fig. 7

 7 Fig. 7 presents the permittivity and dielectric loss obtained by the open ended coaxial probe

4. 4 .

 4 DISCUSSIONThree techniques have been used to characterize absorbing composites. They present different dielectric characteristic in terms of permittivity and dielectric loss, as summarized in Supplementary data 4, for the 0.5 wt.% carbon fibers loaded composite.

2 and 12 . 4

 124 GHz. The simulation made with the free space shows a reflection coefficient less than -40 dB from 3.5 to 18 GHz; below, it increases but remains inferior to -10 dB. Fig. 10 also presents the measured reflection coefficient of the prototype. It shows a low reflection coefficient, lower than -35 dB, in the frequency range between 4 and 18 GHz. The reflectivity increases for lowest frequencies but it is still below the limited value of -10 dB.

Fig. 1 .

 1 Fig. 1. (a) Schematic system of measurement and (b) Anechoic chamber of Institute of

Fig. 2 .

 2 Fig. 2. X-band waveguide measurement setup.

Fig. 3 .

 3 Fig. 3. (a) Photo of the Keysight dielectric coaxial probe (b) configuration of the measurement

Fig. 4 .

 4 Fig. 4. (a) Reflection coefficient for the material alone, (b) Reflection coefficient for the

Fig. 5 .

 5 Fig. 5. (a) Reflection coefficients, (b) tranmission coefficients from waveguide, (c) Permittivity

Fig. 6 .

 6 Fig. 6. Representation of the air holes inside the carbon fiber (CF) loaded composite samples.

Fig. 7 .

 7 Fig. 7. (a) Permittivity and (b) tan obtained using the dielctric coaxial probe measurement for

Fig. 8 .

 8 Fig. 8. Needed samples for the caracterization techniques in: (a) free space (b) X-band

Fig. 9 .

 9 Fig. 9. (a) The pyramid geometry used for simulations and (b) The photo of the achieved

Fig. 10 .

 10 Fig. 10. Simulations and measurement results of the pyramidal absorber prototype made of 0.5

Figure. 1 Fig. 1 .Figure. 2 Fig. 2 .Fig. 3 .Fig. 4 .

 112234 Figure. 1

Figure. 5 Fig. 5 .

 55 Figure. 5

Fig. 6 .

 6 Figure. 6

Figure. 7 Fig. 7 .

 77 Figure. 7

Figure. 8 Fig. 8 .

 88 Figure. 8

Fig. 9 .

 9 Figure. 9

Figure. 10 Fig. 10 .

 1010 Figure. 10

result confirms that the free space technique is the adapted technique for heterogeneous composite characterization, especially for low frequencies (< 5 GHz).

ACKNOWLEDGMENT

The authors would like to thank the Direction Générale de l'Armement (DGA), France for their financial support.


Samir Adly
email: adly@thera@unilim.fr

Emil Ernst
email: emil.ernst@univ.u-3mrs.fr

Michel Théra

Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions

Keywords: barrier cone, convex analysis, Fenchel conjugate, recession analysis, semi-coercive functional, support functional, variational inequality

We characterize the class of those closed convex sets which have a barrier cone with a nonempty interior. As a consequence, we describe the set of those proper extended-real-valued functionals for which the domain of their Fenchel conjugate has a nonempty interior. As an application, we study the stability of the solution set of a semi-coercive variational inequality.

Introduction

Closed convex sets for which the barrier cone has a nonempty interior, as well as proper extended-real-valued functionals for which the domain of their Fenchel conjugate is nonempty, are mathematical objects currently encountered in various areas of optimization theory and variational analysis (see for instance the class of well-behaved functions introduced by Auslender and Crouzeix in [START_REF] Auslender | Well-behaved asymptotical convex functions[END_REF]).

This article provides, in the framework of general reflexive Banach spaces, geometric and analytical characterizations for this type of sets and functionals and extends in this way earlier partial results obtained in [START_REF] Adly | Stability of non-coercive variational inequalities[END_REF] in the context of separable Banach spaces.

Section 2 is dedicated to the study of the class of closed convex sets which have a barrier cone with a nonempty interior. Theorem 2.1 states that this class is identical to the class of closed convex sets having a geometrical property called well-positionedness. An analytical characterization of well-positioned sets -Proposition 2.1 -is equally obtained.

Similar results are deduced in Section 3 for functionals. A geometric characterization, valid for all the proper extended-real-valued functionals, and an analytical one, valid for convex lower semi-continuous functionals constitute the main results of the section.

The article is completed by an application of Theorems 3.1 and 3.2 to the theory of variational inequalities. More precisely, the above mentioned results allow us to describe the class of all semi-coercive variational inequalities which have a nonempty solution set for any sufficiently small uniform perturbations of their data.

Well-positioned Sets

Unless otherwise stated we suppose that X is a reflexive Banach space and we denote by • • the duality pairing between X and its continuous dual X * , by • and • * the norm and the dual norm on X and X * , respectively, and by j X * → X the duality mapping given by f j f = f 2 * j f = f * , (see for example [START_REF] Zeidler | Nonlinear Functional Analysis and its Applications II[END_REF]). Due to a well-known renorming Theorem of Troyanski (see e.g. [START_REF] Dieste1 | Geometry of Banach Spaces[END_REF]) we can (and will) assume that the norms on X and X * are locally uniformly rotund. This implies that the duality mapping j is single-valued and norm-to-norm continuous. As standard, co A, co A span A are the convex, the closed convex hull and the closed linear span of the set A ⊂ X. Finally, we use the symbols X X * and ' ' for the open unit balls in X and X * and the weak convergence, respectively. Following the usual terminology used in convex analysis (see Rockafellar [START_REF] Rockafellar | Convex Analysis[END_REF] as a reference book), we recall that the recession cone of a closed convex C is the closed convex cone C defined by

C = v ∈ X ∀ > 0 x 0 ∈ C x 0 + v ∈ C If X → ∪+
is an extended real-valued function, dom is the set of all x ∈ X for which

x is finite, and we say that is proper if dom = ∅. When is a proper lower semi-continuous convex function, the recession function of is the proper lower semi-continuous convex function whose epigraph is the recession cone of the epigraph of , i.e., epi = epi . Equivalently

x = lim t→+ x 0 +tx t where x 0 is any element such that x 0 is finite. We denote by Ker = u ∈ X u = 0 , and we define the barrier cone of a set C in X as the set of all linear continuous functionals bounded from above on C, i.e.,

C = g ∈ X * sup x∈C g x < +
Finally, C = g ∈ X * g x 0 ∀x ∈ X will denote the negative polar cone of the convex cone C which is C ⊥ when C is a closed subspace. As well-known, C = C and by the Bipolar Theorem

C = C . A convex set C is called linearly bounded if C = 0 .
DEFINITION 2.1. We say that a set C ⊂ X is well-positioned, if there exists x 0 ∈ X and g ∈ X * such that:

g x -x 0 x -x 0 ∀x ∈ C (2.1)
Equivalently, if we denote by C g = x ∈ X g x

x , C is well-positioned if and only if C ⊂ x 0 +C g for some x 0 ∈ X and g ∈ X * . Remark 2.1. If a nonempty convex set C is well-positioned, then Int C is nonempty. However, the converse fails to be true in infinite dimension. Indeed, by definition we know that C ⊂ u 0 +C g for some u 0 ∈ X and g ∈ X * . Hence,

C ⊂ u ∈ X g u u = C g This implies that C g ⊂ C
and, as -g + X * ⊂ C g we deduce that Int C = ∅. Finally, remark that every unbounded linearly bounded closed convex set provides a counterexample for the converse.

LEMMA 2.1. Let C be a closed convex set containing no lines, and y ∈ C. For every R > 0, let us define

M C y R = x -y x -y x ∈ C x -y R
The following two facts are equivalent:

(i) C is well-positioned; (ii) there is R > 0 such that 0 ∈ co M C y R . Proof of Lemma 2 1 i ⇒ ii .
Let C be a well-positioned closed convex set, y an element of C, and x 0 and g two elements of X and X * such that

g x -x 0 x -x 0 ∀x ∈ C (2.2)
The first part of the proof consists of proving that if R satisfies R > 2 1+ g * y -x 0 then 0 does not belong to co M C y R . Indeed, for every x ∈ C such that x -y > 2 1+ g * y -x 0 , relation (2.2) implies that g

x -y

x -y

x -x 0 x -y -g y -x 0 x -y 1- y -x 0 x -y -g * y -x 0 x -y = 1-1+ g * y -x 0 x -y 1 2
This yields,

x ∈ co M C y R
⇒ g x 1 2 and therefore 0 co M C y R . ii ⇒ i Now, let us consider a closed convex set C and fix y ∈ C such that 0 co M C y R for some R > 0. We claim that if z is the element of minimal norm in the set coM C y R and if h = j -1 z , then 3h

z 2 x -y - 2R z z x -y - 2R z z ∀x ∈ C (2.3)
Indeed, for every x ∈ C such that x -y R, the vector x-y x-y belongs to co M C y R , and by the definition of z we have, h x-y x-y z 2 . Accordingly, for each x ∈ C such that x -y R, we have

3h z 2 x -y - 2R z z 3 x -y +3 2R z 3 x -y + 2R z z x -y - 2R z z If x -y R, then 3h z 2 x -y - 3 z 2 h * x -y - 3R z Hence, 3h z 2 x -y - 2R z z - 3R z + 6R z = 3R z 3R = R+2R x -y + 2R z z x -y - 2R z z ∀x ∈ y +R X
The two previous relations prove (2.3). Setting g = 3h z 2 and x 0 = y -2R z z in (2.3) we obtain (2.1), completing the proof of Lemma 2.1.

The following result provides an analytical definition of well-positioned sets.

PROPOSITION 2.1. A nonempty closed convex set C of a reflexive Banach space

X is well-positioned if and only if the following two assumptions are satisfied:

(a) C contains no lines; (b) ∃ x n n∈ ⊂ C x n → + , such that x n x n 0.
Remark 2.2. Proposition 2.1 subsumes the fact that when X is finite dimensional, a nonempty closed convex set is well-positioned if and only if C is pointed, i.e., C ∩-C = 0 . In particular, every compact and convex set is well-positioned in n .

Proof of Proposition 2.1. Let us first prove that every well-positioned convex closed set C satisfies assumptions a and b . As C is well-positioned, there are x 0 in X and g in X * such that C ⊂ x 0 +C g . By the way of obtaining a contradiction, suppose that C contains at least a line, that is, there are x 1 and v in X v = 1, such that

x 1 + v ∈ C ∀ ∈ Accordingly, g v + g x 1 -x 0 = g x 1 + v -x 0 x 1 + v -x 0 -x 1 -x 0 ∀ ∈
For = n and = -n, we obtain respectively

n g v + g x 1 -x 0 n-x 1 -x 0 and -n g v + g x 1 -x 0 n-x 1 -x 0
Summing up the two above relations, we deduce that

g x 1 -x 0 + x 1 -x 0 n ∀n ∈ *
a contradiction establishing assumption (a). Now, let us suppose that the assumption (b) fails, i.e., there exists a sequence x n n∈ * such that x n → and x n / x n 0. Since C is well-positioned, we obtain

0 = lim n→ g x n / x n = lim n→ g x n -x 0 x n lim n→ x n -x 0 x n = 1
a contradiction. Hence, assumption (b) is satisfied, establishing the fact that every well-positioned closed convex set fulfills assumptions (a) and (b).

Conversely, let us consider a closed convex set C satisfying relations (a) and (b), and suppose that C is not well-positioned. Pick y ∈ C; Lemma 2.1 implies that 0 ∈ co M C y n for every n ∈ * . For every integer n, select U n , a finite or countable subset of M C y n such that 0 ∈ co U n , and thus R n ⊂ C\ y +n X where R n is such that

U n = x x x ∈ R n . Let X 1 = span y ∪ n∈ * R n and K = C ∩X 1 .
The definitions of X 1 and K imply that, for every n ∈ * , U n ⊆ M K y n , and therefore

0 ∈ co U n ⊆ co M K y n As M K y R 1 ⊆ M K y R 2 whenever R 1 R 2 , the previous relation yields that 0 ∈ co M K y R
for every R > 0; from Lemma 2.1 it follows that the closed convex set K = C ∩X 1 is not well-positioned. As X 1 is separable, and K contains no lines (being a subset of C which, by virtue of assumption (a), contains no lines), a well-known result of Klee ([9]) implies that there is f ∈ K such that

f w < 0 ∀w ∈ K w = 0 (2.4)
Let us prove the following technical result.

LEMMA 2.2. Let C be a closed convex set of a reflexive Banach space X. Suppose that for some g ∈ X * \ 0 and t ∈ such that t < sup x∈C g x , the set

C g t = x ∈ C g x t is bounded. Then, the set C is well-positioned.
Proof of Lemma 2.2. Let x be an element of C such that t < g x . Since C g t is bounded, there is r >0 such that C g t ⊂ r X , and thus C g t ⊂ x +2r X . Accordingly, for every x ∈ C\ x +2r X , we have g x < t and consequently,

0 < t -g x g x -x < 1
Thus,

z x = t -g x g x -x x + g x -t g x -x
x is a convex combination of x and x and, accordingly, an element of C. Moreover, g z x = t and therefore z x necessarily belongs to C g t . Consequently,

0 < x -z x 2r that is g x -t g x -x x -x 2r ∀x ∈ C\ x +2r X (2.5) Relation (2.5) implies that g x -x x -x t -g x 2r < 0 ∀x ∈ C x -x 2r
Consequently, 0 co M C x 2r , and therefore (see Lemma 2.1) C is well-positioned. The proof of Lemma 2.2 is thereby completed.

As K is not well-positioned, from Lemma 2.2 it follows that the set

K f t = x ∈ K t f x
is unbounded for every t < sup x∈K f x . Accordingly, there is a sequence n n∈ * ⊂ K f t such that n → . Let w be a weak cluster point of the bounded sequence n n n∈ * . As n → , we have w ∈ K f t , and w = 0 by virtue of assumption (b). For every s > 0 y +sw ∈ K f t , so

t f y +sw sup x∈K f x ∀s > 0 that is t -f y s f w sup x∈K f x -f y s ∀s > 0
relation which implies that f w = 0. The contradiction between the previous equality and relation (2.4) completes the proof of Proposition 2.1.

We are now in position to state the main property of the well-positioned sets.

THEOREM 2.1. Let C be a nonempty subset of a reflexive Banach space X. The following two conditions are equivalent:

(1) The barrier cone of C has a nonempty interior;

(2) C is well-positioned.

Moreover, if Int C = ∅, then

Int C = Int C (2.6) Proof of Theorem 2.1. 2 ⇒ 1 we prove that ∅ = Int C ⊆ C .
By Remark 2.1, pick g ∈ Int C . In order to prove that g ∈ C , we first need to establish a technical result. LEMMA 2.3. Suppose C is well-positioned. Then, for every g ∈ Int C , there are R g g > 0 such that

g x R g -g x ∀x ∈ C (2.7) Proof of Lemma 2.3. Let us denote by C f = x ∈ X f x
x . Let x 0 and f such that C ⊂ x 0 +C f , and take g such that g +2 f * g X * ⊂ C ( g exists since g ∈ Int C ). For the purpose of obtaining a contradiction, suppose that there exists a sequence x n n∈ * in C such that g x n ng x n (2.8)

Noticing that

g * + g x n g x n + g x n n
we deduce that lim n→ x n = + . We may therefore define t n = 1

x n

. Let w be a weak cluster point of the sequence t n x n n∈ . Multiplying relation (2.8) by t n and passing to the limit as n → + , we obtain g w lim inf n→ nt ngg (2.9)

Since g +2 f * g X * ⊂ C and w ∈ C , we derive

g +h w 0 for all h ∈ X * h * 2 f * g This yields g w -2 f * g w (2.10)
We combine (2.9) and (2.10) to obtain

-g g w -2 f * g w that is 1 2 f * w (2.11)
On the other hand, when multiplied with t n , relation

f x n -x 0 x n -x 0 x n -x 0 yields f t n x n -t n x 0 1-t n x 0
Passing to the limit as n → + we obtain

f
f u R f -f u ∀u ∈ C (2.15)

Well-positioned Functionals

When endowed with the standard norm

• X× X × → + x X× = x 2 + 2 ∀ x ∈ X ×
the linear space X × becomes a reflexive Banach space whose continuous dual is X * × endowed with the standard norm:

• X * × X * × → + f X * × = f 2 * + 2 ∀ f ∈ X * ×
the duality pairing is given by

f x X * × X× = f x + ∀ f ∈ X * × x ∈ X ×
and J X * × → X × , given by

J f = j f ∀ f ∈ X * ×
is the duality mapping between X * × and X × . Given an extended-real valued function X → ∪ + , recall that the Fenchel conjugate of is the function

* X * → ∪ + given by * f = sup x∈X f x -x
Obviously, the domain of * is connected to the barrier cone of the epigraph of through the following equivalence

g ∈ Int dom * ⇐⇒ g -1 ∈ epi
This yields, dom * × -1 is the intersection of the barrier cone of epi with the hyperplane

X * × -1 of X * × dom * × -1 = epi ∩ X * × -1 (3.16)
Standard convex analysis techniques allow us to prove that the domain of the Fenchel conjugate is nonempty if and only if the barrier cone of the epigraph has a nonempty interior.

DEFINITION 3.1. We say that a proper convex lower semicontinuous functional

X → ∪ + is well-positioned if the epigraph of , epi = x ∈ X × x is a well-positioned subset of X × .
Theorem 2.1 may now be rephrased into the following geometrical characterization of functionals for which the domain of their Fenchel conjugate has a non void interior.

L epi = epi ∩ -epi = epi ∩ -epi as is bounded from below, 0. Hence, epi ∩ -epi = Ker × 0 ∩ -Ker × 0 Consequently, L epi = L Ker × 0
Hence, L epi = 0 if and only if L Ker = 0 . Thus (a) is equivalent to a * . In order to prove the equivalence between (b) and b * , let us first consider a sequence x n n n∈ * ∈ epi such that

x n n X× → and x n n x n n X× 0 As lim n→ f x n x n n X× = lim n→ f 0 x n n x n n X× X * × X× = 0 for every f ∈ X * ,
> 0 such that f -1 u X * × X× R f -f u X× ∀u ∈ dom u
As u X× u , by setting = u in the previous inequality we derive

f ∈ Int dom * ⇒ ∃ f R f > 0 s t f u + f u R f R f + u ∀u ∈ X (3.26)
We have thus proved the following result.

Stability of the Existence of the Solution for Semi-coercive Variational Inequalities

In various problems in optimization and in variational analysis (see the case of well-behaved functionals, for instance), it is a natural question to ask under which conditions the interior of the domain of the Fenchel conjugate of a given functional is nonempty. One of the problem leading to such conditions concerns (see [START_REF] Adly | Stability of non-coercive variational inequalities[END_REF] and [START_REF] Adly | Recession mappings and non-coercive variational inequality[END_REF]) the stability of the solution set of a semi-coercive variational inequality

VI A f K : find u ∈ K ∩ dom such that Au-f v-u + v -u 0 ∀v ∈ K (4.27)
where K is a closed convex set in a reflexive Banach space X, f is a continuous linear functional on X, X → ∪ + is a proper lower semi-continuous and convex functional that we assume to be bounded from below, K ∩ dom = ∅, where dom = x ∈ X

x < + , and A is a semi-coercive operator from X to X * , that is

Av -Au v -u dist U v -u 2 ∀u v ∈ X A x +u = A x ∀x ∈ X and u ∈ U and A X ⊆ U ⊥ (4.28)
for some positive constant and some closed subspace U of X. We suppose furthermore that A is pseudomonotone in the sense of Brezis ([6], p. 142).

In other words, we characterize all data A f K for which there is some > 0 such that the variational inequality VI A f K has solutions for every instance involving a bounded and semi-coercive operator A , a linear continuous functional f , a proper lower semi-continuous and convex functional that is bounded from below, and a closed convex set K such that K ∩ dom = ∅, and

A x -A x * < ∀x ∈ X f -f * < K ⊂ K + X and K ⊂ K + X x - x x + ∀x ∈ X
In this framework, it was proved ([3] Proposition 3.1) that a sufficient and necessary condition ensuring the uniform stability of the solution set of the given variational inequality is that f belongs to the interior of the domain of the Fenchel conjugate of an energy-like functional:

Int R A K = Int dom * where R A K = f ∈ X * V I A f K has at least a solution
is the resolvent set, and x = dist U x 2 +I K x + x ∀x ∈ X I K denoting the indicator function of K, i.e., I K x = 0 if x ∈ K and + else. Theorem 3.1 may now be used to obtain the following analytical characterization of the stability of the solution set.

The variational inequality V I A f K has solutions for every sufficiently small uniform perturbations A f K , if and only if the following three conditions hold: (i) The set U ∩K ∩Ker contains no lines;

(ii) There is no sequence x n n∈ ∈ K such that

x n x n 0 and dist U x n 2 + x n x n → 0 and (iii) f u < u ∀u ∈ K ∩U u = 0.

From Theorem 3.2, it follows ([3], Corollary 5.1) that the stability of the existence of a solution is ensured if and only if the functional -f is coercive.

This result relates the stability of the solution of a variational inequality and the coerciveness of an associated energy-type functional.

We conclude by noticing that the stable states for a semi-coercive inequality are precisely those which, due to the coercivity of the associated energy, are closer to the coercive case. It is our opinion that this link between stability and coercivity (already remarked in other contexts, see [START_REF] Adly | A characterization of convex and semicoercive functionals[END_REF] for example) is not casual, being rather a very general feature.

PROPOSITION 3 . 1 .

 31 Let be a proper lower semicontinuous convex function on a reflexive Banach space. Then,Int dom * = ∅ ⇐⇒ is well-positionedIn the case of proper lower semi-continuous convex extended-real-valued functionals which are bounded from below, Proposition 2.1 provides an analytical characterization of the well-positionedness which is easier to use.

THEOREM 3 . 1 .

 31 A proper lower semi-continuous convex functional defined on a reflexive Banach space X is well-positioned if and only if the two following assumptions hold: (a) Ker contains no lines; (b) ∃ x n n∈ * Cdom x n → + , such that x n Proof of Theorem 3 1. By virtue of Proposition 2.1, we have only to prove that statement (a) is equivalent to a * epi contains no lines, and that statement (b) is equivalent to b * ∃ x n n ∈ epi x n n X× → , such that x n n x n n X×0. Given a closed convex set C, let L C denote the maximal closed linear space contained in C, i.e., L C = C ∩ -C For C = epi , we have

  By contradiction, suppose that Int C = ∅ and C fails to be well-positioned. Pick g in Int C and t ∈ such that t < g x for some x ∈ C. By Lemma 2.2, the set C g t = x ∈ C t g x is unbounded. Hence, by the Banach-Steinhaus Theorem, there exists h ∈ X * and x n ∈ C g t such that:

	h x n n ∀n ∈
	For every fixed > 0, we have
	g + h x n t + n
	Therefore,
	g + h	C	∀ > 0
	Hence,
	w 1		(2.12)
	Combining relations (2.11) and (2.12) gives
	1 2	f * w	f w 1
	a contradiction, and the proof of Lemma 2.3 is achieved.

Inequality (2.7) implies that for all x ∈ C we have g x R g , which yields g ∈ C , and therefore

Int C ⊂ C (2.13)

As a result, Int C = ∅, and the first part of the proof is completed.

Now, let us prove that 1 ⇒ 2 :

g Int C

and the proof of the equivalence between (

1

) and (

2

) of Theorem 2.1 is thereby completed. In order to prove equality (2.6), let us remark that C ⊆ C , so Int C ⊆ Int C (2.14) for every set C. If Int C = ∅, then C is well-positioned, so relation (2.13) holds; relation (2.6) follows from relations (2.14) and (2.13). Theorem 2.1 and Lemma 2.3 have the following immediate consequence. COROLLARY 2.1. For every f ∈ Int C , there are R f f > 0 such that

  fulfills assumption b * , the sequence x n n∈ * satisfies (b); consequently, (b) implies (b * ). Let now x n n∈ * ∈ dom such that x n → + , fulfills b * provided that the sequence x n n∈ * satisfies (b). Accordingly, b * implies (b), and the proof of Theorem 3.1 is completed.

	Hence, Consequently,
	lim n→ x n x n	x n x n n X× x n x n X×	= 1 0	(3.19) (3.25)
	Since x n n X× → + , it follows from (3.19) that lim n→ x n = + from (3.17) and (3.19) we obtain Relations (3.23) and (3.25) show that the sequence x n x n n∈ Let f ∈ Int . By virtue of Corollary 2.1, we deduce the existence of two (3.20)
	x n x n constants R f and f = x n n X× x n	x n n X× x n	1 0 = 0	(3.21)
	The functional		is bounded from below, so there is k ∈ such that
	k			x n		n
	Finally, from (3.18) and (3.21) we get
	0 = lim n→	k x n	lim n→	x n x n	lim n→	n x n	= 0	(3.22)
	Relations (3.20), (3.21) and (3.22) prove that whenever the sequence
	x n n n∈ x n x n 0 and x n x n	→ 0. As x n	x n X×	x n , it follows that
	lim n→	x n		x n X× = +	(3.23)
	as x n x n	→ 0, we derive
	lim n→		x n		x n x n X×	= 1	(3.24)
	Let f		∈ X * × ; from (3.23) and (3.24) it follows that
	it follows that x n x n x n x n X× x n n X× lim n→ f x n 0 = lim n→ f x n x n x n	x n x n X×	+	(3.17)
	On the other side, lim n→ n x n n X× + lim n→	= lim n→ x n x n	0 1 x n x n x n X× x n n x n n X× X * × X× = 0 1+ •0•1 = 0 = 0	(3.18)

* *


Inertial Gradient-Like Dynamical System

Controlled by a Stabilizing Term1

A. CABOT 2Abstract. Let H be a real Hilbert space and let F: H fi R be a C 1 function that we wish to minimize. For any potential U: H fi R and any control function e: R + fi R + which tends to zero as t fi +O, we study the asymptotic behavior of the trajectories of the following dissipative system:

(S) € x x(t) + g _ x x(t) + rF(x(t)) + e(t)rU(x(t)) = 0, g > 0:

The (S) system can be viewed as a classical heavy ball with friction equation (Refs. 1-2) plus the control term e(t)rU(x(t)). If F is convex and e(t) tends to zero fast enough, each trajectory of (S) converges weakly to some element of argmin F. This is a generalization of the Alvarez theorem (Ref. [START_REF] Alvarez | On the Minimizing Property of a Second-Order Dissipative System in Hilbert Space[END_REF]. On the other hand, assuming that e is a slow control and that F and U are convex, the (S) trajectories tend to minimize U over argmin F when t fi +O. This asymptotic selection property generalizes a result due to Attouch and Czarnecki (Ref. 3) in the case where U(x) = |x| 2 =2. A large part of our results are stated for the following wider class of systems:

(GS) € x x(t) + g _ x x(t) + r x Y(t, x(t)) = 0, where Y: R + • H fi R is a C 1 function.

Key Words. Dissipative dynamical systems, nonlinear oscillators, optimization, convex minimization, heavy ball with friction.

Introduction

Let H be a real Hilbert space, with scalar product and corresponding norm respectively denoted by AE . , . ae and | . |. Let F: H fi R be a given C 1 realvalued function, called the potential function. In order to reach the minima of F, a powerful method consists in following the trajectories of a dissipative dynamical gradient-like system as tfi O.

Let SÌH • R be the manifold defined by S := Graph(F) and let us consider the motion of a material point with unit mass, subjected to stay on S and which moves under the action of the gravity force, the reaction force, and the friction force (g > 0 is the friction parameter). A simple application of the fundamental principle of dynamics and a reasonable approximation (Ref. [START_REF] Cabot | Motion with Friction of a Heavy Particle on a Manifold: Applications to Optimization[END_REF]) lead to the so-called ''heavy ball with friction'' system (Refs. 1-2) (HBF) €

x x(t) + g _ x x(t) + rF(x(t)) = 0, t $ 0:

The (HBF) system is dissipative and can be studied in the classical framework of the theory of dissipative dynamical systems [see for example, Hale (Ref. [START_REF] Hale | Asymptotic Behavior of Dissipative Systems[END_REF]) and Haraux (Ref. 6)]. It enjoys most of the nice properties of the steepest descent method. To quote only one of them, when F is convex, the Brezis-Bruck Theorem (Refs. [START_REF] Bre ´zis | Asymptotic Behavior of Some Evolution Systems, Nonlinear Evolution Equations[END_REF][START_REF] Bruck | Asymptotic Convergence of Nonlinear Contraction Semigroups in Hilbert Space[END_REF] has been extended by Alvarez to (HBF).

The main interest of (HBF) lies in the fact that it has more exploration properties than a first-order in time dynamical system. Coming back to the mechanical interpretation, assume that the material point is subjected to extra forces deriving from a time-dependent potential of the form e(t)U, where U: H fiR is a C 1 function and e: R + fi R + tends to zero as t fi +O. In such a situation, (HBF) turns into (S) €

x x(t) + g _ x x(t) + rF(x(t)) + e(t)rU(x(t)) = 0:

In this paper, we are concerned mainly with the asymptotic behavior of the (S) trajectories. The function e acts as a control on the trajectories of (S). When e tends to zero fast enough, more precisely if

ð +O 0 e(t)dt<+O,
the study of (S) is quite similar to that of (HBF), which corresponds to the noncontrolled case (e " 0). In particular, one can extend the Alvarez theorem (Ref. expresses that e(t) tends to zero slowly enough to allow the Tikhonov regularization term e(t)x(t) to be effective asymptotically.

In another direction, Cabot and Czarnecki consider in Ref. 10 a particular case of the (S) system where

H = H 2 , F(x, y) = f(x) + f( y), U(x, y) = V (x -y), thus leading to € x x(t) + g _ x x(t) + rf(x(t)) + e(t)rV (x -y)(t) = 0, € y y(t) + g _ y y(t) + rf( y(t)) -e(t)rV (x -y)(t) = 0:
They focus their attention on the case where the potential V modelizes a repulsion force. Such a potential allows a better exploration of the minima of f than a (simple) (HBF) method. In Ref. plays once more an essential role in order to make the repulsion efficient.

Coming back to the (S) system and assuming that e is a slow control [i.e., Ð +O 0 e(t)dt = +O], we prove in this paper a general asymptotic selection result: under convex conditions on F and U, the trajectories of (S) tend to minimize the function U over the set argmin F when t fi +O. This last property shows the potential interest of the (S) system in convex constrained optimization.

The study of (S) can be placed in a more general setting by introducing the function Y(t, x) = F(x) + e(t)U(x); then, the system (S) can be rewritten as

(GS) € x x(t) + g _ x x(t) + r x Y(t, x(t)) = 0,
where G stands for ''generalized''. This system is very interesting from a theoretical point of view in the sense that it recovers a large number of situations. A first-order in time version of (GS) has been studied by Attouch and Cominetti (Ref. 11), who have coupled approximation methods with the steepest descent system. When dealing with (GS), one can define for each fixed t the default of convexity w (t) of the map Y(t, . ). This quantity can play partially the part of the term e(t) of (S) in the study of (GS). In this paper, we will state our first convergence results in the frame of (GS) and then, when necessary, we will particularize to (S). The paper is organized as follows. In Section 2, we develop the mechanical interpretation of the (S) system. In Section 3, we establish general existence properties for (S) and (GS). In Section 4, we give our first convergence results in the case of a fast control (i.e., tending to zero sufficiently fastly). Section 5 is devoted to the asymptotic behavior of the (S) trajectories when the control e is slow, giving rise to the minimization of U over the set argmin F. The proofs of the main results of the paper are postponed to Section 6.

(S) and (GS) Systems in Mechanics and Optimization

(HBF) € x x(t) + g _ x x(t) + rf(x(t)) = 0, t $ 0,
where g >0 is a positive damping parameter. Let us consider now N $ 2 heavy balls coupled by an interaction force. We assume that the ith oscillator exerts on the jth one a force f ij equal to

f ij = -e(t)rV (x j -x i ),
where V: H fi R is a C 1 function such that rV (x) = -rV (x), for all x ˛H, so that f ij =f ji , and e: R + fiR is a time-dependent function. Then, the trajectories x i ( . ) 1#i#N satisfy the relations

€ x x 1 (t) +g _ x x 1 (t) +rf(x 1 (t)) +e(t) å j " 1 rV (x 1 (t) -x j (t)) = 0, . . . . . . . . . . . . . . . € x x i (t) +g _ x x i (t) +rf(x i (t)) +e(t) å j " i rV (x i (t) -x j (t)) = 0, . . . . . . . . . . . . . . . € x x N (t) +g _ x x N (t) +rf(x N (t)) +e(t) å j " N rV (x N (t) -x j (t)) = 0:
Setting

H = H N , x = (x 1 , . . ., x N ) ˛H, F(x) = å N i=1 f(x i ), U(x) = å N i, j=1 i<j V (x i -x j ),
we can rewrite easily the previous system as a unique equation on H, € x x(t) + g _ x x(t) + rF(x(t)) + e(t)rU(x(t)) = 0:

2.2. (S) and (GS) Systems. Let H be a real Hilbert space with scalar product and corresponding norm respectively denoted by AE . , . ae and | . |. Let F: H fi R and U: Hfi R be functions of class C 1 and let e: R + fi R + be a map which tends to 0 when t fi +O. In the whole paper, we are interested in the following dynamical system:

(S) € x x(t) + g _ x x(t) + rF(x(t)) + e(t)rU(x(t)) = 0, t $ 0:
A first example of the (S) system comes from the mechanical model developed in Section 2.1. This corresponds to the situation where the global interaction is the sum of the interaction between pairs. Such a system with a repulsion force has been studied by Cabot and Czarnecki (Ref. 10) in the case N = 2. The interest in optimization of such a repulsion force comes from the fact that it allows a better exploration of the local minima. Another important situation corresponds to the case where U(x) = |x| 2 =2. Then, the (S) system reduces to

€ x x(t) + g _ x x(t) + rF(x(t)) + e(t)x(t) = 0:
As a matter of fact, the term e(t)x(t) has a stabilizing effect and is linked closely to the Tikhonov regularization (Ref. 12). This system has been studied intensively by Attouch and Czarnecki (Ref. 3).

The study of (S) can be placed in a more general setting by introducing the function Y(t, x) = F(x) + e(t)U(x): Then, the system (S) can be rewritten as

(GS) € x x(t) + g _ x x(t) + r x Y(t, x(t)) = 0,
where G stands for ''generalized''. In the following section, we investigate the general properties of (S) and (GS) by using classical techniques in the study of differential equations.

General Properties of (S) and (GS)

Let H be a real Hilbert space. Let us consider a mapping Y: R + • H fi R of class C 1 which satisfies the following conditions:

(A1) Y( . , x) is a nonincreasing function; i.e., ¶Y= ¶t# 0. (A2) Y is bounded from below on R + • H.
Moreover, assume the following Lipschitz-like conditions on r x Y:

(A3) For every bounded set B of H, there exists K B $ 0 such that jr x Y(t, y) -r x Y(t, x)j #K B jyxj, 8t˛R + , 8x, y˛B:

(A4) For every bounded set B of H, there exists a function L B : R + fi R such that sup R + L B <+O and

jr x Y(t + h, x) -r x Y(t, x)j # L B (t + h) -L B (t), 8x ˛B, 8t, h ˛R+ :
Then, let us consider the following dynamical system:

(GS) € x x(t) + g _ x x(t) + r x Y(t, x(t)) = 0:
The mechanical energy of (GS), defined by

E(t) = (1=2)j _ x x(t)j 2 + Y(t, x(t)),
provides us with a natural Lyapunov function. Indeed, a simple computation gives

_ E E(t) = AE _ x x(t), € x x(t)ae + ( ¶= ¶t)Y(t, x(t)) + AE _ x x(t), r x Y(t, x(t))ae: = -g j _ x x(t)j 2 + ( ¶= ¶t)Y(t, x(t)):
From Assumption (A1), we have

( ¶= ¶t)Y(t, x(t)) # 0; hence, _ E E(t)# 0;
i.e., E is nonincreasing. The existence of a Lyapunov function for a differential equation is useful in the study of the asymptotic stability of its equilibria. Lyapunov methods and other power tools (like the Lasalle invariance principle) have been developed to study such a question. We refer to the abundant literature on this subject (Ref. 6 and Refs. [START_REF] Arnold | Equations Diffe ´rentielles Ordinaires[END_REF][START_REF] Hirsch | Differential Equations, Dynamical Systems, and Linear Algebra[END_REF][START_REF] Lasalle | Stability by Lyapunov Direct Method with Applications[END_REF][START_REF] Reinhardt | Equations Diffe ´rentielles: Fondements et Applications[END_REF]. The central result of this section is given by the following theorem, whose proof relies mainly on the existence of the Lyapunov function E.

Theorem 3.1. Let us assume that Y: R + • H fiR satisfies Assumptions (A1) to (A4) and that the parameter g is positive. Then, the following properties hold:

(i) For all (x 0 , _ x x 0 ) in H • H, there exists a unique maximal solution x: R + fi H of (GS) which is of class C 2 on [0, +O[ and satisfies the initial conditions x(0) = x 0 and _ x x(0) = _ x x 0 . (ii) For every trajectory x( . ) of (GS), the following estimation holds:

_ x x˛L O ([0, +O); H) ˙L2 ([0, +O); H): (iii) Moreover, assuming that x is bounded [i.e., x ˛LO ([0, +O); H )],
then we have lim

t fi +O _ x x(t) = 0 and lim t fi +O € x x(t) = 0, lim t fi +O r x Y(t, x(t)) = 0:
Proof. The proof of (i) is a classical application of the Cauchy-Lipschitz theorem. The L O and L 2 estimates of _

x x are immediate consequences of the decay of the Lyapunov function E. The details of these proofs are left to the reader. The proof of (iii) is postponed to Section 6.1. u Corollary 3.1. Under the assumptions of Theorem 3.1, assume moreover that the function F( . ) = inf t$0 Y(t, . ) is coercive, i.e., lim jxj fi +O F(x) = +O. Then, x˛L O ([0, +O); H ) and the conclusions of Theorem 3.1 hold.

Proof. Using

F(x(t)) # Y(t, x(t))
and the decay of the function E, we obtain

(1=2)j _ x x(t)j 2 + F(x(t)) # E(t)# E(0)
and we deduce that

F(x(t)) # E(0):
This majorization on F(x(t)) and the coerciveness of F imply that x ˛LO ([0, +O); H ). u

Applications to the (S) System. As an application of the previous results, we consider now the case where Y is of the form

Y(t, x) = F(x) -inf F + e(t)(U(x) -inf U):
The functions F: H fi R, U: H fi R, and e: R + fi R + are of class C 1 , and satisfy respectively:

(B1) F is bounded from below on H. (B2) rF is Lipschitz continuous on the bounded subsets of H. (C1) U is bounded from below on H. (C2) rU is Lipschitz continuous on bounded subsets of H. (D1) lim t fi +O e(t) = 0. (D2)
The function e is nonincreasing; i.e., _ e e(t)#0, 8t˛R + .

Then, a simple application of Theorem 3.1 gives the following proposition.

Proposition 3.1. Let us consider the differential system

(S) € x x(t) + g _ x x(t) + rF(x)(t) + e(t)rU(x(t)) = 0,
where F, U, e satisfy respectively Assumptions (B), (C), (D). Then, we have the same conclusions (i), (ii), (iii) as in Theorem 3.1. Moreover, if

x ˛LO ([0, +O); H ), then lim t fi +O rF(x(t)) = 0.

Proof. Setting

Y(t, x) = F(x) -inf F + e(t)(U(x) -inf U),
it is sufficient to prove that Assumptions (A1) to (A4) of Theorem 3.1 are satisfied. This is immediate and is left to the reader. u

When F is strongly convex, the convergence of the gradient toward 0 yields immediately the strong convergence of the trajectory. Proposition 3.2. In addition to the assumptions of Proposition 3.1, let us assume that F is strongly convex; that is, for any R > 0, there exists a function b R : R

+ fi R + , with b R (t n ) fi 0 Þt n fi0, such that AErF(x) -rF( y), x -yae$ b R (jx -yj), 8x, y ˛H with jxj< R, jyj< R: (1)
If the trajectory x( . ) of (S) is bounded [i.e., x ˛LO ([0, +O); H )], then it is norm convergent as t goes to +O to the unique global minimizer x ¯of F.

Proof. The proof is an immediate consequence of the fact that lim

t fi +O rF(x(t)) = 0,
combined with the strong convexity property (1) of F. u

Convergence of (S) and (GS) Trajectories: Case of a Fast Control

In the previous section, we have shown the global existence of (GS) trajectories and the convergence of the gradients toward zero. When the potential F is convex, Alvarez has proved in Ref. 1 that the trajectories of the system (HBF) €

x x(t) + g _ x x(t) + rF(x)(t) = 0 weakly converge to a minimum of F. To extend this result to (GS), we appeal to the notion of convexity default. More precisely, we assume that:

(A5) For every bounded set B of H, there exists w B : R + fiR + such that for every t $0,

AEr x Y(t, y) -r x Y(t, x), y -xae $ -w B (t)jy -xj 2 , 8x, y˛B: (2)
The quantity w B (t) is called the convexity default of Y(t, . ) on B and can be characterized equivalently by the following inequality: for every l ˛[0, 1] and every x, y˛B,

(1 -l)Y(t, x) + lY(t, y) $ Y(t, (1 -l)x + ly) -[w B (t)=2]l(1 -l)jy -xj 2 , (3)
which in turn is equivalent to

Y(t, y)$ Y(t, x) + AEr x Y(t, x), y -xae -[w B (t)=2]jy -xj 2 : (4)
These characterizations of the convexity default will be useful in the sequel. The proof of their equivalence is classical and is left to the reader.

We assume that the convexity default tends to zero fast enough, more precisely:

(A6) For every bounded set B of H,

lim t fi +O w B (t) = 0 and ð +O 0 w B (t)dt <+O: Remark 4.1. If the map Y(t, .
) is convex for each fixed t, one can take w B " 0 and then Assumptions (A5) and (A6) are satisfied automatically.

Assumptions (A5) and (A6) imply that the function F( . ) = lim t fi +O Y(t, . ) is convex continuous on H, as stated by the following proposition. Proposition 4.1. Under Assumptions (A1) to (A6), the function

F( . ) = lim t fi +O Y(t, . ) is convex continuous on H.
Proof. Let B be a bounded convex set of H. Using the characterization (3) of the convexity default of Y(t, . ) on B and passing to the limit when tfi +O by taking into account the fact that lim

t fi +O w B (t) = 0, then we obtain (1 -l)F(x) + lF( y) $F((1 -l)x + ly),
8l ˛[0, 1], 8x, y ˛B;

i.e., F is convex on B. This being true for any bounded convex set B, the map F is convex on H.

Let us now prove that F is continuous on H. From Assumption (A1), the map F is the lower envelope of the functions (Y(t, . )) t$0 . Therefore, F is upper semicontinuous on H, hence locally bounded from above on H. From a classical result, the convex function F is then continuous on H. u

We assume lastly that:

(A7) argmin F"; and, for every z˛argmin F,

Ð +O 0 jr x Y(t, z)jdt <+O.
We can state now the following theorem.

Theorem 4.1. Let us assume that Y: R + • H fi R satisfies Assumptions (A1) to (A7). Let F denote the convex function defined by F( . ) = lim t fi +O Y(t, . ) and let x( . ) be the unique solution of (GS). If x ˛LO ([0, +O); H ), then there exists x O ˛argmin F such that w -lim

t fi +O x(t) = x O :
The proof of Theorem 4.1 uses mainly the Opial lemma and is given in Section 6.2.

Applications to the (S) System. We apply now the previous result to the case where

Y(t, x) = F(x) -inf F + e(t)(U(x) -inf U):
In this framework, Assumptions (A5)-(A6) are equivalent to the fact that F is convex and that e tends to 0 fast enough, Proof. Setting

Y(t, x) = F(x) -inf F + e(t)[U(x) -inf U],
we have just to verify that Assumptions (A5) to (A7) of Theorem 4.1 are satisfied. This simple verification is left to the reader. u

Let us remark that, in the previous result, the boundedness assumption on x( . ) takes a fundamental place. If F is coercive, this assumption is verified automatically. We are going now to examine a sufficient condition on U which ensures that the map x is bounded. Let us state the following proposition.

U(x) = jxj 2 =2, thus leading to € x x(t) + g _ x x(t) + rF(x(t)) + e(t)x(t) = 0: (5)
When F is convex and e: R + fi R + tends to zero slowly, i.e., ð +O 0 e(t)dt = +O, they have proved that each trajectory of (5) converges strongly to the point of minimal norm of the set S = argmin F.

Our purpose in the following theorem is to generalize the result of Attouch and Czarnecki (Ref. 3) to the case where U is a convex function such that argmin S U is reduced to a singleton {p}, p ˛H. For example, this situation occurs when U is strictly convex. Then, the following theorem shows the convergence of the (S) trajectories toward the point p. corresponds to the fact that e(t) does not tend to zero too rapidly, thus allowing the regularization term e(t)rU(x(t)) to be effective asymptotically.

This result shows an asymptotic selection property: the effect of such a slow control e forces all the trajectories to converge to the same equilibrium p.

This situation contrasts sharply with the noncontrolled situation or fast control, where the limits of the trajectories depend on the initial data, and may also be difficult to identify.

Beside the regularization role of the term e(t)rU(x), the (S) system can be useful in the framework of convex constrained minimization. Indeed, given a convex set S Ì H and a convex function U: H fi R, consider the following general minimization problem:

(P) min{U(x), x ˛S}:

We define now a convex function F: 2 ] and a control parameter e: R + fi R + such that ð +O 0 e(t)dt = +O: When (P) has a unique solution p, Theorem 5.1 tells us that the trajectories of (S) converge to p. Figure 1 illustrates this situation with S = R + 2 and

H fi R satisfying argmin F = S [for instance, F = (1=2)d S
U(x, y) = (1=2)[4(x + y + 1) 2 + (x -y -1) 2 ]:
The (S) trajectory is computed with the following data:

F(x, y) = (1=2)d 2 S (x, y), e(t) = 1=(1 + t),
(x 0 , y 0 ) = (5, 3), ( _ x x 0 , _ y y 0 ) = (0, 0), g = 1:

When (P) has a set of nonisolated solutions, the behavior at infinity of the (S) trajectories is still an open problem. From this point of view, Theorem 5.1 can be viewed as a first step when considering the (S) system as a tool for convex constrained minimization. We give now a sufficient condition which ensures that the (S) trajectory is bounded. The proof of Proposition 5.1 is postponed to Section 6.5.

Toward a Nonconvex Potential.

Let us consider now the mechanical model developed in Section 2.1. In the particular case N = 2, it leads to the following system on H = H • H:

€ x x(t) + g _ x x(t) + rf(x(t)) + e(t)rV (x(t) -y(t)) = 0, (6a)
€ y y(t) + g _ y y(t) + rf( y(t)) -e(t)rV (x(t)y(t)) = 0: (6b) Moreover, assume that V modelizes a repulsion potential; i.e., AErV (z), zae H # 0, 8z ˛H:

It is easy to verify that, in general, such a function V and the associated function

U(x, y) = V (x -y)
are not convex. As a consequence, Theorem 5.1 does not apply and one has to perform a specific study. In this direction, Cabot and Czarnecki (Ref. [START_REF] Cabot | Asymptotic Control of Pairs of Oscillators Coupled by a Repulsion, with Nonisolated Equilibria[END_REF] have proved an interesting one-dimensional result. More precisely, when H = R, assume that f: R fi R is convex, that V: R fi R satisfies [START_REF] Bre ´zis | Asymptotic Behavior of Some Evolution Systems, Nonlinear Evolution Equations[END_REF], and that e is a slow control. Setting

argminf = [a, b], -O # a # b #+O,
Cabot and Czarnecki have proved that the solution (x, y) of ( 6) satisfies one of the following cases:

(i) lim t fi +O (x(t), y(t)) = (a, b); (ii) lim t fi +O (x(t), y(t)) = (b, a); (iii) lim t fi +O x(t) = lim t fi +O y(t)˛[a, b].
Case (i) [resp. (ii)] just expresses that (lim t fi +O x(t), lim t fi +O y(t)) ˛R2 is a local minimum point on S = [a, b] 2 for the function (x, y)7 fi U(x, y) = V (xy):

Case (iii) is the pathological one and, after numerical experiments, Cabot and Czarnecki have conjectured that the set of initial conditions leading to this case is negligible. As a consequence, we may conjecture the following generalization of Theorem 5.1: when e is a slow control, for almost every initial condition (x 0 , _ x x 0 ) ˛H2 , the trajectory x( . ) of (S) tends to minimize at least locally the (possibly) nonconvex function U on the set S.

Proof of the Main Results

6.1. Proof of Theorem 3.1(iii). We assume that x ˛LO ([0, +O);H ); i.e., there exists a bounded set B such that x( . ) Ì B. Let us first notice that it is sufficient to prove that lim

t fi +O € x x(t) = 0:
Indeed, in that case, the function h := _ x x satisfies both the relations h ˛L2 ([0, +O); H) and _ h h ˛LO ([0, +O); H):

According to a classical result, these two properties imply lim t fi +O h(t) = 0;

i.e., lim

t fi +O _ x x(t) = 0:
Then, in view of (GS), we infer immediately that lim

t fi +O r x Y(t, x(t)) = 0:
Let us now prove that lim

t fi +O € x x(t) = 0:
For any h > 0, let us define

u h (t) = (1=h)[ _ x x(t + h) -_ x x(t)]:
The proof consists in showing that u h (t) converges to 0, uniformly with respect to h. Let us write the (GS) equation at the points t and t + h; let us make the difference and divide by h. We obtain

_ u u h (t) + g u h (t) = f h (t), (8)
where

f h (t) = -[r x Y(t + h, x(t + h)) -r x Y(t, x(t))]=h:
The function f h can be decomposed as

f h = f 1h + f 2h ,
where

f 1h (t) = -[r x Y(t + h, x(t + h)) -r x Y(t + h, x(t))]=h, f 2h (t) = -[r x Y(t + h, x(t)) -r x Y(t, x(t))]=h:
Let us now multiply ( 8) by e g s and integrate between t¢ and t, obtaining

u h (t) = u h (t¢)e -g (t -t¢) + e -g t ð t t¢ e g s f h (s)ds, implying that ju h (t)j # ju h (t¢)je -g (t -t¢) + e -g t ð t t¢ e g s j f 1h (s)jds + e -g t ð t t¢ e g s j f 2h (s)jds: (9)
In view of (A3), we have

j f 1h (s)j #K B [jx(s + h) -x(s)j=h] #(K B =h) ð s + h s j _ x x(u)jdu,
and from the Cauchy-Schwarz inequality we obtain

j f 1h (s)j 2 # (K 2 B =h) ð s + h s j _ x x(u)j 2 du:
Using again the Cauchy-Schwarz inequality, we can find now an upper bound for e -g t b t¢ t e g s | f 1h (s)|ds, specifically,

e -g t ð t t¢ e g s j f 1h (s)jds# e -g t ð t t¢ e 2g s ds 1=2 ð t t¢ j f 1h (s)j 2 ds 1=2 # (K B = ffiffiffiffiffi ffi 2g p ) ð +O t¢ (1=h) ð s + h s j _ x x(u)j 2 du ds 1=2 : (10)
On the other hand, Assumption (A4) implies

j f 2h (s)j #[L B (s + h) -L B (s)]=h,
and hence,

e -g t ð t t¢ e g s j f 2h (s)jds # ð +O t¢ {[L B (s + h) -L B (s)]=h}ds: (11)
We need now the following lemma.

Lemma 6.1. Let g: R + fi R be a nondecreasing function satisfying sup R + g <+O. Then, for every h > 0 and t$ 0,

ð +O t {[g(s + h) -g(s)]=h}ds# lim s fi +O g(s) -g(t): Proof. Let T $t. Then, ð T t {[g(s + h) -g(s)]=h}ds = (1=h) ð T+h t+h g(s)ds -(1=h) ð T t g(s)ds = (1=h) ð T+h T g(s)ds -(1=h) ð t+h t g(s)ds:
Since g is nondecreasing, the last equality implies ð T t {[g(s + h)g(s)]=h}ds # g(T + h)g(t):

Taking the limit when T fi +O, we obtain the result. u

Combining ( 9)-( 11) and applying Lemma 6.1 to the functions s 7 fi L B (s) and s7 fi b 0 s | _ x x(u)| 2 du, we obtain finally

ju h (t)j# ju h (t¢)je -g (t -t¢) + (K B = ffiffiffiffiffi ffi 2g p ) ð +O t¢ j _ x x(s)j 2 ds 1=2 + lim s fi +O L B (s) -L B (t¢):
Notice that u h (t¢) is uniformly bounded with respect to h. Indeed, from part (ii), the map _ x x is bounded and lim

h fi 0 u h (t¢) = € x x(t¢):
Then, taking the supremum in h and the upper limit when t fi +O, we deduce from the last inequality that lim sup

t fi +O sup h > 0 ju h (t)j ! # (K B = ffiffiffiffiffi ffi 2g p ) ð +O t¢ j _ x x(s)j 2 ds 1=2 + lim s fi +O L B (s) -L B (t¢):
Taking the limit when t¢fi+O, we obtain lim

t fi +O sup h > 0 ju h (t)j ! = 0:
Since, for all t$ 0,

j € x x(t)j # sup h > 0 ju h (t)j,
we conclude that lim

t fi +O € x x(t) = 0: u 6.2. Proof of Theorem 4.1.
The central idea is to prove the weak convergence of the trajectory x by using the Opial lemma (Ref. 9). Lemma 6.2. See Ref. 9. Let H be a Hilbert space and let x: [0, +O[ fiH be a function such that there exists a nonvoid set S Ì H which verifies:

(i) 8z˛S, lim t fi +O jx(t) -zj exists. (ii) 8t n fi +O with x(t n ) fix O weakly in H, we have x O ˛S.
Then, x(t) weakly converges as t fi +O to some element x O of S.

Let us apply the Opial lemma with S = argmin F.

(i) For any z ˛argmin F";, let us set

h(t) = AE _ x x(t), x(t) -zae + (g =2)jx(t) -zj 2 :
Then, we have

_ h h(t) = j _ x x(t)j 2 + AEx(t) -z, € x x(t)ae + g AEx(t) -z, _ x x(t)ae = j _ x x(t)j 2 -AEx(t) -z,r x Y(t,x(t))ae = j _ x x(t)j 2 -AEx(t) -z,r x Y(t,x(t)) -r x Y(t,z)ae -AEx(t) -z,r x Y(t,z)ae:
By Assumption (A5), for every t $ 0,

AEx(t) -z, r x Y(t, x(t)) -r x Y(t, z)ae $ -w B (t)jx(t) -zj 2 ,
where B is a bounded set of H containing z and the trajectory x( . )

[x˛L O ([0, +O); H ), by assumption]. Hence, _ h h(t) # j _ x x(t)j 2 + w B (t)jx(t) -zj 2 + jx(t) -zjjr x Y(t, z)j # j _ x x(t)j 2 + C 2 w B (t) + Cjr x Y(t, z)j,
where C is an upper bound for |x(t) -z|.

From Theorem 3.1, we have

j _ x xj 2 ˛L1 (0, +O):
On the other hand, because of (A6) and (A7), w B ( Á )˛L 1 (0, +O) and jr x Y(:, z)j ˛L1 (0, +O):

Taking the positive part of the previous relation, we obtain finally

_ h h + (t)# j _ x x(t)j 2 + C 2 w B (t) + Cjr x Y(t, z)j ˛L1 (0, +O):
Then, we can apply the following lemma.

Lemma 6.3. Let h: R + fi R be a function of class C 1 which is bounded from below and such that _ h h + ˛L1 (0, +O). Then, lim t fi +O h(t) exists. Hence,

lim t fi +O h(t) = lim t fi +O AE _ x x(t), x(t) -zae + g =2 ð Þjx(t) -zj 2 exists: Since x ˛LO ([0, +O); H) and lim t fi +O _ x x(t) = 0,
we conclude that lim t fi +O jx(t)zj exists:

(ii) Let us now consider a sequence (t n ) such that w -lim

n fi +O x(t n ) = x O :
We have to prove that

x O ˛argmin F:

Let us fix y˛H and consider a bounded set B containing y and the trajectory x( . ). Using the characterization (4) of the convexity default of Y(t, . ) on B, we deduce that

Y(t n , y) -Y(t n , x(t n )) $ AEr x Y(t n , x(t n )), y -x(t n )ae -[w B (t n )=2]jy -x(t n )j 2 : (12)
Passing to the upper limit in [START_REF] Tikhonov | Me ´thodes de Re ´solution de Proble `mes Mal Pose ´s[END_REF], we obtain Using [START_REF] Arnold | Equations Diffe ´rentielles Ordinaires[END_REF], we find that

F( y) -lim inf n fi +O Y(t n , x(t n )) $0, (13)
F( y)$ lim inf n fi +O Y(t n , x(t n )) $ lim inf n fi +O F(x(t n ))$ F(x O ):
The last inequality comes from the fact that F is convex continuous (cf. Proposition 4.1), hence l.s.c. for the weak topology on H. This being true for any y ˛H, we have proved that

x O ˛argmin F:

The Opial lemma allows us to conclude that there exists x O ˛argmin F such that w -lim

t
On the other hand, we have

h(t) = (g =4)jx(t) -zj 2 + (g =4)[jx(t) -zj 2 + 2AE(2=g ) _ x x(t), x(t) -zae] $ (g =4)jx(t) -zj 2 -(g =4)[(4=g 2 )j _ x x(t)j 2 ] $ (g =4)jx(t) -zj 2 -k _ x xk 2 O =g : (15)
Combining ( 14) and ( 15), we obtain We have to prove that h, converges to 0. By differentiating h, we find that

(g =4)jx(t) -zj 2 # k _ x xk 2 O =g + h(0) + ð +O 0 j _ x x(s)j 2 ds + C ð +O 0 e(s)ds + C¢ ð t 0 e(s)jx(s) -zj 2 ds: Setting C † = k _ x xk 2 O =g + h(0) + ð +O 0 j _ x x(s)j 2 ds + C ð +O 0 e (
_ h h(t) = j _
x x(t)j 2 -AEx(t) -p, rF(x(t))ae -e(t)AEx(t)p, rU(x(t))ae:

From the convexity of F, p˛argmin F implies AE px(t), rF(x(t))ae # 0; hence, _ h h(t) # j _

x x(t)j 2 + e(t)AE px(t), rU(x(t))ae:

The main idea of the proof is now to distinguish respectively the cases where AE px(t), rU(x(t))ae0> and AE px(t), rU(x(t))ae # 0:

Precisely, we distinguish three cases:

(a) 9T $ 0, 8t $T, AE px(t), rU(x(t))ae # 0;

(b) 9T $ 0, 8t$ T, AE px(t), rU(x(t))ae> 0; (c) 8T $0, 9t$ T, AE px(t), rU(x(t))ae > 0:

Case (c) obviously contains Case (b), but the main points of the proof are made clearer with this distinction.

In the whole proof, we use the following notation:

E p := {x ˛H, AE p -x, rU(x)ae $ 0}:
Let us state a first claim which will be useful in the sequel. Case (a). We assume that there is T $0 such that, for every t $ T, AE px(t), rU(x(t))ae# 0; hence, from ( 16), we deduce that, for every t$ T,

_ h h(t) # j _ x x(t)j 2 :
Taking the positive part of the previous inequality, we obtain _ h h + ˛L1 (0, +O) and, using Lemma 6. Applying the following claim to the sequence (x(t n )), we obtain that lim

n fi +O x(t n ) = p,
which concludes the proof of Case (b).

Claim 6.2. Let (x n ) be a bounded sequence in E p such that lim n fi +O rF(x n ) = 0: Then, lim n fi +O x n = p: Proof. Since (x n ) is bounded, there exist x ¯˛E p and a subsequence of (x s(n) ), still denoted by (x s(n) ), such that lim

n fi +O x s (n) = x ¯:
In view of lim n fi +O rF(x s (n) ) = 0, we get that x ¯˛S. On the other hand, from Claim 6.1, E p ˙S = { p}, so that we obtain x ¯= p. Since p is the limit of every convergent subsequence of (x n ) n$0 , we deduce that the sequence (x n ) n$0 converges to p. u Case (c). We assume now that, for every T $ 0, there exists some t $ T such that AE px(t), rU(x(t))ae> 0: Take any sequence (t n ) Ì R + such that lim n fi +O t n = +O, and let us prove that lim

n fi +O x(t n ) = p:
First, assume that there is a subsequence (t n ¢) of (t n ) such that x(t n ¢) ˛Ep . Since the map x is bounded, from Proposition 3.1, we have We assume now that there is a subsequence (t n †) of (t n ) such that x(t n †) ˇEp and we prove that lim

n fi +O x(t † n ) = p:
We need the following claim.

We distinguish now the same cases (a), (b), (c) as in the proof of Theorem 5.1.

Case (a). From (17), lim t fi +O jx(t)pj exists, which implies that x˛L O ([0, +O); H ).

Case (b). The trajectory {x(t), t $T} is contained in E p ; hence, from (18) and (19), it is contained in the set [U #min S U] ˙[F# M], which is bounded in view of condition (G); as a consequence, the map x is bounded; i.e., x ˛LO ([0, +O); H ). Case (c). Let T $0 such that x(T ) ˛Ep and consider t $ T.

  can deduce from Theorem 4.1 the following proposition. Proposition 4.2. In addition to Hypotheses (B), (C), (D), let us assume that (E1) The map F is convex and argmin F";. (E2) Ð +O 0 e(t)dt<+O (fast control). Let x( . ) be the unique solution of (S). If x˛L O ([0, +O); H ), then there exists x O ˛argmin F such that w -lim t fi +O x(t) = x O :

Proposition 4 . 3 .

 43 Under the assumptions of Proposition 4.2, let x be the unique solution of the (S) system. Moreover, we assume that rU has at most a linear growth, i.e., jrU(x)j # A + Bjxj, 8x ˛H, for some positive A and B. Then, x ˛LO ([0, +O); H ); hence, the conclusions of Proposition 4.2 apply.The proof of Proposition 4.3 uses Gronwall type arguments and is given in Section 6.3.5. Case of a SlowControl: Minimization of U over argmin F 5.1. Case of a Convex Potential. Attouch and Czarnecki (Ref. 3) have considered the particular case of (S) where

Theorem 5 . 1 .

 51 In addition to Hypotheses (B), (C), (D), let us assume that H = R n and that: (F1) F: H fiR and U: H fi R are convex and S := argmin F";. (F2) The set argmin S U is reduced to a singleton {p}, p˛H. (F3) Ð +O 0 e(t)dt = +O (slow control). If the trajectory x( . ) of the (S) system is bounded [i.e., x˛L O ([0, +O); H )], then it converges to p, namely, lim t fi +O x(t) = p. Theorem 5.1 is proved in Section 6.4. The condition ð +O 0 e(t)dt = +O

Fig. 1 .

 1 Fig. 1. Convergence of the (S) trajectory toward argmin S U.

Proposition 5 . 1 .

 51 Under the assumptions of Theorem 5.1, let x( . ) be the unique solution of the (S) system. Assume that, for every M ˛R, the following condition holds: (G) The set {x˛H, U(x)# min S U} ˙{x ˛H, F(x)# M} is bounded. Then, x ˛LO ([0, +O); H ); hence, in view of Theorem 5.1, lim t fi +O x(t) = p: Moreover, condition (G) is satisfied in each of the following cases: (a) F is coercive; i.e., lim |x| fi +O F(x) = +O. (b) U is coercive; i.e., lim |x| fi +O U(x) = +O.

  because lim t fi +O r x Y(t, x(t)) = 0 (cf: Theorem 3:1) and lim t fi +O w B (t) = 0 [cf: (A6)]:

: 6 . 4 .

 64 s)ds, we can apply now the Gronwall lemma to the function |x -z| 2 and we find thatjx(t)zj 2 #(4C †=g ) exp (4C¢=g )The right side is finite and hencex ˛LO ([0, +O); H): u Proof of Theorem 5.1. The proof is an extension of the arguments of Attouch and Czarnecki (Ref. 3). In our situation, it relies on the study of the function h defined byh(t) = AE _ x x(t), x(t)pae + (g =2)jx(t)pj 2 ,where p := argmin S U:

Claim 6 . 1 .

 61 Under the assumptions of Theorem 5.1, we have E p ˙S = {p}. Proof. Let x ˛Ep ˙S; from the definition of E p , we have AE px, rU(x)ae$ 0, which combined with the convexity of U yields U( p)$ U(x) i:e:, min S U $ U(x): Since x ˛S, we conclude that x˛argmin S U = {p}. u

  lim t fi +O rF(x(t)) = 0, and hence, lim n fi +O rF(x(t¢ n )) = 0: Since x(t n ¢)˛E p , from Claim 6.2, we deduce that lim n fi +O x(t¢ n ) = p:

6 . 5 .

 65 n fi +O jx(t † n )pj = 0, which concludes the proof of Case (c). u Proof of Proposition 5.1. We keep here the notations of the proof of Theorem 5.1. Setting p := argmin S U, we still use the function h defined byh(t) = AE _ x x(t), x(t)pae + (g =2)jx(t)pj 2and the setE p = {x ˛H, AE px, rU(x)ae $0}:From the decay of the functionE(t) = (1=2)j _ x x(t)j 2 + (F(x(t)) -min F) + e(t)(U(x(t)) -min U),there exists M ˛R such that, for every t$ 0,F(x(t)) #M; i.e., {x(t), t $ 0} Ì [F#M]:(18)On the other hand, from the convexity of U, we have that, for every x ˛H, AE px, rU(x)ae $0 Þ U(x)# U( p) = min

  Some variants of the (S) system have been studied by several authors. For example in Ref. 3, Attouch and Czarnecki couple the Tikhonov regularization with the (HBF) dynamical system. They prove that, when F is convex and e: R + fi R + is a C 1 control function such that

			Ð +O 0	e(t)dt = +O,
	then each trajectory of the system
	((S) with U(x) = jxj 2 =2)	€ x x(t) + g _ x x(t) + rF(x(t)) + e(t)x(t) = 0
	strongly converges to the point of minimal norm of argmin F. The condition
	ð +O	e(t)dt = +O
	0
	1) with arguments relying on the Opial lemma (Ref. 9). The situation
	is much more involved when dealing with a slow control, namely,
	ð +O	e(t)dt = +O:
		0

Let f: H fi R be a given function of class C 1 that has to be minimized. The heavy ball with friction method (Ref. 2) consists in studying the following system:

  2.1. Mechanical Origin. Let H be a real Hilbert space with scalar product and corresponding norm respectively denoted by AE . , . ae H and | . | H .

  On the other hand, we have |x ¯-p| = a > 0, a contradiction. As a conclusion, Thus, by integrating between t 1 and t and passing to the limit when t fi +O,

	we get t fi +O lim lim h(t) t fi +O h(t) + l 3, this implies that exists: t 1 ð +O e(s)ds#h(t 1 ) +	t 1 ð +O	j _ x x(s)j 2 ds,	(17)
	We now prove that which contradicts the fact that
		lim t fi +O j _ x xj 2 ˛L1 (0, +O) and eˇL 1 (0, +O): h(t) = 0:
	Assume that it is not true. Then, lim t fi +O h(t) exists and is strictly positive. Hence,
	Let us set	lim t fi +O	h(t) = lim t fi +O	AE _ x x(t), x(t) -pae + (g =2)jx(t) -pj 2 = 0:
	Since	a := lim t fi +O	jx(t) -pj > 0:
	Let us first prove that lim t fi +O _ x x(t) = 0,
	lim sup we conclude that
			lim t fi +O
		lim sup	AE p -x(t), rU(x(t))ae < 0;
			t fi +O
	i.e., there exists l > 0 and t 1 > 0 such that, for every t $ t 1 ,
		AE p -x(t), rU(x(t))ae # -l:
	Hence,

t fi +O AE px(t), rU(x(t))ae < 0: Indeed, if it is not true, there exists a sequence (t n ) such that lim n fi +O t n = +O and lim

n fi +O AE px(t n ), rU(x(t n ))ae = 0:

Since the function x is bounded, without loss of generality we may assume that there is x ¯˛H such that x(t n ) converges to x ¯. At the limit, when nfi +O, we obtain that AE px ¯, rU(x ¯)ae = 0, which implies x ¯˛E p . From Proposition 3.1, we get that lim n fi +O rF(x(t n )) = 0, and hence x ¯˛S. Finally,

x ¯˛E p ˙S = { p}:

_ h h(t)# j _

x x(t)j 2 -e(t)l:

jx(t)pj = 0:

Case (b). We assume that there is T $0 such that, for every t $ T,

AE px(t),

rU(x(t)

)ae> 0 and hence, for every t$ T, x(t) ˛Ep : Since x( . ) is bounded, from Proposition 3.1, we get that lim t fi +O rF(x(t)) = 0: Considering a subsequence x(t n ), then we have lim n fi +O rF(x(t n )) = 0:

  Proof. For every u˛]t (t), t], x(u) ˇEp , that is, We now come back to the proof of Case (c). Since x(t n †) ˇEp , let t (t n †) be defined by Claim 6.3. We notice first that lim ˛Ep , for n large enough. Let n be large enough. Since x(t (t n †)) ˛Ep , from Claim 6.2, =2) lim sup n fi +O jx(t † n )pj 2 :

	Then, we have	h(t) # h(t (t)) + lim sup h(t †	ð +O	j _ x x(u)j 2 du:
		n fi +O			t (t)
	Hence, we deduce that
		lim
		AE p -x(u), rU(x(u))ae# 0:
	From (16), we deduce that
		_ h h(u)# j _ x x(u)j 2 :
	By integrating between t (t) and t, we obtain
		h(t) -h(t (t)) #	ð t	j _ x x(u)j 2 du,
					t (t)
	which yields immediately the expected formula.	u
		n fi +O	t (t †
	we have
		lim n fi +O	x(t (t † n )) = p;
	i.e.,
		lim n fi +O	h(t (t † n )) = 0:
	Since _ x x ˛L2 ([0, +O), H ), we have also
		lim n fi +O	t (t † n ) ð +O	j _ x x(u)j 2 du = 0:
	Hence, in view of Claim 6.3, we deduce that
		lim sup	h(t † n ) #0:
		n fi +O
	On the other hand, it is clear that, because of the definition of h and the
	fact that
		lim n fi +O	_ x x(t † n ) = 0,

Claim 6.3. Let t $ 0 such that x(t) ˇEp and let

t (t) = inf{u˛[0, t] j x([u, t]) ˙Ep = ;}: n ) = +O and that x(t (t † n )) n ) = (g

  If x(t) ˛Ep , then |x(t) -p| # r, where r is the diameter of the bounded set [U # min S U][ F#M]. If x(t)ˇE p , let t (t) be defined by Claim 6.3. Clearly T # t (t) < t and x(t (t)) ˛Ep , which implies thatjx(t (t))pj # r: (t)) = AEx(t (t))p, _ x x(t (t))ae + (g =2)jx(t (t))pj 2 # rk _ x xk O + (g =2)r 2 :(20)In view of (20) and Claim 6.3, we deduce thath(t) # rjj _ x xjj O + (g =2)r 2 +

	Hence,
	h(t ð +O

T j _ x x(u)j 2 du:

This boundedness property of h clearly proves that

x ˛LO ([0, +O); H): u

The author expresses his gratitude to Marc-Olivier Czarnecki for help during the realization of this work. The author thanks the referees for their careful reading of the paper. Their valuable suggestions, critical remarks, and pertinent comments led to numerous improvements throughout.

Assistant Professor, Laboratory LACO, Faculty of Science, University of Limoges, Limoges, France.