Jean-Marie Fontmorin

Florence Fourcade

Florence Geneste

Isabelle Soutrel

Didier Floner

J.-M Fontmorin

A Amrane

Direct electrochemical oxidation of a pesticide, 2,4-Dichlorophenoxyacetic acid, at the surface of a graphite-felt electrode -Biodegradability improvement

Keywords: 2,4-D, Combined process, Electrochemical process, Biological treatment, Carbon felt electrode

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Direct electrochemical oxidation of a pesticide, 2,4-dichlorophenoxyacetic acid, at the surface of a graphite felt electrode: Biodegradability improvement

Introduction

The large accumulation of endocrine disruptors, in continental and marine natural waters are the consequence of the worldwide general application of intensive agricultural methods, the large-scale development of the agrochemical and food industry and the high levels found in some specific effluents (textile, pharmaceutical, hospital waste…). For instance, in the Brittany region (France), the Regional Direction of the Environment (DIREN) observes a contamination of the Brittany Rivers by phytosanitary products, including pesticides, which can interfere with hormone systems of living beings (endocrine disruptors) [START_REF] Ferragu | Pesticides et Santé: Etat des connaissances sur les effets chroniques en 2009[END_REF]. Partly responsible for this pollution, low volumes containing high concentrations of persistent organic pollutants, in the range of concentrations found in some specific industrial and agricultural effluents (unused treatment solution, spray, machine and container washing….) [START_REF] Chiron | [END_REF], can result in large polluted volumes which are difficult to treat owing to their low concentrations. One solution would be therefore to treat pollution at source, as intended in this study.

Pesticides impact on the environment is complex and varied according to various factors, such as toxicity and ecotoxicity of the parent molecule or by-product metabolites, synergistic effects with other pollutants, length of the half-life, exposure time and dose, etc. Various acute or chronic poisoning effects on human health have been described [3][4][5][6]. There is therefore an urgent need for efficient processes for their removal and owing to the possible toxicity of the by-product metabolites, total mineralization is mainly targeted.

For this purpose, biological processes, the most cost-effective for wastewater treatment, which are destructive and have been extensively studied [7][8][9][10][11] do not always appear relevant for the removal of recalcitrant compounds, owing to their low biodegradability.

Contrarily, physico-chemical techniques have proved their efficiency for their removal.

Among them, Advanced Oxidation Processes constitute the most important and widely documented group [START_REF] Chiron | [END_REF][START_REF] Oppenländer | Photochemical Purification of Water and Air[END_REF][START_REF] Badawy | [END_REF], owing to the high reactivity of the free  OH radicals produced.

However and due to the lack of selectivity of the free radicals, possible toxic by-products can be generated, which in some cases can appear more toxic than the parent compounds [14,15]. Consequently, the mineralization time is of major importance; too low process time can result in toxic by-products, while long process time to ensure total mineralization can induce high energy costs.

It is noteworthy that pesticides biorecalcitrance can be related to the presence of complex aromatic chain or to the presence of specific bonds, such as nitro-or halogenated bonds. For instance, among the most used pesticides, a three-quarter of them contains halogenated bonds; and the presence of chlorine atoms on phenyl ring is a factor that favors the toxicity of aryl compounds [16]. From this, the development of processes targeting a selective attack of specific functional groups to improve the biodegradability of a given effluent can constitute another approach to treat effluents containing pesticides. Indeed, if agricultural effluents are for instance considered, pesticides levels can reach 500 mg L -1 [START_REF] Chiron | [END_REF], as it is the case in farm bottom tanks. Hence, a selective attack of specific functional groups, can constitute a relevant solution to relieve AOPs drawbacks. The generation of possible toxic by-products could thus be avoided owing to the expected control of the resulting by-products as well as the high energy costs since total mineralization is not the objective owing to the expected improvement of biodegradability.

For this purpose and in the case of an electroactive target compound, its electrochemical oxidation or reduction can be carried out for its degradation. Total mineralization can be subsequently completed during biological treatment, since the potential advantages of the strategy of combining physico-chemical and biological processes to treat contaminants in wastewater were previously underlined [17][18][19][20]. However, the literature dealing with the use of direct electrochemical oxidation/reduction for effluent pre-treatment remains scarce. Doan et al. [21] and Ghafari et al. [22] coupled an electrochemical process and a biological treatment for the removal of heavy metals and nitrate, respectively; while regarding organic pollutants, up to now our work seem to be the only available dealing with the combination of a direct electrochemical process and a biological treatment. It was investigated for the removal of phosmet, an organophosphorous insecticide [23], some antibiotics, tetracycline [24] and sulfamethazine [25], and a chlorinated phenoxy herbicide 2,4-dichlorophenoxyacetic acid [26,27].

The promising results obtained regarding 2,4-D should be underlined [28,29], since halogenated pesticides are the most widespread pesticides. Indeed, it was shown that mild oxidation of the target compound can be sufficient to improve biodegradability [28] allowing subsequent biological treatment [29]. However in view of the targeted selectivity, the nature of the electrochemical process should be clearly elucidated, namely the involvement or not of free hydroxyl radials in pesticide"s oxidation; it is discussed in this study.

However, the production of this highly reactive species is closely linked to the electrode material used and to some operating parameters such as oxygen overvoltage. If some electrodes (boron-doped diamond electrodes for instance) are well-known as powerful candidates for • OH generation, it is not at all obvious for some other materials, such as graphite felt, the material used in the laboratory [23,[25][26][27]. Indeed, the use of a graphite felt working electrode with a high specific area in a flow electrochemical cell [30] allows the electrochemical reaction of the electroactive species at macroscale level with low electrolysis times. Even if mechanisms involving • OH have been previously suggested for studies based on the use of graphite felt [31], to our knowledge such production at the surface of such electrode material has never been demonstrated up to now. However, this question appears to be of major importance to understand the electrochemical oxidation phenomena occurring during the use of such material and to confirm the specificity and selectivity of the considered processes, namely a direct reaction at the electrode surface. For this purpose, three points have to be considered, the nature of the working electrode, the experimental conditions and the indirect determination of the hydroxyl radicals; these points are investigated in this study.

In addition and to complete mechanism knowledge, the electrochemical reaction of 2,4-D at the electrode was also examined.

Biological treatment involving activated sludge was then considered to examine the efficiency of the electrochemical pretreatment.

Materials and Methods

Chemicals

2,4-dichlorophenoxyacetic acid (2,4-D) (98%) was purchased from Alfa Aesar (Schiltigheim, France). Chlorohydroquinone (85%) was purchased from Sigma-Aldrich (Saint-Quentin Fallavier, France). Acetonitrile (ACN) and formic acid were LC/MS grade from JT Baker (Deventer, Netherlands). All standards were prepared with ultra pure water (PurelabOptions-Q7/15, Elga, 18.2 MΩ cm).

Electrochemical pre-treatment

The electrochemical pre-treatment was based on a home-made flow cell [28]. Graphite felt used as working electrode was supplied by Mersen (RVG 4000 -Mersen, Paris La Défense, France) [27]. The dimensions of the graphite felt were 48 mm diameter and 12 mm width.

Two interconnected stainless steel plates were used as counter-electrodes and compartments were separated by cationic exchange membranes (Ionac 3470 -Lanxess SAS, Courbevoie, France). The reference electrode (Saturated Calomel Electrode -SCE) was positioned in the middle of the graphite felt. To ensure a good homogeneity of the potential distribution in the three dimensional working electrode, the felt was located between the two counter-electrodes [30]. The electrolyte solution percolated the porous electrode and the flow rate was monitored by a Gilson minipuls 2 peristaltic pump (Middleton, WI, USA). A VersaSTAT 3 potentiostat (Ametek/Princeton Applied, Elancourt, France) was used to control the potential.

Biological process

After only one pass through the electrochemical flow-cell, the effluent was collected for the subsequent biological treatment, which was carried out in aerobic conditions, using activated sludge purchased from the local wastewater treatment plant (Station de Beaurade, Rennes, France). Before use and to avoid any residual carbon or mineral nutrient, it was treated as previously detailed [26].

Experiments were carried out in 250 mL erlenmeyer flasks containing 100 mL of medium, stirred at 250 rpm, kept at 30°C and triplicated to ensure the reproducibility of the results. The electrolyzed solution (initially 500 mg L -1 2,4-D) was diluted (five times dilution) to allow a direct comparison to non-treated 2,4-D (100 mg L -1 - [26]). Minerals were spiked in the medium as highly concentrated solutions to reach the following initial composition (mg FeCl 3 .6H 2 O, 0.26; and the initial pH was adjusted to 7.0 ± 0.2. Activated sludge was added in order to have initial concentration of 0.5 g L -1 of dry matter. Samples (5 mL) were taken regularly and filtered through 0.45 m-syringe filters for measurements.

L -

Analysis

Measurements of the residual 2,4-D and chlorohydroquinone concentration were performed by HPLC (High Pressure Liquid Chromatography) (Milford, USA) system involving a pump Waters 600, fitted with a Phenomenex Kinetex ® C18 2.6 µm column (4.6 mm × 100 mm), along with a Waters 996 Photodiode array detector, a Waters 717 ph Autosampler and controlled through an Empower ® 2 program. The mobile phase consisted of acetonitrile and trifluoroacetic acid (TFA) 0.1% in ultra-pure water with a ratio of 30/70 at 1 mL min -1 .

Dissolved Organic Carbon (DOC) measurements

Solutions were filtered on Satorius Stedim Minisart 0.45 µm GF prefilters (Goettingen, Germany). DOC was measured by means of a TOC-V CPH/CPN Total Organic Analyzer Schimadzu. Reproducible DOC values were always obtained using the standard NPOC (Non Purgeable Organic Carbon) method.

Chemical Oxygen Demand (COD) and Biological Oxygen Demand (BOD 5 )

Chemical Oxygen Demand (COD) was measured by means of a Test Nanocolor ® CSB 160 from Macherey-Nagel (Düren, Germany). BOD 5 measurements were carried out in Oxitop IS6 (WTW, Alès, France). Activated sludge provided by a local wastewater treatment plant (Rennes Beaurade, Bretagne, France) was used to inoculate duplicate flasks and the initial microbial concentration was set to 0.5 g L -1 .

More details regarding the experimental procedure can be found in previous papers [26,27].

Results and Discussion

Electrochemical behavior of 2,4-D

Electrochemical behavior of 2,4-D in acidic medium (pH  pKa = 2.8, naturally obtained when 2,4-D was dissolved in water) showed an electroactivity in reduction and in oxidation [27]. 2,4-D electrolysis in reduction showed a disappearance of the first reduction signal, which was however not confirmed by HPLC analysis, which indicated no 2,4-D degradation.

The disappearance of the reductive signal was found to be closely related to its anionic or molecular form, and thus to the pH of the medium, since the reductive signal disappeared at alkaline final electrolysis pH (11), while it was recovered after medium acidification to the initial pH value (2.8) (Fig. 1).

This electrochemical behavior closely related to the pH and coupled to the analytical results obtained by liquid chromatography led to exclude the initial assumption of a C-Cl bond cleavage and to consider instead a reduction of the carboxylic acid function of 2,4-D. Indeed, the carboxylic acid can undergo a one-electron reaction to yield the corresponding carboxylate anion, which is accompanied by hydrogen release (Reaction 1 -Fig. 2); and the reduction of the formed carboxylate appears difficult in the considered solvent, water. The carboxylic acid can also undergo 2-, 4-or 6-electron reductions to give the corresponding aldehyde, alcohol or alcane, respectively (Reactions 2 to 4 -Fig. 2); these reactions can however occur in acidic conditions avoiding the carboxylate reduction [START_REF] Hammerich | Organic electrochemistry[END_REF].

The lack of reduction wave was expected at alkaline pH, since its anionic form largely predominated owing to the 2,4-D pKa (2.8); this characteristic wave can only be observed in the presence of the protonated form of the compound, namely for pH values below 4.

The transformation of the carboxylic acid into its ester form should allow to avoid its carboxylate reduction, which would be helpful to validate the above assumption. Thus, the diethyl ester of 2.4-D was prepared by the reaction of 2.4-D with oxalyl chloride to form the acyl chloride derivative and then with ethanol. However, it was not possible to analyze the synthesized ester in water owing to its low solubility, leading to perform cyclic voltammetry analyses in acetonitrile. As shown in Fig. 3, the synthesized ester was reduced at a lower potential (-1.75 V/ECS) than the initial compound (-1.5 V/ECS). Consequently, the wave observed during 2,4-D reduction was linked to the reduction of the acid to acetate instead of the aromatic ring dechlorination.

The electrochemical behavior of 2,4-D was then examined in oxidation, and hence a similar approach was considered, namely cyclic voltammetry analysis prior to electrolysis to confirm pretreatment feasibility. An oxidation wave was observed at 1.6 V/ECS (Fig. 4).

To confirm 2,4-D electroactivity in oxidation, the absence of pH impact on its electrochemical behavior was checked and confirmed, since the oxidation wave was observed at both acidic and alkaline pH (Fig. 4).

Nature of the electrochemical process

Since no 2,4-D degradation was detected during cathodic reduction experiments in aqueous medium, electrolyses were therefore performed in oxidation. However, one major question remained, which was the nature of the electrochemical process involved in 2,4-D oxidation: are free radicals involved in 2,4-D degradation? Electrodes can be classified as "actives" and "non-actives" according to their reactivity vis-avis the reaction of oxygen formation [START_REF] Bard | Encyclopedia of Electrochemistry[END_REF]. In a first step, plane graphite electrode was used to allow a fine observation of oxygen formation on the material (graphite). The obtained results were then compared to well-known electrode materials, such as platinum or vitreous carbon. At pH 2.7, namely close to the experimental conditions in the presence of 2,4-D, it can be observed an oxygen release from about 1.5-1.6 V/SCE on graphite and platinum and from 1.6 et 1.7 V/SCE for vitreous carbon (it should be remembered that on a BDD it is usually observed near 2.5 V/SCE) [START_REF] Mccreery | [END_REF]. The shape of the curve recorded on graphite suggested a resistive character of the material before oxygen release, which may be related to the nature of the electrode. This phenomenon was systematically observed, irrespective of meticulous electrode polishing, long degassing time or an electrochemical surface cleaning, namely a chronoamperometry at 0 V/SCE before cyclic voltammetry analyses. This relatively early oxygen release on graphite suggested an "active" electrode nature, which was not in favor of • OH production in the considered experimental conditions.

The nature of the working electrode

Current densities can vary according to the considered surface as highlighted in Fig. 5. A cylindrical graphite felt of 10 mm diameter and 12 mm height was considered. The total geometric surface of the cylinder was 5.34 cm 2 . From BET analysis and the felt density (given by the manufacturer), 0.7 m² g -1 and 0.088 g cm -3 , respectively, the BET surface was 580 cm 2 (the cylinder volume was 0.942 cm 3 ), namely a ratio close to 109 between both surfaces. However, it can be observed that the surface given by the BET is most likely overestimated, since deduced from the adsorbed nitrogen amount; while and owing to its low size, nitrogen can penetrate pores which are not accessible to water. Another and probably more realistic approach is to use cyclic voltammetry and especially the oxidation (or reduction) current of potassium ferricyanide (K 3 [Fe(CN) 6 ], 4.9 mM) to estimate the electrochemical electrode surface. Indeed, the ferricyanide peak current is linked to the electrode surface through the following relation [START_REF] Bard | Electrochemical methods: Fundamentals and applications[END_REF]:

With i p the peak current intensity, n the number of exchanged electrons (1), A the electrode surface (cm 2 ), D 0 the ferricyanide diffusion coefficient (0.76 10 -5 cm 2 s -1 in KCl 0.1 M), C 0 the ferricyanide concentration (mol cm -3 ) and ν 0 the sweep rate (0.1 V s -1 ). 0.5 M phosphate buffer at pH 2.7 was used and an approximate diffusion coefficient value of 0.76 × 10 -5 cm 2 /s was considered for ferricyanide (in KCl 0.1 M at 25°C) (273, 274). From this, the electrochemical surface was 55.05 cm 2 , namely between the geometric surface and that deduced from the BET measure.

If compared to plane graphite electrode, the more linear oxygen release observed on the working electrode highlighted a lower conductivity (Fig. 6). The geometric surface seems to minimize the active surface area of the electrode owing to the high current density obtained (Fig. 6). In addition to the external volume, the internal felt volume seems therefore involved in the electrochemical reaction. Contrarily, the BET surface seems to overestimate the active surface area owing to the negligible current density obtained. The electrochemical surface appears consistent if the beginning of oxygen release, occurring at 1.5-1.6 V/SCE, namely close to that observed on a plane graphite felt, is considered.

From this, the oxidation potential of water on graphite felt remains low (1.5-1.6 V/SCE),

showing that it can be considered as an "active" electrode and hence the nature of the working electrode was also in favour of a direct 2,4-D oxidation rather than • OH generation.

The operating conditions

The electrochemical oxidation of 2,4-D was performed at 1.6 V/SCE. Only few studies involving chronoamperometry are available in the literature. However, analytical studies show that the potentials corresponding to the currents applied to BDD electrodes in view of herbicides degradation can easily exceed 3.2 V/SCE [START_REF] Boye | [END_REF], namely highly above the oxidation potential of water on a graphite electrode.

The applied current densities varied widely according to the electrode materials used, the operating conditions (electrolyte and pollutant concentrations, temperature….), but usually remain in the range some tens to a hundred mA cm -2 [37][38][START_REF] Panizza | Importance of electrode material in the electrochemical treatment of wastewater containing organic pollutants[END_REF]. The difficulty to determine the current density applied on the graphite felt has been previously discussed. Indeed, for the graphite felt implemented in the percolation flow-cell (48 mm × 12 mm), the geometric surface was 54.3 cm 2 and those given by the BET and electrochemically were 13300 and 1270 cm 2 . From this and for a usual current of 200 mA, the current density values were 3.7, 0.015 and 0.157 mA cm -2 for the geometric, the BET and the electrochemical surfaces, respectively. Since the active surface should be considered with caution, the current densities appeared low if compared to those generally reported in the literature regarding AOPs, namely involving hydroxyl radicals formation (in the magnitude of some tens mA cm -2 ).

Hydroxyl radicals' dosage

The above arguments are based on analytical observations and assumptions; while the best proof of radicals formation remains their dosage. • OH are known for their strong reactivity with benzene (kinetic rate constant k = 8.10 9 L mol -1 s -1 ), showing the relevance of this compound as a radical scavenger [START_REF] Kochany | [END_REF]41]. Since phenol is the sole product of benzene hydroxylation, the amount of • OH produced can be deduced from the HPLC monitoring of benzene degradation ( =200 nm) linked to phenol production ( =270 nm). The absence of benzene oxidation at the electrode at 1.6 V/SCE was first confirmed. Electrolysis was then performed in the flow-cell with recycling of the electrolyzed solution. Neither noticeable benzene degradation nor phenol formation was observed, confirming the absence of • OH formation in the considered experimental conditions, despite the recycling of the electrolysis solution through the cell to favor phenol accumulation.

In the light of the above results, possible involvement of • OH can be rejected in the mechanism of 2,4-D oxidation. Consequently, the proposed pretreatment can be considered as a "direct" electrochemical process instead of an advanced electrochemical oxidation process.

Feasibility of the combined process

The BOD 5 /COD ratio increased from 0.04 initially to 0.25 for the oxidized solution. Owing to this biodegradability improvement, and even if the limit of biodegradability (0.4 [19,42]) was not reached, a biological treatment of the pretreated solution was performed.

Prior to the biological treatment step, possible biosorption on activated sludge of 2,4-D on the one hand and the major by-product, chlorohydroquinone [27], for the electrolyzed effluent on the other hand was assessed and was found to be negligible [26].

Evolution of the mineralization, as well as 2,4-D and Chlorohydroquinone, during activated sludge culture are displayed in Figure 7; the low 2,4-D amount (Fig. 7) should be related to its almost total removal at the end of the electrochemical pretreatment (96% removal), while mineralization level remained limited, 34% [26]. Contrarily to the behavior found for nonpretreated 2,4-D solutions, for which an acclimation period of seven days was needed before noticeable 2,4-D degradation can be observed [26], no significant lag phase can be noticed in the case of the electrolyzed effluent. Indeed, a 66% decrease of the initial DOC amount in the pretreated effluent was measured after only two days of culture (Fig. 7), without significant involvement of biosorption; illustrating a readily assimilation of some degradation products.

Mineralization continued until day 5 of culture, since 77% mineralization yield was measured.

Beyond this time, no really significant decrease of the initial dissolved organic carbon was measured since only less than 10% further mineralization was obtained after 21 days of culture, leading to a final mineralization yield during the biological treatment of 85%, namely 93% overall mineralization yield by means of the combined process. It should be observed that a biological treatment carried out in similar conditions but lacking of any carbon source, especially 2,4-D, was used as a "blank" test; its DOC value, namely 17 mg L -1 , was considered as the reference (100% mineralization) [26]. Residual 2,4-D and the main degradation product, chlorohydroquinone, were also monitored showing their total removal within the first two days of biological treatment for the former and after 14 days for the latter, in agreement with the total 2,4-D removal also observed after 14 days culture for the nonpretreated effluent [26]. Owing to the presence of non-negligible residual DOC amount, the presence of refractory degradation products in the oxidized effluent can be assumed, which was not assimilated even after 21 days of activated sludge culture. Activated sludge acclimation to the by-products should be therefore subsequently considered to improve the biological mineralization of the electrolyzed solution.

Conclusion

The feasibility of an electrochemical process for pesticide pretreatment was shown, since it improved significantly the mineralization rate and thus significantly shortened the length of the biological treatment if compared to non-pretreated 2,4-D solutions. The absence of hydroxyl radical formation was demonstrated showing that the pretreatment can be considered as a "direct" electrochemical process instead of an advanced electrochemical oxidation process. However, its selectivity was not as high as expected owing to the various degradation products identified [27] and to the biorefractory degradation products remaining at the end of activated sludge culture. Improving selectivity constitutes the main objective to continue this work. For this purpose and since most of the persistent organic pollutants being halogenated compounds, the specific removal of halogen groups from the substrate could be considered.

Works are in progress in the laboratory regarding the electrocatalytic reduction of carbonhalogen bonds of some halogenated target compounds, as well as the influence of the dehalogenation on the biodegradability of these compounds.

Figure 1 .

 1 Figure 1. Cyclic voltammetry of a 2,4-D solution (500 mg L -1 ) at pH 2.8 (dots), at pH 11 (dashes) and at pH 2.8 after re-acidification (full line).Voltammograms recorded at 100 mV s - 1 in Na 2 SO 4 0.1 mol L -1 on a glassy carbon electrode (7 mm²).

Figure 2 .

 2 Figure 2. Possible reactions for the reduction of the carboxylic function of 2,4-D.

20

 20

Figure 3 . 1 -

 31 Figure 3. Cyclic voltammetry of 2,4-D (20 mmol L -1dots) and its ester form (20 mmol L -1 full line) in acetonitrile (KPF 6 0.1 mol L -1blank: dashes) on a glassy carbon electrode (7 mm²). Voltammograms recorded at 100 mV s -1 .

Figure 4 .

 4 Figure 4. Cyclic voltammetry of Na 2 SO 4 0.1 mol L -1 as supporting electrolyte (dots), in the presence of 100 mg L -1 2,4-D at pH 3.6 (full line), at pH 11 (empty dashes) and at pH 2.8 after re-acidification (full dahes).Voltammograms recorded at 100 mV s -1 on a glassy carbon

Figure 5 .

 5 Figure 5. Oxygen evolution observed in phosphate buffer (pH 2.7) on various electrode

Figure 6 .

 6 Figure 6. Oxygen evolution observed in phosphate buffer (pH 2.7) on plane graphite

Figure 7 .

 7 Figure 7. Time-courses of 2,4-D (circles), chlorohydroquinone (squares), and Dissolved

  1 ): Na 2 HPO 4 , 334; K 2 HPO 4 , 208; KH 2 PO 4 , 85; CaCl 2 , 27.4; MgSO 4 .7H 2 O, 22.6; NH 4 Cl, 75;

Acknowledgments

We thank Dr Elisabet Dunach-Clinet for helpful discussions.


Ben Faker

Belgacem

Mahmoud Sidi

Kaber

Ill-Conditioning versus Ill-Posedness for the Boundary Controllability of the Heat Equation

Ill-posedness and/or Ill-conditioning are features users have to deal with appropriately in the controllability of diffusion problems for secure and reliable outputs. We investigate those issues in the case of a boundary Dirichlet control, in an attempt to underline the origin of the troubles arising in the numerical computations and to shed some light on the difficulties to obtain good quality simulations. The exact controllability is severely ill-posed while, in spite of its well-posedness, the null-controllability turns out to be very badly ill-conditioned. Theoretical and numerical results are stated on the heat equation in one dimension to illustrate the specific instabilities of each problem. The main tools used here are first a characterization of the subspace where the HUM control lies and the study of the spectrum of some structured matrices, of Pick and Löwner type, obtained from the Fourier calculations on the state and adjoint equations.

keywords: boundary controllability, Dirichlet control, Structured matrices, ill-conditioning, ill-posedness.

Introduction

The subject is the null and exact controllability problems of the heat equation where the control variable is a boundary Dirichlet data. We focus on the instabilities and their effects users should cope with in the numerical computations. The null-controllability is well-posed and the exact-controllability is ill-posed. The purpose is to investigate the one-dimensional model to underline the specific difficulties generated by the instabilities of both problems. Following [START_REF] Russel | Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions[END_REF], we use a Fourier analysis of the heat equation and we show how the Gramians of the exact and null-controllability problems may be expressed by structured matrices of infinite dimensions. The study of the spectrum of these matrices provides the desired information on the instabilities of both problems. This procedure has been first exposed by the authors in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF].

To be more specific, let I be the segment (0, π) of the real axis and T > 0 be a fixed real-number. We set Q = I×]0, T [. The generic point in I is denoted by x and the generic time variable is t. Let y 0 = y 0 (x), x ∈ I be an initial state and v = v(t), t ∈ (0, T ) be a Dirichlet boundary data. Then we consider y v the solution of the heat equation

∂ t y v -∂ xx y v = 0 in Q, (1)
y v (0, •) = 0, y v (π, •) = v on (0, T ), (2)
y v (•, 0) = y 0 on I. (3)
The null-controllability is when the final state is zero. The problem is then to : find Dirichlet condition u(•) such that

y u (x, T ) = 0, ∀x ∈ I (4)
The exact controllability problem may be expressed as follows: find u(•) such that y u (x, T ) = y T (x), ∀x ∈ I,

where y T is a state fixed in advance. Whenever we deal with problem (5), we set the initial condition to zero (y 0 = 0). This is not a restriction and is only chosen for simplification.

HUM method applied to the null-controllability problem allow to prove the existence of the v solution of (4). This control depends continuously on the initial data y 0 (see [START_REF] Fattorini | Russell -Exact controllability theorems for linear parabolic equation in one space dimension[END_REF][START_REF] Fursikov | Imanuvilov -Controllability of Evolution Equations[END_REF][START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: The linear case[END_REF][START_REF] Werner | Boundary value controllability and abservability problems for the wave and heat Equation[END_REF]). We refer to [START_REF] Micu | Regularity issues for the null-controllability of the linear 1-d heat equation[END_REF][START_REF] Micu | On the regularity of null-controls of the linear 1-d heat equation[END_REF][START_REF] Munch | Zuazua -Numerical approximation of null controls for the heat equation: Ill-posedness and remedies[END_REF][START_REF] Carthel | Lions-On exact and approximate boundary controllabilities for the heat equation: A numerical approach[END_REF][START_REF] Rasmussen | Boundary Control of Linear Evolution PDEs[END_REF][START_REF] Munch | Pedregal -Numerical null controllability of the heat equation through a variational approach[END_REF] for valuable discussion about related numerical issues. Studies for the problem [START_REF] Engl | Regularization of inverse problems[END_REF] are exposed in [START_REF]Lions -Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF][START_REF] Lattès | Lions -Méthode de quasi-réversibilité et applications[END_REF][START_REF] Li | Zhang -Recent progress on Controllability/Observability for systems governed by Partial Differential Equations[END_REF] and conclude to the failure of the failure of the exact-controllability. Some states y T are not reachable. Moreover, recent developments realized in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF] gives a better insight about its severe ill-posedness. The basic tool for the proof is an precise spectral analysis of structured matrices of Pick, Löwner and Cauchy types (see [START_REF]Vavrín -Inverses of Löwner matrices[END_REF][START_REF] Schechter | On the inversion of certain matrices[END_REF]).

The aim of this contribution is to take benefit of the ideas initiated in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF] to investigate further the instabilities related to exact and null controllability problems and to identify their precise nature. This way to proceed is based on a use of the elegant structured matrices and is an alternative to the analysis of bi-orthogonal sequences used in a wide literature. We put therefore the controllability problems ( 4) and ( 5) under algebraic form. The infinite-dimensional-vectors, solutions of each of the resulting equations allow the reconstruction of the HUM control, the one with minimal norm in L 2 (0, T ), the space of square integrable functions defined on (0, T ). The process we follow is constructive and consists in expressing the Gramians of both problems by means of Pick and Löwner matrices. To make so, we use a Fourier basis in L 2 (I) and a system of exponential functions that forms a total system in a closed subspace of L 2 (0, T ), where the active controls are sought for. Each of those matrices enjoys a particular structure, the one corresponding to (4) is of a Löwner type and the one associated to (5) is a Pick matrix which a sum of two Cauchy matrices. Their 'instability' can be clarified after realizing a sharp study of their spectrum. Writing them as the solution of a Sylvester or a Lyapunov equation, and using estimates of Penzl's type on their eigenvalues (see [START_REF] Sabino | Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method[END_REF]) allow us to illustrate the (extremely) high ill-conditioning of the null-controllability and the severe ill-posedness of the exact-controllability. These instabilities, though fundamentally of different nature, are causes of tremendous complications in the numerical simulations to approximate the control.

An outline of the paper is as follows. In Section 2, we describe the exactand null-controllability of the heat equation and we provide an accurate representation of the HUM control that has an important role in our analysis. Section 3 summarizes and completes the results recently elaborated in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF] and [START_REF] Micu | Regularity issues for the null-controllability of the linear 1-d heat equation[END_REF] about the exact controllability problem [START_REF] Engl | Regularization of inverse problems[END_REF]. Following the lines of [START_REF] Fattorini | Russell -Exact controllability theorems for linear parabolic equation in one space dimension[END_REF][START_REF] Russel | Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions[END_REF], we put the controllability problem under an algebraic form with an infinite structured matrix. The ill-posedness degree is therefore derived after studying the eigenvalues of that matrix. Applying Penzl's type estimate to this matrix yields the severe ill-posedness of the exact-controllability problem. Section 4 is dedicated to the null-controllability problem (4) which is well posed. That result is known [START_REF] Fattorini | Russell -Exact controllability theorems for linear parabolic equation in one space dimension[END_REF], [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF]. The objective we have in mind here is to investigate the spectrum of the Gramian of the null-controllability problem so to eventually evaluate the condition number of the problem. Tools we use have been developed for the matrices that are solutions of Lyapunov or Sylvester equations (see [START_REF]Penzl -Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case[END_REF][START_REF] Sabino | Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method[END_REF] and references therein). The analysis conducted here concludes to a very high ill-conditioning of the problem. This confirms why solving the null-controllability causes high numerical troubles. To close this work, some Matlab computations in Section 5 allow us to check out numerically the theoretical results developed here.

Notation -Let X be a Banach space endowed with its norm • X . We denote by C (0, T ; X) the space of continuous functions v from

[0, T ] in X such that v C (0,T ;X) = sup t∈[0,T ] v(t) X < +∞.
We need also in some places the Hilbert space ℓ 2 (R) of square summable real sequences. Finally, we consider the Sobolev space H 1 (I) of all the functions that belong to L 2 (I) together with their first derivatives. The space H 1 0 (I) is then the closure in H 1 (I) of the space D(I) of infinitely differentiable functions with a compact support in I. We denote by H -1 (I) its dual space.

Control's Representation

Let the Dirichlet data v be given in L 2 (0, T ). The parabolic problem (1)-(3) has a unique solution y v in L 2 (Q) which belongs to C ([0, T ]; H -1 (I)) (see [START_REF]Lions -Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués[END_REF]Theorem 9.1]). It may occur that for some v ∈ L 2 (0, T ), the state y v (•, T ) does not belong to L 2 (I), we are therefore led to work with the unbounded operator B, defined by Bv = y v (•, T ).

Its domain

D(B) = v ∈ L 2 (0, T ); y v (•, T ) ∈ L 2 (I)
is dense in L 2 (0, T ) and following [1, Lemma 2.4], B is closed. The adjoint operator B * is well defined and is specified as follows (see [START_REF] Lattès | Lions -Méthode de quasi-réversibilité et applications[END_REF][START_REF] Ben Belgacem | El Fekih-Dirichlet boundary control for a parabolic equation with a final observation I: A space-time mixed formulation and penalization[END_REF]): let ψ be given in L 2 (I) and q ψ the solution of the backward problem (see [START_REF] Lions | Magenes -Problèmes aux limites non homogènes et applications[END_REF]Chap. 4])

-∂ t q ψ -∂ xx q ψ = 0 in Q, (6) q ψ (0, •), q ψ (π, •) = 0 on (0, T ), (7) q ψ (•, T ) = ψ on I. (8)
The domain of

B * D(B * ) = ψ ∈ L 2 (I); ∂ x q ψ (π, •) ∈ L 2 (0, T ) ,
is dense and we have

B * ψ = -∂ x q ψ (π, •).
The properties of both operators B and B * have been listed in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF]. The operator B * turns out to be injective:

Ker B * = {0}.
The direct consequence is that the range R(B) is dense in L 2 (I). The set of reachable states is thus dense in L 2 (I) ; this is the approximate controllability (see [START_REF] Lattès | Lions -Méthode de quasi-réversibilité et applications[END_REF]). The set of non-reachable states L 2 (I) \ R(B) is also dense in L 2 (I), see [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF]Lemma 3.4]. B has therefore a non-closed range and cannot be continuously invertible. This means that any deviation of a reachable state y T ∈ R(B), may produce a new state which is not reachable ( ∈ R(B)).

Then, equation (5) may fail. The determination of the closure of the range of B * and by then of the kernel of B is a little bit more complicated. It will be obtained after carrying out some Fourier computations and using a sharp variant of Müntz theorem.

The methodology followed here has been used earlier in [START_REF] Fattorini | Russell -Exact controllability theorems for linear parabolic equation in one space dimension[END_REF][START_REF] Micu | Regularity issues for the null-controllability of the linear 1-d heat equation[END_REF]. For a given ψ ∈ L 2 (I), we consider the Fourier expansion,

ψ(x) = k≥1 ψ k sin(kx) ∀x ∈ I.
Plugging this series into problem ( 6)-( 8) yields that B * may be expressed as

(B * ψ)(t) = k≥1 (-1) k+1 ke -k 2 (T -t) ψ k , ∀t ∈ (0, T ).
As a consequence, we deduce another characterization of the domain of

B * D(B * ) = ψ ∈ L 2 (I); k≥1 m≥1 (-1) k+m km k 2 + m 2 (1-e -(k 2 +m 2 )T )ψ k ψ m < ∞ .
Notice that the family (e -k 2 (T -t) ) k≥1 is total in R(B * ). Actually the following important result holds. We refer to [2, Lemma 3.4, Remark 4.2], for the proof.

Lemma 2.1 The co-dimension of R(B * ) in L 2 (0, T ) is infinite.
Moreover, all the functions in R(B * ) are analytic in [0, T [ and we have the following representation

R(B * ) = v = k≥1 v k e -k 2 (T -t) , k≥1 m≥1 1 -e -(k 2 +m 2 )T k 2 + m 2 v k v m < ∞ .
The next step consists in deriving a closed form of the operator B itself. It may be obtained by duality (see [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF]). Indeed, let v ∈ D(B). We set for convenience v T (•) = v(T -•) and denote by v T the trivial extension of v T to the semi-axis R + . The operator B may be written under the following form

(Bv)(x) = 2 π k≥1 (-1) k+1 k T 0 v(t)e -k 2 (T -t) dt sin(kx), = 2 π k≥1 (-1) k+1 kL (v T )(k 2 ) sin(kx). (9)
The symbol L (v T ) denotes the Laplace transform of the extended function

v T ∈ L 2 (R + ). An explicit form of D(B) is therefore D(B) = v ∈ L 2 (0, T ); k≥1 kL (v T )(k 2 ) 2 < ∞ .
Above all, the kernel of B is given as follows

Ker B = v ∈ D(B); L (v T )(k 2 ) = 0, ∀k ≥ 1 .
Thus, Ker B has a countably infinite dimension. Moreover, if y T is a reachable state, then the HUM control u † ∈ R(B * )), solution of (5), is analytic in [0, T [. The same result obviously holds for the HUM control u † , solution of the null-controllability (4).

Exact-Controllability: Ill-posedness

The issue of ill-posedness degree of problem ( 5) has been substantially clarified recently in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF] witch concludes to the exponential ill-posedness. The proofs here are based on the properties of some structured matrices. We summarize and complete in this section the ideas exposed in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF]. We introduce a direct transformation of problem ( 5) which leads to the analysis of a linear system of infinite dimension. To proceed with the details we fix the initial condition to zero, y 0 = 0 for simplification. Using expression ( 9) for an expansion of y T ∈ L 2 (I)

y T (x) = k≥1 (y T ) k sin(kx) in I,
we derive an algebraic system on the control u

L (u T )(k 2 ) = (-1) k+1 π 2 (y T ) k k , ∀k ≥ 1. (10)
This is a moment problem (see [START_REF] Fattorini | Russell -Exact controllability theorems for linear parabolic equation in one space dimension[END_REF]). Users are most often interested in the HUM control u † , the unique (hypothetic) one that lies in R(B * ) and is the cheaper control with respect to the energy cost. Owing to Lemma 2.1, we may write

u † (t) = m≥1 (u † ) m e -m 2 (T -t) , ∀t ∈ (0, T ).
Replacing in equation ( 10) implies

m≥1 1 -e -(k 2 +m 2 )T k 2 + m 2 (u † ) m = (-1) k+1 π 2 (y T ) k k , ∀k ≥ 1. (11)
Let us thus define the infinite vectors u † = ((u † ) m ) and y T = ((y T ) m ) in R N * , and the infinite symmetric matrices

C T and D in R N * × R N * with entries c k,m = (1 -e -(k 2 +m 2 )T ) k 2 + m 2 , ∀k, m ≥ 1, d k,m = (-1) k+1 π 2 δ k,m k , ∀k, m ≥ 1.
δ k,m is the Kroenecker symbol. The matrix C T is a Pick type matrix (see [START_REF] Kailath | Sayed -Displacement structure: Theory and applications[END_REF]). With these new notations, using [START_REF] Kailath | Sayed -Displacement structure: Theory and applications[END_REF] we obtain formally

u † = (C T ) -1 Dy T . (12)
To assign a mathematical sense to it, we have to specify vectors u † that are representative of functions u † ∈ L 2 (0, T ). The necessary and sufficient condition for this to hold is

u † 2 L 2 (0,T ) = k≥1 m≥1 1 -e -(k 2 +m 2 )T k 2 + m 2 (u † ) m (u † ) k = (Cu † , u † ) < ∞.
This may be reworded in terms of the data vector y T as follows

(Dy T , (C T ) -1 Dy T ) ℓ 2 (R) = (y T , D(C T ) -1 Dy T ) ℓ 2 (R) < ∞.
The infinite matrix D(C T ) -1 D is unbounded, positive definite and is not of a closed range. The inverse operator A T = D -1 C T D -1 is actually the Gramian of the boundary controllability of the heat equation and is also unbounded. It has been checked out in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF] that it may be viewed as representing the operator BB * . The failure of the exact-controllability can be explained in terms of the properties of this Gramian matrix A T . Indeed, there holds that sup

y∈ℓ 2 (R) (y, (A T ) -1 y) ℓ 2 (R) y 2 ℓ 2 (R) = ∞.
Or equivalently inf

y∈ℓ 2 (R) (y, A T y) ℓ 2 (R) y ℓ 2 (R) = 0. ( 13
)
This may be established after investigating the spectrum of the principle sub-matrices (a k,m ) 1≤k,m≤N of dimension N . The entries are provided by

a k,m = (-1) k+m 4 π 2 km k 2 + m 2 (1 -e -(k 2 +m 2 )T ), ∀k, m) ≥ 1.
From now on, we drop the alternating terms (-1) k+m and the resulting matrix is still called A N . We denote by ((µ N ) k ) 1≤k≤N the eigenvalues of A N ordered decreasingly. Their asymptotic expansions are given in the following Proposition which improves the one in [2, Proposition 4.1].

Proposition 3.1 We have the following bounds for the largest eigenvalue

2(1 -e -2T ) π 2 ln N ≤ (µ N ) 1 ≤ 2(1 -e -2N 2 T ) π 2 N.
Moreover, the eigenvalues ((µ N ) k ) 1≤k≤N satisfy

(µ N ) 2k+1 ≤ γ T (µ N ) 1 exp - π 2 k 2 ln(2N ) , 1 ≤ k ≤ [(N -1)/2]. ( 14
)
The symbol [•] stands for the integer part. The constant γ T depends on T .

Proof: First of all, we have

(µ N ) 1 ≤ A N ∞ = max 1≤k≤N N m=1 a k,m .
The upper bound follows from

km k 2 + m 2 (1 -e -(k 2 +m 2 )T ) ≤ 1 2
(1 -e -2N 2 T ).

For the lower bound, we use the Collatz-Wielandt formula to get

(µ N ) 1 ≥ 4 π 2 (1 -e -2T ) min 1≤k≤N N m=1 km k 2 + m 2 .
Comparing integrals and sums, we get the lower bound. The proof of ( 14) is given in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF]. The proof is complete.

Remark 3.1 Lemma 3.1 together with Cauchy's Interlace Theorem for symmetric matrices (see [START_REF] Golub | Loan -Matrix Computations[END_REF]) that is

(µ N +1 ) 1 > (µ N ) 1 > (µ N +1 ) 2 > (µ N ) 2 > • • • > (µ N +1 ) N > (µ N ) N > (µ N +1 ) N +1 .
yields that the spectrum of A T is actually a positive sequence stretched from zero to infinity and clustered at these extreme points. The exponential illposedness degree mainly comes from the fact that for a fixed k, the eigenvalue (µ N ) N -k decays exponentially fast towards zero when N grows to infinity. In the other hand, the unboundedness is caused by the fact that (µ N ) k grows up to infinity, at worst like a constant times N , when N tends to +∞.

Null-Controllability: Ill-conditioning

The instability for the null-controllability problem ( 4) is of a completely different nature. In spite of its well-posedness established in [START_REF] Fattorini | Russell -Exact controllability theorems for linear parabolic equation in one space dimension[END_REF], the problem turns out to be intrinsically highly ill-conditioned. This equally implies heavy complications to handle it numerically (see [START_REF] Micu | Regularity issues for the null-controllability of the linear 1-d heat equation[END_REF]). The aim now is to find a control that drives the solution state variable y u from an initial condition y 0 to zero at a given final time T . Resuming the approach exposed in the previous section, we obtain a problem that looks like [START_REF]Güichal -A lower bound of the norm of the control operator for the heat equation[END_REF], where the final state y T corresponds to a trajectory initiated at y(•, 0) = y 0 ∈ L 2 (I), with a homogeneous Dirichlet condition at x = π. All computations achieved, we derive the equation

L (u T )(k 2 ) = (-1) k π 2 e -k 2 T k (y 0 ) k , ∀k ≥ 1.
The construction of the HUM control is here again based on the representation in Lemma 2.1, and the vector u † = (u † m ) m∈N * solves the following equation

m≥1 1 -e -(k 2 +m 2 )T k 2 + m 2 (u † ) m = (-1) k π 2 e -k 2 T k (y 0 ) k , ∀k ≥ 1.
Put under a condensed form, it may be rewritten as

C T u † = Q T y 0 (15)
with C T ∈ R N * × R N * defined in the previous section and Q T is the diagonal matrix with entries

q k,m = (-1) k π 2 e -k 2 T k δ k,m ∀k, m ≥ 1.
As indicated earlier, C T is compact in ℓ 2 (R) and so is the operator Q T .

Remark 4.1 Problem (15) may be also reworded in the same form as (12)

u † = (L T ) -1 Dy 0 .
The matrix L T = -diag (e λ k T )C T is a Löwner type matrix.

The null controllability problem (4) is well posed (see [START_REF] Fattorini | Russell -Exact controllability theorems for linear parabolic equation in one space dimension[END_REF][START_REF] Russel | Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions[END_REF]). A particular consequence is the following inclusion on the range subspaces R(Q T ) ⊂ R(C T ). Equation ( 15) has thus a unique solution

u † = (C T ) -1 Q T y 0 .
Recall that u † is the HUM control and according to [START_REF] Fattorini | Russell -Exact controllability theorems for linear parabolic equation in one space dimension[END_REF] and [START_REF] Zuazua | Controllability of Partial Differential Equations[END_REF] we have

u † L 2 (0,T ) ≤ α T y 0 L 2 (I) .
This stability is equivalent to

(y 0 , Q T (C T ) -1 Q T y 0 ) ℓ 2 (R) ≤ α T y 0 ℓ 2 (R) .
The smallest constant α T is the control cost. Be aware that (C T ) -1 is unbounded as the inverse of a compact operator. Nevertheless, the estimate above is possible because of the compactness of Q T which compensates and even dominates the unboundedness of C T . Setting (B T )

-1 = Q T (C T ) -1 Q T
which is a symmetric positive definite operator, the above stability reads as

τ := inf y∈ℓ 2 (R) (y, B T y) ℓ 2 (R) y 2 ℓ 2 (R) > 0.
The constant τ coincides with 1 (α T ) 2 and is dependent on the final time T . It may be too small for fast controls. Indeed, it is proved in [START_REF]Güichal -A lower bound of the norm of the control operator for the heat equation[END_REF] (see also [START_REF]Seidman -How violent are fast controls, III[END_REF]), that α T increases exponentially fast with 1 T for small T .

We explain now the causes of the instabilities of the null-controllability, despite its well-posed. The point then is to prove the high ill-conditioning of the operator B T . Actually, the result we pursue is concerned with the principal sub-matrices B N of dimension N . We aim at showing that their condition numbers blow-up exponentially fast with respect to N . We start by writing down the coefficients of B T ,

b k,m = (-1) k+m 4 π 2 km k 2 + m 2 (e (k 2 +m 2 )T -1), ∀k, m ≥ 1.
The important issue is to investigate the spectrum of B N . Remark that the matrix B N is equivalent to the one with positive entries, obtained as the absolute values of (b k,m ) k,m≥1 . We subsequently consider this new matrix that we still denote abusively B N . Its coefficients are provided by

b k,m = 4 π 2 km k 2 + m 2 (e (k 2 +m 2 )T -1), ∀k, m ≥ 1.
Let ((τ N ) k ) 1≤k≤N be the eigenvalues of B N ordered decreasingly. Using the min-max principle, it is possible to state that the smallest eigenvalue (τ N ) N ≥ τ decreases towards τ , with large N . As a result (τ N ) N ≥ τ is uniformly bounded with respect to N . Next, to figure out the asymptotics of the largest eigenvalues of B N , we need to define some auxiliary matrices. Let us define the diagonal matrix D N = diag {k 2 } 1≤k≤N and the vectors ℓ N = (ke k 2 T ) 1≤k≤N and ℓ ′ N = (k) 1≤k≤N . It is easily checked out that the matrix B N satisfies the Sylvester or Lyapunov equation

D N B N + B N D N = L N , (16)
with

L N = ℓ N ℓ * N -(ℓ ′ N )(ℓ ′ N ) * .
The asymptotics of the eigenvalues of B N depend on the properties of the diagonal matrix D N and the displacement rank of equation ( 16) which is the rank of L N . Estimates established by T. Penzl in [START_REF]Penzl -Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case[END_REF]2000], and then improved by several authors are well-fit to provide the desired estimates. We refer to J. Sabino [START_REF] Sabino | Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method[END_REF]2006] and to the wide bibliography therein. The following result holds

(τ N ) 2k+1 (τ N ) 1 ≤ γ T exp - π 2 k 2 ln(2N ) , 1 ≤ k ≤ [(N -1)/2].
Proof: After observing that the displacement rank of equation ( 16) equals two, we apply the estimate in [25, Theorem 2.1.1] (see also [START_REF] Sabino | Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method[END_REF]Formula (2.14) in p. 46]). We get the following bounds (with κ(D N ), the condition number of D N ):

(τ N ) 2k+1 (τ N ) 1 ≤ γ T exp - π 2 k ln(4κ(D N ))
with a constant γ T independent of N . The displacement rank of the Lyapunov equation influences the counting of the eigenvalues. The condition number κ(D N ) being equals to N 2 , we derive the bound.

(τ N ) N (τ N ) N -2k ≤ γ T exp - π 2 k 2 ln(2N ) , 1 ≤ k ≤ [(N -1)/2].
The constant γ T is independent of N .

Proof: The proof is a fac-simile of the previous one. The difference is that we need to transform the Lyapunov equation ( 16) as follows

D N (B N ) -1 + (B N ) -1 D N = (B N ) -1 L N (B N ) -1 .
The displacement rank is not modified and remains equal to two. Applying once again [25, Theorem 2.1.1] yields that

[(τ N ) N -2k ] -1 [(τ N ) N ] -1 = (τ N ) N (τ N ) N -2k ≤ γ T exp - π 2 k 2 ln(2N )
.

The proof is complete.

Remark 4.2 An immediate consequence is that the condition number κ B N of matrix B N blows up at least exponentially fast with N . This is why solving numerically the null-controllability is a hard computational task. To bound κ B N from below we observe that, in view of the symmetry and the positivity of the matrix B N , we have that

κ B N = (τ N ) 1 (τ N ) N .
Selecting k = [(N -1)/2] in Lemma 4.1, we obtain the following preliminary bound

κ B N ≥ γ T exp π 2 N 4 ln(2N )
.

The general theory initiated by T. Penzl [START_REF]Penzl -Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case[END_REF] about the spectrum of the solutions of Sylvester equations with a low displacement rank predicts that, for a given N , the eigenvalues (τ N ) N -k grows faster than γ T exp π 2 k 4 ln(2N ) . This result is not optimal. Actually, more direct algebraic tools enable to derive an accurate result, at least about (τ N ) 1 . Proposition 4.1 We have

1 2N (e (N 2 +1)T -1) ≤ (τ N ) 1 ≤ N 2 (e 2N 2 T -1).
Proof: Bounding (τ N ) 1 may be obtained owing to the double formula (see [START_REF] Meyer | Matrix analysis and applied linear algebra[END_REF]Chapter 8]), min

1≤k≤N 1≤m≤N km k 2 + m 2 (e (k 2 +m 2 )T -1) ≤ (τ N ) 1 ≤ max 1≤k≤N 1≤m≤N km k 2 + m 2 (e (k 2 +m 2 )T -1).
We deduce that

N 1 + N 2 (e (N 2 +1)T -1) ≤ (τ N ) 1 ≤ N 2 (e 2N 2 T -1).
The proof is complete.

Remark 4.3 A direct a result of the proposition is that the condition number κ B N satisfies the more accurate minoration

κ B N ≥ γ T N e N 2 T .

Numerical Discussion

We conduct a numerical investigation of the Gramians of the controllability problems from the spectra point of view. The aim is to assess the theoretical predictions and to illustrate the instabilities that arise in the computations for the (boundary) controllability of diffusion process. It is also the opportunity to underline the important role the structured matrices have in elucidating those instabilities. We run first some computations on the eigenvalues (µ k ) k∈N of A T , the Gramian for the exact-controllability and (τ k ) k∈N of B T , the Gramian for the null-controllability. Next, we consider the exact-controllability problem where the final target y T is non-smooth and the null-controllability with a given non-smooth initial condition. The control u † is obtained by the inversion of the systems ( 15) and [START_REF]Güichal -A lower bound of the norm of the control operator for the heat equation[END_REF]. All these computations are made within Matlab.

Spectra of the Gramians. We rather check the truncated matrices A N and B N for some values of N . Computations are obtained for cut-off parameter N ranging from 3 to 15. Figure 1 depicts the spectra of A N and B N for different control times T = 0.25, 1 and 1.5. The panels to the left are related to the exact-controllability and those to the right correspond to the nullcontrollability. The main feature of the spectra for the exact-controllability Gramian has already been pointed at in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF]. Most of the eigenvalues are clustered around zero when N grows higher. Actually, as shown in the estimate [START_REF]Lions -Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués[END_REF], they decrease rapidly toward zero. This is a typical sign of severe illposedness which causes strong instabilities for the exact-controllability problem. Notice that the largest eigenvalues grow slowly towards infinity but has a tiny incidence on the instabilities of the problem. The final conclusion here is that the ill-posedness degree of the exact-controllability is exponential. A glance to the results for T = 0.25, shows that ill-posedness is aggravated when a fast controllability is aimed at.

For the null-controllability, the observations concerning the spectra are different , even though their structure is also cause of heavy lack of stability. For a moderate control time that is T = 1 or 1.5, the smallest eigenvalue are clearly bounded away from zero. This is a consequence (and also an indication) of the well-posedness of the null-controllability problem. More significantly, the information one may draw from the distribution of those eigenvalues are twofold. In one hand side, despite the well-posedness, the Gramian and then the problem (15) are highly ill-conditioned. The condition number of the principal sub-matrices (B N ) N of B T (with T = 1) increases tremendously fast to reach extremely high values; for instance κ B 14 comes close to 10 200 . This predicts that the corresponding algebraic problem is too hard to invert. In the other side, when a fast control is aimed at, as for T = 0.25, a fraction of the eigenvalues become dangerously small, and things happen as if the null-controllability were ill-posed. This sounds in agreement with the result found in [START_REF]Güichal -A lower bound of the norm of the control operator for the heat equation[END_REF] (see also [START_REF]Seidman -How violent are fast controls, III[END_REF]), where the control cost is proved to blow-up exponentially fast in 1 T . Actually the singular behavior seems highly stiff with respect to 1 T . To make an accurate estimation of that behavior, Figure 2 depicts in a semi-logarithmic scale the curve of the variations of the smallest eigenvalue (τ N ) N (N = 15) with respect to 1 T for T ranging in [0.3, 1.5]. The linear regression yields that (τ N ) N ≈ 70 exp(- 7.5 T ). This means that the control cost is such that γ T = 1 √ (τ N ) N ≈ 0.12 exp( 3.75 T ). Before closing the chapter on the spectra of the Gramians there remains to find out the behavior of the eigenvalues (µ N ) k and (τ N ) k with respect to k, when N is fixed. In Figure 3 are plotted, again in semi-logarithmic scales, the curves of those eigenvalues with respect to k when N is fixed to 15, and for two final control times T = 0.25 and T = 1.0. The panel to the left is for the exact-controllability spectra (µ 15 ) 1≤k≤12 and the one to the right is for the null-controllability eigenvalues (τ 15 ) 1≤k≤15 . The shape of the curves suggests that (τ 15 ) k increases as exp(η T k 2 ) while (µ 15 ) k decreases like exp(-η T k) with η T , η ′ T > 0. The slight bending of both curves of the exactcontrollability makes us feel that (µ 15 ) k rather decreases like k α exp(-η ′ T k) for some α, η ′ T > 0.

Exact controllability. We start by the investigating the exact-controllability problem [START_REF] Engl | Regularization of inverse problems[END_REF]. As already noticed in [2, Section 6], due to the exponential illposedness, incautiously processing the truncated system [START_REF] Kailath | Sayed -Displacement structure: Theory and applications[END_REF] without any regularization produces erratic results. The Lavrentiev procedure is therefore used for stabilizing the computations. We are hence lead to compute the solution u † ̺ of the regularized problem (compare with [START_REF] Kailath | Sayed -Displacement structure: Theory and applications[END_REF])

̺(u † ̺ ) k + m≥1 1 -e -(k 2 +m 2 )T k 2 + m 2 (u † ̺ ) m = (-1) k+1 π 2 (y T ) k k , ∀k ≥ 1. ( 17
)
This system is symmetric positive definite as long as ̺ > 0. A judicious selection of ̺, so to ensure a balancing between the accuracy and the stability of the computations, is an important point and has been widely addressed for ill-posed algebraic systems. We refer for instance to [START_REF] Engl | Regularization of inverse problems[END_REF] for a wide exposition about that issue. Now, we seek for a Dirichlet control function u(•) that allows to realize a prescribed target y T after a time duration T = 1. That state, we hope to reach (numerically), is represented in the left diagram of Figure 4 and is given by y T (x) = χ(x), with

χ(x) = x, ∀x ∈ [0, π 2 ], π 2 , ∀x ∈ [ π 2 , π],
The singularity at π/2, the mid point of the interval, contained in the state y T makes it non-exactly reachable. However, owing to the approximate controllability result, we may steer the trajectory arbitrarily close to that y T . The curves to the left in Figure 4 plot the numerical states computed by the Lavrentiev method as in equation ( 17). The cut-off frequency is fixed to N = 150 which guarantees a rich representation of u(•) in the exponential basis given in Proposition 2.1. Various values of the parameter ̺ are experienced. Here we retain ̺ = 10 -3 , 10 -4 and 10 -5 . Actually, ̺ = 10 -4 seems to provide the best numerical simulation. Nevertheless, one may consider that, in all these cases, the state y T is approximated with a satisfactory accuracy.

In the contrary, a glance to the shape of the controls u † ̺ , depicted in the right panel shows that it is sensitive to the variations of the parameter ̺. The L 2 -norm of u † ̺ , recorded in Table 1, corroborates such an observation. This has to do with the severe ill-posedness. Before switching to the next example, let us point out the smoothness of the computed controls away form the final time T , which is in agreement with the discussion in Section 2.

̺

10 -3 10 -4 10 -3 u † ̺ L 2 (0,T ) 6.17 7.25 7.75

Table 1: Dependence of the norm of the computed controls with respect to ̺, for the exact-controllability.

Null controllability. Consider now the null-controllability problem (4). The initial state coincides with χ, the one used in the previous simulation as the final state. Actually, we intend the construction of a Dirichlet control u(•) that drives the trajectory to zero after a time T = 0.5. The results are obtained by solving equation ( 15) after a Lavrentiev regularization as in [START_REF] Meyer | Matrix analysis and applied linear algebra[END_REF].

In the left panel of Figure 5, are plotted the initial state, the one related to the absence of control (i.e. with null Dirichlet condition at x = π ) and the controlled states for three values of the regularizing parameter ̺ = 10 -3 , 10 -4 and 10 -5 . The controls represented in the right panel and their norms provided in Table 2 illustrate the important sensitivity of the control u † ̺ to ̺. The variations of the norm with respect to ̺ is more significant here than in the first example, due to a shorter controllability time: here, T = 0.5 versus T = 1 previously. Before closing the numerical chapter, let us draw the attention that matlab fails in inverting the (Löwner) matrix (Q N ) -1 C N for N ≥ 38, despite the fact that the exact-counterpart (Q T ) -1 C T is arithmetically continuously invertible. The reason is that the condition number of this matrix grows up violently with N as predicted in Proposition 4.1. For an illustration, matlab evaluates the condition number of that matrix to 0.24 × 10 28 for N = 10 and to 0.5 × 10 305 when N = 37. ̺ 10 -3 10 -4 10 -3 u † ̺ L 2 (0,T ) 7.41 13.18 22.41

Table 2: The norm of the computed controls versus ̺, for the null-controllability.

Conclusion

We aim to know a little more about the instabilities of the (boundary) controllability of the heat equation. This may help adapting, improving and strengthening (already) cautious numerical approximations. Understanding the exponential ill-posedness of the exact-controllability, initiated in [START_REF] Ben Belgacem | On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree[END_REF], is deepened and sharpened. The analysis of the null-controllability problem unveils its intrinsic instability. Despite its well-posedness, this problem turns out to be badly-conditioned in a very high extent. The spectrum of the Gramian grows violently. Moreover, when fast controls are sought for, the smaller fraction of the spectrum is substantially shifted toward zero. The problem suffers then from two drawbacks. In addition to the stiffening of the condition number, it reacts somehow like a severe ill-posed problem as predicted in [START_REF]Güichal -A lower bound of the norm of the control operator for the heat equation[END_REF][START_REF]Seidman -How violent are fast controls, III[END_REF]. The feasible approaches used nowadays by engineers for the null-controllability consist in tackling it as an exact-controllability problem with a reachable state. The alternative that is to design performing algorithms that would specifically take profit of the well-posedness is likely doomed to fail. The most credible approach, at least so far, seems to be the one exposed in [START_REF] Micu | Regularity issues for the null-controllability of the linear 1-d heat equation[END_REF], for instance.

Lemma 4 . 1

 41 The eigenvalues ((τ N ) k ) 1≤k≤N of B N satisfy the bound

Lemma 4 . 2

 42 The eigenvalues ((τ N ) k ) 1≤k≤N of B N satisfy the bound

Figure 1 :

 1 Figure 1: Eigenvalues of A N (panels in the left) and B N (panels in the right) for various cut-offs N . The top diagrams are for T = 0.25, the middles for T = 1 and the bottoms for T = 1.5. Notice the scales in the horizontal axis, 10 16 for A N and 10 260 for B N .

Figure 2 :

 2 Figure 2: The variation of the smallest eigenvalue (τ N ) N with respect to 1 T (we fix N = 15).

Figure 3 :

 3 Figure 3: The left panel depicts the dependence of (µ 15 ) k with respect to k. The right panel is for the variation of (τ 15 ) k in k.

Figure 4 :

 4 Figure 4: The exact target y T and the numerical final states obtained by Lavrentiev's method with various regularization parameters ̺ (left panel). The related controls (right panel).

Figure 5 :

 5 Figure 5: The null-controllability example. The initial state y 0 (plain line), the one obtained after a time T = 0.5, without any control and the controlled states obtained by various regularization parameters ̺ (left panel). The related controls (right panel).