C Julien

P Paulmier
email: pascal.paulmier@onera.fr

F Laurin

DAMAGE AND STRENGTH ANALYSIS OF OPEN-HOLE LAMINATED PLATES UNDER TENSILE, COMPRESSIVE AND BENDING LOADINGS

Keywords: Damage, Failure, Laminate, open-hole plates, Bending

The present paper deals with an experimental study on the damage and failure mechanisms encountered in laminated open-hole plates subjected bending loading. An original experimental device has been proposed and consists in applying compression with pivots, as an alternative to fourpoint bending tests. Its main advantage is that there is no contact between cylindrical bars and the sample and thus no local indentation. Moreover, the tests have been multi-instrumented with stereodigital images correlation, acoustic emission, strain gauges and LVDT sensor to establish the damage and failure scenarios under bending loading. The influence of the hole diameter and of the total thickness on the failure load have been experimentally studied. The present experimental configurations are interesting test cases for advanced modelling because two stress gradients are generated in the sample, one through-the-thickness due to the bending and another one in-the-plane due to the geometrical singularities.

Introduction

In the latest generation of civil aircraft (B787 Dreamliner or A350-XWB), some primary structures (such as the centre wing box, the wings or the fuselage) are manufactured with laminated composite materials. These composite components present complex geometries, with geometrical singularities, but are also subjected to complex multiaxial loadings. In the literature, the analysis of damage and failure of open-hole laminated plates under uniaxial tensile loading with different configurations (hole diameter, thickness of the ply,…) has been widely studied both from an experimental and numerical point of view [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF][START_REF] Hallett | Modelling the interaction between matrix cracks and delamination damage in scaled quasi-isotropic specimens[END_REF][START_REF] Wisnom | The role of delamination in strength, failure mechanism and hole size effect in open hole tensile tests on quasi-isotropic laminates[END_REF]. Moreover, few experimental studies have been performed to study the global response of open-hole plates subjected to uniaxial compressive loadings, especially for different hole diameters [START_REF] Lee | Thickness effect on the compressive strength of T800/924C carbon fibre-epoxy laminates[END_REF][START_REF] Lee | Measuring the notched compressive strength of composite laminates: Specimen size effects[END_REF]. However, to our knowledge, the strength analysis of open-hole plates subjected to bending loadings has not been deeply studied in the existing literature. These test configurations on open-hole plates are still challenging for the current advanced damage and failure approaches. Indeed, a stress gradient is generated in the vicinity of the perforation and an additional stress gradient through-the-thickness is induced by the bending loading. Therefore, a large test campaign on laminated open-hole plates subjected to tension, compression and bending loading has been performed at Onera to establish clearly the damage and failure scenario. In the section 2.1, an original experimental device to apply bending to open-hole plates, which is an alternative to the classical four-point bending loading device, is presented. In the section 2.2, the tested specimens, presenting different hole diameters, different total thicknesses and different lay-ups, are detailed. All the tests have been multi-instrumented (see section 2.3) using different measurement techniques, such as stereo-digital images correlation, acoustic emission, LVDT sensor to establish the damage and failure scenarios for such a loading. The test results are presented in section 3. Firstly, the damage and failure scenario are presented in section 3.1 for specimens manufactured only with UD plies. Secondly, the damage and failure scenario for multi-layered specimens manufactured with 0°, ±45° and 90° plies, are presented in section 3.2. Finally, in section 3.3, for the different tested lay-ups, the influence of the hole diameter and of the total thickness are discussed.

Specimens and experimental devices

Presentation of the compression with pivots experimental device

Three different test devices have been used in this study to subject laminated open-hole plates respectively to tension, compression and bending. An electro-mechanical Schenck machine (150 kN maximum capacity) is used for the three kinds of loading. The tests are performed in the machine controlled displacement mode and a constant displacement rate is imposed at 0.01 mm/s. The tension and compression experimental devices ensure the existing norms [START_REF] Aitm | Airbus Test Method: Determination of Plain, Open Hole and Filled Hole Tensile Strength[END_REF][START_REF]Airbus Test Method: Determination of Plain, Open Hole and Filled Hole Compression Strength[END_REF] and are quite similar to those used in previous experimental studies [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF][START_REF] Lee | Measuring the notched compressive strength of composite laminates: Specimen size effects[END_REF]. Nevertheless, for bending loadings, there are some alternatives to the classical four-point bending experimental device. Fig. 1 presents an original experimental device of compression with pivots. The idea consists in applying a uniform compression to an open-hole plate clamped in two jaws with pivots. The eccentricity of a given test relates to the offset of the mid-surface of the plate, imposes the ratio of compression versus bending that is imposed to the specimen. It is controlled through the use of a couple of wedges that are inserted adequately in the jaws of the tabs. For a large eccentricity, the specimen is subjected to a quasi-pure bending moment (as encountered with the four-point-bending test device), while for small eccentricity the test tends to a pure compression test. In the present study, large eccentricities, equal to half of the total thickness of the tested specimen, are used in order to obtain quasi-pure bending moment in the gauge length. One of the main advantage of this experimental device, compared to the four-point bending device, consists in avoiding contact between cylindrical bars and the sample and thus preventing local indentation which leads to premature final failure of the specimen. Moreover, another advantage consists in observing easily, during the loading, the two faces of a specimen, subjected respectively to tension and compression, and therefore to establish in a precise manner the damage and failure scenario for such complex bending loading.

Presentation of the tested open-hole plates

The specimens have been manufactured from T700GC/M21 UD prepreg plies with an aerial weight of 268 g/m 2 . The nominal ply thickness is assumed equal to 0.262 mm. The choice of the lay-ups has been made in order to cover the usual stacking sequences used in industries. Four different symmetrical lay-ups, constituted of 0°, ±45° and 90° plies, have been tested: (i) a unidirectional [0 8 °]ns ply noted 100/0/0, (ii) a quasi-isotropic [(45°/90°/135°/0°) 2n ] s laminate noted 25/50/25 usually used in order to be subjected to complex multiaxial loadings, (iii) a highly oriented [(45°/0 2 °/135°/90°/45°/0°/135°/0°) n ] s laminate noted 50/40/10 usually used to be subjected to high uniaxial tensile loading, and (iv) a highly disoriented [(45°/90°/135 2 °/ 0°/45 2 °/90°/135°/0°) n ] s laminate noted 20/60/20,designed mainly to be subjected to in-plane shear loading. The parameter n is associated with the number of repetition of the main sequence in each laminate. The different considered stacking sequences are noted X/Y/Z with X% of 0°plies, Y% of ±45° plies and Z% of 90° plies. For each lay-up, two different thicknesses have been considered with n={1,2}, meaning that the 100/0/0 and 25/50/25 laminates are constituted with either 16 plies or 32 plies and the 50/40/10 and 20/60/20 laminates with 20 or 40 plies.

For each lay-up and each total thickness, unnotched specimens are firstly tested as a reference. Then, tests on the open-hole plates, with two different hole diameters (6.35mm and 12.7mm) and a w/d ratio constant and equal to 5, are performed. Each test configuration on unnotched and open-hole specimens is repeated, respectively three or four times, in order to obtain an estimate of scattering, meaning that 72 specimens have been subjected to bending loading. It is worth mentioning that the length of the specimens has been optimised through the analysis of preliminary finite element simulations in order to avoid large deflection (superior to 15mm) at failure (to simplify the analysis of the tests). For the thinnest specimens (n=1), the length is equal to 130 mm while it is equal to 210 mm for the thickest specimens (n=2).

Instrumentation of the bending tests

Different measurement techniques have been used to improve the understanding of the involved physical mechanisms and to obtain global and local information: (i) LVDT sensor for measuring the evolution of the maximum deflection of the unnotched specimens, (ii) acoustic emission for the detection of damage events during loading, (iii) strain gauges, bonded on the upper and lower faces of the unnotched specimens, for strain measurements, (iv) stereo Digital Image Correlation (performed with the Vic3D system) on the two faces of the specimen with black and white paint speckle for tracking the global displacement, the curvature and to obtain an estimate of the in-plane strains and finally (v) post-mortem micrographs in order to study the failure pattern of the specimens. This multiinstrumentation is an absolute necessity in order to establish the damage and failure scenario for the tested specimens under bending.

Test results

In this section, the damage and failure scenarios are established for the laminates manufactured only with 0° plies and then for the other multi-layered laminates manufactured with 0°, ±45° and 90° plies, representative of industrial applications.

Damage and failure scenario for the 100/0/0 laminate

The 100/0/0 unnotched specimens failed due to fibre failure in compression that is due to kinking of the fibres. The first ply failure in fibre mode is catastrophic for the tested structures. The load/displacement curves are very non-linear. Nevertheless, no significant acoustic event (EA) is recorded prior to the failure. Moreover, stereo Digital Images Correlation (DIC) has also shown that no splitting cracks appear in the upper ply subjected locally to tensile loading before the final failure. The measured non-linear load/displacement curves can easily be described with finite element simulations using (i) shell elements, (ii) assuming a linear elastic behaviour for the UD plies until the final failure and (iii) by taking into account the geometrical non linearity for the current test case. This last point is essential to describe accurately these tests.

For the 100/0/0 open-hole plates, damage and failure scenario is rather different, as reported in Fig. 2. The non-linear response of the structure prior to 80% of the failure load can be again described accurately only by taking into account the geometrical non linearity. For higher levels of applied loading, significant acoustic events are recorded. Splitting cracks are observed on the tension face in the vicinity of the hole due to the induced local in-plane shear stresses. The propagation of the splitting cracks can be observed and measured with DIC through the analysis of the discontinuity of the displacement field U y . Moreover, using the out-of-plane displacement field U z measured on the two faces, it is possible to estimate the delaminated area which is here strongly linked to the splitting cracks. It can also be noticed that no transverse cracks are observed on the other face subjected to compression (the measured displacement with DIC prior the final failure presents no discontinuity on this face). The absence of splitting cracks on this face is due to the apparent reinforcement of the interfibre strengths for combined compression and in-plane shear stresses [START_REF] Puck | Failure analysis of frp laminates by means of physically based phenomenological models[END_REF]. The failure occurs when the upper 0° ply fails in longitudinal compression. For the 100/0/0 laminate, the damage and failure scenario is the same for both thicknesses and both hole diameters. The macroscopic stress / displacement curves for all the tested 100/0/0 open-hole plates are reported in Fig 3a . The macroscopic stress is defined by the load divided by the real section. The influence of the hole diameter on the strength is rather limited as compared to the influence of the total thickness as illustrated in Fig. 3a. Moreover, as illustrated in Fig. 3b, which reports the evolution of the cumulated acoustic energy as a function of the applied stress, the onset of matrix damage occurs around 80% of the failure stress for all the tested configurations. Therefore, the influence of the stress gradient through the thickness seems to be predominant as compared with the in-plane stress gradient due to the presence of the perforation. Finite element simulations have to be performed to confirm these preliminary conclusions.

To conclude, the damage and failure scenario observed for 100/0/0 open-hole plates subjected to bending loading is quite complex and constitutes an interesting test case to evaluate the predictive capabilities of the existing damage approaches [START_REF] Kaddour | A comparison between the predictive capability of matrix cracking, damage and failure criteria for fibre reinforced composite laminates: Part A of the third world-wide failure exercise[END_REF].

Damage and failure scenario for the multi-layered laminates

For the multi-layered unnotched specimens (noted 50/40/10, 25/50/25 and 20/60/20 laminates), the final failure of the specimen is due to the first ply failure in compression fibre mode. The observed non linearity of the load/displacement curves is again due to the geometrical non linearity of the problem. Indeed, no significant acoustic event (EA) is recorded prior to the failure while none of the faces show cracking before failure. The damage and failure scenario for the UD or the multi-layered laminates are rather similar for unnotched specimens.

Nevertheless, for open-hole plates, very little damage is observed prior to the final failure for multilayered laminates. Indeed, because of the insertion of ±45° plies, the splitting cracks induced by shear stresses around the hole are very limited and occur only at 90% of the failure load for the most disoriented laminates, as illustrated in Fig. 4b. Therefore, the non-linear response of the disoriented laminated open-hole plates, observed in Fig. 4a, is again due to the geometrical non linearity. For the oriented and the quasi-isotropic laminates, the onset of damage occurs after 95% of the failure load. In order to analyse the influence of the hole diameter and of the total thickness on the global response of the structure, the macroscopic stress / displacement curves are reported in Fig. 4a. For the 20-ply specimens, as well as the 40-ply specimens, the global response is hardly dependent on the hole radius. Therefore, it is mainly the total thickness that drives the global response of the structure subjected to bending. Those conclusions are consistent with those determined previously on 100/0/0 laminates. Moreover, very few localised transverse cracks are observed prior the final failure through the analysis of the strain field  xx in the vicinity of the hole as proposed in [START_REF] Furtado | Selective ply-level hybridisation for improved notched response of composite laminates[END_REF]. No significant delamination can be detected using the out-of-plane displacement field U z prior the final failure. Moreover, the face under compression shows no sign of transverse cracking.

The failure pattern of the tested multi-layered open-hole plates is detailed in Fig. 5. On one hand, the face, subjected locally to compressive stress, fails due to fibre kinking in the 0° ply closest to the surface. This failure mechanism induces a catastrophic decrease of the global rigidity and the observed failure pattern is oriented normal to the compressive loading. On the other hand, due to load transfers, the face, subjected to tension, also fails but due to both transverse cracking (especially in the upper 45° ply as observed in Fig. 5), and tensile fibre failure in 0° plies close the surface. Large delamination cracks are also observed for these specimens. The failure patterns are very similar for every test configuration, as reported in Fig. 5a and Fig. 5b, excepted for 100/0/0 open-hole plates. Indeed, for the latter specimens, on the face subjected to tension, many long splitting cracks and large associated delamination cracks can be observed. On the other face, failure of the 0° plies in compression induces the final failure of the ligament as illustrated in Fig. 5c.

Influence of the hole diameter and of the total thickness on the macroscopic stress at failure

This section is dedicated to the influence of the hole diameter on the strength of open-hole plates subjected to bending. The influence of the total thickness on the apparent strength is also considered. Fig. 6 presents the evolution of the apparent stress at failure as a function of the hole diameter for the two different considered total thicknesses. Firstly, the influence of the hole on the apparent strength is clearly observed experimentally, especially for the most oriented laminates and is quite similar to that observed for open-hole plates subjected to pure compressive loading. Moreover, the thicker the specimen is, the higher the apparent stress at failure is. The influence of the total thickness on the stress at failure can explain why the apparent strengths of the disoriented laminate (20 or 40 plies) are higher than those of the quasi-isotropic laminate (16 or 32 plies), which seems counterintuitive. Finite element simulations have to be performed in order to improve the explanation of the influence of the hole diameter and of the total thickness on the apparent strength of open-hole plates subjected to bending loading.

Figure 1 .

 1 Figure 1. a) Presentation of the compression with pivots experimental device, b) zoom on the jaws with pivots, c) general principle of the TTS offset compression test with pivots.

Figure 2 .

 2 Figure 2. a) Time / load curve for a thin 100/0/0 open-hole plate with a hole diameter equal to 6.35mm and (b) observation of the different damage mechanisms using the displacement fields measured by DIC on the two faces at 8670N (99% of failure load).

Figure 3 .

 3 Figure 3. a) Macroscopic stress / displacement curves and (b) evolution of the cumulated acoustic energy as a function of the applied macroscopic stress for the different tested 100/0/0 open-hole plates subjected to bending.

Figure 4 .

 4 Figure 4. a) Macroscopic stress / displacement curves and (b) cumulated acoustic energy as a function of the macroscopic stress for the tested 20/60/20 open-hole plates subjected to bending.

Figure 5 .

 5 Figure 5. Failure pattern observed on the tension and compression faces a) for a thick disoriented 20/60/20 open-hole plate with d=6.35 mm, b) for a thick oriented 50/40/10 open-hole plate with d=12.7 mm and c) for a thin 100/0/0 open-hole plate with d=6.35 mm.

Figure 6 .

 6 Figure 6. Macroscopic stress at failure as a function of the hole diameter (a) for the thinnest laminates (n=1) and b) for the thickest ones (n=2).

4 .

 4 Conclusions / PerspectivesA large experimental test campaign on laminated open-hole plates has been performed at Onera. Multi-layered laminates have been manufactured with T700GC/M21 unidirectional plies. To apply a bending loading, an original experimental device has been proposed as an alternative to the classical four-point bending setup. It consists in applying a compressive load, offset with respect to the midplane, using jaws with pivots and permits to avoid premature failure under the cylindrical bars and to observe during loading the damage mechanisms occurring at the two different faces of the tested specimen. Open-hole plates have been tested with different stacking sequences, different total thicknesses and different diameters of perforation. These tests have been multi-instrumented with LVDT sensor, stereo-digital images correlation, acoustic emission, and in-situ optical observations to understand the damage and failure mechanisms. Further works should consist in validating the damage and failure scenarios established through the analysis of these tests through finite element simulations using an advanced damage and failure approach already developed at Onera[START_REF] Laurin | A multiscale hybrid approach for damage and final failure predictions of composite structures[END_REF] and validated on uniaxial tensile or compressive tests on open-hole plates.

F. Laurin, C. Julien and P. Paulmier

Acknowledgments

This work was carried out in the framework of the STAF project, directed by Airbus and funded by DGAC (Directorate General of Civil Aviation), which is gratefully acknowledged.


Stella Zevio

Nordine El Hassouni

Manuel Ruiz

Pierre Larmande
email: pierre.larmande@ird.fr

Nordine El

AgroLD indexing tools with ontological annotations

Keywords: Plant Molecular Biology, Linked Data, Elastic

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Agronomy is an overarching field constituting various research areas such as genetics, plant molecular biology, ecology and earth science. The last several decades has seen the successful development of high-throughput technologies that have revolutionized and transformed agronomic research. The application of these technologies have generated large quantities of data and resources over the web. In most cases these sources remain autonomous and disconnected. The Agronomic Linked Data project (AgroLD) is a Semantic Web knowledge base designed to integrate data from various publicly available plant centric data sources. These include Gramene, Oryzabase, TAIR and resources from the South Green platform among many others. The conceptual framework for the knowledge in AgroLD is based on well-established ontologies: Gene Ontology, Plant Ontology, Plant Trait Ontology (TO) and Plant Environment Ontology (EO). The current phase (phase one) covers information on genes, proteins, ontology associations, homology predictions, metabolic pathways, plant traits, and germplasm. Information on the integrated databases, ontologies and identifiers can be found in the documentation page (http://www.agrold.org/ documentation.jsp).

The RDF knowledge bases are accessed via SPARQL endpoints but these endpoints are more suitable for programmatic access. This requires at the minimum a moderate knowledge of SPARQL which is not usually suitable to the nontechnical users (biologists). Consequently, the Semantic Web resources not being exploited completely. Alternatively, the AgroLD website provides four entry points to access the underlying knowledge:

1. Quick Search, a faceted search plugin made available by Virtuoso, that allows users to search by keywords to browse the knowledge contained in AgroLD.

2. SPARQL Query Editor, that provides an interactive environment to formulate SPARQL queries. The SPARQL editor is based on YASQE and YASR tools [START_REF] Rietvelda | The YASGUI Family of SPARQL Clients[END_REF].

3. Explore Relationships visualizer, an implementation of RelFinder [START_REF] Heim | RelFinder: Revealing Relationships in RDF Knowledge Bases[END_REF] that allows the user explore and visualize existing relationships between entities.

4. Advanced Search, a query form providing entity (e.g. gene) specific information retrieval. The Advanced Search query form is based on the REST API suite developed under the AgroLD project.

To further enable the Quick Search functionality by retrieving more textual information and hiding the technical details, we developed an indexation tool that enables to communicate easily with Elastic clusters. Thus, this tool enables to index Json files and to manage indexes (i.e. update, delete) on Elastic clusters without using cURL. Starting from RDF graphs in AgroLD, we used this tool to automate the indexation task and set up Elastic clusters. Furthermore, we developed a generic annotation tool that enables communication with NCBO Annotator [START_REF] Jonquet | The Open Biomedical Annotator[END_REF] to annotate JSON files with ontologies available from such portals (i.e. BioPortal [START_REF] Noy | BioPortal: ontologies and integrated data resources at the click of a mouse[END_REF] and AgroPortal [START_REF] Jonquet | AgroPortal : a proposition for ontology-based services in the agronomic domain[END_REF]). We use this tool with AgroPortal Annotator to enrich and index the Json files with additional information from ontological terms such as labels, synonyms, parent and child terms, etc.


D Locq

Q Barrès

C Ramusat

J.-M Franchet

P Sallot

P Caron

Etude d'alliages Co-Ni à durcissement γ′

Les alliages à base de cobalt durcis par précipitation de phase γ′ sont des matériaux prometteurs pour des applications dans les moteurs d'avions. Des travaux préliminaires ont permis d'étudier leur aptitude à être élaborés par une voie métallurgique conventionnelle et leur capacité de durcissement par application d'un revenu après trempe à partir d'un traitement supersolvus.

Introduction

Les superalliages à base de cobalt actuellement utilisés dans les moteurs d'avions sont durcis par solution solide et par précipitation de carbures. Ils sont principalement choisis pour la fabrication de chambres de combustion du fait, entre autres, de leur facilité de mise en forme ainsi que de leur bonne soudabilité. Les nombreuses tentatives pour faire précipiter une phase durcissante ordonnée L1 2 dans des alliages binaires Co-X (où X = Ti, Al, Nb, Ta ouW), à l'instar des superalliages à base de nickel, n'ont pas abouti essentiellement pour des problèmes de stabilité des phases obtenues [1][2][3]. En ce qui concerne les alliages ternaires, Lee [2] a attribué le durcissement d'un alliage Co-Al-W à des précipités de structure cubique de type Cu 3 Au (L1 2 ). Toutefois, ce sont les travaux de Sato et al. [4] qui ont récemment déclenché un fort regain d'intérêt pour les alliages à base de cobalt durcis par précipitation de phase γ′ de type Co 3 (Al,W). On peut également citer les travaux encore plus récents menés par un groupe de recherche indien qui étudie des alliages quaternaires Co-Al-Mo-Nb [5] ou Co-Al-Mo-Ta [6,7] également durcis par des précipités de type L1 2 .

Démarche

L'objectif de ces travaux est d'élaborer des alliages de structure γ/γ′ et d'étudier leur microstructure au cours des différentes étapes de mise en oeuvre. L'étude d'une coupe du diagramme de phase ternaire proposé par Sato [4] indique qu'à 900°C le domaine biphasé γ/γ′ est restreint. Ainsi le risque de voir apparaître des phases parasites telles que β-CoAl de structure cubique B2 et χ-Co 3 W de structure hexagonale compacte DO 19 est grand si la composition chimique de l'alliage n'est pas parfaitement maitrisée (Figure 1). Shinagawa et al. [8] ont montré qu'il était possible d'étendre le domaine γ/γ′ par une substitution partielle du cobalt par du nickel (Figure 1). [8]. Les traits gris indiquent les limites des phases dans le système ternaire Co-Al-W d'après [4] et la zone grisée le domaine biphaséγ/γ′ du système quaternaire.

Figu re 1. Coupe isotherme à 900°C du diagramme Co-30Ni-Al-W

Matériaux

La composition de chacun des alliages étudiés (Tableau I), inspirée des travaux de [8], découle des choix suivants : -substitution partielle du cobalt par du nickel (20, 30 et 40 % at. de Ni), -taux de phase γ′ visé compris entre 40 et 60%, -faibles ajouts de bore et de carbone pour renforcer les joints de grains.

Tableau I. Composition atomique visée des trois alliages CoNi-0x

Alliage Co Ni Al W C B CoNi-01 base 20 10 7 0,11 0,06 CoNi-02 base 30 10 7 0,11 0,06 CoNi-03 base 40 10 7 0,11 0,06 Un lingot de 3,5 kg de chaque alliage a été synthétisé dans un four à induction sous vide. Un lopin issu de chaque lingot a subi un traitement thermique à 1300°C durant 24 heures afin d'améliorer l'homogénéité chimique et de mettre en solution la phase γ′ ayant éventuellement précipité durant la solidification du lingot. Les lopins traités ont été encapsulés dans une gaine en acier doux et préchauffés à 1200°C durant une à deux heures pour être ensuite filés (rapport de filage de 8). Après dégainage, les barres filées d'alliages CoNi-0x ont un diamètre de 13 à 14 mm et une longueur utile de l'ordre de 540 mm. Des observations par microscopie électronique à balayage (MEB) des échantillons dans l'état brut de filage n'ont pas permis de statuer avec certitude sur la présence de précipités de phase γ′. La microscopie électronique en transmission devrait permettre de lever cette incertitude. En effet, Meher et al. [9] ont pu observer par cette technique des précipités γ′ d'un diamètre de 4 à 12 nm dans un alliage ternaire Co-10Al-10W (% at.) traité 12 heures à 1320°C et trempé à l'eau.

Influence du revenu

Des échantillons des trois alliages filés puis traités (1200°C/1h) ont subi un revenu de 24 ou 96 heures dans la gamme 600-1000°C. Les résultats des essais de dureté (moyenne de 48 indentations) sont présentés dans la Figure 2. Tous les revenus conduisent à un niveau de dureté nettement supérieur à celui des matériaux sans revenu. La dureté augmente avec la teneur en nickel, mais cet effet est très atténué au-delà de 700°C. Cette hiérarchie est à nouveau constatée à 1000°C, très certainement du fait de la proximité de la température de revenu vis-à-vis de celle du solvus γ′ de chaque alliage. Dans les superalliages polycristallins à base de nickel tel que le N18, des précipités γ′ de trempe de taille supérieure à 100 nm sont observés pour des vitesses de refroidissement supérieures à 600°C/min après traitement à haute température [START_REF] Raisson | High Temperature Materials for Power Engineering[END_REF]. Pour les alliages CoNi-0x, de très fins précipités n'ont pu être observés avec certitude qu'à partir des échantillons ayant subi un revenu de 24 heures à 700°C. Au-delà de cette température, des précipités sont observés et leur taille varie de < 30 nm (700°C/24h) jusqu'à 400-450 nm (1000°C/96h). Un exemple de précipitation γ′ est présenté dans la Figure 3 pour le revenu de 96 heures à 900°C. La taille des précipités est de l'ordre de 200-250 nm et la fraction surfacique d'environ 53%.

Conclusion

Cette étude a démontré la possibilité d'obtenir des alliages à base de cobalt durcis par précipitation de phase γ′ par la voie coulé-filé. Un traitement thermique a été nécessaire pour atteindre une recristallisation totale du matériau filé. Le niveau de dureté atteint après un revenu à 700-800°C est proche de celui des superalliages à base de nickel coulés-forgés et traités supersolvus. Les résultats des essais de dureté et les premières analyses du phénomène de précipitation de la phase γ′ ouvrent des perspectives prometteuses en termes de contrôle de la précipitation et donc de maîtrise de la trempabilité et de la soudabilité de pièces forgées à hautes caractéristiques mécaniques à chaud.

Figure 2 .

 2 Figure 2. Influence du revenu sur la dureté.

Figure 3 .

 3 Figure 3. Précipitation de la phase γ′ dans l'alliage CoNi-02 après un revenu de 96h à 900°C (MEB/électrons secondaires -attaque électrochimique).

  La masse volumique et les températures de transformation des alliages ont été mesurées respectivement par pesée/pycnométrie hélium et par analyse thermique différentielle et sont comparées dans le Tableau II.

	Tableau II. Masse volumique et températures de
		transformation des alliages CoNi-0x
	Alliage	d (g.cm -3 )	T solvus γ′ (°C)	T solidus (°C)	T liquidus (°C)
	CoNi-01	9,29	1004	1423	1459
	CoNi-02	9,30	1031	1419	1460
	CoNi-03	9,31	1061	1416	1459

La substitution partielle de Co par du Ni a peu ou pas d'influence sur la masse volumique et sur les températures de solidus et de liquidus de ces alliages. Par contre, ces résultats confirment que cette substitution influe sur la température de solvus de la phase γ′ (de l'ordre de 2-3°C/ % at. Ni)

[8]

.

Avant filage, la structure granulaire des lopins traités est caractérisée par une taille de grains supérieure au millimètre. Après filage dans les conditions testées, une recristallisation partielle est observée. Un traitement thermique d'une heure à 1200°C (suivi d'une trempe à l'eau glacée) est nécessaire pour atteindre une recristallisation totale. La taille moyenne des grains est alors d'environ 230 µm et des macles sont très fréquemment observées dans les grains.

Remerciements

L'Onera remercie Safran Tech pour le financement de cette étude.


Marc Thomas
email: marc.thomas@onera.fr

Dr Alexander Kirchner

Stefan Drawin
email: stefan.drawin@onera.fr

Role of Metal Powder Characteristics in Additive Manufacturing

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For quality assessment, it is essential to identify the different non-optimum powder characteristics which can be summarized as follows: minor composition variations from batch to batch, non spherical particles, presence of "satellites", inadequate particle size distribution, oxide film, inclusions and entrapped gas porosity. In addition, material defects can also originate from process conditions, build strategy, orientation in the build, thereby possibly resulting in unmelted particles or cracks. Therefore, it is important to establish the traceability of raw materials together with the process conditions and to standardise powder specifications. Powder quality criteria such as the minimum amount of pores, the pore size, morphology and distribution, a low level of surface contamination, a narrow particle size distribution with spherical shape for good flowability and good packing density should be determined. The most influential parameters can also depend on the sensitivity of the given metallic material. For instance, factors such as the volatility of specific alloying elements or interstitial contamination are related to the alloy composition and to the AM process used. Hence, LPW Technology has developed an effective quality assurance for powder transport and storage by incorporating a number of sensors to monitor moisture, temperature, pressure, vibration and oxygen levels [2]. Careful control and analysis of such physico-chemical powder characteristics are indeed essential to guarantee the finished product quality and reproducible mechanical properties with the aim of a higher productivity and a better efficiency in manufacturing parts.

A coordinated strategy is starting to move forward worldwide regarding powder manufacturing. In the United States, the Atomization Technology Innovation Consortium (ATIC) has been established in May 2015 by ASM International, in collaboration with the Metal Powder Industries Federation, Thermal Spray Society, America Makes, the AMES Laboratory, the Institute for Liquid Atomization and Spray Systems, and the University of California-Irvine Combustion Laboratory to develop an advanced technology roadmap for the US atomization industry. Furthermore, the ASTM F3049-14 standard can provide guidance for measuring the metal powder properties with standard methods [START_REF] Astm F | -14[END_REF]. In France, the NF E 67-010 standard deals with the technical characteristics of commonly used additive manufacturing powders to be provided by the supplier upon delivery [START_REF] Nf E | Fabrication additive -Poudres -Spécifications techniques[END_REF].

2-Experimental results

First of all, standard metallographic procedure, including electro-discharge machining, polishing, optical and SEM observations, image analysis, was used to characterize the different samples.

The first case study deals with the Ti64 alloy used for benchmarking different AM platforms: Powder Bed Laser Beam (PB-LB), Powder Bed Electron Beam (PB-EB), Powder Feed Laser Beam (PF-LB), Wire Feed Laser Beam (WF-LB), Wire Feed Electron Beam (WF-EB) and Wire Feed Plasma Source (WF-PS). The idea was to use different powder suppliers and different facilities in order to evaluate the impact on the product quality. Although it is not important for the purpose of this paper, the different machines under investigation can be mentioned: EOS, ARCAM, OPTOMEC, SCIAKY, Norsk Titanium and a laboratory setup for WF-LB. Chemical analysis, microstructure inspection and mechanical characterization have been carried on coupons to compare these Ti64 AM-built materials.

As illustrated in Table I, chemical analyses for the different platforms reveal that EB techniques (under vacuum) induce some aluminium loss. The fact that the different samples from LB techniques exhibit a lower level of Fe and C contents together with a higher level of N content than other techniques should be related to a particular powder particle size which limits the possibility of having different powder suppliers. Furthermore, the WF-PS technique induces the highest oxygen content, even though this remains below specifications. A slight nitrogen pick-up can be evidenced for the PF-LB platform. Due to different interstitial contents, it is expected that powder handling did not always take place in an inert-gas atmosphere throughout the overall process for some platforms. 1 is the large variations in UTS, typically from 720 MPa to 1180 MPa, depending on the platform and on the specimen built-up direction (Z versus X). Except for the WF-LB in Z direction and for the WF-EB, all platforms exhibit higher tensile strengths than that of the reference forging plate. The lowest tensile elongation for the PF-LB platform is related to the higher interstitial content. In general, a higher ductility is found in the Z direction for WF techniques while it is found in the X direction for PB and PF techniques.

PB-LB Z PB-LB X PB-EB Z PB-EB X PF-LB Z PF-LB X WF-LB Z WF-LB X WF-EB Z WF-EB X WF-PS Z WF-PS X

El.

[%]

Tensile stress

[MPa]

Specimen built-up direction YS UTS El.

Figure 1: Tensile tests after stress relaxation heat treatment for the different platforms (X: parallel to laser/EB direction. Z: vertical direction)

The second case study deals with the comparison of two different powder suppliers (EOS and TLS) for MS1 maraging steel. Figure 2 emphasizes the fact that the EOS powder exhibits slightly finer powders with a bimodal particle size distribution. A median size of 25 is obtained for the EOS and TLS powders respectively. This difference in particle size distribution is also enhanced by a higher ratio of coarser particles (> 40 concomitantly by a higher ratio of finer particles (< 20 powder (19%). In parallel, scanning electron micrographs for both powders did not reveal any significant differences in terms of shape and internal microstructure (Figure 3). It is also instructive to compare the chemistries of both powders. As emphasized in Table II, the EOS powder exhibits slightly lower amount of Ti than that of the TLS powder, which might affect the potential of subsequent precipitation hardening. Moreover, the Si content is an order of magnitude lower in the EOS powder than in the TLS powder. Finally, P, S, C and N minor elements are in average in half quantities in the EOS powder.

World PM2016 -Special Aspects in AM Powders

It is also worth mentioning that chemical analysis performed on SLM samples leads to similar results. The next step was to compare tensile properties of samples that were cut from 9 mm thick walls. Our purpose here was not to put forward the fact that one powder exhibits better properties than the other, but to show that slight chemical differences for the same alloy composition can lead to significant differences in mechanical properties.

Table III summarizes these tensile results for two built-up orientations using both EOS and TLS powders and also by changing some process parameters to accelerate the build. All the test specimens received the same over-aged anneal at 560°C for 6 hours, knowing that the peak aging condition was identified from micro-hardness measurements to be at 490°C for 6 hours. In average, the X orientation yields better tensile properties with an absolute increase of 32% in tensile ductility and of about 8% in yield stress. This is attributed to some built-up defects related to the layer-by-layer process along the Z orientation. Furthermore, the build using more rapid process conditions leads to a ductility drop of approximately 30% over the normal process conditions. Finally, the test samples made from the TLS powder exhibit a ductility drop of 46% while the yield stress was slightly increased by 4.4%. This result can be related to the higher contents in Ti, Si and minor elements for the TLS powder, thus participating to additional precipitation hardening. The systematically lower ductility for the TLS samples was rather attributed to such chemical differences than to defect-related of the TLS powder. The third case study which involves aluminium 6061 powders was reported by the UTBM research team. The initial powder which was characterized in powder size and morphology, reveals spherical powders with a few satellites and a normal powder particle size distribution with a d 50 of 31 Chemical analysis was carried out both on the initial powder and on subsequent SLM parts (Table IV).

Maraging steel

Specimen

From such analysis, significant losses of the Mg and Si elements were identified to occur during the SLM process, thereby leading to a higher susceptibility to cracks. Therefore, it could be recommended to adjust chemical contents of such elements to anticipate a subsequent loss from the laser heating source during SLM.

3-Conclusion

The establishment of a clear relationship between metal powders, materials properties and machine performance was a key objective of this work. In particular, the importance of metal powder feedstocks for finished products and of metal powder characterization for quality control in Additive Manufacturing has been addressed in this paper. A key issue for AM products should pass through standardisation of powders. It has also been demonstrated that it may be difficult to separate material-induced defects from process-induced defects. There is still a need to determine the most sensitive parameters for the improvement of process robustness. From the exposed case studies, it has been shown that slight differences in powder chemistries, major element loss and interstitial contamination are materialdependent factors which significantly affect mechanical properties after AM. A clear understanding of feedstock powders characteristics can therefore pave the way towards the adjustment of the raw material to customer-specific requirements suitable for AM applications.

Figure 2 :Figure 3 :

 23 Figure 2: Particle size distribution for the EOS and TLS powders

Figure 3 :

 3 Figure 3: Tensile tests after aging treatment for the maraging steel

Table I :

 I Chemical analysis of Ti64 for the different platforms Room temperature tensile tests have been performed on comparative test specimens after receiving the same stress relaxation heat treatment. The major trend which emerges from Figure

	Ti64	Al	V	Fe	C	N	O	H
		(%)	(%)	(µg/g)	(µg/g)	(µg/g)	(µg/g)	(µg/g)
	Nominal (AMS 4928R forging)	5.5-6.75	3.5-4.5	3000	800	500	2000	200
	PB-LB	5.97	4.14	560	120	300	1300	26
	PB-EB	5.74	4.09	1500	210	230	1100	23
	PF-LB	6.02	4.09	1300	260	440	1300	61
	WF-LB	6.06	3.99	520	140	330	720	26
	WF-EB	4.36	4.05	510	47	63	690	41
	WF-PS	6.00	4.02	1500	260	110	1900	62

Table II :

 II Chemical analysis for the EOS and TLS powders

	wt%	EOS	TLS	EOS part	TLS part
		powder	powder
	Fe	67.28	66.46	67.33	66.54
	Ni	17.69	17.76	17.63	17.93
	Co	9.05	8.85	9.04	8.96
	Mo	5.01	5.31	5.04	5.10
	Ti	0.64	0.86	0.61	0.84
	Cr	0.13	0.15	0.13	0.085
	Al	0.088	0.12	0.081	0.11
	Si	0.044	0.40	0.081	0.35
	Mn	0.064	0.090	0.057	0.085
	P (µg/g)	110	210	110	230
	S (µg/g)	61	96	71	120
	C (µg/g)	130	270	110	230
	N (µg/g)	410	890	540	700

Table IV :

 IV Chemical analysis for the aluminium 6061 powder

	World PM2016 -Special Aspects in AM Powders
	Al 6061	Al	Cr	Cu	Fe.	Mg	Si
		(wt%)	(wt%)	(wt%)	(wt%)	(wt%)	(wt%)
	Powder	Bal.	0.11	0.34	0.28	1.28	0.83
	SLM part	Bal.	0.11	0.36	0.29	1.17	0.73

Acknowledgements

L. Dembinski from UTBM is greatly acknowledged for helpful discussion and for providing data related aluminium 6061. EADS-IW is also acknowledged for making the tensile tests for the Ti64 material.


G Tenenbaum

M Tucsnak

New blow-up rates for fast controls of Schrödinger and heat equations

Keywords: null-controllability, Schrödinger equation, heat equation, series of exponentials AMS subject classifications : 93C25, 93B07, 93C20, 11N36

New blow-up rates for fast controls of Schrödinger and heat equations

Gerald Tenenbaum, Marius

Introduction

In this work we consider the boundary control of systems governed by the Schrödinger or by the heat equation. These systems can be written as an abstract infinite-dimensional linear control system described by the equations (1.1) ẇ = Aw + Bu, w(0) = ψ, where w denotes the state. Here, a dot denotes differentiation with respect to the time t, A is the generator of a strongly continuous operator semigroup on the state space X, B is an admissible control operator for this semigroup (the notion of admissible control operator will be recalled in Section 2) and ψ ∈ X is the initial state of the system. The system receives the input function (also called control function) u.

Assume the linear system (1.1) is null-controllable in arbitrarily small time, i.e., for every T > 0 and every initial state ψ, the set U T,ψ , composed of all controls in L 2 ([0, T ]) such that the corresponding state trajectory satisfies w(T ) = 0, is not empty. Then, as shown in Section 2, U T,ψ contains a unique minimal norm element, which we denote by u(T, ψ). The null-controllability operator in time T , denoted by F T , is defined by F T ψ = u(T, ψ). It is clear that the norm of F T (sometimes called the controllability cost, as in Zuazua [START_REF] Zuazua | Some results and open problems on the controllability of linear and semilinear heat equations[END_REF] and Miller [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF]) must increase unboundedly when the available time decreases to zero. We make the terminological choice of calling control cost the norm of the null-controllability operator. Thus, we write (1.2)

C T := F T and consider the natural question of studying the blow up of C T as the control time T tends to zero. In the case of finite dimensional systems, this question has been investigated by Seidman [START_REF]How violent are fast controls?[END_REF] and Seidman-Yong [START_REF] Seidman | How violent are fast controls? II[END_REF], who showed that, as T tends to zero, C T behaves like 1/T k+1/2 , for suitable k ∈ N. In the infinite dimensional case, a similar analysis has to be limited to systems which are null-controllable in arbitrarily small time, such as systems governed by the Schrödinger or by the heat equations-clearly, delay systems or systems governed by hyperbolic partial differential equations cannot be considered from the above perspective. In the case of the boundary control for the one dimensional heat equation with constant coefficients on the space interval [0, 1], it has been shown by Güichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF] that α * := lim inf

T →0
T ln C T > 0.

This result has been extended and made more precise in [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] and [START_REF]Controllability cost of conservative systems: resolvent condition and transmutation[END_REF], where it is shown that, for the constant coefficients Schrödinger and heat equations on the interval [0, a], we have

(1.3) α * 1 4 a 2 .
On the other hand, Seidman showed in [START_REF] Seidman | Two results on exact boundary control of parabolic equations[END_REF] that

α * := lim sup T →0 T ln C T < ∞.
More recently (see, for instance, Seidman, Avdonin and Ivanov [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF] and Miller [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF], [START_REF]Controllability cost of conservative systems: resolvent condition and transmutation[END_REF]) the above estimate on α * has been extended to the Schrödinger and heat equations with variable coefficients and effective upper bounds have been provided. To our knowledge, the best upper bound for α * in the case of the one dimensional Schrödinger equation has been obtained in [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF] and can be stated as (1.4) α * 4 36 37 2 µ, where µ is a constant depending only on the space interval in which the Schrödinger equation holds and on its coefficients: in the case of constant coefficients, µ reduces to the square of the length of the interval.

For systems governed by a variable coefficients heat equation with boundary control, the upper bound in (1.4) becomes (see [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF]) (1.5) α * 2 36 37 2 µ.

Although originally dealing with partial differential equations in space dimension one, the above mentioned results have been used in [START_REF] Zuazua | Some results and open problems on the controllability of linear and semilinear heat equations[END_REF], [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF] and [START_REF]Controllability cost of conservative systems: resolvent condition and transmutation[END_REF] to derive similar estimates for the Schrödinger and heat equations in several space dimensions.

Our main results provide new upper bounds for the control cost in the case of systems governed by the Schrödinger or the heat equation. Precise statements require some preliminaries, so they are postponed to Section 3. However, we can state at the outset that our upper bounds for C T = F T are valid for every T > 0 and imply that

(1.6) α * 3 2 µ,
for the Schrödinger equation in one space dimension, and

(1.7) α * 3 4 µ, for the heat equation, thereby improving upon (1.4) and (1.5).

The results described above yield new estimates, of independent interest, on "window problems" for series of exponentials as described in Seidman, Avdonin and Ivanov [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF]. They also imply new upper bounds for the control costs of the Schrödinger and heat equations in several space dimensions. Another contribution brought in by our work consists in giving a new proof of the lower bound in (1.3).

The plan of this paper is as follows. In Section 2, we give some background on infinitedimensional systems with emphasis on the null-controllability property. Section 3 is essentially dedicated to the statement of the main results. In Section 4, we establish two lemmas which are essential for the proofs of our theorems, given in Section 5. In Section 6, we apply the earlier obtained results to control problems in several space dimensions. Finally, in Section 7, we provide a simple proof of (1.3).

Notation and preliminaries

Notation

In the sequel, we freely use, according to display convenience, Landau's O-symbol or Vinogradov's -notation. Thus f (x) g(x) (x ∈ X) indicates that, for all x in the set X, the inequality |f (x)| C|g(x)| holds for some suitable constant C > 0 which may depend on some implicit parameters. In the latter case, dependence may be indicated by inserting appropriate subscripts. We write f (x) g(x)

to indicate that both relations f (x) g(x) and g(x) f (x) hold simultaneously.

Throughout this section, U , Y and X are complex Hilbert spaces, identified with their duals. The inner product and the norm in X are denoted by •, • and • , respectively. If P ∈ L (X; Y ) then the null-space and the range of P are the subspaces of X and Y respectively defined by

Ker P = {x ∈ X : P x = 0}, Ran P = {P x : x ∈ X}.
We denote by W = (W t ) t 0 a strongly continuous semigroup on X generated by an operator A : D(A) → X with resolvent set (A). The notation X 1 stands for D(A) equipped with the norm z 1 := (βI -A)z , where β ∈ (A) is fixed, while X -1 is the completion of X with respect to the norm z -1 := (βI -A) -1 z . We use the notation A and W t also for the extensions of the original generator to X and of the original semigroup to X -1 . Recall that X -1 is the dual of D(A * ) with respect to the pivot space X. For B ∈ L (U ; X -1 ) and T > 0 we define

Φ T ∈ L (L 2 ([0, T ], U ); X -1 ) by (2.8) Φ T u = T 0 W T -σ Bu(σ)dσ.

Some background on null-controllability

Definition 2.1. With the above notation, the operator B ∈ L (U ; X -1 ) is called an admissible control operator for W if Ran Φ τ ⊂ X for some τ > 0.

It is known (see Weiss [START_REF]Admissibility of unbounded control operators[END_REF]) that, if B is an admissible control operator for W, if T > 0,

if u ∈ L 2 ([0, T ], U
) and if ψ ∈ X, then the solution of the initial value problem (1.1), viz.

(2.1)

w(t) = W t ψ + Φ t u, satisfies w ∈ C ([0, T ],
X) and we have

Φ T ∈ L (L 2 ([0, T ], U ); X).
Definition 2.2. Given T > 0 and B ∈ L (U ; X -1 ), an admissible control operator for W, the pair (A, B) is said null-controllable in time T if, for any ψ ∈ X, there exists a u in L 2 ([0, T ]; U ) such that the solution w of (1.1) satisfies w(T ) = 0.

It is easy to see that the null-controllability of the pair (A, B) in time T is equivalent to the property Ran Φ T ⊃ Ran W T .

Definition 2.3. For T > 0 and B ∈ L (U ; X -1 ), an admissible control operator for W,

the pair (A, B) is said exactly controllable in time T if Ran Φ T = X.
It is clear that if (A, B) is exactly controllable in time T then (A, B) is null-controllable in time T . The converse is false in the general case but holds if A generates a strongly continuous group on X-this last condition being satisfied for systems governed by a Schrödinger equation. Therefore, in all the statements below concerning Schrödinger type equations, one can replace the term null-controllability by exact controllability.

The next proposition is essential for defining the null-controllability operator correctly. Since we did not find in the literature the required version valid for unbounded input operators (see, for instance, [START_REF] Zabczyk | Mathematical control theory: an introduction, Systems & Control: Foundations & Applications[END_REF] for the bounded case), we provide below a precise statement and a short proof, with no claim of originality.

Proposition 2.4. Suppose that (A, B) is null-controllable in time T . Then there exists an operator

F T ∈ L (X; L 2 ([0, T ], U )) such that 1. W T + Φ T F T = 0. 2. If u ∈ L 2 ([0, T ], U ) is a control driving the solution (2.1) of (1.1) to rest in time T , then u L 2 ([0,T ],U ) F T ψ L 2 ([0,T ],U ) .
Proof. Let ψ ∈ X. Then we have -W T ψ ∈ Ran W T ⊂ Ran Φ T , so there exists a unique y ∈ (Ker Φ T ) ⊥ such that Φ T y = -W T ψ. By setting F T ψ = y, we have that W T ψ + Φ T F T ψ = 0. We still have to prove that F T is bounded from X to L 2 ([0, T ], U ). Since F T is defined on all of X, it suffices to show that F T has a closed graph. Let (ψ n , y n ) be a sequence in the graph of

F T such that lim(ψ n , y n ) = (ψ, y) in X × L 2 ([0, ∞), U ), then lim W T ψ n = W T ψ and lim Φ T y n = Φ T y. Thus, W T ψ = Φ T y and, since (Ker Φ T ) ⊥ is closed, y ∈ (Ker Φ T ) ⊥ , so F T ψ = y.
It remains to show that the minimality property in the second assertion of the proposition also holds. If u ∈ L 2 ([0, T ], U ) is a control driving the solution w of (1.1) to rest in time T , then (2.2)

W T ψ + Φ T u = 0.
Let u = u 1 + u 2 be the orthogonal decomposition of u with u 1 ∈ Ker Φ T and u 2 ∈ (Ker Φ T ) ⊥ . From (2.2) and the definition of F T , we deduce that u 2 = F T ψ, hence

u 2 L 2 ([0,T ],U ) = u 1 2 L 2 ([0,T ],U ) + F T ψ 2 L 2 ([0,T ],U ) F T ψ 2 L 2 ([0,T ],U ) ,
so F T does satisfy the second required condition.

Proposition 2.4 says that, for any ψ ∈ X, F T ψ is the control of minimal norm driving the system (1.1) to rest in time T . We refer to F T as the null-controllability operator in time T .

The admissibility and null-controllability properties of a control operator are respectively dual to the admissibility and final state observability properties of an observation operator. We now recall the definitions of the latter concepts.

Definition 2.5. The operator C ∈ L (X 1 ; Y ) is called an admissible observation operator for W if, for some T > 0, there exists a constant K T > 0 such that

(2.3) T 0 CW t ψ 2 U dt K 2 T ψ 2 X (ψ ∈ D(A)).
Definition 2.6. Let T > 0 and let C ∈ L (X 1 ; Y ) be an admissible observation operator for W. The pair (A, C) is final state observable in time T if there exists k T > 0 such that

W T ψ 2 k 2 T T 0 CW t ψ 2 dt (ψ ∈ D(A)).
The duality mentioned above is made precise in the following result, essentially due to Dolecki and Russell [START_REF] Dolecki | A general theory of observation and control[END_REF].

Proposition 2.7. Suppose that B ∈ L (U ; X -1 ). Then B is an admissible control operator for W if, and only if, B * is an admissible observation operator for W * . The pair (A, B) is null-controllable in time T if, and only if, the pair (A * , B * ) is final state observable in time T . Moreover, if (A, B) is null-controllable in time T , then the norm of the associated null-controllability operator coincides with the greatest lower bound of the set of those numbers C T satisfying

W * T ψ 2 C 2 T T 0 B * W * t ψ 2 dt (ψ ∈ D(A * )).

Systems with self-adjoint or skew-adjoint generator and one dimensional input

Here, we specialize the notions and results of the two previous subsections to the case of systems with self-adjoint or skew-adjoint generator A and with one dimensional input space U -i.e. we take U = C.

Let A 0 : D(A 0 ) → X be a strictly negative self-adjoint operator, with non-empty resolvent set (A 0 ) and with compact resolvents. We denote by (ϕ k ) k∈N * an orthonormal basis of X consisting of eigenvectors of A 0 . For every k ∈ N * , we denote by -λ k the eigenvalue associated to the eigenvector ϕ k . Since A 0 is self-adjoint, λ k is real for all k ∈ N * . We assume that λ k > 0 for every k ∈ N, so that A 0 is a strictly negative operator. According to the Lummer-Phillips theorem, A 0 generates a C 0 contraction semigroup in X. This semigroup, denoted by S = (S t ) t 0 , acts on X according to the formula (2.4)

S t ψ = k∈N * ψ, ϕ k e -λ k t ϕ k (t 0, ψ ∈ X).
On the other hand, the operator iA 0 is skew-adjoint in X so, according to Stone's theorem, it generates a strongly continuous group of linear isometries in X. The action of this group, denoted U = (U t ) t∈R , is described by the formula (2.5)

U t ψ = k∈N * ψ, ϕ k e -iλ k t ϕ k (t ∈ R, ψ ∈ X).
We introduce the scale of Hilbert spaces X α , α ∈ R, as follows: for every α 0, we set

X α := D((-A 0 ) α ), equipped with the norm ψ 2 α := k∈N * λ 2α k | ψ, ϕ k | 2 .
For α > 0, the space X -α is defined as the dual space of X α with respect to the pivot space X. Equivalently, X -α is the completion of X for the norm

ψ 2 -α = k∈N * λ -2α k | ψ, ϕ k | 2 .
The operator A 0 and the semigroups S and U can be extended (or restricted) to each X α , in such a way that A 0 becomes a bounded operator

A 0 : X α → X α-1 (α ∈ R),
and U (respectively S) becomes a C 0 group of isometries (respectively a C 0 contraction semigroup) on X α-1 with generator iA 0 (respectively A 0 ).

Assume that the control space U is one dimensional (i.e. that U = C) and that the control operator B ∈ L (U ; X -1 ) is given by

(2.6) Bu = ub (u ∈ C),
where b is a fixed element of X -1 . For b as above and ψ ∈ D(A 0 ), the notation b, ψ stands for the duality product of b and ψ. For every k ∈ N * we put

(2.7) b k := b, ϕ k .
Sufficient conditions for the admissibility of a control of the form (2.6) are given in the result below, which is a particular case of the admissibility conditions given in Ho and Russell [START_REF] Ho | Admissible input elements for systems in Hilbert space and a Carleson measure criterion[END_REF] and Weiss [START_REF] Weiss | Admissibility of input elements for diagonal semigroups on l 2[END_REF].

Proposition 2.8. With the above notation, assume that sup k∈N * |b k | < ∞ and that the sequence Λ = (λ n ) n∈N is regular, i.e., that

(2.8) γ = γ(Λ) := inf m,n∈N * m =n |λ m -λ n | > 0.
Then B defined by (2.6) is an admissible control operator for S and for U.

The control cost can be interpreted in terms of a window problem for a sequence of complex exponentials. More precisely we may derive from Proposition 2.7 the following statement where, for a := (a n ) n∈N ∈ 2 (C), we denote by f a,Λ and g a,Λ the elements of L 2 loc (R) defined almost everywhere by (2.9)

f a,Λ (t) := n∈N a n e iλnt , g a,Λ (t) := n∈N a n e λn(T -t) , (t ∈ R).
Proposition 2.9. Let Λ be a regular sequence of real numbers, let B be given by (2.6), and assume that b k 1 (k 1).

For the pair (iA 0 , B), we have

F T sup a∈ 2 (C)\{0} a 2 (C) f a,Λ L 2 ([-T /2,T /2],C) (T > 0).
For the pair (A 0 , B), we have

F T sup a∈ 2 (C)\{0} a 2 (C) g a,Λ L 2 ([-T /2,T /2],C) (T > 0).
In both cases, the implicit constants depend only on inf |b k | and sup |b k |.

Proof. From (2.4) and (2.5) it follows that

(S t ) * ψ = k∈N * ψ, ϕ k e -λ k t ϕ k (t 0, ψ ∈ X). (U t ) * ψ = k∈N * ψ, ϕ k e iλ k t ϕ k (t ∈ R, ψ ∈ X).
The above relation and the fact that

B * ψ = k∈N * b k ψ, ϕ k (ψ ∈ X 1 ), imply that B * (S t ) * ψ = k∈N * b k ψ, ϕ k e -λ k t (t 0, ψ ∈ X), and
B * (U t ) * ψ = k∈N * b k ψ, ϕ k e iλ k t (t ∈ R, ψ ∈ X). Since ψ → b k ψ, ϕ k ) k∈N * maps X onto 2 (N * , C
), the desired conclusions follow from the last two formulas and Proposition 2.7.

3 Statement of the main results

Results on Schrödinger type equations and complex exponentials

In this subsection and in the following one we use the notation introduced in the previous sections. Recall, in particular, that X is a Hilbert space, A 0 : D(A 0 ) → X is a selfadjoint strictly negative operator with compact resolvents and with eigenvalues

(-λ k ) k 1 , b ∈ X -1 , the sequence (b k ) is given by (2.7
) and that S is the semigroup generated by A 0 .

Our first result gives an estimate (with explicit constants) for the norm of the control operator in the case of a system governed by an abstract Schrödinger equation.

Theorem 3.1. Assume that |b k | 1 (k 1)
, that the sequence Λ := (λ n ) n∈N is regular and it satisfies

(3.1) |λ n -rn 2 | Cn (n 1).
for some r > 0, C 0. Then the pair (iA 0 , B), with B given by (2.6), is null-controllable in time T and, for every κ > 3 2 π 2 , the control cost C T = F T satisfies the estimate

(3.2) C T e κ/(rT ) (T > 0),
where the implicit constant depends only on κ and Λ.

The result above can be applied for the control of the one-dimensional Schrödinger equation with variable coefficients and with various boundary conditions. In the particular case of Dirichlet boundary control we obtain:

Corollary 3.2. Let a > 0, p ∈ C 2 ([0, a], R), q ∈ C ([0, a], R). Assume that p(x) > 0 for all x ∈ [0, a] and write µ := a 0 p(x)dx 2 . Let α > 3 2 .
Then, for every ψ ∈ H -1 (]0, a[) and every T > 0, there exists u ∈ L 2 [0, T ] verifying

(3.3) u L 2 [0,T ] e αµ/T ψ H -1 (]0,a[) ,
(the implicit constant being independent of T ) such that the solution w of

(3.4)            -i ∂w ∂t (x, t) = ∂ ∂x p(x) ∂w ∂x (x, t) + q(x)w(x, t) (0 < x < a, 0 < t < T ) w(0, t) = u(t) (0 < t < T ) w(a, t) = 0 (0 < t < T ) w(x, 0) = ψ(x) (0 < x < a),
satisfies w(x, T ) = 0 for all x ∈]0, a[. Therefore, the system (3.4) has control cost e αµ/T for every α > 3 2 , which implies (1.6). This improves upon Theorem 4.1 in [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF], where a similar assertion is established under the stronger condition α > 4 36 37 2 .

The duality viewpoint in Theorem 2.7 suggests that Theorem 3.1 can be equivalently stated in terms of a window problem for a sequence of complex exponentials. We actually derive the following statement where, for a := (a n ) n∈N ∈ 2 (C), we denote by f a,Λ the element of L 2 loc (R) defined almost everywhere by

f a,Λ (t) := n∈N a n e iλnt (t ∈ R). Corollary 3.3. Let κ > 3 2 π 2 .
Assume that Λ := (λ n ) n∈N is a regular sequence satisfying (3.1). Then, uniformly for (a n ) n∈N ∈ 2 (C) and T > 0, we have

n 0 |a n | 2 e 2κ/(rT ) T /2 -T /2 n 0 a n e iλnt 2 dt.
For real sequences (λ n ) n∈N satisfying (3.1), this yields an improvement of the constants in the corresponding estimates of [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF].

Results on heat type equations and real exponentials

The analogue of Theorem 3.1 for abstract heat equations is the following statement.

Theorem 3.4. Let κ > 3 4 π 2 .
Then, under the assumptions in Theorem 3.1, the pair (A 0 , B), with B given by (2.6), is null-controllable in any time T > 0. Moreover, for every ψ ∈ X there exists u ∈ C [0, T ] driving the system (1.1) (with A = A 0 and Bu = ub) to rest in time T and satisfying

(3.5) u C ([0,T ]) e κ/(rT ) S T /2 ψ (T > 0, ψ ∈ X),
the implicit constant being independent of T . In particular, the control cost of the pair (A 0 , b) satisfies

(3.6) C T e κ/(rT ) (T > 0).
The above result can be applied to parabolic equations in one space dimension, with various boundary conditions. In the case of Dirichlet boundary control of the one-dimensional heat equation with variable coefficients, it yields the following statement.

Corollary 3.5. Let a > 0, p, q, µ, T be as in Corollary 3.2. Let α > 3 4 . For every

ψ ∈ H -1 (]0, a[), there exists u ∈ C [0, T ] verifying u C [0,T ] e αµ/T ψ H -1 (]0,a[) ,
(the implicit constant being independent of T ) and such that the solution w of

(3.7)            ∂w ∂t (x, t) = ∂ ∂x p(x) ∂w ∂x (x, t) + q(x)w(x, t) (0 < x < a, 0 < t < T ) w(0, t) = u(t) (0 < t < T ) w(a, t) = 0 (0 < t < T ) w(x, 0) = ψ(x) (0 < x < a), satisfies w(x, T ) = 0 for all x ∈]0, a[.
The above result, implying estimate (1.7), improves Theorem 4.1 in [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], where a similar assertion is shown to hold under the stronger hypothesis α > 2 36 37 2 . Another improvement brought in by our Corollary 3.5 is that our estimate involves u C [0,T ] and ψ H -1 ]0,a[ instead of u L 2 [0,T ] and ψ L 2 [0,a] employed in [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF].

The dual version of Theorem 3.4 may be stated as follows.

Corollary 3.6. Let κ > 3 4 π 2 . Assume that Λ := (λ n ) n∈N is a regular sequence satisfying (3.1). Then, uniformly for (a n ) n∈N ∈ 2 (C) and T > 0, we have

(3.8) n 0 |a n | 2 e -λnT e 2κ/(rT ) T 0 n 0 a n e -λnt 2 dt.
Note that a direct application of Theorem 2.7 would provide only (3.8) with

n 0 |a n | 2 e -2λnT
in the left-hand side.

For real sequences (λ n ) n∈N satisfying (3.1), this result improves the constants obtained in [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF]-see [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] for detailed comments on this issue.

Two lemmas

The proofs of our main results rest upon two lemmas. The first one furnishes sharp estimates for the exponential type, and growth on the real axis, for a sequence of entire functions defined by certain infinite products. Recall that a sequence of real numbers is said to be regular if it satisfies (2.8).

Throughout, we denote by x the integer part of the real number x. Also, we use Kronecker's symbol

δ nk := 1 if n = k, 0 if n = k.
Lemma 4.1. Let (λ n ) n 1 be a regular sequence of positive real numbers satisfying

(4.1) |λ n -n 2 | Cn (n 1), for some C > 0. Let (a n ) ∈ 2 (N * , C). For n ∈ N * , define Φ n (z) := k =n 1 - z λ k -λ n (z ∈ C). (4.2)
Then, for suitable B = B(C), we have, uniformly with respect to n 1,

Φ n (z) Λ e π √ |z| (1 + |z|) B (z ∈ C), (4.3) Φ n (-ix -λ n ) Λ (λ n + |x|) B e π √ |x|/2 (x ∈ R). (4.4)
The implicit constants depend at most upon C and γ, as defined in (2.8).

Proof. Let δ = δ n := inf k =n |λ k -λ n | γ(Λ) := inf k =j |λ k -λ j |. We have (4.5) ln |Φ n (z)| k 1 ln 1 + |z| |λ k -λ n | = k 1 |z| 0 dt t + |λ k -λ n | = |z| 0 k 1 1 t + |λ k -λ n | dt = |z| 0 k 1 ∞ |λ k -λn| ds (t + s) 2 dt = |z| 0 ∞ δ L n (s) (t + s) 2 dsdt with L n (s) := |λ k -λn| s 1.
From assumption (4.1), we readily get

(4.6) L n (s) λ n + s -(λ n -s) + + O(1),
where the implicit constant depends on C.

The contribution of the term

√ λ n + s -(λ n -s) + to the right hand side of (4.5) is |z| 0 ∞ δ √ λ n + s -(λ n -s) + (t + s) 2 ds dt = |z| ∞ δ √ λ n + s -(λ n -s) + s(s + |z|) ds |z| √ λ n U |z| λ n + V |z| λ n , with U (x) := 1 0 √ 1 + v - √ 1 -v v(v + x) dv = 1 0 2 dv (v + x) √ 1 + v + √ 1 -v , V (x) := ∞ 1 √ v + 1 v(v + x) dv.
Since the global contribution of the term O(1) from the right hand side of (4.6) to the right hand side of (4.5) is ln 1 + |z|/δ , the conclusion of our lemma follows provided that we show the inequality

(4.7) √ x{U (x) + V (x)} π (x 0).
Since (4.7) can be easily verified numerically for x 3, we assume x > 3 henceforth. Denote

a := 1 0 2 dv √ 1 + v + √ 1 -v b := ∞ 1 dv v{ √ v + √ v + 1} • We notice that U (x) a x and V (x) = ∞ 1 √ v v(v + x) dv + ∞ 1 dv v(v + x){ √ v + √ v + 1} = 1 √ x ∞ 1/ √ x 2 dt 1 + t 2 + ∞ 1 dv v(v + x){ √ v + √ v + 1} π √ x - 2 √ x arctan 1 √ x + b x + 1 π √ x - 2 x + 2 3x 2 + b x - b x(x + 1) • Therefore (4.8) √ x{U (x) + V (x)} π + a + b -2 √ x + 2 3x 3/2 - b √
x(x + 1) .

We shall see that

a = 2 √ 2 -2 ln 1 + √ 2 ≈ 1.0656, b = 2 + 2 ln 1 + √ 2 -2 √ 2 ≈ 0.9343.
Thus a + b = 2 and the sum of the last two terms in (4.8) is negative for x > 3, which yields estimate (4.7).

It remains to establish the above formulae for a and b. From the successive changes of variables v = 1 -2(sin ϑ) 2 and t = tan(ϑ/2), we obtain

a = 4 √ 2 π/4 0 sin ϑ cos ϑ sin ϑ + cos ϑ dϑ = 8 √ 2 1 0 (1 -t 2 )t (1 + t 2 ) 2 (1 + 2t -t 2 )
dt.

This furnishes the announced value for a after routine calculations.

Similarly, writing v = (tan ϑ) 2 and then t = tan(ϑ/2)

yields b = π/2 π/4 2 dϑ (1 + sin ϑ) sin ϑ = 2 1 √ 2-1 1 + t 2 t(1 + t) 2 dt,
from which the stated formula for b stems by standard calculus.

The proof of (4.4) is similar but easier. We have

Φ n (-ix -λ n ) := k =n 1 + ix/λ k 1 -λ n /λ k .
Now, similarly to (4.5), writing λ 0 := min k 1 λ k , we have

k 1 ln 1 + x 2 /λ 2 k = |z| 2 /λ 2 0 0 M (t) 1 + t dt with M (t) := λ k |z|/ √ t 1 |x|t -1/4 + O(1).
The bounded remainder term contributes ln(1 + |x|) and the main term does not exceed

|x| ∞ 0 dt (t + 1)t 1/4 = π 2|x|, since ∞ 0 dt (t + 1)t 1/4 = ∞ -∞ 2w 2 dw 1 + w 4 = 2πi Res(J; e iπ/4 ) + Res(J; e 3πi/4 ) = π √ 2,
with J(w) := 2w 2 /(1 + w 4 ).

Thus, we have shown so far that, for a suitable constant D, we have

|Φ n (-ix -λ n )| B n (1 + |x|) D e π √ |x|/2 with B n := k =n 1 -λ n /λ k -1 .
It remains to bound |B n | from above. We may plainly assume that n is sufficiently large.

Put m := C + 1, so that (k -m) 2 λ k (k + m) 2 for all k > m. For n > 3m, we have from Euler's product formula for sin(πz) (see, for instance, Ahlfors [1, p.195])

|B n | k<n-2m λ n (k + m) 2 -1 -1 n-2m k n+2m λ n λ n -λ k k>n+2m 1 - λ n (k -m) 2 -1 (λ n /γ) 4m+1 1 k m 1 - λ n k 2 n-m k n+m 1 - λ n k 2 k 1 1 - λ n k 2 -1 Λ λ 7m+1 n 1 -λ n / √ λ n 2 √ λ n sin π √ λ n Λ λ 7m+2 n .
Remark 4.2. Euler's product formula for sin(πz) shows the optimality of the exponent π in (4.3).

In our second lemma, we construct an entire function with fast decay on the real line. This will be essentially obtained as the Fourier transform of the C ∞ function defined by (4.9)

σ ν (t) := exp - ν 1 -t 2 if |t| < 1, 0 if |t| 1,
where ν is a positive constant. We note straightaway that, for every η ∈]0, 1[, we have

1 -1 σ ν (t)dt 2η exp - ν 1 -η 2 . Selecting η := 1/ √ ν + 1 readily yields (4.10) 2e -ν-1 √ ν + 1 1 -1 σ ν (t)dt 2e -ν .
The following result furnishes the required fast decay property-see Bombieri, Friedlander and Iwaniec [START_REF] Bombieri | Primes in arithmetic progressions to large moduli[END_REF] and Jaffard and Micu [START_REF] Jaffard | Estimates of the constants in generalized Ingham's inequality and applications to the control of the wave equation[END_REF] for related estimates.

                 H β (0) = 1, H β (ix) e β|x|/(2 √ ν+1) 11 √ ν + 1 (x ∈ R) |H β (z)| e β|y| (z = x + iy, x, y ∈ R), |H β (x)| √ ν + 1 e 3ν/4-(π+δ/2) √ |x| (x ∈ R).
Proof. Conditions (4.12) and (4.14) immediately follow from the definition of C ν .

To show (4.13), we may assume x 0. We first note that, from(4.10), we have (4.16)

1 2 e ν C ν 3 2 √ ν + 1 e ν .
Then, since σ ν (t) e -ν-1 for 1 2 η t η with η := 1/ √ ν + 1, we may write

H β (ix) 1 2 C ν ηe -ν-1+βxη/2 1 11 ηe βηx/2 ,
as required.

Thus, it only remains to establish condition (4.15). Since H β is even, we restrict to the case x > 0. Since σ ν ∈ C ∞ (R), σ ν (-1) = σ ν (1) = 0, we obtain by partial integration (4.17)

|H β (x)| C ν σ (j) ν ∞ (βx) j (x > 0, j ∈ N).
For t ∈] -1, 1[ we set = 1 -t and z = t + e iϑ , with ϑ ∈] -π, π]. We have

2 1 -z 2 = 1 1 -z + 1 1 + z = 1 2 + 1 -(sin ϑ/2) 2 2 -2 (2 -)(sin ϑ/2) 2 .
Since the last term is an increasing function of (sin ϑ/2) 2 , we obtain Applying Cauchy's integral formula, we obtain that

2 1 -z 2 1 2 + 1 2 (|z -t| = ).
|σ (j) ν (t)| e -ν/4 sup >0 j!e -ν/4 j (j ∈ N, t ∈ [-1, 1]),
which, in view of the elementary inequality j! > j j e -j (j 1), yields

(4.19) |σ (j) ν (t)| e -ν/4 2 j j! 2 ν j (j ∈ N, t ∈ [-1, 1]).
From this, (4.16), (4.17) and the fact that H β is even, we get that

|H β (x)| 3 2 √ ν + 1 e 3ν/4 2 j j! 2 (βνx) j (x > 0, j ∈ N).
Selecting j := 0 when 0 x 1 and j := 1 2 √ βνx otherwise, we readily check that (4.15) holds as required. Indeed, we deduce from the above that, for x > 1,

|H β (x)| √ ν + 1 e 3ν/4 2 j j! 2 (2j) 2j e -2j j e -(π+δ) √ x √ x e -(π+δ/2) √ x .
This concludes the proof.

Proofs of the main results

Proof of Theorem 3.1. A simple change of variables shows that it suffices to prove the result for any given special value of r. For simplicity, we choose r = 1. The proof, following the strategy in Fattorini and Russell [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF], is divided into two steps.

First step: construction of a family bi-orthogonal to (e iλnt ) n 1 . For n ∈ N * , we define

Ψ n (z) := Φ n (z -λ n ) = k =n 1 - z -λ n λ k -λ n = k =n 1 -z/λ k 1 -λ n /λ k ,
where (Φ n ) n∈N * is the sequence of entire functions constructed in Lemma 4.1. By (4.3), we have

Ψ n (z) e π √ |z-λn| 1 + |z -λ n | B .
Let T > 0, κ > 3 2 π 2 , 3π 2 T /(4κ) < β < T /2 and select δ > 0 so small that (5.1)

ν := (π + δ) 2 /β (4 -δ)κ/(3T ).
We next define the functions

g n (z) := Ψ n (-z)H β (z + λ n ),
where H β is the entire function constructed in Lemma 4.3. Since Ψ n (λ k ) = δ kn and H β (0) = 1, we have

(5.2) g n (-λ k ) = δ kn (k, n ∈ N * ).
Moreover, it follows from (4.3), (4.15) and (5.1) that

(5.3) |g n (x)| Λ K T (1 + |x + λ n |) B e (δ/2) √ |x+λn| K T 1 + |x + λ n | 2 (x ∈ R), with (5.4) K T := √ ν + 1 e 3ν/4
e κ/T .

Finally, since β < T /2, we infer from (4.3) and (4.14) that (5.5)

g n (z) n e T |z|/2 (z ∈ C).
By the Paley-Wiener theorem (see, for instance, Rudin [19, p.375]), g n is, for every n ∈ N * , the Fourier transform of a function f n ∈ L 2 (R) with support included in -1 2 T, 1 2 T , i.e.

(5.6)

g n (z) = T /2 -T /2 f n (t)e -itz dt (z ∈ C).
Since (5.2) and (5.6) imply

(5.7)

T /2 -T /2 f n (t)e iλ k t dt = δ kn (k, n ∈ N * ),
we see that the sequence

(g n ) n 1 is, in L 2 [-T /2, T /2],
biorthogonal to the the family (e iλnt ) n 1 .

Fast controls of Schrödinger and heat equations

Second step: construction of the control. Given ψ ∈ X, we define u ∈ L 2 [0, T ] by

(5.8) u(t) = - k∈N * a k e -iT λ k /2 f k (t -T /2) (0 t T ),
where a k := ψ, ϕ k /b k (k ∈ N). We deduce from (5.3) and (5.4) that

T 0 |u(t)| 2 dt e 2κ/T m∈N * n∈N * |a m a n | R dx (1 + |x + λ m | 2 )(1 + |x + λ n | 2 )
.

Inserting the elementary inequality

R dx (1 + |x + λ m | 2 )(1 + |x + λ n | 2 ) 4π 1 + |λ m -λ n | 2 (m, n ∈ N * ), yields T 0 |u(t)| 2 dt e 2κ/T m∈N * n∈N * |a m a n | 1 + (λ m -λ n ) 2 .
We plainly have from (2.8), that |λ m -λ n | γ|m -n| for all positive integers m, n. So we derive from the above estimate that (5.9) In view of (5.10), this yields that w(T ) = 0. By (5.9), this implies in turn that the pair (iA 0 , b) is null controllable in time T and that the control cost satisfies (3.2).

u L 2 [0,T ] e κ/T ψ (T > 0).
Proof of Corollary 3.2. It is easily checked that, without loss of generality, we may assume q 0-see, for instance [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF]. In order to apply Theorem 3.1, we write X := H -1 (]0, a[) and we consider the linear operator A 0 : X 1 → X defined by (5.11)

X 1 = H 1 0 (]0, a[), A 0 ϕ = d dx p dϕ dx + qϕ (ϕ ∈ X 1 ).
That A 0 is self-adjoint and strictly negative readily follows from our assumptions on p and q. Let U stand for the group of isometries on X generated by iA 0 . We select U = C

as input space and we consider b ∈ X -1 defined by

(5.12) b = δ 1 ,
where δ 1 is the Dirac distribution supported at x = 1. The input operator B ∈ L (C; X -1 ) is defined by Bu = ub for all u ∈ C. It is known (see, for instance, Curtain and Pritchard [START_REF] Curtain | Infinite dimensional linear systems theory[END_REF] and Curtain and Weiss [START_REF] Curtain | Well posedness of triples of operators (in the sense of linear systems theory)[END_REF]) that the system (3.4) can be written in the form (1.1) with X, A 0 and B chosen as above and A := iA 0 . Therefore, in order to prove the theorem, it suffices to show that X, A and B satisfy the assumptions in Theorem 3.1 with a suitable constant r.

According to classical estimates on Sturm-Liouville operators (see, for instance, Courant and Hilbert [4, p.415]), the sequence (λ k ) k∈N * formed by the eigenvalues of -A 0 , is regular and satisfies (5.13) λ n -n 2 π 2 /µ 1 (n 1).

Moreover, let (ϕ k ) k∈N * be an orthonormal basis in X = H -1 (Ω) consisting of eigenvectors of A 0 . Using standard estimates on the eigenvectors of Sturm-Liouville operators (see, for instance, [4, Section V.5]), we can check that b k = b, ϕ k = -ϕ k (1) 1. Therefore, we can apply Theorem 3.1 (with r = π 2 /µ) to conclude that the system (3.4) is null-controllable in any time T > 0 and that its null-controllability operator satisfies F T e αµ/T for every α > 3 2 .

Proof of Corollary 3.3. Put X := 2 (N, C) and let A 0 : D(A 0 ) → X be the diagonal operator defined by

D(A 0 ) = ψ ∈ 2 (N, C) : k∈N (1 + |λ k | 2 )|ψ k | 2 < ∞ , (A 0 ψ) k = -λ k ψ k (k ∈ N).
Note that the canonical basis (e k ) k∈N of 2 (N, C) is the sequence of eigenvectors of A 0 , with corresponding sequence of eigenvalues (-λ k ). The operator A := iA 0 generates a (diagonal) group of isometries defined by

(5.14) (U t ψ) k = e -iλ k t ψ k (k ∈ N, t ∈ R). Let b ∈ X -1 be defined by b k = 1 (k ∈ N).
The operator B ∈ L (C; X -1 ) corresponding by (2.6) to the above choice of b is defined by

(Bu) k = u (u ∈ C, k ∈ N),
and its adjoint B * ∈ L (X 1 ; C) is given by (5.15)

B * ψ = k∈N * ψ k (ψ ∈ D(A)).
The operators A and B clearly satisfy the assumptions in Theorem 3.1, so the pair (A, B) is null-controllable in any time T > 0 and the control cost C T = F T satisfies (3.2) for every κ > 3 4 π. Combined to (5.14), (5.15) and to Proposition 2.7, this implies the required conclusion.

Proof of Theorem 3.4. Arguing in a similar way than for Theorem 3.1, we first choose r = 1 and then proceed in two steps. First step: construction of a family of functions bi-orthogonal to (e λnt ) n 1 . Let T > 0 and κ > 3 4 π 2 . We choose β ∈]3π 2 T /(4κ), T [ and δ > 0 so small that ν := (π + δ) 2 /β satisfies (5.16) ν < (4 -δ)κ/(3T ).

For n ∈ N * and z ∈ C, we set (5.17)

G n (z) := Φ n (-iz -λ n ) H β (z/2) H β (iλ n /2) ,
where

(Φ n ) n∈N * is
G n ∈ L 1 (R) ∩ L 2 (R) (n ∈ N * ); (5.18) G n (z) n e T |z|/2 (z ∈ C); (5.19) G n (iλ k ) = δ nk (n ∈ N * ). (5.20)
Moreover, (4.4), (4.13) and (4.15) readily yield that

G n (x) (ν + 1)(λ n + |x|) B e 3ν/4-βλn/(4 √ ν+1)-(δ/2) √ |x|/2 (x ∈ R),
so that, in view of (5.16), we plainly have

(5.21) G n L 1 (R) λ -1 n e κ/T .
By the Paley-Wiener theorem, G n is, for each n ∈ N * , the Fourier transform of a function

F n ∈ L 2 (R) with support included in -1 2 T, 1 2 T , i.e. G n (z) = T /2 -T /2 F n (t)e -itz dt (z ∈ C).
By (5.20), this implies that, for all n, k ∈ N * , we have (5.22)

T /2 -T /2 F n (t)e λ k t dt = T /2 -T /2
F n (t)e -i(iλ k )t dt = δ nk .

The family

(F n ) n 1 is therefore bi-orthogonal, in L 2 [-T /2, T /2],
to (e λnt ) n 1 .

Second step: construction of the control. Given arbitrary ψ ∈ X, we define u ∈ L 2 [0, T ] by

(5.23) u(t) := - k∈N * a k e -T λ k /2 F k (t -T /2) , with a k := ψ, ϕ k /b k (k 1).
We claim that u satisfies (3.5). Indeed, from (5.23) and (5.21), we have

u C [0,T ] e κ/T k∈N * |a k |e -T λ k /2 λ -1 k ,
whence, by the Cauchy-Schwarz inequality, (5.24)

u 2 C [0,T ] e 2κ/T k∈N * |a k | 2 e -T λ k m∈N * λ -2 m e 2κ/T k∈N * |a k | 2 e -T λ k .
By the choice of a k and the estimate |b k | 1, this implies that

u 2 C [0,T ] e 2κ/T k 1 | S T /2 ψ, ϕ k | 2 = e 2κ/T S T /2 ψ 2 .
Therefore, u satisfies (3.5).

It remains to show that u as defined in (5.23) drives the solution w of (1.1) (with A = A 0 and Bu = ub) to rest in time T . From (2.8), (2.1), (2.4) and (5.23), we have (5.25)

w(T ) = k∈N * ψ, ϕ k + b k T 0 u(s)e λ k s ds e -λ k T ϕ k .
From (5.22) and (5.23) it follows that

T 0 u(s)e λ k s ds = - m∈N * a m e -T λm/2 T 0 F m (s -T /2) e λ k s ds = -a k .
Substituting in (5.25) yields w(T ) = 0. Hence the control u drives the system w to rest in time T . This implies the required conclusion.

Corollary 3.5 may be derived from Theorem 3.4 by following step by step the proof of Corollary 3.2. We omit the details.

Proof of Corollary 3.6. Consider the operators A 0 and B introduced in the proof of Corollary 3.3. According to Theorem 3.4 the pair (A 0 , B) is null-controllable in any time T > 0. Let u ∈ L 2 [0, T ] be the control given by (5.23) with b k = 1 and -a k in place of a k . Writing a n e -λnt (t 0).

h(t) := u(T -t) = k∈N * a k e -T λ k /2 F k (T /2 -t) (0 t T ),

We have

T 0 h(t)f (t)dt = n, k 1 a k a n e -T (λn+λ k )/2 T 0 F k (T /2 -t)e λn(T /2-t) dt = k 1 |a k | 2 e -λ k T ,
where the last equality follows from (5.22). In view of (5.26), this yields the required conclusion by the Cauchy-Schwarz inequality.

Results in several space dimensions

As noted in the introduction, the results of the previous sections have consequences on nullcontrollability problems in several space dimensions. The passage from one dimensional results (as Corollaries 3.4 and 3.7) to several space dimensions estimates has been studied in [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF], [START_REF]Controllability cost of conservative systems: resolvent condition and transmutation[END_REF] by the control transmutation method.

Let Ω ⊂ R n be an open bounded set and let Γ be an open non empty subset of ∂Ω. We consider the initial and boundary value problem: (6.1)

           i ẇ + ∆w = 0 (x ∈ Ω, t 0), w = u (x ∈ Γ, t 0), w = 0 (x ∈ ∂Ω Γ, t 0), w(x, 0) = ψ(x) (x ∈ Ω).
We also introduce a corresponding initial and boundary value problem for the heat equation (6.2)

           ẇ -∆w = 0 (x ∈ Ω, t 0), w = u (x ∈ Γ, t 0), w = 0 (x ∈ ∂Ω \ Γ, t 0), w(x, 0) = ψ(x) (x ∈ Ω).
It is classical knowledge that, under some regularity assumptions on Ω and Γ, each of the systems (6.1) and (6.2) determines a well-posed linear system with input space L 2 (Γ) and state space H -1 (Ω). A sufficient condition for the null-controllability of these systems is that Γ satisfies the generalized geodesics condition of Bardos, Lebeau and Rauch [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF]. In our case, this means, roughly speaking, that any light ray travelling in Ω and reflected according to geometrical optic laws when it hits ∂Ω, will intersect Γ (see [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF] or [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] for more details on this condition). Proposition 6.1. Assume that Ω ⊂ R n is a bounded domain with smooth frontier and that Γ ⊂ ∂Ω satisfies the generalized geodesics condition. Then, the system determined by (6.1) is null-controllable in any time T > 0 and the control cost satisfies

(6.3) lim sup T →0 T ln C T 3 2 L 2 Γ ,
where L Γ is the length of the longest generalized geodesic in Ω not intersecting Γ.

Proof. Consider the system with state space L 2 (Ω) × H -1 (Ω) and input space L 2 (Γ) determined by the equations (6.4)

           ẅ -∆w = 0 (x ∈ Ω, t 0), w = u (x ∈ Γ, t 0), w = 0 (x ∈ ∂Ω \ Γ, t 0), w(x, 0) = ψ 0 (x), ẇ(x, 0) = ψ 1 (x) (x ∈ Ω).
According to [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF], our assumptions imply that there exists T 0 > 0 such that the system (6.4) is exactly controllable in time T 0 . This fact, combined to Theorem 3.1 of [START_REF]Controllability cost of conservative systems: resolvent condition and transmutation[END_REF] and to our Corollary 3.2, implies the required result.

If Ω is the rectangle ]0, a[×]0, b[, with a, b > 0, a sufficient condition ensuring that Γ satisfies the generalized geodesics condition is that Γ ⊃ ([0, a] × {0}) ∪ ({0} × [0, b]). The result in Proposition 6.1, although not directly applicable to Ω and Γ as above (since Ω does not have a smooth boundary), suggests that the control cost satisfies lim sup

T →0 T ln C T 3 2 (a 2 + b 2 ).
The result below improves the above estimate inasmuch the constant a 2 + b 2 is replaced by a smaller one and we obtain an effective estimate valid for every T > 0. Proof. Appealing to Proposition 2.7, it can be checked (see, for instance, [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF]) that F T = C T is the smallest quantity satisfying (6.5)

∇ψ 2 L 2 (Ω) C 2 T T 0 Γ ∂ϕ ∂ν 2 dΓdt ψ ∈ H 2 (Ω) ∩ H 1 0 (Ω) ,
where ϕ is the solution of the initial and boundary value problem

(6.6)      φ + i∆ϕ = 0 (x ∈ Ω, t 0), ϕ = 0 (x ∈ ∂Ω, t 0), ϕ(x, 0) = ψ(x), (x ∈ Ω).
Define

ϕ mn (x, y) = 2 √ ab π √ b 2 m 2 + a 2 n 2 sin mπx a sin nπy b (m, n ∈ N * ).
The above family forms an orthogonal basis in L 2 (Ω) and the family (∇ϕ mn ) is orthonormal in L 2 (Ω). Let (c mn ) be the components of ψ with respect to this basis so that

ψ = m,n∈N c mn ϕ mn , with (c mn ) ∈ 2 (C). It follows that T 0 Γ ∂ϕ ∂ν 2 dΓdt 4a b T 0 a 0 m,n 1 nc mn e iπ 2 (m 2 /a 2 +n 2 /b 2 ) t √ b 2 m 2 + a 2 n 2 sin mπx a 2 dxdt + 4b a T 0 b 0 m,n 1 mc mn e iπ 2 (m 2 /a 2 +n 2 /b 2 ) t √ b 2 m 2 + a 2 n 2 sin nπy a 2 dydt.
The above formula, combined to the orthogonality of the family (sin (mπx/a)) m 1 (respectively (sin (nπy/b)

) n 1 ) in L 2 [0, a] (respectively in L 2 [0, b]), implies that T 0 Γ ∂ϕ ∂ν 2 dΓdt a b m 1 T 0 n 1 nac mn e iπ 2 n 2 t/b 2 √ b 2 m 2 + a 2 n 2 2 dt + b a n 1 T 0 m 1 mbc mn e iπ 2 m 2 t/a 2 √ b 2 m 2 + a 2 n 2 2 dt.
By applying Corollary 3.3 it follows that there exists C T > 0 such that C T e αµ/T for every α > 3 2 and

C 2 T T 0 Γ ∂ϕ ∂ν 2 dΓdt m,n 1 |c mn | 2 ,
which is exactly (6.5). This ends up our proof.

Remark 6.3. It has been recently shown in Ramdani, Takahashi, Tenenbaum and Tucsnak [START_REF] Ramdani | A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator[END_REF] that if Ω is a square in R 2 then the system determined by (6.1) is null-controllable even when the controlled part Γ of the boundary is arbitrarily small. It would be interesting to prove that this property holds in arbitrarily small time and to estimate the corresponding control cost.

The result in Proposition 6.1 has the following counterpart for the heat equation.

Proposition 6.4. Assume that Ω ⊂ R n is a smooth bounded domain and that Γ satisfies the generalized geodesics condition. Then, the system determined by (6.2) is null-controllable in any time T > 0 and the control cost satisfies

(6.7) lim sup T →0 T ln C T 3 4 L 2 Γ ,
where L Γ is the length of the longest generalized geodesic in Ω not intersecting Γ.

Proof. The result follows directly from Corollary 7.6 and Theorem 6.2 in [START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF]. Proposition 6.2 also has a counterpart for the heat equation. Since the proof is identical, we omit it. Proposition 6.5. Let Ω, a, b > 0, Γ and µ be as in Proposition 6.2 and let α > 3 4 . Then then the system (6.2) is null-controllable in any time T > 0 and the control cost C T satisfies C T e αµ/T (T > 0).

Lower bounds

The question of giving lower bounds of the control cost for the Schrödinger and the heat equations has also been investigated in the literature. The first result in this direction, due to Güichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF], concerns the heat equation. It asserts that, for p ≡ 1 and q ≡ 0, the control cost for the parabolic system (3.7) satisfies the condition

lim inf T →0 T ln C T > 0.
This is extended to the heat equation in several space dimensions and with internal control in [START_REF] Zuazua | Some results and open problems on the controllability of linear and semilinear heat equations[END_REF]-see also Fernandez-Cara and Zuazua [START_REF] Fernández-Cara | The cost of approximate controllability for heat equations: the linear case[END_REF]. These results are improved in [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], where it is shown, in particular, that for p ≡ 1 and q ≡ 0, the control cost for the system (3.7) satisfies the condition

(7.1) lim inf T →0 T ln C T 1 4 a 2 .
As far as we know, the only available lower bound for the control cost of the Schrödinger equation appears in [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF], where the inequality (7.1) is proved to hold also for the system (3.4), with p ≡ 1 and q ≡ 0. Moreover, in the case of Schrödinger and heat equations in several space dimensions (with internal control), lower bounds for the control cost are provided in [START_REF] Zuazua | Some results and open problems on the controllability of linear and semilinear heat equations[END_REF], [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] and [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF].

In this section, we give, for both the Schrödinger and heat equations in one space dimension, a simple proof of the estimate (7.1). Moreover, we use the same method to establish a lower bound for the Schrödinger equation in a rectangular domain in R 2 (this estimate is slightly different of that obtained in the general case). Our approach is based on classical properties of the Jacobi theta function, while the arguments of [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] and [START_REF]How violent are fast controls for Schrödinger and plate vibrations?[END_REF] rest upon a deep formula of Varadhan. Proposition 7.1. Let r > 0 and let Λ := (λ n ) be the sequence defined by λ n = rn 2 for all n ∈ N. Assume that, for every T ∈]0, 1], there is a real number K T such that the inequality

(7.2) n 0 |a n | 2 K 2 T T /2 -T /2 n 0 a n e iλnt 2
dt, holds uniformly for (a n ) ∈ 2 (C). Then K T r T 1/4 e π 2 /(4rT ) .

Proof. We may plainly restrict to proving the result for any given special value of r. Let us select r = π. Let ξ > 0 be a parameter which will be specified later. We choose (7.3) a 0 = 1 2 , a n := (-1) n e -πn 2 ξ (n 1).

Denoting z := ξ -it, we can write

f (t) = n 0
a n e -2πn 2 ti = 1 2 n∈Z (-1) n e -πn 2 z .

We next introduce the Jacobi theta function defined by It is known (see, for instance, [START_REF] Whittaker | A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions[END_REF], §21.51) that ϑ satisfies the functional equation

ϑ(z) = 1 √ z ϑ 1 z ,
where the square root is chosen to be positive for real positive z. Therefore It follows that for T, ξ ∈]0, 1[, |t| 1 2 T , we have f (t) e -πξ/(4ξ 2 +T 2 ) ξ + T so that I |f (t)| 2 dt T e -2πξ/(4ξ 2 +T 2 ) ξ 2 + T 2 .

Since

n 0 |a n | 2 = 1 2 ϑ(2ξ) 1 √ ξ ,
we see that by selecting ξ = T /2 that if (7.2) holds then

1 √ T K 2 T e -π/2T T •
The required estimate follows immediately.

The above result yields, at least for the case λ n = rn 2 , a lower bound for the control cost of the system in Theorem 3.1. More precisely, we obtain the following statement as a consequence of Propositions 2.7 and 7.1. Let B ∈ L (C; X -1 ) be the operator defined by

(Bu) k = u (u ∈ C, k ∈ N).
Then the control cost for the pair (A, B) satisfies C T T 1/4 e π 2 /(4rT ) (0 < T 1).

Applying the above to the system (3.4), with p ≡ 1 and q ≡ 0 we obtain the following lower bound for the constant α appearing in (3.3). e αa 2 /T ψ H -1 (]0,a[) ,

(the implicit constant being independent of T ) and such that the solution w of (7.6)

            
-i ∂w ∂t (x, t) = ∂ 2 w ∂x 2 (x, t) (0 < x < a, 0 < t < T ) w(0, t) = u(t) (0 < t < T ) w(a, t) = 0 (0 < t < T ) w(x, 0) = ψ(x) (0 < x < a), satisfies w(x, T ) = 0 for all x ∈]0, a[. Then α 1 4 a 2 .

In order to obtain a corresponding lower bound for the control cost of the heat equation, we need the following counterpart for real exponentials of the estimate in Proposition 7.1. Proposition 7.4. Let r > 0 and set λ n = rn 2 for all n ∈ N. Assume that, for each T ∈]0, 1], there is a number K T such that the inequality Proof. As before, we may select any convenient value for r and we pick up r = π: the general result follows by a change of variables. Let ξ > 0 to be specified later. Consider the the sequence (a n ) n 0 with a n = (2 -δ 0n )(-1) n e -πn 2 ξ , so that (7.8)

f (t) := n 0
a n e -πn 2 t = n∈Z (-1) n e -πn 2 (ξ+t)

= 2ϑ(4ξ + 4t) -ϑ(ξ + t) Let B ∈ L (C; X -1 ) be the operator defined by

= 1 √ ξ + t ϑ 1 4ξ + 4t -ϑ 1 ξ + t .
(Bu) k = u (u ∈ C, k ∈ N),
Then the control cost for the pair (A, B) satisfies the condition C T T 1/4 e π 2 /(4rT ) (0 < T 1).

Lemma 4 . 3 . 1 - 1 σ

 4311 Let β > 0, δ > 0, and set ν := (π + δ) 2 /β. The function σ ν being defined as in (4.9), put C ν := 1/ σ ν 1 and denote by H β the entire function defined by(4.11) H β (z) := C ν ν (t)e -iβtz dt.

Therefore ( 4 . 18 )

 418 |σ ν (z)| e -ν/4 -ν/4 (|z -t| = ).

Now, ( 2 . 8 )

 28 and (2.1) (with W = U), together with (2.5) and (5.8), imply that the solution w of (1.1) (with A = iA 0 and Bu = ub) satisfies (5.10) w(T ) = k∈N * ψ, ϕ k + b k T 0 u(s)e iλ k s ds e -iλ k T ϕ k . We deduce from (5.7) and (5.8) that, for every k ∈ N * , we have T 0 u(s)e -iλ k s ds = -m∈N * a m e -iT λm/2 T 0 f m (s -T /2) e iλ k s ds = -a k .

1 |a k | 2 e

 12 -λ k T . Put f (t) := n 1

Proposition 6 . 2 .

 62 Let Ω be the rectangle ]0, a[×]0, b[, with a, b > 0 and assume that Γ ⊃ ([0, a] × {0}) ∪ ({0} × [0, b]). Denote µ := max (a 2 , b 2) and let α >3 2 . Then the system (6.1) is null-controllable in any time T > 0 and the control cost satisfies the bound C T e αµ/T (T > 0).

(7. 5 )

 5 f (t) = 1 2 n∈Z e -4πn 2 z -n∈Z e -πn 2 z -n∈Z e -4πn 2 z

Corollary 7 . 2 .

 72 Let r > 0. Put X = 2 (N, C) and let A : D(A) → X be the diagonal operator defined byD(A) = ψ ∈ 2 (N, C) : k∈N (1 + k 2 )|ψ k | 2 < ∞ , (Aψ) k = irk 2 ψ k (k ∈ N).

Corollary 7 . 3 .

 73 Let a > 0, T ∈]0, 1[ and assume that, for everyψ ∈ H -1 (]0, a[), there exists u ∈ L 2 [0, T ] verifying u L 2 [0,T ]

  (a n ) n 0 ∈ 2 (C). Then K T r T 1/4 e π 2 /(4rT ) .

  n | 2 e -πn 2 T = 2ϑ(2ξ + T ) -1 1 √ ξ + T ,we see, by selecting for instance ξ = T 2 , that two corollaries, which are the counterparts for the heat equation of the results in corollaries 7.2 and 7.3. More precisely, we have the following statements.Corollary 7.5. Put X = 2 (N, C) and let A : D(A) → X be the diagonal operator defined by(Aψ) k = rk 2 ψ k , D(A) = ψ ∈ 2 (N, C) : k∈N (1 + k 2 )|ψ k | 2 < ∞ .

  as defined in Lemma 4.1 and H β is the entire function constructed in Lemma 4.3. Clearly, G n is, for each positive integer n, an entire function. It immediately follows from (4.2), (4.3), (4.12), (4.13) and (4.14) that

Corollary 7.6. Let a > 0, T ∈]0, 1[ and assume that, for every ψ ∈ H -1 (]0, a[), there exists u ∈ L 2 [0, T ] verifying

(the implicit constant being independent of T ) and such that the solution w of (7.9)

We omit the proofs of the two above corollaries, since they are almost identical to those of Corollaries 7.2 and 7.3.

We end up with the remark that the estimate

easily established by Poisson's summation formula, readily provides a lower bound for the control cost of a system governed by a Schrödinger or a heat equation in a rectangular domain. We omit the proof, which is very similar to that of Proposition 7.4.