Amin Chabchoub
email: a.chabchoub@imperial.ac.uk

Norbert Hoffmann

Hubert Branger

Christian Kharif

Nail Akhmediev

Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model

come   L'archive ouverte pluridisciplinaire

Ocean RWs 1 are known to appear suddendly from nowhere and to disappear without a trace. Within the framework of exact breather solutions of the NLS on finite background [START_REF] Akhmediev | Exact first-order solutions of the nonlinear Schrödinger equation[END_REF][START_REF] Peregrine | Water waves, nonlinear Schrödinger equations and their solutions[END_REF][START_REF] Kharif | Rogue waves in the ocean[END_REF] RWs have indeed this typical feature. Recent observations of doubly-localized solutions of the NLS [START_REF] Kibler | The Peregrine soliton in nonlinear fibre optics[END_REF][6][7][START_REF] Chabchoub | Super rogue waves: observation of a higher-order breather in water waves[END_REF] confirmed the fact that the long period limit of modulation instability (MI), also referred to as the Benjamin-Feir instability [START_REF] Benjamin | The disintegration of wave trains on deep water. Part 1. Theory[END_REF] , may be the key object to discuss within the framework of the NLS, a weakly nonlinear evolution equation describing the dynamics of wave packets in nonlinear dispersive media. Breather solutions are therefore under intensive study due to analogies that can be established for several nonlinear dispersive media [START_REF] Kharif | Rogue waves in the ocean[END_REF][START_REF] Osborne | Nonlinear ocean waves and the inverse scattering transform[END_REF]11 .

Being localized in time and in space, the Peregrine breather solution of NLS 3 is considered to be an appropriate model to describe the dynamics of RWs since in addition to its double localization, it amplifies the carrier by a factor of three. Lately, wave flume experiments 6 proved its physical validity in water waves and it is nowadays used as a model to analyse the impact of RWs on ships [START_REF] Onorato | Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test[END_REF] . Here, we present experimental results showing the evolution of the Peregrine breather in the presence of weak as well as strong wind forcing. The results show the robustness of the Peregrine solution with respect to either of these perturbations. Furthermore, it is shown that the shape of the main frequency peak in the frequency spectrum is invariant to the level of forcing. This work may motivate further analytical and numerical work in order to confirm these experimental findings.

The effect of wind blowing over packets of surface water waves is twofold: (i) it produces a variation in the pressure exerted on the surface that results in a flux of energy from the wind to the waves and (ii) it generates a rotational current in the water. Generally, the MI is considered independently of the effects of wind. The question is: How the presence of wind may modify this kind of instability? Note that in addition to these two effects, the wind produces randomness at the sea surface, too. It was shown that the effect of randomness characterized by the spectral bandwidth is to reduce the growth rate and extend of the instability compared to the regular case, using a simplified nonlinear spectral transport equation [START_REF] Alber | The effects of randomness on the stability of two-dimensional surface wavetrains[END_REF] . These results were confirmed later using the Zakharov equation 14 . Later on, a series of experiments on mechanically generated waves with wind blowing over them [START_REF] Bliven | Experimental study of the influence of wind on Benjamin-Feir sideband instability[END_REF][START_REF] Li | Effects of velocity shear on the stability of surface deep water wave trains[END_REF] confirmed that gentle wind tends to enhance sideband instability whereas stronger wind tends to reduce or suppress MI. An experimental investigation on the initial instability of nonlinear deep-water wave trains [START_REF] Waseda | Experimental study of the stability of deep-water wave trains including wind effects[END_REF] were not in agreement with the previous results since it was shown that wind did not inhibit the growth of the sideband. Nevertheless, for a quite strong wind a disappearance of the sideband was observed and the fact that the wind may reduce the naturally developed initial sideband energy, while the growth rate is not necessarily suppressed, was reported. The latter authors emphasized that further study of the natural selection process of the sideband in the tank is crucial for estimating what may happen in the ocean. Numerical simulations on the interaction of wind and extreme gravity waves due to MI were reported in [START_REF] Touboul | On the interaction of wind and extreme gravity waves due to modulational instability[END_REF] , using the modified Jeffreys sheltering mechanism as wind modelling. Contrary to the case without wind, it was observed that the wind sustains the maximum of modulation due to the MI. Another experimental and numerical investigation on the influence of wind on extreme wave events due to dispersive focusing and MI as well, by use of the modified Jeffreys sheltering theory as wind-model, demonstrated that wind blowing over modulated wave trains may increase the time duration and the amplitude of extreme wave events [START_REF] Kharif | Influence of wind on extreme wave events: experimental and numerical approaches[END_REF] . The theoretical analysis on the MI in deep-water under the action of wind and dissipation by deriving a perturbed NLS [START_REF] Kharif | Modulational instability in deep water under the action of wind and dissipation[END_REF] showed that the MI of the wave group depends on both: the frequency of the carrier wave and the strength of the friction velocity.

Using a fully nonlinear approach the theoretical work has been developed to the case of strongly nonlinear waves [START_REF] Kharif | Under which conditions the Benjamin-Feir instability may spawn an extreme wave event: A fully nonlinear approach[END_REF]22 . A few papers have been devoted to describe the influence of the mentioned vorticity in water on the MI, for instance [START_REF] Johnson | On the modulation of water waves on shear flows[END_REF] . Although the condition of linear stability of the nonlinear plane wave solution is provided in a complicated way, the latter author did not develop a detailed stability analysis as a function of the vorticity and depth. Considering the instability properties of weakly nonlinear wave packets to three dimensional disturbances in the presence of shear, the system of equations reduces to the familiar NLS equation, when confining the evolution to be purely two dimensional, and a stability analysis for the case of a linear shear was as well illustrated in [START_REF] Oikawa | The propagation of nonlinear wave packets in a shear flow with a free surface[END_REF] . Within the framework of deep-water, the MI of a Stokes wave train in uniform velocity shear has been studied [START_REF] Li | Effects of velocity shear on the stability of surface deep water wave trains[END_REF] . The coefficient of the nonlinear term of the NLS equation the authors derived was erroneous, as noted in [START_REF] Baumstein | Modulation of gravity waves with shear in water[END_REF] . The latter author investigated the effect of piecewise-linear velocity profiles in water of infinite depth on sideband instability of a finite-amplitude gravity wave, while The coefficients of the derived NLS equation were computed numerically for specific values of the vorticity and depth of shear layer. In 26 , the author considered the MI of a modulated wave train in both positive and negative shear currents within the framework of the fully nonlinear water wave equations. For a given wave steepness, the results have been compared with the irrotational case and it has been shown that the envelope of the modulated wave train grows faster in a positive shear current and slower in a negative shear current. Using the fully nonlinear equations, a numerical investigation of some instability characteristics of two-dimensional finite amplitude surface waves on a linear shearing flow to three-dimensional infinitesimal rotational disturbances was reported in [START_REF] Okamura | The linear stability of finite amplitude surface waves on a linear shearing flow[END_REF] . Very recently, a NLS equation in the presence of a vertically sheared current of constant vorticity was derived [START_REF] Kharif | A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity[END_REF] . The results of the study prove that the presence of vorticity modifies significantly the MI properties of weakly nonlinear plane waves, namely the growth rate and bandwidth.

As we can see from the above references, there is a solid body of research on the influence of wind on MI. Some results contradict the others and further work is needed to clarify the issue. However, here, we do not intend to do this. Instead, we concentrate just on the limiting case of MI when its period goes to infinity. Our aim is to study, experimentally, the influence of wind on the evolution of the Peregrine breather once it was initiated with the small amplitude perturbation.

The MI can be described within the framework of deep-water waves which are described by the NLS [START_REF] Zakharov | Stability of periodic waves of finite amplitude on a surface of deep fluid[END_REF] given by:

i ∂ψ ∂t + c g ∂ψ ∂x - ω 0 8k 2 0 ∂ 2 ψ ∂x 2 - ω 0 k 2 0 2 |ψ| 2 ψ = 0, (1)
where ψ denotes the wave envelope, while ω 0 and k 0 are the wave frequency and wave number, respectively. They are linked through the dispersion relation: ω 0 = √ gk 0 , where g is the gravitational acceleration. It is well known that the plane wave solution ψ (x, t) = a 0 exp -i k 2 0 a 2 0 ω 0 t 2 of the NLS with arbitrary amplitude a 0 is unstable relative to periodic modulations within certain range of wavenumbers [START_REF] Osborne | Nonlinear ocean waves and the inverse scattering transform[END_REF] . The MI can be also discussed in terms of breather solutions on finite background 2,3 . First-order family of breather solutions, describing the exact Benjamin-Feir mechanism, are known as Akhmediev breathers 2 . These solutions are periodic in space and have specific growth-decay evolution in time. For each specific Akhmediev breather, i.e. for each specific modulation period, the maximal amplitude amplification of the carrier is bounded by one and three [START_REF] Akhmediev | Exact first-order solutions of the nonlinear Schrödinger equation[END_REF]11 . The limiting case of the Peregrine breather solution 3 corresponds to the case when the modulation period becomes infinite and is given in dimensional units by:

ψ P (x, t) = a 0 exp -i k 2 0 a 2 0 ω 0 2 t    -1 + 4 1 -ik 2 0 a 2 0 ω 0 t 1 + 2 √ 2k 2 0 a 0 (x -c g t) 2 + k 4 0 a 4 0 ω 2 0 t 2    .
This doubly-localized breather solution 6 amplifies the carrier by an exact factor of three. As such, it is considered to be an appropriate prototype to describe ocean rogue waves. As we show in the present work, the Peregrine breather once initiated is a robust formation and can survive even in a chaotic wave field generated by the wind.

Experiments have been conducted in a closed wind wave facility installed at IRPHE/Luminy Marseille. A schematic illustration is shown in Fig. 1 4 . The tank is 40 m long and 3 m wide. The water is 0.9 m deep, and the air channel is 1.5 m high. The tank is equipped with a controlled recirculating wind tunnel which can generate wind speeds between 1 and 14 ms -1 . A computer-controlled wave-maker can generate waves in a frequency range from 0.6 Hz to 2.5 Hz. It is entirely submerged under the upstream beach to avoid any perturbation of the air flow which could be induced by its displacement. The upwind end of the tank is specially profiled to ensure minimum disturbance to either the generation of mechanical waves or the turbulent boundary layer in the airflow above the waves. The tunnel roof is carefully profiled to create an airflow boundary layer of zero pressure gradient along the test section. At the downwind end of the tank a permeable absorbing beach was installed to minimize wave reflection. A complete description of the tank can be found in [START_REF] Coantic | Wind-water tunnel simulation of small-scale ocean-atomsphere interactions[END_REF]

η(x * , t) = Re (ψ P (x * , t) • exp [i (k 0 x * -ω 0 t)]) . (2)
All experiments have been conducted for a carrier amplitude of a 0 = 0.75 cm, while the wave frequency and wave number were chosen to be ω 0 = 10.68 rad • s -1 and k 0 = 11.63 rad • m -1 , respectively.

The first set of experiments was conducted in order to reproduce the Peregrine breather the same way as reported in 6 but in the water tank described above. Namely, first, we excited measurements at the gauge 5 confirmed that the maximal amplification of three is reached.

The group velocity is calculated from the Hilbert transform of the cross-correlation function of the measured water levels at two different locations. In the absence of wind, we measured the group velocity to be c g = 0.49 ms -1 , which nearly coincides with the theoretical group velocity given by the dispersion relation to second-order in steepness. Here, we concentrate our attention on the advanced stages of the evolution of the breather measured by the gauges 2, 3, 4, 5, 6 and 7, installed at the positions between 8 and 12 meters from the wave generator. In the following experiments, we increased, step by step, the constant wind speed U blowing in the direction of wave propagation for the same intitial conditions. The wind naturally distorts the background wave train as we can see from Fig. 3. The higher the wind speed, the higher are these distortions, which are generally chaotic. Nevertheless, the initial small modulation grows and its highest amplitude prevails in the chaotic wave field of the background. Furthermore, Fig. 3 shows that even in the presence of the strongest wind with the speed of 9 ms -1 , the Peregrine breather dynamics engenders the formation of a large amplitude wave, which stands out against the chaotic background wave train. In order to confirm this fact, which perhaps cannot be clearly appreciated from the relatively low resolution traces in Fig. 3 (d), we show separately the trace measured by the gauge 3 from the above set in Fig. 4. We can see clearly, that the Peregrine breather appears as a large amplitude wave, which is significantly higher than the rest of the surrounding waves. The maximal breather amplitude changes, of course, when the wind is present. The maximal amplitude of the waves as well as the standard deviation for the corresponding wind speed are shown in Table 2. Remarkably, they are higher than in the absence of the wind, which is providing an additional amplification of the breather. Here, it has to be mentioned that these amplifications may be much more significant since recurrent breaking is observed during the experiments [START_REF] Iafrati | Modulational Instability, Wave Breaking, and Formation of Large-Scale Dipoles in the Atmosphere[END_REF] and it cannot be confirmed that the wave gauges, fixed at a specific position, measured the breather's amplitude at its maximal local steepness, i.e. just before breaking. These observations are in line with the analytical results reported in [START_REF] Touboul | On the interaction of wind and extreme gravity waves due to modulational instability[END_REF][START_REF] Kharif | Modulational instability in deep water under the action of wind and dissipation[END_REF][START_REF] Onorato | Approximate rogue wave solutions of the forced and damped nonlinear Schrödinger equation for water waves[END_REF] , which predicted that the wind accelerates the evolution of the MI. Furthermore, the presented experimental results confirm the work of [START_REF] Waseda | Experimental study of the stability of deep-water wave trains including wind effects[END_REF] and validate the fact that the effect of even strong winds on the MI is not dominant and indeed small. Another visible effect caused by the wind is the increase of the group velocity. Measured group velocities are for U = 5 ms -1 ; c g = 0.56 ms -1 for U = 7 ms -1 ; c g = 0.59 ms -1 for U = 9 ms -1 . As explained by [START_REF] Touboul | On the interaction of wind and extreme gravity waves due to modulational instability[END_REF] , the increase of velocity is due to the wind-induced surface-drift current which is proportional to the wind drug velocity. Another unique feature of the Peregrine breather potentially important for its prediction is its characteristic triangular spectrum co-located on top with the main frequency component of the background wave [START_REF] Akhmediev | Rogue wave early warning through spectral measurements?[END_REF] . The latter is a deltafunction in frequency domain in case of an infinite monochromatic background. If we ignore or eliminate the central peak caused by the background, the Peregrine breather dynamics can be easily identified from the experimental triangular spectra [START_REF] Chabchoub | Spectral properties of the Peregrine soliton observed in a water wave tank[END_REF] . Our present experiments show these spectra which are also slightly distorted by the wind. Fig. 5 shows the power spectra of several wave train records, measured with the first gauge. In the absence of the wind, i.e. for U = 0 ms -1 , we can clearly notice the main triangular spectrum at 1.7 Hz. For nonzero wind speeds up to 9 ms -1 the main triangular feature of spectrum at 1.7 Hz does not disappear. When increasing the wind speed, the total energy of the wave train simply grows but the triangular shapes are preserved. This demonstrates that the Peregrine breather is present in the wave train no matter if the wind is blowing or not. These experiments justify once again the promising idea of using the spectral measurements of the doubly-localised breather solutions for a possible early prediction of rogue waves even in presence of winds and a chaotic background makes their measurements viable candidates for the rogue wave predictions [START_REF] Akhmediev | Early detection of rogue waves in a chaotic wave field[END_REF] .

To summarize, we report an experimental investigation of the Peregrine breather evolution in a wave tank in presence of wind. Our study shows that slow winds in propagation direction do not have significant influence on the evolution of the breather. Furthermore, under the action of strong winds, a formation of a Peregrine breather-like large wave amplitude can be clearly observed. The spectral measurements at the beginning of the breather evolution show a typical triangular spectra co-located with the main frequency component of the background wave. This spectral feature does not disappear when the wind speed is increased. Thus, the MI dynamics of the Peregrine breather dynamics remains valid.

FIG. 1 .

 1 FIG. 1. Schematic illustration of the wind wave facility 4

the• s - 1 FIG. 2 .

 12 FIG.2. Evolution of the Peregrine breather for the carrier parameters a 0 = 0.75 cm and ε := a 0 k 0 = 0.08 without wind action (blue lines) starting from small amplitude amplification as initial conditions (red line). The group velocity measured experimentally is c g = 0.49 ms -1 .

FIG. 3 .

 3 FIG. 3. Advanced stages of the evolution of the Peregrine breather for (a) a wind speed value of U = 3 ms -1 , (b) U = 5 ms -1 , (c) U = 7 ms -1 and (d) U = 9 ms -1 . These observations show that despite the presence of a strong wind and related chaotization of the background, the theoretical Peregrine breather dynamics still reasonably well defines the formation of the centrally located RW.

FIG. 4 .

 4 FIG.4. The measurements shown in Fig.3 (d) recored by the gauge 3 with higher resolution along the vertical axis. The largest wave appears at this position at t = 55 s.

2 Hz - 1 )U = 0 ms - 1 U = 3 ms - 1 U = 5 ms - 1 U = 7 ms - 1 U = 9 ms - 1 FIG. 5 .

 21111115 FIG. 5. Power spectra of the first gauge record for the different wind speed values

  A. C. would like to thank Efim Pelinovsky for helpful discussions. N. H. and N. A. acknowledge the support of the Volkswagen Stiftung. N. A. acknowledges partial support of the Australian Research Council (Discovery Project No. DP110102068). A. C., C. K. and N. A. acknowledge support from the Fields Institute. N. A. acknowledges support from the Alexander von Humboldt Foundation.

  . Wind speed was measured with a Pitot tube. We used one capacitance wave gauge and six resistive wave gauges to measure the water elevation at different locations inside the tank in order to see in the advanced stages of wave evolution in more detail, see Table1. The accuracy of

	GN	1	2	3	4	5	6	7
	P (m) 2.06 8.09 8.81 9.50 10.41 11.08 11.71

TABLE I. Positions (P) of the gauge number (GN) along the tank

the water elevation measurement is about 0.3 mm. The boundary condition, applied to the flap, is determined as the surface elevation, given in the first-order in steepness at a specific position x * :

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett. 106, 204502 (2011).

H. Bailung, S. K. Sharma, and Y. Nakamura, Observation of Peregrine solitons in a mul-


Bruno Lemiere
email: b.lemiere@brgm.fr

Pascale Michel

Claire Alary

Laurence Haouche

Hervé Brequel

Jérome Jacob

Nathalie Gineys

Agnès Laboudigue

Bruno Lemière

Hervé Bréquel

Jérôme Jacob

Novel scenarios for sustainable waterway sediments management deduced from a decision-support tool

come   L'archive ouverte pluridisciplinaire

Introduction: Sediments accumulating in waterways represent a triple threat: for fluvial navigation, for flooding hazards, and for their pollutant contents. Waterways dredging releases millions of m3 of sediments, from which a large part is contaminated or even polluted enough to be considered as hazardous waste. Temporary or final storage on land is no longer a sustainable option.

Methods:

The GeDSeT decision support tool (DST) aims to provide sediment management options with quantitative data, in order to evaluate various scenarios taking into account cost and sustainability and consequently to highlight good practice. It allows to evaluate different scenarios of sediment management based on several indicators, in order to take into account all the consequences (effects, fig. 1) of the chosen options ("what-if" tools, [START_REF] Lemière | The GeDSeT project: constitution of a decision support tool (DST) for the management and material recovery of waterways sediments in Belgium and Northern France[END_REF]). As it is aimed at assessing all consequences of a chosen option (environmental, economic…), the tool is based on multicriteria analysis [START_REF] Laboudigue | The GeDSeT Project: coupling multi-criteria analysis and knowledge improvement on sediment for a close-to-the-field Decision Support Tool[END_REF]. Results: Scenarios were developed using databases and focused research results [START_REF] Laboudigue | The GeDSeT Project: coupling multi-criteria analysis and knowledge improvement on sediment for a close-to-the-field Decision Support Tool[END_REF] through discussions with operators, communities and industries [START_REF] Lemière | L'outil d'aide à la décision GeDSeT[END_REF].

Scenario 1: selective dredging is a 2-phase scheme in which pollution hotspots are removed before bulk dredging, to improve reusability of sediment (fig. 2).

Scenario 2: on-site treatment implies processing as much as possible the sediment at a ship-borne plant. On-site dehydration benefits include output material with easier handling properties, and reduction of the dredged volume to be managed. Water can be returned to the waterway after treatment.

Scenario 3: selective treatment refers to directing sediment loads to a treatment procedure adequate for their pollutant contents (inorganic and/or organic). This scenario is applicable to sediment treatment facilities currently developed near canals. Treatment may be aimed at reducing contamination under critical levels for less polluted sediments, or at concentrating the pollutants in a low volume fraction for safe disposal. Reusable sediment loads may be shipped to reuse sites.

Dechargement

Mesure sur site Mesure sur site Mesure sur site Scenario 4: alternative use of sediment

Selectively dredged or treated sediments may be directed to reuse according to contamination level and regulatory constraints. Potential uses comprise:

-bulk use where applicable (landfill cover, civil works, excavation backfill), -composite use (mix with concrete demolition aggregate), -use as an alternative mineral resource (cement production). Benefits include the reduction of primary minerals extraction and of sediment storage, hence increase of possible waterways dredging operations.

Scenario 5: alternative use of disposal sites. Sediment deposits are highly vegetated due to abundant organic matter, but they are unfit for food crops. Their use for energy crops (wood pellets, seeds) would reduce undesirable land use and allow energy crops on fertile soil without competition for land with food crops.

All these scenarios are aimed at increasing the reuse of sediments, and reducing their disposal as waste. Scenarios 1 and 3 require the availability of field analytical methods, currently in development.

Discussion: The benefits of alternative and novel scenarios (environmental, land use, employment and economic activity) are not properly accounted for if dredging projects are evaluated through a tendering process. Benefits are identified by enlarging system boundaries [START_REF] Lemière | The GeDSeT project: constitution of a decision support tool (DST) for the management and material recovery of waterways sediments in Belgium and Northern France[END_REF]. The benefits of sediments reuse, at constant budget, are to reduce land pressure and to improve waterways maintenance, offering therefore more possibilities to sustainable fluvial transport.

Fig. 1 :

 1 Fig. 1: Data and indicators for the GeDSeT tool.

Fig. 2 :

 2 Fig.2: Selective dredging and detail characterisation. Crosses: sampling points. Boxes: contour of composites currently used for lab characterisation for bulk dredging. Field characterisation of individual samples would allow selective dredging of hotspots (pink) and subsequent valorisation of less contaminated sediment (blue).

Fig. 3 :

 3 Fig.3: Selective processing of dredged sediment loads (left) in canal-side treatment plant: uncontaminated, direct shipping to reuse site; low contamination: dehydration or landfarming until reuse (1); medium contamination: treatment and separation (2); high contamination: concentration and disposal as hazardous waste (3).

Acknowledgements: This research was supported by the European InterReg IV program, by the Walloon region of Belgium, by the French Ministry of Research, the Nord Pas de Calais region, and the French Waterways (VNF).


Christian Mühl

Guillaume Chanel
email: guillaume.chanel@unige.ch

Brendan Allison
email: ballison@ucsd.edu

Anton Nijholt
email: a.nijholt@utwente.nl

Third Workshop on Affective Brain-Computer Interfaces: Introduction

The human mind -and hence the brain -is fascinating in its complexity, and its exploration holds many promises for the development of mankind. For computer science and especially human-computer interaction, brain-computer interfaces (BCI) enable direct access to the information represented in the brain, allowing scientists and engineers to use this information to help users interact with applications and devices. Over the last decade, BCI research developed into a major research field establishing methodological frameworks, guidelines and a community of dedicated researchers. Besides proving valuable as a communication channel that bypasses impeded muscular pathways, these technologies are also spurring novel applications for able-bodied users in fields such as entertainment, health and lifestyle.

Affective BCI systems allowing users to control computer games, support relaxation training, or trigger your alarm clock during a shallow sleep stage have been proposed, implemented, and sold. Moreover, the affordable hardware and software tools also encouraged artists to play with the idea of a direct access to people's most private information: their affective and cognitive states. From these explorations followed a number of interesting installations, suggesting novel ways of human-computer as well as human-human interaction: neurotechnology-based systems that encourage affective self-reflection, the synchronization and empathizing between or the competition of different minds, and the collaborative creation and manipulation of multimedial content.

Following the first and second workshop on affective brain-computer interfaces, held in conjunction with ACII in Amsterdam ( 2009) and Memphis (2011), the third workshop explores the advantages and limitations of using neurophysiological signals for the automatic recognition of affective and cognitive states, and the different ways to use this information about the user in applications within the health, arts, and entertainment domains. The goal is to bring researchers, artists, and practitioners together to present stateof-the-art progress, discuss pitfalls and limitations and share and create visions, and thereby encourage the development of guidelines and frameworks for affective BCI.

The contributions featured a large range of interesting topics. The most works explore the classification of affective states via different neurophysiological measurements, such as Electroencephalography (EEG), functional near-infrared spectroscopy (fNIRS), and functional magnetic resonance imaging (fMRI). Other works study the inclusion of additional physiological signals, such as muscular activity, electrodermal measurements, or heart rate, for the detection of emotions. In this context techniques for the identification of different electrophysiological signal sources, multimodal data fusion methods, and non-linear feature extraction approaches are discussed. Other contributions treat methodological problems, like the generalization of a (workload) classifier from the specific context in which it was trained to a more complex task and the search for suited evaluation criteria for affect classifiers. An unusual but valuable perspective is taken by works that look at the influence of affect on active BCI performance: Is the emotional state of BCI users a critical factor for their capability to control thoughtbased interaction and if so, what can we do to put them in the optimal state? Finally, theoretical contributions elucidate the value of BCI for the arts and for industry.

ACKNOWLEDGMENT

We thank the members of the program committee who provided their insightful and critical comments, contributing to the high quality of the chosen works: Egon L. van den Broek (TNO Technical Sciences, Delft, The Netherlands), Anne-Marie Brouwer (TNO Perceptual and Cognitive Systems, Soesterberg, the Netherlands), Stephen Fairclough (John Moores University, Liverpool, UK), Didier Grandjean (Swiss Center for Affective Sciences, Geneva, Switzerland), Hayrettin Gürkök (University of Twente, Enschede, the Netherlands), Jonghwa Kim (University of Augsburg, Germany), Brent Lance (Army Research Laboratory/TNB, Aberdeen Proving Ground, USA,) Fabien Lotte (INRIA Bordeaux -Sud-Ouest, France), Winfried Menninghaus (Freie Universität Berlin, Germany), Gary Garcia Molina (Philips Research North America, Briarcliff, USA), Christopher Honey (Princeton University, USA), Olga Sourina (NanYang Technological University, Singapore), Ioannis Patras (Queen Mary University, London, UK), Mannes Poel (University of Twente, Enschede, the Netherlands), Ed Tan (University of Amsterdam, the Netherlands), Aleksander Valjamae (University of Graz, Austria), Thorsten Zander (Technische Universität Berlin, Germany), and many subreviewers. We thank Hendri Hondorp, who took care of the aBCI website.


Yavar Kian

Quang Sang Phan

Eric Soccorsi

Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains

Keywords: AMS 2010 Mathematics Subject Classification: 35R30 Inverse problem, Schrödinger equation, scalar potential, Carleman estimate, infinite cylindrical domain

We consider the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain from one boundary Neumann observation of the solution. We prove Hölder stability by choosing the Dirichlet boundary condition suitably.

Statement of the problem and results

We continue our analysis of the inverse problem of determining the scalar potential q : Ω → R in a unbounded quantum cylindrical domain Ω = ω × R, where ω is a connected bounded open subset of R n-1 , n ≥ 2, with no less than C2 -boundary ∂ω, from partial Neumann data. This may be equivalently reformulated as to whether the electrostatic disorder occuring in Ω, modelling an idealized straight carbon nanotube, can be retrieved from the partial boundary observation of the quantum wave propagating in Ω. We refer to [START_REF] Kian | Carleman estimate for infinite cylindrical quantum domains and application to inverse problems[END_REF][ §1.2] for the discussion on the physical motivations and the relevance of this model.

Namely we seek stability in the identification of q from partial Neumann measurement of the solution u to the following initial boundary value problem    -iu ′ -∆u + qu = 0, in Q := (0, T ) × Ω u(0, x) = u 0 (x), x ∈ Ω u(t, x) = g(t, x), (t, x) ∈ Σ := (0, T ) × Γ.

(1.1)

Here T > 0 is fixed, Γ := ∂ω × R and the ' stands for ∂ ∂t . Since Γ is unbounded we make the boundary condition in the last line of (1.1) more explicit. Writing x := (x ′ , x n ) with x ′ := (x 1 , . . . , x n-1 ) ∈ ω for every x ∈ Ω we extend the mapping

C ∞ 0 ((0, T ) × R; H 2 (ω)) -→ L 2 ((0, T ) × R; H 3/2 (∂ω))) v → [(t, x n ) ∈ (0, T ) × R → v(t, •, x n ) |∂ω ],
to a bounded operator from L 2 ((0, T) × R; H 2 (ω)) into L 2 ((0, T) × R; H 3/2 (∂ω)), denoted by γ 0 . Then for every u ∈ C 0 ([0, T ]; H 2 (Ω)) the above mentioned boundary condition reads γ 0 u = g.

Throughout the entire paper we choose g := γ 0 G 0 , with G 0 (t, x) := u 0 (x) + it(∆ -q 0 )u 0 (x), (t, x) ∈ Q,

where q 0 = q 0 (x) is a given scalar function we shall make precise below.

In the particular case where q is a priori known outside some given compact subset of Ω and on the boundary Γ, it is shown in [START_REF] Kian | Carleman estimate for infinite cylindrical quantum domains and application to inverse problems[END_REF] that the scalar potential may be Lipschitz stably retrieved from one partial Neumann observation of the solution to (1.1) for suitable initial and boundary conditions u 0 and g. This result is similar to [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][Theorem 1], which was derived by Baudouin and Puel for the same operator but acting in a bounded domain. The main technical assumption common to [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][START_REF] Kian | Carleman estimate for infinite cylindrical quantum domains and application to inverse problems[END_REF] is that

u ∈ C 1 ([0, T ]; L ∞ (Ω)).
(

In this paper we pursue two main goals. First we want to analyse the direct problem associated to (1.1)-(1.2) in order to exhibit sufficient conditions on q and u 0 ensuring (1.3). Second, we aim to weaken the compactness condition imposed in [START_REF] Kian | Carleman estimate for infinite cylindrical quantum domains and application to inverse problems[END_REF] on the support of the unknown part of q, in the inverse problem of determining the scalar potential appearing in (1.1) from one partial Neumann observation of u.

The following result solves the direct problem associated to (1.1)-(1.2). Here and in the remaining part of this text we note w j,O , j ∈ N, for the usual H j -norm of w in any subset O of R m , m ∈ N * , where H 0 (O) stands for L 2 (O).

Theorem 1.1. Let k ≥ 2, assume that ∂ω is C 2(k+1) , and pick

(q 0 , u 0 ) ∈ W 2k,∞ (Ω) ∩ C 2(k-1) (Ω; R) × H 2(k+1) (Ω), such that (-∆ + q 0 ) 2+j u 0 = 0 on ∂Ω for all j ∈ N k-2 := {0, 1, . . . , k -2}. (1.4) Then for each q ∈ W 2k,∞ (Ω) ∩ C 2(k-1) (Ω) obeying the condition ∂ m x q = ∂ m x q 0 on ∂Ω for all m := (m j ) n j=1 ∈ N n with |m| := n j=1 m j ≤ 2(k -2), (1.5)
there is a unique solution u ∈ ∩ k j=0 C j ([0, T ]; H 2(k-j) (Ω)) to the boundary value problem (1.1)-(1.2). Moreover, we have the estimate

k j=0 u (j) C 0 ([0,T ];H 2(k-j) (Ω)) ≤ C u 0 2(k+1),Ω , (1.6)
where C > 0 is a constant depending only on T , ω, k, and max( q 0 W 2k,∞ (Ω) , q W 2k,∞ (Ω) ).

Remark 1.2. It can be seen from the reasonning developped in section 2 that Theorem 1.1 may be generalized to the case of more general boundary conditions than (1.2). Nevertheless, in order to avoid the inadequate expense of the size of this article, we shall not go into this matter further.

We now consider the natural number

ℓ ∈ N * ∩ (n/4, n/4 + 1] . (1.7)
Then, applying Theorem 1.1 with k = ℓ + 1, we get that u ∈ C 1 ([0, T ]; H 2ℓ (Ω)) and the estimate

u C 1 ([0,T ];H 2ℓ (Ω)) ≤ C u 0 2(ℓ+2),Ω . Since 2ℓ > n/2 then H 2ℓ (Ω) is continuously embedded in L ∞ (Ω).
This assertion, that is established by Lemma 2.7 in section 2.4, extends the corresponding well known Sobolev embedding theorem in R n (see e.g. [START_REF] Brezis | Analyse Fonctionnnelle. Théorie et applications, Collection mathématiques appliquées pour la maîtrise[END_REF][Corollary IX.13] or [START_REF] Evans | Partial Differential Equations[END_REF][ §5.10, Problem 18]) to the case of the unbounded cylindrical domain Ω. This immediately entails the: Corollary 1.3. Under the conditions of Theorem 1.1 for k = ℓ + 1, where ℓ is defined by (1.7), the solution u to (1.1)-(1.2) satisfies (1.3) and the estimate

u C 1 ([0,T ],L ∞ (Ω)) ≤ C u 0 2(ℓ+2),Ω .
Here C > 0 is a constant depending only on T , ω and max( q 0 W 2(ℓ+1),∞ (Ω) , q W 2(ℓ+1),∞ (Ω) ).

For q 0 (and u 0 ) as in Theorem 1.1, we aim to retrieve real-valued scalar potentials q verifying

|q(x ′ , x n ) -q 0 (x ′ , x n )| ≤ ae -b xn dǫ , (x ′ , x n ) ∈ Ω, (1.8)
where a > 0, b > 0, ǫ > 0 and d ǫ ∈ (2(1 + ǫ)/3, +∞) are a priori fixed constants. Here and henceforth the notation t stands for (1 + t 2 ) 1/2 , t ∈ R. Notice that (1.8) weakens the compactness condition imposed in [START_REF] Kian | Carleman estimate for infinite cylindrical quantum domains and application to inverse problems[END_REF][Theorem 1.1] on the support of the unknown part of q. Namely, we introduce the set of admissible potentials as A ǫ (q 0 ) := {q ∈ W 2(ℓ+1),∞ (Ω) ∩ C 2ℓ (Ω; R) verifying (1.5) for k = ℓ + 1 and (1.8)}.

Our main result on the above mentioned inverse problem is as follows.

Theorem 1.4. Let ∂ω, q 0 and u 0 obey the conditions of Theorem 1.1 for k = ℓ + 1, where ℓ is the same as in (1.7). Assume moreover that there are two constants υ 0 > 0 and ǫ > 0 such that we have

|u 0 (x ′ , x n )| ≥ υ 0 x n -(1+ǫ)/2 , (x ′ , x n ) ∈ Ω.
(1.9)

For M > 0 fixed, we consider two potentials q j , j = 1, 2, in A ǫ (q 0 ), such that q j W 2(ℓ+1),∞ (Ω) ≤ M , and we note u j the solution to (1.1)-(1.2) where q j is substituted for q, given by Theorem 1.1. Then, for all δ ∈ (0, b), where b is the same as in (1.8), there exists a subboundary γ * ⊂ ∂ω and a constant C > 0, depending only on ω, T , M , u 0 2(ℓ+2),Ω , δ, ǫ, a, b and υ 0 , such that the estimate

q 1 -q 2 0,Ω ≤ C ∂ ν u ′ 1 -∂ ν u ′ 2 θ 0,Σ * , (1.10) holds for Σ * := (0, T ) × γ * × R and θ := (b -δ)/(2b -δ).
It is evident that Theorem 1.4 yields uniqueness in the identification of the scalar potential in A ε (q 0 ) from the knowledge of partial Neumann data for the time-derivative of the solution to (1.1)-(1.2):

∀(q 1 , q 2 ) ∈ A ε (q 0 ) 2 , (∂ ν u ′ 1 (t, x) = ∂ ν u ′ 2 (t, x), (t, x) ∈ Σ * ) =⇒ (q 1 (x) = q 2 (x), x ∈ Ω) .
Further, it is worth mentioning that Theorem 1.4 applies for any subboundary

Γ * = γ * × R such that γ * ⊃ {x ′ ∈ ∂ω, (x ′ -x ′ 0 ) • ν ′ (x ′ ) ≥ 0},
for some arbitrary x ′ 0 ∈ R n-1 \ ω, where ν ′ (x ′ ) ∈ R n-1 denotes the outgoing normal vector to ∂ω computed at x ′ . See Assumption 3.1 in subsection 3.2 for more details.

Moreover we stress out that there are actual scalar potentials q 0 and initial values u 0 fulfilling the conditions of Theorem 1.4. For the sake of completeness a whole class of such matching q 0 's and u 0 's is indeed exhibited in the concluding remark ending the paper.

The remaining part of the paper is organized with two main sections. Section 2 contains the proofs of Theorem 1.1 and Corollary 1.3, while Theorem 1.4 is shown in section 3.

Analysis of the direct problem

In this this section we examine the direct problem associated to (1.1). To this end we introduce the Dirichlet Laplacian

A 0 := -∆, D(A 0 ) := H 1 0 (Ω) ∩ H 2 (Ω), (2.11) which is selfadjoint in H := L 2 (Ω) (see e.g. [5][Lemma 2.2]
), provided ∂ω is C 2 . We perturb A 0 by the scalar potential q ∈ L ∞ (Ω). Since q is assumed to be real then the operator

A q := A 0 + q, D(A q ) := D(A 0 ) = H 1 0 (Ω) ∩ H 2 (Ω), is selfadjoint in H = L 2 (Ω).

Abstract evolution problem

To study the direct problem associated to (1.1) we consider the abstract evolution problem

-iv ′ (t) + A q v(t) = f (t), t ∈ (0, T ) v(0) = v 0 , (2.12)
with initial data v 0 and source f . We shall derive smoothness properties of the solution to (2.12) with the aid of the following technical result, which is borrowed from [START_REF] Choulli | Stable determination of time dependent scalar potential from boundary measurements in a periodic quantum waveguide[END_REF][Lemma 2.1]:

Lemma 2.1. Let X be a Banach space and let M be an m-accretive operator in X with dense domain D(M ). Assume that v 0 ∈ D(M ) and h ∈ C 1 ([0, T ]; X). Then there is a unique solution

v ∈ C 1 ([0, T ]; X) ∩ C 0 ([0, T ]; D(M )) to the following evolution problem v ′ (t) + M v(t) = h(t), t ∈ (0, T ) v(0) = v 0 . (2.13) Moreover the estimate v C 0 ([0,T ];D(M)) + v ′ C 0 ([0,T ];X) ≤ C v 0 D(M) + h C 1 ([0,T ];X) , (2.14)
holds for some positive constant C depending only on T .

Assume that ∂ω is C 2 , let q ∈ L ∞ (Ω) be real and choose v 0 ∈ D(A q ) and f ∈ C 1 ([0, T ]; H). Since A q is selfadjoint in H then the operator iA q is m-accretive so Lemma 2.1 applies with X = H, M = iA q and h = if : there is a unique solution v ∈ C 1 ([0, T ]; H) ∩ C 0 ([0, T ]; D(A q )) (2.15)
to (2.12), such that

v C 0 ([0,T ];D(Aq)) + v ′ C 0 ([0,T ];H) ≤ C 1 v 0 D(Aq) + f C 1 ([0,T ];H) , (2.16)
for some constant C 1 > 0 depending only on T . We shall improve (2.15)-(2.16) upon assuming higher regularity on and ∂ω, q, v 0 and f . This requires that the following notations be preliminarily introduced. First, for the sake of definitiness we set A 0 q := I and D(A 0 q ) := H, where I denotes the identity operator in H. Then we put

A k q v := A q (A k-1 q v), v ∈ D(A k q ) := {v ∈ D(A k-1 q ), A q v ∈ D(A k-1 q )},
for each k ∈ N * , and recall that the linear space D(A k q ) endowed with the scalar product

v, w D(A k q ) := k j=0 A j q v, A j q w 0,Ω , is Hilbertian. Proposition 2.2. For k ≥ 2 fixed, assume that ∂ω is C 2(k+1) and let q ∈ W 2(k-1),∞ (Ω), v 0 ∈ D(A k q ) and f ∈ ∩ k j=1 C j ([0, T ]; D(A k-j q
)). Then (2.12) admits a unique solution

v ∈ ∩ k j=0 C j ([0, T ]; D(A k-j q )),
which satisfies the estimate

k j=0 v (j) C 0 ([0,T ];D(A k-j q )) ≤ C k   v 0 D(A k q ) + f C 0 ([0,T ];D(A k-1 q )) + k j=1 f (j) C 0 ([0,T ];D(A k-j q ))   ,
(2.17) where C k is a positive constant depending only on T .

Proof. The proof is by induction on k ≥ 2. a) Suppose that k = 2. We consider the space H := D(A q ), which is Hilbertian for the scalar product v, w H := v, w D(Aq) , and define the operator Ãq , acting in H, by

Ãq v := A q v, v ∈ D( Ãq ) := D(A 2 q ).
Since q is real, then the sesquilinear form a q [w] := ∇w 2 H + qw, w H ∈ R for every w ∈ H 1 0 (Ω), hence

Re i Ãq v, v H = Re (i(a q [v] + a q [A q v])) = 0, v ∈ D( Ãq ).
Moreover, for all ϕ ∈ H it is clear that the vector v ϕ := (iA q + 1) -2 (i Ãq + 1)ϕ ∈ D( Ãq ) verifies (i Ãq + 1)v ϕ = ϕ, proving that Ran(i Ãq + 1) = H. Thus i Ãq is m-accretive in H. Therefore, applying Lemma 2.1 for X = H, M = i Ãq and h = if , we see that there is a unique function

v ∈ C 1 ([0, T ]; D(A q )) ∩ C 0 ([0, T ]; D(A 2 q )), that is solution to -iv ′ (t) + Ãq v(t) = f (t), t ∈ (0, T ) v(0) = v 0 , (2.18)
and fulfills

v C 0 ([0,T ];D(A 2 q )) + v ′ C 0 ([0,T ];D(Aq)) ≤ C 1 v 0 D(A 2 q ) + f C 1 ([0,T ];D(Aq)) . (2.19)
Since v is solution to (2.18), it is also solution to (2.12). Hence v is the solution to (2.12), by uniqueness. Further, since A q is linear bounded from

D(A q ) into H and v ∈ C 1 ([0, T ]; D(A q )), we have A q v ∈ C 1 ([0, T ]; H) with (A q v) ′ (t) = A q v ′ (t). (2.20)
In light of (2.12) this entails that v

′ (t) = i(f (t) -A q v(t)) ∈ C 1 ([0, T ]; H). As a consequence we have v ∈ C 2 ([0, T ]; H) and -iv ′′ (t) + A q v ′ (t) = f ′ (t), t ∈ (0, T ), (2.21)
according to (2.20). Moreover (2.21) yields

v ′′ C 0 ([0,T ];H) ≤ A q v ′ C 0 ([0,T ];H) + f ′ C 0 ([0,T ];H) ≤ v ′ C 0 ([0,T ];D(Aq)) + f ′ C 0 ([0,T ];H) . From this and (2.19) then follows that 2 j=0 v (j) C 0 ([0,T ];D(A 2-j q )) ≤ max(1, 2C 1 ) v 0 D(A 2 q ) + f C 1 ([0,T ];D(Aq)) ,
which entails (2.17) for k = 2. b) Let us now prove the result for k ≥ 3. We suppose that the assertion holds true for all nonzero natural number smaller or equal to k -1 and choose

v 0 ∈ D(A k q ) and f ∈ ∩ k j=1 C j ([0, T ]; D(A k-j q
)). We know from part a) that the solution v of (2.12) is lying in

C 2 ([0, T ]; H) ∩ C 1 ([0, T ]; D(A q )) and satisfies (2.21). Putting w = v ′ we thus find out that w ∈ C 1 ([0, T ]; H) ∩ C 0 ([0, T ]; D(A q )) is solution to the problem -iw ′ (t) + A q w(t) = f ′ (t), t ∈ (0, T ) w(0) = i(f (0) -A q v 0 ). Since f (0) -A q v 0 ∈ D(A k-1 q ) and f ′ ∈ ∩ k-1 j=1 C j ([0, T ]; D(A k-1-j q
)), we deduce from the induction

hypothesis that w ∈ ∩ k-1 j=0 C j ([0, T ]; D(A k-1-j q
)) and that

k-1 j=0 w (j) C 0 ([0,T ];D(A k-1-j q
)) is majorized, up to the multiplicative constant C k-1 , by the following upper bound:

f (0) -A q v 0 D(A k-1 q ) + f ′ C 0 ([0,T ];D(A k-2 q )) + k-1 j=1 f (j+1) C 0 ([0,T ];D(A k-1-j q )) ≤ v 0 D(A k q ) + f C 0 ([0,T ];D(A k-1 q )) + f ′ C 0 ([0,T ];D(A k-2 q )) + k j=2 f (j) C 0 ([0,T ];D(A k-j q
)) .

This entails

k-1 j=0 w (j) C 0 ([0,T ];D(A k-1-j q )) ≤ C k-1   v 0 D(A k q ) + f C 0 ([0,T ];D(A k-1 q )) + k j=1 f (j) C 0 ([0,T ];D(A k-j q ))   .
As a consequence we have

v ∈ ∩ k j=1 C j ([0, T ]; D(A k-j q )), (2.22)
and

k j=1 v (j) C 0 ([0,T ];D(A k-j q )) ≤ C k-1   v 0 D(A k q ) + f C 0 ([0,T ];D(A k-1 q )) + k j=1 f (j) C 0 ([0,T ];D(A k-j q ))   .
(2.23) Now it remains to show that v ∈ C 0 ([0, T ]; D(A k q )). This can be seen from (2.12) and (2.22), entailing

A q v = f + iv ′ ∈ C 0 ([0, T ]; D(A k-1 q )). Therefore we have v C 0 ([0,T ];D(A k q )) ≤ f C 0 ([0,T ];D(A k-1 q )) + v ′ C 0 ([0,T ];D(A k-1 q
)) , which together with (2.23), yields (2.17).

Remark 2.3. In light of (2.15)-(2.16) it is clear that the statement of Proposition 2.2 holds true for k = 1 as well.

Characterizing

D(A k q ) for k ∈ N *
In light of Proposition 2.2 we need to describe more explicitely the domain of the operator A k q , k ∈ N * . Namely, under suitable assumptions on ∂ω and q we shall characterize D(A k q ) as a subset of H 2k (Ω). The main tool we use in the derivation of this result, stated in Proposition 2.5, is the elliptic boundary regularity property in Ω established in Lemma 2.4.

Elliptic boundary regularity in Ω

We now extend the classical elliptic boundary regularity result for the Dirichlet Laplacian, which is well known in any sufficiently smooth bounded domain of R n (see e.g. [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF][Chap. 2, Theorem 5.1] or [START_REF] Evans | Partial Differential Equations[END_REF][ §6.3, Theorem 5]), to the case of the unbounded cylindrical waveguide Ω. Namely we consider the following problem

-∆v = ϕ in Ω v = 0 on ∂Ω, (2.24)
and establish the:

Lemma 2.4. Let ℓ ∈ N, assume that ∂ω is C 2(ℓ+1) and choose ϕ ∈ H 2ℓ (Ω). Then (2.24) admits a unique solution v ∈ H 2(ℓ+1) (Ω), obeying v H 2(ℓ+1) (Ω)) ≤ C ℓ ϕ H 2ℓ (Ω) , (2.25)
where C ℓ > 0 is a constant depending only on ω and ℓ.

Proof. The proof is by induction on ℓ ∈ N and relies essentially on the following decomposition of the Dirichlet Laplacian A 0

F A 0 F -1 = ⊕ R A 0,p dp. (2.26)
Here F denotes the partial Fourier transform with respect to x n , i.e.

(F w)(x ′ , p) = w(x ′ , p) = 1 (2π) 1/2 R e -ipxn w(x ′ , x n )dx n , (x ′ , p) ∈ Ω, and A 0,p := -∆ x ′ + p 2 , p ∈ R, (2.27)
is the selfadjoint operator in L 2 (ω) generated by the closed quadratic form

a 0,p [w] := ω (|∇ x ′ w(x ′ )| 2 + p 2 |w(x ′ )| 2 )dx ′ , w ∈ D( a 0,p ) := H 1 0 (ω).
Since ω is a bounded domain with boundary no less than C 2 , we have

D( A 0,p ) = H 1 0 (ω) ∩ H 2 (ω), p ∈ R, (2.28)
from [START_REF] Agmon | Lectures on Elliptic Boundary Value Problems[END_REF]. For further reference we notice for all p ∈ R that

A 0,p w, w 0,ω = a 0,p [w] ≥ (c 0 (ω) + p 2 ) w 0,ω , w ∈ H 1 0 (Ω) ∩ H 2 (Ω),
where c 0 (ω) > 0 is the constant appearing in the Poincaré inequality associated to the bounded domain ω. As a consequence the operator A 0,p is boundedly invertible in L 2 (ω) and it holds true that

A -1 0,p B(L 2 (ω)) ≤ (c 0 (ω) + p 2 ) -1 , p ∈ R. ( 2
v(p) 2,ω ≤ C(ω) g(p) 0,ω ≤ C(ω)( ϕ(p) 0,ω + p 2 v(p) 0,ω ), p ∈ R. (2.31)
Here and henceforth C(ω) denotes some generic positive constant depending only on ω. On the other hand, since v(p) = A -1 0,p ϕ(p) by (2.30), we have

v(p) 0,ω ≤ (c 0 (ω) + p 2 ) -1 ϕ(p) 0,ω , p ∈ R,
according to (2.29). From this and (2.31) then follows that

v(p) 2,ω ≤ C(ω) ϕ(p) 0,ω , p ∈ R. (2.32) Since p → ϕ(p) ∈ L 2 (R; L 2 (ω)) as ϕ ∈ L 2 (Ω), then we have v = F -1 v ∈ L 2 (R; H 2 (ω)) and v L 2 (R;H 2 (ω)) ≤ C(ω) ϕ 0,Ω , (2.33)
by (2.32). Further, taking into account that

A 0,p v(p) 2 0,ω = ∆ x ′ v(p) 2 0,ω + 2p 2 ∇ x ′ v(p) 2 0,ω + p 4 v(p) 2 0,ω = ϕ(p) 2 0,ω , p ∈ R,
from (2.27) and (2.30), we get that R p 4 v(p) 2 0,ω dp ≤ ϕ 2 0,Ω . This entails that v ∈ H 2 (R;

L 2 (ω)) obeys v H 2 (R;L 2 (ω)) ≤ ϕ 0,Ω . (2.34) Applying [8][Chap. 1, Theorem 7.4] to v ∈ L 2 (R; H 2 (ω)) ∩ H 2 (R; L 2 (ω)) we find that v ∈ H 2 (Ω). Moreover, the norm in L 2 (R; H 2 (ω)) ∩ H 2 (R; L 2 (ω)) being equivalent to the usual one in H 2 (Ω), (2.33)-(2.34) yields v 2,Ω ≤ C(ω) ϕ 0,Ω .
b) For ℓ ≥ 1 fixed, assume that ∂ω is C 2(ℓ+1) and let ϕ ∈ H 2ℓ (Ω). We suppose in addition that the claim obtained by substituting k ∈ N ℓ-1 for ℓ in Lemma 2.4, and the following estimate

v(p) 2(k+1),ω ≤ C(ω) k j=0 p 2j ϕ(p) 2(k-j),ω , p ∈ R, ( 2
v(p) 2(ℓ+1),ω ≤ C(ω) ℓ j=0 p 2j ϕ(p) 2(ℓ-j),ω , p ∈ R,
(2.37) from (2.36). As a consequence we have v ∈ L 2 (R; H 2(ℓ+1) (ω)) and the following estimate:

v L 2 (R;H 2(ℓ+1) (ω)) ≤ C(ω) ℓ j=0 ϕ H 2j (R;H 2(ℓ-j) (ω)) ≤ C(ω) ϕ 2ℓ,Ω . (2.38)
Further, multiplying (2.30) by p 2ℓ yields p 2(ℓ+1) v(p) = p 2ℓ ( ϕ(p) + ∆ x ′ v(p)) for a.e. p ∈ R, so we get

p 2(ℓ+1) v(p) 0,ω ≤ p 2ℓ ( ϕ(p) 0,ω + v(p) 2,ω ) ≤ C(w)p 2ℓ ϕ(p) 0,ω , p ∈ R,
by applying (2.35) with k = 0. From this then follows that v ∈ H 2(ℓ+1) (R; L 2 (ω)), with

v H 2(ℓ+1) (R;L 2 (ω)) ≤ C(ω) ϕ H 2ℓ (R;L 2 (ω)) ≤ C(ω) ϕ H 2ℓ (Ω) .
(2.39)

On the other hand, each v ∈ H 2k (Ω) such that A j q v ∈ H 1 0 (Ω) for all j ∈ N k-1 , fulfills

A m q v ∈ H 2(k-1) (Ω) and A j q (A m q v) ∈ H 1 0 (Ω), j ∈ N k-2 , m = 0, 1, so we get that v ∈ D(A k-1 q ) (resp., A q v ∈ D(A k-1 q
)), by applying the induction assumption to A m q v with m = 0 (resp., m = 1). As a consequence we have v ∈ D(A k q ), showing that the inclusion relation may be reversed in (2.42), which proves the desired result. This immediately entails the: Corollary 2.6. For k ≥ 2, assume that ∂ω is C 2(k+1) and let q 0 , q ∈ W 2(k-1),∞ (Ω) ∩ C 2(k-2) (Ω) fulfill (1.5). Then we have D(A k q ) = D(A k q0 ).

Proof of Theorem 1.1

Evidently, u is solution to

(1.1)-(1.2) iff v := u -G 0 is solution to the following boundary value problem    -iv ′ -∆v + qv = f in Q v(0, x) = 0, x ∈ Ω v(t, x) = 0, (t, x) ∈ Σ, (2.43)
where

f := iG ′ 0 + (∆ -q)G 0 = it(-∆ + q 0 ) 2 u 0 -(q -q 0 )G 0 , (2.44)
and G 0 is the function defined by (1.2). We first prove that

f ∈ C ∞ ([0, T ]; D(A k-1 q )), (2.45)
and

f C 1 ([0,T ];D(A k-1 q )) ≤ C u 0 2(k+1),Ω , (2.46)
where C denotes some generic positive constant depending only on T , ω and M . To do that we notice from (1.2) that G 0 ∈ C ∞ ([0, T ]; H 2k (Ω)), with

G 0 C 1 ([0,T ];H 2k (Ω)) ≤ C u 0 2(k+1),Ω . (2.47)
As a consequence we have (q -q 0 )G 0 ∈ C ∞ ([0, T ]; H 2(k-1) (Ω)). Hence Proposition 2.5 yields

(q -q 0 )G 0 ∈ C ∞ ([0, T ]; D(A k-1 q )), (2.48)
since every (-∆ + q 0 ) j (q -q 0 )G 0 , j ∈ N k-2 , vanishes on ∂Ω, by (1.5). Here we used the identity

D(A k-1 q ) = D(A k-1 q0
), arising from (1.5) and Corollary 2.6. Similarly, since (-

∆ + q 0 ) 2 u 0 ∈ H 2(k-1) (Ω), we deduce from (1.4) that (-∆ + q 0 ) 2 u 0 ∈ D(A k-1 q
). This, (2.44) and (2.48), entails (2.45), while (2.46) follows from (2.47) and the basic estimate (-∆ + q 0 ) 2 u 0 2(k-1),Ω ≤ C u 0 2(k+1),Ω .

Further, refering to (2.43), we see that v is solution to (2.12). Since

f ∈ C ∞ ([0, T ]; D(A k-1 q )), by (2.45), then v is uniquely defined in ∩ k j=0 C j ([0, T ]; D(A k-j q
)) from Proposition 2.2. Moreover, bearing in mind that f is affine in t, we get

k j=0 v (j) C 0 ([0,T ];D(A k-j q )) ≤ C u 0 2(k+1),Ω , (2.49)
by (2.17) and (2.46). Now, upon recalling that G 0 is an affine function of t, the claim of Theorem 1.1 follows readily from the identity u = v + G 0 , (2.47), (2.49) and Proposition 2.5.

A Sobolev embedding theorem in Ω

The derivation of Corollary 1.3 from Theorem 1.1 boils down to the following result.

Lemma 2.7. Let k ∈ N * satisfy k > n/2 and assume that ∂ω is C k . Then we have

H k (Ω) ⊂ L ∞ (Ω).
Moreover there is a constant c > 0, depending only on n, k and ω, such that the estimate

h L ∞ (Ω) ≤ c h k,Ω , (2.50)
holds for all h ∈ H k (Ω).

Proof. Since ω is a bounded domain of R n-1 with C k boundary, there exists an extension operator

P ∈ B(H j (ω); H j (R n-1 )), j ∈ N k , (2.51)
such that

(P f ) |ω = f, f ∈ L 2 (ω), (2.52)
according to [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]

[Chap. 1, Theorem 8.1]. Next, for all f ∈ L 2 (Ω), put Pf (x ′ , x n ) = [P f (•, x n )](x ′ ), (x ′ , x n ) ∈ Ω. It is apparent from (2.51) that P ∈ B(H m (R; H j (ω)); H m (R; H j (R n-1
)) for all natural numbers m and j such that m + j ≤ k. As a consequence we have

P ∈ B(H k (Ω); H k (R n )). (2.53)
Moreover, it follows readily from (2.52) that

(Ph) |Ω = h, h ∈ L 2 (Ω). (2.54)
Pick h ∈ H k (Ω). Since Ph ∈ H k (R n ) by (2.53), then the Sobolev embedding theorem [START_REF] Brezis | Analyse Fonctionnnelle. Théorie et applications, Collection mathématiques appliquées pour la maîtrise[END_REF][Corollary IX.13] yields Ph ∈ L ∞ (R n ) and the estimate

Ph L ∞ (R n ) ≤ C Ph k,R n , (2.55)
where the constant C > 0 is independent of h. From this and (2.54) then follows that h ∈ L ∞ (Ω), with

h L ∞ (Ω) ≤ Ph L ∞ (R n ) . ( 2

Stability estimate

In this section we prove (1.10) by adapting the Bukhgeim-Klibanov method introduced in [START_REF] Bukhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF]. It is by means of a Carleman estimate specifically designed for the system under consideration.

Linearization and time symmetrization

Set ρ := q 1 -q 2 so that u := u 1 -u 2 is solution to the boundary value problem

   -iu ′ -∆u + q 1 u = -ρu 2 in Q u(0, x) = 0, x ∈ Ω u(t, x) = 0, (t, x) ∈ Σ. (3.57) Since u ∈ C 2 ([0, T ]; H 2(ℓ-1) (Ω)) ∩ C 1 ([0, T ]; H 2ℓ (Ω))
  -iv ′ -∆v + q 1 v = -ρu ′ 2 in Q v(0, x) = -iρu 0 , x ∈ Ω v(t, x) = 0, (t, x) ∈ Σ, (3.58)
where v := u ′ . Since q j ∈ A ǫ (q 0 ), j = 1, 2, we have

ρu 0 ∈ H 1 0 (Ω) ∩ H 2 (Ω). Consequently v ∈ C 1 ([0, T ]; L 2 (Ω)) ∩ C 0 ([0, T ]; H 1 0 (Ω) ∩ H 2 (Ω)
) by applying Remark 2.3 and Proposition 2.5 for k = 1. Further, putting u 2 (-t, x) = u 2 (t, x) for all (t, x) ∈ [-T, 0) × Ω and bearing in mind that u 0 and q j , j = 1, 2, are real-valued, we deduce from (3.58) that the function v, extended on [-T, 0) × Ω by setting v(t, x) := -v(-t, x), is the

C 1 ([-T, T ]; L 2 (Ω)) ∩ C 0 ([-T, T ]; H 1 0 (Ω) ∩ H 2 (Ω))-solution to the system    -iv ′ -∆v + q 1 v = -ρu ′ 2 in Q := (-T, T ) × Ω v(0, x) = -iρu 0 , x ∈ Ω v(t, x) = 0, (t, x) ∈ Σ := (-T, T ) × Γ.
(3.59)

The main tool needed for the derivation of (1.10) is a global Carleman inequality for the Schrödinger equation in (3.59). We use the estimate derived in [START_REF] Kian | Carleman estimate for infinite cylindrical quantum domains and application to inverse problems[END_REF][Proposition 3.3], that is specifically designed for unbounded cylindrical domains of the type of Ω.

Global Carleman estimate for the Schrödinger equation in Ω

Given the Schrödinger operator acting in (

C ∞ 0 ) ′ ( Q), L := -i∂ t -∆, (3.60)
we introduce a function β ∈ C 4 (ω; R + ) and an open subset γ * of ∂ω, satisfying the following conditions:

Assumption 3.1. (i) ∃C 0 > 0 such that the estimate |∇ x ′ β(x ′ )| ≥ C 0 holds for all x ′ ∈ ω;
(ii) ∂ ν β(x ′ ) := ∇ x ′ β(x ′ ).ν(x ′ ) < 0 for all x ′ ∈ ∂ω\γ * , where ν is the outward unit normal vector to ∂ω;

(iii) ∃Λ 1 > 0, ∃ǫ > 0 such that we have λ|∇ Notice that there are actual functions β verifying Assumption 3.1, such as ω ∋

x ′ → |x ′ -x ′ 0 | 2 , for an arbitrary x ′ 0 ∈ R n-1 \ ω and a subboundary γ * ⊃ {x ′ ∈ ∂ω, (x ′ -x ′ 0 ) • ν ′ (x ′ ) ≥ 0}. Next, for all x = (x ′ , x n ) ∈ Ω, put β(x) := β(x ′ ) + K, where K := r β ∞ for some r > 1, (3.61)
and define the two following weight functions for λ > 0:

ϕ(t, x) := e λβ(x) (T + t)(T -t) and η(t, x) := e 2λK -e λβ(x) (T + t)(T -t) , (t, x) ∈ Q. (3.62)
Finally, for all s > 0, we introduce the two following operators acting in (C ∞ 0 ) ′ ( Q):

M 1 := i∂ t + ∆ + s 2 |∇η| 2 and M 2 := isη ′ + 2s∇η • ∇ + s(∆η). (3.63)
It is apparent that M 1 (resp. M 2 ) is the adjoint (resp. skew-adjoint) part of the operator e -sη Le sη , where L is given by (3.60).

We may now state the following global Carleman estimate, that is borrowed from [START_REF] Kian | Carleman estimate for infinite cylindrical quantum domains and application to inverse problems[END_REF][Proposition 3.3]. Proposition 3.2. Let β and γ * obey Assumption 3.1, let β, ϕ and η be given by (3.61)-(3.62), and let the operators M j , j = 1, 2, be defined by (3.63). Then there are two constants s 0 > 0 and C > 0, depending only on T , ω and γ * , such that the estimate

s e -sη ∇ x ′ w 2 0, Q + s 3 e -sη w 2 0, Q + j=1,2 M j e -sη w 2 0, Q ≤ C s e -sη ϕ 1/2 (∂ ν β) 1/2 ∂ ν w 2 0, Σ * + e -sη Lw 2 0, Q ,
holds for all s ≥ s 0 and any function w ∈ L 2 (-T, T ; H 1 0 (Ω)) verifying Lw ∈ L 2 ( Q) and ∂ ν w ∈ L 2 (-T, T ; L 2 (Γ * )). Here Γ * (resp., Σ * ) stands for γ * × R (resp., (-T, T ) × γ * × R).

Proof of Theorem 1.4

We use the same notations as in §3.1 and, for the sake of notational simplicity, we denote the various positive constants appearing in the derivation of Theorem 1.4 by C. Following [START_REF] Kian | Carleman estimate for infinite cylindrical quantum domains and application to inverse problems[END_REF][Lemmae 3.3 & 3.4], we start by establishing the coming technical result with the aid of Proposition 3.2. Lemma 3.3. For all s > 0, we have the estimate: This, along with (3.64) and the identity φ(0, •) 0,Ω = e -sη(0,•) v(0, •) 0,Ω entail e -sη(0,•) v(0, •) 2 0,Ω = 2Im (-T,0)×Ω M 1 φ(t, x)φ(t, x)dtdx

e -sη(0,•) ρu 0 2 0,Ω ≤ C s -3/2 e -sη(0,•) ρu ′ 2 2 0,Q + s -1/2 e -sη(0,•) ∂ ν v 2
≤ 2 M 1 φ 0, Q φ 0, Q ≤ s -3/2 s 3 e -sη v 2 0, Q + M 1 e -sη v 2 0, Q .
Finally, the desired result follows from this upon recalling (3.59) and applying Proposition 3.2 to v.

Fix y > 0. In virtue of Lemma 3.3, it holds true that e -sη(0,•) ρu 0 2 0,ω×(-y,y) ≤ C s -3/2 e -sη(0,•) ρu ′ 2 2 0,Q + s -1/2 µ , s > 0,

where

µ := ∂ ν u ′ 1 -∂ ν u ′ 2 2
0,Σ * . This entails (υ 2 0 y -(1+ǫ) -Cs -3/2 ) e -sη(0,•) ρ 2 0,ω×(-y,y) ≤ C s -3/2 ρ 2 0,ω×(R\(-y,y)) + s -1/2 µ , s > 0, (3.65) since |u 0 (x)| ≥ υ 0 y -(1+ǫ)/2 for all x ∈ ω × (-y, y), by (1.9), u ′ 2 L ∞ (Q) ≤ C by Corollary 1.3, and η(0, x) ≥ 0 for all x ∈ Ω. Taking s = (υ 2 0 /(2C)) -2/3 y 2(1+ǫ)/3 in (3.65) and bearing in mind that η(0, .) L ∞ (Ω) ≤ e 2K /T 2 , we thus obtain that ρ 2 0,ω×(-y,y) ≤ Ce C y 2(1+ǫ)/3 ρ 2 0,ω×(R\(-y,y)) + y 2(1+ǫ)/3 µ . Therefore, (1.10) follows from this and (3.70).

Concluding remark

In this subsection we build a class of scalar potentials q 0 and initial conditions u 0 , fulfilling the conditions of Theorem 1.4. To do that we first introduce two functions u b , q b : R → R, defined by where c > 0 is some fixed constant. Evidently, we have u b ∈ H 2(ℓ+2) (R) and q b ∈ W 2(ℓ+1),∞ (R) ∩ C 2ℓ (R). (3.72)

  x ′ β(x ′ ) • ζ| 2 + D 2 β(x ′ , ζ, ζ) ≥ ǫ|ζ| 2 for all ζ ∈ R n-1 , x ′ ∈ ω and λ > Λ 1 , where D 2 β(x ′ ) := ∂ 2 β(x ′ ) ∂xi∂xj 1≤i,j≤n-1 and D 2 β(x ′ , ζ, ζ) denotes the R n-1 -scalar product of D 2 β(x ′ )ζ with ζ.

2 0

 2 0,Σ * . Proof. Put φ := e -sη v. In light of (3.61)-(3.62) it holds true that lim t↓(-T ) η(t, x) = +∞ for all x ∈ Ω, hence lim t↓(-T ) φ(t, x) = 0. As a consequence we have φ(0, •) 2 0,Ω = (-T,0)×Ω (|φ| 2 ) ′ (t, x)dtdx = 2Re (-T,0)×Ω φ ′ (t, x)φ(t, x)dtdx . (3.64) On the other hand, (3.63) and the Green formula yield Im (-T,0)×Ω M 1 φ(t, x)φ(t, x)dtdx = Re (-T,0)×Ω φ ′ (t, x)φ(t, x)dtdx + W, with W := Im (-T,0)×Ω ∆φ(t, x)φ(t, x)dtdx + s 2 (∇η)φ

  (R\(-y,y)) ≤ 4a 2 |ω| 2 R\(-y,y) e -2b xn dǫ dx n ≤ 4a 2 |ω| 2 R e -δ xn dǫ dx n e -(2b-δ) y dǫ .From this and (3.66) then follows that ρ 2 0,ω×(-y,y) ≤ C y 2(1+ǫ)/3 e C y 2(1+ǫ)/3 e -(2b-δ) y dǫ + µ .(3.67)Set µ δ := e -(2b-δ) . We examine the two cases µ ∈ (0, µ δ ) and µ ≥ µ δ separately. If µ ∈ (0, µ δ ) we takey = y(µ) := -(-y,y) ≤ C y 2(1+ǫ)/3 e C y 2(1+ǫ)/3 -(2b-δ) y dǫ . Since d ǫ > 2(1 + ǫ)/3, this entails that ρ 2 0,ω×(-y,y) ≤ C sup t>1 t 2(1+ǫ)/3 e Ct 2(1+ǫ)/3 -δt dǫ e -2(b-δ) y dǫ ≤ Cµ 2θ , µ ∈ (0, µ δ ). (3.68) On the other hand, it follows from (1.8) that ρ 2 0,ω×(R\(-y,y)) ≤ C R e -2δ xn dx n e -2(b-δ) y dǫ ≤ Cµ 2θ , µ ∈ (0, µ δ ). (3.69) Putting (3.68)-(3.69) together, we obtain that ρ 2 0,Ω ≤ Cµ 2θ , µ ∈ (0, µ δ ). (3.70) Finally, if µ ≥ µ δ , we use the estimate ρ 2 0,Ω ≤ 4a 2 |ω|( R e -2b xn dǫ dx n ), arising from (1.8), and find: ρ 2 0,Ω ≤ 4a 2 |ω| R e -2b xn dǫ dx n µ 2θ δ µ 2θ ≤ Cµ 2θ , µ ≥ µ δ .

u

  b (y) := c y -(1+ǫ)/2 , q b (y) := u ′′ b (y) u b (y) , y ∈ R,(3.71)

  by Theorem 1.1, we may differentiate (3.57) w.r.t. t, getting 

CPT, CNRS UMR 7332, Université d'Aix-Marseille, 13288 Marseille, France & Université du Sud-Toulon-Var, 83957 La Garde, France.

Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University,

30-348 Kraków, Poland. 3 CPT, CNRS UMR 7332, Université d'Aix-Marseille, 13288 Marseille, France & Université du Sud-Toulon-Var, 83957 La Garde, France.

Summing up, we have v ∈ L 2 (R; H 2(ℓ+1) (ω)) ∩ H 2(ℓ+1) (R; L 2 (ω)). Interpolating with [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF][Chap. 1, Theorem 7.4], we thus find out that v ∈ H 2(ℓ+1) (Ω). Finally, (2.25) follows from (2.38)-(2.39) and the equivalence of norms in L 2 (R; H 2(ℓ+1) (ω)) ∩ H 2(ℓ+1) (R; L 2 (ω)) and H 2(ℓ+1) (Ω). This and (2.37) proves the assertion of the lemma. Lemma 2.4 we are now in position to prove the following:

More on

,∞ (Ω), and assume that ∂ω is C 2(k+1) . Then we have

(2.40)

is equivalent to the usual norm in H 2k (Ω). Namely, if q W 2(k-1),∞ (Ω) ≤ M for some M > 0, then we may find a constant c k = c k (M, ω) > 0, such that we have

, it is enough to prove the right inequality in (2.41). This can be done by noticing that v ∈ D(A q ) is solution to (2.24) with ϕ = A q v -qv ∈ H, which entails

and then applying (2.25) with ℓ = 0. b) Let k ≥ 2 be fixed and suppose that (2.40)

), which is embedded in H 2(k-1) (Ω) by induction assumption, so we have ϕ ∈ H 2(k-1) (Ω). Therefore, v being solution to (2.24), it holds true that v ∈ H 2k (Ω), from Lemma 2.4. Moreover the r.h.s. of (2.41), and hence both sides, since the l.h.s. is completely straightforward, follows by applying (2.25) with ℓ = k -1. Further, the induction hypothesis combined with the fact that v ∈ D(A k-1 q

). Summing up, we have

Then, upon setting for all x = (x ′ , x n ) ∈ Ω,

it is apparent from (3.72) that q 0 ∈ W 2(ℓ+1),∞ (Ω) ∩ C 2ℓ (Ω) and u 0 ∈ H 2(ℓ+2) (Ω). Moreover, it follows readily from (3.71) and (3.73)-(3.74) that

Finally, to show that (q 0 , u 0 ) satisfies (1.4), we invoke (3.74), getting

and consequently

by (3.71). This immediately yields (-∆ + q 0 ) 2+j u 0 (x) = (-∆ + q 0 ) 1+j (-∆ + q 0 )u 0 (x) = 0, x ∈ Ω ∩ (O × R), j = 0, 1, • • • , ℓ -1, which, in turn, entails (1.4), since O × R is a neighborhood of ∂Ω in R n .