Morad Chadni
email: morad.chadni@utc.fr

Nabil Grimi

Isabelle Ziegler

Nicolas Brosse

Olivier Bals

High voltage electric discharges treatment for high molecular weight hemicelluloses extraction from spruce

Introduction

Polymeric hemicelluloses are polysaccharides with an important structural diversity and functionality. Although cellulose is the most widely used polysaccharide, the potential of application of polymeric hemicelluloses has been demonstrated by several authors [START_REF] Bai | Influences of Configuration and Molecular Weight of Hemicelluloses on Their Paper-Strengthening Effects[END_REF][START_REF] Lindqvist | Galactoglucomannan Derivatives and Their Application in Papermaking[END_REF]Mikkonen and Tenkanen 2012;[START_REF] Alvira | Pretreatment Technologies for an Efficient Bioethanol Production Process Based on Enzymatic Hydrolysis: A Review[END_REF]Hartman, Albertsson, and Sjöberg 2006). The physicochemical properties of hemicelluloses have made them today an important untapped resource that can be used in several fields.

Spruce hemicelluloses are mainly composed of acetylated galactoglucomannan (AcGGM). AcGGM has a backbone of β-D-(1→4) -glucopyranosyl and β-D-(1→4)-mannopyranosyl units in a linear or slightly branched chain. Single α-D-galactopyranose units are attached to the main chain by (1→6)bonds. The mannopyranosyl units are acetylated at C-2 or C-3 with a degree of acetylation of about 0.5 [START_REF] Capek | Galactoglucomannan from the Secondary Cell Wall of Picea Abies L. Karst[END_REF]. An approximate ratio of galactose-to-glucose-to-mannose is around 0.5-1.1:1:3.5-4.5 for water-soluble AcGGM [START_REF] Timell | Recent Progress in the Chemistry of Wood Hemicelluloses[END_REF]. The average weight molar mass Mw of native AcGGM has been reported to be between 20 and 50 kDa and the degree of polymerization varies from 100 to 400 [START_REF] Timell | Recent Progress in the Chemistry of Wood Hemicelluloses[END_REF][START_REF] Willför | Spruce-Derived Mannans -A Potential Raw Material for Hydrocolloids and Novel Advanced Natural Materials[END_REF]).

GGMs have the potential to become high added-value products with various applications in food, health, papermaking, textile and cosmetic industries (Mikkonen et al. 2012(Mikkonen et al. , 2010)). Different applications request GGMs in different forms (different molar masses). High molecular masses GGMs can be used as such or mixed with other polysaccharides to form a raw material for biodegradable films with promising oxygen barrier properties (Hartman et al. 2006).

The extraction of spruce hemicelluloses in general has been investigated for years. The common studies were focused on the extraction of monomeric form of wood polysaccharides for ethanol bioconversion. However, due to the discovery of the high potential of application of polymeric hemicelluloses, many authors have investigated the extraction of the polymeric form of hemicelluloses from wood. Extraction of softwood hemicelluloses has been studied by hydrothermal treatment at high temperatures (≥170 °C) [START_REF] Krogell | Intensification of Hemicellulose Hot-Water Extraction from Spruce Wood in a Batch Extractor -Effects of Wood Particle Size[END_REF]Song, Pranovich, and Holmbom 2012), by chemical extraction (alkaline) [START_REF] Capek | Galactoglucomannan from the Secondary Cell Wall of Picea Abies L. Karst[END_REF][START_REF] Curling | Development of a Method for the Production of Hemicellulosic Gels from Sitka Spruce[END_REF][START_REF] Alvarez-Vasco | Alkaline Hydrogen Peroxide Pretreatment of Softwood: Hemicellulose Degradation Pathways[END_REF], ionic liquids [START_REF] Anugwom | Selective Extraction of Hemicelluloses from Spruce Using Switchable Ionic Liquids[END_REF], by pressured hot water [START_REF] Song | Extraction of Galactoglucomannan from Spruce Wood with Pressurised Hot Water[END_REF][START_REF] Gallina | Hemicellulose Extraction by Hot Pressurized Water Pretreatment at 160oC for 10 Different Woods: Yield and Molecular Weight[END_REF], and by microwave heat-fractionation ( > 180 °C) [START_REF] Gulbrandsen | Extracting Hemicelluloses from Softwood and Bagasse as Oligosaccharides Using Pure Water and Microwave Heating[END_REF][START_REF] Palm | Extraction of Hemicellulosic Oligosaccharides from Spruce Using Microwave Oven or Steam Treatment[END_REF][START_REF] Lundqvist | Isolation and Characterization of Galactoglucomannan from Spruce (Picea Abies)[END_REF].

The fractionation of wood at high temperatures leads to high solubilization of all macromolecules, including hemicelluloses. However, the extraction at higher temperatures led to a lower pH level, causing deacetylation and partly hydrolytic cleavage of GGMs, and consequently, decreasing the solubility, the yield and the molar mass of extracted GGMs.

Nowadays, electrical pretreatments that provoke electroporation and cell walls destabilization have also been used for biomolecules extraction [START_REF] Grimi | Effect of a Pulsed Electric Field Treatment on Expression Behavior and Juice Quality of Chardonnay Grape[END_REF][START_REF] Boussetta | Intensification de l'extraction Des Polyphénols Par Électrotechnologies Pour La Valorisation Des Marcs de Champagne[END_REF][START_REF] Boussetta | Extraction of Soluble Matter from Grape Pomace by High Voltage Electrical Discharges for Polyphenol Recovery: Effect of Sulphur Dioxide and Thermal Treatments[END_REF][START_REF] Loginova | Acceleration of Soluble Matter Extraction from Chicory with Pulsed Electric Fields[END_REF]. High voltage electric discharges (HVED) have shown very interesting results

for the biomolecules extraction from biomass. HVED showed its effectiveness for polyphenols extraction from vine shoots [START_REF] Rajha | A Comparative Study of Physical Pretreatments for the Extraction of Polyphenols and Proteins from Vine Shoots[END_REF]) and grape pomace [START_REF] Boussetta | Extraction of Soluble Matter from Grape Pomace by High Voltage Electrical Discharges for Polyphenol Recovery: Effect of Sulphur Dioxide and Thermal Treatments[END_REF][START_REF] Boussetta | Scale-up of High Voltage Electrical Discharges for Polyphenols Extraction from Grape Pomace: Effect of the Dynamic Shock Waves[END_REF]) lignin from rapeseed straw [START_REF] Brahim | Delignification of Rapeseed Straw Using Innovative Chemo-Physical Pretreatments[END_REF], flavonoids from peanut shells based [START_REF] Yan | Continuous High Voltage Electrical Discharge Extraction of Flavonoids from Peanut Shells Based on 'Annular Gap Type' Treatment Chamber[END_REF], polyphenol and protein extraction from sesame cake [START_REF] Sarkis | Effect of Pulsed Electric Fields and High Voltage Electrical Discharges on Polyphenol and Protein Extraction from Sesame Cake[END_REF], polyphenols and lignans from flaxseed cake (Boussetta et al. 2013) and sugars from orange peels [START_REF] El Kantar | High Voltage Electrical Discharges Combined with Enzymatic Hydrolysis for Extraction of Polyphenols and Fermentable Sugars from Orange Peels[END_REF]. Indeed, when applied in a liquid medium, HVED have physical features leading to the extraction of biomolecules from raw materials located inside the cell walls. The main effect of HVED is the fragmentation and destruction of raw materials. This destruction occurs through a certain number of physical phenomena, which will be discussed later. These physical phenomena are complex and the way they lead to extraction of raw material is insufficiently delighted.

There are numerous phenomena that could lead to the fragmentation of biomass during the application of HVED in liquid medium. Such as electrical breakdown which occurs when the streamer crosses the two electrodes, the breakdown induces the formation of a powerful shock wave (hundreds of MPa) in only a few micro-or nanoseconds. Shock waves are known to be responsible of mechanical cell membrane damages [START_REF] Boussetta | Extraction of Soluble Matter from Grape Pomace by High Voltage Electrical Discharges for Polyphenol Recovery: Effect of Sulphur Dioxide and Thermal Treatments[END_REF]. Directly after shock waves, the apparition and the collapse of microcavitation bubbles create an intense and very localized shock waves that can lead to the fragmentation of raw material. The electric arc also emits high-intensity UV light and induces a large variety of chemical reactions like radical formation, radical molecule reaction, photochemical reaction, ozone molecules reaction and aqueous electron reaction etc. [START_REF] Adda | High-Voltage Electrical Discharge Assisted Extraction of Phenolic Compounds from Grape Seeds[END_REF]. UV light in the range of 200-400 nm is mutagenic to cells [START_REF] Boussetta | Extraction of Valuable Biocompounds Assisted by High Voltage Electrical Discharges: A Review[END_REF] and hydroxyl radicals lead to oxidative cell damage.

To our knowledge, the HVED have not yet been applied to the extraction of hemicelluloses from sawdust wood. The originality of this study is to suggest an original method to extract high molecular weight hemicelluloses and reducing the severity of conventional treatments by combining HVED with chemical and thermal treatment. The main objective of this work is to extract high molecular weight hemicelluloses at mild conditions (low temperature and suitable pH). Mild conditions were achieved by combining HVED with chemical extraction and autohydrolysis.

Materials and methods

The objective of this study is to evaluate the effect of HVED pretreatment on the extraction of high molecular weight hemicelluloses from spruce sawdust. For this purpose, numerous pretreatments at different impulsions (0-4 ms) were carried out in two different mediums (1M NaOH and water). The temperature generated during HVED pretreatment was between 40 and 50 °C.

Raw material

Spruce sawdust with a particle size of (1-5 mm) was produced from a spruce tree from the Lorraine region in France and stored at room temperature in a sealed polyethylene bags in the dark until extraction. The average moisture content of the raw material was 10% w/w. The hemicellulose content of the sawdust (~30 % w/w dry biomass) was calculated by analyzing no-cellulosic monosaccharides of spruce wood according to NREL/TP-510-42618 method. Carbohydrate composition of sawdust includes 72 % glucan, 7,5 % xylan, and 16,3 % of mannan and other sugars.

Extraction procedure

HVED pretreatment

HVED pretreatment was carried out using a high voltage pulsed power 40 kV-10 kA generator (Basis, Saint Quentin, France), 1-liter cylindrical batch treatment chamber. The HVED treatment chamber was equipped with needle-plate geometry electrodes. The diameters of stainless-steel needle and the grounded disk electrodes were 10 and 35 mm, respectively. The distance between the electrodes was 10 mm. The energy was stored in a set of low-inductance capacitors, which were charged by the highvoltage power supply. The electrical discharges were generated by electrical breakdown in the medium of extraction (liquid) at the peak pulse voltage (U) of 40 kV. (1)

Where n (=1-400) is the number of pulses and (≈10 ± 1 μs) is the pulse duration.

The specific energy input W (kJ/kg) was obtained from Eq. ( 2):

(2)

Where is the pulse energy (kJ/pulse) and m is sawdust mass (kg). is determined from Eq. ( 3).

(3)

Treatment chamber

(a) (b)
Where U is the voltage (V) and I is the current strength (A)

Diffusion

The goal of the diffusion step is to facilitate the transfer of hemicelluloses chains from wood matrix to the extracellular medium. Temperature during HVED treatment was between 40 and 50 °C. In order to maintain mild conditions of extraction (low temperature, low decrease of pH), the diffusion was done under reflux at 50°C during 3 h with agitation.

Separation of extracted hemicelluloses

After each experiment, the extract was filtered on qualitative filter paper with retention particle size of 5-13 µm (VWR International, Fontenay-sous-Bois, France). The volume and end-pH of the extract solutions were measured at room temperature shortly after the extraction. The liquid fraction from 1M

NaOH impregnated sawdust was neutralized to a pH of 5.5-6 by a solution of 72% H 2 SO 4 . 10 mL were taken from the filtered extract for analyses. The extract solution was concentrated by vacuum evaporation using a water bath at 50 °C. To precipitate high-molecular weight hemicelluloses, four volumes of 95% ethanol were added to the liquid extracts. The liquid phase was separated from the precipitated hemicelluloses by settling at room temperature and decantation. Hemicelluloses were further separated from the remaining liquid by centrifugation at 4000 rpm for 10 min at room temperature. The precipitated hemicelluloses were re-dissolved in distilled water and dialyzed with a dialysis membrane with a cut-off of 3.4 kDa (Spectrum Laboratories, Inc., Rancho Dominguez, CA, USA). Finally, dialyzed hemicelluloses solutions were freeze-dried.

Measurement of monosaccharides and uronic acids using HAPE-PAD

The monosaccharide content of the soluble fraction was analyzed according to NREL/TP-510-42618 NaOH (20 mM) 40% 37-41 min. Each elution was followed by a wash and subsequent equilibration step. External sugar and uronic acid standards were used for calibration (7 points per curve). The reagents and standards are provided by Sigma-Aldrich and Fisher Scientific.

Size-exclusion chromatography (SEC)

A Shimadzu Prominence HPLC system is used to determine molecular weight distribution of hemicelluloses by gel filtration chromatography. The stationary phase is composed of two columns linked in series: Phenomenex Polysep GFC-P as a guard column (7.8 x 35 mm), Phenomenex Polysep GFC-P3000 and GFC-P2000 analytical columns (7.8 x 300 mm). Sodium nitrate (0.25 M) is used as an eluent with a 0.5 mL.min -1 flow rate. Samples are filtered with a 0.45 µm membrane and 20 µL are injected. Samples are separated at 30 °C and detection is done with a Refractive Index Detector (Shimadzu RID-20A). Each analysis runs for 70 min. aqueous solutions (1 mg/mL) of dextran standards with molecular weights of 1000; 5000; 12000; 24000; 50000 and 80,000 Da were used to prepare the calibration curve.

FTIR

FTIR analysis for extracted hemicelluloses were performed by a FTIR spectrophotometer (Thermo Scientific iD3ZnSe ATR-Nicolet iS 5 FTIR Spectrometer unit). Hemicelluloses powder was placed on a ZnSe crystal plate. Each IR spectrum was obtained over the wave number range of 400-4000 cm -1 with eight scans at resolution of 4 cm -1 . The spectra were analyzed using ThermoScientific OMNIC software.

2D 13C-1H HSQC NMR spectroscopy

2 H NMR and 1 H-13C correlation 2D NMR (HSQC) spectra were recorded on a Bruker III 400

MHz spectrometer equipped with a 5-mm BBFO probe. Experiment was conducted with 20 mg sample dissolved in 1 ml D2O. The number of collected complex points was 1024 for the 1Hdimension with a relaxation of 1.5-and 256-time increments were recorded in 13C-dimension.

HSQC spectra were taken at room temperature.

Thermal gravimetric analysis (TGA)

The thermal behavior of extracted hemicelluloses was investigated by TGA (Thermal Gravimetric Analysis) (Setaram Setsys). The samples were dried before thermal analysis. The heating rate in the experiments was 10 C.min-1 up to 600 C. A platinum cup was used in these tests. The weight of the sample was recorded during the temperature rise under azote flow (30 ml / min).

Scanning microscopy

The surface morphology of sawdust fibers was studied by scanning electron microscopy (SEM). An environmental high-resolution electron scanning microscope QUANTA 250 FEG (FEI Company) was used in low-vacuum mode.

Results and discussion

Yield of hemicellulose extraction

Fig. 2 (A andB), shows the yield of extraction of non-monomeric hemicellullose chains from spruce sawdust in the liquid fraction in distilled water and in basic media (1M NaOH) respectively. As expected higher hemicelluloses yields were obtained in basic conditions (≈ + 80 %). This observation is be explained by the fact that alkaline solutions increase the water absorption capacity, swell up cellulosic fibers and decrease their crystallinity, and thus increase the specific surface area by increasing the volume of the solid [START_REF] Wen | Isolation and Characterization of Hemicellulose and Cellulose from Sugar Beet Pulp[END_REF]. HVED pretreatments increase significantly the yield of extraction in both media. In water, at 4 ms of HVED pretreatment (376 J/g DM), the yield of extraction was increased by 38 % compared to the extraction without HVED activation (0 ms) (5.8 mg/g of DM for 0 ms pretreatment and 8 for 4 ms pretreatment). In basic medium, the advantage of HVED pretreatment step was also apparent. The yield was about 15.8 and 19.7 mg/g of dry wood for 0 and 4 ms of pretreatment respectively.

The application of HVED-treatment results in generation of shock waves. These shockwaves may play an important role in the raw material fragmentation, thus facilitating hemicelluloses extraction. The formation of the shock waves in liquid medium is often followed by the apparition of cavitation bubbles. The combined effect of shock waves and cavitation bubbles may improve the disruption and the destabilization of cell walls [START_REF] Boussetta | Extraction of Valuable Biocompounds Assisted by High Voltage Electrical Discharges: A Review[END_REF][START_REF] Sarkis | Effect of Pulsed Electric Fields and High Voltage Electrical Discharges on Polyphenol and Protein Extraction from Sesame Cake[END_REF][START_REF] Boussetta | Intensification de l'extraction Des Polyphénols Par Électrotechnologies Pour La Valorisation Des Marcs de Champagne[END_REF].

The pH of extraction is also an important parameter for the extraction of hemicelluloses chains. After extraction, the final pH of the suspension treated in water had dropped from 7 to 6.1 for untreated sawdust (0 ms) and from 7 to 5.2-5.3 for HVED pretreated sawdust. The drop of pH in the suspension treated in water is due to the release of carboxylic acids, mainly acetic acid from GGMs [START_REF] Casebier | Chemistry and Mechanism of Water Prehydrolysis on Southern Pine Wood[END_REF]. Acetic acid released in the suspension treated in 1M NaOH solution are neutralized by hydroxide anions and pH remains stable at 13.5 for both untreated and HVED-treated samples.

Fig.2 (A and B)

show also the composition of the extracted hemicelluloses in water and 1M NaOH solution respectively. Polysaccharides extracted in water are composed of mannose (28%), glucose (16%), galactose (23%) xylose (2.1 %), arabinose (6.5%), galacturonic acid (2.25 %) and glucuronic acid (20%). The relative sugar composition of these polysaccharides is the same as for a typical GGM [START_REF] Timell | Recent Progress in the Chemistry of Wood Hemicelluloses[END_REF]. Indeed, in water 64 % of extracted sugars are GGMs. In 1M NaOH, the sugar composition of liquid extract is as follows: mannose (22%), glucose (11.3%), galactose (12%), xylose arabinose (≈ 8%). The fraction extracted in water is reach in mannose, galactose and glucose (40 %, 27 % and 13 % respectively). The relative sugar composition of these polysaccharides is the same as for a spruce galactoglucomannans [START_REF] Timell | Recent Progress in the Chemistry of Wood Hemicelluloses[END_REF]. On the other hand, hemicelluloses extracted in 1M

NaOH solution seem to be mainly composed of arabinoxylans. From these results, it is remarkable to note that the medium of extraction (pH) strongly affects the selectivity of the extraction (solubilization of hemicelluloses). The degree of acetylation is calculated as the molar Ac/Man ratio. As shown in Fig. 6, all extracted hemicelluloses in water exhibited an acetylation degree (AD) in the range of 0.05-0.08, lower compared to that of native GGM in spruce wood (0.45 -0.50) [START_REF] Timell | Recent Progress in the Chemistry of Wood Hemicelluloses[END_REF]). It can be observed from Compared to 4 ms -1M NaOH, the weight loss mainly occurred between 250 and 335 °C. It was concluded that 4 ms-1M NaOH was thermally more stable at this phase. This higher thermal stability canbe attributed to their higher molecular weights (see table 1). Outstandingly, this thermal decomposition process can be attributed to the cracking and abscission of C-C and C-O bonds connected with the main branch of hemicelluloses, and the decarboxylation and decarbonylation may also occur [START_REF] Sun | Successive Alkali Extraction and Structural Characterization of Hemicelluloses from Sweet Sorghum Stem[END_REF]. The third phase involved a gradual loss in weight at temperature above 350 °C, which may be attributed to decomposition of cellulose and continued up to 400 °C. Since the thermal and chemical condition during extraction of hemicelluloses are so mild??? (low temperature, low decrease of pH). The presence of a very small quantities of cellulose in the extracted hemicelluloses owing to the physical and electrical effect of HVED on the wood matrix. Which could be due to the propagation of high-energy shock waves which could provoke the cleavage of the βglycosidic linkage. This observation is in accordance with [START_REF] Brahim | Impact of Ultrasounds and High Voltage Electrical Discharges on Physico-Chemical Properties of Rapeseed Straw's Lignin and Pulps[END_REF], who found that HVED induced a fragmentation of cellulose macromolecules from rapeseed straw mostly when HVED is associated to the soda treatment. The fourth phase exhibited weight loss at a temperature above 400 °C resulted from the decomposition of lignin. It is observed that a higher lignin content was co-extracted in basic conditions (4 ms -1M NaOH).

Fig. 1 .

 1 Fig. 1. HVED setup and treatment chamber: (a) before HVED pretreatment; (b) during HVED

  method. The sugar concentrations were measured by HAPE-PAD (ICS-3000 Dionex, Dionex Corp., Sunnyvale, USA) equipped with a Dionex CarboPac TM PA-20 (3 × 150 mm) analytical column. Filtered samples (20 μL) were eluted at 35 °C and at 0.4 mL/min with the following composition: pure water 99.2%/250 mM NaOH 0.8%: 0-20 min; pure water 75%/250 mM NaOH 20%/ NaOCOCH 3 (1 M)-NaOH (20 mM) 5% 20-37 min; and pure water 40%/250 mM NaOH 20%/NaOCOCH 3 (1 M)-

  Fig. 2. Yield of extraction and sugar composition of extracted hemicelluloses versus HVED treatment

Fig. 3 .Fig. 4

 34 Fig.3. Total monomers and polymers extracted versus HVED-treatment time in (a) water and (b) 1M

Fig. 4 .

 4 Fig. 4. Sugar composition of hemicellulose chains recovered in water and 1 M NaOH.

  Holmbom 2011) and by saponification during the extraction in basic solutions. At higher pH levels Song et al., 2011 have found that deacetylation was very fast, thus decreasing the extraction of GGM.

Fig x that

  Fig x that the AD increased with the increase of HVED pretreatment time. This observation can be

Fig. 5 .

 5 Fig. 5. Acetylation degree of extracted hemicelluloses in water after HVED pretreatment

Fig. 6 .

 6 Fig. 6. FTIR spectra of extracted hemicelluloses in different conditions

Fig. 7 .

 7 Fig. 7. 2D HSQC NMR spectral analysis of extracted hemicelluloses in 1M NaOH (A) and Water (B)

Fig. 8 .

 8 Fig.8. Thermograms of hemicellulosic fractions

Fig. 9 .

 9 Fig.9. Scanning electron microscopy images of sawdust submitted to HVED pretreatment: (A) in 1M

Table 1 )

 1 . With increasing the time of pretreatment by HVED, the average molecular weight of high molecular weight hemicelluloses increased. Pretreatment for 4 ms by HVED gave the highest Mw (55 kDa and 47 kDa in 1M NaOH solution and water respectively). As expected, the Mw of the extracted carbohydrates clearly increased with 1M NaOH solution compared to extraction in water.

	The polydispersity index (PI) of hemicelluloses is also given in table 1. As shown in table 1, all
	hemicelluloses fractions show a moderate low polydispersity index (1.5-2.8), suggesting a chemical
	and structural homogeneity.
	Table.1 molecular weight distribution of recovered hemicelluloses

Deacetylation of hemicelluloses occurs principally during hot-water extraction

(Song, Pranovich, and

Acknowledgment

This work was financially supported by SOFREN Company and ANRT "Agence National de la Recherche et de la Technologie". The authors appreciate the financial support from SOFREN and ANRT. LERMAB is supported by the French National Research Agency through the Laboratory of Excellence ARBRE (ANR-12-LABXARBRE-01).

Abbreviations

HVED :


Jiuchun Gao
email: jiuchun.gao@imt-atlantique.fr

Anatol Pashkevich
email: anatol.pashkevich@imt-atlantique.fr

Fabien Claveau
email: fabien.claveau@imt-atlantique.fr

Philippe Chevrel
email: philippe.chevrel@imt-atlantique.fr

Real Time Motion Generation for Mobile Robot

Keywords: mobile robot, real time control, motion generation, time-optimal systems 

The paper proposes a new real time motion generation technique for a mobile robot, which is able to find time-optimal motions along a curved path taking into account capabilities of the driving motors and ensuring the wheels rolling without skidding. The problem is converted to a time-optimal control of a second-order dynamic system under constraints on the control input, the first derivative of output, and mixed constraint on the control variable and the output derivative. After the state space discretization, the original problem is presented as a combinatorial one where the desired robot trajectory corresponds to a shortest path on the relevant graph. To find this path and ensure its real time implementation, a moving window strategy combined with dynamic programming is proposed. Advantages of this approach and its suitability to real time control are illustrated by a case study dealing the fastest motion of the mobile robot along a sinusoidal path.

INTRODUCTION

Currently, due to significant developments in computational geometry and computer vision, essential progress has been achieved in the area of mobile robot path planning. Existing on-board robot controllers are able to generate the path in real time for very complex and partially unknown environment [START_REF] González | A review of motion planning techniques for automated vehicles[END_REF]. In the meantime, the problem of time-optimal trajectory planning along the generated path received less attention. In literature, existing techniques are mostly based on the classical phase-plane method that requires some non-trivial actions in the case of complex statedependent constraints [START_REF] Petrinić | Time-optimal velocity planning along predefined path for static formations of mobile robots[END_REF]. On the other side, the problem of time-optimal trajectory planning along a specified path has been intensively studied in robotics for many years. Relative works focused on serial manipulators and the desired optimal end-effector motions were generated taking into account the velocity/acceleration bounds in the joints and physical constraints of the actuating motors [START_REF] Bobrow | Timeoptimal control of robotic manipulators along specified paths[END_REF][START_REF] Pfeiffer | A concept for manipulator trajectory planning[END_REF][START_REF] Shiller | Computation of path constrained time optimal motions with dynamic singularities[END_REF][START_REF] Yang | Fast Algorithms for Near-Minimum-Time Control of Robot Manipulators: Communication[END_REF][START_REF] Butler | A general algorithm for time-optimal trajectory generation subject to minimum and maximum constraints[END_REF]. These works employ the phase-plane technique and are based on the fact that the time-optimal solution is "bang-bang", i.e. at any time instant one of the actuators works at the limit of its velocity/acceleration. Because of the geometric limits imposed by the specified path, the state vector contains only two variables, the distance and the velocity along the path. However, the non-linear robotic manipulator kinematics and dynamics transform the actuators limits into state-dependent constraints on velocity and acceleration along the path. Thus, the original problem is converted to the time-optimal control of a second-order linear system with nonlinear statedependent constraints on velocity/acceleration, where the acceleration is also treated as a control input. In the above mentioned papers, the authors have presented numbers of examples and obtained some important properties of the time-optimal control, i.e. multiple-switching between acceleration and deceleration, which are not common for the second-order dynamic systems studied in the classical control theory. Nonetheless, in the frame of the phase-plane method, there is no general approach that allows generating multipleswitching optimal control for the considered problem.

It is also worth of mentioning that several works dealing with time-optimal control of multi-axes CNC machines where it is necessary to specify the feedrate variation along a curved path yielding minimum traversal time subject to prescribed acceleration bounds along each axis. Here, the optimal solution also relies on the ''bang-bang'' principle [START_REF] Timar | Algorithms for time-optimal control of CNC machines along curved tool paths[END_REF]. More recent results focus on smooth minimumtime trajectory generation with higher order constraints (i.e. axes velocities, accelerations, jerks, etc.) [START_REF] Bharathi | Feedrate optimization for smooth minimum-time trajectory generation with higher order constraints[END_REF]. However, an essential practical difficulty exists in this application area, which is caused by common industrial practice of approximating free-form curved paths with short linear/circular G-code segments. It makes the realization of continuously-varying feedrates problematic. The difficulty is gradually disappearing when the new CNC machines allow the real time interpolation for analytic curves (with complete information of the path curvature and the feedrate variation).

For wheeled mobile robots, the time-optimal control problem is usually presented in a slightly different way. It is assumed that the path is not known in advance and it is obtained at the first stage using relevant geometric algorithms and taking into account constraints imposed by dynamic workspace obstacles, required path smoothness and nonholonomy of the mobile robot [START_REF] Jacobs | Planning smooth paths for mobile robots[END_REF]Canny, 1993, Soueres and[START_REF] Soueres | Shortest paths synthesis for a car-like robot[END_REF]. Then at the second stage, a time-optimal velocity profile is generated under dynamic constraints from the motor physical capacities and also must ensure the wheel rolling without skidding [START_REF] Weiguo | Optimal motion planning for a wheeled mobile robot[END_REF]. The nonskidding condition can be presented as a non-linear constraint imposed on the velocity/acceleration, which leads to the problem of optimal control with multiple switchings. In most of existing works, the desired time-optimal profiles are obtained using the phase-plane technique, which in the case of multiple switchings is difficult for algorithmization and can be hardly implemented in mobile robot controllers. In our previous work [START_REF] Gao | Optimal motion generation for mobile robot with nonskidding constraints[END_REF], it was proposed a combinatorial technique allowing generating time-optimal trajectory with multiple switchings. But, its time efficiency is not high enough for real time control. For this reason, this work concentrates on improvement of this technique and development of relevant algorithms that allow generating in real time the optimal motions for any combinations of linear/nonlinear constraints on velocity and acceleration.

PROBLEM STATEMENT

Let us assume that the desired mobile robot path is presented in the parametric form and is defined by two continuously differentiable functions max { ( ), ( ) ; [0, ] } x s y s s s 

(1) describing evolution of Cartesian coordinates of the reference point while driving along the given path, where s is the distance from the origin to the current position (treated here as the parameter). Also, let us define the path curvature described by the radius function

max ( ), [0, ] r s s s  (2)
which can be computed from () xs , () ys using standard formulas. The problem is to find the motion law along the path () The first two of the above constraints describe physical limitations of the robot driving motors whose velocity () st and torque (transformed into the linear acceleration () st ) are bounded and cannot exceed allowable limits max v and max a . The third constraint takes into account physical properties of the contact "robot wheel road surface" where the total inertia force cannot exceed the friction force. It can be easily proved that after some simplifications such condition can be presented in the form of

2 2 22 () ( ) ( ) () st s t g rt        (5)
where the components on the left-hand side are the lateral and longitudinal accelerations respectively, g is the gravity acceleration, and  is the friction coefficient.

Using terminology from the optimal control theory and treating the acceleration as a control input, the considered problem can be present as minimization of the cost functional If the last of the above constraints is not active, the optimal solution is trivial and can be obtained using conventional methods based on "bang-bang" principle. In this case the desired motion is obtained by sequentially applying accelerations max max ( , 0, ) aa  that produce either the trapezoidal or triangular velocity profile. However, generally, these constraints may compete to each other and the structure of the optimal solution becomes more complicated, which creates essential difficulties when generating time-optimal motion in real time. It should be mentioned that usually the constraint ( 9) is applied at the reference point of the mobile robot, which is the robot model centroid as shown in Fig. 1. However in practice, it is necessary to verify this constraint for each wheel, so the non-skidding constraint becomes slightly stronger. This particularity will be taken into account in the proposed motion generation algorithms.

REAL TIME MOTION GENERATION STRATEGY

In classical control theory, the time-optimal motions for the second-order dynamic systems are generated using the conventional phase-plane method, taking into account specific properties of the optimal control input () ut. It was proved that in the simplest case, when only the input constraint max ua  is applied to the dynamic system (7), the optimal control has a "bang-bang" form, where the amplitude is constant . For these simple cases, the phase-plane method is efficient. But, as follows from relevant studies [START_REF] Bobrow | Timeoptimal control of robotic manipulators along specified paths[END_REF][START_REF] Petrinić | Time-optimal velocity planning along predefined path for static formations of mobile robots[END_REF], application of the additional non-linear constraint similar to (9) can crucially change the structure of the optimal control function () ut , which may include multiple switchings. To take into account approximately the non-linear constraint (9) and assume that at any time instant the time-optimal control operates at the limit of either the control input u or the admissible velocity s , let us present the modification of the classical phase-plane method. A basis idea here is to replace the primary upper bounds on the velocity and acceleration ( 8) by new values depending on the current state of the dynamic system ( , )

ss. Such way of combination of the inequalities (8) and ( 9 that are included in the time-optimal motion planning algorithm. It is worth of mentioning that this modified phaseplane technique works perfectly well if the non-linear constraint ( 10) is not too hard, and it is satisfied even for the trajectory segments corresponding to 0 u  . However, in the general case, the constraints ( 8) and ( 9) may compete strongly and the obtained forward and backward trajectories are not feasible [START_REF] Gao | Optimal motion generation for mobile robot with nonskidding constraints[END_REF]).

An alternative approach to the time-optimal motion generation, which is proposed in [START_REF] Gao | Optimal motion planning in redundant robotic systems for automated composite lay-up process[END_REF], is based on the discrete dynamic programming. Because of its universal nature, this technique is able to take into account all considered constraints in a similar way and to generate optimal trajectories of complex structure that include multiple segments separated by so-called switching points (also known as discontinuity points, tangent points and singular critical points [START_REF] Timar | Algorithms for time-optimal control of CNC machines along curved tool paths[END_REF]). In contrast to the phaseplane method, the dynamic programming considers a set of feasible trajectories at each time instant, from which the best one is selected at the final stage. However, as follows from our study, this approach is rather time consuming while applied to the entire displacement to be implemented. For example, in the case of grid 630×100, the total computation time for the time-optimal trajectory generation in Matlab R2014b was about two minutes (Intel i5 2.67GHz). For this reason, a "moving-window" strategy combined with dynamic programming for the real time optimal motion generation is proposed in this paper. Similar to the technique presented in [START_REF] Gao | Optimal motion generation for mobile robot with nonskidding constraints[END_REF], this strategy is able to generate feasible solutions while taking into account all the constraints, but it is significantly faster.

Fig. 2. Moving window scheme of real-time trajectory planning for mobile robot

This proposed approach breaks down the full-size problem into a set of sub-problems. As is shown in Fig. 2, a moving window is applied segment by segment. For each window, a developed trajectory planning algorithm (related information presented in next section) is used to generate the local timeoptimal trajectory. The solution previously generated is already able to be implemented while the following computation is still ongoing. This allows implementing the feasible motion in real-time. In more details, an outline of the moving window scheme is presented below in the form of pseudo-codes. The input includes the arrays of displacement and path curvature { ( ), ( ) 1, 2,... } S i R i i n 

, velocity and acceleration limits, window size "width" and cut point "cutid", as well as the discretization step of the velocity "dv". The algorithm output is the optimal trajectory "VS". The total procedure is composed of three basic steps. The first step defines the initial state, including the position, velocity and the distance to the end point. At the second step, a window with the desired size is applied starting from the second path point (since the starting point is known [ vs ] = DP_ planning ( ini, dis, dv, rs, vm, am); VS = [VS; vs]; planning function "DP_ planning" is invoked to generate the optimal solution for this window. Then, the generated trajectory is split into two parts at the "cutid" position. The former section is remained and the latter is removed. After that, the window is moved to the "cutid" position and the same operation is applied again. This process is repeated if the robot is not approaching to the final point. At the third step, the last window is applied to generate the trajectory ending with zero velocity at the final position.

MOTION PLANNING USING DISCRETE DYNAMIC RROGRAMMING FOR ONE SEGMENT

For one window (one segment), the motion generation problem is solved using discrete dynamic programming technique.

i k i k i s v s k    C .
While considering the path points are ordered in time, the original sequence of i s described by presented equations may be converted into a directed graph presented in Fig. 3. It should be noted that some of the states generated by previous equation should be excluded from the graph. These inadmissible states are not connected to any neighbor. It is clear that due to time-irreversibility, the allowable connections between the graph nodes are limited to the subsequent configuration states Using the discrete search space, the considered problem can be transformed to the searching of the shortest path on the above presented graph. In the frame of this notation, the desired solution can be represented as the sequence

21 (0,1) ( , 2) ( , 1) (0, ) { } { } ...{ } { } n k k n n      C C C C
. The distance between subsequent nodes can be evaluated as the displacement time in the following way:

11 ( , ) ( , 1) 1 ( , ) 2

( ) ( )

i i i i k i k i i i k k dist s s v v      CC (12)
The latter allows us to present the objective function (travelling time) as follows

1 1 ( , ) ( , 1) 1 ( , ) ii n k i k i i T dist       CC (13) that depends on the indices 1 2 1 , ,... , nn k k k k  .
It should be noted that the applied method of search space discretization automatically takes into account the velocity constraints, but the acceleration constraints must be examined as follows

1 max () ii v v t a     (14) where 1 ( , ) ( , 1) ( ,
) ii k i k i t dist    CC .
Additionally, the nonskidding condition (5) for each wheel should be also verified for computing the edge weights as follows

2 2 2 () i i a v R g     (15)
where can be estimated from ( 14).

After discretization, the problem is converted to a combinatorial one, which can generally be transformed to the classical shortest path search on the graph. However, the straightforward approach is extremely more time-consuming than dynamic programming based technique [START_REF] Gao | Optimal motion planning in redundant robotic systems for automated composite lay-up process[END_REF].

The developed algorithm breaks down the full-size problem into a set of sub-problems, aiming at finding the shortest path

Algorithm: DP based technique

Input:

Matrix of states -C(k,i) of size m×n Array of radius -R(i) of size 1×n Output: Minimum path length -D min Optimal path indices -k 0 (i), i = 1,2,…n Notations: Distance matrix -D(k,i) of size m×n Pointer matrix -P(j,i-1) of size m×n Functions: Distance between nodes -dist ( C(k 1 ,i 1 ), C(k 2 ,i 2 ) )

Skidding test for a node -skid ( C(k,i), R(i) ) Acceleration test for nodes -acc (C(k 1 ,i 1 ), C(k 2 ,i 2 )) (1) Set D(k,1):=0; P(k,1):=null; ∀k=1,2,…m (2) For i = 2 to n do For k = 1 to m do For j = 1 to m do (a)

If (acc (C(k,i), C(j,i-1)) = 0) & (skid ( C(k,i), R(i) )=0 ) & (skid ( C(j,i-1), R(i-1) )=0 ) r(j) := D(j,i-1)+dist ( C(k,i), C(j,i-1) ) else r(j) := Inf end (b) Set D(j,i) := min(r); P(j,i) := argmin(r); (3) Set D min := min( D(k,n) ); k 0 (n) := argmin(r) (4) For i = 2 to n do Set k 0 (i-1) := P(k 0 (i), i) from the initial 1 ( ,1) 1 { , } k k  C to the current ( , ) { , } i k i i k  C
. To present the basic idea, let us denote , ki d as the length of the shortest path connecting the initial node to the current node . The algorithm operates with two tables ( , ) D k i and ( , ) P k i that include the minimum distances for the sub-problem of lower size (for the path 1i) and the pointers to the previous locations respectively. The procedure is composed of four basic steps. The first step initializes the distance and pointer matrices. In step (2), the recursive formula ( 16) is implemented. The computing start from the second layer and it tries all possible connections between the nodes in the current layer and the previous one. It includes verifications of the non-skidding constraint and acceleration limit in the substep (2a). The sub-step (2b) finds the minimum path from the current node ( , ) C k i to the first layer { ( ,1), } C j j  and records the reference to { ( , 1), } C j i j  in the pointer matrix. In steps (3) and ( 4), the optimal solution is finally obtained and corresponding path is extracted by backtracking.

AN APPLICATION EXAMPLE

To evaluate the efficiency of the proposed technique, let us apply it to the generation of time-optimal motions along a sinusoidal path whose curvature varies essentially. This path is presented in Fig. 4 For this path of length max 152.72 m s  , the radius of curvature varies essentially from +10.00 m to the positive infinity and from the negative infinity to -10.00 m, which makes the non-skidding constraint (9) active at the path segments closest to the peak and the bottom. . So, with the reduced grid 60×50, the computing time for the following windows is less than 1.0 sec. In total, the computation time for the entire displacement was about 18.0 sec. It is much more efficient compared to the computational effort of our previous technique (more than 2 min). In order to meet the requirement of real time, this Matlab code was also transformed into the C code. In the environment of C, the computing time for entire displacement was 0.51 sec and for one window was 0.13 sec.

The obtained time-optimal trajectory is presented in Fig. 5, which shows that the desired motion is composed of multiple acceleration/deceleration segments separated by the relevant number of the switchings, which separate segments with different active constraints. This result is in good agreement with physical nature of the problem, which requires reducing the speed in advance when closing to the path segments with high curvature to ensure pure rolling of the robot wheels. Hence, the proposed method allows user obtaining optimal trajectories in real time for any combination of linear/nonlinear constraints describing limitations of the robot driving motors without skidding.

CONCLUSIONS

Recent advances in mobile robotics and increase of their achievable driving speeds require essential improvement of existing motion generation algorithms. For this reason, in addition to the conventional velocity and acceleration constraints issued from the limited capacity of the driving motors, the "non-skidding" constraints imposed by the physical properties of the wheels contacts with the moving surface must be also considered. Because of the complicated description, they can be hardly integrated into existing motion planning algorithms that rely on the phase-plane method. To overcome this difficulty, the paper proposes a new technique that allows finding time-optimal motions along a curved path that takes into account limitations of the driving motors and also ensures the robot wheels rolling without skidding. After the state space discretization, the original problem is presented as a combinatorial one, where the desired robot trajectory is the shortest path on the relevant graph. To obtain this path and ensure its implementation in real time, a moving window strategy combined with dynamic programming is proposed. This technique allows taking into account all considered constraints in a similar way and generating optimal trajectories of complex structure that include multiple segments separated by switching points. Advantages of this technique and its suitability to real time control were confirmed by a case study dealing the fastest motion of the mobile robot along a sinusoidal path.

Fig. 1 .

 1 Fig. 1. Schematic diagram of a mobile robot model (planar)

  considered, the optimal control is also discontinuous, but it includes an intermediate interval with ( ) 0 ut  ensuring satisfaction of the velocity constraint. So, there are two switchings here, from max

  ) yields the modified bounds of the control input u and the velocity s :

  Function: Motion planning for one window -DP_ planning (1) Set d2e:= max(S); VS:= [0,0]; ini:= [VS, R(1)]; i:= 2; ds: = 0.5*vm*vm/am; (2) While d2e>1.5*ds Set dis: = S (i:width+i-1); rs: = R(i:width+i-1); [ vs ] = DP_ planning ( ini, dis, dv, rs, vm, am); VS : = [VS; vs(1:cutid)]; ini : = [vs(cutid), rs(cutid)]; d2e := d2e-(s(i+cutid -1)-s(i-1)); i:= i+cutid; End (3) Set dis := S(i:end); rs := R(i:end);

  to the minimum travelling time.

Fig. 3 .

 3 Fig. 3. Graph-based presentation of the discrete search space for real-time trajectory planning

   . The parameters of the mobile robot included in the constraints (8) and (9) are assumed to be as follows:

Fig. 4 .

 4 Fig. 4. A sinusoidal path shape used for motion generation For this case study, the width of the window was about 15 m, which is higher than the minimum accelerating/decelerating distance 6.25 m. The displacement step 0.28m s  and the velocity step 0.10m s v  . So, the search space of the first window was discretized with the grid 60×100. The computing time for the first window in Matlab R2014b was around 2.1 sec (Intel i5 2.67GHz). Then, all the next windows started from the 50 th point of the previous one, and the sampling of [0, 10] v  was replaced by [5, 10] v . So, with the reduced grid 60×50, the computing time for the following windows is less than 1.0 sec. In total, the computation time for the entire displacement was about 18.0 sec. It is much more efficient compared to the computational effort of our previous technique (more than 2 min). In order to meet the requirement of real time, this Matlab code was also transformed into the C code. In the environment of C, the computing time for entire displacement was 0.51 sec and for one window was 0.13 sec.

Fig. 5 .

 5 Fig. 5. Time optimal motion generated for sinusoidal path using moving window and discrete dynamic programming: (a) non-skidding constraint for reference point only; (b) non-skidding constraint for reference point and wheels.

  To present the problem in a discrete way, let us sample the allowable domain of the velocity and displacement

		max vv  [0,	]	and	max ss  [0,	]	with the steps
	v  and s 
		() () k i v s	; k v i s      	;	k i	 	0,1,... 0,1,... n m	(11)
	where	max v v 	m	and	max s s 	n	. Then, for each path
	point we can generate a number of the mobile robot states
	with all possible velocities, i.e.	( , )	( ) ( , ); ( )

   corresponding to the final layer and applying the backtracking, one can get the desired optimal path in graph. It is described by the recorded indices 1 2

	( , ) {} ki  C	. Then, taking into account the additivity of the
	objective (13), the shortest path for the nodes belong to the
	next layer	( , 1) ki  C {	, } k 	can be found by combining the
	optimal solutions for the previous layer	( , ) ki  C {	,	k 	} 	and the
	distances between the nodes with the indices i and	1 i  . The
	latter corresponds to the formula
		d	 d   , 1 , ( , ) ( , 1) min ( , k i k i k i k i k dist   CC	 )	(16)
	that is applied sequentially starting from the second layer, i.e.
	1, 2,... 1 in  . Finally, after selection of the minimum length
	,1 ki
					{ , ,... k k	1 nn , } k k 	.
	In more details, an outline of the developed algorithm is
	presented as follows in the form of pseudo-codes. The input
	includes the state matrix { ( , ) C k i k	1, 2,... ; 1, 2,... } m i n 
	containing the information of displacement and velocity, and
	also the array of path radius { ( ) R i i		1, 2,... } n

d


Joackim Bernier
email: joackim.bernier@ens-rennes.fr

Benoit Erwan Faou
email: erwan.faou@inria.fr

Beno Ît

G R Ébert

LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE D-DIMENSIONAL TORUS

Keywords: 2000 Mathematics Subject Classification. 35L05, 37K55, 35B40, 35Q55 Wave equation, Birkhoff normal form, Resonances, Hamiltonian PDEs

HAL is

Introduction

Let us consider the nonlinear wave equation set on the d-dimensonal torus

T d = (R/2πZ) d with d ≥ 1, u tt -∆u + mu + f (u) = 0, x ∈ T d , (NLW)
satisfied by a real valued function u(t, x) with given initial data u(0) ≡ u(0, •) and u(0) ≡ u(0, •) = ∂ t u(0, •). The function f ∈ C ∞ (R, R) is at least of order 2 at the origin, i.e. f (0) = f (0) = 0. For small and smooth initial data u(0) ∈ H s (T d ) and u(0) ∈ H s-1 (T d ) with large s, we are interested in a description of the long time behavior of u(t) ≡ u(t, •) solution of (NLW).

In dimension d = 1, it is known that if ε measures the size of the initial data, the solution is controlled for arbitrary polynomial times with respect to ε, and for almost all m away from zero. More precisely, it has been proved (see for instance [START_REF]Birkhoff normal form for PDE's with tame modulus[END_REF] and also [START_REF] Bambusi | Birkhoff normal form for some nonlinear PDEs[END_REF]) Theorem 1.1. ( [START_REF]Birkhoff normal form for PDE's with tame modulus[END_REF]) Let d = 1 and r ≥ 2. For almost any m > 0 there exists s * (r) such that for s > s * (r) there exists ε 0 (r, s) > 0 such that for all ε < ε 0 (u(0), u(0))

H s ×H s-1 ≤ ε =⇒ (u(t), u(t)) H s ×H s-1 ≤ 2ε, t ≤ ε -r .
(

The crucial tool to obtain this result is to show that for a large set of parameters m, the frequencies ω j = |j| 2 + m of the linear wave operator satisfy a non-resonance condition of the following form: Fix r ≥ 3, there exists γ > 0 such that for k = (k 1 , • • • , k p ) ∈ (Z d ) p , = ( 1 , • • • , q ) ∈ (Z d ) q with p + q ≤ r we have

|ω k 1 + • • • + ω kp -ω 1 -• • • -ω q | ≥ γ µ 3 (k, ) α , (µ 3 ) unless (|k 1 |, • • • , |k p |) and (| 1 |, • • • , | q |)
are equal up to a permutation, and where µ 3 (k, ) denotes the third largest number amongst the collection (|k i |, | j |) i,j , and α depends on r. This condition, introduced in [START_REF]Birkhoff normal form for PDE's with tame modulus[END_REF], allows to eliminate (or normalize) all the terms in the Hamiltonian of the perturbation (depending on F , a primitive of f in (NLW)) involving at most two high Fourier modes via a Birkhoff normal form procedure. On the other hand, it is known since [START_REF] Bambusi | Birkhoff normal form for some nonlinear PDEs[END_REF] that we can neglect all the monomials involving more than three high modes (see for instance [START_REF] Grébert | Birkhoff normal form and Hamiltonian PDEs[END_REF] or [START_REF] Bambusi | A Birkhoff normal form theorem for some semilinear PDEs[END_REF] for a simple presentation of these two facts). So once we have (µ 3 ) we can expect a control of the Sobolev norms similar to (1). Notice that (µ 3 ) is close from the so-called second-order Melnikov nonresonance condition 1 that says in a formulation allowing comparison with 1 This terminology refers to the original papers [START_REF] Mel'nikov | On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function[END_REF][START_REF] Mel'nikov | A family of conditionally periodic solutions of a Hamiltonian system[END_REF] where similar conditions where introduced for proving the stability of low-dimensional invariant tori in Hamiltonian dynamics, and popularized in the KAM literature [Mos67, Eli88, Bou97, XY01] and later while extending these results to Hamiltonian PDEs [START_REF] Kuksin | Nearly Integrable Infinite Dimensional Hamiltonian Systems[END_REF][START_REF] Pöschel | A KAM-theorem for some nonlinear partial differential equations[END_REF].

(µ 3 ): Fix n ≥ 2, there exists γ > 0 such that for k

= (k 1 , • • • , k p ) ∈ (Z d ) p , = ( 1 , • • • , q ) ∈ (Z d ) q with |k i |, | i | ≤ n and for (j 1 , j 2 ) ∈ (Z d ) 2 with |j 1 |, |j 2 | > n |ω k 1 + • • • + ω kp -ω 1 -• • • -ω q + ω j 1 -ω j 2 | ≥ γ r α , unless (|k 1 |, • • • , |k p |) and (| 1 |, • • • , | q |)
are equal up to a permutation, where r = p + q. We note that in the Melnikov case, the "length" of the resonance (r + 2) is free but the number of "interior" modes (here (2n + 1) d ) is fixed while it is exactly the converse in (µ 3 ). So the two conditions are not equivalent but similar.

The condition (µ 3 ) applies to many situations, including the one dimensional wave equations, the one dimensional nonlinear Schrödinger equation with an external potential [Bam03, Bam07, BG06], the d-dimensional nonlinear Schrödinger equation with a convolution potential [START_REF] Bambusi | Forme normale pour NLS en dimension quelconque[END_REF][START_REF]Birkhoff normal form for PDE's with tame modulus[END_REF], plane waves stability for non-linear Schrödinger equation [START_REF] Faou | Lubich Sobolev stability of plane wave solutions to the cubic nonlinear Schr dinger equation on a torus[END_REF], wave equations on Zoll manifolds [START_REF] Bambusi | Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds[END_REF] or quantum harmonic oscillator on

R d [GIP09].
The main difficulty of the higher dimensional case for (NLW) is that the frequencies do not satisfy the second-order Melnikov condition for a large set of parameters m, as already noted for instance in [START_REF] Bourgain | Construction of periodic solutions of nonlinear wave equations in higher dimension[END_REF][START_REF] Delort | On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus[END_REF]. In fact, in dimension d ≥ 2 (µ 3 ) is not satisfied either: let for instance

|k 1 | ≥ | 1 | ≥ µ 3 (k,
) then in the left hand side of (µ 3 ) the quantity ω k 1 -ω 1 depends on indices that are not controlled by µ 3 (k, ). For d = 1, if k 1 and 1 are large, the difference ω k 1 -ω 1 is close to be an integer and the parameter m can be chosen in a large set so that (µ 3 ) holds. However, in dimension d ≥ 2, ω k 1 -ω 1 describes a dense set at the scale |k 1 | which prevents (µ 3 ) to hold. Actually we can only prove that for a large set of parameters m, the following condition (which is related to the so-called first-order Melnikov condition) holds

|ω k 1 + • • • + ω kp -ω 1 -• • • -ω q | ≥ γ µ 2 (k, ) α (µ 2 ) unless (|k 1 |, • • • , |k p |) are (| 1 |, • • • , | q |)
equal up to a permutation where µ 2 (k, ) is the second largest index in the multi-index (k, ). Despite this known problem, some results have been proved concerning the existence of quasi-periodic solutions, where the loss of derivative is controlled by the use of KAM-Newton schemes, see [START_REF] Bourgain | Construction of periodic solutions of nonlinear wave equations in higher dimension[END_REF]. Concerning the control of large open sets of solutions, some results can be found in [START_REF] Delort | Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds[END_REF] and [START_REF] Delort | On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus[END_REF] but the time control depends on the shape on the nonlinearity inducing restriction on the index r (essentially driven by the annulation degree of the nonlinearity in 0).

Another situation where (µ 2 ) appears in a natural way is given by numerical discretization of Hamiltonian PDEs. For example standard splitting methods applied to wave equations in dimension d = 1 induces numerical resonances destroying the property (µ 3 ) and degenerating to (µ 2 ) even for generic time discretization parameters 2 . In this case, it is however possible to control the solution by playing with the time integrator or with the space discretization, see [START_REF] Cohen | Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations[END_REF][START_REF] Faou | Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization[END_REF][START_REF] Faou | Paturel Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part II: Abstract splitting[END_REF][START_REF] Faou | Geometric numerical integration and Schrödinger equations[END_REF][START_REF] Faou | Lubich Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation Forum of Math[END_REF].

In this paper, we propose a new method to overcome this difficulty by a careful examination of the normal form induced by (µ 2 ) and a control in mixed Sobolev norm inspired by some tools used in numerical analysis. In particular it can be seen as a nonlinear extension of [START_REF] Dujardin | Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential[END_REF], here in a continuous in time setting.

As byproduct of this method, we prove the following.

Theorem 1.2. Let d ≥ 2, r ≥ 2 and s > d/2. For almost any m > 0 there exists s * = s * (r, s) and ε 0 (r, s) > 0 such that for all ε < ε 0 the solution to (NLW) satisfies

(u(0), u(0)) H s * ×H s * -1 ≤ ε =⇒ (u(t), u(t)) H s ×H s-1 ≤ 2ε, t ≤ ε -r . (2)
In other words, (1) holds up to a finite loss of derivative in the initial condition 3 . Concretely we can prove that s * (r, s) is of order less than O(r 5 s 5 ) but this is certainly not optimal.

The previous property (2) is in fact a corollary of a stronger abstract result proved in Theorem 2.9. The main idea is to decompose the dynamics into low and high frequencies according to some large threshold depending on ε, and then to try to conjugate the system to a normal form whose dynamics can be described and controlled.

When the (µ 3 ) condition holds, this normal form approach allows to conjugate all the (NLW) flow to a flow preserving the H s norm up to terms that are arbitrarily small. When only (µ 2 ) is satisfied, this cannot be done, and linear terms remains in the dynamics of high modes, coupled with the low modes. Fortunately, these terms can be put under a symmetric Hamiltonian quadratic form. Hence, despite the linear nature of their dynamics, the L 2 norm of the high modes can be proved to be preserved over long times in the normal form analysis. This crucial information allows to initiate an effective decomposition between low and high modes. To prove the almost preservation of higher Sobolev norms of the high modes we use a sort of pseudo-differential argument (or commutator Lemma) that allows to gain one derivative (see (49) and (50)). Then for given indices s and s 0 with s s 0 > d/2, we can control the low modes in a Sobolev norm H s and 2 The fundamental reason is that time discrete numerical schemes require the control of small divisors of the form e ihΩ(k, ) -1 instead of Ω(k, ) as defined in (µ3), where h is the time discretization parameter. Hence numerical resonances can occurs when for instance h(ω k 1 -ω 1 ) is close to an arbitrary large multiple of 2π (see for instance [START_REF] Shang | Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems[END_REF][START_REF] Hairer | Geometric Numerical Integration[END_REF] for a finite dimensional analysis of symplectic integrators).

3 Historically, a similar result with loss of derivative was first given under the condition (µ3) in [START_REF] Bourgain | Construction of approximative and almost-periodic solutions of perturbed linear Schrödinger and wave equations[END_REF].

we show a polynomial growth in time of a Sobolev norm H s 0 of the high modes of order O(t s 0 ). By choosing a smoother initial condition such that (u(0), u(0)) H 2s ×H 2s-1 ≤ ε we then obtain (see Theorem 3.2)

u(t) ≤Nε H s ≤ 2ε and u(t) >Nε H s 0 ≤ ε r , t ≤ ε -r s 0 +1 , (3)
in a regime where s -s 0 is large with respect to r. Here u(t) ≤Nε and u(t) >Nε denote the low and high modes parts, according to the threshold

N ε = ε -r
s-s 0 . When s is large, we obtain (2) after a change of indices, but Theorem 3.2 (see also Theorem 2.9) gives more precise informations. It shows that the dynamics of the low modes preserves the super-actions4 , i.e. the quantities J n (t) = |k|=n |u k (t)| 2 , n ≤ N ε over very long times, where u k (t), k ∈ Z d denote the Fourier coefficients of u(t). Such a result does not hold for high modes since the interaction between two close large modes cannot be eliminated but only controlled. Theorem 2.9 expresses the fact that the condition (µ 2 ) -much more general than (µ 3 )-is enough to ensure a decoupling of the dynamics of low and high modes for very long times.

In the previous estimate, s 0 has only to satisfy the condition5 s 0 > d/2. It also typically corresponds to what is numerically observed for initial data taken as trigonometric polynomials6 for which the dynamics does not exhibit energy exchanges over long times. Such two-stage norms with different Sobolev scaling were previously used in the context of numerical analysis of splitting methods for Schrödinger equations in the linear case, see [START_REF] Dujardin | Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential[END_REF] and [START_REF] Faou | Debussche Modified energy for split-step methods applied to the linear Schrödinger equation[END_REF] where again the preservation of the L 2 norm of the high-modes was crucial to obtain a global control of the dynamics.

We also believe that results of the form (2) or (3) involving mixed Sobolev norms provides a natural setting for numerical discretization, for which (µ 2 ) is the generic control of non-resonance condition. It might for instance allow to weaken the usual CFL conditions required, or to derive low-order integrators following the analysis of [START_REF] Hofmanová | An exponential-type integrator for the KdV equation[END_REF][START_REF] Ostermann | Low regularity exponential-type integrators for semilinear Schrödinger equations[END_REF].

Abstract statement

2.1. Hamiltonian formalism. We recall in this subsection the formalism used in [START_REF] Grébert | Birkhoff normal form and Hamiltonian PDEs[END_REF][START_REF] Faou | A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus[END_REF][START_REF] Bernier | Rational normal forms and stability of small solutions to nonlinear Schrödinger equations[END_REF] to deal with infinite dimensional Hamiltonians and flows depending on a infinite set of symplectic variables (q, p) = (q a , p a )

a∈Z d ∈ R Z d × R Z d equipped with the usual 2 s (Z d , R 2 ) norm defined 7 as (q, p) 2 s = a∈Z d a 2s (p 2 a + q 2 a ), a 2 = 1 + |a| 2 .
As explained in [START_REF] Grébert | Birkhoff normal form and Hamiltonian PDEs[END_REF], for U an open set of 2 s , for a function

H(q, p) such that H ∈ C ∞ (U, R), with 2 gradient ∇ (q,p) H ∈ C ∞ (U, 2
s ), we can define the flow of a Hamiltonian system

∀ a ∈ Z d , qa = ∂H ∂p a (q, p), ṗa = - ∂H ∂q a (q, p). (4)
To easily deal with normal form transformations, it is convenient to use the complex representation (ξ a )

a∈Z d = ( 1 √ 2 (q a + ip a )) a∈Z d in C Z d equipped with the 2 s (Z d , C) norm. Then with the notations ∂ ∂ξ a = 1 √ 2 ∂ ∂q a -i ∂ ∂p a and ∂ ∂ ξa = 1 √ 2 ∂ ∂q a + i ∂ ∂p a ,
the real Hamiltonian system is equivalent to the complex system,

∀ a ∈ Z d , ξa = -i ∂H ∂ ξa (ξ, ξ) =: (X H (ξ, ξ)) a (5)
where H(q, p) = H(ξ, ξ) ∈ R is a called a real Hamiltonian. The notation X H (ξ, ξ) = (X H (ξ, ξ)) a∈Z d thus denote the Hamiltonian vector field associated with the Hamiltonian H. If we associate with (ξ, ξ) a complex function ψ on T d , through the formula

ψ(x) = a∈Z d ξ a e iax , (6)
then the Sobolev norm ψ H s is equivalent to the norm

ξ 2 s = a∈Z d a 2s |ξ a | 2 .
The symplectic structure is given by

a∈Z d dq a ∧ dp a = -i a∈Z d dξ a ∧ d ξa . ( 7
)
and the Poisson bracket in complex notation reads

{F, G} = -i a∈Z d ∂F ∂ξ a ∂G ∂ ξa - ∂F ∂ ξa ∂G ∂ξ a . ( 8
)
7 with the usual notation

|a| 2 = a 2 1 + • • • + a 2 d for a = (a1, . . . , a d ) ∈ Z d .
Definition 2.1. For a given s > d/2 and a domain U containing 0 in 2 s := 2 s (Z d , C), we denote by H s (U) the space of real Hamiltonians P (ξ, ξ) ∈ R satisfying P ∈ C ∞ (U, R), and X P ∈ C ∞ (U, 2 s ), where X P is defined in (5). We will use the shortcut F ∈ H s to indicate that there exists a domain U containing 0 in 2 s such that F ∈ H s (U). Notice that for F and G in H s the formula (8) is well defined in a neighborhood of 0. With a given Hamiltonian function H ∈ H s , we associate the Hamiltonian system (5), and we can naturally defines its flow.

C Z d × C Z d C U 2 ×Z d where U 2 = {±1} and use the convenient notation (ξ, ξ) = z = (z j ) j∈U 2 ×Z d ∈ C U 2 ×Z d where j = (δ, a) ∈ U 2 × Z d =⇒ z j = ξ a if δ = 1, z j = ξa if δ = -1. (9)
We define the 2 s norm of an element z = (ξ, ξ) to be

z 2 s := j∈U 2 ×Z d j 2s |z j | 2 = 2 a∈Z d a 2s |ξ a | 2 = 2 ξ 2 s .
where for j = (δ, a) ∈ U 2 × Z d we set j 2 = a 2 . With this notation, the Hamiltonian system (5) can be written

ż = X H (z), where (X H (z)) (δ,a) := -iδ(X H (ξ, ξ)) a , z = (ξ, ξ) .
Here X H (ξ, ξ) denote the vector field in (5). Another way to formulate this notation is to say that with the identification (9), the vector field X H (ξ, ξ) is naturally extended as (X H (z)

) j = -iδ ∂H ∂z j (z), for j = (δ, a) ∈ U 2 × Z d ,
where j = (-δ, a).

For k = (k 1 , . . . , k m ) = (δ i , a i ) m i=1 ∈ (U 2 × Z d ) m we denote the momentum M(k) = m i=1 δ i a i ,
and we define the conjugate k

= (-δ i , a i ) m i=1 ∈ (U 2 × Z d ) m . We set M m = {k ∈ (U 2 × Z d ) m | M(k) = 0} (10)
the set of zero-momentum multi-indices. For a given k ∈ M m we write

z k = z k 1 • • • z km .
We also define

R m = {(δ j , a j ) m j=1 ∈ M m | ∃σ ∈ S m , ∀j = 1, . . . , m, δ j = -δ σ j and a j = a σ j } (11)
the set of resonant multi-indices. Note that by construction if m is odd then R m is empty and that if k = (δ j , a j ) m j=1 ∈ R m is associated with a permutation σ then we have

z k = 1≤j≤m δ j =1 ξ a j ξaσ j . ( 12
)
Definition 2.3. We say that P (z) is a homogeneous polynomial of order m if it can be written

P (z) = P [c](z) = j∈Mm c j z j , with c = (c j ) j∈Mm ∈ ∞ (M m ), ( 13
)
and such that the coefficients c j satisfy c j = c j .

Note that the last condition ensures that P is real valued, as the set of indices are invariant by the application j → j. Following [START_REF] Faou | A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus[END_REF][START_REF] Bernier | Rational normal forms and stability of small solutions to nonlinear Schrödinger equations[END_REF] but in a 2 s framework, we get the following Proposition. It turns out to be a consequence of the more general Lemma 5.1 proved below. (i) A homogeneous polynomial, P [c], of degree m ≥ 2 belongs to H s ( 2 s ) and we have

X P [c] (z) s ≤ (C s ) m c ∞ z m-1 s , z = (ξ, ξ), ( 14
)
for some constant C s depending only on s.

c ∞ ≤ 2mn c ∞ c ∞ . (15)
We end this section with a result concerning Lie transformations. We recall that to a Hamiltonian function F we associate, if it exists, the Lie transformation Φ 1 F which is the time one flow generated by F . This transformation is automatically symplectic.

Lemma 2.5. Assume that s > d/2 and let

P = P [c 3 ] + • • • + P [c r ], c m ∈ M m ,
be a polynomial of order at least 3 at the origin, then for all ν ≤ κ r,s max( c 3 ∞ , . . . , c r ∞ )

-1 , the Lie transformation τ = Φ 1 P is well defined on a ball B s (0, ν) of 2 s with values in B s (0, 2ν) and we have

τ (z) -z s ≤ C r,s max m c m ∞ ν 2 , ∀ z ∈ B s (0, ν), (16)
where C r,s and κ r,s are some positive constants depending only on r and s.

Proof. In view of Proposition 2.4 assertion (i) we deduce from the Cauchy-Lipschitz Theorem that the flow Φ t P is locally well defined on 2 s . Furthermore we have for z ∈ B s (0, ν) and as long as z(t) = Φ t P (z) ∈ B(0, 2ν)

Φ t P (z) -z s ≤ t 0 X P (z(w)) s dw ≤ r m=3 C m,s c m ∞ |t|(2ν) m-1 ≤ C r,s |t| max m c m ∞ ν 2 .
Thus we conclude by a bootstrap argument that, taking ν small enough, the flow is defined for all z ∈ B(0, ν) up to time t = 1 and that z(t) remains in B(0, 2ν) for t ≤ 1 and satisfies (16).

2.3. Non resonance condition.

Definition 2.6. A family of frequencies

ω = {ω a , a ∈ Z d } is non resonant, if there exist (α(r)) r≥1 ∈ (R * + ) N and (γ r ) r≥1 ∈ (R * + ) N , such that for all r ≥ 1, all N ≥ 1 and all k = (δ i , a i ) r i=1 ∈ (U 2 × Z d ) r satisfying a i ≤ N, for i = 1, • • • , r, we have |δ 1 ω a 1 + • • • + δ r ω ar | ≥ γ r N -α(r) , when k / ∈ R r , (H1
)

|δ 1 ω a 1 + • • • + δ r ω ar + ω b | ≥ γ r+1 N -α(r+1) , ∀ b > N with M(k) + b = 0, ( H2
)
|δ 1 ω a 1 + • • • + δ r ω ar + ω b 1 + ω b 2 | ≥ γ r+2 N -α(r+2) , ∀ b 1 , b 2 > N (H3) with M(k) + b 1 + b 2 = 0.
We notice that conditions (H1)-(H2) are equivalent to condition (µ 2 ) introduced in the introduction while conditions (H1)-(H2)-(H3) are not equivalent to (µ 3 ) since in (H3) we are not considering the case where the two high frequencies have opposite sign.

Remark 2.7. Note that in (H2) using the zero momentum condition (see (10)), b is in fact bounded by rN . Hence (H2) is a trivial consequence of (H1). Similarly, as in many applications ω a ∼ |a| ν when a → ∞, for some ν > 0, (H3) is also a consequence of (H1) as we can restrict (H3) to a set of (b 1 , b 2 ) that are bounded by C(r)N . Similarly, we note z ≥N and z >N the decomposition induced by the notation (9). We now make our hypothesis on the Hamiltonian H that we will consider.

Hypothesis 2.8. The Hamiltonian H can be written

H = H 2 + P = a∈Z d ω a |ξ a | 2 + P ( 17
)
where ω = (ω a ) a∈Z d is a non resonant family of real numbers in the sense of Definition 2.6, P belongs to H s for some s > d/2 and P is of order at least 3 at the origin (which means that P and its differentials up to the order 2 vanish at 0).

Note that in general the frequencies ω a will not be uniformly bounded with respect to a, and hence the quadratic part a∈Z ω a |ξ a | 2 , does not belong to H s . Nevertheless it generates a continuous flow which maps 2 s into 2 s explicitly given for all time t and for all indices a by ξ a (t) = e -iωat ξ a (0). Furthermore this flow has the group property. By standard arguments (see for instance [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and [BFG] in a similar framework), this is enough to define the local symplectic flow, ż = X H (z), in 2 s . Theorem 2.9. Let H be a Hamiltonian satisfying Hypothesis 2.8. Then for all r ≥ 2 and all s > s 0 > d/2 satisfying s -s 0 ≥ s * (r) := 6r 2 α(3r) + 2dr (18)

there exists ε 0 (r, s, s 0 , ω) > 0 such that for all ε < ε 0 (r, s, s 0 , ω) the solution ξ(t), generated by the flow of H issued from an initial datum ξ(0) ∈ 2 2s satisfying ξ(0) 2s ≤ ε, exists for all time t ≤ ε -r s 0 +1 and satisfies

∀ a ≤ N ε := ε -r s-s 0 , ∀ t ≤ ε -r s 0 +1 , b∈Z d b = a |ξ b (t)| 2 - b∈Z d b = a |ξ b (0)| 2 ≤ ε 3 a -2s (19) and        ξ(t) ≤Nε 2 s = a ≤Nε a 2s |ξ a (t)| 2 ≤ 4ε 2 ξ(t) >Nε 2 s 0 = a >Nε a 2s 0 |ξ a (t)| 2 ≤ ε 2r for t ≤ ε -r s 0 +1 .
Note that by playing with the indices, we can obtain the following corollary (in the same vein as Theorem 1.2) which essentially proves that arbitrary high regularity small solutions to (5) are controlled over arbitrary long times.

Corollary 2.10. Let H be a Hamiltonian satisfying Hypothesis 2.8. For all r ≥ 2 and s > d/2, there exists s * (r, s) and ε 0 (r, s, ω) > 0 such that for all ε < ε 0 (r, s, ω),

ξ(0) s * ≤ ε =⇒ ξ(t) s ≤ 2ε for t ≤ ε -r .
Proof. It is a consequence of the previous Theorem by replacing r by r(s 0 + 1), s 0 by s and assuming that s * is large enough with respect to r and s.

3. Application to (NLW) on T d .

Introducing v = u t ≡ u we rewrite (NLW) as u = v, v = -Λ 2 u -f (u) , (20)
where Λ = (-∆ + m) 1/2 and f ∈ C ∞ having a zero of order at least two at the origin. When m > 0, we can define

ψ = 1 √ 2 (Λ 1/2 u + iΛ -1/2 v), (21)
and we get that (u, v)

∈ H s (T d , R) × H s-1 (T d , R) is solution of (20) if and only if ψ ∈ H s+1/2 (T d , C) is solution of i ψ = Λψ + 1 √ 2 Λ -1/2 f Λ -1/2 ψ + ψ √ 2 . (22)
Then, endowing the space L 2 (T d , C) with the standard (real) symplectic structure -idψ ∧ d ψ equation ( 22) reads as a Hamiltonian equation

i ψ = ∂H ∂ ψ
where H is the hamiltonian function

H(ψ, ψ) = 1 (2π) d T d (Λψ) ψ dx + 1 (2π) d T d F Λ -1/2 ψ + ψ √ 2 dx,
and F is a primitive of f with respect to the variable u, i.e. f = ∂ u F . The linear operator Λ is diagonal in the complex Fourier basis8 {e ia•x } a∈Z d , with eigenvalues

ω a = |a| 2 + m, a = (a 1 , . . . , a d ) ∈ Z d , |a| 2 = a 2 1 + • • • a 2 d . (23)
Decomposing ψ in Fourier variables with Fourier coefficients (ξ a ) a∈Z d as in (6), ( 22) takes the form (5) where the Hamiltonian function H is given by H = H 2 + P with,

H 2 (ξ, ξ) = a∈Z d ω a |ξ a | 2 , P (ξ, ξ) = 1 (2π) d T d F   a∈Z d ξ a e ia•x + ξa e -ia•x √ 2ω a   dx.
As F is smooth, the function P is in H s and we can define its flow. Finally

(u, v) ∈ H s (T d , R)×H s-1 (T d , R) is a solution of (20) if and only if ξ ∈ 2 s-1/2
is a solution of the Hamiltonian system associated with the Hamiltonian H.

In order to apply Theorem 2.9, we need the following result A direct proof of this proposition can also be done by using the arguments given in [START_REF] Bambusi | Birkhoff normal form for some nonlinear PDEs[END_REF][START_REF] Eliasson | KAM for non-linear beam equation[END_REF] or [START_REF] Faou | Lubich Sobolev stability of plane wave solutions to the cubic nonlinear Schr dinger equation on a torus[END_REF]. By following these proofs, one can verify that α(r) is of order O(r 3 ).

As a consequence, Theorem 2.9 applies. By scaling back to the variable (u, v), we obtain Theorem 3.2. Let f be a C ∞ function with zero of order at least 2 at the origin. Then for almost all m ∈ (0, +∞) and for all r ≥ 2, and all s 0 > (d + 1)/2, there exists s 1 (r, s 0 ) such that for all s ≥ s 1 (r, s 0 ), there exists ε 0 (r, s 0 , s, m) and for all ε < ε 0 (r, s 0 , s, m), if (u(0), u(0)) ∈ H s × H s-1 (T) satisfies (u(0), u(0)) H s ×H s-1 ≤ ε, then the system (NLW) admits a solution over a time T ≥ ε -r s 0 +1 , and we have

   u(t) ≤Nε s ≤ 2ε u(t) >Nε s 0 ≤ ε r for t ≤ ε -r s 0 +1
where

N ε := ε -r s-s 0 .
Using that α(r) = O(r 3 ) we can verify that s 1 (r, s) (rs) 5 . Then Theorem 1.2 is just a reformulation of Corollary 2.10.

Normal form

The strategy used to prove Theorem 2.9 is to put the original Hamiltonian (17) into normal form eliminating most of interactions between the low and high frequencies. By using Taylor expansion at the origin, the Hamiltonian H can be written

H = H 2 + r m=3 P m + R r+1 , with H 2 = a∈Z d ω a |ξ a | 2 , ( 24
)
where P m is a homogeneous polynomial of degree m, and where R r+1 ∈ H s is of order r + 1 which means that its differentials vanish up to the order r.

In particular, it is small in the sense that we have

X R r+1 (z) s ≤ (C s ) r z r s (25)
for some constant depending only on s and for z small enough in 2 s .

For j ∈ M m let us denote by µ n (j) the n-th largest number amongst the collection j i m i=1 :

µ 1 (j) ≥ µ 2 (j) ≥ • • • ≥ µ m (j).
By convention, we will also set µ 0 (j) = +∞.

Let N be a fixed number and let H be a Hamiltonian satisfying Hypothesis 2.8. For a given r we decompose the Hamiltonian H in (24) as follows:

H = H 2 + r m=3 (P (•) m + P (i) m + P (ii) m + P (iii) m ) + R r+1 ,
where for all m,

• the polynomial

P (•) m = P [c (•)
m ], depends only of low modes:

∀j ∈ M m , (c (•) m ) j = 0 =⇒ µ 1 (j) ≤ N. ( 26
)
• P (i)

m = P [c (i)
m ] contains only one high mode:

∀j ∈ M m , (c (i) m ) j = 0 =⇒ µ 1 (j) > N ≥ µ 2 (j), (27)
• P (ii)

m = P [c (ii)
m ] contains only two high modes:

∀j ∈ M m , (c (ii) m ) j = 0 =⇒ µ 2 (j) > N ≥ µ 3 (j), (28)
• and P

(iii) m = P [c (iii)
m ] contains at least three high modes:

∀j ∈ M m , (c (iii) m ) j = 0 =⇒ µ 3 (j) > N. ( 29
)
Following [START_REF]Birkhoff normal form for PDE's with tame modulus[END_REF], we know that polynomials of the form P

(iii) m = P [c (iii)
m ] are already small in the sense that

X P (iii) m s ≤ CN -s z m-1 s .
On the other hand, thanks to our non resonance condition, polynomials of the form P (i)

m = P [c (i)
m ] can be killed by a symplectic change of variables since it cannot be resonant (only one high mode) and polynomials of the form P (•)

m = P [c (•)
m ] can be normalized by a standard Birkhoff normal form procedure. In addition to these two known facts, the following Theorem says that we can also symmetrize polynomials of the form P (ii)

m = P [c (ii) m ]:
Theorem 4.1. Assume that the frequencies ω = (ω a ) a∈Z d are non resonant in the sense of Definition 2.6, and let r ≥ 2 be given. There exists a constant C depending on r, such that for all N ≥ 1, there exists a polynomial Hamiltonian

χ = r m=3 (χ (•) m + χ (i) m + χ (ii) m ) (30) such that χ (•) m = P [a (•) m ], χ (i)
m = P [a (i) m ], χ (ii)
m = P [a (ii)
m ], contain zero, one and two high modes respectively (i.e. satisfy (26), ( 27) and (28) respectively), with coefficients satisfying

a (•) m ∞ + a (i) m ∞ + a (ii) m ∞ ≤ CN rα(r) , (31)
and such that the Lie transformation Φ 1 χ , whose existence is guaranteed in a neighborhood of the origin of 2 s for all s > d/2 by Lemma 2.5, puts H in normal form:

H • Φ 1 χ = H 2 + r m=3 (Z (•) m + S (ii) m + P (iii) m ) + Rr+1 , (32)
where

Z (•) m = P [b (•) m ], S (ii)
m = P [b (ii) m ], P (iii)
m = P [b (iii)
m ], contain zero, two and at least three high modes respectively (i.e. satisfy (26), ( 28), (29)), with coefficients satisfying

b (•) m ∞ + b (ii) m ∞ + b (iii) m ∞ ≤ CN rα(r) . Moreover • Z (•)
m contains only resonant monomials, which means that

∀ j ∈ M m , j / ∈ R m =⇒ (b (•) m ) j = 0. • S (ii)
m contains terms that are symmetric in the high modes which means that if b (ii) j = 0 for j ∈ M m , the two highest modes are of opposite signs: they are of the form (δ, a) and (-δ, b) for some a and b ∈ Z d .

• The remainder term Rr+1 is of the form

Rr+1 = R r+1 • Φ 1 χ + 1 0 (1 -s) r+1 P [b r+1 ] • Φ s χ ds (33)
where P [b r+1 ] defines a homogeneous polynomial of order r + 1 with coefficients bounded by b r+1 ∞ ≤ CN rα(r) .

Proof. The proof is standard and use the non resonant Birkhoff normal form procedure (see [START_REF]Birkhoff normal form for PDE's with tame modulus[END_REF][START_REF] Bambusi | Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds[END_REF][START_REF] Grébert | Birkhoff normal form and Hamiltonian PDEs[END_REF]). We follow here the construction made in [START_REF] Faou | A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus[END_REF]. By using the formal series expansions H = H 2 + m≥3 P m and χ = m≥3 χ m in homogeneous polynomials, the formal normal form problem is to find χ and X (under normal form) such that

H • Φ 1 χ = k≥0 ad k χ (H 2 + P ) = H 2 + X.
In the formal series algebra, this problem is equivalent to a sequence of homological equation of the form

∀ m ≥ 3, {H 2 , χ m } = Q m -X m ,
where Q m depends on the function P k , χ k and X k , k < m previously constructed. It is obtained by iterated Poisson brackets preserving the homogeneity of polynomial and boundedness of coefficients (see ( 15)). Formula for Q m can be found in [START_REF] Faou | A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus[END_REF], Eq. (3.4). Now assume that Q m = j∈Mm q j z j is given. For a given N we can decompose it into terms containing zero, one, two and at least three high modes:

Q m = Q (•) m + Q (i) m + Q (ii) m + Q (iii)
m and a similar decomposition for the coefficients q j . The normal form term X m is then the sum of the resonant terms in Q

{H 2 , z k } = iΩ(k)z k with Ω(k) = - m i=1 δ i ω a i , k = ((δ i , a i )) m i=1 ∈ M m ,
we then solve the other terms by setting

χ m = j∈Mm\Rm q (•) j iΩ(j) z j + j∈Mm q (i) j iΩ(j) z j + j∈Mm\Sm q (ii) j iΩ(j) z j ,
where S m denote the set of indices with two symmetric high modes. Note that when j ∈ M m \S m , the two highest modes (larger that N ) have the same sign, and the denominator Ω(j) is controlled by (H3). Similarly, the first term can be controlled using (H1) and the second using (H2). We then observe that we loose a factor N α(m) after each solution of the homological equation, yielding a bound of order N rα(r) after r iterations. Note also that all the operations (solution of the Homological equation and Poisson brackets) preserve homogeneity and the reality of the global Hamiltonians.

It is easy to see that for all s > s 0 for z small enough (such that C r,s N rα(r) z s ≤ 1 for some constant C r,s by using ( 14)) the flow Φ 1 χ (z) is well defined and locally invertible in 2 s (its inverse being Φ -1 χ ), see Lemma 2.5.

Finally to obtain (32) we use a Taylor expansion of the term H • Φ t χ for t ∈ (0, 1).

For ξ ∈ 2 s , we define the pseudo-actions:

J a (ξ, ξ) = b∈Z d a = b |ξ b | 2 (and J j = 1 2 ∈U 2 ×Z d = j |z | 2 , j ∈ U 2 × Z d ). ( 34
)
By definition of the resonant set R m (see (11)) and the corresponding resonant monomials (see ( 12)) we see that for all m, the normal form terms Z (•) m can be written 9

Z (•) m (ξ, ξ) = k∈Rm ∀i, k i ≤N c k z k = a,b∈(Z d ) m 2 , ∀i, a i = b i ≤N c ab m 2 i=1 ξ a i ξb i , z = (ξ, ξ).
We notice that a polynomials in normal form commutes with all the pseudoactions:

{Z (•) m , J a } = 0, ∀a ∈ Z, (35)
in such a way the flow generated by Z (•) m will not modify the 2 s norms since z 2 s = (1 + a 2 ) 2s J a .

Proof of the main Theorem

To prove the main Theorem, we will need the following Lemma which controls polynomial vector fields in mixed Sobolev norms, as high and low modes are not controled at the same Sobolev scale.

Lemma 5.1. For all s ≥ s 0 > d/2, there exists a constant C such that for all m ≥ 3, for all N ≥ 1 and for all c ∈ ∞ (M m ), if the polynomials P [c] contains at least n-th high modes, n = 0, 1, 2, 3 (i.e. c j = 0 =⇒ µ n (j) > N ) then we have for z = (ξ, ξ) with z ≤N ∈ 2 s and z >N ∈ 2 s 0

X P [c] (z) ≤N s ≤    C m c ∞ z ≤N m-1 s if n = 0 C m c ∞ N n(s-s 0 ) z ≤N m-n-1 s z >N n s 0 if n ≥ 1, ( 36
) and

X P [c] (z) >N s 0 ≤            0 if n = 0, C m c ∞ N s 0 -s z ≤N m-n-1 s if n = 1, C m c ∞ z ≤N m-n s z >N n-1 s 0 if n ≥ 2. ( 37
)
Proof. For z = (ξ, ξ) ∈ 2 s , we have

X P [c] (z) ≤N 2 s ≤ ≤N 2s ∂P [c] ∂z (z) 2 ≤ m 2 c 2 ∞ ≤N 2s 0 P (z) 2 ,
where

P (z) = s-s 0 M(j)=0 µn(j, )>N |z j 1 | • • • |z j m-1 |. ( 38
)
Let j = (j 1 , . . . , j m ) be given such that M(j, ) = 0. We have

s-s 0 ≤ ( j 1 + • • • + j m-1 ) s-s 0 ≤ m s-s 0 j 1 s-s 0 • • • j m-1 s-s 0 .
9 Recall that Z (•) m = 0 when m is odd.

If P [c] contains no high modes, i.e. if n = 0 then we can define zj = j s-s 0 z j and we obtain

P (z) ≤ M(j, )=0 µ 1 (j)≤N |z j 1 | • • • |z j m-1 |.
For a function g on the torus, we will denote ĝb = (1/2π) d

T d g(x)e -ib•x dx, b ∈ Z d , the Fourier coefficients of g. Then denoting f (x) = ≤N |z |e iδ a•x we have M(j, )=0 µ 1 (j)≤N |z j 1 | • • • |z j m-1 | = ( f m-1 ) -δa , = (δ, a) ∈ U 2 × Z d
for some constant C depending on m. So we get for some generic constant A depending on s and s 0 but independent on m

X P [c] (z) ≤N 2 s ≤ Am s-s 0 +2 c 2 ∞ a∈Z d a 2s 0 |(f m-1 ) a | 2 = Am s-s 0 +2 c 2 ∞ f m-1 2 H s 0 ≤ AC m c 2 ∞ f 2m-2 H s 0 ,
where g H s 0 , s 0 > d/2 is the usual Sobolev norm on T d equivalent to the 2 s 0 norm of the Fourier coefficients of g. Here C depends on s and s 0 but not on m. We then note that f H s 0 = z ≤N s which shows the first equation in (36).

To prove the second in the case n ≥ 1, we simply bound s-s 0 by N s-s 0 and we obtain

P (y) ≤ N s-s 0 M(j, )=0 µn(j)>N |z j 1 | • • • |z j m-1 | = ( f m-n g n ) -δa
for = -δa with the same notation as before, where

f (x) = ≤N |z |e iδ a•x and g(x) = >N |z |e iδ a•x ,
and we conclude as in the previous case.

To show (37) we use

X P [c] (z) >N 2 s 0 ≤ m 2 c 2 ∞ >N 2s 0 P (z) 2 , where P (z) = M(j)=0 µn(j, )>N |z j 1 | • • • |z j m-1 |, (39)
In the case n = 1, we have no high mode in the sum. However due to the zero momentum condition, there exists one mode greater than N/(2m),

we have for ε small enough and for all w ∈ (0, 1)

Φ w χ (y) ≤N s ≤ 2ε and Φ w χ (y) >N s 0 ≤ 2ε r+1 . (43)
Now let us write the Hamiltonian (32) as H • Φ 1 χ = H 2 + Z (•) + S (ii) + P (iii) + R3r+1 by gathering together the terms with different homogeneity.

Control of the low modes y(t) ≤N . For j ∈ U 2 × Z d , let J j (t) = J j (y(t)) with the definition (34). As Z (•) and H 2 commute with J a for all a ∈ Z d , we have for t ≤ min(T, ε -r s 0 +1 ) and j ≤ N ,

j 2s | Jj | = j 2s |{J j , H}| ≤ = j s ∂(S (ii) + P (iii) + R3r+1 ) ∂y¯ (z) s |y |.
By summing in j ≤ N , and using Cauchy-Schwarz inequality, we obtain

j ≤N j 2s | Jj | ≤ X S (ii) + P (iii) + R2r+1 (y) ≤N s y ≤N s
Now under bootstrap hypothesis (42), we conclude using Lemma 5.1

X S (ii) + P (iii) (y) ≤N s ≤ CN 3rα(3r) N s-s 0 ε 2r+2 ≤ Cε r+ 3 2 .
On the other hand since R 3r+1 is a Hamiltonian in H s of order at least 3r + 1, we have using (25), (33) and Lemma 2.5 X R3r+1 (y) ≤N s ≤ CN 3rα(3r) ε 3r .

Hence we obtain

j ≤N j 2s | Jj | ≤ C(ε 3r+1 + ε r+ 5 2 ).
Therefore for t ≤ min(T, ε -r+1 ) we conclude

j ≤N j 2s |J j (t) -J j (0)| ≤ ε 7 2
for ε small enough which in turn implies that

y(t) ≤N s ≤ 11 8 ε. (44)
Furthermore by using estimates on the vector field X χ , we deduce that z(t) = Φ 1 χ (y(t)) satisfies (19).

Control of the high modes. The Hamiltonian S (ii) can be written as ( P (iii) + R3r+1 ). Using (37) we have

S (ii) m = a,
Q(y) >N s 0 ≤ CN 3rα(3r) (ε 2r+2 ε + ε 3r ). ( 47
)
So using the fact that the operator A is hermitian, we get for t ≤ T (recall that we assume that y(0) >N 0 ≤ ε 2r+1 ) y(t) >N 0 = 2 ξ(t) >N 0 ≤ 2 ξ(0) >N 0 + tCε 2r+ Then we note that using the zero momentum condition (45) we have B ab (t) = 0 when |a -b| > 3rN . Hence, since s 0 > 1/2, we have

) s 0 + Cε 3r+ 5 2 . ( 52
)
We are going to apply the following elementary lemma Lemma 5.2. Let α ∈ (0, 1), f : R → R + a continuous function, and x : R → R + a differentiable function satisfying the inequality

∀ t ∈ R, d dt x(t) ≤ 1 1 -α f (t)(x(t)) α .
Then we have the estimate ∀ t ∈ R, x(t) 1-α ≤ x(0) 1-α + t 0 f (s) ds.

Considering (52), we are going to apply Lemma 5.2 with α = 1 -1/s 0 and x(t) = y(t) >N 2 s 0 + ε ι(s 0 ,r) where ι(s 0 , r) = (3r + 5 2 -2r+1 s 0 )(1 -1 s 0 ) -1 . Indeed, x naturally satisfies the estimate d dt x(t) ≤ C(1 + t)

1 s 0 ε 2r+1 s 0 x(t) 1-1 s 0 .
Now applying Lemma (5.2), we get x(t) ≤ Cx(0) + Cε 2r+1 (1 + t) s 0 +1 .

Since we have assumed that y(0) >N s 0 ≤ ε 2r+1 and since a straightforward estimate proves that ι(s 0 , r) ≥ 2r + 1, we deduce that y(t) >N s 0 ≤ Cε 2r+1 (1 + t) s 0 +1

and hence for t ≤ min(T, ε -r (s 0 +1) ) and ε small enough

y(t) >N s 0 ≤ 1 2 ε r+1 . ( 53
)
Hence combining (44) and (53), we conclude by continuity argument that T ≥ ε -r (s 0 +1) which finishes the proof.

  Proposition 2.4. Let s > d/2.

(

  ii) For two homogeneous polynomials, P [c] and P [c ], of degree respectively m and n, the Poisson bracket is a homogeneous polynomial of degree m + n -2, {P [c], P [c ]} = P [c ] and we have the estimate

2. 4 .

 4 Statement. Let us start with the following notation: For ξ ∈ 2 s and a given number N , we decompose ξ = ξ ≤N + ξ >N where for all j ∈ U 2 × Z d , (ξ ≤N ) j = ξ j for j ≤ N, 0 for j > N, and ξ >N = ξ -ξ ≤N . Given a function ψ ∈ H s (T d ) with Fourier coefficients ξ a , a ∈ Z d , and a number N ≥ 1 the previous decomposition induces naturally the decomposition ψ = ψ ≤N + ψ >N with ψ ≤N (x) = a ≤N ξ a e ia•x and ψ >N (x) = a >N ξ a e ia•x .

  Proposition 3.1 ([Del09], Theorem 2.1.1). For almost all m ∈ (0, +∞) the family of frequencies ω a (m) = |a| 2 + m, a ∈ Z d associated with (NLW) is non resonant in the sense of Definition 2.6.

  b∈Z d a >N, b >N B ab (y ≤N )ξ a ξb , y = (ξ, ξ). As the Hamiltonian is real we have B ab (y ≤N ) = B ba (y ≤N ), i.e. the operator A = (B ab (y ≤N ) a , b >N acting on 2 s (Z d >N ) is Hermitian. Moreover, we have B ab (y ≤N ) = 2r-2 m=1 j∈Mm µ 1 (j)≤N M(j)=a-b b ab,j y j (45) where the coefficients b ab,j are uniformly bounded by CN 3rα(3r) . Hence for s > d/2 and as soon as y satisfies the bootstrap assumption (42), we obtain that ∀ a , b > N |B ab (y ≤N )| ≤ CN 3rα(3r) y ≤N s . (46) When writing the dynamics of y = (ξ, ξ), we get ξa = -iω a ξ a -i b∈Z d B ab (y ≤N )ξ b -iQ a (y), a > N where Q a = ∂ ∂ ξa

  3 2 ≤ C(1 + t)ε 2r+1 . (48) Let us define D s 0 the diagonal operator from 2 s (Z d >N ) into 2 s-2s 0 (Z d >N ) given by D s 0 = diag( a 2s 0 , a > N ). With the notation (η, ξ) >N = a >N ηa ξ a , we have d dt (ξ, D s 0 ξ) >N = -i(ξ, [D s 0 , A(t)]ξ) >N + Im(Q(y), D s 0 ξ) >N where [A, B] denotes the commutator of A and B: [A, B] = AB -BA. Hence by bootstrap hypothesis (and using (47)) d dt (ξ, D s 0 ξ) >N ≤ (ξ, [D s 0 , A(t)]ξ) >N + Cε 3r+ 5 2 .

≤

  (ξ, [D s 0 , A(t)]ξ) >N = a , b ≥N |a-b|≤3rN B ab (y)( a 2s 0 -b 2s 0 ) ξa ξ b ≤ a , b ≥N |a-b|≤3rN 2r|B ab (t)|| a -b |( a 2s 0 -1 + b 2s 0 -1 )| ξa ξ b |.Using the bound (46) and the fact that| a -b | ≤ a -b ≤ CN , we get (ξ, [D s 0 , A]ξ) ≤ CN 3rα(3r) ε a , b ≥N |a-b|≤3rN a s 0 -1 b s 0 | ξa ξ b |(49)where we used that for |a -b| ≤ 3rN we have a ≤ b + 3rN . Now we apply the Cauchy-Schwarz inequality to geta , b ≥N |a-b|≤3rN a s 0 -1 b s 0 | ξa ξ b | ≤     a , b ≥N |a-b|≤3rN a 2s 0 -2 | ξa | 2 CN d y >N s 0 -1 y >N s 0where we used again that for |a -b| ≤ 3rN we have a ≤ b + 3rN with a ∈ Z d . Thus, using (40), we obtain d dt

  Proposition 2.2. Let s > d/2. Any Hamiltonian in H s defines a local flow in 2 s which is a symplectic transformation. 2.2. Polynomial Hamiltonians. To algebraically deal with polynomials depending on (ξ, ξ), we identify

Already used in[START_REF]Birkhoff normal form for PDE's with tame modulus[END_REF] for (NLW) in dimension d = 1 in a periodic setting.

Condition that could probably be refined in the critical case using Galigardo-Nirenberg inequality.

Note that in the present result, all the constants depends on the Sobolev indices s and s0. An optimization of these constant could be done in an analytic context, following the technics used in[START_REF] Faou | A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus[END_REF].

Here for a = (a1, . . . , a d ) ∈ Z d and x = (x1, . . . x d ) ∈ T d we set a•x = a1x1 +• • •+a d x d

Acknowledgement. During the preparation of this work the three authors benefited from the support of the Centre Henri Lebesgue ANR-11-LABX-0020-01 and B.G. was supported by ANR -15-CE40-0001-02 "BEKAM" and ANR-16-CE40-0013 "ISDEEC" of the Agence Nationale de la Recherche.

hence with the same notation as before, we have

and we conclude as before. The proof of the last estimate in (37) is similar as the proof of (36) for n ≥ 1 .

Proof of Theorem 2.9. Let ε > 0, s ≥ s 0 > d/2 and r be given, and z(0) ∈ 2 2s such that z(0) 2s ≤ ε. We apply the normal form Theorem (4.1) at the order 3r and set

s-s 0 . Under the hypothesis (18), we have the bound

In particular, all the coefficients of the polynomials Hamiltonian χ, Z • m , S (ii) m and P (iii) m in (32) are bounded by CN 3rα(3r) ≤ Cε -1 2 . Hence using Lemma 2.5 we have for ε small enough (in a way depending on r and s) that y(0) = Φ -1 χ (z(0)) is in 2 2s and satisfies y(0) 2s ≤ 5 4 ε. This implies in particular that y(0) ≤N s ≤ 5 4 ε and

Now we need to control the dynamics of y(t) the solution of the Hamiltonian system associated with the Hamiltonian H • Φ 1 χ given by (32). We define

We notice that y(0) 2s ≤ 5 4 ε < 3 2 ε and y(0) >N s 0 ≤ 2ε 2r+1 < ε r+1 . Therefore by classical results for the definition of mild solutions of semilinear problems in Sobolev spaces with index greater than d/2, T is strictly positive. Let us prove that if t ≤ min(T, ε -r s 0 +1 ) then y(t) ≤N s ≤ 11 8 ε and y(t) >N s ≤ 1 2 ε r+1 . We will then conclude by a continuity argument that T ≥ ε

Control of the transformation. In view of Lemma 5.1, under bootstrap hypothesis, the vector field X χ (y(t)) satisfies the estimates

where we used N s-s 0 = ε -r , and

Hence this shows that as soon as y satisfies the bootstrap hypothesis, i.e. y(t) ≤N s ≤ 3 2 ε and y(t) >N s 0 ≤ ε r+1 , (42)