Louisiane Verger

Varun Natu

Michael Ghidiu

Michel W Barsoum
email: barsoumw@drexel.edu

Effect of Cationic Exchange on the Hydration and Swelling Behavior of Ti 3 C 2 T z MXenes

MXenes, two dimensional (2D) transition metal carbides, have showed great promise as materials for capacitive energy storage, electromagnetic interference shielding, transparent electrodes, among many other applications. Colloidal suspensions are usually processed to design materials for these applications. However, the mechanisms underlying the MXene exfoliation into single or few nanosheets after etching with hydrochloric acid and lithium fluoride are not well understood. Herein the structural changes induced by cationic exchange in the interlayer space of Ti3C2Tz during washing are investigated. We also show, mostly by X-ray diffraction of Ti3C2Tz multilayers, that cationic exchange between H3O + ions, on the one hand, and Li + , K + or Mg 2+ , on the other is occurring reversibly. The presence of Li + is a necessary condition for the crystalline swelling observed to ≈ 16 Å, and the subsequent exfoliation of Ti3C2Tz into single layers.

Introduction

Layered materials are comprised of stacked atomically thin sheets held together by forces weaker than the intraplanar forces. This feature allows for the modification of the interlayer space by the intercalation and deintercalation of molecules and/or ions. Understanding the exchange mechanism in the interlayer space is mandatory to fully understand the surface chemistry, 1 structure 2 and electronic properties 3 of these materials. Some examples of layered materials with tuneable interlayer spaces are clay minerals, graphite and multi-layered MXenes. [START_REF] Nicolosi | Liquid Exfoliation of Layered Materials[END_REF] Discovered in 2011, MXenes are two-dimensional (2D) transition metal carbides and/or nitrides with a general formula Mn+1XnTz (or M1.33XTz in special cases), where M is an early transition metal, X is carbon and/or nitrogen, n = 1, 2 or 3 and Tz represents variable surface terminations (such as O, OH, F). [START_REF] Naguib | Two-Dimensional Nanocrystals Produced by Exfoliation of Ti3AlC2[END_REF][START_REF] Naguib | Two-Dimensional Transition Metal Carbides[END_REF] To date, by far the most studied MXene is Ti3C2Tz. A good understanding of what occurs during its etching and subsequent washing is required to scale up the synthesis of this promising material that has shown great potential in a host of applications including energy storage, [START_REF] Lukatskaya | Cation Intercalation and High Volumetric Capacitance of Two-Dimensional Titanium Carbide[END_REF] structural composites, [START_REF] Ling | Flexible and Conductive MXene Films and Nanocomposites with High Capacitance[END_REF] electromagnetic interference shielding, [START_REF] Han | Ti3C2 MXenes with Modified Surface for High-Performance Electromagnetic Absorption and Shielding in the X-Band[END_REF][START_REF] Shahzad | Electromagnetic Interference Shielding with 2D Transition Metal Carbides (MXenes)[END_REF] and many others. [START_REF] Anasori | 2D Metal Carbides and Nitrides (MXenes) for Energy Storage[END_REF][START_REF] Pang | Applications of 2D MXenes in Energy Conversion and Storage Systems[END_REF] One of the most common ways to synthetize Ti3C2Tz is to first selectively etch the Al layers from Ti3AlC2. [START_REF] Naguib | Two-Dimensional Nanocrystals Produced by Exfoliation of Ti3AlC2[END_REF] The multi-layered (ML) material obtained is composed of 2D titanium carbide layers that are weakly bonded to each other. If hydrofluoric acid (HF) is used to etch the Al layers, [START_REF] Naguib | Two-Dimensional Nanocrystals Produced by Exfoliation of Ti3AlC2[END_REF] the use of an intercalant such as dimethyl sulfoxide (DMSO) is required to exfoliate the MLs into few or single layers. [START_REF] Mashtalir | Intercalation and Delamination of Layered Carbides and Carbonitrides[END_REF] If the etchant is a mixture of hydrochloric acid (HCl) and lithium fluoride (LiF), there is no need for an exfoliant because after etching and during the washing steps, a spontaneous exfoliation is observed. [START_REF] Ghidiu | Conductive Two-Dimensional Titanium Carbide "clay" with High Volumetric Capacitance[END_REF] In this case, colloidal suspensions with concentrations as high as 25 g.L -1 of 2D flakes can be obtained. [START_REF] Alhabeb | Guidelines for Synthesis and Processing of Two-Dimensional Titanium Carbide (Ti3C2Tx MXene)[END_REF] Cations easily exchange in the interlayer space of MXenes. [START_REF] Lukatskaya | Cation Intercalation and High Volumetric Capacitance of Two-Dimensional Titanium Carbide[END_REF][START_REF] Mashtalir | Intercalation and Delamination of Layered Carbides and Carbonitrides[END_REF][START_REF] Ghidiu | Ion-Exchange and Cation Solvation Reactions in Ti3C2 MXene[END_REF][START_REF] Ghidiu | Alkylammonium Cation Intercalation into Ti3C2 (MXene): Effects on Properties and Ion-Exchange Capacity Estimation[END_REF] When cations intercalate the structure, they also carry interlayer solvating water (H2O) molecules with them. When the interlayer space (d-spacing) increases uniformly with the intercalation of H2O and cations, the phenomenon, at least in the clay literature is called crystalline swelling. [START_REF] Norrish | Crystalline Swelling of Montmorillonite: Manner of Swelling of Montmorillonite[END_REF] Ghidiu el al., showed that the hydration behaviour of Ti3C2Tz depends on the nature of the cations between the layers. [START_REF] Ghidiu | Ion-Exchange and Cation Solvation Reactions in Ti3C2 MXene[END_REF] For example, while the d-spacing of K + intercalated MLs is mostly independent of the relative humidity and remains at 12.5 Å, the d-spacing of Li + or Mg 2+ intercalated MXenes increases to 16.8 Å and 15.5 Å with increasing humidity, respectively.

Anions also impact the hydration behaviour; Voigt et al. showed that the presence of anions larger than F -is required to observe the swelling to ≈ 16 Å. [START_REF] Voigt | Anion Adsorption, Ti3C2Tz MXene Multilayers, and Their Effect on Claylike Swelling[END_REF] Typically, a d-spacing of 12.5 Å was attributed to a planar cationic hydration shell, and the d-spacing of 16.8 Å or 15.5 Å corresponds to a hydration shell where the H2O molecule are stacked. [START_REF] Ghidiu | Ion-Exchange and Cation Solvation Reactions in Ti3C2 MXene[END_REF][START_REF] Muckley | Multimodality of Structural, Electrical, and Gravimetric Responses of Intercalated MXenes to Water[END_REF][START_REF] Célérier | Hydration of Ti3C2Tx MXene: An Interstratification Process with Major Implications on Physical Properties[END_REF] The focus of this work is to understand the mechanisms at the origin of the spontaneous exfoliation observed during washing after etching Ti3AlC2 powders with HCl/LiF solutions. In this work, we probed the nature of the interlayer cations as a function of washing cycles and chemistry of the washing solutions, by measuring the d-spacing using X-ray diffraction (XRD).

We show that cationic exchange can explain the structural changes observed.

Experimental section

Sample preparation

Synthesis of Ti3AlC2. Ti3AlC2 powder was synthesized by heating a ball-mixed mixture of TiC, Al and Ti (Alfa Aesar, 99.5% purity) powders in a molar ratio of 2:1.05:1, respectively, in argon, Ar, atmosphere at a rate of 5 °C / min to 1350 °C and holding time of 2 h. The resulting solid is milled and sieved (-400 mesh) to obtain a powder with particle size under 38 μm.

Washing procedure. In the rest of the experimental section, a washing step consists of the following procedure: 40 mL of the washing solution is added to the MLs, the mixture is shaken for 1 min and centrifuged at 3500 rpm for 2 min. The pH of the supernatant is measured, and then discarded.

Synthesis of Ti3C2Tz intercalated with ions. Ti3C2Tz intercalated with Li + , K + and Mg 2+ were synthesized following Ghidiu et al. procedure. [START_REF] Ghidiu | Ion-Exchange and Cation Solvation Reactions in Ti3C2 MXene[END_REF] Here, 3.27 g of lithium chloride (LiCl) was added to 30 mL of a 10 wt.% HF solution. Then, 3 g of the Ti3AlC2 powders (sieved to < 38 μm) was slowly added, and the solution was left to stir at room temperature (between 20 and 25 °C) for 24 h. The resulting sediment was divided into 6 different batches and washed three times with 6 M HCl to remove unreacted LiCl, and at least three times with distilled water until the supernatant reaches a pH ~ 5. Lastly, 40 mL of aqueous salt solutions (1 M LiCl, 1M KCl, or 0.5 M MgCl2) were added to the sediment. The mixtures were shaken for 2 mins and allowed to sit.

After 1 h, the salt solutions were replaced with fresh ones, and the mixtures were allowed to sit for a further 24 h. After centrifugation, the sediment was washed twice with distilled water.

Using this procedure, MXenes intercalated with Li + , K + or Mg 2+ were obtained. Different washing procedure were then implemented on these starting materials, crucially, before they were allowed to dry. Schematics of the washing procedure can be found in Figures 1 and2.

Additional washing procedures can be found in Supplementary information.

•

For all the samples, the first wash (wash 1) was with a 0.5 M HCl solution. The second wash (wash 2) was with either H2O (samples M-a, M = Li, K or Mg) or 1 M of KCl, LiCl, or 0.5 M MgCl2 (samples M-b). In the two latter cases, a third wash was carried out (wash 3). Figure 1A shows a schematic of these steps.

•

For the Energy dispersive spectroscopy (EDS) and X-ray photoelectron spectroscopy (XPS) analyses, the same washing procedure as that shown in Figure 1A was used, except the samples washed with KCl or MgCl2 were washed twice with water to remove unreacted salts (see Figure 2A).

Synthesis of Ti3C2Tz in HCl/LiF. 1.5 g of lithium fluoride (LiF, Alfa Aesar, 98% purity) was dissolved in 15 mL of 9 M hydrochloric acid (HCl, Fisher Scientific). 1.5 g of Ti3AlC2 (sieved to < 38 μm) was slowly added. The solution was stirred at 35 °C for 24 h. The sediment was washed 10 or 8 times with either distilled water or 0.5 M HCl, respectively. Between every washing step, X-ray diffraction (XRD) patterns were acquired on the wet sediment (Figure 3).

The samples collected for XRD was then placed back in the centrifuge tube before continuing the washing procedure.

In a second experiment (Figure 5), the sediment, after etching, was washed 10 times with 0.5 M HCl. The sediment was then split in two: one part was washed four times with water, and the other part was washed four times with a 1 M aqueous LiCl solution.

Characterization techniques

XRD patterns on wet samples were acquired using a Rigaku SmartLab powder diffractometer setup in the Bragg-Brentano geometry with Cu Kα radiations in the 4 -9° 2θ range using a 0.02° step size and a dwell time of 1 s per step. It is particularly important to ensure that the samples remain wet during the XRD acquisition, and therefore the scans durations were kept to a minimum (5 min). No additional water was added to the sample, in order to not modify the pH of the environment. XRD patterns on dry samples were acquired using a Miniflex Rigaku diffractometer in the 4 -80 ° 2θ range using a 0.02° step size and a dwell time of 2 s per step.

X-ray Photoelectron Spectroscopy (XPS) was performed using a VersaProbe 5000 instrument (Physical Electronics, Chanhassen, Minnesota). All samples were sputtered using an Ar-ion gun (1 kV) for 2 min. A pass energy of 23.5 eV was used for all scans, with an energy step of 0.025 eV and step time of 0.5 s. The number of repeats per scan was set to 25.

Energy dispersive spectroscopy (EDS) measurements in a scanning electron microscope (SEM)

were performed on a Zeiss Supra 50VP (Carl Zeiss AG, Germany) with an acceleration voltage of 15 kV. ML powders were fixed on a carbon tape for these measurements.

Results and discussion

Structural changes induced by cation exchange in Ti3C2Tz

As noted above, the washing procedure after etching in HCl/LiF is typically carried out with water to remove the etchants -HCl and LiF -and any reaction by-products, such as AlF3. To decouple the effects of HCl and intercalated cations, we first studied the influence of HCl on the d-spacings, as a function of the latter. For this, Ti3C2Tz MLs intercalated with Li + , K + or Mg 2+

were synthesized using the procedure of Ghidiu et al. [START_REF] Ghidiu | Ion-Exchange and Cation Solvation Reactions in Ti3C2 MXene[END_REF] A series of washing steps outlined in The main d-spacings of the initial materials synthesized are the same as previously reported in the literature: 16.2 Å, 12.8 Å and 15.4 Å for the Li + , K + and Mg 2+ intercalated Ti3C2Tz MLs, respectively. [START_REF] Ghidiu | Ion-Exchange and Cation Solvation Reactions in Ti3C2 MXene[END_REF] For the Li-intercalated sample, a peak is also detected at 12.8 Å, implying the sample was not fully hydrated. [START_REF] Muckley | Multimodality of Structural, Electrical, and Gravimetric Responses of Intercalated MXenes to Water[END_REF] However, when the starting materials were first washed with HCl and then with salt solutions containing the initial intercalated cations, the d-spacings evolved quite differently.

With Li, a distribution of d-spacings (pattern labelled Li b -wash 2 LiCl in Figure 1B) was observed between 16.2 Å and 12.8 Å, consistent with Li + in the interlayer space with two and one H2O layers, respectively. [START_REF] Célérier | Hydration of Ti3C2Tx MXene: An Interstratification Process with Major Implications on Physical Properties[END_REF] With an additional wash with 1 M LiCl, the intensity of the 16.2 Å peak increased, at the expense of the one at 12.8 Å. This additional wash clearly facilitated the hydration of the interlayer space. For K + , the d-spacing returns to 12.8 Å upon washing with KCl (pattern labelled K b -wash 2 KCl in Figure 1C). In the Mg 2+ case two peaks at higher d- Energy dispersive spectroscopy (EDS) in a scanning electron microscope (SEM) were performed on the K-and Mg-samples (Figure 2 and Table S1). For the measurements on samples washed with KCl or MgCl2 solution (K b -wash 3 and Mg b -wash 3, respectively), two additional washes with H2O were carried out to remove any un-reacted salts without modifying the d-spacing (Figure 2 A and B). For the HCl washed samples, neither Mg 2+ nor K + was detected by EDS or after subsequent washing with H2O (see spectra M a -wash 1 HCl and wash 2 H2O, M is Mg or K). For the salt washed MLs, even after the two additional H2O washes, K and Mg were detected in the EDS spectra which confirms the re-intercalation of these elements when exposed to KCl or MgCl2 solutions, respectively. Additional measurements on the samples with K by X-ray photoelectron spectroscopy (XPS) confirmed these results (Figure S1). The same series of experiments were repeated on K-and Mg-intercalated MLs using hydrobromic acid (HBr). The XRD and XPS results showed the same trends (Figures S2 andS3) seen when HCl was used (Figure 1). It follows that the role of anions on the d-spacings is much weaker than the cations during washing.

To summarize the results at this juncture: cation exchange between Li + , K + or Mg 2+ , on the one hand, and H3O + on the other, occurs when MLs Ti3C2Tz are washed with HCl. This cation exchange result in a modification of the d-spacing. Similar exchanges were reported in clay minerals (such as montmorillonite), [START_REF] Norrish | The Swelling of Montmorillonite[END_REF][START_REF] Wang | Synthesis of Novolac/Layered Silicate Nanocomposites by Reaction Exfoliation Using Acid-Modified Montmorillonite[END_REF][START_REF] Hendricks | Hydration Mechanism of the Clay Mineral Montmorillonite Saturated with Various Cations1[END_REF] and layered titanates. [START_REF] Izawa | Ion Exchange and Dehydration of Layered [Sodium and Potassium] Titanates, Na2Ti3O7 and K2Ti4O9[END_REF] From the XRD and EDS data, we can conclude that a d-spacing of 13.4 Å corresponds to the presence of only H3O + in the interlayer space. Further evidence that interlayer H3O + is being replaced by Li + , K + or Mg 2+ is the systematic lowering of the supernatant's pH after washing with the salt solutions, as compared to samples washed with H2O (compare the pH value of Li, K or Mg a -wash 2 H2O with Li, K, or Mg b -wash 2 -LiCl, KCl or MgCl2). Here H3O + leached into the bulk solution causes the observed drop in pH. This set of experiments also shows that cation exchange is a reversible process: the d-spacing of samples washed with HCl and then washed with their corresponding salt solutions return to their original value (Figure 1 B-D). It is also possible to directly exchange K + with Mg 2+ , or to exchange K + with H3O + , and then Mg 2+ (see Figure S4 in Supporting Information). Lastly, contrary to previous reports in which one day of soaking was used to ensure complete exchange, [START_REF] Ghidiu | Ion-Exchange and Cation Solvation Reactions in Ti3C2 MXene[END_REF] herein the physical effects of cation exchange (d-spacing change) are shown to be quite rapid, < 10 min.

Structural changes during washing Ti3C2Tz etched in HCl/LiF

We now turn our attention to the effect of washing after etching in HCl/LiF solutions. For this, XRD patterns on wet samples were acquired between every wash (Figure 3 A). Just after etching (wash 1), the d-spacing was at ≈ 13.8 Å. The peak was broad, with a full width at half maximum (FWHM) of Δ(2θ) = 0.9 °, which is evidence for structural disorder along the c direction. [START_REF] Célérier | Hydration of Ti3C2Tx MXene: An Interstratification Process with Major Implications on Physical Properties[END_REF] As discussed above, this position corresponds to the presence of H3O + between the layers. Upon further washing with H2O, the d-spacing increases to 16.2 Å and the FWHM decreases to Δ(2θ) = 0.4 °, reflecting a strong ordering of the interlayer space. The removal of some etching by-products between the layers could help to increase the structural order. The pH increases until wash 6, and then stabilizes around pH = 4 (inset in Figure 3 A). Spontaneous delamination was observed during wash 10. The bimodal distribution observed in d-spacings corresponds to a discontinuous structural expansion along c. If no cationic exchange were occurring, then washing with H2O after etching with HCl/LiF should not alter the d-spacing. However, d increases from ≈ 13 Å to 16.5 Å (Figure 3 A). The following hypothesis can then be made: both H3O + and Li + are present in the interlayer space after etching, and H3O + keeps the d-spacing at ≈ 13 Å (Figure 4 A). As the material is washed, cation exchange from excess salt occurs, and Li + substitutes H3O + , enabling the interlayer space to swell to 16.5 Å (Figure 4 B). The source of this Li is most probably, the large excess of LiF used during the synthesis (LiF:Ti3AlC2 = 7.5:1 molar ratio). Not all the Li + intercalates the MLs during etching, and several washing steps are needed to remove unreacted LiF. Indeed, LiF is detected by XRD on dry samples just after etching and after the first wash (Figure S5). To further confirm this important conclusion, the MLs after etching in HCl/LiF, were washed with 0.5 M HCl instead of H2O. The pH was maintained below 2 (Figure 3 B) and the dspacing remained at ≈ 13.8 Å throughout the washing procedure. The FWHM decreased from Δ(2θ) = 0.7° to Δ(2θ) = 0.3°, again indicating an increase in structural order. Since the d-spacing

was not modified, it can be concluded that the H3O + remains in the interlayer space.

Furthermore, the Li + cations present after etching are probably exchanged by H3O + due to the high concentration in H3O + in the washing solution (0.5 M HCl).

After washing 10 times with 0.5 M HCl, the batch was divided into two parts as outlined in Figure 5 A. One part was washed with H2O until a pH close to 4 was reached, and the other part was washed with a 0.5 M LiCl solution. After washing with H2O, the d-spacing remained fixed at ≈ 13.8 Å (pattern labelled a -wash 14 H2O) because no cationic exchange occurred.

However, when washing with LiCl, crystalline swelling was observed to ≈ 16.8 Å at a pH as low as 2 (pattern labelled b -wash 12 LiCl). The shoulder at higher 2θ indicates a distribution of dspacings. This again evidences that cationic exchange is necessary to modify the d-spacing. No swelling is induced when H3O + intercalated MLs are washed with H2O, while crystalline swelling to 16.8 Å is observed when H3O + intercalated MLs are washed with LiCl. Our results provide a better understanding of several experimental procedures already reported in the literature. For example, Liu et al. [START_REF] Liu | Preparation of Ti3C2 and Ti2C MXenes by fluoride salts etching and methane adsorptive properties[END_REF] studied the impact of the nature of cations during etching Ti3AlC2 in different HCl/MF mixtures, where M is Li, Na, K or NH4. After their washing procedure, residual Li was detected on the MLs etched with HCl/LiF. However, almost no Na or K was detected after washing MLs etched in HCl/NaF or HCl/KF, respectively. Given that the solubilities of NaF and KF in water are higher than LiF, may result in the easier removal of unreacted salts during the first few washes, that in turn impeded or reduced cationic exchange during washing. Moreover, the samples were washed several times with 2 M HCl, which probably resulted in some exchange of Na + and K + by H3O + .

Voigt et al. reported a d-spacing of ≈ 13.5 Å after etching in HF and washing with H2O in Ti3C2Tz. [START_REF] Voigt | Anion Adsorption, Ti3C2Tz MXene Multilayers, and Their Effect on Claylike Swelling[END_REF] In light of our results, we can conclude that H3O + ions intercalate the MLs when the etching solution is HF alone, but no cationic exchange is possible during washing and no swelling is observed. However, by washing in the presence of a cation (Li + in Ref. [START_REF] Voigt | Anion Adsorption, Ti3C2Tz MXene Multilayers, and Their Effect on Claylike Swelling[END_REF] ), an exchange occurs, resulting in a d-spacing of ≈ 12.4 Å or ≈ 16 Å. The latter d-spacing value is reached only if larger anions than F -are present during etching, which enables a full hydration of the interlayer space.

Summary and Conclusions

Multilayered Ti3C2Tz intercalated with different cations (Li + , K + and Mg 2+ ) were prepared and structural modifications due to cation exchange reactions were investigated. When Ti3C2Tz MLs are hydrated with only H3O + between the layers, a d-spacing of 13.4 Å is obtained. Furthermore, cations such as Li + , K + or Mg 2+ can reversibly substitute for H3O + and for each other, with this exchange occurring in a single washing step and quite rapidly (< 10 mins). It has long been established that etching Ti3AlC2 powders in HF, resulted in MLs that did not spontaneously delaminate, while those etched in HCl/LiF solutions, after several washes did. The reasons for this difference in behavior have remained unclear. Using mostly XRD, and various washing procedures, we conclude that for Ti3C2Tz MLs: i) Just after etching Ti3AlC2 with HCl and LiF, both H3O + and Li + are present in the interlayer space. When Li + cations are present during washing with water (excess in etching solution), they replace their H3O + counterparts and the interlayer space swells to ≈ 16 Å. It is this swelling, in turn, that results in spontaneous delamination.

ii) When washing with HCl, the interlayer space remains at ≈ 13.5 Å and dominated by H3O + cations.

This work thus provides new insight on the crystalline swelling observed after washing MXenes etched in HCl/LiF mixtures. With this new understanding, even better control of both the interlayer chemistry and spacing is possible. Similar studies need to be carried out for other MXene compositions as well.

ASSOCIATED CONTENT Supporting Information. EDS, XPS and additional XRD data referred in the article (PDF).

Table of Contents Graphic

Figure 1 A

 1 Figure 1 A and explained in the experimental section were then executed. XRD patterns were

Figure 1 B

 1 Figure 1 B-D), even after two additional washes (XRD patterns not shown).

  Figure 1D).

Figure 1 :

 1 Figure 1: Influence of washing procedures on d-spacing of Ti3C2Tz MLs intercalated with Li + , K + or Mg 2+ . A) Schematic of washing procedures. B-D) XRD patterns on wet samples, for bottom to top for each panel: initial

Figure 2 :

 2 Figure 2: (A) Schematic of washing procedure, which was identical to that shown in Figure 1, except two additional washes with water were performed on the samples K-b and Mg-b after washing with KCl or MgCl2 to remove unreacted salts. (B) XRD pattern on samples K b -wash 5 and Mg b -wash 5, the d-spacing remains at 12.5 and 15.5 Å, respectively as in K b -wash 3 and Mg b -wash 3. (C-D) EDS data for (from bottom to top): initial materials in blue or orange for MXenes intercalated with K + or Mg 2+ , respectively; K a and Mg a samples first wash with 1 M HCl in blue and orange dotted lines, respectively; K a and Mg a samples after second wash with water shown by grey patterns; K b and Mg b samples after fifth wash with KCl or MgCl2 salt solutions and then water, in blue and red, respectively. Peak indexed with a star corresponds to an unknown impurity.

Figure 3 :

 3 Figure 3: Effect of pH on d-spacing during washing Ti3C2Tz MLs after etching with HCl/LiF mixture. XRD patterns acquired on wet MLs between every wash with, (A) distilled H2O, and (B) 0.5 M HCl. Insets show pH evolution of supernatant after every wash.

Figure 4 :

 4 Figure 4: Schematic of effect on washing of the d-spacing of Ti3C2Tz MLs. (A) After etching, both H3O + and Li + are present between the layers. (B) As the MLs are washed in the presence of excess Li + , originating from the excess LiF used, the H3O + are replaced by Li + , with the hydration of Li + , eventually leading to swelling and exfoliation to 16.5 Å.

Figure 5 :

 5 Figure 5: Influence of washing procedures on d-spacing of Ti3C2Tz MLs etched with HCl/LiF. A) Summary of washing procedures. B) XRD patterns on wet samples, from bottom to top: a -wash 10 after 10 washes with 0.5 M HCl in green (the same acquisition on b is similar), a -wash 14 H2O after 4 additional washes with H2O in black, and b -wash 12 and wash 14 after 2 and 4 additional washes with 1 M LiCl in purple, respectively. Inset shows the evolution of the supernatant's pH between every wash with HCl (green open symbols), H2O (black symbols) and 1 M LiCl (purple open symbols).

ACKNOWLEDGMENT

This work has been funded by the National Science Foundation (Grant # 1740795).

AUTHOR INFORMATION

Corresponding author: *Email: barsoumw@drexel.edu

Notes

The authors declare no competing financial interests.


Nicolas Le Sommer
email: nicolas.le-sommer@univ-ubs.fr

Frédéric Guidec
email: frederic.guidec@univ-ubs.fr

Hervé Roussain
email: herve.roussain@univ-ubs.fr

A Context-Aware Middleware Platform for Autonomous Application Services in Dynamic Wireless Networks

Dynamic wireless networks, and especially mobile ad hoc networks, impose new constraints regarding the design and the implementation of service-oriented middleware platforms dedicated to mobile computing. Indeed from now, these platforms must be able to capture the context in which they operate in order to provide the services they host with an abstraction of their running context, and to notify them of the variations occurring in this context. These middleware platforms must also implement communication means that make it possible for services to support the frequent and unpredictable connection disruptions. In this paper, we present the design of a serviceoriented middleware platform capable of environmental context introspection and of asynchronous communications.

I. INTRODUCTION

In a near future, thanks to the recent technological advances realised in the wireless networking and computing hardware domains, the number and the diversity of the computing devices populating the physical environment in which we leave should increase significantly, thus giving to this environment some smartness capacities. Sensors and actuators are some examples of these new types of devices. The prospect of integrating such devices in pre-existent infrastructure-based networks, or in ad hoc networks, to provide people with information about their environment (e.g. forecast weather, road conditions, accident warnings) appears as an attractive one. However, the mobility, the volatility, the limitation of the communication range of wireless interfaces and the massive decentralisation of devices impose new constraints regarding the design and the implementation of the services deployed on such devices, as well as on the middleware platforms hosting these services. Indeed, the context in which these services and platforms operate suffers from frequent and unpredictable changes, requiring therefore from these middleware and services some capabilities that traditional services and platforms do not support, such as context-awareness, autonomous behaviour, adaptivity, proactivity and collaboration capabilities. Furthermore, in view of the dynamicity of this context, the synchronous communication paradigm appears to be not suitable, since it assumes that the involved services are present at the same time during the interactions. In contrast, with asynchronous communications, messages can be stored temporarily on certain hosts while being routed or disseminated in the network, and can be forwarded later when circumstances permit. Such a communication paradigm thus makes it possible for a message to eventually reach a destination that was not reachable at the time the message was sent originally.

In this paper, we present the design of a middleware platform with which we address the issues raised by the discovery and the access of resources in dynamic networks composed of heterogeneous devices. This middleware is able to perform a proactive and reactive introspection of the context in order to discover what resources are available in the environment (e.g. neighbouring devices, remote services), and to dynamically reify some abstractions modelling this context in order to provide the services it hosts with facilities for adapting their behaviour to the variations occurring in their running context. It also implements a collaborative and asynchronous communication paradigm to support the frequent and unpredictable transmission disruptions.

The remainder of the paper is organised as follows. Section II describes an application example, brings to the fore some fundamental issues using this example, and outlines our approach to address them. Section III presents the asynchronous communication service we have implemented in our middleware platform, and Section IV details the context-aware functionalities provided by this platform. Section V presents related work, and finally, Section VI concludes the paper.

II. MOTIVATIONS AND BACKGROUND

In this section, we present an application example involving both mobile and static devices operating within a dynamic ubiquitous environment. On the basis of this example we bring out some issues that must be considered when designing and implementing the application services and the middleware platforms that are likely to be used in such a context.

A. Motivating example

For the sake of illustration, let us consider the scenario presented in Figure 1. This figure illustrates a dynamic ubiquitous environment on a city scale. This environment includes distinct or discontinuous infrastructure-based networks and of mobile ad hoc networks. These networks are composed of both static devices, such as sensors and IEEE 802.11 (a.k.a. Wi-Fi) access points providing wireless access to wired domains (including the Internet), and mobile devices equipped with Wi-Fi communication interfaces, such as the portable devices used by nomadic people (e.g. personal digital assistants, smartphones) and those embedded in vehicles. The standard IEEE 802.11 makes it possible for mobile devices to be connected to the access points of infrastructure-based networks or to communicate directly in ad hoc mode (i.e. without resorting to any kind of infrastructure-based network). The ad hoc networks considered in this scenario can appear and evolve spontaneously as mobile devices themselves appear, move and disappear dynamically in/from the network. In this scenario, we mainly focus on the person who arrives by plane in the city, and who wants to go to its hotel all by herself. We also suppose that this person has a PDA (Personal Digital Assistant) equipped with an IEEE 802.11 communication interface, and that she has installed on her device a cartography and positioning application service that is able to display the map images it downloads from the Internet and to indicate the position of the user on these images thanks to the information it obtains from remote positioning services.

B. Context-awareness and discovery of resources and services

The environment in which the nomadic person operates is brought to evolve dynamically throughout its journey because networks, devices and services are likely to appear and disappear spontaneously. So as to provide acceptable quality of service to its user and, consequently, to improve the satisfactions of the later, the cartography and positioning application service has to be context-aware and able to adapt its behaviour to context changes. For instance, by performing a proactive and/or reactive introspection of the context, this application service should be able to discover itself the wireless network deployed in the airport when the person arrives in, and to identify what neighbouring devices offer a positioning service. Another application, also having communication needs with third devices, should be able to achieve the same context introspection task. Consequently, it is suitable that the introspection of the context be performed by the middleware rather than by the services themselves.

Such an approach requires from the middleware the ability to provide the services it hosts with abstractions modelling their running context and to notify these services of the variations occurring in it, and from services the ability to exhibit their non-functional properties regarding their running conditions. By providing such information, services ask their platform for not being notified of all variations systematically, but only of the variations they are interested in. For example, at startup, the cartography and positioning application service could indicate to its hosting middleware platform that it requires a positioning service for running. In return, this middleware platform is expected to notify this application service when it discovers a positioning service in the environment.

C. Selection and provision of services

The devices populating an environment comparable to that presented in Figure 1 can be extremely heterogeneous, and can potentially act as clients and/or providers of services. Consequently, a service provider can hardly found a uniform quality of service for all end clients because these ones can have radically different characteristics. Similarly, it can be difficult for clients to found and to select services that fit their own software and hardware capacities. We thus believe that client services should be able to provide information about the hardware and software capabilities of the devices on which they run, and that providers should be able to exhibit both their functional and non-functional properties. For example, by providing service providers with information about the hardware characteristics of the PDA used by the nomadic person considered in Figure 1, the cartography and positioning application service helps these providers to deliver images whose resolution is compatible with the display capabilities of the device. By specifying the image resolution they can provide, service providers are expected to help the cartography and positioning application service to select suitable services.

Service access continuity is another important issue in dynamic networks. Indeed, in contrast of the wireless standards fostering the "Always On, Always connected" scenario, and where mobile devices are actually considered as terminals with which users can access services offered by infrastructure-based networks, the dynamic networks we consider involve static and mobile devices equipped with communication interfaces whose transmission range is limited. Consequently, any device can only communicate directly with neighbouring devices. Multi-hop transmissions can sometimes be obtained by implementing a dynamic routing algorithm on each device, but it is observed that a realistic ad hoc network often presents itself as a fragmented network, and that devices that belong to distinct islands cannot communicate together, because no transmission is possible between islands. An additional problem with dynamic ad hoc networks is that in many realistic scenarios such networks present themselves as disconnected networks. As a consequence, direct transmissions between any pair of devices are not always feasible, as such transmissions require that both devices are active simultaneously in the network, and that a connected-path can be established between these devices at transmission time. Asynchronous communications should help at overcoming these constraints and at improving service access. Indeed, as mentioned in the Section 1, messages are not simply routed and disseminated within the network, they can be stored temporarily on certain hosts while travelling from host to host, and be forwarded later when circumstances permit. This propagation model is inherently a probabilistic one, as eventual message delivery cannot be guaranteed. However, theoretical studies based on simulations (on variants of this model [START_REF] Vahdat | Epidemic Routing for Partially-Connected Ad Hoc Networks[END_REF], [START_REF] Lindgren | Probabilistic Routing in Intermittently Connected Networks[END_REF]) show that is many realistic scenarios, the probability that messages actually reach their destination can be quite high. For example, before leaving the network island where the owner of the PDA and the bus reside, the car may have collected some messages that must be sent in the Internet (e.g. request sent by the cartography and positioning application service for downloading map images). When, it comes in the area covered by the access points of the infrastructure-based network deployed in the Town hall, this car can sent these messages in the Internet and broadcast the responses in the network, thus allowing other devices to store these responses. Thus, the Jeep should be able to deliver the data requested by the cartography and positioning application service when it joins the ad hoc network formed by the bus and the PDA owned by the nomadic person. With such a approach, the application service can have an Internet access even if it is not in a network providing such a access.

III. ASYNCHRONOUS COMMUNICATION SERVICE

In the recent years, message delivery issues in disconnected mobile ad hoc networks have been considered several times and following different ways. For instance, a new network architecture relying on the general principle of message switching in a store-and-forward mode has been proposed in [START_REF] Fall | A Delay-Tolerant Architecture for Challenged Internets[END_REF]. In this approach data are transported as so-called bundles between bundle forwarders, which are capable of storing messages (or bundles) before sending them again in the network. In epidemic routing [START_REF] Poupyrev | Analysis of Wireless Message Broadcast in Large Ad Hoc Networks of PDAs[END_REF], [START_REF] Vahdat | Epidemic Routing for Partially-Connected Ad Hoc Networks[END_REF], [START_REF] Qayyum | Multipoint Relaying; an Efficient Technique for Flooding in Mobile Wireless Networks[END_REF], messages are buffered in mobile hosts, and the exchanges of messages between these hosts are expected to allow message delivery in partially-connected networks. The models proposed in these papers mostly address the problem of message delivery in disconnected networks from a theoretical point of view. Indeed, they propose new algorithms and heuristics for delivering messages in such networks, and they report the results of simulations that are meant to demonstrate how these algorithms should run in realistic conditions. In contrast, the asynchronous communication service we present in this section is more practical, since it consists in an implementation of a middleware-level service for message dissemination in dynamic mobile networks. This service is meant to be used by application services running on our middleware platform.

The general architecture of our communication service is presented in Figure 2. With this message-oriented communication service, any message sent in the network is maintained as long as possible in a local cache by as many devices as possible, so it can remain available for devices that could not receive it at the time it was sent originally. The underlying idea is that the dissemination of multiple copies of the same message may contribute to overcome the volatility of devices, while the mobility of these devices can itself help transport information between islands in a fragmented network or in distinct networks. Besides providing a caching system where messages can be maintained in mobile devices, our service also provides facilities for message advertisement, message discovery, and message transport between neighbouring devices. For example, a device can sporadically or periodically notify its neighbours about all or part of the messages stored in its cache. It can also look for specific messages in its neighbourhood, and either push messages toward or pull messages from its neighbours. Like JMS [START_REF] Microsystems | Java Message Service Specification -Version 1[END_REF], our communication service provides basic primitives for sending and receiving messages, and for looking for messages from the cache. It also offers facilities for application services for being notified when a new message concerning them is inserted in the cache.

A. Message structure

A message descriptor is associated with a each message. The descriptor of a message is meant to provide information about the message (type, client service, keywords, content, etc.). A message may encapsulate its own descriptor, but the descriptor can also be handled separately (which means it can for example be sent and stored separately).

<transfer-descriptor message-id="fb0097820f0b371" origin="172.17.96.37/map_service destination=" * /gps_service" number-of-hops="5" date="Jan 30 08:26:32 CET 2004" lifetime="12:00:00" advertisement-period="00:20:00" /> Fig. 3. An XML-formatted transfer descriptor of a message exchanged by application-level services.

When an application-level message must be sent in the network, it must be encapsulated in a transfer message, whose descriptor specifies transmission parameters for this message. Figure 3 shows a typical XML-formatted transfer descriptor. The value of attribute message-id in a descriptor should be unique. This condition makes it possible for a device receiving a message from the network to verify if a copy of this message is already available in its local cache.

The message descriptor in Figure 3 also specifies that this message has been sent by the application service map_service (attribute origin) and that it is addressed to all devices running the service gps_service (attribute destination). The attribute number-of-hops plays approximately the same role as the field TTL (Time-To-Live) in IP packets. It ensures that a message will not be propagated eternally in the network. The attribute date specifies the date when the message was originally sent in the network, and the attribute lifetime specifies that this message should be considered as being valid for only twelve hours after this date. The last attribute advertisement-period indicates that once the message has been put in the local cache of a device, this device may periodically announce that this message is available, with a periodicity of 20 minutes.

Advertisement and request messages make it possible to implement on each device some procedures ensuring proactive and/or reactive behaviours. A device behaving as a proactive consumer can send requests in order to ask for the direct transmission of specific messages, or in order to discover that some of its neighbours own messages it may be interested in. After receiving an advertisement for a message it is interested in, a reactive consumer can ask for the immediate transmission of this message. Similarly, a device behaving as a proactive producer can send advertisement messages in order to provide its neighbours with descriptors of the messages it can send them on demand.

It must be notice that a device that considers that its cache needs to be cleaned up has no obligation to maintain a message in this cache until it becomes obsolete. Likewise, a device has no obligation to announce that it owns a message, even if the descriptor of the message suggests an announcement period. In any case, attributes specifying a message's lifetime or suggesting a period for announcing the availability of this message are meant to serve as guidelines about how this message should be handled by devices. The actual behaviour of each device with respect to the messages it maintains in its cache can be guided by such suggestions, but it can also conform to default strategies defined by higher-level services, or according to user-defined global strategies applying equally to all messages.

B. Implementation details

The asynchronous communication service presented in this section was implemented in Java as an OSGi service [START_REF] Alliance | Osgi service platform, release 3[END_REF]. Messages and message descriptors are thus reified as standard Java objects, which can be sent in the network either as serialised Java objects, or as XML-formatted documents. In the current implementation, message dissemination is performed by broadcasting UDP datagrams. We thus assume, for the time being, that each message is small enough to fit in a single datagram. In the future we plan to improve this communication service so as to address issues pertaining on message segmentation and reassembly. Both implementation details and utilisation examples of this service can be found in [START_REF] Guidec | Asynchronous Document Dissemination in Dynamic Ad Hoc Networks[END_REF].

IV. INTROSPECTION AND MODELLING OF THE CONTEXT

As mentioned previously, our work aims to foster the design and the implementation of application services endowed with context-awareness and autonomous behaviour capabilities. Seen from this point of view, we have designed and implemented a middleware-level service capable of proactive and reactive context introspection. Likewise the previous service, this service is implemented in Java as an OSGi service. This section presents this service. Software and hardware resources populating dynamic pervasive environments, such as that considered in Figure 1, can be of various nature. Consequently, the service responsible of performing context introspection must be able to define and to handle dynamically some abstractions modelling these different kinds of resources. These abstractions must also be used by the application-level services hosted by the middleware platform in order to adapt their behaviour to their running context. Three kinds of abstractions can be dynamically reified as first-class objects: resources, resource descriptors and resource observation reports. A resource object implements functionalities to use and control the resource it reifies. Resource descriptor provides "static" information about the resource, whereas the resource observation report gives information about the current state of the resource.

A. Modelling of resources

All resource objects implement the interface Resource presented in Figure 4. This interface extends the interface Observable, which defines a method observe() that returns an observation report (i.e. an object implementing the interface ObservationReport). A resource object is also designed to return a resource descriptor (i.e. an object implementing the interface ResourceDescriptor) when the method getDescriptor() is called. A resource descriptor provides a reference to the considered resource, a textual description of this resource and a set of attributes (objects implementing the interface ResourceAttribute) that describe the properties of this one (e.g. hardware characteristics for devices, functional and nonfunctional properties for services). Furthermore, in order to make resource reification easier, our framework provides a set of interfaces reflecting a resource taxonomy. This taxonomy makes it possible to classify resources following their resource category (e.g. hardware, conceptual) and depending on whether they are observable, listenable, etc. These interfaces are used to define the operations that should be provided by the resource objects (i.e. by the objects reifying resources). For instance, an object modelling a CPU resource should implement the interfaces Observable and HardwareResource. This implementation makes it possible to observe the CPU resource by invoking appropriate methods on the CPU object. Similarly, conceptual resources -which mostly make sense at application level (sockets, files, threads, the software packages and the services, etc.)-should implement the interfaces Con-ceptualResources, Observable and Listenable. Further details about the reification of resources are given in [START_REF] Sommer | Towards Dynamic Resource Contractualisation for Software Components[END_REF].

In many circumstances, resource descriptors and observation reports must be sent in the network (e.g. at the resource advertisement process time). So as to facilitate both their dissemination within the network and their treatment, these objects are transformed into XML documents. An XML resource descriptor the identifier of the resource it describes, the object class of the descriptor, and a description of the resource. A resource descriptor can be reified dynamically from a XML document. For example, such descriptors can be used to provide an XML-formatted description of the hardware characteristics of a personal digital assistant. Similarly, to describe services running on this PDA, it is possible to use the simple XML description of application services provided by our middleware platform.

This context introspection service was designed so as to be highly extensible. Hence new classes can be developed in the future in order to model new resources, new observation reports and new resource descriptors.

B. Tracking and selection of resources

Resources can appear and disappear frequently in the environment. It is thus suitable to have a mean to identify and to keep track of available resources. Our middleware provides such a mean via the resource objects and a resource register. Indeed, all resource objects are designed to register a description of themselves (i.e. their own resource descriptor) with a resource register at creation time (see Figure 5). By consulting this register periodically, it is thus possible for an adaptive service deployed on a mobile device to obtain information about its local execution context. For instance in Figure 5, the service Service#1 can call the method getResources() of the resource register in order to obtain a list of the local resources. Such a service can also act as listener of the resource register, and thus can be notified of the variations of the local context (i.e. of the appearance and disappearance of resources). As shown in Figure 5, the service Service#2 has registered itself as listener with the resource register in order to be notified when a new resource appears in the environment (e.g. the resource Resource#2 in Figure 5). In certain circumstances, it could be suitable to focus only on a specific resource -or on a specific kind of resources. For example, a service such as the service Service#2 in Figure 5 should be notified only when a given resource become available in the environment. Such a need requires to be able to select resources according to a predefined criterion.

Our framework provides such a functionality with the notion of "resource pattern". A resource pattern defines a function isMatchedby(), which takes a resource descriptor object as a parameter, and returns a boolean whose value depends on whether this object satisfies the considered selection criterion or not. In a simple case resource selection can rely on the actual type of the resource which is submitted to the test. But one can also implement more sophisticated selection mechanisms by taking resource type and resource attributes into account.

C. Introspection of the local system

In the approach we propose, the introspection of the network context is fundamental, but it is not sufficient. Indeed, services must also have information about the resources offered by the device on which they run. The introspection of the local system can be easily performed by obtaining a description of the local resources from the resource register and by monitoring the local resources. These resources can be monitored following either a synchronous or an asynchronous mode. The synchronous monitoring is obtained by implementing a callback mechanism in resource objects. Any resource objects can admit one or several listeners. Whenever a variation of the resource occurs -or when a method is called on a conceptual resource-, the resource object informs all its registered listeners (see Figure 5). All resources cannot be monitored using the synchronous approach, though. For example, access to the CPU is not achieved by calling methods explicitly. Instead it is controlled directly by the scheduler of the underlying operating system. In order to deal with system resources such as the CPU (system memory, network interfaces, etc.), which can hardly be monitored synchronously, we propose to do with asynchronous monitoring. Monitoring a resource asynchronously consists in consulting the state of this resource explicitly every now and then, in such a way that the time of the observation does not necessarily coincide with the time of an attempt to use the resource. This observation can be performed by calling the method observe() on the object that models the resource that must be observed. This method gives an observation report (i.e. an object implementing the interface ObservationReport described in Figure 4) in return.

D. Introspection of the network neighbourhood

The introspection of the network neighbourhood aims to discover what resources are available in the physical environment. In contrast of the introspection of the local system, which can be simply achieved using the resource register and the objects modelling the local resources, the introspection of the network neighbourhood requires a more sophisticated mechanism to discover automatically the resources available in the environment. Indeed, as brought out in the Section II, the network context in which mobile and static devices operate suffers from frequent and unpredictable changes.

The context introspection service implemented in our middleware platform provides facilities to address this issues. First, it provides facilities to discover the close wireless networks, to configure the wireless communication interface of mobile devices automatically, and to identify what hosts can be reached in these networks. Obviously, owners of mobile devices can configure this functionality in order to select themselves the network they want to join. In its current implementation, this service relies on the system commands of the Linux operating system (e.g. wireless tools). The discovery of neighbouring hosts and of remote services is achieved using the asynchronous communication service presented in the previous section.

The discovery of hosts can thus be made either proactively by sending host discovery requests in the network, or reactively by analysing the unsolicited advertisements sent by other devices in the network (each device is responsible to manage its perception of its environment). When the service responsible of the context introspection receives an advertisement from a remote host, it reifies this advertisement as a resource object (i.e. as a host descriptor object) and registers it in the resource register dedicated to remote resources.

The discovery of the remote services is performed following the same mode. In the proactive approach the middleware sends in the network an XML document containing the service request. This request is formally describe using a resource pattern. The service providers, which have a service whose descriptor fits the pattern specified in the request, send in response an XML document containing the description of the service they can offer. The service discovery can also be performed following a reactive mode. In this mode, service providers send unsolicited advertisements (i.e. service descriptors).

V. RELATED-WORK

A. Asynchronous Communication in mobile networks

The asynchronous communication service presented in Section III can obviously be compared with that proposed by Vahdat and Becker in [START_REF] Vahdat | Epidemic Routing for Partially-Connected Ad Hoc Networks[END_REF], where they introduce the concept of Epidemic Routing. In their model, messages are buffered in mobile hosts and are exchanged randomly among hosts. This model makes the delivery of messages possible in partiallyconnected ad hoc networks. Although Vahdat and Becker are mostly concerned with unicast transmissions in their paper, they suggest that Epidemic Routing is also appropriate for supporting multicast traffic. A variant of Epidemic Routing has been proposed in [START_REF] Lindgren | Probabilistic Routing in Intermittently Connected Networks[END_REF]. This variant introduces probabilistic heuristics in order to increase the chances that a message is routed toward its destination.

The approach we investigate with our asynchronous communication service can also be compared with that of IRTF1 's group working on delay-tolerant networks (DTNRG: Delay-Tolerant Network Research Group2 ). This group focuses on the deployment of networks in high dynamic environments in which end-to-end connectivity cannot be guaranteed. It has defined a new network architecture relying on the general principle of message switching in a store-and-forward mode. This approach consists in transporting pieces of informationcalled bundles-and implementing forwarders that are capable of storing messages (i.e. the bundles) before they can be sent againin the network [START_REF] Fall | A Delay-Tolerant Architecture for Challenged Internets[END_REF].

B. Middleware Support for Adaptive Services and Pervasive Computing

The middleware Centaurus [START_REF] Kagal | Centaurus: A Framework for Intelligent Services in a Mobile Environment[END_REF] and Mobishare [START_REF] Valavanis | MobiShare: Sharing Context-Dependent Data and Services from Mobile Sources[END_REF] provides owners of mobile devices with facilities to discover the services that are offered spontaneously within an infrastructure-based network, as well as to access to these services remotely. Both Centaurus and Mobishare implement a centralised approach of service management: service as are deployed on predefined servers, service discovery relies on the utilisation of a unique service repository, etc. Such middleware are designed to be used in close smart environments (e.g. home, office) including low mobile devices. Since service clients do not suffer from frequent and unpredictable service disconnection, these middleware do not implement mechanisms to ensure service access continuity for nomadic people. Moreover, the centralised approach implemented by these middleware do not allow to take advantages of wireless communication. Indeed, in contrast of a peer-to-peer approach, a centralised approach does not favour the collaboration of mobile devices to achieve service provision.

Many recent middleware platforms designed to be used in mobile dynamic environments (e.g. ad hoc networks) implement peer-to-peer technologies in order to make it possible for mobile devices to collaborate spontaneously. These technologies provide means to describe, to advertise, to request, and to retrieve services. For instance, in domotic environments, such technologies permit people to control their domestic devices (e.g. light, tv). UPnP (Universal Plug and Play) [START_REF]UPnP Device Architecture[END_REF], SLP (Service Location Protocol) [START_REF] Guttman | Service Location Protocol, Version 2[END_REF] are some example of these technologies. The middleware Konark [START_REF] Helal | Konark : Service Discovery and Delivery Protocol for Ad-hoc Networks[END_REF] relies on an implementation of the UPnP technology in order to describe, to announce and to discover services in mobiles ad hoc networks. The services considered in Konark are Web Services that can be invoked using the protocol SOAP. Like Centaurus and Mobishare, Konark is designed to be used in wireless environments including low mobile devices.

The middleware 7DS [START_REF] Papadopouli | Seven Degrees of Separation in Mobile Ad Hoc Networks[END_REF] and XMIDDLE [START_REF] Mascolo | XMIDDLE: A Data-Sharing Middleware for Mobile Computing[END_REF] are not designed to provide services, but are designed to find, to relay, to disseminate and to share information in peer-to-peer mobile ad hoc networks. 7DS implements sophisticated prefetching and collaborative mechanisms that could be used to relay, and to provide access to services asynchronously in high dynamic mobile networks, such as delay-tolerant ad hoc network. The middleware XMIDDLE is dedicated as for it, to the sharing of XML documents in mobile networks composed of heterogeneous devices. It implements replication and reconciliation mechanisms in order to allow nomadic people to have copies of documents on their own devices, and thus to make it possible for them to access to these documents autonomously. The reconciliation mechanisms aims at ensuring the update of the copies when devices are connected.

None of the above-presented works are designed to support adaptive services, and thus do not the services deployed on the mobile devices with mechanisms that allow them to perform an introspection of both their network context and their local context. Yet, as shown in this paper, it could be relevant to deploy adaptive services on mobile devices in order to offer service access continuity to owners of mobile devices. In the remainder of this section we present some works designed to support adaptive services.

The project Molène [START_REF] Andre | A Framework for Dynamic Adaptation in Wireless Environments[END_REF] aims at offering an object-oriented framework to implement adaptive applications. This framework defines a notification system of context variations based on a set of resource monitors organised in two levels, and a customisable system of reaction. The monitors of the first level are specialised monitors (e.g. battery monitor, network interface monitor). The monitor of the second level are designed to analyse information they received from the monitors of the first level, and to notify applications of the variations of their context according the directives they received at startup.

The aim of the project AMPROS [START_REF] Conan | A Pro-Active Middleware Platform for Mobile Environment[END_REF] is to design and to develop a middleware platform in dynamic wireless networks with the generalisations that can be used for such services as emergency aid, crisis managements etc. Seen from this point of view, AMPROS proposed an approach, that is similar to ours, since it mostly relies on the deployment of disconnected-components on mobile devices in order to ensure service continuity (the disconnected-components provide a service identical to those offer by the remote components). AMPROS also implements reconciliation mechanisms in order to ensure that the state of the local disconnected-components remains the same that that of the remote components. However, the deployment of component -and of services-require to have security mechanisms in order to monitor the components at runtime. Indeed, a component could be introduced in the network by a malevolent device in order to put the system in jeopardy by destroying crucial data files for instance.

Like Molène and AMPROS, the middleware Carisma [START_REF] Capra | CARISMA: Context-Aware Reflexive mIddleware System for Mobile Applications[END_REF] implements some context introspection mechanisms in order to make the development of adaptive application easier. Seen from this point of view, this middleware provides to the application it hosts an abstraction of their running context and a set of reflexive mechanisms in order to act on this context using this abstraction.

VI. CONCLUSION AND FUTURE WORK

In this paper, we have presented key services of a serviceoriented middleware platform we have designed and implemented in order to foster the development of application services with context-awareness, autonomous behaviour, adaptivity, proactivity and spontaneous collaboration properties. The first service presented in this paper permits the asynchronous dissemination of messages in dynamic wireless networks, such as those composed of highly mobile and volatile communicating devices. It proposes an asynchronous, peer-to-peer, message-oriented propagation model, where each message received by a device can be maintained in a local cache in this device, so it can later be sent again in the network, either spontaneously, or after a request for this message has been received from another device. This approach is expected to help do with the volatility of devices, since it permits that messages reach devices that are only active sporadically in the network. It is also expected to permit information dissemination in a fragmented network, taking advantage of the mobility of devices which can serve as carriers between disconnected parts of the network or between distinct networks. The second service presented in this paper is designed to perform a continuous context introspection, to reify dynamically objects modelling the resources populating the context, and to notify the application services hosted by the middleware platform of the variations occurring in their running context.

Since the development of the middleware platform is not complete, no extensive performance evaluation has been performed so far. However, preliminary experiments we have made prove to be promising. Before starting an evaluation process of our middleware, we plan to improve the discovery and delivery of services. Indeed, the asynchronism inherent to delay-tolerant networks raises specific difficulties. For instance, the scope in time and space of the discovery queries and service announcement must be controlled. To achieve this goal we plan to introduce the concept of context attribute as an effective, flexible means to exploit relevant context information during the service discovery and delivery process. Context attributes can express various context information including user's preferences, quality of service, network conditions, service availability, etc. Such attributes aims at helping middleware -or users-to choose the best services that fulfil their needs.

Fig. 1 .

 1 Fig. 1. Illustration of a dynamic ubiquitous environment on a city scale.

Fig. 2 .

 2 Fig. 2. General architecture of the asynchronous communication service.

Fig. 4 .

 4 Fig. 4. Object-oriented modelling of resources

Fig. 5 .

 5 Fig. 5. Tracking of resources

Fig. 6 .

 6 Fig. 6. Monitoring of resources.

IRTF: Internet Research Task Force (http://www.irtf.org).

http://www.dtnrg.org


Mehdi Badra

Takéo Takahashi

Switching controls for parabolic systems

Keywords: controllability, parabolic systems, switching controls 2010 Mathematics Subject Classification. 93B05, 35K10, 93C20

We consider the controllability of an abstract parabolic system by using switching controls. More precisely, we show that under general hypotheses, if a parabolic system is null-controllable for any positive time with N controls, then it is also null-controllable with the property that at each time, only one of these controls is active. The main difference with previous results in the literature is that we can handle the case where the main operator of the system is not self-adjoint. We give several examples to illustrate our result: coupled heat equations with terms of orders 0 and 1, the Oseen system or the Boussinesq system.

where z is the state of the system and uj are controls. We are interested by the problem proposed in [START_REF] Zuazua | Switching control[END_REF] where the idea is to control z by using "switching" controls so that only one control uj is active at the same time. We consider here parabolic systems in the sense defined below. Our aim is to generalize the results already obtained in [START_REF] Zuazua | Switching control[END_REF] and in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF]. In [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF], the authors manage to prove the null-controllability of the system (1.1) with switching controls in the case where A is a self-adjoint operator in a Hilbert space and if (1.1) is null-controllable for all T > 0 without the switching conditions on the controls. They also obtain this result in the case where the state space is finite-dimensional and they conclude with a result for non self-adjoint operator in a Hilbert space but with conditions on A that can be quite restrictive (see the discussion after Theorem 5.1 in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF]). Here, we show a more general result with less restrictive conditions on A and we give at the end of this article several examples showing that we can handle many important situations. We also state a corresponding result for the approximate controllability with switching controls.

We assume that, in (1.1), A : D(A) → H is an unbounded operator in a Hilbert space H, and Bj, j = 1, . . . , N are control operators satisfying Bj ∈ L(Uj, D((A * ) k ) ) for some k 0, where Uj is a Hilbert space. Here and in what follows, we identify H with its dual and for a Hilbert space V ⊂ H such that V is dense in H, V stands for the dual space of V with respect to the pivot space H.

In what follows, we assume that -A is the infinitesimal generator of an analytic semigroup e -tA t 0

.

(H1)

This implies that the spectrum σ(A) of A is contained in a sector of C (see, for instance, [3, Theorem 2.11, p.112]). We assume moreover that σ(A) is only composed by eigenvalues (λj) with finite algebraic multiplicity. (H2)

We recall that one can define the algebraic multiplicity of λj by using a projection operator

P(λj) = 1 2πı Γ (λI -A) -1 dλ, (1.2)
where Γ is a positively oriented simple closed curve enclosing λj but not other point of σ(A) (see [17, pp. 180-181]). Then, the algebraic multiplicity of λj is the dimension of Im P(λj). There exists n ∈ N * such that Im P(λj) = Ker(λjI -A) n and an element of Im P(λj) is called a root vector of A. We denote by mj the smallest n satisfying the above equality (the index of λj). We have similar definitions for A * and we assume that The root vectors of A are complete in H.

(H3)

Our main result is Theorem 1.1. Assume (H1), (H2), and (H3). If (1.1) is null-controllable in any time T > 0, then it is null-controllable in any time T > 0 with the following constraint on the controls (u k ) k=1,...,N :

N j=1 k =j u k (t) U k = 0 a.e. t ∈ (0, T ). (1.
3)

The condition (1.3) means that at most one control u k is active (non null) at the same time. In the case

N = 2, (1.1) writes z + Az = B1u1 + B2u2 in (0, T ), z(0) = z 0 , (1.4)
and the condition (1.3) reduces to the condition

u1(t) U 1 u2(t) U 2 = 0 a.e. t ∈ (0, T ). (1.5) Remark 1.2.
To be more precise in the statement of Theorem 1.1, let us define for r = k + α, k ∈ N, α ∈ [0, 1], the spaces:

Hr def = D(A k+1 ), D(A k ) 1-α , H-r def = D((A * ) k+1 ), D((A * ) k ) 1-α , (1.6)
where [•, •]• denotes the interpolation space obtained with the complex interpolation methods. Then, there exists γ 0 such that Bj ∈ L(Uj, H-γ) (j = 1, . . . , N ).

(1.7)

From standard result on parabolic systems, if uj ∈ L 2 (0, T ; Ui) and if z 0 ∈ [H1-γ, H-γ] 1/2 , then the solution z of (1.1) satisfies

z ∈ H 1 (0, T ; H-γ) ∩ L 2 (0, T ; H1-γ) ∩ C 0 ([0, T ]; [H1-γ, H-γ] 1/2 ) (1.8)
and Theorem 1.1 states that if (1.1) is null-controllable for any time T > 0 in [H1-γ, H-γ] 1/2 with controls in L 2 (0, T ; Uj) then for any z 0 ∈ [H1-γ, H-γ] 1/2 , there exist uj ∈ L 2 (0, T ; Uj) satisfying (1.3) and such that z(T ) = 0. Note that, using the parabolic regularity, if z 0 is in a larger space H -γ , with γ > γ, then taking uj ≡ 0 in (0, ε) we have z(ε) ∈ [H1-γ, H-γ] 1/2 and we deduce the null-controllability with switching controls.

Remark 1.3. With respect to the literature, we generalize here the results obtained in [START_REF] Zuazua | Switching control[END_REF] and in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF]. In [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF], the authors assume that the operator A is self-adjoint or that H is of finite dimension. They have also obtained an extension in the case of a parabolic system (Theorem 5.1 in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF]) but they need an hypothesis on the semigroup e -tA * t 0 and they explain that this hypothesis is difficult to check in practice with some examples. Here we have more general hypotheses on A and there is a typical situation where (H1), (H2) and (H3) hold: if A can be decomposed as A = A0 + A1 with A0 a self-adjoint operator in H with compact resolvent, D(A1) ⊂ D(A α 0 ) for some α ∈ [0, 1) and the eigenvalues µj of A0 satisfy for some 1 p < ∞ and for some J 1, With a similar proof as Theorem 1.1, one can also obtain the following result:

∞ j=J 1 |µj| p < ∞, (1.9
Theorem 1.4. Assume (H1), (H2) and (H3). If (1.1) is approximately controllable for some time T > 0, then it is approximately controllable in time T > 0 with the constraint (1.3).

Remark 1.5. We can make the above statement more precise, using Remark 1.2. If (1.1) is approximately controllable for some time T > 0, and if Bj satisfies (1.7), then for any z 0 , z 1 ∈ [H1-γ, H-γ] 1/2 and for any ε > 0, there exist uj ∈ L 2 (0, T ; Ui) satisfying (1.3) so that the solution z of (1.1) (satisfying

(1.8)) verifies z(T ) -z 1 [H 1-γ ,H -γ ] 1/2 ε.
The outline of this article is as follows: in the next section, we show Theorem 1.1 and give the main ideas to adapt the proof to Theorem 1.4. Then in Section 3, we present some examples to illustrate this result.

Proof of the main result

The proof of Theorem 1.1 follows the same scheme as in [START_REF] Zuazua | Switching control[END_REF] and in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF]. The new part of this proof corresponds to Lemma 2.2. We only repeat all the proof for sake of clarity. To simplify, we only show Theorem 1.1 for N = 2. The details of the proof for N > 2 can be found in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF]. Finally, we can also assume that Bj ∈ L(Uj, H) (the case of "bounded" control operators). Indeed, taking µ0 > -inf

j 1
Re λj, one can apply (µ0 + A) -k to (1.4) and we obtain the control system

z + A z = B1u1 + B2u2 in (0, T ), z(0) = z 0 , (2.1)
where

z def = (µ0 + A) -k z, z 0 def = (µ0 + A) -k z 0 , Bj def = (µ0 + A) -k Bj ∈ L(Uj, H).
If there exist controls uj satisfying (1.5) and such that z(T ) = 0 then it implies Theorem 1.1. First, we consider the adjoint system of (1.1)

ϕ + A * ϕ = 0 in (0, T ), ϕ(0) = ϕ 0 , (2.2) that is ϕ(t) = e -tA * ϕ 0 (t 0). Since (1.
e -τ A * ϕ 0 H C(τ ) τ 0 B * 1 e -tA * ϕ 0 2 U 1 + B * 2 e -tA * ϕ 0 2 U 2 dt 1/2 ϕ 0 ∈ H . (2.3)
Following the proof of [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF], we introduce the function

α(t) = 1 + 1 2 sin(ωt), (2.4)
with ω ∈ R that will be fixed further. As explained in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF], where the authors introduce such a function, many other choices are possible for α. We fix T > 0 and we define the following function

ϕ 0 X def = T 0 max B * 1 e -tA * ϕ 0 2 U 1 , α(t) B * 2 e -tA * ϕ 0 2 U 2 dt 1/2 (2.5)
and due to the bounds of α, we deduce from (2.3) that

e -T A * ϕ 0 H C ϕ 0 X ϕ 0 ∈ H . (2.6)
Using that e -tA * t 0 is an analytic semigroup, the above relation implies that • X is a norm. We consider the closure of H with respect to this norm:

X def = Clos • X H. (2.7)
Remark 2.1. Note that from (2.6), we can extend the application ϕ 0 → e -T A * ϕ 0 into a bounded map of L(X , H). Using (2.3), for any τ ∈ (0, T ), we can also extend the application ϕ 0 → e -τ A * ϕ 0 into a bounded map of L(X , H). For j = 1, 2, we can also extend ϕ 0 → t → B * j e -tA * ϕ 0 into a bounded map of L(X , L 2 (0, T ; Uj)).

We can now define

J(ϕ 0 ) def = 1 2 ϕ 0 2 X + z 0 , e -T A * ϕ 0 H (ϕ 0 ∈ X ).
The function J is convex, continuous and coercive: there exists a minimizer φ 0 ∈ X of J. First, if φ 0 = 0, then let us consider ϕ 0 ∈ H. For any ε > 0,

J(εϕ 0 ) = 1 2 ε 2 ϕ 0 2 X + ε z 0 , e -T A * ϕ 0 H 0 = J(φ 0 )
and similarly,

1 2 ε 2 ϕ 0 2 X -ε z 0 , e -T A * ϕ 0 H 0.
Dividing by ε and taking ε → 0 in the two above relations, we deduce that e -T A z 0 , ϕ 0 = 0 for any ϕ 0 ∈ H and thus e -T A z 0 = 0. In particular, the controls u1 = 0 and u2 = 0 in (0, T ) lead the system to rest and satisfy the switching condition. Second, assume φ 0 = 0. Then, Lemma 2.2 states that there exists ω such that the set

I0 def = t ∈ (0, T ) ; B * 1 e -tA * φ 0 2 U 1 = α(t) B * 2 e -tA * φ 0 2 U 2 (2.8)
is of Lebesgue measure equal to 0. We then define

I1 def = t ∈ (0, T ) ; B * 1 e -(T -t)A * φ 0 2 U 1 > α(T -t) B * 2 e -(T -t)A * φ 0 2 U 2 , and
I2 def = t ∈ (0, T ) ; B * 1 e -(T -t)A * φ 0 2 U 1 < α(T -t) B * 2 e -(T -t)A * φ 0 2 U 2 .
We also define

j(t, ϕ 0 ) def = max B * 1 e -(T -t)A * ϕ 0 2 U 1 , α(T -t) B * 2 e -(T -t)A * ϕ 0 2 U 2
and we note that the differentiate of j(t, •) at the point φ 0 in the direction ϕ 0 satisfies

D φ 0 j(t, ϕ 0 ) =    2 B * 1 e -(T -t)A * φ 0 , B * 1 e -(T -t)A * ϕ 0 U 1 if t ∈ I1, 2α(T -t) B * 2 e -(T -t)A * φ 0 , B * 2 e -(T -t)A * ϕ 0 U 2 if t ∈ I2.
By considering all the possibilities for the maximum, one can show (see the appendix A of [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF] for more details) the existence of C > 0 such that

1 h j(t, φ 0 + hϕ 0 ) -j(t, φ 0 ) C B * 1 e -(T -t)A * φ 0 2 U 1 + B * 2 e -(T -t)A * φ 0 2 U 2 + B * 1 e -(T -t)A * ϕ 0 2 U 1 + B * 2 e -(T -t)A * ϕ 0 2 U 2 (h ∈ (0, 1)).
Using the Lebesgue theorem and the fact that |I0| = 0, we deduce that

D φ 0 J(ϕ 0 ) = I 1 B * 1 e -(T -t)A * φ 0 , B * 1 e -(T -t)A * ϕ 0 U 1 dt + I 2 α(T -t) B * 2 e -(T -t)A * φ 0 , B * 2 e -(T -t)A * ϕ 0 U 2 dt + z 0 , e -T A * ϕ 0 H . (2.9) Setting u1(t) def = 1I 1 (t)B * 1 e -(T -t)A * φ 0 , u2(t) def = 1I 2 (t)α(T -t)B * 2 e -(T -t)A * φ 0 (2.10)
and taking ϕ 0 ∈ H, we deduce that

D φ 0 J = e -T A z 0 + T 0 e -(T -t)A (B1u1(t) + B2u2(t)) dt.
Using that φ 0 is a minimizer we deduce that the solution z of (1.4) with u1 and u2 defined by (2.10) satisfies z(T ) = 0 whereas u1 and u2 satisfy (1.5). This ends the proof of Theorem 1.1, provided we can prove that the set I0 defined by (2.8) is of Lebesgue measure equal to 0. This is the new part of this proof since it is done without assuming that H is finite dimensional or that A is self-adjoint.

Lemma 2.2. Assume that ω / ∈ Im (λ k -λ k ) , 1 2 Im (λ k -λ k ) for all (k, k ) ∈ (N * ) 2 such that Re(λ k ) = Re(λ k ) . (2.11)
If φ 0 = 0 then |I0| = 0.

Before proving the above result, let us recall a result proved in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF] (that we have slightly adapted): Proof. Lemma 2.5 in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF] shows this result if pj are polynomials of degree 0. To prove Lemma 2.3, we consider

d def = N max j=1
deg pj that is 0 if the pj are not all null. Then by taking the limit of the above sum divided by t d , Lemma 2.5 in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF] yields that all the monomials of pj of degree d are 0 which leads to a contradiction.

We can now prove Lemma 2.2.

Proof of Lemma 2.2. Assume that |I0| = 0.

(2.12)

From (2.12), there exists n ∈ N * such that

I0 ∩ T n , T = 0. (2.13)
On the other hand, using Remark 2.1, e -T n A * φ 0 ∈ H. This fact and (2.13) allow us to reduce the proof of Lemma 2.2 to the case φ 0 ∈ H. Using (2.12) and the analyticity of the semigroup, we deduce that

B * 1 e -tA * φ 0 2 U 1 = α(t) B * 2 e -tA * φ 0 2 U 2 (t 0). (2.14)
From the eigenvalues of A, we can define an increasing sequence (σj) of R such that

{σj, j ∈ N * } = {Re λ k , k ∈ N * } .
Let us define

Σ1 def = λ k ; Re λ k = σ1 , K1 def = {k ∈ N * ; Re λ k = σ1} .
From (H1) and (H2) there exists a positively oriented simple closed curve Γ1 enclosing Σ1 but not other point of σ(A * ). Then, we consider the projection operator (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]Theorem 6.17, p.178])

P * 1 def = 1 2πı Γ 1 (λI -A * ) -1 dλ (2.15)
and we define

H+ def = P * 1 (H) , H- def = (I -P * 1 ) (H) . We have that H+ = k∈K 1 Ker A * -λ k I m k ⊂ D(A * ),
where m k is the index of λ k . We also define

A * + ∈ L(H+), A * + z def = Az (z ∈ H+), D(A * -) def = H-∩ D(A * ), A * -z def = Az (z ∈ D(A * -)).
Then, from Theorem 6.17 in [17, p.178], the spectra of A * + and of A * -are respectively Σ1 and σ(A) \ Σ1. In particular, there exists ε > 0 such that

inf {Re λ, λ ∈ σ(A * -)} > σ1 + ε. (2.16)
Then, we set φ+(t)

def = P * 1 e -tA * φ 0 e σ 1 t = e -tA * + P * 1 φ 0 e σ 1 t , (2.17) φ-(t) def = (I -P * 1 ) e -tA * φ 0 e σ 1 t = e -tA * -(I -P * 1 ) φ 0 e σ 1 t . (2.

18)

Using that A * -is the infinitesimal generator of an analytic semigroup and (2.16) (see, for instance, [3, Proposition 2.9, p. 120], we deduce that for some constant C > 0,

B * 1 φ-(t) U 1 + B * 2 φ-(t) U 2 C φ 0 H e -εt (t 0). (2.19)
Moreover, there exists φ k, ∈ H+ ⊂ D(A * ), k ∈ K1 and ∈ {0, . . . , m k } such that

φ+(t) = k∈K 1 e ı Im λ k t m k =0 t φ k, . (2.20)
Thus, we deduce from (2.14) that

B * 1 φ+(t) 2 U 1 -α(t) B * 2 φ+(t) 2 U 2 → 0 if t → ∞, (2.21)
and using (2.4), we can develop the above expression as in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF]:

B * 1 φ+(t) 2 U 1 -α(t) B * 2 φ+(t) 2 U 2 = k,k ∈K 1 e ı(Im λ k -Im λ k )t m k =0 m k =0 t + (B * 1 φ k, , B * 1 φ k , ) U 1 -(B * 2 φ k, , B * 2 φ k , ) U 2 - 1 4ı k,k ∈K 1 e ı(Im λ k -Im λ k +ω)t m k =0 m k =0 t + (B * 2 φ k, , B * 2 φ k , ) U 2 + 1 4ı k,k ∈K 1 e ı(Im λ k -Im λ k -ω)t m k =0 m k =0 t + (B * 2 φ k, , B * 2 φ k , ) U 2 .
This implies that B * 1 φ+(t)

2 U 1 -α(t) B * 2 φ+(t)
2 U 2 has the form N j=1 pj(t)e iµ j t of Lemma 2.3. From this lemma, we deduce that all the polynomials pj = 0 and in particular the ones associated with µj = 0 and µj = ω. From (2.11), these polynomials are respectively

k∈K 1 m k =0 t B * 1 φ k, 2 U 1 - m k =0 t B * 2 φ k, 2 U 2 = 0 and - 1 4ı k∈K 1 m k =0 t B * 2 φ k, 2 U 2 = 0
and we deduce that

∀k ∈ K1, ∀ ∈ {0, . . . , m k }, B * 1 φ k, = 0 and B * 2 φ k, = 0.
The above relation combined with (2.17) and (2.20) implies that

B * 1 e -tA * P * 1 φ 0 = 0 and B * 2 e -tA * P * 1 φ 0 = 0 (t 0).
Thus, P * 1 φ 0 X = 0 and we deduce that P * 1 φ 0 = 0. Thus φ 0 ∈ (Im P1) ⊥ , where

P1 = 1 2πı Γ 1 (λI -A) -1 dλ.
In particular,

φ 0 ∈   k∈K 1 Ker (A -λ k I) m k   ⊥ .
By induction, we deduce that φ 0 is orthogonal to all the root vectors of A. Using (H3), we deduce that φ 0 = 0.

Let us finish this section by giving some ideas to adapt the above proof in order to show Theorem 1.4. The arguments are classical and thus we skip the details. We only consider the case N = 2 and the case of Bj ∈ L(Ui, H), the other cases can be done similarly (note that we would need in particular to replace the space H by the space [H1-γ, H-γ] 1/2 , see Remark 1.2 and Remark 1.5). Assume z 0 , z 1 ∈ H and assume ε > 0. We want to obtain u1 ∈ L 2 (0, T ; U1), u2 ∈ L 2 (0, T ; U2) satisfying (1.5) such that the solution of (1.4) (with initial condition z 0 ) verifies

z(T ) -z 1 H ε.
Then we replace the functional J of the proof of Theorem 1.1 by

Jε(ϕ 0 ) def = 1 2 ϕ 0 2 X + z 0 , e -T A * ϕ 0 H -z 1 , ϕ 0 H + ε ϕ 0 H (ϕ 0 ∈ H).
Then we have that Jε is convex, continuous in H. Moreover, one can show that lim inf [START_REF] Fabre | Approximate controllability of the semilinear heat equation[END_REF]). Consequently, there exists a minimizer φ 0 ∈ H of Jε and the rest of the proof is the same as the proof of Theorem 1.1.

ϕ 0 H →∞ Jε(ϕ 0 ) ϕ 0 H ε so that Jε is coercive in H (see, for instance,

Examples

Here, we present some examples of application of Theorem 1.1. We focus in the case where A is not selfadjoint, and we refer the reader to [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF] for several interesting examples in the self-adjoint case.

Coupled heat equations

Assume Ω is a bounded open set of R d , with regular boundary. Let us consider m 2 and a nonempty open subset ω of Ω. We consider the following coupled system of heat equations:

         ∂tyj -dj∆yj + m k=1 (a j,k • ∇y k + b j,k y k ) = 1ωuj in (0, T ) × Ω, yj = 0 on (0, T ) × ∂Ω, yj(0, •) = y 0 j in Ω. (3.1)
Here dj ∈ R * + , a j,k ∈ R d , b j,k ∈ R (j, k ∈ {1, . . . , m}). Using the Carleman estimates for the heat equation (see, [START_REF] Fursikov | Controllability of evolution equations[END_REF] or [START_REF] Fernández | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]), it is well-known that the above system is null-controllable in L 2 (Ω) m with controls uj ∈ L 2 (0, T ; L 2 (ω)) (1 j m). In the literature, there are lot of works devoted to the possibility to decrease the number of controls while keeping the null-controllability property. One can refer for instance to [START_REF] Ammar-Khodja | Recent results on the controllability of linear coupled parabolic problems: a survey[END_REF] for a survey of results until 2011.

Applying Theorem 1.1, we deduce here the following result: Proof. We set

H def = L 2 (Ω) m , D(A) def = H 2 (Ω) ∩ H 1 0 (Ω) m , A = A0 + A1, where D(A0) def = D(A), D(A1) def = H 1 0 (Ω) m = D(A 1/2 0 ), A0y def = (-dj∆yj) j=1,...,m , A1y def = m k=1 [a j,k • ∇y k + b j,k y k ] j=1,...,m .
We also set Ui where (e1, . . . , em) is the canonical basis of R m . To check (H1), (H2) and (H3), we use Remark 1.3 since the eigenvalues µj of A0 satisfy (1.9) from some 1 p < ∞ (by using Weyl formula, see, for instance, [2, Section 1.6] ).

The Oseen system

Assume Ω is a bounded open set of R d , with regular boundary. Assume Γ is a nonempty open subset of ∂Ω.

Let us consider the controllability of the Oseen system:

         ∂ty -ν∆y + ∇p + y S • ∇ y + (y • ∇) y S = 0 in (0, T ) × Ω, div y = 0 in (0, T ) × Ω, y = u1Γ on (0, T ) × ∂Ω, y(0, •) = y 0 in Ω. (3.2)
Here ν > 0 is the viscosity of the fluid and y S ∈ W 1,∞ (Ω). Usually, for physical motivation, one has d = 2, 3 and y S is a stationary state. Note moreover that for the controllability of (3.2), one can take y S ∈ L ∞ (Ω) (see [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF]), but to simplify the presentation we take y S more regular. In the above system, the control u is located at the boundary ; in the case of distributed controls, there are several works devoted to the controllability of the Stokes or the Navier-Stokes system. Let us quote for instance [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF][START_REF] Coron | Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary[END_REF][START_REF] Yu | On exact controllability for the Navier-Stokes equations[END_REF]. Some of these studies are devoted to the case of controls with some vanishing components: see, for instance, [START_REF] Fernández-Cara | Some controllability results for the N -dimensional Navier-Stokes and Boussinesq systems with N -1 scalar controls[END_REF], [START_REF] Coron | Null controllability of the N -dimensional Stokes system with N -1 scalar controls[END_REF] and [START_REF] Carreño | Insensitizing controls with one vanishing component for the Navier-Stokes system[END_REF]. Let us also quote [START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF] where the authors obtain the local null controllability of the Navier-Stokes system in dimension 3 with a control having two vanishing components. Their method is quite different to the previous works and is based on results of Gromov.

Let us define

H def = y ∈ L 2 (Ω) d ; div y = 0, y • n = 0 on ∂Ω .
We denote by P : L 2 (Ω) d → H the orthogonal projection (the Leray projector). Due to the incompressibility condition, the controls have to satisfy the condition Γ u • n dγ = 0 in (0, T ). such that the solution y of (3.2) satisfies P y(T, •) = 0. In particular, taking u ≡ 0 in (T, ∞), we deduce that the solution y of (3.2) satisfies y ≡ 0 in (T, ∞).

Remark 3.3. As in [START_REF] Chaves-Silva | Switching controls for analytic semigroups and applications to parabolic systems[END_REF], instead of using the canonical basis of R d to decompose u, we could use an orthonormal basis (τ1(x), . . . , τ d-1 (x), n(x)), x ∈ Γ, where n the normal to ∂Ω (so that τj(x) is a tangential vector of ∂Ω at the point x). The proof below would be exactly the same and we would obtain a controllability result for which at each instant of time, the control is either tangential or normal.

Proof of Corollary 3.2. We consider the following operators To check (H1), (H2) and (H3), we use Remark 1.3 since the eigenvalues µj of A0 satisfy (1.9) from some 1 p < ∞ (see [START_REF] Métivier | Valeurs propres d'opérateurs définis par la restriction de systèmes variationnels à des sous-espaces[END_REF]).

1 10 1

 110 Coupled heat equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 The Oseen system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 The Boussinesq system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction and main result Let us consider the following system

Lemma 2 . 3 .

 23 Assume (µj) N j=1 is family of N distinct real numbers and (pj) N j=1 a family of N polynomial functions. Then lim t→∞ N j=1 pj(t)e iµ j t = 0 =⇒ ∀j ∈ {1, . . . , N }, pj = 0.

Corollary 3 . 1 .

 31 Assume y 0 = y 0 j j=1,...,m ∈ L 2 (Ω) m . Then for any T > 0, there exist uj ∈ L 2 (0, T ;L 2 (ω)), j = 1, . . . , m with m j=1 k =j u k (t, •) L 2 (ω) = 0 a.e. t ∈ (0, T )such that the solution y of (3.1) satisfies y(T, •) = 0.

def = L 2

 2 (ω), Bjuj def = 1ωujej (j = 1, . . . , m),

(3. 3 )Corollary 3 . 2 .

 332 Applying Theorem 1.1, we deduce here the following result: Assume y 0 ∈ H. Then for any T > 0, there exists u ∈ L 2 (0, T ; L 2 (Γ)) d satisfying (3.3) andd j=1 k =j u k (t, •) L 2 (Γ) = 0 a.e. t ∈ (0, T )

D=

  (A) def = y ∈ H 2 (Ω) ∩ H 1 0 (Ω) d ; div y = 0 , A def = A0 + A1, S • ∇ y + (y • ∇) y S .To introduce the control operator, we writeUi def = ui ∈ L 2 (Γ) ; Γ uini dγ = 0 ,and we consider the Dirichlet operators Di : Ui → H defined by w def Diui is the solution of     λ0w -ν∆w + ∇π + y S • ∇ w + (w • ∇) y S = 0 in Ω, div w = 0 in Ω, w = uiei1Γ on ∂Ω,(3.4)for some λ0 > 0 large enough. Then we define Bi : Ui → D(A * ) by Bi def = (λ0I -A)P Di. Following[START_REF] Raymond | Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions[END_REF], one can write (3.2) under the form P )Djuj in (0, T ).(3.6)

The Boussinesq system

Assume Ω is a bounded open set of R 3 , with regular boundary. Assume ω is a nonempty open subset of Ω. Let us consider the controllability of the Boussinesq system:

where (e1, e2, e3) is the canonical basis of R 3 . Here ν > 0 is the viscosity of the fluid and θ S ∈ W 3,∞ (Ω). In [START_REF] Carreño | Local controllability of the N -dimensional Boussinesq system with N -1 scalar controls in an arbitrary control domain[END_REF], the author obtained that the above system is null-controllable for any T > 0. We keep the same notation for H as in the previous section. Applying Theorem 1.1, we deduce here the following result:

such that the solution (y, θ) of (3.7) satisfies (y, θ)(T, •) = 0.

The proof is similar to the proofs in the previous sections and we skip it.