Adrian Wolf

Juan-José Cid

Emilie Moulin
email: emoulin@unistra.fr

Frédéric Niess

Guangyan Du

Antoine Goujon

Eric Busseron

Adrian Ruff

Prof. Dr Sabine Ludwigs

Nicolas Giuseppone
email: giuseppone@unistra.fr

Dr A Ruff

Unsymmetric bistable [c2]daisy chain rotaxanes which combine two types of electroactive stoppers

Keywords: rotaxane, triarylamine, perylene, pH-responsive, electroactive

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Introduction

Control of the mechanical bond has been a source of inspiration for molecular and supramolecular chemists over the past thirty years. [1] In particular, the possibility to reach large intramolecular motions by switching interlocked molecules between different stations appears of first interest to access molecular machines. [START_REF] Erbas-Cakmak | [END_REF]3] For instance, in 2000, Sauvage designed the first topologically double-threaded rotaxane which mimics (at nanoscale) the function of a sarcomere in muscular tissues (at microscale). [4] The so-called [c2]daisy chain rotaxane proved to contract or extend by 1.8 nm because of the interplay of transition metal ions binding selectively the two stations with different coordination geometries. This seminal work inspired several other groups (including us) to reproduce the natural contractile motion of muscles up to the macroscopic level. [5][6][7][8][9][10][11][12][13][14] Symmetric [c2]daisy chain rotaxanes were also used towards functional devices such as logic gates [15] or even plasmonic actuators. [16] However, the design of unsymmetric [c2]daisy chain rotaxanes remains unexplored. In this paper, we describe the syntheses and the pH-controlled contraction/extension of unsymmetric [c2]daisy chain rotaxanes with two different electroactive stoppersan electron donor (Figure 1, blue) and an electron acceptor (red). The donor stoppers consist of triarylamine units (TAAs), [17][18][19][20][21][22][23] and the acceptor stoppers of unsymmetrical 1,7-dipyrrolidinyl-substituted perylene diimides (PDIs). [24][25][26][27][28][29] This interlocked structure presents two molecular stationsan ammonium group (orange in Figure 1) and a triazolium group (purple in Figure 1)with different affinities for the macrocycle. The mechanical actuation of the [c2]daisy chain is based on a strategy developed by Coutrot [30] and already used in our laboratory. [8][9][10]15] In the protonated state, the macrocycles reside around the ammonium stations via ion-dipole and hydrogen Scheme 2. Synthetic route towards unsymmetric perylene bisimide 7 and 10.

The reaction pathways to access (non-methylated) triazole rotaxanes 12-15 are shown in Scheme 3. In the one pot version (Scheme 3a), the bisalkynyl-substituted pseudorotaxane 11 [30,31] was reacted with azido-substituted triarylamine 2 or 3 and azido-substituted perylene 7 or 10 via [3+2] Huisgen cycloaddition using a 1:1:1 (11 / (2 or 3) / (7 or 10)) stoichiometric ratio. Unfortunately, all reaction conditions used to couple compound 7 failed probably due to the strong delocalization of electrons in the perylene ring making the azido group unreactive. The reactions involving 10 were carried out in the presence of 2,6-lutidine and tetrakisacetonitrile copper(I) hexafluorophosphate in dichloromethane at room temperature for 3-4 days. They satisfyingly gave the desired triarylamine-perylene diimide (TAA-PDI) systems 14 and 15 in 25 % and 10 % isolated yields, respectively. The difficulty of this one-pot approach to prepare unsymmetrical systems 14 and 15 relied on the purification of the crude reaction mixtures, which contained at least five different species on TLC. Several factors account for this very complex mixture: 1) the presence of stereoisomers of the [c2]daisy chain unit; [31] 2) the formation of mono-coupled cycloadducts between 10 and 11 as well as between 2 or 3 and 11; 3) the formation of bis-coupled cycloadducts between 10 and 11 as well as between 2 or 3 and 11; and 4) the presence of residual amounts of unreacted starting materials. Purification of 14 and 15 was however accomplished by conventional flash column chromatography on silica gel. First, elution with pure dichloromethane was used to remove unreacted triarylamine 2 or 3 and PDI 10. Then, a ternary eluent mixture of dichloromethane, methanol and toluene starting with a ratio of 100 : 0 : 2 and increasing polarity slowly up to 100 : 5 : 2 allowed the sequential removal of 1) [2+11+2] or [3+11+3]; [15] 2) mono-coupled cycloadducts ([2+11] or [3+11]), and 3) desired compound 14 or 15. Further elution with increasing polarity afforded PDI derivatives [10+11] and [10+11+10]. Alternatively, we envisioned a sequential approach involving first the formation of mono-coupled cycloadducts [2+11], [3+11] or [10+11] as intermediates (Scheme 3b for [10+11]). In this stepwise approach, 4.7 equivalents of 11 were first stirred with either one equivalent of 2, 3 or 10 for three days in the presence of one equivalent of tetrakisacetonitrile copper(I) hexafluorophosphate and 2,6-lutidine in dichloromethane to give the intermediates [2+11], [3+11] or [10+11] in 32 %, 19 % and 32 % isolated yields, respectively. For the synthesis of [2+11] and [3+11], the side-products resulted only in excess quantities of 11 and a small amount of [2+11+2] or [3+11+3] (see SI). For the synthesis of [10+11], bis-coupled cycloadducts [10+11+10] could not be detected by LC-MS. The second step was then tested between [10+11] and 2 which were mixed in a 1 : 1 stoichiometric ratio in dichloromethane for three days in the presence of an equimolar amount tetrakisacetonitrile copper(I) hexafluorophosphate and 2,6-lutidine. In this way, compound 15 was obtained in 30% yield after an easier chromatography column. Finally, the triazole groups in 14 and 15 were selectively methylated in the 3 position using a mixture of methyl iodide and dichloromethane over 4 days, followed by anion exchange using ammonium hexafluorophosphate to give the extended systems 16 Ext and 17 Ext in 80 % and 94 % yields, respectively (Scheme 4, top). Deprotonation of these compounds to the contracted form, 16 Cont and 17 Cont , was accomplished using a biphasic mixture of 0.1 M aqueous sodium hydroxide and dichloromethane at room temperature for 1 hour and proceeded in 95 % yield (Scheme 4, bottom). Compounds 16 Cont and 17 Cont proved less soluble in acetonitrile than their extended forms 16 Ext and 17 Ext .

Characterization of [c2] daisy chain rotaxanes 16 and 17

All compounds were characterized by 1 H and 13 C NMR and by either ESI or MALDI-TOF mass spectroscopy (see Experimental Section and SI for 1 H and 13 C NMRs of key compounds). In order to probe the contraction/extension event of the unsymmetric [c2] daisy chain rotaxanes, a precise determination of the 1 H signals of compounds 16 and 17 is necessary. This was achieved thanks to a combination of standard 1D and 2D NMR techniques on most of the synthetic intermediates and particularly on compounds 2, 3, 10, 14 and 15. For compounds 2 and 3, the 1 H NMR spectra of the aromatic region was assigned based mainly on the integration of the aromatic proton signals of the triarylamine unit and on 1 H-1 H COSY NMR (Figure 2 and SI). For both compounds, the aromatic protons of the Bn group showed a multiplet between 7.50 and 7.30 ppm while the two doublet peaks integrating for 4 each at  ~ 7.04 and 6.92 ppm were assigned to the phenyl ring B. [START_REF]From 1D and 2D NMR it was not possible determine proton B3 and B4. However, considering the electron density of nitrogen and oxygen atoms[END_REF] For compound 3, one doublet with an integral of 2 was found strongly shifted towards the low-field region of the spectrum compared to 2 and was thus assigned to proton HA1. The other doublet at  ~ 6.97 ppm in between the two HB protons was assigned to HA2. For compound 10, the use of 1 H and/or 1 H-1 H COSY NMRs on all synthetic intermediates from compound 1,7-5 was necessary to precisely determine the different proton signals (Figure 3 and SI for 1 H and COSY NMRs of compounds 5, 6, 8, 9 and 10). Below, we describe only the 1 H NMR spectrum of compound 10. Two singlets are observed for HA1 and HD1. Compared to proton A1, the chemical shift of proton D1 should not be affected during the whole synthetic sequence from 1,7-5 to 10 and thus, based on 1 H NMR spectra, the lowest field singlet ( = 8.48 ppm) was ascribed to HA1 while the highest field one ( ~ 8.39 ppm) was ascribed to HD1. Protons HB1,B2 and HC1,C2 were identified based on their multiplicity (doublet) and on COSY correlations. Aromatic ortho or bay protons were discriminated based on their proximity to the electron-withdrawing imide groups. Namely, the low-field shifted doublets ( ~ 8.41-8.36 ppm) was assigned to the ortho protons (HB1 and HC1) as they experience a stronger deshielding effect from the imide group compared to the bay protons (HB2 and HC2), which thus appear at higher field ( ~ 7.63, 7.57 ppm). Protons HB2 and HC2 (and respectively HB1 and HC1) were then identified following a similar reasoning to the one made for HA1 and HD1, i.e. HB protons are shifted to higher field compared to HC protons. Protons HE1 and HE2 were identified based mainly on their integrations compared to all protons from the perylene core. The more downfield shifted signal was attributed to protons HE2, because the HE1 ones, closer in space to the free primary amine group, might experience a greater shielding effect resulting in a lower resonance frequency. Finally, the amide proton (Ham) was clearly evidenced when comparing 1 H NMR spectra of compounds 9 and 10. In the aromatic region, this proton signal exhibits an upfield shift ( = -0.16) from 8.44 ppm in PDI 9 to 8.28 ppm in PDI 10. This observation probably results from an increased electron density on the nitrogen from the azide involved in hydrogen bonding compared to the chlorine atom, thus causing the shielding effect that makes the amide proton in 10 resonate upfield shifted compared to PDI 9. This observation is in good agreement with the upfield shift ( = -0.07 ppm; from  = 4.23 to 4.16 ppm) observed for the protons in the  position (H). Our experience with molecules based on [c2]daisy chain rotaxane 11, [8,9,15] together with the precise determination of the 1 H signals of compounds 2 and 10 and complemented by 1 H-1 H COSY NMR allowed us to assign all protons on compound 14 from their chemical shifts in CDCl3 (Figures S16 andS17). However, the solubility of the unsymmetric [c2]daisy chain rotaxane 15 proved to be quite limited in d-chloroform and thus 1 H and 1 H-1 H COSY NMR in CD3CN were used to precisely assign the different proton signals of compound 15 (Figures S19 andS20). Methylation of compounds 14 and 15 did not greatly affect their 1 H NMR spectra, which displayed mainly a downfield shift of the signals corresponding to the protons of the triazole units (from  ~ 7.50 ppm to  ~ 8.20 ppm) and the appearance of new singlets at  ~ 4.20-4.10 ppm corresponding to the methyl groups on the triazole (Figures 4a, S20a, andS21a). We then probed the pH-controlled contraction/extension for 16 and 17 by 1 H NMR. Figure 4 shows a comparison between the 1 H NMR spectra of 16 Ext and 16 Cont in CD3CN. The signals assigned to the vinylic protons of the triazolium groups (T1 and T1') and to the methylene protons in between the amide and triazolium units ( and ') show downfield shifts going from the extended state (16 Ext ) to the contracted state (16 Cont ) as a result of the shuttling motion of the macrocycles from the ammonium stations to the triazolium stations. Furthermore, the benzylic methylene protons (9 and 9') next to the ammonium station and the methylene protons (3 and 3') next to the triazolium station show up-and downfield shifts, respectively (green dashed lines). These observations are in agreement with what was observed for other molecules based on [c2] daisy chain rotaxane 11. [8,9,15,31] Similar observations were made for compounds 17 Ext and 17 Cont , regarding the vinylic protons of the triazolium groups, the methylene protons in between the amide and triazolium units, the benzylic methylene protons next to the ammonium station and the methylene protons next to the triazolium station (Figure S22). We also used 1 H DOSY NMR spectroscopy to confirm the contraction/extension event on compound 16 (Figure 5). DOSY NMR experiments provide information on the diffusion behavior of the molecules and thus on their hydrodynamic radii (RH) using the Stokes-Einstein equation which considers a hard sphere model. For the extended state, a lower diffusion coefficient is expected compared to the contracted state. DOSY NMR spectra confirmed the presence of a single diffusing species in both the contracted and extended forms. Diffusion coefficients (D) of 247 µm 2 s -1 and 343 µm 2 s -1 , which correspond to hydrodynamic radii RH of 2.47 and 1.78 nm, were measured for 16 Ext and 16 Cont , respectively. This result is consistent with the expected extended and contracted states and confirms the occurrence of the contraction/extension event upon pH modulation. This stands also for compound 17 (see Figure S3). We then studied the optical properties of the unsymmetric rotaxanes by means of UV-Vis-NIR absorption and fluorescence experiments. UV-Vis-NIR absorption spectra of compounds 16 Ext and 16 Cont , together with references 3 and 9, were recorded in chloroform, a solvent in which all components display a good solubility (Figure 6 and S1a). We found that reference compound 3 absorbs light below 400 nm while perylene 9 displays a maximum of absorption at 705 nm with a small shoulder at 645 nm, corresponding to the S0-S1 electronic transition, along with a higher energy S0-S2 electronic transition at 436 nm, in agreement with the literature. [24] Overall, we found that the absorption spectra of 16 Cont correspond well to the sum of its reference components in the contracted state. However, for compound 16 Ext , a bathochromic shift of the absorbance corresponding to the S0-S1 electronic transition (725 nm) was observed (Figure 6a). Compared to reference perylene 9 and compound 16 Cont , such behavior can suggest the formation of J-aggregates in the extended state. However, for compounds 17 Ext and 17 Cont in benzonitrile, the actuation did not lead to significant differences by UV-Vis spectroscopy (Figure S1b). Steady-state fluorescence experiments on compounds 9, 16 Ext and 16 Cont were also recorded in chloroform after excitation at 440 nm (Figure 6 andS1c). The fluorescence spectrum of extended compound 16 Ext shows a maximum of emission at 773 nm, which is red-shifted by 39 nm compared to the fluorescence spectrum of reference perylene compound 9. Interestingly, for contracted compound 16 Cont , the maximum of emission was located at around 753 nm, which is blue-shifted by 20 nm compared to its extended counterpart 16 Ext . Furthermore, we can note that the intensity of emission is enhanced for the contracted rotaxane compared to the extended one. These observations are in agreement with the steady-state absorption studies and suggest that, in the extended state, aggregation of the perylene units occurs while in the contracted one, the presence of the crown ether macrocycle in the vicinity of the perylene unit prohibits this phenomenon. Finally, the electrochemical behavior of compounds 16 and 17 including the determination of reduction and oxidation potentials to define their energy levels and electronic properties was probed by cyclic voltammetry in 0.1 M NBu4PF6/CH2Cl2 at a gold disk electrode under planar diffusion conditions (Figures 7 andS2). Cyclic voltammograms of 16 Ext (Figure 7a, black trace) show a chemical reversible reduction process with a mid-point potential E1/2,c centered at -1.30 V vs. Fc/Fc + (Table 1) and two chemically reversible oxidation processes with mid-point potentials (E1/2,a) of +0.19 V and +0.31 V vs. Fc/Fc + . The higher peak amplitude of the first oxidation process indicates an overlap of the oxidation of the TAA unit and the first oxidation of the perylene unit. This becomes obvious by comparing the redox potential of the isolated perylene (S2) and triarylamine (S4) reference compounds (Figure S2) that reveal mid-point potentials for the oxidation of +0.17 and +0.30 V (both S2) and +0.17 V (S4). Due to the blocking of the para-positions in the triarylamine stopper no follow-up reactions such as dimerization occur. [START_REF] Blanchard | [END_REF]37] Moreover, the pronounced current amplitude of the reduction of the perylene unit in the reference compound S2 that is about twice the current amplitude of the first oxidation process in S2 (Figure S2a) strongly suggests a two-electron transfer mechanism for the cathodic process. Evidently, the cyclic voltammetric response of the extended form 16 Ext reflects the voltammetric signatures of the individual reference compounds S2 and S4 and significant effects of the macrocycle on the electrochemical response of the electroactive stoppers are absent. Consequently, the donor and acceptor site can be regarded as isolated redox sites within the daisy chain rotaxane in 16 ext . On the other hand, the contracted form 16 Cont (Figure 7a, red trace) shows a rather broad reduction wave of low chemical reversibility and only a rather broad chemically reversible oxidation wave. The crown ether-based macrocycle seems to significantly alter the electrochemical response of the redox active stoppers in the contracted conformation. In case of the reduction the closer proximity of the crown ether moiety seems to have a destabilizing effect on the reduced perylene stopper form. This is seen from the weak re-oxidation peak in the backward scan of the reduction suggesting lower chemical stability. Comparing the oxidation waves of the extended and contracted forms one sees that the potential of the second oxidation of the perylene unit is shifted to lower potentials which results in a stronger superposition with the first oxidation of the perylene and the TAA stopper. A lower potential for this second oxidation does also indicate easier oxidation in the contracted form compared to the extended one.

The electrochemical response of 17 Ext (Figure 7b, black trace) and 17 Cont (red trace) shows a similar behavior. However, the effect of the crown ether moiety seems to be less pronounced. Cyclic voltammograms of 17 Ext show a single chemically reversible reduction process located at -1.33 V vs. Fc/Fc + (Table 1) in analogy to 16 Ext . However, in the contracted form, a peak splitting was observed that might be attributed to an interaction of the macrocycle with the singly reduced perylene unit. In contrast to 16 Ext , here, the macrocycle seems to have a stabilizing effect on the radical anion and shifts the potential of the second reduction to more negative values (-1.45 V vs. Fc/Fc + ). A similar effect on the potential was observed for the oxidation processes. The mid-point potentials of the oxidation processes (stepwise oxidation of the perylene unit and 1e -oxidation of the triarylamine unit) are shifted to more positive values when going from 17 Ext (+0.13 V, +0.27 V and +0.36 V) to 17 Cont (0.18 V, 0.31 V and 0.41 V). Comparison with the model compound S3 (isolated triarylamine unit, E1/2,a = +0.3 V, Figure S2) suggests that the second oxidation process is most likely located at the triarylamine based stopper. Although not fully deciphered, the difference in the electrochemical behavior between the extended and contracted forms can be attributed to the shuttling of the crown ether in the vicinity of the redox unit and thus: i) change the solvation shell around it; ii) induce van der Waals or H-bonds interactions; and iii) have a general shielding effect on the structural rearrangement upon electron transfer. All these effects will lead to a change in the reorganization energy upon reduction / oxidation and can also change the kinetics of the electron transfer interactions which can induce peak splitting, as for instance observed for the reduction of 17 Cont . The differences in the electrochemical response between the extended and contacted forms should in principle allow for an electrochemical identification of the individual species under acidic or basic conditions and shows the potential applicability of this system in e.g. pH sensitive sensor devices based on an electrochemical readout. It should be here mentioned that such an altered electrochemical response of redox active rotaxane upon protonation/deprotonation was recently reported for a [2]rotaxane equipped with redox active viologen units. [START_REF] Ragazzon | Proc. Natl. Acad. Sci[END_REF] From the estimated mid-point potentials, the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) energy levels (EHOMO and ELUMO) were calculated for the electron donor-acceptor systems 16 and 17 (Table 1, for details on the calculation see Supporting Information). The lowest HOMO value was observed for 16 Ext (5.28 eV). The highest LUMO level was found for 17 (both forms, 3.77 eV). The electrochemical band gaps, determined from the differences in energy levels, ranged from 1.46 to 1.51 eV. These values are even above the values for TAA-fullerenes dyads and triads (ECV, gap = 1.31 -1.34 eV) and demonstrate the strong electron affinity of the perylene unit. [32] Table 1. Mid-point potentials of the oxidation (E 1/2,a ) and reduction processes (E 1/2,c ), energy levels (E HOMO and E LUMO ) and electrochemical band gap (E CV, gap ) values for compounds 16 and 17 determined in 0.1 M NBu 4 PF 6 /CH 2 Cl 2 at a planar Au disk electrode with a nominal diameter of 3 mm. All potentials were given vs. the potential of the Fc/Fc + redox couple. E1/2,a (V) [a] E1/2,c (V) EHOMO (eV) [b] ELUMO (eV) [b] Egap (eV) [

Conclusions

To summarize, we have designed and synthesized two unsymmetrical pH-switchable [c2]daisy chain rotaxanes which combine electroactive triarylamine and perylene units as stoppers. Using 1D and 2D NMR spectroscopies, we have demonstrated the efficiency of the shuttling process upon stepwise addition of base and acid. This advanced synthetic example shows that complex [c2]daisy chains bearing distinct substituents can enrich the number of chemical structures within that family of compounds. In addition, cyclic voltammetry experiments show that the electrochemical properties of such [c2] daisy chain rotaxanes can be altered by the shuttling event occurring upon pH modulation. It can be anticipated that the electrochemical discrimination -although not pronounced but clearly visible -of the two conformations allow for the use of such supramolecular structures in pH sensitive electrochemical sensor devices. Moreover, the unsymmetrical nature of the redox active daisy chain rotaxanes based on the integration of an electron donor (triarylamine) and an electron acceptor (perylene) unit allows for the selective electrochemical triggering of only a single stopper unit and may thus be of particular importance for the fabrication of complex electrochemically active molecular machines.

Experimental Section

General methods. All reactions were performed under an atmosphere of argon unless otherwise indicated. All reagents and solvents were purchased at the highest commercial quality and used without further purification unless otherwise noted. Dry solvents were obtained using a double column SolvTech purification system. Yields refer to purified spectroscopically ( 1 H NMR) homogeneous materials. Thin Layer Chromatographies were performed with TLC silica on aluminium foils (Silica Gel/UV254, Aldrich). In most cases, irradiation using a Bioblock VL-4C UV-Lamp (6 W, 254 nm and/or 365 nm) as well as p-anisaldehyde, phosphomolybdic acid and Cerium ammonium molybdate stainings were used for visualization. Ultra Performance Liquid Chromatographies coupled to Mass Spectroscopy (UPLC-MS) were carried out on a Waters Acquity UPLC-SQD apparatus equipped with a PDA detector (190-500 nm, 80Hz), using a reverse phase column (Waters, BEH C18 1.7 µm, 2.1mm x 50 mm), and the MassLynx 4.1 -XP software. MALDI mass spectra were recorded on a Bruker Daltonics Autoflex II TOF spectrometer. High resolution mass spectra were recorded on a Micro-Q-TOF apparatus from Bruker. 1 H NMR spectra were recorded on a Bruker Avance 400 spectrometer at 400 MHz and 13 C spectra at 100 MHz in CDCl 3 or CD 3 CN at 25°C. The spectra were internally referenced to the residual proton solvent signal (CDCl 3 : 7.26 ppm and CD 3 CN: 1.94 ppm for 1 H spectrum, and CDCl 3 : 77.16 ppm and CD 3 CN: 118.26 and 1.32 ppm for 13 C spectrum). For 1 H NMR assignments, the chemical shifts are given in ppm. Coupling constants J are listed in Hz. The following notation is used for the 1 H NMR spectral splitting patterns: singlet (s), doublet (d), triplet (t), multiplet (m), large (l). Diffusion Order Spectroscopy (DOSY) NMR spectra were recorded on a Bruker Avance I spectrometer at 500 MHz at the Service de Résonance Magnétique Nucléaire at the Faculty of Chemistry at the University of Strasbourg. UV-Vis-NIR spectra were recorded on either Cary 500 Scan Varian or Perkin Elmer -Lambda 25 UV-Vis-NIR spectrophotometers in a quartz glass cuvette with 1.0 cm optical path. Fluorescence emission spectra were recorded on a FluoroMax-4 (Horiba JobinYvon) spectrofluorometer with the following settings: slit width = 5 nm, increment = 1 nm, integration time = 0.1 s in a quartz glass cuvette with a 1.0 cm optical path. Cyclic Voltammetry experiments were recorded according to the protocol reported in the Supporting Information.

Compound 2: To a stirred, sticky suspension of 1 (188 mg, 398 mol) in MeOH (5 mL) was added slowly a 1 M HCl solution (6 mL) followed by two drops of conc. HCl. The mixture was then cooled down to 0°C. After 15 min at this temperature, NaNO 2 (129 mg, 1.87 mmol) in water (0.8 mL) was added dropwise within about 2 min. The yellow-orange suspension immediately turned brown after the first drop. The mixture was further stirred for 35 min. Then, NaN 3 (259 mg, 3.98 mmol) in water (2.6 mL) was added in two portions. The brown suspension was further stirred for 40 min, while allowing the mixture to warm up to room temperature. Dichloromethane (90 mL) was added and the mixture was washed with water (90 mL). The organic phase was washed with sat. NaHCO 3 (25 mL) followed by brine (25 mL), dried over Na 2 SO 4 and concentrated under reduced pressure. The oily brown crude product was purified by flash column chromatography (SiO 2 , dichloro-methane/cyclohexane: 1/1) to afford compound 2 (131 mg, 66 % over two steps) as a brown waxy solid. R f (dichloromethane/cyclohexane: 1/1) = 0. H 2 SO 4 (200 mL) and the solution was heated at 55°C for 19 hours. I 2 (330 mg, 1.30 mmol) was then added to the reaction mixture which was stirred for 6 hours. Br 2 (3.88 mL, 75.3 mmol) was added over 45 min and the mixture was heated at 85°C for 16 hours. After cooling down to room temperature, the reaction mixture was carefully poured into H 2 O (1 L). The precipitate was filtered and washed with copious amounts of H 2 O, followed by MeOH to afford a crude mixture of 1,6-, 1,7-dibromo and 1,6,7-tribromo isomers as a dark red solid (18.4 g). This crude mixture was used without further purification. 1 H NMR analysis in 96-98 % D 2 SO 4 in D 2 O revealed a ratio of about 24:70:6 of 1,6-, 1,7-and 1,6,7-isomers, respectively. Acetic acid (20.0 mL, 349 mmol) was added over a suspension of the mixture of 1,6-, 1,7-dibromo and 1,6,7-tribromo isomers (4.00 g, 7.27 mmol) in N-methylpyrrolidinone (100 mL) under an argon atmosphere. The mixture was stirred at 60°C for 20 minutes. N-Dodecylamine (3.37 g, 18.2 mmol) was then added and the mixture was stirred at 120°C for 12 hours. The resulting dark-red suspension was then poured into water (1 L) to give a precipitate, which was filtered off and washed with MeOH (1 L). The crude product was purified by flash column chromatography (SiO 2 , cyclohexane/ethyl acetate: 9/1) to provide a 1:4 regioisomeric mixture of N,N'-didodecyl-

Figure 1 .

 1 Figure 1. (a) General representation of the mechanical actuation of an unsymmetric [c2]daisy chain rotaxane incorporating electron donor (blue) and electron acceptor (red) stoppers; (b) Molecular structures of the pH-switchable [c2]daisy chain rotaxanes 16 Ext-Cont and 17 Ext-Cont .

Scheme 3 .Scheme 4 .

 34 Scheme 3. a) One-pot synthetic route towards unsymmetrical [c2] daisy chain rotaxanes 12-15; b) Sequential synthetic route towards unsymmetrical [c2] daisy chain rotaxane 15.

Figure 2 .

 2 Figure 2. Partial 1 H NMR spectra (400 MHz, CDCl 3 , 25 °C) of compounds 2 (a) and 3 (b) with chemical shift assignments.

Figure 3 .

 3 Figure 3. Partial 1 H NMR spectra (400 MHz, CDCl 3 , 25 °C) of compound 10 with chemical shift assignments a) in the low-field region from 8.6 to 7.0 ppm and b) in the high-field region from 4.5 to 0.5 ppm.

Figure 4 . 1 H

 41 Figure 4. 1 H NMR spectra (400 MHz, CD 3 CN, 25 °C) of compounds a) 16 Ext and b) 16 Cont showing different chemical shifts between extended and contracted states. The purple, blue and green dashed lines indicate important shifts of protons influenced by the shuttling of the macrocycles.

Figure 5 .

 5 Figure 5. Partial DOSY NMR spectra of compounds 16 Ext and 16 Cont (400 MHz, CD 3 CN, 25 °C) showing differences in the diffusion coefficient between extended and contracted states.

Figure 6 .

 6 Figure 6. Molar absorbance coefficient (plain line) and fluorescence spectra (dashed line) upon excitation at 440 nm of rotaxanes 16 Ext (red) and 16 Cont (black) recorded in chloroform at a concentration of 0.1 mM.

Figure 7 .

 7 Figure 7. Cyclic voltammograms of compounds 16 (a) and 17 (b) in 0.1 M NBu 4 PF 6 /CH 2 Cl 2 at a gold disk electrode (planar diffusion conditions). a) 16 Ext (c = 0.74 mM), 16 Cont (c = 0.77 mM); b) 17 Ext (c = 0.25 mM), 17 Cont (c = 0.28 mM); all voltammograms were recorded with a scan rate of 100 mV.s -1 and were not background corrected; arrows indicate initial scan direction.

[ a ]

 a Values in parentheses correspond to the mid-point potentials of the second and third oxidation. [b] E HOMO (=-(E 1/2,a +5.1)[eV]) and E LUMO (=-(E 1/2,c +5.1)[eV]) values were calculated from the mid-point potentials of the first oxidation or reduction process, respectively; the error in potential values is estimated to be ≤ ± 0.01 V. [c] E gap calculated as the difference between the HOMO and LUMO level.

  , 123.9, 122.2, 121.9, 120.8, 119.2, 118.2, 52.3, 40.7, 32.1, 29.8 (3C),29.6, 29.5, 28.4, 27.4, 25.9, 22.8, 14.3; HRMS: m/z calcd for C 56 H 72 N 4 O 4 : 864.5523 [M+H] + , found 864.5554. A solution of diimide 1,7-5 (420 mg, 485 mol) in tert-butanol (75 mL) was heated up to reflux under an argon atmosphere. KOH (1.91 g, 34.0 mmol) suspended in hot tert-butanol (38 mL) was then added, whereas the dark-green reaction mixture turned blueish green. After heating for 10 minutes, the hot mixture was carefully poured into acetic acid (50 mL) and the mixture turned back dark-green. The reaction mixture was cooled to room temperature and dichloromethane (33 mL) was added. The mixture was washed with H 2 O (3  80 mL) and the aqueous phases were extracted with dichloromethane (100 mL). The combined organic phases were dried over Na 2 SO 4 . The solvent was removed under reduced pressure and the resulting crude product was purified by flash column chromatography (SiO 2 , chloroform/acetone/n-hexane: 10/1/9) to give monoimide 6 (214 mg, 63 %) as a dark-green solid. R f A solution of compound 6 (30 mg, 0.22 mmol) in DMF (1 mL) was added over a solution of 4-aminophenylazide (31 mg, 0.05 mmol) and imidazole (171 mg, 4.15 in toluene (5 mL), and the resulting mixture was refluxed for 12 h. After cooling to room temperature, chloroform (30 mL) was added. The solution was washed successively with HCl (1%) (20 mL), K 2 CO 3 sat. (20 mL), water (20 mL) and the organic phase was concentrated under reduced pressure. The crude product was purified by column chromatography (SiO 2 , cyclohexane/EtOAc: 1/1) to afford pure compound 7 (31 mg, 85%) as a green solid. R f (cyclohexane/EtOAc: 1/1) = 0.20; 1 H NMR (CDCl 3 , 400 MHz, 25°C): δ = 8.50 (s, 1H), 8.46 (s, 1H), 8.44 (d, 3 J = 8.1 Hz, 1H), 8.41 (d, 3 J = 8.1 Hz, 1H), 7.69 (d, 3 J = 8.0 Hz, 1H), 7.68 (d, 3 J = 8.0 Hz, 1H),7.13 (d, 3 J = 8.5 Hz, 2H), 6.84 (d, 3 J = 8.5 Hz, 2H), 4.22 (t, 3 J = 7.5 Hz, 2H), 3.82-3.65 (brm, 4H), 2.91-2.75 Na 2 CO 3 (40 mL). The organic phase was dried over Na 2 SO 4 and concentrated under reduced pressure. The crude product was purified by flash column chromatography (SiO 2 , dichloromethane → dichloromethane/MeOH: 99.5/0.5 → 99/1) to give unsymmetrical diimide 7 (208 mg, quantitative) as a dark-green solid. R f (dichloromethane/methanol: 99/1) = 0.19; 1 H NMR (CDCl 3 , 400 MHz, 25°C):  = 8.47 (s, 1H), 8.41 (s, 1H), 6.39 (d, 3 J = 8.5 Hz, 1H), 6.37 (d, 3 J = 8.2 Hz, 1H), 7.62 (d, 3 J = 8.3 Hz, 1H), 7.60 (d, 3 J = 8.8 Hz, 1H), 7.15 (d, 3 J = 8.4 Hz, 2H), 6.87 (d, 3 J = 8.4 Hz, 2H), 4.22 (t, 3 J = 7.2 Hz, 2H), 3.90-3.50 (brm, 4H), 2.95-2.60 (brm, 4H), 2.20-1.85 (brm, 8H), 1.80-1.69 (m, 2H), 1.52-1.19 (m, 18H), 0.87 (t, 3 J = 6.8 Hz, 3H); 13 C NMR (CDCl 3 , 100 MHz, 25°C):  = 164.8, 164.7, 164.3, 164.2, 146.7, 146.6, 146.4, 134.6, 134.3, 130.2, 130.1, 129.5, 127.1, 126.8, 126.6, 124.1, 123.9, 122.7, 122.2, 122.1, 121.9, 121.2, 120.9, 119.4, 119.3, 118.5, 118.1, 116.1, 52.3 (2C), 32.1, 29.8, 29.8 (2C), 29.7, 29.6, 29.5, 28.4, 27.4, 26.0, 22.8, 14.3; HRMS: m/z calcd for C 50 H 53 N 5 O 4 : 788.4181 [M+H] + , found 788.4170.Compound 9:A solution of diimide 7 (208 mg, 264 µmol) and triethylamine (36.8 L, 264 mol) in dry dichloromethane (7 mL) was added slowly over a solution of 2-chloroacetyl chloride (21.0 µL, 264 µmol) in dry dichloromethane (1.6 mL) at 0°C under an argon atmosphere. The reaction mixture was stirred for 2 hours at room temperature and then further dissolved in Et 2 O (40 mL). The organic phase was extracted with sat. NH 4 Cl (20 mL), brine (20 mL) and dried over Na 2 SO 4 . After concentration under reduced pressure, the crude product was purified by a short column (SiO 2 , ethyl acetate/n-hexane: 1/1) to afford compound 8 (203 mg, 89 %) as a dark-green solid, which was pure enough to be used as such in the next step. R f (dichloro-methane/methanol: 99/1) = 0.18; 1 H NMR (CDCl 3 , 400 MHz, 25°C):  = 8.51 (s, 1H), 8.44 (s, 1H), 8.37-8.31 (m, 3H), 7.76 (d, 3 J = 8.7 Hz, 2H,), 7.55 (d, 3 J = 8.0 Hz, 1H), 7.49 (d, 3 J = 8.0 Hz, 1H), 7.41 (d, 3 J = 8.7 Hz, 2H), 4.23-4.18 (m, 4H), 3.80-3.56 (brm, 4H), 2.89-2.60 (brm, 4H), 2.14-1.86 (brm, 8H), 1.81-1.69 (m, 2H), 1.50-1.15 (brm, 18H), 0.87 (t, 3 J = 6.8 Hz, 3H); 13 C NMR (CDCl 3 , 100 MHz, 25°C):  = 171.1, 164.6, 164.5, 164.3, 164.3, 146.8, 146.6, 137.1, 134.9, 134.2, 132.5, 130.2, 130.1, 129.7, 127.3, 126.8, 124.2, 123.9, 122.7, 122.2, 122.0, 121.7, 121.3, 121.0, 119.3, 118.9, 118.7, 117.9, 52.4, 52.4, 43.1, 40.7, 32.1, 29.8, 29.8, 29.7, 29.6, 29.5, 28.4, 27.3, 25.9, 22.8, 14.2; HRMS: m/z calcd for C 52 H 54 ClN 5 O 5 : 863.3813 [M] + , found 863.3795.Compound 10: A solution of compound 9 (203 mg, 235 µmol) and NaN 3 (214 mg, 3.29 mmol) in dry N,N'-dimethylformamide (8 mL) was heated at 80°C for 25 hours. After cooling down to room temperature, chloroform (65 mL) was added and the mixture was extracted with H 2 O (3  45 mL). The combined aqueous phases were extracted with chloroform (3  45 mL). The combined organic phases were dried over Na 2 SO 4 and concentrated under reduced pressure. The resulting crude product was purified by flash column chromatography (SiO 2 , cyclohexane/ethyl acetate: 1/1) to give azide 10 (143 mg, 70 %) as a dark-green solid. R f (cyclohexane/ethyl acetate: 1/1) = 0.37;

	126.7Compound 6: (chloroform/acetone/n-hexane: 10/1/9) = 0.49; 1 H NMR (CDCl 3 , 400 MHz, 25°C):  = 8.51 (s, 1H), 8.45 (s, 1H), 8.44 (d, 3 J = 8.2 Hz,
	1H), 8.41 (d, 3 J = 8.0 Hz, 1H), 7.74 (d, 3 J = 8.0 Hz, 1H), 7.59 (d, 3 J = 8.0 Hz, 1H), 4.22 (t, 3 J = 7.2 Hz, 2H), 3.90-3.60 (brm, 4H), 3.00-
	2.70 (brm, 4H), 2.20-1.90 (brm, 8H), 1.82-1.68 (m, 2H), 1.50-1.14 (brm, 18H), 0.87 (t, 3 J = 6.8 Hz, 3H); 13 C NMR (CDCl 3 , 100 MHz,
	25°C):  = 164.2, 164.1, 161.5, 161.0, 147.2, 146.5, 136.1, 133.9, 130.6, 130.4, 128.9, 126.7, 124.9, 124.5, 123.9, 122.7, 122.6, 122.3,
	121.1, 120.1, 119.8, 117.5, 117.2, 114.6, 52.7, 52.5, 40.8, 32.1, 31.1, 29.9, 29.8, 29.7, 29.6, 29.5, 28.4, 27.3, 26.0 (2C), 22.8, 14.3;
	HRMS: m/z calcd for C 44 H 47 N 3 O 5 : 697.3516 [M] + , found 697.3524.
	Compound 7:
	1,6-and 1,7-dibromoperylene-3,4,9,10-tetracarboxylic acid diimides 1,6:1,7-4 (3.10 g, 48 % over two steps). R f (cyclohexane/ethyl
	acetate: 9/1) = 0.23 (1,6-4), 0.18 (1,7-4); 1 H NMR (CDCl 3 , 400 MHz, 25°C) of major 1,7 isomer:  = 9.45 (d, 3 J = 8.2 Hz, 2H), 8.89 (s,
	2H), 8.67 (d, 3 J = 8.2 Hz, 2H), 4.20 (t, 3 J = 7.6 Hz, 4H), 1.79-1.70 (brm, 4H), 1.46-1.25 (brm, 36H), 0.87 (t, 3 J = 6.8 Hz,6H); 13 C NMR
	(CDCl 3 , 100 MHz, 25 °C) of major 1,7 isomer:  = 163.0, 162.5, 138.1, 133.1, 132.9, 130.1, 129.4, 128.6, 127.1, 123.3, 122.9, 120.9,
	41.0, 32.1 (2C), 29.8, 29.8, 29.8, 29.7, 29.5, 28.2, 27.3, 22.8, 14.3; MALDI-TOF: m/z calcd for C 48 H 56 Br 2 N 2 O 4 : 885.267 [M+H] + , found
	885.685.
	Compound 1,6-5 and 1,7-5: A solution of a 1:4 regioisomeric mixture of 1,6:1,7-4 (1.30 g, 1.47 mmol) in pyrrolidine (85 mL) was heated
	at 55°C for 26 hours under an argon atmosphere. The solvent was then evaporated under reduced pressure and the crude product
	was purified by flash column chromatography (SiO 2 , dichloromethane/cyclohexane: 2/1 → 4/1) to give compound 1,6-5 (170 mg, 13 %)
	as a dark-blue solid and compound 1,7-5 (481 mg, 38 %) as a dark-green solid.
	Compound 1,6-5: R f (dichloromethane/cyclohexane: 4:1) = 0.28 (1,6-5); 1 H NMR (CDCl 3 , 400 MHz, 25°C):  = 8.59 (d, 3 J = 8.0 Hz,
	2H), 8.24 (s, 2H), 7.78 (d, 3 J = 8.0 Hz, 2H), 4.21 (t, 3 J = 7.6 Hz, 2H), 4.14 (t, 3 J = 7.5 Hz, 2H), 3.80-3.46 (brm, 4H), 2.80-2.50 (brm,
	4H), 2.20-1.82 (brm, 8H), 1.82-1.54 (brm, 4H), 1.50-1.10 (brm, 36H), 0.88-0.82 (m, 6H); 13 C NMR (CDCl 3 , 100 MHz, 25°C):  = 164.3,
	164.1, 149.9, 135.7, 131.1, 130.2, 128.4, 128.3, 123.3, 122.9, 117.9, 117.6, 117.0, 116.9, 52.2, 40.7, 40.5, 32.0, 30.3, 29.7 (3C), 29.6,
	29.5, 29.4, 28.4, 28.3, 27.4, 27.3, 27.0, 25.7, 22.8, 14.2; ESI-MS: m/z calcd for C 56 H 72 N 4 O 4 : 882.56 [M+H 2 O] + , found 882.58;
	Compound 1,7-5: R f (dichloromethane/cyclohexane: 4:1) = 0.22 (1,7-5); 1 H NMR (CDCl 3 , 400 MHz, 25°C):  = 8.39 (s, 2H), 8.33 (d, 3 J
	= 8.0 Hz, 2H), 7.56 (d, 3 J = 8.0 Hz, 2H), 4.22 (t, 3 J = 7.6 Hz, 4H), 3.82-3.50 (brm, 4H), 2.90-2.65 (brm, 4H), 2.15-1.85 (brm, 8H), 1.83-
	1.58 (brm, 4H), 1.52-1.10 (brm, 36H), 0.87 (t, 3 J = 6.9 Hz, 6H); 13 C NMR (CDCl 3 , 100 MHz, 25°C):  = 164.2, 146.6, 134.3, 130.0,

(brm, 4H), 2.16-1.90 (brm, 8H), 1.81-1.70 (brm, 2H), 1.40-1.19 (brm, 18H), 0.87 (t, 3 J = 6.8 Hz, 3H);

13

C NMR (CDCl 3 , 100 MHz, 25°C): δ = 164.8, 164.7, 164.2, 164.2, 146.8, 146.6, 146.6, 134.5, 134.3, 130.2, 130.1, 129.5, 127.0, 126.8, 126.3, 124.1, 123.8, 122.6, 122.2, 122.1, 121.8, 121.2, 120.8, 119.4, 119.2, 118.4, 118.1, 115.8, 52.3, 40.7, 32.1, 29.8, 29.8, 29.7, 29.6, 29.5, 28.4, 27.3, 27.2, 25.9, 22.8, 14.3.

Compound 8:

A solution of p-phenylene diamine (293 mg, 2.71 mmol) in dry N,N'-dimethylformamide (6 mL) was added over a solution of monoimide 6 (189 mg, 271 mol) and imidazole (313 mg, 4.60 mmol) in toluene

(35 mL)

under an argon atmosphere. The mixture was heated up to 150°C for 14 hours. After cooling down to room temperature, chloroform (65 mL) was added. The solution was washed successively with 1 % HCl (40 mL) and sat. 1 H NMR (CDCl 3 , 400 MHz, 25°C): δ = 8.45 (s, 1H), 8.40 (s, 1H), 8.33 (d, 3 J = 8.1 Hz, 1H), 8.29 (d, 3 J = 8.4 Hz, 1H), 8.28 (s, 1H), 7.71

Acknowledgements

This work was performed within the framework of the IRTG "Soft Matter Science: Concepts for the Design of Functional Materials" (A.W., S. L. and N.G.). We wish to thank the international center for Frontier Research in Chemistry (fellowship to A.W.) and the Laboratory of Excellence for Complex System Chemistry. This work was also supported by the Agence Nationale pour la Recherche (ANR-09-BLAN-034-02), the French Ministry of Research and the China Scholarship Council (fellowships to J.J.C., F.N. and G. D. respectively).


Mohamed Yagoubi
email: mohamed.yagoubi@imt-atlantique.fr

Redouane Chaibi

Reflection-based technique for control synthesis under information structure constraints with guaranteed convergence

Keywords: Nearest matrix problems, Hurwitz-stable matrices set, Douglas-Rachford reflection method, output feedback control, information structure constraints

This paper deals with stabilization, H2 and H∞ static feedback synthesis, under information structure constraints. The proposed approach relies on a prominent property of Hurwitz-stable matrices set and a Douglas-Rachford (DR) reflection method type for finding a point in the intersection of two closed sets. The method proposed in this paper emulates the genericity of the Linear Matrix Inequalities framework while keeping the feedback gain separated from any matrix with direct or indirect connection to the Lyapunov function. The link between the sequence of iterates generated by the proposed algorithm and a continuous dynamical system results in the use of Lyapunov stability theory for guaranteeing the convergence. Several examples are given to prove the validity of the proposed method.

Introduction

Several control problems can be recast as an optimization of a linear criteria under one or several LMI constraints [START_REF] Boyd | Linear matrix inequalities in systems and control theory[END_REF][START_REF] Vanantwerp | A tutorial on linear and bilinear matrix inequalities[END_REF]. The solution of the control problem is then derived efficiently using Semi-Definite Programming (SDP). However, not all control problems can be cast as convex optimization ones, especially when it comes to a control problem under information structure constraints [START_REF] Rubio-Massegu | Static output-feedback control under information structure constraints[END_REF] or a control problem involving various specifications such as the mixed H 2 /H ∞ control case [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF]. The rise of LMIs since later 1990s have led to different strategies relaxing these problems. Unfortunately they were not suited for medium or large size problems. Due to the presence of Lyapunov variables, whose number grows quadratically with the system size, LMI programs (and hence BMI ones) [START_REF] Vanantwerp | A tutorial on linear and bilinear matrix inequalities[END_REF] quickly lead to problem sizes where existing numerical algorithms (interior-point algorithm for convex cases, for instance, or cutting-plane algorithm for bilinear programs) fail [START_REF] Lewis | Nonsmooth optimization and robust control[END_REF]. These considerations are true in the case of state feedback and are even more exacerbated in the case of static (with or without information constraints) output feedback design (see the survey paper [START_REF] Sadabadi | From static output feedback to structured robust static output feedback: A survey[END_REF] and references therein).

In that context, it is still of interest to pave the way for an alternative solution that can be numerically competitive to existing methods and takes into account different control objectives as simply as for the Lyapunov/LMI paradigm. It must also keep a separation between the output (or state) feedback gain and any matrix with direct or indirect connection to the Lyapunov function.

The original approach, proposed in this paper, relies among others on Douglas-Rachford method which is a splitting method frequently employed for finding zeros of sums of maximally monotone operators. The success of the DR method (see the survey paper [START_REF] Lindstrom | Survey: Sixty years of Douglas-Rachford[END_REF] and references therein) in the context of closed, convex, non-empty sets is well-known and understood from a theoretical view [START_REF] Bauschke | On the Douglas-Rachford algorithm[END_REF][START_REF] Bauschke | The rate of linear convergence of the Douglas-Rachford algorithm for subspaces is the cosine of the Friedrichs angle[END_REF][START_REF] Bauschke | Finding best approximation pairs relative to two closed convex sets in Hilbert spaces[END_REF]. When one of the sets is non-convex the performance of such algorithm is still impressive [START_REF] Borwein | The Douglas-Rachford algorithm in the absence of convexity, Fixed-Point Algorithms for Inverse Problems in Science and Engineering[END_REF]. In particular, the proposed approach brings together wisely a prominent property of Hurwitz-stable matrices set and the DR iterates to propose a method for stabilization, H 2 and H ∞ control synthesis, under information structure constraints. Besides, the proposed iterative method has a guaranteed convergence. The connection between the continuous limit sequence of the iterates and the trajectory of a continuous dynamical system results in the use of the direct method of Lyapunov stability theory to guarantee the local convergence.

The paper is organized as follows. Section 2 recalls some exiting and also introduces some new nearest matrix problems that are intensively used in the sequel. Section 3 proposes a new Douglas Rachford's like algorithm for stabilization, H 2 and H ∞ control synthesis, under information structure constraints. Section 4 exhibits the convergence analysis and rate of the proposed iterative scheme. Section 5 illustrates the validity and efficiency of the proposed method through several numerical examples before concluding in section 6.

Notations : Hereafter, I n designates the identity matrix of dimensions n × n and vec(Z) denotes the vec-operator that stacks the columns of Z into a vector. Z denotes the transpose of a given real matrix Z and Z † denotes its Moore-Penrose pseudoinverse matrix. Z = ( i,j Z 2 ij )

1 2 designate the Frobenius norm associated to the inner product < Z 1 , Z 2 >= trace(Z 1 Z 2 ). Z > 0 (Z ≥) means that Z is a positivedefinite (positive semi-definite) matrix. In the same vein, Z < 0 (Z ≤) means that Z is a negative-definite (negative semi-definite) matrix. The set of positive semi-definite (definite) matrices is denoted by S + (S + * ). In the same way, S - g (S - * g ) denotes the set of matrices with a negative semi-definite (definite) symmetric part. For a symmetric matrix represented block-wise, off-diagonal blocks are abbreviated with •. Finally, without special stated, all matrices used in this paper are supposed to have appropriate dimensions.

On some nearest matrix problems

In this section, some preliminary definitions and results that will be used intensively in the sequel are presented. Hereafter, the Frobenius norm is chosen as a metric for its sub-multiplicative and invariance under rotations properties.

Definition 2.1 (Nearest matrix problem). A nearest matrix problem is defined as finding the nearest matrix X F ∈ D m×n to a given matrix Z ∈ R m×n under Frobenius 2 norm :

X F = arg min X∈D Z -X (1)
The solution of such problem depends entirely on the set D since Z -X is a smooth function of X.

Nearest matrix problem under LME constraint

Let us consider the matrix set defined by :

D L {X ∈ R n×n /∃ X ∈ R m×p : X = U 0 + U 1 XU 2 } with U i , i ∈ {0,
X F = arg min X∈DL Z -X (2)
Or equivalently,

X F = arg min X∈R m×p Z -(U 0 + U 1 XU 2 ) .
This problem is convex, since it can be recast as a linear constrained Quadratic Program (QP), hence it can be solved efficiently. In fact, using the vec operator notations:

x v vec(X), z v vec(Z)
one can easily establish the equality given below :

Z -X 2 = z v z v -2z v x v + x v x v
which is quadratic in x v . Moreover, according to [START_REF] Jovovic | A note on solutions of the matrix equation AXB = C[END_REF], the LME :

X = U 0 + U 1 XU 2 is consistent if and only if, for some pseudo-inverses U 1 † , U 2 † verifying U 1 U 1 † U 1 = U 1 † and U 2 U 2 † U 2 = U 2 †
, the following equality holds :

U 1 U 1 † (X -U 0 )U 2 † U 2 = (X -U 0 )
Hence, using vec operator properties we obtain :

(U 2 U 2 † ⊗ U 1 U 1 † )(x v -vec(U 0 )) = (x v -vec(U 0 ))
which is a linear constraint in x v .

Remark 1. A zero-non zero structure on the matrix X can be added to the nearest matrix problem under LME (2) using a linear matrix equality of the form : X ∆ = X (where denotes the element wise Hadamard product) with ∆ ∈ R m×p a matrix of either zero or one elements.

On another note, the projections on the set of positive definite matrices S + * and generalized negative definite matrices S - * g are presented hereinafter.

Nearest positive matrix problem

Projection onto the cone of positive semi-definite matrices denoted S + , is equivalent to finding the positive semi-definite approximant using the Frobenius norm metric. Since S + is convex the approximant is unique and is equivalent to the problem, first introduced by Higham in [START_REF] Higham | Computing a nearest symmetric positive semi-definite matrix[END_REF], given below :

X + = arg min X=X ≥0 Z -X (3)
The solution of (3), for real matrices, is given by the following Theorem [START_REF] Higham | Computing a nearest symmetric positive semi-definite matrix[END_REF]:

Theorem 2.3. Let Z ∈ R n×n be a given matrix and Z s = (Z + Z )/2, Z ss = (Z -Z )/2 be the symmetric and skew-symmetric parts of Z respectively. Let Z s = U H be a polar decomposition of Z s with (U

U = I n , H = H ≥ 0), then X + = (Z s + H)/2
is the unique positive semi-definite approximant of Z in the Frobenius norm.

Proof. Let X be a symmetric positive semi-definite matrix.

Z -X 2 = Z s -X 2 + Z ss 2 + 2 < Z s -X, Z ss > (4)
This equality reduces to Z -X 2 = Z s -X 2 + Z ss 2 since Z ss is a skew-symmetric matrix. Hence, the problem consists in approximating Z s . Let Z s = T ΛT be a spectral decomposition with T T = I n , Λ = diag(λ i ), let Y = T XT . Since the Frobenius norm is invariant by rotations we obtain :

Z s -X 2 = Λ -Y 2 = i =j y 2 ij + i (λ i -y ii ) 2 ≥ λi<0 (λ i -y ii ) 2 > λi<0 λ 2 i
Since Y is positive semi-definite, the lower bound λi<0 λ 2 i is reached uniquely for the matrix Y = diag(d i ) where :

d i = λ i , λ i ≥ 0 0, λ i < 0 (5)
that is : [START_REF] Higham | Computing a nearest symmetric positive semi-definite matrix[END_REF].

X F = T diag(d i )T . Finally, since H = T diag(|λ i |)T , we get X + = (Z s + H)/2 ∈ S +
Remark 2. Projection on S + * can be obtained in the same manner as in the last Theorem by substituting the equation ( 5) by the following equation

d i = λ i , λ i > 0 , λ i ≤ 0, > 0 (6)
with a sufficiently small positive real scalar (compared to the condition number associated to Z s ). This projection can be numerically obtained based on the Schur decomposition, for instance, using the spectral decomposition Z s = T ΛT where T is an orthogonal matrix and Λ = diag(λ i ), i = 1, 2, . . . , n. Then,

Z + s = T ΛT T where Λ = diag( λ+ i ) = max{ ; λ i } for i = 1, 2, . . . , n.

Nearest generalized negative matrix problem

Unlike the nearest positive matrix problem solved by Higham, the following generalization is original and introduced for the first time in this paper. Let us consider the following set S - g = {Z ∈ R n×n / Z s = (Z + Z )/2 ≤ 0} including matrices with a negative semi-definite symmetric part. Then we have S -⊂ S - g and projecting onto S - g can be seen as a generalization of the problem given in (3) and is given by :

X g = arg min X∈S - g Z -X (7)
if one consider X s = (X + X )/2, X ss = (X -X )/2 and Z s = (Z + Z )/2, Z ss = (Z -Z )/2 the symmetric and skew-symmetric parts of X and Z respectively, then the function to be minimized in (7) becomes :

Z -X 2 = Z s + Z ss -X s -X ss 2 which leads to Z -X 2 = Z ss -X ss 2 + Z s -X s 2 + 2 < Z ss -X ss , Z s -X s > .
This reduces to : Z -X 2 = Z ss -X ss 2 + Z s -X s 2 since Z ss -X ss is skewsymmetric. Hence (7) can be obtained by solving the following minimization problem:

min Xss=-Xss ,Xs∈S - Z ss -X ss 2 + Z s -X s
Since both Z ss and X ss are skew-symmetric matrices, the solution of ( 7) can be obtained by solving the following minimization problem:

X g = arg min Xs=X T s ≤0 Z s -X s (8)
which is similar to problem (3). Finally, the solution of the nearest generalized negative matrix problem ( 7) is given by X g = Z ss + X s -with X s -is the projection of Z s onto S - * .

Reflection based techniques for constrained control synthesis

Hurwitz-stable matrices set

The set of Hurwitz-stable matrices, henceforth denoted by H s , is a simply connected open cone [START_REF] Duan | A note on Hurwitz stability of matrices[END_REF] since it is the product of two convex open cones. It follows from that, as shown in [START_REF] Duan | A note on Hurwitz stability of matrices[END_REF], any Hurwitz-stable matrix Z can be expressed as a product of a generalized negative definite matrix X ∈ S - * g and a positive definite matrix Y ∈ S + * as stated by the following Theorem.

Theorem 3.1. [START_REF] Duan | A note on Hurwitz stability of matrices[END_REF] A matrix Z ∈ R n×n is Hurwitz-stable if and only if there exist a generalized negative definite matrix X and positive definite matrix Y such that : Z = XY .

Nearest Hurwitz-stable matrix problem

The projection of a given matrix Z 0 ∈ R n×n onto H s is yet another nearest matrix problem given by the following expression :

Z F = arg min Z∈Hs Z 0 -Z (9)
Using the result of Theorem 3.1, one can recast this nearest matrix problem as the following original optimization problem : min

X∈S - * g ,Y ∈S + * Z 0 -XY (10)
To solve this problem a gradient-projection scheme is proposed and presented by iterations of the form :

X k+1 = (X k -δ X ∂ X ) - g Y k+1 = (Y k -δ Y ∂ Y ) + (11)
where (•) - g , (•) + designate the projections on S - * g and S + * respectively. ∂ X and ∂ Y denote the sub-gradients given by :

∂ X = (Z 0 -X k Y k )Y k ∂ Y = X k (Z 0 -X k Y k ) (12)
δ X and δ Y designate the adaptive step-sizes given by :

δ X = ∂ X 2 / ∂ X Y k 2 δ Y = ∂ Y 2 / X k ∂ Y 2 (13)
Moreover, the initial starting points X 0 and Y 0 are computed using polar decomposition of Z 0 = U H that is :

X 0 = (U ) - g , Y 0 = H.

Douglas-Rachford like technique for constrained control synthesis

Douglas-Rachford methods (DR) are frequently used to find a feasible point (i.e. a point in the intersection) of two closed constraint sets in a Hilbert space. In our case, let us consider the following closed subsets1 DL ⊂ D L and Hs ⊂ H s . The stabilization problem under information structure constraint for the linear time-invariant (LTI) system described by :

ẋ = Ax + Bu, x ∈ R nx , u ∈ R nu y = Cx, y ∈ R ny (14)
can be recast as the following original feasibility problem :

Find K ∈ R nu×ny / Z (A + BKC) ∈ DL ∩ Hs (15)
In order to simplify both notation and implementation the projections on each subset, which are no other than the nearest matrix problem under LME associated to DL and the nearest Hurwitz-stable matrix problem associated to Hs , will be noted PD L and P Hs respectively, in the sequel. Hence, the corresponding reflection mappings noted RD L , R Hs are defined by :

RD L µ d PD L + (1 -µ d )I, R Hs µ h P Hs + (1 -µ h )I, (16)
with µ d ∈ (0, 2], µ h ∈ (0, 2]. In [START_REF] Fält | Optimal convergence rates for generalized alternating projections[END_REF], an optimal value of the adaptation parameters µ d , µ h is found in the case of finding a point in the intersection of two (convex) affine subspaces. Let us just remind that the problem tackled with involves a convex and a potentially non-convex set. The (local) convergence, the smoothing effect of the proposed optimization scheme are discussed in the next section. Moreover, DR method is often referred to as reflect-reflect-average algorithm. See survey papers [START_REF] Lindstrom | Survey: Sixty years of Douglas-Rachford[END_REF][START_REF] Ryu | A primer on monotone operator methods survey[END_REF] and references therein) that trace back its origin to the mid-1950s. It admits many variations and has proven its superiority on alternating projections algorithms. In this paper, an original DR-like scheme is proposed and is defined hereafter.

Definition 3.2 (Modified DR method (MDR)). Given an initial matrix K 0 ∈ R nu×ny , the MDR method generates a sequence (K k ) ∞ k=1 as follows :

K k+1 ∈ TD L, Hs (K k ), TD L, Hs (1 -µ)I + µRD L R Hs (17)
µ ∈ (0, 1] is some adaptation parameter.

H ∞ and H 2 characterizations

In light of the stability characterization, original characterizations of H ∞ and H 2 performances are proposed hereafter. For this aim, let us consider the LTI system described by :

     ẋ = Ax + B w w + Bu, x ∈ R nx , w ∈ R nw , u ∈ R nu z = C z x + D z u, z ∈ R nz y = Cx, y ∈ R ny . ( 18
)
The following result presents the new H ∞ performance characterization of the closedloop given by :

ẋ = A c x + B w w, A c A + BKC z = C c x, C c C z + D z KC (19)
Theorem 3.3. The closed-loop system given by ( 19) has an H ∞ norm less than γ > 0 if and only if there exist matrices X ∈ R nx×nx , Y ∈ R nx×nx and G ∈ R nz×nx such that:

A c C c = A C z + B D z KC = X G Y (20) with X G V 1 + V 2 ∈ S - *
g is a generalized definite negative matrix and Y ∈ S + * a definite positive matrix. V 1 and V 2 are some constant matrices given by :

V 1 = I nx 0 nx×ny , V 2 = 1 2 1 γ 2 B w B w 0 0 -I ny .
Proof. According to the bounded real-lemma, system (19) has an H ∞ norm less than γ > 0 if and only if there exist a matrix Q = Q > 0 such that :

A c Q + QA c + C c C c • B 1 Q -γ 2 I < 0 Since A c is Hurwitz-stable : ∃ X ∈ S - * g , Y = Q ∈ S + * / Ac = XY (21)
Moreover, Y = Q is non singular means that :

∃ G ∈ R ny×nx / C c = GY (22)
Hence, the bounded real-lemma writes :

Y (X + X + N N + 1 γ 2 B 1 B 1 )Y < 0
which, by Schur's lemma, leads to

X + X + 1 γ 2 B 1 B 1 • G -I < 0
or equivalently,

X G V 1 + V 2 ∈ S - * g . ( 23
)
Hence, the H ∞ synthesis problem under information structure constraint for the LTI system given by ( 18) can be recast as the following feasibility problem :

Find K ∈ R nu×ny / A c C c ∈ DL ∩ H∞ ( 24
)
with H∞ is the closed subset of matrices Z that can be written as

A c C c = A C z + B D z KC = X G Y with X G V 1 + V 2 ∈ S - *
g is a generalized definite negative matrix and Y ∈ S + * a definite positive matrix.

Remark 3. This problem can be solved by the MDR like method (T m DL, H∞ ) as presented in Definition 3.2. The projections and reflections can be derived by gradientprojection algorithms as for the stability case.

In the same vain, the H 2 performance is presented hereafter.

Theorem 3.4. The closed-loop system given by (19) has an H 2 norm less than γ 2 > 0 if and only if there exist matrices X ∈ R nx×nx , Y ∈ R nx×nx and N ∈ R nz×nx such that :

A c C c = A C z + B D z KC = X G Y (25) with X G V 1 + V 3 ∈ S - *
g is a generalized definite negative matrix and Y ∈ S + * a definite positive matrix verifying in addition the LME constraint trace(B 1 Y B 1 ) < γ 2 2 . V 1 and V 3 are some constant matrices given by :

V 1 = I nx 0 nx×ny , V 3 = 1 2 0 0 0 -I ny .

Local Convergence Analysis

Let us first recall that the proximal mapping (or prox-operator) of a convex function f : R n → R is defined as :

prox f (v) = arg min x (f (x) + 1 2 x -v 2 ) (26) For instance, if f (x) = 0, prox f (v) = v.
Moreover, if f is the indicator function a of closed convex set F, then prox f is non other than the projection on F given by :

prox f (v) = arg min z∈F z -v 2 = P F (v). ( 27
)
Recall also that if f is convex, the prox f (v) exists and is unique for all v. Another important point that should be recalled is that the Moreau-Yosida regularization (known as Moreau envelope) of a function f , [START_REF] Moreau | Proximité et dualité dans un espace hilbertien[END_REF], is defined as :

f m (v) = inf z (f (z) + 1 2λ z -v 2 ), λ > 0 (28)
Or equivalently,

f m (v) = f (prox λf (v)) + 1 2λ prox λf (v) -v 2 . ( 29
)
The Moreau envelope is usually used as an approximation of f , although regularization was not the purpose of the seminal paper [START_REF] Moreau | Proximité et dualité dans un espace hilbertien[END_REF]. Recall that in the nonsmooth case, it consists on a smooth regularization since it is always convex and its gradient ∇f m exists and is given by :

∇f m (v) = 1 λ (v -prox λf (v)) (30)
Moreover, ∇f m is Lipschitz continuous with constant 1 λ . In the MDR based technique context, we suppose that DL admits a quadratic indicator function :z = vec(Z) → d(z) and H, which represents henceforth either Hs , Hm ∞ or Hm 2 , admits a non-smooth indicator function : z = vec(Z) → h(z). Given an initial point z 0 (derived from a randomly chosen tri-diagonal matrix Z 0 ∈ R n×n for instance representing the initial closed-loop matrix A c ), the MDR method generates a sequence (z k ) ∞ k=1 as follows :

z k+1 = (1 -µ)I + µR HRD L (z k ), µ ∈ (0, 1] (31)
where the mappings noted RD L , R H are defined by :

RD L µ d PD L + (1 -µ d )I, µ d ∈ (0, 2] (32) R H µ h P H + (1 -µ h )I, µ h ∈ (0, 2] (33)
Hence, the problem tackled with, in this section, consists in analysing the convergence and the convergence rate of the MDR search scheme given by equations ( 31), ( 32) and ( 33) by means of a Lyapunov based stability analysis of a dynamical system whose trajectory goes through the discrete iterates (z k ). It is shown, hereafter, that the MDR iterates can be interpreted as a gradient-descent method for minimizing a differentiable function, subsequently constructed, namely f mdr . In that case, given an initial point z 0 ∈ R n 2 , the MDR method generates a sequence (z k ) ∞ k=1 that is a discretization of the trajectory of the continuous dynamical system given by

ż(t) = µ R HRD L -I z(t), z(t 0 ) = z 0 (34)
using the forward discretization ż(t) ≈ z k+1 -z k and z(t) ≈ z k approximation (for simplicity of presentation, the sample time is fixed to 1 without loss of generality). Therefore, we aim here to recast (34) as

ż(t) = -∇f mdr (z), z(t 0 ) = z 0 (35)
According to (30) one can rewrite the reflections RD L , R H as follows :

RD L (z) = (I -µ d ∇d m )(z), R H(z) = (I -µ h ∇h m )(v). ( 36
)
This leads to :

µ I -R HRD L (z) = µµ d ∇d m (z) + µµ h ∇h m (z -µ d ∇d m (z)) Since, d is twice continuously differentiable the term (I -µd 2 ∇ 2 d m (z)) exists. Besides, since d is quadratic the term (I -µd 2 ∇ 2 d m (z)) is constant that is (I -µd 2 ∇ 2 d m (z)) Γ.
At this point we are able to construct f mdr as follows :

f mdr (z) µ(Γ -1 (µ d d m (z) + µ h h m (z -µ d ∇d m (z)) - µ d 2 (∇d m (z)) 2 )) (37)
Hence,

ż(t) = µ R HRD L -I z(t) = -∇f mdr (z). ( 38
)
The asymptotic stability of a point z * supposed to be a strict local minimizer of f mdr , follows from Lyapunov theorem by choosing the following Lyapunov function :

V L (z) f mdr (z) -f mdr (z * )
Then, one can note that

VL (z) = < ∇f mdr (z), ż > = -∇f mdr (z) 2 ≤ 0. ( 39
)
Assuming that z * is an isolated local minimizer then ∇f mdr (z) = 0. Therefore, z * is an asymptotically stable equilibrium point of (35) and consequently for the continuoustime dynamical system (34). Furthermore, choosing a Lyapunov function of the form :

V L (z, t) t(f mdr (z) -f mdr (z * )) + 1 2 z -z * 2 , ∀t ≥ 0. ( 40
)
leads to

f mdr (z) -f mdr (z * ) ≤ 1 t V L (z 0 , t 0 ). ( 41
)
which means that the convergence rate is of O( 1 t ). The last inequality writes also :

f mdr (z) -f mdr (z * ) ≤ 1 t c(µ, µ d , µ h ) ( 42
)
where c depends only on parameters µ, µ d , µ h since z 0 is fixed.

Remark 4. Inequality (42) means that the adaptation parameters have a direct effect, as expected, on the convergence rate.

Numerical Examples

Stabilization

An academic example borrowed from [START_REF] Rubio-Massegu | Static output-feedback control under information structure constraints[END_REF] consisting in a LTI system given by ( 14) with the following state-space matrices data :

A =       -4 0 -2 0 0 0 0 2 0 0 0 -2 0 -1 0 -2 0 -1 0 3 0 -2 0 -1       , B =       1 0 0 1 0 0 0 0 0 0 1 0 0 0 1       , C =   1 0 0 0 0 0 1 0 0 0 0 0 0 0 1   ,
is considered to test the MDR method (with µ = 1 2 , µ d = µ h = 2) for static output feedback stabilization. Starting from K 0 = 0 3×3 with σ(A) = {-1.5000±1.9365i, -3.7004± 1.1286i, 0.4009}, only 2 iterations2 and a computation-time less than 1 second to obtain the stabilizing static output feedback K * given by :

K * =   -0.
with σ(A + BK * s C) = {-3.8330 ± 1.0587i, -1.5000 ± 1.9365i, -0.1102}. The MDR method is then tested on the well-known COMPleib library by Friedemann [START_REF] Leibfritz | COMPleib: COnstrained Matrix optimization Problem library a collection of test examples for nonlinear semidefinite programs, control system design and related problems[END_REF]. All the examples are run from a zero gain matrix and no random start is used. In Table I, n x designates the system size, Iter the size of MDR sequence, Dist the distance of Z = A + BK * C to the set of feasible output feedbacks DL , K * the static controller gain solution and t the computation-time in seconds.

H ∞ control

Once more, the same academic example borrowed from [START_REF] Rubio-Massegu | Static output-feedback control under information structure constraints[END_REF] is considered. It consists of a LTI system given by (18) with the following complementary state-space matrices data:

B w = 1 1 1 1 1 , C z = I 5 0 3×5 , D z = 0 5×3 I 3
The aim is to find a stabilizing static output feedback that ensures a closed-loop H ∞ performance less than γ ∞ = 0.78. After 12 iterations and a computation-time of t = 1.2s : is obtained achieving an H ∞ closed-loop performance of T zw (s) ∞ = 0.7712. Moreover, the proposed method was tested intensively on COMPleib systems. For the AC5 system, for instance, 10 iterations and a computation time of t = 0.778s, were required to obtain the following static output feedback gain : -6.5008 -0.1737 -3.6406 1.9473 (46) achieving T zw (s) ∞ = 667.29 which is better than the one found by HIFOO approach (γ ∞ = 669.56) [START_REF] Arzelier | H 2 for Hifoo[END_REF]. For the NN16 system, 3 iterations and a computation time of t = 0.225s, were required to obtain the following static output feedback gain

K ∞ =
K ∞ = 1e6
K ∞ = 1e6    
-0.0453 0.0070 -0.0144 0.0032 0.0069 -0.0313 -0.0032 -0.0011 -0.0144 -0.0032 -0.0453 -0.0070 0.0032 -0.0011 -0.0069 -0.0313

    (47)
achieving T zw (s) ∞ = 0.9909 which is substantially equal to the one found using a Particle Swarm Optimization based technique (γ ∞ = 0.99) [START_REF] Yagoubi | Particle Swarm Optimization for the static H ∞ static output feedback[END_REF].

Conclusion

A reflection-based method for feedback synthesis under information structure constraints is presented. The approach relies on a prominent property of Hurwitz-stable matrices set and a Douglas-Rachford's like reflection optimization method. The convergence of the generated iterates continuous limit is connected to a continuous dynamical system trajectory. Hence, the Lyapunov stability theory is used to prove the local convergence of the proposed scheme. Finally, the efficiency of the proposed approach is demonstrated through several examples.

  0526 0.0167 -0.2446 0.0725 -0.0465 0.1227 -0.3732 0.0956 -0.7068   (43) with σ(A + BK * C) = {-3.8230 ± 1.2702i, -1.4944 ± 1.9661i, -0.1079}. When an additional structure constraint, of the type K ∆ = K

  1, 2} are some given matrices of appropriate dimensions. Let us now define the nearest matrix problem under a Linear Matrix Equality (LME) constraint of the type X = U 0 + U 1 XU 2 .

	Definition 2.2 (Nearest matrix problem under LME). The nearest matrix problem
	under a Linear Matrix Equality (LME) is defined as :

Table 1 .

 1 MDR method applied to static output feedback problems from COMPleib

	System nx	Iter	λmax	Dist								K *	t(s)
	AC5	4	3	-0.203	6.19e-15				1e3		0.3619 -0.5873 -0.0230 -1.5307	0.095
	AC7	9	4	-0.0114	4.5e-15							0.5879 0.1152	0.299
	AC8	9	3	-0.0164	1.876e-15		0.0036 -0.0005 -0.0085 -0.0001 0.0009	0.305
											3	-2	-6	0	1	
	AC9	10	4	-0.0028	1.07e-15			1e -4	  20 -9 -39 -1 6 5 -1 -10 0 1  	0.424
												0	3	4	0	0
	AC10	55	4	-0.0232	1.08e-13				1e -5	-0.0026 0.2386 0.1321 0.2207	6.11
	AC11	5	11	-0.0209	1.6e-14			0.0055 0.0615 0.2154 -0.1586 0.0036 0.0244 0.0961 -0.0628	0.14
									0.5			9.1 -10.1 -2190.4	
	AC12	4	2	-0.6503e-4	2.8e-15				0.4			0.7	-0.7	-313		0.26
								-0.1 0.2	-0.2	-43.9
	NN1	3	16	-0.0454	1.6e-14							2.8647 43.1201	1.03
	NN5	7	25	-6.0232e-5	1.58e-15						11.5909 10.3878	0.48
	NN13	9	12	-0.0545	1.22e-14			0.1044 2.2036 75.5248 6.3875	4.23
	HE1	4	10	-0.0022	5.9e-16								0.0725 0.3695	0.64
							0.035	-0.764 -0.052 -0.010 -1.097 -0.291 
	HE3	8	3	-0.0150	3.18e-16	 	-0.043 0.036	1.663 -0.204 -0.439 -0.364	0.134 0.012	2.729 -0.048 -0.462 0.180  	0.15
							0.014	-0.391 -0.785 -0.096 -0.026	-0.82
											0 0.03 4.56 -0.07 
	WEC1	10	5	-0.002	7e-15			10 -2		0	0	0.09	0		0.65
											0	0	0	0
									-0.0013 -0.0178 0.0025 
	JE2	21	6	-0.053	2e-14				0.0178	0.0095 0.0133 	0.12
									-0.0144 -0.0055 0.0636
	has been imposed we obtain, after 2 iterations, the structured stabilizing static output
	feedback K * s :
						 -0.0023 0 -0.0023 
					K * s =	 -0.0006 0 -0.0006 
						-0.7683 0 -0.7739

An element wise linear constraint can be added, without loss of generality, to ensure the closedness of the considered subsets.

On Matlab@Mathworks, Processor Intel Core i7-4500U CPU @ 1.8GHz


J Mungkalasiri

L Bedel

F Emieux

J Doré

F N R Renaud

F Maury

DLI-CVD of TiO 2 -Cu antibacterial thin films: Growth and characterization

Keywords:

TiO 2 -Cu nanocomposite films were grown by pulsed direct liquid injection chemical vapor deposition (DLI-CVD) on stainless steel, silicon and glass substrates with the goal to produce bactericidal surfaces. Copper bis (2,2,6,6-tetramethyl-3,5-heptadionate), Cu(TMHD) 2 , and titanium tetra-iso-propoxide, TTIP, were used as metalorganic precursors. Liquid solutions of these compounds in xylene were injected in a flash vaporization chamber connected to a cold wall MOCVD reactor. The deposition temperature was typically 683 K and the total pressure was 800 Pa. The copper content of the layers was controlled by the mole fraction of Cu (TMHD) 2 which was adjusted by the injection parameters (injection frequency and concentration of the starting solution). The chemical, structural and physical characteristics of the films were investigated by XRD, XPS, FEG-SEM and TEM. Copper is incorporated as metal particles with a relatively large size distribution ranging from 20 to 400 nm (with a large majority in 20-100 nm) depending on the copper content of the films. The influence of the growth conditions on the structural features and the antibacterial properties of the thin films are reported and discussed.

Introduction

Bacteria essentially exist and multiply in biofilms that grow on various surfaces. They cause significant problems of infection that spread in many areas. Today great efforts attempt to prevent the formation of biofilms. The post treatments of contaminated surfaces as chemical or physical cleanings do not settle the problem because they are curative and not preventive solutions. Long term antibacterial surfaces are required. Many solutions were proposed in the literature using hightechnology surface treatments and growth of antibacterial thin films. For instance, works were reported on the ion implantation of copper or silver [1,2], on the deposition of SiO 2 -Ag [3], TiO 2 [4], TiN-Cu [5], and other thin films containing triclosan [6] or quaternary ammonium salt [7] as well as on glasses and alloys doped with antibacterial elements [8].

Anatase TiO 2 under UV radiation is a well known photocatalytic material [9]. The self cleaning property by decomposition of organic species and antibacterial activity were previously reported [4,10]. This antibacterial coating is very efficient but UV light is necessary and for many applications the use of UV is not possible or could be dangerous. Furthermore, it is not an economic solution since electrical energy is required for UV lamps [11].

The addition of metals (Cu, Ag, Au, Co, Fe, etc.) to TiO 2 was carried out to improve the photocatalytic activity [12][13][14]. For instance, it was found that a suitable amount of silver (2-4 mol%) increased the pho-tocatalytic activity of TiO 2 films grown by sol-gel [15]. Many other deposition processes have also been used for the growth of doped-TiO 2 as CVD and PVD films. The antibacterial properties of copper have been known for centuries. Its efficiency was discussed in many papers [8,16]. Several possible mechanisms of copper activity on bacteria have been discussed. Copper ions exhibit a high affinity for the thiol and amino groups present in proteins. Thus, specialized proteins containing clusters of these functional groups transport and store the copper ions, hampering their potential toxicity [17].

The direct liquid injection chemical vapor deposition process (DLI-CVD) presents advantages for large scale surface treatment such as a high growth rate, a good control of the chemical composition of the films, the deposition under atmospheric or moderately reduced pressure as well as the possibility to coat complex and porous substrates. We have recently used this process for the growth of TiO 2 -Ag films and the results will be published soon. In this paper, TiO 2 -Cu nanocomposite films were deposited in one-step by DLI-CVD for antibacterial applications. The influence of the growth conditions on the main structural features of the films is presented. The antibacterial properties were found to increase with the copper content of these nanocomposite coatings.

Experimental

Film preparation

The TiO 2 -Cu thin films were deposited in a vertical cold wall CVD reactor equipped with a pulsed Direct Liquid Injection system (Fig. 1).

Liquid solutions of the precursors were introduced through injectors at the top of a flash vaporization chamber and then reactive vapors were carried down to the deposition zone under low pressure using a carrier gas (N 2 ). Before reaching the substrates, the vapor passed through a showerhead which distributed the precursors homogeneously on the sample holder. A vacuum pump was used to eliminate the gaseous by-products including the carrier gas and solvent. The total pressure was controlled independently of both the atmosphere and the total flow rate using an automatic system. It was maintained constant at 800 Pa.

The substrates (stainless steel 316 L, Si and glass) were cleaned with acetone (15 min in an ultrasonic bath) prior to use and then rinsed with ethanol before being dried by air stream. The TiO 2 -Cu films were grown using two metalorganic precursors as Cu and Ti sources that we have previously employed in conventional MOCVD: copper bis(2,2,6,6-tetramethyl-3,5-heptadionate) (Cu(TMHD) 2 ) [18] and titanium tetra-iso-propoxide (TTIP) [9]. Here they were dissolved in xylene as solvent. Pure N 2 (99.9999%) and O 2 (99.9999%) were used as carrier gases (N 2 :O 2 = 9:1). The gas flow rates were monitored by mass flow controllers. The mole fractions of Cu(TMHD) 2 and TTIP were controlled by the injection parameters (Table 1). Two injectors were used to feed the reactor: one for the Ti source and the other for Cu. The evaporator and the showerhead were maintained at 473 K. The reactor wall was heated at 540 K to avoid the condensation of the reagents. The deposition temperature was fixed at 683 K except otherwise specified for a series of experiments. The film thickness was controlled by varying the deposition time. The mole fraction of the copper precursor in the input gas phase is the main parameter that controls the Cu content of the films. It was varied either by adjusting the solution concentration or the injection frequency while the other parameters were maintained constant as reported in Table 1. We will focus on the influence of the mole fractions of Cu(TMHD) 2 .

Film characterization

The films grown on Si wafers were used to analyze the composition and the microstructure. The structure was determined by X-ray diffraction (XRD) using a θ-2θ or grazing angle configuration. The morphology and film thickness were investigated with a scanning electron microscope (Leo 1530 FEG-SEM) equipped with an X-ray energy dispersive spectroscopy analyzer (EDX). The size of Cu particles was measured by transmission electron microscopy (TEM). The film composition was analyzed by X-ray photoelectron spectroscopy (XPS). The JIS Z 2801 standard with Staphylococcus aureus was applied as antibacterial test on samples generally prepared on glass. More details on this bacteriological standard test will be reported in a future paper.

Results and discussion

Film morphology

A granular surface morphology is observed for pure TiO 2 DLI-CVD films both on Si and stainless steel substrates. It is constituted of small lamellar grains of 50-100 nm width that agglomerate to form bigger nodular grains (200-300 nm in diameter) leading to a relatively high surface roughness (Fig. 2a). The TiO 2 -Cu nanocomposite films still exhibit a granular morphology but the grains are smaller and less agglomerated. It seems to be more compact. The surface is smoother (Fig. 2b). The copper particles are uniformly distributed in the TiO 2 matrix with particle sizes in the range of 50 to 250 nm as observed by SEM. The largest Cu particles emerge from the surface of the thinnest layers (Fig. 2c). The number of Cu particles visible on the surface seems to increase with the Cu(TMHD) 2 mole fraction. By reducing the deposition temperature from 683 to 623 K, the nanocomposite films exhibit a denser structure due to the quasi amorphous structure of the oxide matrix (Fig. 2d). No significant influence of the substrate was found on the morphology of these nanocomposite layers.

Growth rate

The deposition rate of pure TiO 2 films is high (115 nm min -1 ). It decreases quickly to approximately 60 nm min -1 when the Cu(TMHD) 2 mole fraction is increased and reaches a nearly constant level already at a Cu(TMHD) 2 mole fraction of 0.15% (Fig. 3). This behavior could be due to premature interactions between the two precursors in the gas phase or in the showerhead but no evidence for that was observed at 683 K. So it can be assumed that (i) a competition occurs on the adsorption sites for the two molecular precursors and (ii) the nucleation and growth of Cu particles hinder the growth of TiO 2 crystals that form the matrix. It is frequently observed in CVD that doping influences the kinetics of the process.

For a series of experiments the deposition temperature was increased from 623 to 723 K and the growth rate of TiO 2 -Cu was found to The mole fraction of the Cu precursor was the main parameter varied either by adjusting the solution concentration or the injection frequency.

decrease from 40 to 18 nm min -1 (Fig. 4). This contrasts with the deposition by low pressure MOCVD of pure TiO 2 which was found to be thermally activated [19]. The vapor of solvent (xylene) in this DLI-CVD process probably plays a role on the kinetics but further research work is need to support this hypothesis. On the other hand, by increasing the deposition temperature, premature reactions can occur in the gas phase which consumes reagents. For instance, Fig. 4 also shows that the relative content of copper in the films decreases by increasing the growth temperature, as deduced from the Cu/Ti EDX ratio intensities, indicating a preferential decomposition of the Cu precursor before reaching the deposition zone. This is consistent with the fact that this precursor is the less thermally stable [18]. At this stage, we have not tried to optimize the growth rate or to study the deposition mechanisms. Further experiments are required for such a discussion, in particular because of the great complexity of the reactive gas phase (TTIP/Cu (TMHD) 2 /N 2 /O 2 ). We have also to consider that the growth rate of anatase is generally faster than that of rutile and we will see in the next section that both phases can be formed depending on the growth conditions.

Structure, microstructure and composition of the films

The XRD pattern of a TiO 2 -Cu coating grown at 683 K (300 nm thick) reveals that the TiO 2 matrix crystallizes with the anatase structure and copper is present in the metallic form (Fig. 5). No evidence for copper oxide was found by XRD. This was supported by XPS analysis. The binding energies of Ti 2p 3/2 and O 1s found at 458.9 eV and 530.1 eV, respectively, are in good agreement with TiO 2 . The absence of shake-up satellites between Cu 2p 3/2 and Cu 2p 1/2 and the binding energy of Cu 2p 3/2 found at 932.5 eV indicate the absence of CuO. This is confirmed by comparison with a recent work where CuO was deposited with TiO 2 [20]. The presence of Cu 2 O is more difficult to detect and this is generally achieved from the Auger parameter (A.P. is defined as the difference between the kinetic energy of the Auger and XPS electrons). Such analysis reveals that Cu 2 O exists on the surface of our as-deposited samples (typically A.P. = 1849.7 eV) but tends to disappear after Ar + sputtering for a few minutes to clean the surface contamination. This indicates that only the Cu metal particles emerging from the surface of the film are superficially oxidized due to interaction with the air.

The presence of metal Cu particles (and not of Cu 2 O) was definitively demonstrated by TEM (Fig. 6). The lattice parameter measured for the metallic particles at 0.3615 nm is in good agreement with the cubic fcc structure of metal Cu and far from that of Cu 2 O (0.4245 nm).

The influence of the Cu(TMHD) 2 mole fraction in the input gas phase on the film composition is given in Fig. 7. Both XPS and XRD analyses confirm an increase of the Cu content of the films with the Cu (TMHD) 2 mole fraction. This indicates that the film activity which is certainly related to the Cu content could be controlled only by this growth parameter.

Interestingly, the formation of rutile was observed by XRD when the Cu content increased in the films while the deposition temperature was maintained constant at 683 K (Fig. 8). The correlation between the increase of rutile with Cu content suggests that this metal favors the crystallization of rutile. This was reported for other metals as for TiO 2 -Ag deposited by sol-gel [15]. It is well known that the smaller the crystallites, the broader the XRD peaks. So the average crystallite size (t) can be derived from the broadening of the diffraction peaks using the Scherrer equation:

t = Kλ = ðB cos θÞ ð 1Þ
where K is a dimensionless shape factor (0.9), λ is the wavelength of the X-rays and B is the integral breadth of a reflection (in radians 2θ) located at the Bragg angle 2θ. B is often calculated from the line broadening at half maximum intensity (FWHM) relative to a reference solid with crystallite size N500 nm according to B 2 = B s 2 -B r 2 . The average crystal size of TiO 2 determined from the (101) TiO 2 XRD peak increases with the film thickness to reach typically 40 nm for a 100 nm thick layer. We noted that an increase of the Cu(TMHD) 2 mole fraction decreased the growth rate of the films and we also observed that it decreases the mean crystal size of anatase and rutile. This confirms that the deposition of Cu particles hinders the growth of TiO 2 crystals.

TEM analyses have revealed a uniform distribution of the Cu particles both on the surface and through the layer thickness. No agglomeration of Cu particles was found and different particle sizes with a spherical shape ranging from 20 to 400 nm were observed (with a large majority in the range of 20-100 nm). Fig. 9 shows the size distribution of Cu particles in the nanocomposite film. These particle sizes are in good agreement with the values determined by FEG-SEM. A detailed analysis of the TEM images indicates that the population density of small (20-100 nm) Cu particles (10 8 particles cm -2 ) is significantly higher than the largest ones. These largest Cu particles are typically in the ranges of 100-250 nm and N250 nm as deduced from different samples. It is possible that Cu particles smaller than 20 nm are present but they are not detected by our TEM analyses.

Antibacterial activity

Antibacterial activity was the motivation of this work and copper was selected as bactericidal agent with the goal to use a minimum amount in the films. The mole fraction of Cu(TMHD) 2 was varied since it allows to control the Cu content of the films. According to the JIS Z 2801 standard the antibacterial activity was calculated by the following formula:

Antibacterial activity = Log ðA = BÞ ð 2Þ
where A and B are the number of CFU (colony forming unit) on the surface of the reference and treated samples, respectively. For convenient comparisons between various series, we use also a definition of the relative activity given by:

Relative activity = ½Log ðA = BÞ = Log ðAÞ⁎100:

This relative activity means that when it is equal to 100% the surface is bactericidal, in the range of 0-100% the surface has an antibacterial behavior and zero means the surface is inactive. Fig. 10 shows that an antibacterial activity is detected for mole fractions of Cu(TMHD) 2 higher than 0.05% and a bactericidal efficiency is observed for mole fractions higher than 0.15%. This means that the number of bacteria reduced from 10 7 colonies/mL to 0 colony in 24 h under standard conditions, i.e. without UV irradiation.

Specific experiments to analyze the long term stability of these bactericidal surfaces are currently in progress. At this stage, we have observed a slight decrease of the biocidal property after 5 months of storage under various conditions since samples that exhibited a bactericidal behavior (100% of relative activity) reveal a relative antibacterial activity of approximately 70%. More details will be discussed elsewhere.

Conclusions

TiO 2 -Cu nanocomposite films were deposited by DLI-CVD under low pressure (800 Pa) at 683 K on stainless steel, silicon and glass substrates using Cu(TMHD) 2 and TTIP as Cu and Ti sources, respectively. N 2 and O 2 were used as carrier and reactive gases, respectively. The mole fraction of the Cu precursor influences the morphology, the structure and the growth rate of the TiO 2 -Cu films while the other parameters were maintained constant. At 683 K, TiO 2 crystallizes with the anatase structure but increasing the Cu content of the film favors the rutile nucleation and growth. Cu is incorporated as metallic and spherical particles which are uniformly dispersed on the surface and well distributed through the thickness of the layers. Most of the Cu particle sizes are in the range of 20 to 100 nm. The DLI-MOCVD process is suitable for the growth of multifunctional layers as TiO 2 -Cu with a high output. A more thorough investigation of the antibacterial properties of these films will be described in a future paper in relation with the Cu content of the layers since this appears as a key feature.

Fig. 1 .

 1 Fig. 1. Sketch of the DLI-MOCVD reactor.

Fig. 2 .

 2 Fig. 2. FEG-SEM micrographs of TiO 2 -Cu films deposited on silicon wafer as a function of the Cu(TMHD) 2 mole fraction in the reactive gas phase: (a) 0% at 683 K (pure TiO 2 ); (b) 0.076% at 683 K (surface morphology); (c) 0.076% at 683 K (cross section); (d) 0.076% at 623 K (cross section). The scale bar is the same for each micrograph.

Fig. 3 .

 3 Fig. 3. Influence of the Cu(TMHD) 2 mole fraction on the growth rate of TiO 2 -Cu films (T = 683 K).

Fig. 4 .

 4 Fig. 4. Influence of the growth temperature on the deposition rate and on the content of copper in TiO 2 -Cu films (EDX analyses).

Fig. 5 .

 5 Fig. 5. Grazing angle XRD pattern of a TiO 2 -Cu film showing the presence of anatase and metallic copper. Peaks noted with an asterisk originate from the substrate.

Fig. 6 .

 6 Fig. 6. TEM analyses of a TiO 2 -Cu film: (a) bright field image of one of the biggest Cu particles; (b) dark field image; (c) corresponding selected area electron diffraction pattern.

Fig. 7 .

 7 Fig. 7. Variation of the Cu content in the films determined by XPS (at.%, left axis) and ratio of (111) Cu/(101) TiO 2 XRD intensity (right axis, θ-2θ configuration) as a function of the Cu(TMHD) 2 mole fraction in the input gas phase.

Fig. 8 .

 8 Fig. 8. Influence of the Cu(TMHD) 2 mole fraction in the input gas phase on the relative content of rutile and copper as determined from XRD intensity ratios (110)Rutile/(101) Anatase and (111)Cu/(101)Anatase.

Fig. 9 .

 9 Fig. 9. TEM micrograph of a TiO 2 -Cu film deposited at 683 K directly on a microscopy grid made of amorphous carbon (a) and size distribution of the density of particles for different mole fractions of Cu(TMHD) 2 in the gas phase (b): 0.056%, 0.076% and 0.087%.

Fig. 10 .

 10 Fig. 10. Influence of the Cu(TMHD) 2 mole fraction in the input gas phase on the antibacterial activity with Staphylococcus aureus of TiO 2 -Cu films. Results are presented as standard antibacterial activity according to the JIS Z 2801 standard (right axis; Eq. (2) in the text) and as relative antibacterial activity (left axis; Eq. (3) in the text).

Table 1

 1 Experimental DLI-CVD conditions for the growth of TiO 2 -Cu nanocomposite films.

	Deposition temperature (K)	623-723
	Total pressure (Pa)	800
	N 2 carrier gas (sccm)	324
	O 2 carrier gas (sccm)	36
	TTIP concentration in xylene (mol L -1 )	1
	TTIP injection frequency (Hz)	2
	TTIP injection opening time (ms)	2
	Cu(TMHD) 2 concentration in xylene (mol L -1 )	0.005-0.1
	Cu(TMHD) 2 injection frequency (Hz)	0.5-4
	Cu(TMHD) 2 injection opening time (ms)	2

Acknowledgements

The authors thank Dr. C. Sarantopoulos (CIRMAT, Toulouse) for fruitful discussion on structural characterization and C. Anglade and M. Jouve (CEA, Grenoble) for technical assistance. This work was supported by CEA, CNRS and ANR under contract ANR-06-MAPR-0007-01.