Alexandra Morscher

Matthew S Dyer

Benjamin B Duff

Guopeng Han

Jacinthe Gamon

Luke M Daniels

Yun Dang

T Wesley Surta

Craig M Robertson

Frédéric Blanc

John B Claridge

Matthew J Rosseinsky
email: m.j.rosseinsky@liverpool.ac.uk

Li6SiO4Cl2: A Hexagonal Argyrodite based on anti-perovskite layer stacking

A hexagonal analogue, Li6SiO4Cl2, of the cubic lithium argyrodite family of solid electrolytes is isolated by a computation-experiment approach. We show that the argyrodite structure is equivalent to the cubic anti-perovskite solid electrolyte structure through anion site and vacancy ordering within a cubic stacking of two close-packed layers. Construction of models that assemble these layers with the combination of hexagonal and cubic stacking motifs, both well-known in the large family of perovskite structural variants, followed by energy minimisation identifies Li6SiO4Cl2 as a stable candidate composition. Synthesis and structure determination demonstrate that the material adopts the predicted lithium site-ordered structure with low lithium conductivity of ~10 -10 S cm -1 at room temperature, and the predicted hexagonal argyrodite structure above an order-disorder transition at 469.3(1) K. This transition establishes dynamic Li site disorder analogous to that of cubic argyrodite solid electrolytes in hexagonal argyrodite Li6SiO4Cl2 and increases Li ion mobility observed via NMR and AC impedance spectroscopy.

The compositional flexibility of both argyrodite and perovskite alongside this newly established structural connection which enables the use of hexagonal and cubic stacking motifs identifies a wealth of unexplored chemistry significant to the field of solid electrolytes.

Introduction

Compounds related to the mineral argyrodite (Ag8GeS6) have been the focus of considerable interest for over half a century. The argyrodite family is compositionally flexible, with the general formula 𝐴 (12-𝑛-𝑦) 𝑚 ⁄ 𝑚+ 𝐿 6-𝑦 𝑛+ 𝑋 6-𝑦 2-𝑌 𝑦 -(A = Ag + , Cu + , Cd 2+ etc.; L = Ga 3+ , Si 4+ , Ge 4+ , P 5+ etc; X = S 2-, Se 2-, Te 2-; Y = Cl -, Br -, I -), [START_REF] Kuhs | The argyrodites-A new family of tetrahedrally close-packed structures[END_REF] where A cation content can vary from 3.25 to 9 in order to maintain charge neutrality. This family of compounds contains examples of fast Ag + and Cu + ion conductors (e.g., Cu6PS5Cl, 1 and Ag7NbS6 2 ) as well as materials interesting for their non-linear optical properties (e.g., Cd3.25PS5.5I0.5 [START_REF] Kuhs | The argyrodites-A new family of tetrahedrally close-packed structures[END_REF] ) and thermoelectric properties (e.g., Ag8SnSe6 3 ).

The reported crystal structures of argyrodites are related to the high temperature 𝐹4 ̅ 3𝑚 cubic polymorph of Ag8GeS6, where A cations often demonstrate extended dynamic disorder. Static ordering of cations at low temperatures, along with corresponding structural displacements, can lead to low temperature polymorphs with lower symmetry. [START_REF] Gaudin | Structures and phase transitions of the A7PSe6 (A= Ag, Cu) argyrodite-type ionic conductors. II. β-and γ-Cu7PSe6[END_REF][START_REF] Evain | Structures and phase transitions of the A7PSe6 (A= Ag, Cu) argyrodite-type ionic conductors. I. Ag7PSe6[END_REF] This dynamic disorder in higher symmetry polymorphs has led to extensive investigations of argyrodites as potential solid-state electrolytes in all solid-state lithium ion batteries (ASSBs). [START_REF] Rao | Studies of lithium argyrodite solid electrolytes for all-solid-state batteries[END_REF][START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] In the lithiumcontaining argyrodites, the introduction of a halide anion transforms the room temperature orthorhombic polymorph (Li7PCh6 (Ch = S, Se): space group Pna21) into cubic symmetry (Li6PCh5X (X = Cl, Br, I): space group 𝐹4 ̅ 3𝑚), while charge compensation occurs via reduction of the lithium content. The latter cubic phase exhibits an order-disorder phase transition from static order (LT cubic phase) to extended dynamic disorder (HT cubic phase), with delocalization of the lithium distribution. [START_REF] Deiseroth | PS6 and Li6PS5X (X: Cl, Br, I): Possible Three-dimensional Diffusion Pathways for Lithium Ions and Temperature Dependence of the Ionic Conductivity by Impedance Measurements[END_REF] In cubic oxide analogues such as Li6PO5Br, this phase transition does not occur, and lithium localization within the 𝐹4 ̅ 3𝑚 structure is maintained over a wide temperature range (173-873 K). [START_REF] Kong | 6PO5Br and Li6PO5Cl: The first Lithium-Oxide-Argyrodites[END_REF] In the ordered 𝐹4 ̅ 3𝑚 structure of Li6PO5Br, lithium ions fully occupy a single crystallographic 24g position.

In the dynamically disordered HT 𝐹4 ̅ 3𝑚 structure of the lithium sulphide argyrodites, the additional available lithium sites (Wyckoff position 48h) and extended diffusion of Li + are linked to a dramatic increase in ionic conductivity by six orders of magnitude. [START_REF] De Klerk | Diffusion mechanism of Li argyrodite solid electrolytes for Liion batteries and prediction of optimized halogen doping: the effect of Li vacancies, halogens, and halogen disorder[END_REF][START_REF] Kraft | Influence of lattice polarizability on the ionic conductivity in the lithium superionic argyrodites Li6PS5X (X= Cl, Br, I)[END_REF] The most highly conducting Li-containing argyrodites are based on mixed sulphidehalide compositions e.g., Li6PS5Br [START_REF] Rao | Studies of lithium argyrodite solid electrolytes for all-solid-state batteries[END_REF] and Li6.35P0.65Si0.35S5Br in which Li + is delocalized across 48h and 24g positions. [START_REF] Minafra | Effect of Si substitution on the structural and transport properties of superionic Li-argyrodites[END_REF] The high-temperature 𝐹4 ̅ 3𝑚 argyrodite structure has previously been described in terms of tetrahedral closepacking [START_REF] Kuhs | The argyrodites-A new family of tetrahedrally close-packed structures[END_REF] or as localised pseudo-octahedral cages of A cations surrounding one anion, and separated by the other anions. [START_REF] De Klerk | Diffusion mechanism of Li argyrodite solid electrolytes for Liion batteries and prediction of optimized halogen doping: the effect of Li vacancies, halogens, and halogen disorder[END_REF][START_REF] Kraft | Influence of lattice polarizability on the ionic conductivity in the lithium superionic argyrodites Li6PS5X (X= Cl, Br, I)[END_REF] In this work, we establish a new relationship between the structures of argyrodites and those of cubic anti-perovskites, which are themselves good conductors of lithium ions (i.e., Li3OCl1-xBrx). [START_REF] Zhao | Superionic conductivity in lithium-rich anti-perovskites[END_REF] We extend this relationship, which provides a framework for designing lithium ion conductors, from cubic to hexagonal antiperovskites and thus propose the lithium hexagonal argyrodite family, computationally identifying several new compositional targets within this family. Li6SiO4Cl2 is then successfully synthesized following these predictions based on mixed cubic and hexagonal stacking, and shows analogous dynamical cation site disorder to cubic argyrodites which is required for Li transport.

Experimental Procedures

Computational methods

Periodic plane-wave based density functional theory (DFT) calculations were performed using the VASP code (version 5.4.4). [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF] All calculations were performed with the projector augmented wave method, [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] a plane-wave cutoff energy of 700 eV and a k-point spacing of 0.15 Å -1 . Geometry optimisation of both atomic positions and unit cell parameters was terminated once all forces fell below 0.001 eV Å -1 . The PBE functional [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] was used to calculate relative energies and mechanical properties, and the PBEsol functional [START_REF] Perdew | Restoring the density-gradient expansion for exchange in solids and surfaces[END_REF] used for direct comparison between computational and experimental crystal structures. The convex hull of energies for all reported phases in the Li-Si-O-Cl-Br and Li-Si-O-Cl-I phase fields were calculated using pymatgen. [START_REF] Ong | Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis[END_REF] Normal mode calculations were performed using the harmonic approximation, with finite displacements of 0.01 Å and including distortions of the unit cell. This allowed the calculation of elastic constants, including the bulk and shear modulus. [START_REF] Deng | Elastic properties of alkali superionic conductor electrolytes from first principles calculations[END_REF] For comparison with existing compounds the elastic constants of Li3OBr (Pm3 ̅ m structure) [START_REF] Wortmann | 7O2Br3-Eine weitere Strukturvariante der neuen Alkalimetallchalkogenidhalogenid-Perowskite/Li7O2Br3-A Further Structure Variant of the New Alkali Metal Chalcogenide Halide Perovskites[END_REF] and Li6PS5Br (Cc structure of Li6PS5I) [START_REF] Kong | Lithium argyrodites with phosphorus and arsenic: order and disorder of lithium atoms, crystal chemistry, and phase transitions[END_REF] were computed using the same method.

Synthesis

Materials

Li2CO3 (99.99%), SiO2 (silica gel, technical grade, particle size 40-63 μm) and LiCl (> 99.0%) were purchased from Sigma Aldrich.

Synthesis of Li4SiO4 22

Precursors were dried overnight at 473 K before weighing. Li2CO3 (1.2331 g, ) and SiO2 (0.5013 g) were weighed according to the stoichiometric 2:1 ratio. The powders were ground with an agate pestle and mortar for 15 min, placed into an alumina crucible and heated in air to 1073 K at a ramp rate of 5 K min -1 , held at 1073 K for 12 h and cooled at a ramp rate of 5 K min -1 . The resulting product was ground to obtain a fine powder, which was then used as a precursor in the final synthesis step.

Synthesis of Li6SiO4Cl2-xBrx

Li4SiO4, LiCl and LiBr were vacuum dried overnight (under 10 -4 mbar) before placing them in an Ar-filled glove box.

All precursors and resulting powders were then handled in an Ar-filled glove box. LiCl, LiBr and Li4SiO4 were mixed in the stoichiometric ratio, ground with an agate pestle and mortar for 15 min and transferred to an alumina crucible.

The crucible was placed in a silica tube which was sealed under vacuum (<10 -4 mbar). The tube was heated to 798 K at a ramp rate 5 K min -1 , held at 798 K for 12 h and cooled at a rate of 5 K min -1 . Once cool, the silica tube was opened inside the Ar glovebox, and the powder ground in a pestle and mortar for further characterisation.

Single crystals of Li6SiO4Cl2 were grown by mixing LiCl and Li4SiO4 in the stoichiometric ratio, heating the mixture to 883 K at a ramp rate of 5 K min -1 , annealing it for 3 h and slow cooling at 3 K h -1 to room temperature.

Powder X-ray Diffraction

Routine assessment of sample purity was carried out using a Bruker D8 Discover diffractometer with monochromatic Cu radiation (Kα1, λ = 1.54056 Å) in Debye Scherrer transmission geometry with sample powders loaded into 0.5 mm borosilicate glass capillaries.

Synchrotron X-ray diffraction (SXRD) was performed at Diamond Light Source U.K., on high-resolution beamline I11, [START_REF] Thompson | Beamline I11 at Diamond: A new instrument for high resolution powder diffraction[END_REF] at λ = 0.826552 Å. The pattern was recorded in transmission mode [0° < 2θ < 150°] using a multi analyser crystal (MAC) detector. The sample was introduced into a 1.0 mm diameter borosilicate glass capillary. The experiment was performed at room temperature.

Synchrotron variable temperature X-ray diffraction (VT-XRD) was performed in transmission mode using a position sensitive detector (PSD, λ = 0.82660 Å) on a sample which was introduced into a 1.0 mm diameter silica capillary.

The experiment was performed in the temperature range 298 -798 K in 25 K steps on heating, then cooled directly to room temperature to assess reversibility.

Rietveld refinements were carried out using TOPAS Academic. [START_REF] Coelho | Whole-profile structure solution from powder diffraction data using simulated annealing[END_REF]

Single crystal X-ray diffraction

A twinned crystal was examined with a Rigaku MicroMax-007 HF X-ray generator equipped with a Mo-Kα rotatinganode microfocus source and a Saturn 724+ detector. The data were collected at 100 K. Refinement of the cell parameters, indexing of twin components, and reduction of the data were performed on the obtained diffraction images with use of the software package CrysAlisPro. [START_REF]CrysAlisPro Software System[END_REF] The volume ratio of the twin components was about 1/1.

The degree of overlap of two components is around 1%. The twin operator is 92.82° rotation around the -0.59 0.37 -0.72 reciprocal axis. The detwinned HKLF4 format file was used for structure solution using Olex2. [START_REF] Dolomanov | OLEX2: a complete structure solution, refinement and analysis program[END_REF] The crystal structure was solved with Intrinsic Phasing method provide by the ShelXT 27 structure solution program.

Refinements were carried out with the ShelXL 28 refinement package using Least Squares minimization.

Differential Scanning Calorimetry

Heat flux profiles were measured from 15.1 mg of powdered sample in a 40 µL aluminium crucible cold-welded under Ar atmosphere (<0.1 ppm O2, H2O) using a Netzsch DSC 404 F1 differential scanning calorimeter. Data were recorded on heating to 823 K and then cooling to 323 K using heating and cooling rates of 10 K min -1 under a constant 50 mL min -1 flow of helium. The transition temperature is the average of the values obtained from both the heating and cooling curves, which were extracted through peak fitting.

AC Impedance Spectroscopy and DC polarisation

A pellet of Li6SiO4Cl2 was made by uniaxially pressing ~30 mg of material in an 8 mm cylindrical steel die at a pressure of 125 MPa. The pellet was sintered in an evacuated, flame dried silica tube for 12 h at 848 K. Using this method, a relative density of 84% was achieved.

AC impedance measurements were conducted using an impedance analyser (Keysight impedance analyser E4990A). A sputtered gold coating of 300 nm thickness was used as the ion blocking electrodes. Sputtering was achieved under argon atmosphere using a Q150R sputter coater and temperature-dependent conductivity measurements were performed under an argon atmosphere over a frequency range of 2 MHz -20 Hz (with an amplitude of 100 mV). Measurements were performed in the temperature range 333 -575 K in 20 K steps. The ZView2 program [START_REF] Johnson | A Software Program for IES Analysis 3.5 d[END_REF] was used to fit the impedance spectra with an equivalent circuit.

A pellet of 84% relative density was used for DC potentiation polarisation measurements. DC polarisation data was collected at 300 °C on an Au|Li6SiO4Cl2|Au symmetric cell by applying constant potentials of 0.05, 0.1, 0.5 and 1V for 7200s and monitoring the current variation with time. Once a constant current was achieved the current was recorded and plotted against the applied voltage allowing for the electronic conductivity (σe) to be extracted from following equation

𝜎 𝑒 = 𝑈𝑙 𝐴𝐼
Where 𝑈 refers to the polarisation voltage, 𝑙 to the pellet thickness, 𝐴 to the Au electrode area and 𝐼 to the current, respectively 3.7 Nuclear magnetic resonance (NMR) spectroscopy [START_REF] Johnson | A Software Program for IES Analysis 3.5 d[END_REF] Si magic-angle spinning (MAS) NMR spectra were recorded with a 4 mm HXY MAS probe in double resonance mode on a Bruker 9.4 T Avance III HD spectrometer. [START_REF] Johnson | A Software Program for IES Analysis 3.5 d[END_REF] Si NMR data was obtained using a pulse length of 5 μs at a radio frequency (rf) amplitude of 50 kHz and at a MAS rate of νr = 10 kHz. The sample was packed into a rotor in an Ar-filled glovebox to eliminate exposure to air and moisture. [START_REF] Johnson | A Software Program for IES Analysis 3.5 d[END_REF] Si shifts were externally referenced to Octakis(trimethylsiloxy)silsesquioxane ( 29 Si chemical shifts were externally referenced to the lowest frequency signal of octakis(trimethylsiloxy)silsesquioxane at -109 ppm, [START_REF] Brendler | A pentacoordinate chlorotrimethylsilane derivative: a very polar snapshot of a nucleophilic substitution and its influence on 29 Si solid state NMR properties[END_REF] relative to tetramethylsilane primary reference at 0.0 ppm).

Variable temperature [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li NMR experiments were recorded with a 4 mm HX High Temperature MAS Probe on a 9.4

T Bruker Avance III HD spectrometer under static conditions with the X channel tuned to [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li at ω0/2π ( 7 Li) = 156

MHz. The sample was sealed in glass ampoule and the spectra were recorded with a pulse length of 1.5 µs at a rf field amplitude of ω1/2π = 83 kHz and referenced to 10 M LiCl in D2O at 0 ppm.

The homonuclear dipolar coupling constant between two [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li nuclear spins d7Li7Li (in Hz) can be calculated via the following expression:

where µ0 is the permeability of free space, ℏ the reduced Planck's constant, γ7Li the gyromagnetic ratio of 7Li and rij the Li-Li distance.

Temperature calibrations were performed using the chemical shift thermometers Pb(NO3)2 using 207 Pb NMR [START_REF] Beckmann | A thermometer for nonspinning solid-state NMR spectroscopy[END_REF] and CuI and CuBr using 63 Cu NMR. [START_REF] Becker | Temperature dependence of NMR chemical shifts in cuprous halides[END_REF][START_REF] Wu | Temperature calibration for high-temperature MAS NMR to 913 K: 63 Cu MAS NMR of CuBr and CuI, and 23 Na MAS NMR of NaNbO3[END_REF] The errors associated with this method were calculated using the isotropic peak line broadening and range from 5 -20 K.

Maximum entropy methods

Maximum entropy method (MEM) analysis was performed on SXRD and VT-XRD data using the software Jana2006 [START_REF] Petříček | Crystallographic computing system JANA2006: general features[END_REF] and BayMEM. [START_REF] Smaalen | The maximum-entropy method in superspace[END_REF] Rietveld models from Topas were input into Jana2006, which was used to extract Fobs and generate MEM inputs. BayMEM was used to calculate the electron density distribution using the Fobs and the number of electrons in the nominal stoichiometry. Calculations using the number of electrons associated with a 10% Li deficiency were also performed for all models investigated and found to have negligible impact in the resulting electron density distributions. All MEM results were visualized in the VESTA crystal structure visualization software. [START_REF] Momma | VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data[END_REF] 4 Results and Discussion

Selection of target compounds by structural analogy between argyrodite and perovskite

We describe the relationship between the argyrodite structure, exemplified by Li6PO5Br (Figure 1a), and the cubic anti-perovskite structure. In analogy with perovskites of general formula ABX3, Li6PO5Br can be written as:

[((PO4)0.5Br0.5)(O0.5□0.5)Li3]2, where □ is a vacancy. As such, Li6PO5Br could be considered as an inverse double perovskite. When viewed as a cubic anti-perovskite, the lithium ions in Li6PO5Br replace the anion positions in the conventional cubic perovskite (e.g., SrTiO3, [START_REF] Abramov | The chemical bond and atomic displacements in SrTiO3 from X-ray diffraction analysis[END_REF] Figure 1b). The phosphate polyanions and bromide anions replace the larger A-site cations in a rock-salt ordered fashion. As the cubic perovskite can be represented as a cubic closepacked stacking of AX3 layers along [111], here the cubic stacking is an alternation of Li3Br and Li3PO4 layers that produces the rock-salt anion ordering. The remaining, isolated oxide anions then occupy half of the octahedral Bsite positions between these layers also in an ordered manner, leaving the other half of the B-sites vacant. The P-O bond vector of each tetrahedral (PO4) 3-anion is directed towards the vacant B-sites, coupling the rock salt order of the two A site anions to the anion vacancy defect ordering on the B site. Rock-salt order of 50% B-site vacancies occurs in the structure of well-known K2PtCl6 38 (Figure 1c). When considering these structural similarities between the lithium argyrodites and the lithium anti-perovskites, it is evident that the lithium sites in both structures are topologically equivalent. As a result, both structures share the same network of connectivity between lithium sites (Figure 1d and 1e), suggesting that the underlying mechanism for high ionic conductivity is common to both material

d 7Li7Li = - μ 0 8π 2 ℏγ 7Li 2 r LiLi 3 (Eq. 1)
families. The existence of both cubic and hexagonal perovskites associated with the distinct stackings of the AX3 layers suggests the existence of hexagonal Li argyrodites that are also based on alternating Li3Br and Li3PO4 stackings.

of Although closely related in composition to argyrodite, the compound Cu8GeSe6 has a hexagonal P63mc hightemperature polymorph rather than the cubic 𝐹4 ̅ 3𝑚 argyrodite structure, [START_REF] Jaulmes | Varietes de haute et basse temperature du seleniure double de cuivre et de germanium[END_REF][START_REF] Onoda | Superspace-group approach to the phase transition of Cu8GeSe6[END_REF][START_REF] Jiang | Cu8GeSe6-based thermoelectric materials with an argyrodite structure[END_REF] as do the compounds Ag5PS4I2 42 and Li8SiO6. [START_REF] Hofmann | Ein neues Oxogermanat: Li8GeO6= Li8O2 [GeO4].(Mit einer Bemerkung über Li8SiO6 und Li4GeO4)[END_REF] Analysis of these structures show that they contain similar close-packed layers to those present in the cubic anti-perovskite description of the argyrodite family, but that, instead of a -c-(-a-b-c-) close-packed stacking, they are stacked in a -h-c-h-c-(-a-b-a-c-) manner, as would occur in a hexagonal anti-perovskite. Different closepacked layer stacking motifs are known within the anti-perovskites. [START_REF] Gäbler | Stacking Design of Inverse Perovskites: The Systems (Sr3-xBaxN) E, E= Bi[END_REF] The argyrodite/cubic anti-perovskite structural relationship suggests that Li-containing materials based on the hexagonal, rather than cubic, anti-perovskite stacking of these layers, related to the hexagonal structures of Cu8GeSe6 and Ag5PS4I2, would be analogous to the lithium-containing argyrodites, where the number of Li sites and extent of Li ordering among them control ionic conductivity.

Starting from the cubic Li6PO5X (X = Cl, Br) argyrodites, the phosphate (PO4) 3-(P-O bond length: 1.54 Å) was replaced with silicate (SiO4) 4-(Si-O bond length: 1.65 Å) to increase the A-site cation radius and thus drive a cubic to hexagonal transition through the perovskite tolerance factor (rP(V) = 0.17 Å, rSi(IV) = 0.26 Å). [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF] The isolated oxide ion was replaced by a halide ion (F, Cl, Br, I) to balance charge. Structures were built with compositions Li6SiO4XX΄ Periodic DFT calculations showed that the composition Li6SiO4ClBr in this P63mc structure was the most stable when compared with a stoichiometric combination of lithium orthosilicate Li4SiO4 and the relevant binary lithium halides (LiCl and LiBr) (Table S1). Through phonon calculations, lower energy structures than the high symmetry hexagonal structure were found, arising from rotation of lithium atoms off the mirror plane of P63mc through activation of displacive Γ2, Γ3, Γ5, M2 and M3 modes (Figure 3b), leading to four potential lower symmetry polymorphs in space groups P31c, P63, Pna21 and Pca21. The stabilities of all compositions were re-computed in these four lower symmetry polymorphs, revealing that Li6SiO4Cl2, Li6SiO4Br2, Li6SiO4ClBr and Li6SiO4ClI were predicted to be stable (Figure 3a, Table 1, Table S1). These results suggest that a new lithium hexagonal argyrodite family of compounds may be experimentally accessible, with Li6SiO4Cl2 and Li6SiO4ClBr (with the larger Br ordered on the A site) predicted to be the most stable and chosen as targets for experimental synthesis.

DFT calculations of the elastic constants of Li6SiO4Cl2 in the Pna21 structure result in a computed bulk modulus, BDFT, of 53 GPa and a computed shear modulus, GDFT, of 31 GPa. This material is considerably softer than oxide solid state lithium electrolytes such as Li7La3Zr2O12 (BDFT = 117 GPa, GDFT = 64 GPa), [START_REF] Deng | Elastic properties of alkali superionic conductor electrolytes from first principles calculations[END_REF] and has mechanical properties closer to that of the lithium anti-perovskites (e.g., Li3OBr, BDFT = 50.6 GPa, GDFT = 37 GPa). Li6SiO4Cl2 is stiffer than related sulphide compounds (e.g. Li6PS5Br: BDFT = 27 GPa, GDFT = 14 GPa) and consequently easily meets the Monroe-Newman criterion for the prevention of dendrite growth. [START_REF] Monroe | The impact of elastic deformation on deposition kinetics at lithium/polymer interfaces[END_REF] Future electrolytes based on these mixed oxide/halide compounds may thus be able to overcome some of the processing challenges and mechanical stability issues inherent in the use of pure oxide and sulphide ceramics. [START_REF] Kim | Solid-State Li-Metal Batteries: Challenges and Horizons of Oxide and Sulfide Solid Electrolytes and Their Interfaces[END_REF]

Synthesis, thermal behaviour and structure determination

Compounds were synthesised as powders with compositions Li6SiO4Cl2-xBrx (x = 0, 0.5,1) to explore the computationally predicted targets. Synthesis of powders was attempted at varying reaction temperatures (50 K steps from 723 K to 873 K) and reaction times (12 h, 24h, 48h), and starting materials and resulting powders were handled under argon atmosphere and annealed in alumina crucibles in evacuated silica ampoules (Section 3.2).

Synthesis at 823 K for 48 h yielded phase pure powders for Li6SiO4Cl2. This new phase persists in the compositional range Li6SiO4Cl2-xBrx (0 ≥ x ≥ 1) and displays increased lattice parameters as a function of bromine content (Figure S1). Despite this clear indication of anion substitution all compositions with x > 0 contained impurities; only Li6SiO4Cl2 was synthesised as a phase pure white powder, and as such all further discussion therefore concerns Li6SiO4Cl2. Single crystals of Li6SiO4Cl2 were synthesised by annealing at 883 K for 3 h before slow cooling (3 K h - 1 ) to room temperature. High resolution synchrotron XRD data (SXRD) were collected on the powder from 298 K to 798 K in 25 K steps. At temperatures 473-498 K, convergence of some peaks and disappearance of other small peaks indicates a phase transition to a high temperature, higher symmetry phase (denoted HT-phase) (Figure 4a,4b). This transition is also observed in differential scanning calorimetry (DSC) data measured from Li6SiO4Cl2 powder, which shows endothermic and exothermic events on heating and cooling, respectively, associated with this reversible phase transition (Figure 4c). The exact transition temperature determined from DSC data is 469. improved Rwp from 4.61 to 4.27 and χ 2 from 2.81 to 2.61 (Figure S3). Small impurity phases of LiCl (1.95 %) and Li2SiO3 (2.35 %) were observed (Figure S4a). Maximum entropy method (MEM) analysis did not show any additional electron density, indicating that all lithium sites are accounted for in the structural model described above (Figure S6). The final Rietveld fit is shown in Figure 4a and refined structural parameters are presented in Tables S7 andS8. The refined composition is Li5.957(2)Si1.00O3.986(2)Cl1.985(1) and the RT-phase will therefore be denoted RT-Li6SiO4Cl2 hereafter for simplicity.

Room temperature phase -Structure Description

As described in Section 4.1, the orthorhombic structure arises from rotation of lithium atoms off the mirror plane in P63mc through activation of Γ5 and M2 modes, stabilising the Pna21 symmetry. This leads to a change in anionic and cationic coordination environments compared to the hexagonal defect anti-perovskite parent.

The experimentally-determined RT structure is shown in Figure 5a, highlighting the atomic packing within the structure defined by alternating Li3SiO4 and Li3Cl layers. Comparison of the experimentally determined Pna21 structure to the computed structure shows almost identical atomic positions (Figure S7a). The chloride ions that occupy 50% of the anti-perovskite B-site positions (Cl1) are found in an octahedral environment with bond lengths ranging from 2.415(3) Å to 2.621(3) Å to neighbouring Li. Compared to the idealised octahedral environment expected for B-site anions in anti-perovskites, the octahedra are distorted slightly as a result of the non-rigid rotation of Li ions and the displacement of chlorine towards neighbouring vacant B-sites. The chloride ions that occupy 50% of the A site (Cl2), typically forming cuboctahedral (12-fold) coordination environments in hexagonal perovskites such as 4H-BaMnO3, occupy a distorted octahedral environment in the orthorhombic polymorph with bond lengths from 2.446(3) Å to 2.798(7) (Figure S8). All lithium atoms are found in tetrahedral LiCl2O2 coordination environments. Due to the difference in ionic radii (rCl = 1.81 Å, rO = 1.40 Å), [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF] lithium atoms are displaced towards the oxygen atoms (Figure 5c). Li-O bond lengths vary from 1.829(3) Å to 1.931(3) Å, comparable to distances reported for the cubic argyrodite Li6PO5Cl (1.93 Å) [START_REF] Kong | 6PO5Br and Li6PO5Cl: The first Lithium-Oxide-Argyrodites[END_REF] and shorter than Li-O bonds in Li4SiO4

(1.84 -2.51 Å). [START_REF] Tranqui | Crystal structure of ordered Li4SiO4[END_REF] Li-Cl bond lengths range from 2.415(3) Å to 2.798(3) Å. These values are larger than expected for typical LiCl4 bond lengths, e.g., 2.38 Å in Li2MgCl4. [START_REF] Partik | Kristallstrukturen von MgCr2O4-Typ Li2VCl4 und Spinell-Typ Li2MgCl4 und Li2CdCl4[END_REF] The difference in bond lengths compared to homoleptic In the RT-structure, LiCl2O2 tetrahedra are connected through the corner-sharing of Cl and O vertices, and the sharing of Cl-Cl edges; they are referred to using the central lithium atom, i.e., Li(1)O2Cl2 as a Li1 tetrahedron.

Edge-sharing between Li1-Li3 and Li4-Li6 tetrahedra is facilitated by the lithium ion displacements away from the shared Cl-Cl edges (Figure 6a and 6c), minimising electrostatic repulsions.

The Li3SiO4 layer can be described (Figure 6a) as alternating corner-sharing Li2-Li3 and Li1-SiO4 chains running along b, connected in the a-direction by Li1-Li2, SiO4-Li2, SiO4-Li3 corner-sharing and Li1-Li3 edge-sharing (Figure 6b and6c). Cl2 atoms located in the Li3Cl layers connect Li3SiO4 layers along the stacking c-direction (Figure 6b).

The Li3Cl layer is made up of Li4, Li5 and Li6 tetrahedra connected through Li4-Li5, Li4-Li6, Li5-Li6 corner-sharing and Li4-Li6 edge-sharing (Figure 6c and6d). The Li3Cl layer is connected to the Li3SiO4 layer through cornersharing with Li1, Li2, Li3 and SiO4 tetrahedra, forming the 3D-network of corner-and edge-shared tetrahedra.

High temperature phase -Structure determination

Indexing of the HT phase (above 523 K) with TOPAS resulted in a hexagonal unit cell with lattice parameters a = 6.11 Å and c = 10.02 Å. Systematic absences in the powder pattern were consistent with the P31c and P63mc space groups, agreeing with the computationally-stable hexagonal structures (Section 4.1, Figure 3c). Very similar Pawley fits were obtained using these symmetries, and lattice parameters refined to a = 6.110805 (12) Å and c = 10.02068(3) Å for P63mc (Figure S4b). The higher symmetry computed P63mc structure gave a reasonable starting point for Rietveld refinement. The unit cell contains two distinct crystallographic Cl positions (Wyckoff position 2b), two O positions (Wyckoff positions 6c and 2a), one Si position (Wyckoff position 2a) and three Li positions (Wyckoff positions 6c and 12d). Occupancies of Cl and O sites were refined and the occupancies of Li sites were fixed to ensure charge neutrality. In the computational model, Li2 is located on the mirror plane in special position 6c (Figure S7b). Inspection of the Fourier Density Map (FDM) reveals extra electron density around this position and indicates displacement of Li2 away from the mirror plane onto the general position 12d (Figure S9a). Replacing the 6c site with the 12d position improved Rwp from 3.61 to 2.98%. The FDM also revealed electron density around the Li1 position along the c-direction (Figure S9b). Splitting the Li1 site into two distinct Li1a and Li1b sites along c further improved Rwp to 2.75%. The computed P31c model (Figure 3c) is similar, only allowing two distinct 6c crystallographic sites for Li2 instead of one 12d site. A fit of the diffraction data to this model, with independent refinement of both Li2a and Li2b 6c occupancies, did not result in an improved fit. The P63mc model was therefore used for final refinement. MEM analysis also supports the displacement of Li1 and Li2 positions (Figure S6). The final refinement is shown in Figure 4b and the refined structural parameters are provided in Table S7. The refined composition is Li5.82(4)Si1.00O3.934(6)Cl1.956(4) and the HT-phase will therefore be denoted HT-Li6SiO4Cl2 hereafter for simplicity.

High temperature phase -Structure description

The HT P63mc structure can be described as a 4H hexagonal anti-perovskite of general formula ABX3, with structural similarities to 4H-BaMnO3 (Figure 2b) in which half of the B sites are vacant. This is consistent with cation-deficient hexagonal perovskites with the formula A2BX6, such as Rb2MnF6 with P63mc symmetry. [START_REF] Schippers | Derivation and discussion of crystal structures of compounds ABX3 and A2BX6: Part I. Derivation of the structures[END_REF] The HTstructure of Li6SiO4Cl2, consisting of alternating hexagonally stacked Li3Cl and Li3SiO4 layers, is shown in Figure 5b. The (poly)anion network in the HT-structure remains unchanged through the phase transition from the RTstructure, with the observed symmetry change originating from displacements in Li positions. The lithium environments remain tetrahedral (LiO2Cl2) with lithium atoms displaced towards the oxygens, due to its smaller ionic radius compared to chlorine. Li-Cl bond lengths vary from 2.366(3) Å to 2.908(8) Å, and Li-O bond lengths range from 1.847(4) Å to 1.942(4) Å (Figure 5d).

As shown in Figures 5c and5d, the ordered lithium atoms of RT-Li6SiO4Cl2 are disordered in the higher symmetry HT-structure. Similar to the results from normal mode calculations described in Section 4.1., this order-disorder transition occurs via activation of occupational M2 and Γ5 modes, with a small contribution from displacive Γ5 modes that predominantly involve displacement of the Li1 atoms within the close-packed Li3SiO4 layer (Figure S10). This disorder of the A cation positions is well-known in 𝐴 (12-𝑛-𝑦) 𝑚 ⁄ 𝑚+ 𝐿 6-𝑦 𝑛+ 𝑋 6-𝑦 2-𝑌 𝑦 -argyrodites, [START_REF] Nilges | A structural differentiation of quaternary copper argyrodites: Structure-property relations of high temperature ion conductors[END_REF][START_REF] Albert | Disorder in Ag7GeSe5I, a superionic conductor: temperature-dependent anharmonic structural study[END_REF] and provides a direct comparison of Li disorder between oxide and sulphide argyrodite materials. The delocalisation of Li positions in HT-Li6SiO4Cl2 resembles that observed in high-temperature 𝐹4 ̅ 3𝑚 cubic argyrodites Li6PS5X (X = Cl, Br, I), in which Li partially occupies both tetrahedral and trigonal planar coordination environments (48h and 24g Wyckoff sites, respectively, Figure S9c), and is one of the reasons for reduced energy barriers for bulk Li ion transport in these materials. [START_REF] Deiseroth | PS6 and Li6PS5X (X: Cl, Br, I): Possible Three-dimensional Diffusion Pathways for Lithium Ions and Temperature Dependence of the Ionic Conductivity by Impedance Measurements[END_REF] The order-disorder transition of Li6SiO4Cl2 (Figure 5c and5d) is similar to the behaviour of many argyrodites;

Cu8GeSe6 displays an order-disorder transition between P63cm and P63mc structures, where Cu ions are disordered in the latter similarly to Li on the 6c and 12d Wyckoff positions in Li6SiO4Cl2, [START_REF] Jaulmes | Varietes de haute et basse temperature du seleniure double de cuivre et de germanium[END_REF][START_REF] Jiang | Cu8GeSe6-based thermoelectric materials with an argyrodite structure[END_REF] and analogous behaviour is observed in silver-containing argyrodites. [START_REF] Kuhs | The argyrodites-A new family of tetrahedrally close-packed structures[END_REF] Specifically, cubic Li6PS5I has ordered lithium positions at low temperature (<180 K) and a disordered structure at 298 K in which lithium is delocalised across the 24g and 48h positions (Figure 5e), [START_REF] Kong | Lithium argyrodites with phosphorus and arsenic: order and disorder of lithium atoms, crystal chemistry, and phase transitions[END_REF] resulting in comparable local lithium coordination environments with Li6SiO4Cl2 (Figure 5c and 5d) that are consistent with dynamical disorder. This is distinct from the cubic oxide argyrodite, Li6PO5Br, in which lithium remains localised on the trigonal planar 24g Wyckoff position in the 𝐹4 ̅ 3𝑚 structure over a wide (173-873 K) temperature range (Figure 1d). [START_REF] Kim | Solid-State Li-Metal Batteries: Challenges and Horizons of Oxide and Sulfide Solid Electrolytes and Their Interfaces[END_REF] Stabilisation of this lithium disorder which enables access to higher energy sites in Li6SiO4Cl2 would increase lithium ion migration compared against the room temperature structure.

Transport properties

The ionic conductivity of Li6SiO4Cl2 was investigated through AC impedance spectroscopy on a sintered white pellet of ~84% theoretical density (pellet cold-pressed under 125 MPa and sintered in evacuated silica ampoule for 12 h at 848 K). A typical set of data measured at 533 K in inert atmosphere are shown in Figure 7a. The impedance complex plane plots, Z*, consist of a high-frequency arc with the presence of a small low-frequency inclined spike.

The large single arc is attributed to the sample bulk, as shown by overlapping peaks in the combined Zʹʹ/Mʹʹ spectra (Figure S12a). The associated capacitance of the high-frequency arc, 0.6 pF cm -1 , corresponding to a permittivity of ~6.8, is also consistent with the grain response (Figure S12b). The low-frequency spike is attributed to the double-layer capacitance at the blocking sample-electrode interface, where Li ions cannot pass. To a first approximation, the high-frequency arc could be modelled with an equivalent circuit consisting of a resistor in parallel with a constant phase element (CPE). Impedance data were fitted to the equivalent circuit using ZView software (Figure 7a). In general, the material is homogeneous and shows a total conductivity of 6.2 × 10 -6 S cm -1 at 575 K and ~10 -10 S cm -1 at room temperature. DC polarisation measurements show a low electronic contribution of > 0.5 % at 300 °C to the overall conductivity (Figure S13). From the low-frequency intercept of the impedance arc on the Zʹ axis, values of the total resistance were obtained and are shown in Arrhenius format in Figure 7b. A change in slope can be observed above ~498 K, agreeing with the phase transition observed via VT-XRD and DSC experiments. The orthorhombic phase (red circles) and the hexagonal phase (black squares) have activation energies of 0.560(8) eV and 0.444(9) eV, respectively.

The local ionic mobility of lithium was investigated through [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li solid-state NMR. The temperature dependence of the static [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li NMR spectra of Li6SiO4Cl2 over the temperature range of 331-593 K is shown in Figure 7c. At temperatures where ion mobility is minimal i.e., the rigid lattice regime, the ½ ↔ -½ central transition is broadened by [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li- [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li homonuclear dipolar coupling. For Li6SiO4Cl2, this region is between 350 and 450 K, where the linewidth of the central transition is ~ 5.5 kHz. As the temperature is increased above 450 K, the linewidth decreases drastically over a small temperature range (Figure 7d) due to the increasing motion of the Li spins continuously averaging the dipolar interactions, and clearly evidences an increase in ionic mobility facilitated by the phase transition from the ordered RT-orthorhombic to the higher symmetry HT-hexagonal phase.

Through calculation of the strength of [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li- [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li homonuclear dipolar coupling of both the room temperature and high temperature phases of Li6SiO4Cl2, further insight into the local ion dynamics of the two phases can be determined.

By extracting the shortest Li-Li distance from the SXRD data (Li1-Li5, 2.697(6) Å), the (absolute value) of the dipolar coupling constant can be calculated (Eq. 1) as ~5.8 kHz in RT-Li6SiO4Cl2 in full agreement with the experimental linewidth observed, while ~8.2 kHz is obtained from Li1a and Li2 (2.410(5) Å) for HT-Li6SiO4Cl2. Due to the increase in dipolar coupling after the phase transition, it would be expected that [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li spectra would demonstrate increased linewidths at higher T in the absence of an increase of Li ion mobility. However, the line narrowing observed (Figure 7d) is a clear indication that the phase transition from the RT-orthorhombic phase to the higher symmetry HThexagonal phase facilitates an increase in Li ion mobility, as also observed in AC impedance measurements (Figure 7b). The lithium positions in the orthorhombic RT structure are fully ordered. The delocalisation of lithium in HT-Li6SiO4Cl2 across the 6c and 12d sites (Figure 5c and5d) results in a lowering of the activation energy from RT-Li6SiO4Cl2. The activation energies obtained from Li6SiO4Cl2 are lower than those extracted from Li6PO5Cl across the measured temperature range, despite the latter adopting the higher symmetry cubic 𝐹4 ̅ 3𝑚 structure. 9 of the central [START_REF] Deiseroth | 6PS5X: a class of crystalline Li-rich solids with an unusually high Li + mobility[END_REF] Li NMR transition. The temperature at which the phase change is detected via DSC measurements is also noted (dashed line).

The distinction between the oxide materials Li6SiO4Cl2 and Li6PO5Cl is likely the increased size of the trigonal anion window (Figure 5f) and the associated dynamical Li disorder (absent in cubic Li6PO5Cl, where Li is localised on the 24g position) in hexagonal Li6SiO4Cl2 (partial occupancy of 6c and 12d positions by Li), leading to lower activation energies for ion mobility. Local jumps of Li + between these tetrahedral sites separated by the trigonal anion windows are the most favourable for ionic diffusion in argyrodites. [START_REF] Deiseroth | PS6 and Li6PS5X (X: Cl, Br, I): Possible Three-dimensional Diffusion Pathways for Lithium Ions and Temperature Dependence of the Ionic Conductivity by Impedance Measurements[END_REF] Occupational disorder via chalcogenide-halide mixing on the anion position that forms part of the trigonal window that mobile cations traverse (Figure 5f) is an important route to enhancing Li ion mobility. [START_REF] Minafra | Effect of Si substitution on the structural and transport properties of superionic Li-argyrodites[END_REF][START_REF] Adeli | Boosting solidstate diffusivity and conductivity in lithium superionic argyrodites by halide substitution[END_REF] Unlike Li6PO5Cl, where this window is formed from three oxide anions, the LiCl2O2 environments of Li6SiO4Cl2 yield a window described by one chloride and two oxide anions. The larger Cl (rCl = 1.81 Å, rO = 1.40 Å) [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF] increases the size of this window, further lowering the activation energy of ion transport compared to Li6PO5Cl.

Conclusion

The connection between argyrodite and anti-perovskite structures is established, thus identifying close-packed layers that can be assembled through the diversity of combined hexagonal and cubic stacking operations known to give the perovskite materials family its breadth of scientific and technological importance. We exemplify these mixed stackings in the hexagonal argyrodite materials family through the prediction then isolation of the non-toxic and earth abundant Li6SiO4Cl2. Here the resulting layer stacking gives access to the disordered Li distribution which is observed in fast lithium ion conductors, a distribution not accessible to its nearest known cubic compositional counterpart, Li6PO5Cl. Exploration of the many accessible mixed layer sequences and associated site and chemical orderings afforded within this family opens new routes to the tuning and creation of ion transport pathways through variation of both composition and structure, as well as broader functional outcomes based on the importance of the cubic argyrodite materials.

For Table of Contents Only

  Initially, Pawley fits were performed on SXRD data, refining the lattice parameters and the background using a Chebyshev function with 12 parameters. The peak shape was modeled using a Pseudo-Voigt function (high temperature patterns) and a Thomson -Cox function (room temperature patterns). Refined parameters from final Pawley fits were then used as starting points for Rietveld refinements where the following parameters were refined: (1) scale factor, (2) atomic coordinates (3) isotropic (Li, Si, O) and anisotropic (Cl) displacement parameters, and (4) atomic occupancies: the occupancies of Cl, O and Li were refined whilst the occupancy of the Si site was set to the nominal value.

Figure 1 .

 1 Figure 1. The cubic argyrodite structure of Li6PO5Br (Li: pink, P: light green, O: red, Br: dark green, vacant B-site (V): grey; the arrow shows the direction of the P-O bond vector oriented towards the vacant B-site), showing the relationship with (b) the cubic perovskite SrTiO3 (Sr: green, Ti: dark red, O:red) and (c) the B-site vacancy-ordered K2PtCl6 (K: dark green, Pt: red, Cl: light green, Vacancies: grey). d) The cubic anti-perovskite structure of Li3OBr, highlighting the octahedral coordination environment of Li, coordinated by 4 A-site (Br: dark green) and two B-site (O: red) anions. e) Compared with Li3OBr, the cubic argyrodite structure of Li6PO5Br has a reduced Li coordination number defined by 4 A-site (Br: dark green, PO4: polyanion light green) and one B-site anion (O: red) with half of the B-sites vacant (Vacancy: grey).

  (X, X΄ = F, Cl, Br, I) with the h-c-h-c-(a-b-a-c-) stacking pattern, following Cu8GeS6, in the P63mc space group. This affords a hexagonal anti-perovskite in which half of the A-sites are occupied with silicate polyanions and half with halide anions. Half of the B-sites are then occupied in an ordered manner by the remaining halide anions, with the vacant B-sites chosen to avoid interactions between the halide anions and the corners of the silicate tetrahedra (Figure2a).

Figure 2 .

 2 Figure 2. a) The DFT optimised P63mc structure of Li6SiO4ClBr (Li: pink, Si: light green, O: red, Br: dark green, Cl: blue, vacant B-site: grey) is compared to (b) the 4H-BaMnO3 hexagonal perovskite structure (Ba: green, Mn: blue, O: red). The oxide anions in the conventional hexagonal perovskite are replaced by lithium cations in Li6SiO4ClBr, the A-site barium cations are replaced by an ordered mixture of silicate polyanions and bromide anions, and the B-site manganese cations are replaced with ordered chloride anions and vacancies. In Li6SiO4ClBr, close-packed Li3Br layers (dark pink lithium) alternate along the c direction with close-packed Li3SiO4 layers (light pink lithium) in an a-b-a-c-stacking sequence. Half of the octahedral B-sites between these layers are then occupied by chlorine anions (blue octahedra), leaving half of the B-sites vacant (grey octahedra). c)h) Comparing close-packed AX3 layers in Li6PO5Br cubic argyrodite (viewed along the [111] direction in c and d) and computed Li6SiO4ClBr hexagonal argyrodite viewed along [001] in f and g: c) Li6PO5Br: Li3PO4 layer d) Li6PO5Br: Li3Br layer e) Li6PO5Br: cubic (-a-b-c-) stacking of close packed Li3Br and Li3PO4 layers. Oxygen atoms (red) and vacancies (grey) occupy octahedral B -sites between the close packed layers f) Li6SiO4ClBr: Li3SiO4 layer g) Li6SiO4ClBr: Li3Br layer h) Li6SiO4ClBr: -h-c-h-c (-a-ba-c) stacking of close-packed Li3Br and Li3SiO4 layers. Chlorine atoms (blue) and vacancies (grey) occupy octahedral B-sites between the close packed layers. For all further figures, all (poly) anions occupying the A-site are drawn in green and all Bsite anions are drawn in blue

Figure 3 .

 3 Figure 3. a) The stabilities of the compounds Li6SiO4XX΄ (X/X΄ = F -, Cl -, Br -, I -) against decomposition into Li4SiO4 + LiX + LiX΄, calculated using DFT. Li6SiO4BrCl and Li6SiO4Cl2 are calculated to be the most stable compositions. b) DFT optimised structures of Li6SiO4Cl2 (Li: pink, SiO4: green, Cl: light green and blue). In the high symmetry P63mc structure (centre), half of the lithium ions lie on mirror planes (shown in darker pink). Displacement of these lithium ions off the mirror planes leads to a lowering of symmetry and a reduction in the DFT calculated energy. Triangles of lithium ions are displaced by small rotations about the c axis. These rotations can be in phase or out of phase, leading to the different low symmetry structures shown. Rotations of lithium ions in the Li3SiO4 layers and Li3Cl layers are shown by pink and white arrows, respectively.

3 ( 1 )K

 31 , consistent with the observations from SXRD, NMR and AC impedance measurements (see below). Similarly to the RT-phase, the experimental SXRD pattern of the HT-phase could not be indexed to any known compounds in the Li-Si-O-Cl phase field. The structure of the RT-phase was solved by single crystal (SC) X-ray diffraction, and the HT-phase by Rietveld refinement of a computed starting model against high resolution SXRD data.

Figure 4 . 4 . 2 . 1 .

 4421 Figure 4. a) RT -Li6SiO4Cl2: Rietveld refinement against SXRD of RT-Li6SiO4Cl2 (Diamond Light Source I11 beam line) with Iobs (black circles), Icalc (red line), Iobs -Icalc (grey line) and Bragg reflections (black tick marks for Li6SiO4Cl2, green tick marks for Li2SiO3, blue tick marks for LiCl), inset highlights reflections consistent with orthorhombic Pna21 symmetry. b) HT -Li6SiO4Cl2: Rietveld refinement against SXRD data of Li6SiO4Cl2 (black tick marks for Li6SiO4Cl2, green tick marks for LiAlO2 and blue tick marks for LiCl), inset compares reflections from RT orthorhombic phase at 298 K (black line) with the HT hexagonal phase at 548 K (red line) c) Differential scanning calorimetry (DSC) data showing a reversible thermal event associated with phase transition between Pna21 and P63mc symmetry in Li6SiO4Cl2

9 Figure 5 .

 95 Figure 5. a) RT -Li6SiO4Cl2 unit cell showing alternating Li3Cl and Li3SiO4 layers. Comparing with ABX3 perovskite, SiO4 (green tetrahedra) and Cl2 (light green) occupy the A-sites, Cl1 (blue) ions fill 50% of the B-sites with remaining 50% vacant, Lithium atoms occupy the X-sites (Li1, Li2, Li3 atoms in light pink, Li4, Li5, Li6 atoms in dark pink). Vacant tetrahedral lithium sites in the Li3SiO4 layer are shown in white to facilitate comparison with the HT-structure. b) HT -Li6SiO4Cl2 unit cell showing the same alternating Li3Cl and Li3SiO4 layers. Lithium atoms are disordered over partially occupied sites with respect to the RT-structure (Li1a and Li1b atoms in light pink, Li2 atoms in dark pink). c) RT -Li6SiO4Cl2 lithium tetrahedra showing displacement of lithium (4a Wyckoff site) towards coordinating oxygen. The vacant tetrahedral sites (V), drawn in white, are generated by activation of occupational modes when lowering the symmetry of HT -Li6SiO4Cl2 to Pna21 through ISODISTORT (Figure S10). d) HT -Li6SiO4Cl2 Li1a, Li1b (6c Wyckoff site) and Li2 (12d Wyckoff site) coordination environment, showing partial occupancy of tetrahedral sites that are both fully occupied and vacant at RT. e) This order-disorder behaviour of lithium sites is analogous to

Figure 6 .

 6 Figure 6. a) Li3SiO4 intralayer environments, Li2-Li3 (pink) and Li1-SiO4 (blue and green respectively) chains of corner-and edge-sharing tetrahedra b) Connection of Li3SiO4 layers along the stacking axis (c) through corner sharing of Cl2 atoms (light green). c) Li3SiO4 (top) and Li3Cl (bottom) layers consisting of corner-and edge-sharing tetrahedra (Li1: light blue Li2, Li3: light pink, Li4, Li5, Li6: dark pink Si: green, Cl1 (B sites): blue, Cl2 (A sites): light green, O: red), all shared edges are Cl-Cl. d) Li3Cl layers connected via SiO4 units viewed along the b-direction showing stacking along c, SiO4 tetrahedra are shown to guide the eye.

Figure 7 .

 7 Figure 7. a) Impedance complex plane plots Z* of Li6SiO4Cl2 at 533 K, inset shows equivalent circuit used to model the data. The inset shows the low frequency inclined spike attributed to the capacitance of the blocking electrode. (b) Arrhenius plots of the total conductivity, activation energies derived from the data are shown. A change in Arrhenius behaviour and activation energy can be seen at ~470 K corresponding to the phase change observed in VT-XRD and DSC measurements. c) 7 Li NMR spectra under static conditions as a function of temperature. d) Motional narrowing of the line width (full width at half-maximum)

Table 1 :

 1 Computed decomposition energies of the compounds Li6SiO4XX' in the most stable symmetry

	Composition	Energy	Symmetry
		mev/atom
	Li6SiO4Cl2	-18	Pna21
	Li6SiO4ClBr	-16	Pna21
	Li6SiO4Br2	-6	Pna21
	Li6SiO4ClI	-2	P63mc

Acknowledgements

We are grateful to the ISCF Faraday Challenge project "SOLBAT -The Solid-State (Li or Na) Metal-Anode Battery" [grant number FIRG007] for funding Y. D., the studentship of A. M., and partial studentship support of B. B. D., who is also supported by the University of Liverpool. This work used the Cirrus UK National Tier-2 HPC Service at EPCC (http://www.cirrus.ac.uk) funded by the University of Edinburgh and EPSRC (EP/P020267/1), along with the University of Liverpool HPC cluster, Barkla. We thank EPSRC for funding G. H. and J. G. under EP/N004884, and for funding T. W. S. under EP/R011753/1. This work was carried out with the support of Diamond Light Source, instrument I11 (proposal CY23666).

Supporting information

Computed decomposition energies of the compounds Li6SiO4XX΄ (X = F, Cl, Br, I) for different space groups, PXRD patterns for Li6SiO4Cl2-xBrx, Pawley fits (RT and HT phase), comparison of Rietveld refinement before and after refinement of atomic positions as well as with and without Li atoms (RT phase), MEM analysis (RT and HT phase), comparison of computed and experimental structures (RT and HT phase), Fourier difference map (HT phase), structural information of Li6SiO4Cl2 determined by single crystal diffraction and powder diffraction refinement (refinement details, atomic positions, bond distances and angles), additional [START_REF] Johnson | A Software Program for IES Analysis 3.5 d[END_REF] Si NMR and impedance results are available.

Underlying data are available at http://datacat.liverpool.ac.uk/id/eprint/1158. The crystal structure of Li6SiO4Cl2 is deposited with the CSD 2032155.


M M Cavalcanti

V N Domingos Cavalcanti

C Rosier

L Rosier

Numerical control of a semilinear wave equation on an interval

We are concerned with the numerical exact controllability of the semilinear wave equation on the interval (0, 1). We introduce a Picard iterative scheme yielding a sequence of approximated solutions which converges towards a solution of the null controllability problem, provided that the initial data are small enough. The boundary control, which is applied at the endpoint x = 1, is taken in the space H 1 0 (0,T ) for T = 2. For the linear part, the control input is obtained by imposing a transparent boundary condition at x = 1. Next, we provide several simulations to show the efficiency of the algorithm, using collocation pseudospectral methods on Chebychev grids to discretize the second order derivative in space in the wave equation.

Introduction

The issue of the exact controllability of the wave equation is well understood for a long time [START_REF] Russell | Controllability and stabilization theory for linear partial differential equations: recent progress and open questions[END_REF][START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF]. Unfortunately, the numerical control of the wave equation turns out to be delicate, as the discrete control may not converge towards M.M. Cavalcanti Departamento de Matemática, Universidade Estadual de Maringá 87020-900, Maringá, PR, Brazil. e-mail: mmcavalcanti@uem.br V.N. Domingos Cavalcanti Departamento de Matemática, Universidade Estadual de Maringá 87020-900, Maringá, PR, Brazil. e-mail: vndcavalcanti@uem.br C. Rosier Université du Littoral Côte d'Opale, Laboratoire de Mathématiques Pures et Appliquées J. Liouville, BP 699, F-62228 Calais, France. e-mail: Carole.Rosier@univ-littoral.fr L. Rosier Université du Littoral Côte d'Opale, Laboratoire de Mathématiques Pures et Appliquées J. Liouville, BP 699, F-62228 Calais, France. e-mail: Lionel.Rosier@univ-littoral.fr the continuous control of the wave equation. This is due to the emergence of spurious high-frequencies oscillations for the discrete control problem. Several strategies have been proposed to face this problem: multigrid strategy, filtering of the high frequency modes, introduction of viscous terms, etc. We refer to [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation[END_REF][START_REF] Asch | Geometrical aspects of exact boundary controllability for the wave equation-a numerical study[END_REF][START_REF] Zuazua | Boundary observability for the finite difference space semi-discretizations of the 2-D wave equation in the square[END_REF][START_REF] Münch | A uniformly controllable and implicit scheme for the 1-D wave equation[END_REF][START_REF] Castro | Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method[END_REF][START_REF] Castro | Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square[END_REF][START_REF] Micu | Uniform boundary controllability of a semidiscrete 1-D wave equation with vanishing viscosity[END_REF][START_REF] Pedregal | A numerical method of local energy decay for the boundary controllability of time-reversible distributed parameter systems[END_REF] for some works about the numerical control of the wave equation, and to [START_REF] Ervedoza | Numerical approximation of exact controls for waves[END_REF][START_REF] Zuazua | Propagation, observation, control and numerical approximation of waves[END_REF] for some surveys. See also [START_REF] Cavalcanti | Numerical analysis for the wave equation with locally nonlinear distributed damping[END_REF][START_REF] Cavalcanti | General decay rate estimates and numerical analysis for a transmission problem with locally distributed nonlinear damping[END_REF] for the numerics corresponding to the wave equation with damping. We refer also to [START_REF] Bastin | Stability and Boundary Stabilization of 1-D Hyperbolic Systems[END_REF] for the stabilization of hyperbolic systems, and to [START_REF] Abolinya | Mixed problems for quasi-linear hyperbolic systems in the plane[END_REF][START_REF] Cooke | Differential-difference equations and nonlinear initialboundary value problems for linear hyperbolic partial differential equations[END_REF] for the reduction of initialboundary value problems for hyperbolic PDEs to initial-value problems for integrodifferential-difference equations.

In most of the papers concerned with the theoretical or numerical control of the wave equation, the control strategy is based upon the Hilbert Uniqueness Method (HUM) introduced by J.-L. Lions. However, in dimension one there exists another strategy based upon the use of transparent boundary conditions leading to the finitetime stability of the system. This was first noticed by Majda in [START_REF] Majda | Disappearing solutions for the dissipative wave equation[END_REF], and next used in [START_REF] Komornik | Exact Controllability and Stabilization. The Multiplier Method[END_REF][START_REF] Pedregal | A numerical method of local energy decay for the boundary controllability of time-reversible distributed parameter systems[END_REF] to give explicit control inputs for the wave equation in dimension one. The transparent boundary conditions can be as well used for the finite-time stabilization of systems of conservation laws on networks [START_REF] Leugering | On the modelling and stabilization of flows in networks of open canals[END_REF][START_REF] Perrollaz | Finite-time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF][START_REF] Alabau-Boussouira | Finite-time stabilization of a network of strings[END_REF][START_REF] Gugat | Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms[END_REF]. The idea is to let waves leave the domain in a natural way without bounce (or damping) at the boundary.

We shall see that the trajectory corresponding to the transparent boundary conditions is the same as those obtained by applying homogeneous Dirichlet conditions on a larger domain, thanks to the finite propagation speed of the wave equation. Thus, to design numerically the control, we can replace HUM (unstable) by a numerical method to solve a classical Cauchy problem for the wave equation (stable).

In this paper, we are concerned with the exact boundary controllability of a semilinear wave equation

y t t -y x x = f (y)
on the interval (0, 1) from a numerical perspective. By the time reversibility of the wave equation, it is sufficient to restrict ourselves to the null controllability problem. The aim of the paper is to propose an algorithm at the continuous level which gives a sequence of controls h n → h and a sequence of "trajectories" y n → y such that, at the limit, y satisfies both

y t t -y x x = f (y), (t, x) ∈ (0,T ) × (0, 1) y(t, 0) = 0, y(t, 1) = h(t), t ∈ (0,T ) (y(0, .), y t (0, .)) = (ξ 0 , ξ 1 )
and (y(T, .), y t (T, .)) = (0, 0).

Here, (ξ 0 , ξ 0 ) denotes any small initial data in some appropriate space, and the function f is locally Lipschitz continuous and it vanishes at 0. The state (y, y t ) and the control h will be taken more regular (see [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF]) than in the linear case ( f ≡ 0) in order to give a sense to the nonlinear term f (y).

The control h n is computed numerically by solving several linear control problems, and the state y n , obtained by solving numerically an initial boundary-value problem, provides a numerical approximation of the trajectory y.

The paper is organized as follows. In Section 2, we describe our iterative scheme and prove the convergence of both the trajectories and the control inputs in Theorem 1, the main result in the paper. Section 3 is devoted to the description of the numerical approach used here and Section 4 provides several numerical simulations. Some concluding remarks are displayed in Section 5. Some technical lemmas needed to prove Theorem 1 are gathered in an appendix.

Picard iteration scheme

We first propose, at the continuous level, an exponentially convergent iterative scheme providing at the limit both the controlled trajectory and the control input for the semilinear wave equation. Due to obvious storage limitations, it is preferable to keep in the memory of the computer the control input h(t) computed at the previous step than the corresponding trajectory y(t, x). That choice, needed for a practical implementation of the algorithm, results in a scheme for which the convergence turns to be more complicated to establish than for a scheme based on the storage of the trajectory.

To describe the scheme, we need to introduce a few notations. For the sake of simplicity, we write

H k x = H k (0, 1) (k ∈ Z), L 2 x = L 2 (0, 1), H 1 t = H 1 (0,T )
and introduce the spaces

H 0 := H 1 0 (0, 1) × L 2 (0, 1) ⊂ H := {(y, z) ∈ H 1 x × L 2 x ; y(0) = 0}
endowed with the norm

||(y, z)|| H = ||y x || 2 L 2 x + ||z|| 2 L 2 x 1 2 .
Let S(t) t ∈R denote the unitary group on H 0 generated by the operator A(y, z) = (z, y x x ) with domain

D( A) = (H 2 (0, 1) ∩ H 1 0 (0, 1)) × H 1 0 (0, 1) ⊂ H 0 .
Thus, for given (ξ 0 , ξ 1 ) ∈ H 0 , if (y(t), z(t)) = S(t)(ξ 0 , ξ 1 ), then z(t) = y t (t) and y solves the system

y t t -y x x = 0, (t, x) ∈ R × (0, 1) (1) y(t, 0) = y(t, 1) = 0, t ∈ R (2)
(y(0, .), y t (0, .)) = (ξ 0 , ξ 1 ).

Pick δ 0 > 0 and f ∈ W 1,∞ (-δ 0 , δ 0 ) such that

f (0) = 0, and || f || L ∞ (-δ 0 ,δ 0 ) ≤ η < η 0 < 1 2 . ( 4
)
Let finally Λ : H 0 → H 1 0 (0,T ) denote a continuous map such that for any given (ξ 0,T , ξ 1,T ) ∈ H 0 , the solution y of the controlled system

y t t -y x x = 0, (t, x) ∈ (0,T ) × (0, 1) (5) y(t, 0) = 0, y(t, 1) = h(t) := Λ(ξ 0,T , ξ 1,T )(t), t ∈ (0,T ) (6)
(y(0, .), y t (0, .)) = (0, 0)

satisfies (y(T, .), y t (T, .)) = (ξ 0,T , ξ 1,T ).

From well-known results, Λ is well-defined and continuous from L 2

x × H -1 x into L 2 (0,T ) if T ≥ 2, but Λ is also well-defined and continuous from H 0 into H 1 0 (0,T ) (see e.g. [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF]). (Note that h(0) = 0 by [START_REF] Canuto | Spectral methods in Fluid Dynamics[END_REF] and h(T ) = ξ 0,T (1) = 0, for ξ 0,T ∈ H 1 0 (0, 1).) Thus, there is some constant

C Λ > 0 such that ||h|| H 1 t ≤ C Λ ||(ξ 0,T , ξ 1,T )|| H (8)
Actually Λ may be constructed explicitly by using d'Alembert's formula, and this is precisely the numerical strategy used here.

We are in a position to describe our Picard iterative scheme. Pick T = 2 and let (ξ 0 , ξ 1 ) ∈ H 0 .

First iteration.

We set h 1 (t) = 0 for t ∈ [0,T]. Let y 1 1 solve the nonlinear system

y 1 1, t t -y 1 1, x x = f (y 1 1 ), (t, x) ∈ (0,T ) × (0, 1) (9)
y 1 1 (t, 0) = 0, y 1 1 (t, 1) = h 1 (t) = 0, t ∈ (0,T ) (10)
(y 1 1 (0, .), y 1 1t (0, .)) = (ξ 0 , ξ 1 ).

Next, we set (δh) 1 := -Λ(y 1 1 (T, .), y 1 1t (T, .)),

and introduce the solution of the linear system y 1 2, t ty 1 2, x x = 0, (t, x) ∈ (0,T ) × (0, 1) (13)

y 1 2 (t, 0) = 0, y 1 2 (t, 1) = (δh) 1 (t), t ∈ (0,T ) (14)
(y 1 2 (0, .), y 1 2t (0, .)) = (0, 0).

Set y 1 := y 1 1 + y 1 2 . Then, by construction, y 1 solves

y 1 t t -y 1 x x = f (y 1 1 ), (t, x) ∈ (0,T ) × (0, 1) (16
)

y 1 (t, 0) = 0, y 1 (t, 1) = h 1 (t) + (δh) 1 (t), t ∈ (0,T ) (17)
(y 1 (0, .),

y 1 t (0, .)) = (ξ 0 , ξ 1 ), (18)
and it satisfies (y 1 (T, .),

y 1 t (T, .)) = (0, 0). ( 19
)
Thus, (δh) 1 is a correction term added to h 1 in order that (19) be satisfied.

nth iteration (n ≥ 2).

Assume that h n-1 ∈ H 1 0 (0,T ) and (δh) n-1 ∈ H 1 0 (0,T ) have been constructed. Set

h n (t) := h n-1 (t) + (δh) n-1 (t), (20)
and let y n 1 denote the solution of the nonlinear system

y n 1, t t -y n 1, x x = f (y n 1 ), (t, x) ∈ (0,T ) × (0, 1) (21) y n 1 (t, 0) = 0, y n 1 (t, 1) = h n (t), t ∈ (0,T ) (22)
(y n 1 (0, .), y n 1t (0, .)) = (ξ 0 , ξ 1 ).

Next, set (δh) n = -Λ(y n 1 (T, .), y n 1t (T, .)),

and let y n 2 denote the solution of the linear system

y n 2, t t -y n 2, x x = 0, (t, x) ∈ (0,T ) × (0, 1) (25) y n 2 (t, 0) = 0, y n 2 (t, 1) = (δh) n (t), t ∈ (0,T ) (26)
(y n 2 (0, .), y n 2t (0, .)) = (0, 0).

Let

y n := y n 1 + y n 2 . ( 28
)
Then y n satisfies

y n t t -y n x x = f (y n 1 ), (t, x) ∈ (0,T ) × (0, 1) (29) y n (t, 0) = 0, y n (t, 1) = h n (t) + (δh) n (t), t ∈ (0,T ) (30) (y n (0, .), y n t (0, .)) = (ξ 0 , ξ 1 ), ( 31
) (y n (T, .), y n t (T, .)) = (0, 0). ( 32
)
We are in a position to state the convergence result for the Picard iterative scheme at the continuous level.

Theorem 1 Let T = 2. Then there exist some constants δ ∈ (0, δ 0 ), ε ∈ (0, 1) and C > 0 such that for any (ξ 0 , ξ 1 ) ∈ H 0 with 2 and we have

||(ξ 0 , ξ 1 )|| H < δ (33) the sequence (y n 1 , y n 2 , h n , (δh) n ) n ≥1 is well defined in C([0,T], H 1 x ) 2 × (H 1 t )
||(y n 1 -y, y n 1, t -y t )|| C ([0,T ], H ) +||(y n 2 , y n 2, t )|| C ([0,T ], H ) +||h n -h|| H 1 t +||(δh) n || H 1 t ≤ Cε n , ( 34
) where (y, y t , h) ∈ C([0,T], H ) × H 1 0 (0,T ) satisfies y t t -y x x = f (y), (t, x) ∈ (0,T ) × (0, 1) (35) y(t, 0) = 0, y(t, 1) = h(t), t ∈ (0,T ) (36)
(y(0, .), y t (0, .)) = (ξ 0 , ξ 1 ), ( 37)

and (y(T, .), y t (T, .)) = (0, 0). ( 38
)
Proof We need two lemmas whose proofs are postponed in appendix.

Lemma 1 (Estimates for the linear system) There exists a constant C 1 > 0 such that for any (z 0 , z 1 ) ∈ H 0 , any h ∈ H 1 t with h(0) = 0, and any g ∈ L 1 (0,T, L 2 x ), the solution of the system

z t t -z x x = g, (t, x) ∈ (0,T ) × (0, 1) (39) z(t, 0) = 0, z(t, 1) = h(t), t ∈ (0,T ) (40)
(z(0, .), z t (0, .)) = (z 0 , z 1 ) (41)

fulfills ||(z, z t )|| C ([0,T ], H ) ≤ ||(z 0 , z 1 )|| H + ||g|| L 1 (0,T, L 2 x ) + C 1 ||h|| H 1 t . (42)
Let (z, z t ) = W (h) denote the solution of (39)-(41) for h ∈ H 1 (0,T ) with h(0) = 0 and (z 0 , z 1 , g) = (0, 0, 0). Lemma 2 (Estimate for the nonlinear problem) Let C 1 be as in Lemma 1. Then there exist some constants

δ 1 ∈ (0, δ 0 ), R > 0 and C 2 > 0 such that for (z 0 , z 1 , h) ∈ H 0 × H 1 t with h(0) = 0 and ||(z 0 , z 1 )|| H + C 1 ||h|| H 1 t ≤ δ 1 , (43)
the map

Γ : (z, z) → S(t)(z 0 , z 1 ) + W (h) + t 0 S(t -s)(0, f (z(s)))ds (44)
has a fixed-point (z, z) in the ball B(0, R) ⊂ C([0,T], H ). Furthermore, z = z t , and z solves the nonlinear system

z t t -z x x = f (z), (t, x) ∈ (0,T ) × (0, 1) (45) z(t, 0) = 0, z(t, 1) = h(t), t ∈ (0,T ) (46)
(z(0, .), z t (0, .)) = (z 0 , z 1 ), (47)

and we have

||(z, z t )|| C ([0,T ], H ) ≤ C 2 ||(z 0 , z 1 )|| H + C 1 ||h|| H 1 t . ( 48
)
The proof of Theorem 1 is done in two steps. In the first step, we show by induction on n that ||h n || H 1 t is uniformly bounded, and that for some ε ∈ (0, 1)

||(δh) n+1 || H 1 t ≤ ε||(δh) n || H 1 t .
In the second step, we combine the above estimates to prove the exponential convergence of all the sequences. Pick any δ ∈ (0, δ 1 ) and set (for notational convenience) (δh) 0 (t) := sin(πt/T ). Let

ε := 2C Λ C 1 π -2η η, C 3 = C Λ C 2 1 -ε • (49)
where C Λ is as in [START_REF] Castro | Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method[END_REF]. We have ε < 1 and C 3 > 0 if 0 < η < η 0 < 1 2 with η 0 small enough.

Step 1. Estimates of the control inputs.

Proposition 1 Let ε and C 3 be as in (49). Then we have

||h n || H 1 t ≤ C 3 δ, ∀n ≥ 1, (50)
||(δh) n || H 1 t ≤ ε||(δh) n-1 || H 1 t , ∀n ≥ 1. ( 51
)
Proof of Proposition 1. We proceed by induction on n ≥ 1. Assume first that n = 1. Then (50) is obvious, for h 1 = 0. To prove (51), we estimate (δh) 1 in H 1 t as follows:

||(δh) 1 || H 1 t = ||Λ(y 1 1 (T, .), y 1 1t (T, .))|| H 1 t ≤ C Λ ||(y 1 1 (T, .), y 1 1t (T, .))|| H ≤ C Λ C 2 ||(ξ 0 , ξ 1 )|| H ≤ C Λ C 2 δ (52) ≤ ε||(δh) 0 || H 1 t if δ is small enough.
Assume that (50) and ( 51) are valid for any integer n < n, where n ≥ 2. Let us check that (50) and (51) are also true for n. Since h n = n-1 k=1 (δh) k , we have that

||h n || H 1 t ≤ n-1 k=1 ε k-1 ||(δh) 1 || H 1 t ≤ C Λ C 2 1 -ε δ = C 3 δ,
where we used (51) for k < n and (52). Let us proceed to the proof of (51). Let

w n := y n-1 -y n 1 = (y n-1 1 + y n-1 2 ) -y n 1 . Then w n satisfies w n t t -w n x x = f (y n-1 1 ) -f (y n 1 ), (t, x) ∈ (0,T ) × (0, 1) w n (t, 0) = 0, w n (t, 1) = 0, t ∈ (0,T ) w n (0, .) = w n t (0, .) = 0, (w n (T, .), w n t (T, .)) = -(y n 1 (T ), y n 1t (T )). (53)
Thus, by [START_REF] Russell | Controllability and stabilization theory for linear partial differential equations: recent progress and open questions[END_REF], ( 42) and (53), we have

||(δh) n || H 1 t ≤ C Λ ||(y n 1 (T, .), y n 1t (T, .))|| H ≤ C Λ ||(w n , w n t )|| C ([0,T ], H ) ≤ C Λ || f (y n-1 1 ) -f (y n 1 )|| L 1 (0,T, L 2 x ) ≤ C Λ η||y n-1 1 -y n 1 || L 1 (0,T, L 2 x ) . (54)
For the last line, we used (4) and the estimate sup

(||y n-1 1 || L ∞ t, x , ||y n 1 || L ∞ t, x
) ≤ δ 0 valid if δ is small enough. Indeed, using (48) and (50) for n -1 and n,

||y n 1 || L ∞ ((0,T )×(0,1)) ≤ ||(y n 1 ) x || L ∞ (0,T, L 2 (0,1)) ≤ C 2 ||(ξ 0 , ξ 1 )|| H + C 1 ||h n || H 1 t ≤ C 2 δ + C 1 C 3 δ ≤ δ 0 and similarly ||y n-1
1 || L ∞ ((0,T )×(0,1)) ≤ δ 0 , for δ small enough. It remains to estimate

||y n-1 1 -y n 1 || L 1 (0,T, L 2 x ) in terms of ||(δh) n-1 || H 1 t . Let v n = y n-1 1 -y n 1 . Then v n solves v n t t -v n x x = f (y n-1 1 ) -f (y n 1 ), (t, x) ∈ (0,T ) × (0, 1) (55) v n (t, 0) = 0, v n (t, 1) = -(δh) n-1 (t), t ∈ (0,T ) v n (0, .) = v n t (0, .) = 0.
An application of Lemma 1 yields

||(y n-1 1 ) x -(y n 1 ) x || C ([0,T ], L 2 x ) ≤ ||(y n-1 1 -y n 1 , y n-1 1, t -y n 1, t )|| C ([0,T ], H ) ≤ || f (y n-1 1 ) -f (y n 1 )|| L 1 (0,T, L 2 x ) + C 1 ||(δh) n-1 || H 1 t ≤ η||y n-1 1 -y n 1 || L 1 (0,T, L 2 x ) + C 1 ||(δh) n-1 || H 1 t • (56) It follows from Poincaré's inequality f L 2 x ≤ π -1 f x L 2 x (valid for f ∈ H 1 x with f (0) = 0) that ||y n-1 1 -y n 1 || L 1 (0,T, L 2 x ) ≤ T π ||(y n-1 1 ) x -(y n 1 ) x || C ([0,T ], L 2 x ) (57)
which, combined with (56), gives

||y n-1 1 -y n 1 || L 1 (0,T, L 2 x ) ≤ (1 - T η π ) -1 C 1 T π ||(δh) n-1 || H 1 t . (58)
Gathering together (54), (58) and the fact that T = 2, we conclude that

||(δh) n || H 1 t ≤ ε||(δh) n-1 || H t 1
with ε as in (49). The proof of Proposition 1 is complete.

Step 2. Exponential decay Let us start with the convergence of the sequence of controls (h n ) n ≥1 . From (51) and (52), we infer that

||h n+1 -h n || H 1 t = ||(δh) n || H 1 t ≤ C Λ C 2 δε n-1 , ∀n ≥ 1, (59)
so that for n, p ≥ 1

||h n+p -h n || H 1 t ≤ C Λ C 2 δ(ε n-1 + • • • + ε n+p-2 ) ≤ C Λ C 2 δ 1 -ε ε n-1 . (60)
Thus (h n ) n ≥1 is a Cauchy sequence in H 1 t , and its limit h

∈ H 1 t satisfies ||h -h n || H 1 t ≤ C Λ C 2 δ 1 -ε ε n-1 (61)
by letting p → ∞ in (60). It follows from ( 25)-( 27), (42), (51) and (52) that

||(y n 2 , y n 2, t )|| C ([0,T ], H ) ≤ C 1 ||(δh) n || H 1 t ≤ C 1 ε n-1 ||(δh) 1 || H 1 t ≤ C Λ C 1 C 2 δε n-1 =: C 4 δε n-1 ,
and hence (y n 2 , y n 2, t ) → (0, 0) in C([0,T], H ) as n → ∞. From (55), ( 58) and (59), we infer that

||(y n+1 1 -y n 1 , y n+1 1, t -y n 1, t )|| C ([0,T ], H ) ≤ C 1 +η(1- T η π ) -1 C 1 T π ||(δh) n || H 1 t ≤ C 5 δε n-1 with C 5 = C 1 + η(1 -T η π ) -1 C 1 T π C Λ C 2 . This yields ||(y n+p 1 -y n 1 , y n+p 1, t -y n 1, t )|| C ([0,T ], H ) ≤ C 5 δ 1 -ε ε n-1 ,
so that the sequence (y n 1 , y n 1, t ) converges towards some function (y, y t ) in C([0,T], H ) with y(t, 0) = 0 and

||(y -y n 1 , y t -y n 1, t )|| C ([0,T ], H ) ≤ C 5 δ 1 -ε ε n-1 .
Thus (34) is proved for some constant C > 0.

It remains to check that the pair (y, h) satisfies ( 35)-( 38). ( 35)-(37) follow from ( 21)-( 23) and (34), while (38) follows from (28), ( 32) and (34).

Numerical schemes

Let us describe the numerical schemes used to compute y n 1 , y n 2 and (δh) n . To simplify the notations, we remove the index n (corresponding to the n t h Picard iteration) in y n 1 , y n 2 and (δh) n . We focus on the computation of y 1 and h, as y 2 is not needed to approximate y.

Chebychev collocation spectral method

We have to solve numerically several times the linear (or semilinear) wave equation on a bounded domain. After an obvious change of spatial variables, we can assume that x ∈ (-1, 1), incorporating a coefficient c in the wave operator. Consider the Cauchy problem for the semilinear wave equation:

u t t -c 2 u x x = g(t, x,u), (t, x) ∈ (0,T ) × (-1, 1), (62) u(t, -1) = h -1 (t), u(t, 1) = h 1 (t), t ∈ (0,T ), (63) u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ (-1, 1), ( 64
)
where g, h -1 , h 1 , u 0 and u 1 are given functions. In this paper, we shall use the Chebychev collocation spectral method to solve (62)-(64), as described e.g. in [START_REF] Trefethen | Spectral Methods in Matlab[END_REF].

Let N ∈ N * be given. We introduce the Chebychev points x j = cos(θ j ), where θ j = ( j -1)π/N for 1 ≤ j ≤ N + 1. Note that x 1 = 1 and x N +1 = -1. Any smooth function v = v(x) defined on [-1, 1] is conveniently approximated by the Lagrange polynomial p ∈ R[x] of degree at most N defined by p(x i ) = v(x i ) for 1 ≤ j ≤ N + 1. It turns that a good approximation of du/dx can be obtained by using the Chebychev differentiation matrix D = (D i j ) 1≤i, j ≤ N +1 defined as

D j j = N +1 k = 1 k j (x j -x k ) -1 , 1 ≤ j ≤ N + 1, D i j = 1 a j N +1 k = 1 k i, j (x i -x k ), 1 ≤ i j ≤ N + 1,
where

a j = N +1 k = 1 k j (x j -x k ) for 1 ≤ j ≤ N + 1. Indeed, introducing the vector U = (U j ) 1≤ j ≤ N +1 with U j = u(x j ) (1 ≤ j ≤ N + 1
) and setting Ũ = DU, Ũ = ( Ũj ) 1≤ j ≤ N +1 , then we notice that the Lagrange polynomial q of degree ≤ N defined by q(x j ) = Ũj for 1 ≤ j ≤ N + 1 provides a good approximation of du/dx; namely, we have for some constants C > 0 and K > 1

| du dx (x j ) -Ũj | ≤ CK -N , ∀N ≥ 1, ∀j ∈ {1, ..., N + 1}.
(See e.g. [START_REF] Canuto | Spectral methods in Fluid Dynamics[END_REF][START_REF] Trefethen | Spectral Methods in Matlab[END_REF].) Let M ∈ N * and ∆t = T/M, where T = 2. For 1 ≤ j ≤ N + 1, 0 ≤ m ≤ M, let U m j stand for the numerical approximation of u(m∆t, x j ). To solve (62)-(64) numerically, we use a leap-frog formula in t and Chebychev spectral differentiation in x to get

(∆t) -2 (U m+1 j -2U m j + U m-1 j ) -c 2 (D 2 U) j = g(m∆t, x j ,U m j ), 2 ≤ j ≤ N, 0 ≤ m ≤ M -1, U m+1 1 = h 1 ((m + 1)∆t), U m+1 N +1 = h -1 ((m + 1)∆t), 0 ≤ m ≤ M -1, U 0 j = u 0 (x j ), U -1 j = u 0 (x j ) -(∆t)u 1 (x j ), 2 ≤ j ≤ N.

Numerical scheme for the linear control problem

To compute (δh) n as in [START_REF] Russell | Controllability and stabilization theory for linear partial differential equations: recent progress and open questions[END_REF], we have to solve numerically the following linear control problem: for given (y 1 (T, .), y 1, t (T, .)) ∈ H 0 , find h ∈ H 1 0 (0,T ) such that the solution y 2 of the system y 2, t ty 2, x x = 0, (t, x) ∈ (0,T ) × (0, 1) (65) y 2 (t, 0) = 0, y 2 (t, 1) = h(t), t ∈ (0,T ) (66) y 2 (0, x) = y 2, t (0, x) = 0, x ∈ (0, 1) (67) satisfies

y 2 (T, x) = -y 1 (T, x), y 2, t (T, x) = -y 1, t (T, x), x ∈ (0, 1). ( 68
)
The change of unknown function z(t, x) = y 2 (Tt, x) transforms the exact controllability problem ( 65)-( 68) into the following null controllability problem

z t t -z x x = 0, (t, x) ∈ (0,T ) × (0, 1) (69) z(t, 0) = 0, z(t, 1) = h(T -t), t ∈ (0,T ) (70) z(0, x) = z 0 (x) = -y 1 (T, x), z t (0, x) = z 1 (x) = y 1, t (T, x), x ∈ (0, 1) (71) z(T, x) = z t (T, x) = 0, x ∈ (0, 1). ( 72
)
Instead of considering a discretization of the control problem ( 69)-( 72) and solving a finite-dimensional control problem, we design a control for the problem ( 69)-( 72) based upon the finite-time stabilization method widely used for systems of conservation laws (see e.g. [START_REF] Leugering | On the modelling and stabilization of flows in networks of open canals[END_REF][START_REF] Perrollaz | Finite-time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF][START_REF] Alabau-Boussouira | Finite-time stabilization of a network of strings[END_REF][START_REF] Gugat | Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms[END_REF]). It is well known that the transparent boundary condition z x = -z t taken at x = 1 (replacing the condition z(t, 1) = h(Tt)), which is a control in feedback form, leads to the finite-time stability of ( 69)-( 72). The idea is to let the waves leave the domain (0, 1) in a free way, as they do naturally when the spatial domain is R instead of (0, 1). To achieve such a natural control, we proceed as follows. We extend z 0 and z 1 as odd functions with compact support. Namely, we set

z 0 (x) =        -y 1 (T, x) if 0 < x < 1, y 1 (T, -x) if -1 < x < 0, 0 if x ∈ (-∞, 1) ∪ (1, +∞), (73)
and

z 1 (x) =        y 1, t (T, x) if 0 < x < 1, -y 1, t (T, -x) if -1 < x < 0, 0 if x ∈ (-∞, 1) ∪ (1, +∞). (74)
Let z = z(t, x) solve now the system

z t t -z x x = 0, 0 < t < T, x ∈ R (75) z(0, x) = z 0 (x), z t (0, x) = z 1 (x), x ∈ R. (76)
Since y 1 (T, .) ∈ H 1 0 (0, 1) and

y 1, t (T, .) ∈ L 2 (0, 1), (z 0 , z 1 ) ∈ H 1 (R) × L 2 (R) and (z, z t ) ∈ C([0,T], H 1 (R) × L 2 (R))
. Furthermore, z is odd in x, for z 0 and z 1 are odd in x. Thus z(0,t) = 0 for all t ∈ (0,T ). Finally, it follows from D'Alembert's formula

z(t, x) = 1 2 [z 0 (x + t) + z 0 (x -t)] + 1 2 x+t x-t z 1 (s)ds (77)
and from the choice of z 0 , z 1 , and T = 2 that

z(T, x) = z t (T, x) = 0, for -1 < x < 1. (78)
We infer from (77) that

supp z(t, .) ⊂ [-1 -t, 1 + t] ⊂ [-3, 3] for 0 ≤ t ≤ T = 2.
Thus, we can actually limit the computations to the spatial domain [-3, 3] by imposing homogenous Dirichlet boundary conditions at x = ±3, and use e.g. the scheme described in Subsection 3.1. The control h(t) = z(Tt, 1) is numerically obtained by computing z(m∆t, 1) for 0 ≤ m ≤ M. Note that 1 may not be a Chebychev point 3x j for some j ∈ [1, N + 1]. Assuming that 1 ∈ [3x j+1 , 3x j ], then a linear interpolation between z(m∆t, 3x j ), z(m∆t, 3x j+1 ) gives an approximation of z(m∆t, 1). Another way to compute numerically h(t) = z(Tt, 1) is to use (77) for x = 1, which gives for t > 0

z(t, 1) = 1 2 z 0 (1 -t) + 1 2 1 1-t z 1 (s)ds. (79)
(This expression was already derived in [START_REF] Pedregal | A numerical method of local energy decay for the boundary controllability of time-reversible distributed parameter systems[END_REF].) The integral term in (79) can be estimated by using the trapezoid rule.

A third way to compute numerically h(t) = z(Tt, 1) is to go back to the transparent boundary condition z x (t, 1) = -z t (t, 1) and to introduce the Riemann invariants s := z t +z x and d := z t -z x . Set d 0 := z 1 -z 0

x . Then the transport equation d t + d x = 0 is integrated as d(t, x) = d 0 (xt). On the other hand z t = (s + d)/2 and s(t, 1) = 0, for z x (t, 1) = -z t (t, 1). It follows that

z(t, 1) = t 0 z t (s, 1)ds = 1 2 t 0 d(s, 1)ds = 1 2 t 0 d 0 (1 -s)ds (80)
which is nothing but (79). Numerically,

h(T -m∆t) = z(m∆t, 1) is approximated by 1 4 m-1 n=0 [d 0 (1 -n∆t) + d 0 (1 -(n + 1)∆t)].
The main avantage of this way to compute the control in the linear problem is that there is no need to solve the wave equation for (t, x) ∈ (0,T ) × (-3, 3). What is needed is the computation of d 0 , first on the Chebychev points (x j ) 1≤ j ≤N +1 (using the Chebychev differentiation matrix to evaluate z 0

x ), and next on the fine grid (1 -n∆t) 0≤n ≤ M by using the associated Lagrange polynomial.

Numerical simulations 4.1 Linear problem

We first test the boundary control of the linear wave equation by using the numerical integration of the linear wave equation ( 75)-(76) supplemented with the boundary conditions z(t, ±3) = 0. We take as initial data z 0 (x) = sin(πx) and z 1 (x) = 0 for x ∈ (0, 1) and we extend z 0 and z 1 to R as in (73)-(74). We use the numerical scheme in Subsection 3.1 with N = 200 and ∆t = 0.1/N 2 . The simulation is displayed in Figure 1. As expected, the solution z vanishes at T = 2 for x ∈ (-1, 1). If we let e(z 0 ) denote the sup-norm of the vector of the values of z 0 at the Chebychev points inside (-1, 1), which gives an estimate of z 0 L ∞ (-1,1) , then we obtain as relative error e(z(T, .)) e(z 0 ) ≈ 2.5 10 -3 .

Nonlinear control problem

From now on, we compute numerically the control h in the linear control problem (69)-(72) by using a discretization of (80).

The general case

We are concerned with the numerical simulation of the control problem (35)- [START_REF] Cavalcanti | Numerical analysis for the wave equation with locally nonlinear distributed damping[END_REF], where f is as in [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF]. We shall take f (u) = Cu 2 , where C ∈ R is some parameter. We shall see that the parameter C has to take its values in a bounded interval for the control problem to be numerically solved. We take as initial data ξ 0 (x) = 1.5 sin(3πx), ξ 1 (x) = x 2 for x ∈ (0, 1).

For C = 4, we apply our Picard iterative scheme with n = 5 iterations, N = 200, ∆t = 1/N 2 . The simulation is displayed in Figure 2 (bottom). The relative error is found to be e(z (T, .

)) e(z 0 ) ≈ 6.9 10 -3 . Another simulation is performed with C = 0 (linear case) and is also displayed in Figure 2 (top). The presence of the nonlinear term 4u 2 in the PDE leads to a visible change of the control input.

The case of an odd function f

When f is odd, then the solution of the system y t ty x x = f (y), t ∈ (0,T ), x ∈ (-1, 1), y(t, ±1) = ±h(t), t ∈ (0,T ), y(0, x) = sign(x)ξ 0 (|x|), y t (t, 0) = sign(x)ξ 1 (|x|), x ∈ (-1, 1).

is odd, and hence it vanishes for x = 0. This allows to do the computations on the interval (-1, 1). We first test our numerical scheme with f (u) = Cu 3 . It is numerically proved to work well for C ∈ [-2 , 1.8]. For C = 2, we have a blow-up, while for C ≤ -3 we never reach the null state at T = 2. This suggests that the Lipschitz condition (4) cannot be replaced by the passivity-like sector condition f (y)y > 0. Note that, even for small values of C, the best result is achieved for a "small" number of iterations: the relative error reaches its minimal value and next increases as the number of iterations tends to infinity. We take ξ 0 (x) = 2 sin(2πx) and ξ 1 (x) = 0 as initial data, and N = 200 and ∆t = 4/N 2 as parameters. For C = 1.5 we obtain as relative error 5.7 10 -3 with 5 iterations (see Figure 3). For C = -4, the "best" relative error is 0.54 and it is obtained with 4 iterations (see Figure 4). Finally, with the choice f (u) = C sin u (which is both Lipschitz continuous and bounded), we obtain with C = 6 as relative error 0.26 with 3 iterations (see Figure 5). This suggests that our algorithm based on a perturbative argument does not work for large (Lipschitz continuous and bounded) perturbations.

Conclusion

An iterative algorithm was proposed to derive a boundary control for the null controllability of a semilinear wave equation in dimension one. Designing the control of the linear part by using a transparent boundary condition and using a collocation pseudospectral method on a Chebychev grid to compute the second order derivative in x in the wave equation, we provided several numerical simulations showing the efficiency of the method, at least for small nonlinearities and small initial data. Of course, further are required to provide a numerical analysis of the algorithm. It would be also interesting to see whether the method can be extended to the wave equation in dimension two or three. This will be done elsewhere.

Appendix

Proof of Lemma 1

The estimate for h ≡ 0 follows from classical semigroup theory, S(t) t ∈R being a unitary group on H 0 . Assume now that h 0 and that (z 0 , z 1 , g) = (0, 0, 0 where z denotes the solution by transposition of (39)-( 41) with (z 0 , z 1 , g) = (0, 0, 0). Introduce w := z t . Then w solves the system w t tw x x = 0, (t, x) ∈ (0,T ) × (0, 1) w(t, 0) = 0, w(t, 1) = h (t), t ∈ (0,T ) (w(0, .), w t (0, .)) = (z 1 , z 0 x x ) = (0, 0).

Since h ∈ H .

We shall apply the contraction mapping principle in the ball B(0, R) ⊂ C([0,T], H ), where R > 0 will be chosen later on. Note that

||z|| L ∞ x (0,1) ≤ ||z x || L 2 x if z ∈ H 1
x with z(0) = 0. Assume that z 0 ∈ H 1 0 (0, 1), and that

||(z 0 , z 1 )|| H + C 1 ||h|| H 1 t =: δ ≤ δ 0 . ( 83

2 Fig. 1

 21 Fig.1Free evolution of the linear wave equation.

Fig. 2

 2 Fig. 2 Control of the nonlinear wave equation: f (u) = Cu 2 .

Fig. 3

 3 Fig. 3 Control of the semilinear wave equation: f (u) = 1.5 u 3 .

Fig. 4 Fig. 5

 45 Fig. 4 Control of the semilinear wave equation: f (u) = -4u 3 .

1 |||Γ(z 1 , 0 S 1 = ( 1 -

 11011 ) R < δ 0 . (84)Note that, for (z, z) ∈ B(0, R),||z(t)|| L ∞ x (0,1) ≤ ||z x (t)|| L 2 x ≤ |||(z, z)||| < δ 0 ∀t ∈ [0,T].Pick two pairs (z 1 , z1 ) and (z 2 , z2 ) in B(0, R). Then by Lemma z1 )-Γ(z 2 , z2 )||| = ||| t (ts)(0, f (z 1 (s))f (z 2 (s)))ds||| ≤ || f (z 1 )f (z 2 )|| L 1 (0,T, L 2 x ) ≤ η||z 1z 2 || L 1 (0,T, L 2 x ) ≤ T η|||(z 1z 2 , z1 -z2 )|||. (85)Note that Γ contracts in B(0, R), for T = 2 and η as in (4). On the other hand, for any (z, z) ∈ B(0, R), we have by Lemma 1 and (85)|||Γ(z, z)||| ≤ |||S(t)(z 0 , z 1 ) + W (h)||| + ||| t 0 S(ts)(0, f (z(s)))ds||| ≤ δ + 2η|||(z, z))|||. (86) Thus Γ(B(0, R)) ⊂ B(0, R) if δ + 2η R ≤ R. 2η)δ 0 < δ 0 .Then Γ has a unique fixed point (z, z) in B(0, R) by the contraction mapping principle. Clearly, z = z t and (48) holds with C 2 = (1 -2η) -1

  ). Since ||(z, z)|| H and ||z|| H 1 x + || z|| L 2 x are equivalent norms on H , it is sufficient to prove ||z||C ([0,T ], H 1 x ) + ||z t || C ([0,T ], L 2 x ) ≤ C||h|| H 1 t .(Here, C denotes some constant that may vary from line to line.) It is well-known that||z|| C ([0,T ], L 2 x ) + ||z t || C ([0,T ], H -1 x ) ≤ C ||z 0 || L 2 x + ||z 1 || H -1 x + ||h|| L 2

t

(81)

  1 t , we infer from (81) that||w|| C ([0,T ], L 2 x ) + ||w t || C ([0,T ], H -1 x ) ≤ C||h|| H 1 t ,(82)so that the estimate for||z t || C ([0,T ], L 2 x ) is established. For ||z|| C ([0,T ], H 1 x ), it is sufficient to notice that z x x = z t t = w t and to combine||z x x || C ([0,T ], H -1 x ) ≤ C||h|| H 1 t with the elliptic estimate ||z(t, .)|| H 1 x ≤ C ||z x x (t, .)|| H -1 x + |z(t, 0)| + |z(t, 1)| ≤ C(||z x x (t, .)|| H -1 x + |h(t)|).For (z, z) ∈ C([0,T], H ), let |||(z, z)||| = max

	Proof of Lemma 2
	t ∈[0,T ]	||(z(t), z(t))|| H = max t ∈[0,T ]	||z x (t)|| 2 L 2 x	+ || z(t)|| 2 L 2 x	1 2

Acknowledgments

The authors are partially supported by CAPES: n o 88881.520205/2020-01 and MATH AMSUD: 21-MATH-03.