S Komaba

L Croguennec

F Tournadre

P Willmann

C Delmas

Thermal Behavior of the Layered Oxide Li 2/3 Co 2/3 Mn 1/3 O 2 Obtained by Ion Exchange from the P2-Type Na 2/3 Co 2/3 Mn 1/3 O 2 Phase

Thermal evolution of the layered oxide Li2/3Co2/3Mn1/3O2, showing a T # 2 stacking and prepared by a Na/Li ion exchange in P2-Na2/3Co2/3Mn1/3O2, was investigated by thermal analyses and X-ray diffraction. A thermal expansion of the T # 2 orthorhombic unit cell is observed from 25 to 350 °C; from 350 °C the T # 2 stacking is destabilized to the benefit of an O6-type stacking obtained from the former through slab gliding. The T # 2 to O6 phase transformation is allowed to occur from a stacking with larger interlayer distances and the lithium ions in tetrahedral sites to a stacking with smaller interlayer distances and the lithium ions in octahedral sites. This phase transition from T # 2 to O6 is reversible, even though its kinetic can be very slow: the thermal treatment of the T # 2-type Li2/3Co2/3Mn1/3O2 phase at 450 °C with a quenching in air has shown to stabilize the O6HT-Li2/3Co2/3Mn1/3O2 phase. At temperatures higher than 450 °C, the layered oxide Li2/3Co2/3Mn1/3O2 is gradually decomposed into Li2MnO3 and Co3O4. First electrochemical tests performed in lithium batteries have revealed that O6HT-Li2/3Co2/3Mn1/3O2 delivers as positive electrode material a high reversible capacity of ∼230 mAh•g -1 over two voltage domains around 3 and 4 V vs Li + /Li.

Introduction

In the past thirty years, LiMO2 (M = transition metal) layered oxides have attracted the attention of many research groups from materials science and battery application points of view. Among these materials, the LiMO2 layered oxides obtained by lithium ion exchange from high-temperature sodium containing phases demonstrated interesting properties as positive electrode materials for Li-ion batteries. [START_REF] Delmas | A New Variety of LiCoO2 with an Unusual Oxygen Packing Obtained by Exchange Reaction Mater[END_REF][START_REF] Paulsen | Layered Li-Mn-Oxide with the O2 Structure: A Cathode Material for Li-Ion Cells Which Does Not Convert to Spinel[END_REF][START_REF] Paulsen | Layered LiCoO2 with a Different Oxygen Stacking (O2 Structure) as a Cathode Material for Rechargeable Lithium Batteries[END_REF][START_REF] Carlier | On the Metastable O2-type LiCoO2[END_REF][START_REF] Carlier | Lithium Electrochemical Deintercalation from O2-LiCoO2 Structure and Physical Properties[END_REF][START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part II: Its Electrochemical Behavior[END_REF][START_REF] Berthelot | Synthesis and Investigations on an O4-LiCoO2 Polytype[END_REF][START_REF] Komaba | Structural and Electrochemical Behaviors of Metastable Li2/3[Ni1/3Mn2/3O2 Modified by Metal Element Substitution Electrochim[END_REF] Note that all the phases will be named here according to the packing designation commonly used for the layered oxides: the letters P, T, and O describe the alkali ion environment (prismatic, tetrahedral, and octahedral, respectively) and the numbers 1, 2, 3, ... give the number of slabs required to describe the unit cell. [START_REF] Delmas | Structural Classification and Properties of the Layered Oxides[END_REF] The polymorph of LiCoO2 showing an O2-type oxygen packing was the first metastable lithiated oxide synthesized by ion-exchange reaction. [START_REF] Delmas | A New Variety of LiCoO2 with an Unusual Oxygen Packing Obtained by Exchange Reaction Mater[END_REF] Recently, a new polymorph of LiCoO2 showing an O4 structure [START_REF] Berthelot | Synthesis and Investigations on an O4-LiCoO2 Polytype[END_REF][START_REF] Komaba | A New Polymorph of Layered LiCoO2[END_REF] was obtained from the high-temperature Na0.36Li0.43CoO1.96 phase first reported by Balsys et al. [START_REF] Balsys | The Structure of Li0.43Na0.36CoO1.96[END_REF] and recently revisited in our lab. [START_REF] Berthelot | Reinvestigation of the OP4-(Li/Na)CoO2-Layered System and First Evidence of the (Li/Na/Na)CoO2 Phase with OPP9 Oxygen Stacking[END_REF] In the past few years ion-exchange reaction was also shown to be an attractive synthesis method to stabilize new polyanionic phases showing interesting properties as electrode materials for lithium batteries. [START_REF] Marx | Structural and Electrochemical Study of a New Crystalline Hydrated Iron(III) Phosphate FePO4.H2O Obtained from LiFePO4(OH) by Ion Exchange[END_REF][START_REF] Satya Kishore | Topotactic Insertion of Lithium in the Layered Structure Li4VO(PO4)2: The Tunnel Structure Li5VO(PO4)2[END_REF] The systematic investigations performed by Dahn and co-workers have shown that only a few well-crystallized phases could be obtained by ion-exchange reaction in NaxMO2: for instance, T # 2-Li2/3Ni II 1/3Mn IV 2/3O2 [START_REF] Paulsen | O2-Structure Li2/3[Ni1/3Mn2/3]O2: A New Layered Cathode Material for Rechargeable Lithium Batteries: Part III: Ion Exchange[END_REF] and T # 2-Li2/3Co III 2/3Mn IV 1/3O2, [START_REF] Lu | 2 and O2 Li2/3[CoxNi1/3-x/2Mn2/3-x/2O2 Electrode Materials[END_REF] the sign # showing that their oxygen packing cannot be described anymore in the classical packings of triangular lattices. The crystal structure analysis and battery performance of the former [START_REF] Komaba | Structural and Electrochemical Behaviors of Metastable Li2/3[Ni1/3Mn2/3O2 Modified by Metal Element Substitution Electrochim[END_REF] and latter phases [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part II: Its Electrochemical Behavior[END_REF][START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part I: Its Structural Characterization[END_REF] were studied in detail by some of us. Figure 1a gives a perspective view of the T # 2-type structure described in the orthorhombic Cmca space group, with two (Co2/3Mn1/3)O2 slabs per unit cell and in between, in the interslab space, the three tetrahedral sites (8e, 8fedges, and 8fface) available for the lithium ions. Nevertheless, for T # 2-Li2/3Co2/3Mn1/3O2 it was shown from nuclear density Fourier difference map calculations from neutron diffraction data that lithium ions are distributed among the 8e and 8fedges tetrahedral sites only (with a statistical distribution according to the average ratio 8e:8fedges ≈ 1:2). [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part I: Its Structural Characterization[END_REF] No ordering was evidenced between cobalt and manganese ions in the (Co2/3Mn1/3)O2 slabs, nor between lithium ions and vacancies in the interslab space, on the contrary to observations made in T # 2-LixCoO2 [START_REF] Shao-Horn | Lithium and Vacancy Ordering in T # 2-LixCoO2 Derived from O2 Type LiCoO2[END_REF] and T # 2-Li2/3Ni1/3Mn2/3O2. [START_REF] Lu | 2 and O2 Li2/3[CoxNi1/3-x/2Mn2/3-x/2O2 Electrode Materials[END_REF] As described in previous work [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part II: Its Electrochemical Behavior[END_REF] lithium deintercalation from T # 2-Li2/3Co2/3Mn1/3O2 occurs at an average potential of 4 V vs Li + /Li whereas lithium intercalation in T # 2-Li2/3Co2/3Mn1/3O2 is observed at a significantly lower potential, i.e., around 2.7 V vs Li + /Li. Stepwise formation of T # 2′, O61, and O62 phases is observed as a function of lithium content, the former corresponding to the Li1/2Co2/3Mn1/3O2 composition with a lithium/vacancy ordering in the interslab space and the latter two phases possessing an unusual O6-type oxygen packing. As shown by the perspective view given in Figure 1b, the O6 structure can be described in the R3̅ m space group by six MO2 slabs packed along the c-axis. All the LiO6 octahedra share on one side a face with an MO6 octahedron and on the other side three edges with three MO6 octahedra, but two different types of MO2 slabs are observed in the O6 structure: one with MO6 octahedra sharing only faces with LiO6 octahedra and another with MO6 octahedra sharing only edges with LiO6 octahedra. As shown in Figure 2, the phase transition between T # 2-and O6-type structures is simply achieved by MO2 slab gliding without breaking any M-O bond. [START_REF] Carlier | Lithium Electrochemical Deintercalation from O2-LiCoO2 Structure and Physical Properties[END_REF][START_REF] Carlier | On the Structure of T # 2-LixCoO2[END_REF] In this paper, we study the structural evolutions observed for T # 2-Li2/3Co2/3Mn1/3O2 upon increasing temperature combining thermal analyses and high-temperature powder X-ray diffraction. We will show that a new O6-type Li2/3Co2/3Mn1/3O2 phase can be obtained with interesting electrochemical properties and also, that as expected from ab initio theoretical calculations, the T # 2 and O6-type structures are very close in their formation energy. [START_REF] Carlier | First-Principles Investigation of Phase Stability in the O2-LiCoO2 System[END_REF] Figure 2. Comparison of the a and b cell parameters for the orthorhombic T # 2 and hexagonal O6 stackings and phase transition mechanism between the two oxygen stackings.

Experimental Section

P2-Na2/3Co2/3Mn1/3O2 was synthesized by the coprecipitation method. [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part I: Its Structural Characterization[END_REF] A solution of transition metal nitrates (1 M Co(NO3)2•6H2O:1 M Mn(NO3)2•6H2O = 2:1 by volume) was slowly dripped into a basic solution containing 1 M NaOH and 3 M NH4OH (with a 5% Na excess) while the solution was stirred vigorously. The obtained precipitate suspension was aged for 1 h, dried under vacuum at 80-90 °C for 3 h to evaporate water, and dried in air at 110 °C overnight. The precipitate was then ground and heated at 950 °C under O2 during 12 h. The powder was finally quenched in air. The lithiumcontaining phase was prepared from the sodium-containing one by ion exchange of Na + /Li + in molten salts. For ion exchange, P2-Na2/3Co2/3Mn1/3O2 powder was thoroughly mixed with LiNO3 and LiCl powder (88:12 by weight, melting point ca. 244 °C, with Li/Na = 7) and heated at 280 °C for 1 h in air. After the exchange reaction the mixture was cooled, then the product was washed with demineralized water and the T # 2-Li2/3Co2/3Mn1/3O2 phase was recovered after filtration and drying at 80 °C overnight. This synthesis method allows obtaining a pure phase whose composition and structure were fully characterized as described in details in ref [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part I: Its Structural Characterization[END_REF] . To study the thermal stability of this T # 2 phase, it was thermally treated at 400 or 450 °C for 10 h in air, then it was cooled at several cooling rates or was quenched in air directly at the exit of the furnace into another alumina crucible already at room temperature. Thermogravimetric (TGA) and differential thermal (DTA) analyses (SDT Q600, TA Instrument) were carried out under oxygen stream, at a heating rate of 20 deg•min -1 and from room temperature to 700 °C. The typical weight of samples used was 80 mg. Differential scanning calorimetry (DSC) experiments (Pyris Diamond DSC, Perkin-Elmer) were also performed to evaluate heat exchange of the materials; the temperature was scanned from room temperature to 500 °C at a rate of 40 deg•min -1 . To directly observe high-temperature phases derived from T # 2-Li2/3Co2/3Mn1/3O2 powder X-ray diffraction (XRD) patterns were recorded under air from 25 to 600 °C, using a PANalytical X'Pert Pro diffractometer equipped with an alumina sample holder settled in an HTK1200 Anton Paar furnace, Co Kα radiation, and an X'celerator detector. When T # 2-Li2/3Co2/3Mn1/3O2 powder was heated at a rate of 60 deg•min -1 (respectively 1 deg•min -1 ), XRD patterns were collected at a given temperature for 20 min (respectively 4 min.) in the angular ranges of 5-120° (2θ) (respectively 19-25° (2θ)) in order to determine changes in lattice parameters and phase transition temperatures. Note that for each technique different heating rates were tested; no significant changes were observed in the phase diagram obtained upon thermal treatment of T # 2-Li2/3Co2/3Mn1/3O2. The experiments reported here are those giving for each of them the data (weight loss, heat flow, and XRD pattern resolution) that are the most accurate. The XRD patterns classically recorded to identify the phases formed were collected at room temperature from 5 to 80 °C (2θ) with a 0.02 deg step and a 12 s counting time by step, using a Siemens D5000 powder diffractometer with Cu Kα radiation and a graphite diffracted beam monochromator. Scanning electron micrographs were collected with a Hitachi S4500 field emission electron microscope (FE-SEM) with an accelerating voltage of 5.0 kV. To avoid charge accumulation during observation, a 2-nm-thick layer of platinum was deposited by sputtering on the surface of the materials. The electrochemical properties were tested in lithium batteries defined as Li/(liquid electrolyte)/(positive electrode).

Results and Discussion

1. Thermal Analysis of T # 2-Li2/3Co2/3Mn1/3O2

Thermal analyses of T # 2-Li2/3Co2/3Mn1/3O2 were carried out by using coupled TGA-DTA and DSC techniques in order to obtain more insight about the stability of T # 2 and the possible phase transitions occurring upon its thermal treatment. It was difficult to observe any heat exchange peak for the T # 2 material at a typical heating rate of 5 deg•min -1 since the absolute heat flux is very small. When the TGA-DTA analysis was carried out at a faster scanning rate of 20 deg•min -1 in order to enhance the heat exchange signals from the material, the signals became observable and distinguishable. As shown in Figure 3a no significant weight loss occurs below 450 °C (less than 0.5 wt %) although a small weight loss is observed up to 200 °C, probably due to evaporation of adsorbed water. In the DTA curve also given in Figure 3a 1 /18O2 is expected to be lost (i.e., 1.85 wt %) whereas less than 1 wt % is effectively lost in these conditions. To identify the nature of the phases formed during this thermal treatment, a thermal X-ray diffraction study was performed. 2. High-Temperature X-ray Diffraction Study of T # 2-Li2/3Co2/3Mn1/3O2

Figure 4 shows changes in the powder X-ray diffraction pattern of T # 2-Li2/3Co2/3Mn1/3O2 during its thermal treatment between 25 and 600 °C. The XRD pattern recorded at 25 °C is typical of a pure T # 2-Li2/3Co2/3Mn1/3O2 phase. [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part I: Its Structural Characterization[END_REF] This phase is found to be stable up to 200 °C as no additional diffraction lines were observed. Nevertheless, upon increasing temperature all the peaks shift to lower diffraction angles, in good agreement with a thermal expansion of the T # 2-Li2/3Co2/3Mn1/3O2 unit cell. Above 250 °C, the T # 2 and O6-type stackings clearly begin to coexist and the O6 phase is obtained as a single phase according to Figure 5 in the temperature range [385-450 °C]. As already mentioned in Figure 2, the phase transition from a T # 2 to an O6 stacking is accomplished through (Co2/3Mn1/3)O2 slab gliding of a vector (0, 1/6, 0)Cmca. [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part II: Its Electrochemical Behavior[END_REF][START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part I: Its Structural Characterization[END_REF] To our knowledge this is the first observation of thermal transformation into O6-type structure even if theoretical calculations had already shown for Li1/2CoO2 that a difference in energy of only a few millielectronvolts exists between the T # 2 and O6 stackings. [START_REF] Carlier | First-Principles Investigation of Phase Stability in the O2-LiCoO2 System[END_REF] Until now the O6 stacking was either directly obtained by lithium exchange from the corresponding sodium P2 phase or during a lithium deintercalation or intercalation process. [START_REF] Carlier | Lithium Electrochemical Deintercalation from O2-LiCoO2 Structure and Physical Properties[END_REF][START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part II: Its Electrochemical Behavior[END_REF][START_REF] Komaba | Structural and Electrochemical Behaviors of Metastable Li2/3[Ni1/3Mn2/3O2 Modified by Metal Element Substitution Electrochim[END_REF][START_REF] Paulsen | Layered T2, O6 and P2 type A2/3[M′1/3M2/3O2 bronzes A = Li, Na; M′ = Ni, Mg; M = Mn[END_REF] At higher temperatures, above 460 °C, the layered O6 stacking was decomposed into Co3O4 and Li2MnO3. The (002) peak of the T # 2 phase (at ∼20.6° (2θCo)) continuously shifts to smaller diffraction angles upon an increase in temperature between 25 and 250 °C showing the expansion of the unit cell along the c-axis. Moreover, in the temperature range from 250 to 385 °C, the (006) diffraction line of the O6 phase is getting progressively more intense in parallel with a decrease in intensity of those associated to T # 2. At 385 °C the phase transformation from T # 2 into O6 is achieved. From 460 °C, the Co3O4 and Li2MnO3 phases appear gradually resulting from a decomposition of O6-Li2/3Co2/3Mn1/3O2. Figure 5 also shows that the diffraction peak associated to Co3O4 at 22.1° (2θCo) appears during the heating process from 460 °C, whereas that of Li2MnO3 at 21.7° (2θCo) is clearly observed from 520 °C. Intergrowths between the two phases are probably formed as intermediate phases. 3. O6HT-Li2/3Co2/3Mn1/3O2: Synthesis, Morphology, and Electrochemical Properties Based on the above X-ray diffraction analysis performed upon increasing temperature, the compounds obtained after a thermal treatment of T # 2 at 400 and 450 °C for 10 h followed by a quenching in air were recovered at room temperature and analyzed by XRD as shown in Figure 6. The material obtained after heating at 400 °C consisted surprisingly mainly of the T # 2 phase with only traces of O6, whereas the XRD pattern of that obtained after a thermal treatment at 450 °C can be fully indexed as an O6 phase in the R3̅ m space group without any additional phases, as highlighted in Figure 6. These results suggest the following: (i) the T # 2 → O6 phase transformation is reversible at 400 °C whereas it is not at 450 °C and (ii) as a consequence the O6 phases formed at 400 and 450 °C are different in structural defects and/or in composition.

Even if no direct comparison can be made, as it was observed at room temperature, it is interesting to mention at this point that the study of the phase diagrams observed upon cycling for O2-LiCoO2 [START_REF] Carlier | Lithium Electrochemical Deintercalation from O2-LiCoO2 Structure and Physical Properties[END_REF] and T # 2-Li2/3Co2/3Mn1/3O2 [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part II: Its Electrochemical Behavior[END_REF] has also revealed reversible transformation between O2, T # 2, and O6 stackings depending on the competition between the composition in lithium and the electrostatic repulsions: indeed, when lithium ions play sufficiently their role of screening ions in the structure the 8e tetrahedral sites are such that they can accommodate them, then slabs gliding occurs in order to stabilize other (octahedral) sites for lithium ions and minimize the electrostatic repulsion between the oxygen layers (with as a consequence a decrease in the interlayer distance). In addition, irreversible phase transformation between T # 2 and O6 stackings was also observed in LixCo2/3Mn1/3O2 (x > 2/3) and was associated to the migration of cobalt ions from the slab to the interslab space in tetrahedral sites (Phase O62 in Figure 5 in ref 6). The O6-type phase obtained here by quenching the T # 2-Li2/3Co2/3Mn1/3O2 phase after a thermal treatment at high temperature will be denoted in the following as O6HT to discriminate with the O6 phases obtained upon cycling (O61 and O62 in ref [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part II: Its Electrochemical Behavior[END_REF] ). The two phases T # 2 and O6HT were shown to be similar in composition combining Li, Mn, and Co chemical analyses, as well as redox titration of the average transition metal oxidation state. To check more carefully the (ir)reversibility of the phase transformation between T # 2 and O6, the formation of the O6HT phase was examined according to the cooling rate, after maintaining the temperature at 450 °C for 10 h. Figure 7 shows the XRD patterns of the materials recovered after a cooling at a rate of 1, 6, or 60 deg•h -1 , or a quenching in air. Obviously, the nature of the materials obtained is highly dependent on the cooling rate. Pure O6HT phase is obtained only after a quenching. The slower the cooling the more T # 2 is recovered: at a rate of 1 deg•h -1 the material formed is mainly the T # 2 phase, with a small amount of the O6HT phase remaining. A rather similar T # 2-O6HT mixture seems to be obtained after a quenching from 400 °C (as shown by the comparison of Figures 6 and7), but with much sharper diffraction peaks showing that the T # 2 phase recovered after a thermal treatment at lower temperature shows a more ordered structure. As described previously, [START_REF] Carlier | On the Structure of T # 2-LixCoO2[END_REF] the change in the broadening of the diffraction lines (a sharpening of the O6HT lines in parallel with a broadening of the T # 2 lines with increasing cooling rate) is directly in relation with the relative size along the c-axis of the O6HT and T # 2 domains within the crystallites. A thermal treatment at 450 °C would induce a partial migration of Co and/or Mn ions directly through a face shared by the octahedral sites of the MO2 slabs (initially fully occupied) and the octahedral sites of the interslab spaces only partially occupied (□1/3Li2/3O2) (see the description of the structure in Figure 1). By quenching from 450 °C these transition metal ions would remain in the interslab spaces, acting as pillars between the slabs and preventing the reverse transformation from the O6 to the T # 2 structures. With a very slow cooling, the phase transition from O6 to T # 2 occurs very slowly but reversibly, in good agreement with a higher stability of the Co 3+ and Mn 4+ ions in the octahedral sites of the slabs due to smaller M-O distances (∼1.91 Å vs <2.17 Å in the interslab spaces depending on the position of M, at the center or shifted from the center) that implies the further stabilization of the more stable phase, the quasi-lamellar O6 phase, and thus the possibility for a reverse transformation from the O6 to the T # 2 phases. The thermal behavior of T # 2-Li2/3Co2/3Mn1/3O2 is schematically summarized in Figure 8. As reported in ref [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part II: Its Electrochemical Behavior[END_REF] the pristine T # 2 phase undergoes a reversible phase transition (through slab gliding) into O61 by lithium deintercalation at room temperature. Similar slab gliding is involved in the formation of O6 from T # 2 upon increasing temperature and after a thermal expansion of T # 2 up to 350 °C, and the expansion leading to enlargement of the interslab distance makes the tetrahedral sites unstable for the lithium ions. Then, at approximately 450 °C, partial migration of transition metal ions occurs from their original octahedral sites in the slabs into the interslab spaces, decreasing the kinetics of the reversible transition from O6 to T # 2 as Co 3+ and Mn 4+ ions are not stable in the tetrahedral sites from the interslab space of T # 2. Only very slow cooling rates (less than 1 deg•h -1 ) allow recovering the T # 2-type phase after a thermal treatment at 450 °C. At higher temperatures than 450 °C, the material is gradually decomposed into cobalt and manganese oxides (Co3O4 and Li2MnO3).

It is interesting to compare the morphology and electrochemical properties of this new phase O6HT-Li2/3Co2/3Mn1/3O2 with that of the pristine T # 2. As clearly shown in Figure 9 the size and shape of the particles are maintained during the phase transition from T # 2-to O6HT-Li2/3Co2/3Mn1/3O2, which agrees well with the fact that it is simply due to slab gliding as described in Figure 2. The particles are characterized by a strong anisotropy with a platelet-like shape, a diameter ranging from 2 to 6 μm, and a thickness from 0.5 to 1 μm. Figure 10a gives a typical cycling curve obtained at C/20 between 2.0 and 4.8 V vs Li + /Li for lithium cells using O6HT as positive electrode material. O6HT-Li2/3Co2/3Mn1/3O2 exhibits comparable electrochemical properties to the pristine T # 2 phase, with two potential domains: one at an average voltage of 4 V vs Li + /Li corresponding to Li deintercalation from O6HT-Li2/3Co2/3Mn1/3O2 with most probably the oxidation of Co 3+ to Co 4+ and another at an average voltage around 2.7 V vs Li + /Li corresponding to Li intercalation in O6HT-Li2/3Co2/3Mn1/3O2 and thus to reduction of the transition metal ions (Mn and/or Co). Combined XPS and XANES measurements are currently underway in order to follow the oxidation and redox processes involved in this material upon cycling. As shown in Figure 10a, 0.8 Li + ions can be reversibly removed and inserted from/into this host structure. Figure 10b gives a comparison of the GITT curves obtained for O6HT-Li2/3Co2/3Mn1/3O2 and the pristine T # 2. In the high-voltage domain the charge curves are very similar for the two compounds, with two plateaus at 4.13 and 4.22 V vs Li + /Li. In the low-voltage domain, their discharge curves are very different with a continuous change in potential for O6HT upon lithium intercalation and the observation of two distinct potential domains for T # 2 in the composition ranges [2/3 to ∼0.8] and [∼0.8 to 1]. The structure of O6HT will be described in detail in a forthcoming paper, with also an in-depth study of the structural changes occurring upon its cycling as positive electrode material in lithium batteries. These processes involved upon cycling will be compared to those observed for the T # 2 material.

Conclusions

The layered oxide Li2/3Co2/3Mn1/3O2, showing an original T # 2 stacking obtained by a Na/Li ion exchange in P2-Na2/3Co2/3Mn1/3O2, was transformed by thermal treatment and through slab gliding into another original O6 stacking. The thermal expansion of the interlayer distance destabilizes the T # 2 stacking with lithium ions in tetrahedral sites, and promotes the formation of an O6 stacking with lithium ions in octahedral sites and a reduction of the interlayer distance.

The reversibility of the phase transition T # 2 to O6 strongly depends on both the temperature and cooling rate at which O6 is obtained, in other words on the amount of defects formed (i.e., the amount of transition metal ions present in the interslab spaces). This study highlights first that the difference in energy between the two stackings T # 2 and O6 is effectively very small (as already suggested by ab initio theoretical calculations, i.e. a few meV [START_REF] Carlier | First-Principles Investigation of Phase Stability in the O2-LiCoO2 System[END_REF] ), and then that the T # 2 stacking is definitely destabilized to the benefit of the O6 stacking as soon as the interlayer distance becomes too long: it was already observed upon lithium deintercalation with an increase of the electrostatic repulsions between the oxygen layers, [START_REF] Tournadre | The T # 2-Li2/3Co2/3Mn1/3O2 System -Part II: Its Electrochemical Behavior[END_REF][START_REF] Carlier | First-Principles Investigation of Phase Stability in the O2-LiCoO2 System[END_REF] and as revealed by this study through a thermal expansion of the stacking axis. The structure of these layered oxides is a result of a compromise between a minimization of the electrostatic repulsions and the stabilization of the cations (alkali and/or transition metals) in the sites available in the oxygen packing. Finally, it is interesting to stress the unique electrochemical properties delivered by these layered oxides obtained through a Na + by Li + ion-exchange reaction, with a reversible capacity of ∼230 mAh/g in the potential window [2-4.8 V].

Figure 1 .

 1 Figure 1. Description (a) of the T # 2-type stacking and (b) of the O6-type oxygen stacking for Li2/3Co2/3Mn1/3O2.

  three endothermic peaks are observed in the temperature ranges of 100-200, 200-400, and 400-500 °C. DSC measurement performed at a rate of 40 deg•min -1 also reveals three endothermic peaks in very similar temperature ranges, as shown in Figure3b. From these DTA and DSC data we can safely conclude that they are not background drifts but endothermic signals from the material. Signal 1 around 100-200 °C is most probably attributed to the evaporation of adsorbed water at the surface of the powder. Signals 2 and 3 are due to phase transformations as confirmed below by high-temperature XRD; the latter signal is rather sharp and intense and appears to be not reversible (at least at 40 deg•min -1 ), as shown on the DSC curve given in Figure3b. At higher temperature, above 500 °C in Figure3a, an exothermic peak associated to a weight loss is observed on the DTA curve and attributed to the decomposition into Li2MnO3 and Co3O4 (as further confirmed by XRD):

P a g e | 5 Figure 3 .

 53 Figure 3. (a) TGA-DTA and (b) DSC curves obtained for T # 2-Li2/3Co2/3Mn1/3O2 by a thermal treatment performed at 20 and 40 deg•min - 1 under oxygen, respectively. The baselines for DTA and DSC are shown by blue dashed lines and the heat exchange peaks are marked by blue arrows.

Figure 4 .

 4 Figure 4. Powder XRD patterns recorded in situ during the thermal treatment of T # 2-Li2/3Co2/3Mn1/3O2 from 25 to 600 °C. The heating rate was 60 deg•min -1 . When the target temperature was reached, it was maintained for 30 min: 10 min before the acquisition and 20 min during the acquisition. The O6 diffraction peaks are highlighted by red dashed lines in the angular range (40-80° (2θCo)).

Figure 5 .

 5 Figure 5. First Bragg reflections of the XRD patterns obtained in situ during the thermal treatment of T # 2-Li2/3Co2/3Mn1/3O2 from 25 to 600 °C. Temperature was raised at 1 deg•min -1 between each target temperature. The XRD patterns were collected during 4 min while the temperature was maintained.

Figure 6 .

 6 Figure 6. XRD patterns of materials recovered at room temperature after the thermal treatment of T # 2-Li2/3Co2/3Mn1/3O2 at 400 and 450 °C for 10 h and a quenching in air, compared with that of the pristine T # 2-Li2/3Co2/3Mn1/3O2.

Figure 7 .

 7 Figure 7. XRD patterns of the materials recovered after the thermal treatment of T # 2-Li2/3Co2/3Mn1/3O2 at 450 °C for 10 h and its cooling in air at various rates: quenching, 60, 6 , 1 deg•h -1 , compared with that of the pristine T # 2-Li2/3Co2/3Mn1/3O2.

Figure 8 .

 8 Figure 8. Schematic illustration of the thermal evolution and phase transition observed for T # 2-and O6HT-Li2/3Co2/3Mn1/3O2.

Figure 9 .

 9 Figure 9. SEM micrographs of (a) T # 2-and (b) O6HT-Li2/3Co2/3Mn1/3O2 prepared instead of heat treatment of T # 2 at 450 °C for 10 h followed by a quenching to room temperature in air.

Figure 10 .

 10 Figure 10. (a) Galvanostatic charge-discharge curves obtained for an Li//O6HT-Li2/3Co2/3Mn1/3O2 cell between 4.8 and 2.0 V vs Li + /Li at a rate of C/20 and (b) comparison of GITT curves obtained for T # 2 and O6HT starting in charge and in discharge from the Li2/3Co2/3Mn1/3O2 composition.

  by volume) was used as electrolyte. A foil of lithium metal was used as the negative electrode. The cells were assembled in an argon-filled drybox and cycled at room temperature, either at C/20 between 2 and 4.8 V vs Li + /Li or in galvanostatic intermittent titration technique (GITT) conditions (i.e., at a rate of C/100 rate during the exchange of 0.006 e -with alternative periods of relaxation such as ΔV/Δt < 1 mV•h -1 ). C rate corresponds to a theoretical exchange of one electron in one hour during charge and discharge.

The positive electrode consisted of a mixture of 88 wt % active material, 10 wt % carbon black/graphite (1:1), and 2 wt % binder (PTFE for polytetrafluoroethylene). LiPF6 (1 M) dissolved in a mixture of propylene carbonate (PC), ethylene carbonate (EC), and dimethyl carbonate (DMC) (1:1:3

Acknowledgment

The authors thank Philippe Dagault and Cathy Denage (ICMCB) for technical assistance. This work was financially supported by CNES and Région Aquitaine (CPER Véhicule Electrique 21-13). S.K. thanks MEXT (Japan) for financial support for staying in ICMCB.

Author Information


Corbinian Schlosser
email: cschlosser@laas.fr

Converging approximations of attractors via almost Lyapunov functions and semidefinite programming

Keywords: Global attractor, Lyapunov function, outer approximations, dynamical systems, infinite-dimensional linear programming, sum-of-squares, semidefinite programming

 are based on convex optimization via sum-of-squares techniques the same is true for our proposed method. The method is easy to use and numerical examples illustrate the procedure.

Introduction

Global attractors and Lyapunov functions are intimately connected (see for example [START_REF] Nam | Dynamical systems: stability theory and applications[END_REF]). Hence it is only natural that Lyapunov functions play a central role in control and asymptotic analysis of dynamical systems. Computing Lyapunov functions is therefore an intriguing but complicated task and often restricted to the case of finding a Lyapunov function for an apriori given set. That finds excessive application in verification of asymptotic stability of an equilibrium point x * which lays at the base of many control problems. There are several techniques available to search for Lyapunov functions for a given set A (such as A = {x * }), see [START_REF] Giesl | Review on computational methods for lyapunov functions[END_REF]. Among these are sum-of-squares (SOS) techniques that recently gained more and more popularity, beginning with the work of Parillo [START_REF] Parrilo | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[END_REF], and have further developed since [START_REF] Anderson | Advances in computational lyapunov analysis using sum-of-squares programming[END_REF]. SOS techniques typically result in feasibility problems for linear matrix inequalities for which fast numerical methods exist.

In the case of exponentially asymptotically stable fixed points searching for polynomial Lyapunov functions is sufficient [START_REF] Anderson | Advances in computational lyapunov analysis using sum-of-squares programming[END_REF]. But already dropping the assumption of exponentially fast decay causes that global asymptotic stability of a single equilibrium point can not any longer be verified by a polynomial Lyapunov function, i.e. there exists a dynamical system with globally asymptotically stable equilibrium point for which no polynomial Lyapunov function exists [START_REF] Amir | A globally asymptotically stable polynomial vector field with no polynomial lyapunov function[END_REF]. That shows that the general use of SOS polynomial Lyapunov functions is limited and extensions of the SOS methods are needed in order to treat more general systems. This gets even more drastic when we search for the attractor of the system, that is we do not a-priori know for which set A we want to check if it is an attractor.

Except for Lyapunov based techniques, several methods for computing/approximating the global attractor have been proposed. Some approximate the attractor by following trajectories for long but finite time T ∈ [0, ∞) or using set oriented methods as in [START_REF] Dellnitz | The algorithms behind gaio-set oriented numerical methods for dynamical systems[END_REF]. Another approach motivated by a relaxation of the notion of Lyapunov functions was given in [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF]. There the authors showed that the weaker concept of Lyapunov function they use is still sufficient to envelope the attractor by positively invariant sets while at the same time SOS polynomials can be used. In order to make their method computationally tracktable -via a SOS hierarchy of finite dimensional (convex) semidefinite programs (SDPs) -the authors relaxed their problem at the cost of guaranteed convergence. Related to this approach is [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] where the global attractor is characterized by an infinite dimensional linear programming problem (LP), also solved via a hierarchy of finite dimensional SDPs. In [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] the authors showed convergence but the obtained sets lack the good property of being positively invariant. In this paper we merge both methods into one, maintaining both their advantages (using positively invariant sets and guaranteed convergence). We use that, for both, the basis is built by an infinite dimensional optimization problem. This similarity in nature can be well combined, results in an infinite dimensional LP and can be solved by sequence of finite dimensional SPDs.

Notations

We denote by N the natural numbers. The non-negative real numbers are denoted by R + . We denote the euclidean inner product of two vectors a, b ∈ R n by a • b. The function dist(•, K) denotes the Euclidean distance function to a set K ⊂ R n and dist(K 1 , K 2 ) denotes the Hausdorff distance of two subsets of R n . The space of polynomials (in n variables) is denoted by R[x 1 , . . . , x n ] or shorter R[x]. The degree of p ∈ R[x] is denoted by deg(p) and the set polynomials of degree at most k ∈ N by R[x] d . The space of continuous functions on X is denoted by C(X) and the space of continuously differentiable functions on R n by C 1 (R n ) and we denote the gradient of g ∈ C 1 (R n ) by ∇g. The Lebesgue measure is denoted by λ.

Setting and preliminary definitions

We consider ordinary differential equations

ẋ = f (x), x(0) = x 0 ∈ R n . ( 1
)
for a locally Lipschitz continuous vector field f : R n → R n . Further we restrict to a compact constraint set X ⊂ R n . By ϕ t (x 0 ) we denote the solution of (1) at time t.

Next we define what it means for a set to be positively invariant for (1).

Definition 1 A set S is called positively invariant for (1) if for all x ∈ S and t ∈ R + the solution ϕ t (x) exists and is located in S.

The main object in this paper are attractors. There are several different notions of attractors, we use the concept of global attractors from [START_REF] Robinson | Infinite-Dimensional Dynamical Systems[END_REF].

Definition 2

The set

M + = {x 0 ∈ X : ϕ t (x 0 ) ∈ X for all t ∈ R + } (2)
is called the maximum positively invariant (MPI) set. A compact set A ⊂ X is called a global attractor if it is minimal uniformly attracting, i.e., it is the smallest compact set A such that lim t→∞ dist(ϕ t (M + ), A) = 0.

Since X, and hence also M + is compact, the global attractor exists and is unique [START_REF] Robinson | Infinite-Dimensional Dynamical Systems[END_REF].

It remains to define (smooth) Lyapunov functions.

Definition 3 Let M + be the maximum positively invariant set. A non-negative function

V ∈ C 1 (R n
) is called a smooth Lyapunov function for the dynamical system induced by f if for all x ∈ M + we have

∇V (x) • f (x) ≤ -V (x). ( 3
)
We say V is a Lyapunov function for a set A ⊂ M + if V is a Lyapunov function and

A := V -1 ({0}) ∩ M + .
Lyapunov functions and global attractors are intimately related, one reason is the following theorem.

Theorem 1 ([3] Theorem 2.7.1, [START_REF] Andrew | A smooth lyapunov function from a class-estimate involving two positive semidefinite functions[END_REF]) A compact set A ⊂ M + is asymptotically stable for (1) with constraint set X if and only if there exists a smooth Lyapunov function V for A. On the contrary, if V is a (smooth) Lyapunov function then V -1 ({0}) is asymptotically stable and contains the global attractor.

Two infinite dimensional optimization problems for the global attractor

We begin with presenting the infinite dimensional optimization problems for global attractors from [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] and [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF].

The first one is the LP presented in [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] which reads

p * 1 = inf X w(x) dx s.t. (w, v 1 , v 2 ) ∈ C(R n ) × C 1 (R n ) × C 1 (R n ) -v 1 -v 2 + w ≥ 1 on X w ≥ 0 on X βv 1 -∇v 1 • f ≥ 0 on X βv 2 + ∇v 2 • f ≥ 0 on X (4)
where β > 0 is a parameter. In [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] the authors showed λ(A) = p * 1 where λ denotes the Lebesgue measure. Each feasible triple (w,

v 1 , v 2 ) induces a set A := w -1 ([1, ∞)) that satisfies A ⊂ A and λ(A \ A) ≤ X w dλ -p * 1 , (5)
and hence, as (w, v 1 , v 2 ) gets optimal, A converges to A with respect to the Lebesgue measure [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF].

The optimization problem from [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] reads

p * 2 = inf λ (J -1 ([0, 1])) s.t. J ∈ C 1 (R n ) J(x) ≥ 0 for all x ∈ X ∇J • f ≤ -(J -1) on X ∅ = J -1 ([0, 1]) ⊂ X (6)
The reason for replacing the Lyapunov constraint ∇J • f ≤ -J by ∇J • f ≤ -(J -1) is that there always exist polynomials satisfying the second constraint but, as mentioned before, this might not be the case for the first. In [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] the authors showed that p * 2 = λ(A) and that the set J -1 ([0, 1]) ⊂ X is positively invariant, contains the global attractor and converges towards the global attractor with respect to the Lebesgue measure when J gets optimal.

Compared to the method from [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] this has the advantage that the sets obtained as in [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] are always positively invariant. On the other hand ( 6) is not linear and the evaluation of the cost is difficult, while evaluating the cost in (4) is easy. A heuristic for the cost term in [START_REF] Fantuzzi | Bounding extreme events in nonlinear dynamics using convex optimization[END_REF] was proposed in order to simplify the cost term but this came at the cost of exactness and convergence could not be guaranteed anymore.

We will formulate an optimization problem that combines the advantages of both methods without sacrificing convergence. Afterwards we solve the optimization problem as in [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] and [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] using sum-of-squares methods.

A combined approach

Here we will merge the idea of the relaxed Lyapunov equation

∇J • f ≤ -(J -1) (7)
and the linear structure of ( 4) with an easy evaluation of the cost term.

We first note the following result, which is only a reformulation of [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF]Proposition 5], about a perturbed Lyapunov condition.

Lemma 1 Let ε > 0 and 0 ≤ J ∈ C 1 (R n ) satisfy ∇J • f ≤ -(J -ε) = -J + ε on M + ( 8
)
then the set S := J -1 ([0, ε]) is positively invariant and contains the global attractor (for X).

Further, if S ∩ X ⊂ X then S ∩ X ⊂ M + .
Proof: This follows from [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF]Proposition 5] by considering the function ε -1 V which satisfies [START_REF] Giesl | Review on computational methods for lyapunov functions[END_REF].

We refer to almost Lyapunov functions for functions 0 ≤ J that satisfy [START_REF] Goluskin | Bounds on mean energy in the kuramotosivashinsky equation computed using semidefinite programming[END_REF]. Because of the close relation between Lyapunov functions and functions solving (8) we start with an infinite dimensional LP motivated by ( 4) and ( 6) but involving only Lyapunov functions before turning to the perturbed version (note that the following LP considers M + and not X yet)

p * 3 = inf X w(x) dx s.t. (w, V ) ∈ C(M + ) × C 1 (R n ) w + V ≥ 1 on M + w ≥ 0 on M + V ≥ 0 on M + ∇V • f ≤ -V on M + (9)
This is also an infinite dimensional LP and clearly, any feasible V is a Lyapunov function and hence by Theorem 1 we have V = 0 on the global attractor A. In particular from w + V ≥ 1 on M + it follows w ≥ 1 on A and by non-negativity of w,

M + w(x) dx ≥ λ(A).
That means p * 3 ≥ λ(A). On the other hand let V be a Lyapunov function for A, i.e.

V -1 ({0}) = A, satisfying ∇V • f ≤ -V , according to Theorem 1. For k ∈ N define the function w k := max{0, 1 -k • V } ∈ C(M + ). The pair (w k , k • V ) is feasible for all k ∈ N and for all x ∈ M + w k (x) → 1, x ∈ A 0, else.
By the Beppo-Levi monotone convergence theorem it follows

M + w k (x) dx → λ(A), hence p * 3 ≤ λ(A). We have shown Proposition 1 We have p * 3 = λ(A).
In the next step we want to replace the last constraint in the LP ( 9) by ( 8) because this will allow us to search for feasible J that are polynomial. Since ( 8) is a relaxation of the Lyapunov condition (3) we penalize a function for not being a Lyapunov function. The corresponding LP reads

p * 4 = inf X w(x) dx + ελ(X) s.t. (w, J, ε) ∈ C(R n ) × C 1 (R n ) × [0, ∞) w + J ≥ 1 on M + w ≥ 0 on M + J ≥ 0 on M + ∇J • f ≤ -J + ε on M + (10)
Since any feasible pair (w, V ) for the LP (9) induces a feasible triple (w, V, 0) for [START_REF] Jones | Converse lyapunov functions and converging inner approximations to maximal regions of attraction of nonlinear systems[END_REF] with the same cost we get p * 4 ≤ p * 3 . On the other hand for any feasible (w, J, ε) we have by Lemma 1 that V ≤ ε on A and hence by the first constraint in [START_REF] Jones | Converse lyapunov functions and converging inner approximations to maximal regions of attraction of nonlinear systems[END_REF] we have w ≥ 1 -ε on A and w ≥ 0 by the second constraint, so that we get

X w(x) dx + ελ(X) ≥ (1 -ε)λ(A) + ελ(X) ≥ (1 -ε)λ(A) + ελ(A) = λ(A).

This proves

Proposition 2 We have p * 4 = λ(A).

The last obstacle that we have to overcome is that we do not want to include M + explicitely because the set M + is typically not known a-priori. Therefore we make use of what already entered the LP (4) (note that in the LP (4) only X but not M + appears) where the MPI set enters implicitly through the constraint βv -∇v • f ≥ 0, see [START_REF] Korda | Convex computation of the maximum controlled invariant set for polynomial control systems[END_REF]. That leads to the final LP with discounting factor β > 0

p * 5 = inf X w(x) dx + ελ(X) s.t. (w, J, ε, v) ∈ C(R n ) × C 1 (R n ) × [0, ∞) × C 1 (R n ) w + J -v ≥ 1 on X w ≥ 0 on X J ≥ 0 on X ∇J • f + J + v ≤ ε on X βv -∇v • f ≥ 0 on X (11)
We will show that also for this LP we have that the optimal value is given by the volume of the global attractor λ(A).

Theorem 2 Let X be compact and f : R n → R n be a locally Lipschitz continuous vector field. Let A be the global attractor for the dynamical system induced by f with constraint set X. Then we have for any β > 0 in (11)

p * 5 = λ(A).
Further for any feasible (w, J, ε, v) we have J -1 ([0, ε]) is positively invariant and

A ⊂ K := J -1 ([0, ε]) ∩ v -1 ([0, ∞)) ∩ X (12) with λ(K \ A) ≤ X w(x) dx + ελ(X) -p * 5 ( 13
)
which converges to zero as (w, J, ε, v) gets optimal for [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF].

Proof: The essential observation is that any feasible (w, J, ε, v) satisfies v ≥ 0 on M + . This is implied by the last constraint in [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] [13, Lemma 3]. It follows that ∇J • f ≤ -J + ε on M + as well as w + J ≥ 1 on M + , i.e. that (w, J, ε) is feasible for the LP (10), p * 5 ≥ p * 4 = λ(A) and that the set J -1 ([0, ε]) is positively invariant and contains A by Lemma 1. To show the remaining inequality, λ(A) ≤ p * 5 , we use a construction from [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] to find a function

v ∈ C 1 (R n ) with βv -∇v • f = 0, v = 0 on M + and v < 0 on X \ M + . (14)
In [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] it was shown that such a v exists for β larger than Lip(f, X) the Lipschitz constant of f on X. If β is smaller than Lip(f, X) then there exists still a function v having properties (arbitrarily) close to [START_REF] Korda | Moments and convex optimization for analysis and control of nonlinear partial differential equations[END_REF] (in the sense that βv -∇v • f ≥ 0, v ≤ r on M + and v < 0 on a relative open set U ⊂ X \ M + where r > 0 is arbitrary and U can be chosen arbitrarily close to X \ M + , see [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF]. The following arguments then remains the same, just gets a bit more technical -therefore we assume to have a funciton v satisfying [START_REF] Korda | Moments and convex optimization for analysis and control of nonlinear partial differential equations[END_REF]. We show that for any point (w, J, ε) feasible for [START_REF] Jones | Converse lyapunov functions and converging inner approximations to maximal regions of attraction of nonlinear systems[END_REF] and δ > 0 we can find k ∈ N such that (w + δ, J, ε + δ, kv) is feasible for [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF]. The corresponding cost for (w + δ, J, ε + δ, ṽ) differs from the cost of (w, J, ε) for the LP [START_REF] Jones | Converse lyapunov functions and converging inner approximations to maximal regions of attraction of nonlinear systems[END_REF] only by 2δλ(X). Then λ(A) = p * 4 ≤ p * 5 follows once we have found a k ∈ N such (w + δ, J, ε + δ, kv) is feasible for [START_REF] Jones | Converse lyapunov functions and converging inner approximations to maximal regions of attraction of nonlinear systems[END_REF]. The non-negativity of w and J is not (negatively) affected by δ we focus on the first and fourth constraint in the LP [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF]. For any k ∈ N and all x ∈ X where

w(x) + J(x) + δ ≥ 1 and ∇J(x) • f (x) + J(x) ≤ ε + δ (15)
we have w(x) + δ + J(x) -kv ≥ 1 and ∇J(x) • f (x) + J(x) + kv ≤ ε + δ as well, by nonpositivity of kv. Therefore let us focus on the points where (15) is not satisfied, i.e. on the set

U = {x ∈ X : w + J < 1 -δ or ∇J • f + J > ε + δ}. ( 16
)
By feasibility of (w, J, ε) for [START_REF] Jones | Converse lyapunov functions and converging inner approximations to maximal regions of attraction of nonlinear systems[END_REF] we have U ∩ M + = ∅ and from δ > 0 and continuity of w and J, ∇J and f we even have U ∩ M + = ∅. Hence there exists ρ > 0 such that -v > ρ on

U . Let now k ∈ N with k ≥ ρ -1 max x∈U {1 -δ -w(x) -J(x), ∇J(x) • f (x) + J(x) -ε}.
For this choice of k we get for all x ∈ U w(x) + δ + J(x) -kv(x) ≥ w(x) + δ + J(x) -kρ ≥ 1 and

∇J(x) • f + J(x) + kv(x) ≤ ∇J(x) • f + J(x) -kρ ≤ ε.
Hence (w + δ, J, ε + δ, ṽ) is feasible and because δ > 0 was arbitrary we conclude p * 5 ≤ p * 4 = λ(A). It remains to show [START_REF] Korda | Convex computation of the maximum controlled invariant set for polynomial control systems[END_REF]. The first constraint in the LP [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] gives for K from ( 12)

w ≥ 1 -J + v ≥ 1 -ε on K (17)
which contains A due to A ⊂ M + ⊂ v -1 ([0, ∞) as mentioned at the beginning of the proof. Non-negativity of w now implies

X w(x) dx + ελ(X) ≥ K 1 -ε dx + ελ(X) = (1 -ε)λ(K) + ελ(X) ≥ λ(K).
Subtracting p * 5 = λ(A) on both sides finishes the proof. From Lemma 1 we derive the following corollary.

Corollary 1 In Theorem 2, if J -1 ([0, ε]) ⊂ X then the sets X and v -1 ([0, ∞) can be omitted in [START_REF] Korda | Computing controlled invariant sets from data using convex optimization[END_REF].

Proof: By Lemma 1 we have J -1 ([0, ε]) ∩ X ⊂ M + and the crucial estimate (17) used to show (13) holds even on J -1 ([0, ε]).

Note that by [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] there exist minimizing sequences for [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] with almost Lyapunov functions J such that J -1 ([0, ε]) ⊂ X whenever A ⊂ X.

Remark 1

The dual problem of the LP [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] acts on the space of Borel measures on X. We did not include the dual problem here for two reasons: First, it demands further technical notation. Second, the dual problem gives less insight into the global attractor -this is pointed out in [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] -as for instance When the global attractor has vanishing Lebesgue volume then one optimal solution to the dual problem is trivial, i.e. all decision variables are zero.

Remark 2 Discrete time systems can be handled in the same way -only the continuous time objects have to be replaced by their discrete counterparts. We refer to [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] for details about a similar treatment for discrete time systems. A discrete time system is of the form x + = f (x), and the notions of (maximum) positive invariant sets and global attractors adapt ( [START_REF] Robinson | Infinite-Dimensional Dynamical Systems[END_REF], [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF]). The LP corresponding to the LP [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] has the following form for discounting factor α and decay rate γ with α, γ ∈ (0, 1)

p * discrete = inf X w(x) dx + ελ(X) s.t. (w, J, ε, v) ∈ C(R n ) × C(R n ) × [0, ∞) × C(R n ) w + J -v ≥ 1 on X w ≥ 0 on X J ≥ 0 on X γJ • f -J + v ≤ ε on X v -αv • f ≥ 0 on X ( 18
)
Similar to Theorem 2 we obtain that p * discrete = λ(A) for discrete time systems.

Solving the linear programs

Here we assume algebraic structure of the problem in order to get access to the polynomial optimization framework that has also been used in [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] and [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] to solve the corresponding optimization problems by a hierarchy of finite dimensional semidefinite programs. The idea is to replace the decision variables w, J, v by polynomials (this will be justified by the Stone-Weierstraß theorem) and the non-negativity will be algebraically certified by a SOS condition. This is a standard procedure and we refer to [START_REF] Bernard | Moments, positive polynomials and their applications[END_REF] and [START_REF] Korda | Convex computation of the maximum controlled invariant set for polynomial control systems[END_REF] for details.

Assumption 1:

The vector field f is polynomial and X is a compact basic semi-algebraic set, that is, there exist polynomials g 1 , . . . , g j ∈ R[x 1 , . . . , x n ] such that X = {x ∈ R n : g i (x) ≥ 0 for i = 1, . . . , j}. Further we assume that one of the p i is given by g

i (x) = R 2 X -x 2 2 for some large enough R X ∈ R.
For each fixed degree k of the appearing polynomials we get the following SDP

d k := inf w l + εl 1 s.t. w, J, v ∈ R[x] k , ε ∈ R, p 0 , q 0 , r 0 , s 0 ∈ R[x]k /2 p i , q i , r i , s i ∈ R[x](k-deg(g i )) /2 for i = 1, . . . , j ε ≥ 0 w + J -v -1 = p 2 0 + j i=1 p 2 i g i w(x) = q 2 0 + j i=1 q 2 i g i ∇J • f + J + v -ε = t 2 0 + j i=1 t 2 i g i βv -∇v • f = s 2 0 + j i=1 s 2 i g i (19)
where w is the vector of coefficients of the polynomial w and l is the vector of the moments of the Lebesgue measure over X (i.e., l α = X x α dx, α ∈ N n , i α i ≤ k), both indexed in the same basis of R[x] k such that l 1 corresponds to α = 0, i.e. l 1 = λ(X), which gives w l = X w(x) dx.

The following theorem shows that the properties of the infinite dimensional LP are inherited to the finite dimensional SPDs and that their optimal values converge to λ(A) as well.

Theorem 3 Under Assumption 1 we have

d k λ(A) as k → ∞, for any feasible (w k , J k , ε k , v k ) (and corresponding p i , q i , r i , s i ) we have J -1 k ([0, ε k ]
) is positively invariant and

J -1 k ([0, ε k ]) ∩ v -1 k ([0, ∞) ∩ X ⊃ A (20)
with convergence with respect to the Lebesgue measure as k → ∞ when (w k , J k , ε k , v k ) get optimal for [START_REF] Robinson | Infinite-Dimensional Dynamical Systems[END_REF].

Proof: Since the SPDs [START_REF] Robinson | Infinite-Dimensional Dynamical Systems[END_REF] are tightenings of the LP [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] we only need to show convergence. The proof is very similar to the proof of [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF]Theorem 5 and 6]. The important difference to the LP [START_REF] Henrion | Convex computation of the region of attraction of polynomial control systems[END_REF], where the semidefinite relaxation is doomed to fail when there exists no polynomial Lyapunov function, is that the additional perturbation parameter ε guarantees that we can always find (almost) optimal polynomials J satisfying the constraint ∇J •f +J ≤ ε by increasing ε slightly if necessary. To see this let (w, J, ε, v) be feasible for the LP [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF].

Then for any δ > 0 the quadrupel (w + δ, J + δ, ε + 3δ, v + δ) is strictly feasible, that is

(w + δ) + (J + δ) -(v + δ) ≥ 1 + δ, w + δ ≥ δ, J + δ > δ, ∇(J + δ) • f + (J + δ) + v + δ ≤ ε + 2δ < ε + 3δ, (21) β
(v + δ) -∇(v + δ) • f = βv -∇v • f + βδ ≥ βδ0.
and the cost only changes by 4δλ(X). By the Stone-Weierstraß theorem we can find polynomials (p w , p J , p v ) close enough to (w + δ /2, J + δ /2, v + δ /2) in the topology of C 1 such that (p w , p J , ε + 3δ, p v ) is still strictly feasible for the LP (11) (due to [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF]). An SOS representation of (p w , p J , p v ) follows then from Putinar's positivstellensatz [START_REF] Putinar | Positive polynomials on compact semi-algebraic sets[END_REF] which shows that (p w , p J , ε + 3δ, p v ) is feasible for the SPD [START_REF] Robinson | Infinite-Dimensional Dynamical Systems[END_REF] for large enough k with the same cost of (w, J, ε, v) plus the additional term 4δλ(X). Since δ > 0 was arbitrary it follows that d k → p * 5 = λ(A). An convergence estimate as in Theorem 2 and similarly an extension as in Corollary 1 can be obtained by the same arguments.

Corollary 2 In Theorem 3, for k ∈ N let (w k , J k , ε k , v k ) be optimal for (19) then we have

λ( J -1 k ([0, ε k ]) ∩ v -1 k ([0, ∞)) ∩ X \ A ≤ d k -λ(A) → 0 as k → ∞. Further if J -1 k ([0, ε k ]) ⊂ X then also λ(J -1 k ([0, ε k ]) \ A) ≤ d k \ λ(A).
Proof: This holds by the same arguments as in Theorem 2 and Corollary 1 because the SPD ( 19) is a tightening of the LP [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF].

The cost of having guaranteed bounds and finding the global optima using SOS methods comes at the price of expensive scaling, that is, high degree d or dimension n makes the SDP intracktable for current solvers/memory/computation power. Therefore, further structure should be exploited, such as sparsity [START_REF] Schlosser | Sparse moment-sum-of-squares relaxations for nonlinear dynamical systems with guaranteed convergence[END_REF], [START_REF] Tacchi | Approximating regions of attraction of a sparse polynomial differential system[END_REF], [START_REF] Wang | Exploiting term sparsity in moment-sos hierarchy for dynamical systems[END_REF] or symmetry [START_REF] Goluskin | Bounds on mean energy in the kuramotosivashinsky equation computed using semidefinite programming[END_REF].

Numerical examples

We illustrate our approach by three numerical examples that have been used in [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] and [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF], namely by the following globally asymptotically stable system with attractor A = {(0, 0)}, which does not allow for a polynomial Lyapunov function [START_REF] Amir | A globally asymptotically stable polynomial vector field with no polynomial lyapunov function[END_REF] ẋ = -2y -x 4 + 2x 2 y 2 + y 4 -2x(x 2 + y 2 )

x 4 + 2x 2 y 2 -y 2 ẏ = 2x x 4 + 2x 2 y 2 -y 4 -2y(x 2 + y 2 ) -x 4 + 2x 2 y 2 + y 4 , (22)
the Van-der-Pol oscillator ẋ = 2y, ẏ = -0.8x -10(x 2 -0.21)y [START_REF] Andrew | A smooth lyapunov function from a class-estimate involving two positive semidefinite functions[END_REF] and the Hénon map as an example of discrete time systems, given by

x m+1 = 2 3 (1 + y m ) -2.1x 2 m , y m+1 = 0.45x m . (24)
Van-der-Pol oscillator we observe as in [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] and [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] that the proposed method works very well and comparable with the method in [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF], see Figure 1. We notice that the approximation from [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] seems to perform slightly better than our approach in terms of volume discrepancy with the real attractor. For the Hénon map the situation is similar (Figure 2) compared to [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] -it takes higher degree polynomials to capture the complex topology of the Hénon attractor via (18) compared to [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF].

In the case of the system [START_REF] Tacchi | Approximating regions of attraction of a sparse polynomial differential system[END_REF] we notice some numerical instabilities in particular with the decision variable ε in [START_REF] Robinson | Infinite-Dimensional Dynamical Systems[END_REF], that penalizes the deviation from a Lyapunov function. We observed that the solutions obtained using Yalmip [START_REF] Lfberg | A toolbox for modeling and optimization in matlab[END_REF] and Mosek showed some robustness issues. The optimal value ε * obtained by the solver was too small in some cases, while the function J still showed the correct behaviour. The set on the right in Figure (3) gives an outer approximation consisting of a positively invariant set and is consistent with the set obtained in [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF]. The picture on the left illustrates the numerical problem concerning the decision variable ε in the SDP (19) -namely, that the optimal ε * found by the solver was too small (order 10 -7 ) and created the holes that can be observed in the picture on the left in Figure [START_REF] Nam | Dynamical systems: stability theory and applications[END_REF]. A correction, by increasing ε * by 8 times leads to the picture on the right in Figure [START_REF] Nam | Dynamical systems: stability theory and applications[END_REF], which shows that the function J found by the solver provides the correct geometry.

The appearance of numerical issues at the example of ( 22) can be explained by reminding that the polynomial J tries to mimic a Lyapunov functions for the global attractor. Since there exists no polynomial Lyapunov function for [START_REF] Tacchi | Approximating regions of attraction of a sparse polynomial differential system[END_REF], naturally, the problem gets numerically difficult. Remark 3 The discounting parameter β > 0 can be tuned and several solutions corresponding to different values of β can be intersected to improve the quality of the approximation [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF].

Conclusion

We presented a linear programming approach to outer approximations of global attractors via positively invariant sets. This builds on the recent works in [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] and [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF]. We combine both methods by keeping their fundamental advantages. That is: We use the approximation via positively invariant sets from [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] by using their method of perturbed Lyapunov equation and we overcome their difficulty in evaluating the cost function by maintaining the linear structure of the optimization problem from [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF].

This leads to an infinite dimensional linear programming problem characterizing (up to Lebesgue measure discrepancy zero) of global attractors via certain preimages. Applying sum-of-squares techniques as in the [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] and [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF] allows us to formulate a converging hierarchy of semidefinite programs. This gives rise to convergent outer approximations of the global attractor by positively invariant semialgebraic sets.

We illustrated the approach by numerical examples, including one of a vector field which does not allow for a polynomial Lyapunov function. For this case we observed numerical issues which can be explained by the antagonizing structure of non-existence of polynomial Lyapunov functions but the search for a polynomial "almost" Lyapunov function.

Therefore, with regard to applications we think that our approach should be understood as a practical extension of [START_REF] Jones | A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems[END_REF] and as a qualitative extension of the previous work in [START_REF] Schlosser | Converging outer approximations to global attractors using semidefinite programming[END_REF], where it seems that the global attractor is approximated with less Lebesgue measure discrepancy but not necessarily by positively invariant sets are produced.

Appart from approximating the global attractor this work can be extended to bounding extreme events on attractors based on the work [START_REF] Fantuzzi | Bounding extreme events in nonlinear dynamics using convex optimization[END_REF].

Future work could be devoted to use a similar technique to merge the works [START_REF] Jones | Converse lyapunov functions and converging inner approximations to maximal regions of attraction of nonlinear systems[END_REF] and [START_REF] Henrion | Convex computation of the region of attraction of polynomial control systems[END_REF] in a similar way. Another possible direction could be towards data based algorithms for attractors as in [START_REF] Korda | Computing controlled invariant sets from data using convex optimization[END_REF] or towards attractors of partial differential equations -where Lyapunov functions also provide a powerful tool -via for instance [START_REF] Korda | Moments and convex optimization for analysis and control of nonlinear partial differential equations[END_REF], [START_REF] Chernyavsky | Convex relaxations of integral variational problems: pointwise dual relaxation and sumof-squares optimization[END_REF].

Figure 1 :

 1 Figure 1: Outer approximations (black) of the attractor (red) for the Van-der-Pol oscillator for X = {x : 0.4 ≤ x 2 ≤ 2}. Left: approximation, degree 12 polynomials and β = 0.2. Right: approximation for polynomials up to degree 16 and β = 0.2.

Figure 2 :

 2 Figure 2: Outer approximations (black) of the attractor (red) for the Hénon map for X = [0, 1] 2 . Left: approximation, degree 6 polynomials and α = 0.002, γ = 0.05. Right: approximation for polynomials up to degree 14 and with the same α, γ.

Figure 3 :

 3 Figure 3: Outer approximations (black) of the attractor A = {(0, 0)} and trajectories starting from(1, 1), (1, -1), (-1, 1), (-1, -1) (red) for (22) for X = [-1, 1] 2 . Left: approximation by degree 16 polynomials and β = 0.2, the obtained ε * in (19) is too small and causes incorrect behaviour of the set J -1 ([0, ε * ]), see white "holes". Right: Outer approximation for polynomials up to degree 16 and β = 0.2 obtained by J -1 ([0, 8 • ε * ]).