Jean-Claude Bouret
email: jean-claude.bouret@lam.fr

Coralie Neiner

Ana I Gómez De Castro

Chris Evans

Boris Gaensicke

Steve Shore

Luca Fossati

Cécile Gry

Stéphane Charlot

Frédéric Marin

Pasquier Noterdaeme

Jean-Yves Chaufray

The science case for POLLUX, a high-resolution UV spectropolarimeter onboard LUVOIR

Keywords: LUVOIR, high resolution spectroscopy, ultraviolet, polarimetry, magnetic fields, POLLUX

POLLUX is a high-resolution, UV spectropolarimeter proposed for the 15-meter primary mirror option of LUVOIR 1 . The instrument Phase 0 study is supported by the French Space Agency (CNES) and performed by a consortium of European scientists. POLLUX has been designed to deliver high-resolution spectroscopy (R ≥ 120,000) over a broad spectral range (90-390 nm). Its unique spectropolarimetric capabilities will open-up a vast new parameter space, in particular in the unexplored UV domain and in a regime where high-resolution observations with current facilities in the visible domain are severely photon starved.

POLLUX will address a range of questions at the core of the LUVOIR Science portfolio. The combination of high resolution and broad coverage of the UV bandpass will resolve narrow UV emission and absorption lines originating in diffuse media, thus permitting the study of the baryon cycle over cosmic time: from galaxies forming stars out of interstellar gas and grains, and stars forming planets, to the various forms of feedback into the interstellar and intergalactic medium (ISM and IGM), and active galactic nuclei (AGN). UV circular and linear polarimetry will reveal the magnetic fields for a wide variety of objects for the first time, from AGN outflows to a diverse range of stars, stellar explosions (both supernovae and their remnants), the ISM and IGM. It will enable detection of polarized light reflected from exoplanets (or their circumplanetary material and moons), characterization of the magnetospheres of stars and planets (and their interactions), and measurements of the influence of magnetic fields at the (inter)galactic scale. In this paper, we outline the key science cases of POLLUX, together with its high-level technical requirements. The instrument design, its estimated performances, and the required technology development are presented in a separated paper 2 .

INTRODUCTION

The major challenge of contemporary astrophysics is to advance our understanding of the origin and evolution of galaxies, stars and planets that make up our Universe, and the life within it. The Large Ultraviolet/Optical/Infrared Surveyor (LUVOIR) proposed as one of the four "flagship" mission concept studies led by NASA for the 2020 Decadal Survey is designed to address this challenge and the related science cases. A versatile suite of instruments is envisioned to shed new light on the fascinating questions at the core of the LUVOIR vision. Under the leadership of LAM and LESIA (France), European institutes have come together to propose an instrument that would be onboard the 15-meter primary mirror option of LUVOIR. This instrument, POLLUX, is a high-resolution (R ≥ 120,000) spectropolarimeter, operating at UV wavelengths (90-400 nm). The Phase 0 study for POLLUX funded by CNES, started in January 2017.

The UV wavelength range is a treasure trove for astrophysics as it provides many of the most powerful probes of the hot and cold ISM and IGM gas, protostellar and protoplanetary disks, (exo)-planetary atmospheres, the peak emission from hot stellar atmospheres, star-forming galaxies, and AGN outflows. The major diagnostics for these objects are typically in the rest-frame UV between 100-200 nm and remain below the atmospheric cutoff (at 310 nm) for redshifts z ≲ 1-2. Futher, extreme highly-ionized species formed in hot, diffuse gas produce emission lines in the EUV range that can be redshifted into the near-UV domain. Only with a space-borne facility can we benefit from this unrivaled diagnostic power of the UV. POLLUX will observe over the spectral range from 90 to 390 nm, covering a wide range of species and temperatures that probe most astrophysical processes, for a large variety of objects and structures in the Universe.

POLLUX will deliver R ≥ 120,000 throughout its spectral range, enabling resolution of the numerous and narrow UV emission and absorption lines (with FHWMs of a few km s -1 ). This includes important molecules such as H2 and CO, which are intrinsically far stronger than the molecular emission bands accessible in the near-IR (e.g. with the James Webb Space Telescope). Access to these molecular bands is required to infer the density profile of planet-forming disks, their temperature, kinematics, and mass-transfer rates. Similarly, POLLUX will identify the pattern of these molecules (plus possibly H2O) in the upper atmosphere of (transiting) exoplanets, thereby revolutionizing our understanding of these objects. At such high spectral resolution, we can even imagine studies of winds on exoplanets if the atmospheric Doppler shift they induce differs from the shift due to the planet's orbital motion. Other topics for which high-resolution UV spectroscopy with POLLUX is essential include studies of the atmospheric dynamics of cool stars, their circumstellar medium and astrospheres within 100 pc around the Sun, as well as studies of the baryon cycle within galaxies, the ISM and the IGM, which requires resolving velocity structures at 2-5 km s -1 .

The most innovative characteristic of POLLUX is its unique spectropolarimetric capability. This will enable studies of a broad range of important astrophysical environments, spanning most of cosmic time, that are out of reach of current high-resolution spectroscopy. Spectropolarimetry enables detection and characterization of the magnetic fields (B-fields) and local environments of astrophysical objects. In addition, linear polarimetry will provide information on deviations from spherical symmetry, providing an extension of interferometry into a domain that is not restricted by the angular size of the objects but by their flux. This aspect of POLLUX will be a very powerful tool in studies of contemporary stellar physics, addressing topics in star formation, stellar structure and evolution, and the nature of circumstellar environments, including light echoes of supernovae, which might be discernible in high redshift systems by the polarization across specific features in the spectrum. By providing linear and circular polarization (giving access to the four Stokes parameters) it will also probe the physics of accretion disks in many astrophysical situations, from supermassive black holes in AGNs to white dwarfs, detecting the polarized light reflected from Earth-like exoplanets or from their circumplanetary material (e.g. rings) and moons, and constrain the properties (sphericity) of stellar ejecta and explosions. The magnetospheric properties of planets in the solar system will be accessible in exquisite detail, while the influence of magnetic fields at the Galactic scale and in the IGM will also be within reach.

The parameter space opened by POLLUX is uncharted territory, so its potential for ground-breaking discoveries is high. It will also neatly complement and enrich some of the cases advanced for LUMOS [START_REF] France | The LUVOIR Ultraviolet Multi-Object Spectrograph (LUMOS): instrument definition and design[END_REF] , the multi-object spectrograph for LUVOIR.

SCIENCE OVERVIEW 2.1 Stellar magnetic fields across the Hertzsprung-Russell diagram

From the ground in the visible domain, we can obtain sophisticated tomographic mapping of structures on the surfaces of stars and their magnetic fields. However, to understand the formation and evolution of stars and their accompanying planets it is necessary to also explore their circumstellar environments. POLLUX will provide a powerful, highresolution, full-Stokes (IQUV) spectropolarimetric capability to uniquely trace these structures. When combined with the immense light-gathering power of LUVOIR, it will deliver unprecedented views of the disks, winds, chromospheres and magnetospheres around a broad range of stars, as highlighted below (also see Figure 1).

Figure 1: Left: Sketch of a hot star with its fossil magnetic field lines, channeled polar wind, surface spots, equatorial magnetosphere, corotating interaction regions, and a stellar companion. Right: Sketch of a cool star, with its dynamo magnetic field, surface faculaes and plages, wind, a coronal mass ejection, and a bow shock between the star and its planet. (© S. Cnudde)

Magnetic fields play a significant role in stellar evolution, while also being a key factor in both planet and star formation. We need POLLUX spectropolarimetry to identify and characterize the B-fields across all stellar masses, to address central questions such as:

• How do B-fields develop in the pre-main-sequence (PMS) phase? POLLUX will characterize the strong Bfields arising at the sheared interface between star and disk in the PMS phase (via MgII, FeII, and CII UV lines from the interface region), and will trace the flows via polarization measurements of the nearby continuum.

• How does the stellar dynamo build-up and evolve? As stellar rotation decouples from the young planetary disk, the B-field is predicted to get stronger and more complex [START_REF] Emeriau-Viard | Origin and Evolution of Magnetic Field in PMS Stars: Influence of Rotation and Structural Changes[END_REF] . POLLUX will investigate propagation of magnetic energy through the stellar atmosphere into the uppermost coronal layers and stellar wind.

• How do stellar dynamos form and evolve in cool stars? What is their impact on planet formation and evolution? POLLUX will reveal the magnetically mediated interaction between young cool stars and their planetary disks during dynamo formation and stabilization when planets and planetary atmospheres form.

• A small but significant fraction of massive stars have strong B-fields (~10%) [START_REF] Fossati | B fields in OB stars (BOB): Low-resolution FORS2 spectropolarimetry of the first sample of 50 massive stars[END_REF] . However, very weak fields could be ubiquitous in massive stars. POLLUX will uniquely detect sub-Gauss B-fields in massive stars, providing a much-needed understanding of weak fields, field-decay mechanisms, and their impact on stellar evolution.

• POLLUX will enable studies of stellar B-fields beyond the Milky Way for the first time. These will open-up observations in the metal-poor Magellanic Clouds (MCs), probing the evolutionary processes linked to B-fields at metallicities equivalent to those at the peak of star-formation in the Universe.

POLLUX will also enable breakthroughs in other areas of contemporary stellar astrophysics, including:

• Jet formation: The mechanism driving the powerful jets from the interaction between disks around PMS stars and their magnetospheres is poorly constrained; POLLUX will transform our understanding of this dramatic part of star formation.

• Winds & Outflows: A large uncertainty in evolutionary models of massive stars is the link between rotation and mass lost via their winds, which limits our understanding of the progenitors of gamma-ray bursts, pair-instability SNe, and gravitational-wave sources. POLLUX observations in the Galaxy and MCs will map the density contrasts with stellar latitude required to calibrate these models.

• Novae & Supernovae (SNe): Both classical novae and SNe remnants provide excellent opportunities to learn about the production and lifecycle of dust. POLLUX spectropolarimetry is required to identify sites in the ejecta where dust condenses -allowing us to also infer the effects of kicks imparted by core-collapse SNe -and the shapes/sizes of the grains involved. Furthermore, polarization measurements of the unresolved ejecta during the earliest stages of the expansion will reveal its structural properties, especially sphericity, for novae and supernovae.

• White Dwarfs (WDs): A minority of WDs is strongly (>1 MG) magnetic [START_REF] France | The LUVOIR Ultraviolet Multi-Object Spectrograph (LUMOS): instrument definition and design[END_REF] , but little is known of the incidence of weaker fields. POLLUX UV spectropolarimetry will explore this regime for the first time to investigate the ubiquity of B-fields in WDs, as well as the processes generating their fields (e.g., amplification of fossil fields and/or binary mergers?)

A 100-hour POLLUX observing program would be transformational in our understanding of B-fields in stellar evolution.

For instance, we could include UV monitoring of two low-mass sources (PMS T Tauri, evolved T Tauri) to directly characterize their B-fields and magnetic-energy transport, combined with observations of a first sample of tens of Galactic massive stars to assess the presence/strength of weak B-fields.

What are the characteristics of exoplanets atmospheres and how do planets interact with the host stars?

The characterization of exoplanets is key to our understanding of planets, including those in the Solar System. POLLUX unique, simultaneous, high-resolution and polarimetric capabilities in the UV are essential to unveiling the origins of the huge range of chemical and physical properties found in exoplanetary atmospheres [START_REF] Sing | A continuum from clear to cloudy hot-Jupiter exoplanets without primordial water depletion[END_REF] and to understand the interaction between planets and their host stars [START_REF] Cuntz | On Stellar Activity Enhancement Due to Interactions with Extrasolar Giant Planets[END_REF] .

The line and continuum polarization state of starlight that is reflected by a planet is sensitive to the optical properties of the planetary atmosphere and surface and depends on the star-planet-observer phase angle (Figure 2). Atmospheric gases are efficient scatterers at UV wavelengths and deviations from the expected polarization across the UV would reveal the presence and microphysical properties of aerosol and/ or cloud particles. This information is crucial for getting insight into a planet's climate. It would also strongly complement transit observations. POLLUX can significantly detect polarization signatures for close-in gas giants (and brown dwarfs) orbiting stars out to distances of 70 pc, which currently comprises over a dozen targets, two of them transiting.

Star-planet interactions (SPI) could generate detectable signatures in exoplanetary systems. Searching for SPI has developed into a very active field [START_REF] Shkolnik | Evidence for Planet-induced Chromospheric Activity on HD 179949[END_REF][START_REF] Kashyap | Extrasolar Giant Planets and X-Ray Activity[END_REF][START_REF] Miller | A Comprehensive Statistical Assessment of Star-Planet Interaction[END_REF] . Evidence of SPI has been observed in UV stellar emission lines (e.g., NV [START_REF] France | The MUSCLES Treasury Survey. I. Motivation and Overview[END_REF] ). POLLUX will be capable of studying the intensity of UV stellar emission lines forming in a wide range of temperatures as a function of planetary orbital phase, uniquely combining this information with measurements of the magnetic field derived from the Stokes V profiles of these lines. This will allow for the first time to identify the stellar regions mostly affected by SPI and hence to understand their origin.

At UV wavelengths, a hot Jupiter orbiting a Sun-like star with a two-day orbital period presents a maximum polarization of ≈30% (Figure 3), leading to a maximum polarization signal of the order of 7 x 10 -5 , when observed unresolved from its star. With POLLUX mounted on a 15-m LUVOIR telescope and a binning size of 50 Å, the signal-to-noise ratio (S/N) necessary to detect such a signal can be obtained in less than 3 hours of shutter time for systems as far as 70 pc (Figure 3). Measuring and modeling the variation of the polarization signal as a function of the planet's orbital phase requires at least eight measurements spread across the orbit. A POLLUX 100-hour observing program would allow the detection and characterization of the UV polarimetric signature for over a dozen targets.

The various phases of the ISM and extragalactic IGM

Matter in the interstellar space is distributed in diverse, but well-defined phases that consist of (1) the hot, ionized (T~10 6-7 K) ISM that emits soft X-rays, (2) the warm neutral or ionized medium (6000 < T < 10 4 K), and (3) the cold (T~10-200 K) neutral medium and molecular star-forming clouds. Boundaries between the different phases play a fundamental role in the cooling of the gas and by consequence in Galactic evolution. However, it is not yet clear how these different phases trade matter and entropy. We need to address the multi-phase aspect of the ISM and the influence of different factors such as magnetic field, thermal/pressure (non-)equilibrium, metallicity, shocks, etc.

Gaining fundamental insights into the processes that influence the different phases requires the wealth of interstellar absorption features appearing in the UV spectra of hot stars and requires observing them at R ≥ 120,000 to measure individual clouds and disentangle the different phases contributions. POLLUX will, for the first time, give access to all phases at the required high spectral resolution. It will enable simultaneous study of tracers of hot-gas (OVI, CIV), warm gas (OI, NI, and many singly-ionized species), as well as cold-HI gas like CI, and of the molecular phases through H2 and CO lines. POLLUX will resolve the velocity profiles of the different H2 rotational levels, yielding temperature and turbulence, and information on the formation of H2 and its role as a coolant, e.g., in turbulence dissipation.

ISM studies also need to be generalized to external galaxies. POLLUX will enable tomography of chemical abundances in several hundred nearby galaxies using neutral gas absorption lines toward individual stars across the entire (stellar) body of galaxies or toward background quasars. Abundance variations across and between galaxies of different types and metallicities will document the dispersal/mixing spatial/time scales of newly produced elements and their nucleosynthetic origin, the role of infalling and outflowing gas in the metallicity build-up, and the dust production/composition through depletion patterns. Furthermore, cooling rates, physical conditions, and the molecular gas content can be determined spatially in the HI reservoir (with implications for the regulation of star formation and for galaxy evolution at large) and in neutral shells of HII regions (with implications for the star-formation process itself). A huge step forward in the knowledge of the Galactic magnetic field in terms of sensitivity, sky coverage and statistics has been made from recent dust polarization measurements. However, they tell us nothing on the distance of the magnetic field and its distribution in the different components or phases. The POLLUX spectropolarimeter will provide this information by its ability to detect linear polarization in UV absorption lines through Ground State Alignment [START_REF] Yan | Tracing magnetic fields with ground state alignment[END_REF] . It will open up access to the magnetic field 3-D, by measuring its orientation in different clouds, on limited distances along sight lines and on small scales (Figure 4). In a 100-hour program, observing several hundreds of hot stars with a S/N of 500 to measure linear polarization in optically thin absorption lines at a level of a few % will help understand the interplay of magnetic field and the different ISM phases. By discerning magnetic fields in gas with different dynamical properties, POLLUX will allow the first study of interstellar magnetic turbulence [START_REF] Zhang | SDSS-IV MaNGA: the impact of diffuse ionized gas on emission-line ratios, interpretation of diagnostic diagrams and gas metallicity measurements[END_REF] .

Dust composition and magnetic field strength of unresolvable regions of AGNs

POLLUX will offer unique insight into the still largely unknown physics of Active Galactic Nuclei (AGNs), which are believed to arise from accretion of matter by supermassive black holes in the central regions of galaxies. Some key signatures of accretion disks can be revealed only in polarized light, and with higher contrast at ultraviolet than at longer wavelengths. Specifically, high-resolution UV polarimetry will provide geometric, chemical and thermodynamic measurements of accretion disks in unprecedented detail. By probing the ubiquitous magnetic fields expected to align small, non-spherical dust grains on scales from the accretion disk out to the extended dust torus, POLLUX will be able to reveal the mechanisms structuring the multi-scale AGN medium.

The key information encoded into the polarized light will allow determinations of the mineralogy, structure and alignment of the smallest dust grains, together with on-line-of-sight magnetic-field strengths. Measurements of magnetic field strengths will also make constraints on the structure of magneto-hydrodynamic winds in nearby, broad absorptionlines systems accessible to the instrument. On larger scales, UV polarimetric studies of young star-forming regions will provide unprecedented insight into the enigmatic relation between onset of star formation and triggering of nuclear activity. Once activated, an AGN is expected to feed vast amounts of energy back into the interstellar medium of its host galaxy through radiation and shocks. This feedback can decrease or increase the star formation activity of the host, the physical conditions of which observational constraints from UV polarization with POLLUX will greatly help understand.

In addition to the extended environment of AGNs, POLLUX will provide insight into the nature and lifetime of particles in relativistic jets, which are other key factors to fully understand AGN feedback on star formation in galaxies. By focusing on bright, low-redshift galaxies, it will be possible to obtain, for the first time, high-resolution spectra providing striking details on the structure and physics of AGNs. Complemented with the optical and infrared instruments on board LUVOIR, POLLUX will constitute a groundbreaking means of assessing the important role played by AGNs on galaxy evolution (Figure 5). which an effective area similar to that of HST/COS was adopted). POLLUX will enable observations of hundreds of AGN over a wide UV spectral domain, and with much larger spectral resolution than ever achieved.

Testing fundamental physics and cosmology using absorption lines towards quasars

Physics and cosmology as we know today have been remarkably successful in reproducing most of the available observations with only a small number of parameters. However, it also requires that 96% of mass-energy content of the Universe is in mysterious forms (dark energy and dark matter) that has never been seen in the laboratory. This shows that our canonical theories of gravitation and particle physics may be incomplete, if not incorrect. Improving the sensitivity of current observational constraints is therefore of utmost importance, irrespective of whether it is consistent with the current standard physics-in which case it will reject other scenarios-or whether it will instead favor new physics.

Absorption-line systems produced by intervening gas in the spectra of background sources provide original sensitive probes of fundamental physics and cosmology. The high UV spectral resolution of POLLUX on LUVOIR will open a unique window on such probes, in particular (1) the measurement of the primordial abundance of deuterium, (2) the stability of fundamental constants over time and space and (3) the redshift evolution of the cosmic microwave background (CMB) temperature. The D/H ratio can be estimated from the DI and HI Lyman series lines. Any change in the proton-to-electron mass ratio (m=mp/me) translates into a relative wavelength change of the H2 Lyman and Werner lines. Finally, the CMB radiation excites the CO molecules so that the relative population in different rotational levels (measured through the electronic bands in the UV) is an excellent thermometer for the CMB temperature.

We remark that, while these are independent probes, they are intimately related by the underneath physics. For example, models involving varying scalar-photon couplings 14 also affect the temperature-redshift relation so that constraining this relation is complementary to a search for varying fundamental constants. The Big Bang nucleosynthesis calculations of the D/H ratio are also dependent on the fundamental constants and can be altered if new physics is at play [START_REF] Olive | Higher D or Li: probes of physics beyond the standard models[END_REF] .

The baseline specifications of LUVOIR/POLLUX are S/N~100 per resolution element in 1h for F𝜆=10 -14 erg s -1 cm -2 Å -1 . This means that for known quasars with low-z H2 absorbers as observed by HST/ COS [START_REF] Avgoustidis | Cosmological effects of scalar-photon couplings: dark energy and varying-α Models[END_REF] , we can reach a precision of a ∆μ/μ ~ a few 10 -7 in 20-30h (Figure 6). This is about an order of magnitude better than the current best limits in less observing time than typical used to achieve those (around 5 x 10 -6 with UVES on the Very Large Telescope). Similarly, the achieved precision on TCMB scales directly with the S/N ratio (0.1 K at S/N~100 and R~100,000 when current limits are DT~1K). The same is again true with the D/H ratio, although a very high resolution is less critical since the corresponding lines are typically thermally broadened above 10 km s -1 .

Figure 6: Expected error on Δµ/µ as a function of signal-to noise ratio (red band, abscissa scale) and minimum wavelength (dashed line, absorber's rest-frame, top axis) covered by a R=120,000 spectrum.

Figure 7: Simulated CO electronic absorption profile at R=120,000 and different S/N ratios. The error on the measured excitation temperature is indicated below each panel.

Solar system: surfaces, dust scattering and auroral emissions

UV observations uniquely probe the surface of telluric bodies of the solar system. They diagnose their volcanic and plume activity, their interaction with the solar wind and their composition in the frame of space weather and exobiology/habitability fields. Very few UV polarimetric observations were obtained so far with WUPPE [START_REF] Fox | Solar System Observations by the Wisconsin Ultraviolet Photopolarimeter Experiment.II.The First Linear Ultraviolet Spectropolarimetry of Io[END_REF] , revealing in particular the Io surface as spatially covered by 25% SO2 frost with polarization variations associated to different volcanic regions. POLLUX will primarily characterize volcanism and/or plume activity of icy moons from polarized solar continuum reflected light and spectral UV albedo. Its high sensitivity is necessary to track any organic and ice composition of the crust of comets and Kuiper Belt objects from their UV spectrum.

The giant planets' UV aurorae are mainly radiated from H and H2 atmospheric species, collisionally excited by accelerated charged particles precipitating along the auroral magnetic field lines. Aurorae thus directly probe complex interactions between the ionosphere, the magnetosphere, the moons and the solar wind. Precipitation of auroral particles is additionally a major source of atmospheric heating, whose knowledge is needed to assess the energy budget, the dynamics and the chemical balance of the atmosphere. The narrow Field-of-View of POLLUX will measure the bright complex aurorae of Jupiter and Saturn, the fainter ones of Uranus and catch those of Neptune, only seen by Voyager 2 [START_REF] Lamy | Earth-based detection of Uranus' aurorae[END_REF] . The high spectral resolution will be used to finely map the energy of precipitating electrons from partial spectral absorption of H2 by hydrocarbons [START_REF] Ménager | H Lyman α line in Jovian aurorae: electron transport and radiative transfer coupled modelling[END_REF][START_REF] Gustin | Statistical study of Saturn's auroral electron properties with Cassini/UVIS FUV spectral images[END_REF] and the thermospheric wind shear from the H Ly-α line (Figure 8, left panels [START_REF] Chaufray | Asymmetry in the Jovian auroral Lyman-α line profile due to thermospheric high-speed flow[END_REF] ).

The asymmetric profile of the H Ly-α line in the Jovian auroral region (Figure 8, left panels) should be resolved by POLLUX: for an exposure time of 10 minutes, and considering the brightness of the wings of the Ly-α line measured by HST/STIS, the two wings should produce 68±8 cts/px and 34±6 cts/px respectively, implying a velocity larger than 4 km/s which can be measured precisely by POLLUX.

The albedo of a region of 100 km 2 can be derived with an accuracy of 0.1% with a spectral resolution of 1 nm with a 5min exposure only near 320 nm. The linear polarization of Io surface varies between 1 and -10% between 220 and 320 nm; near 320 nm, the linear polarized signal observed by POLLUX for a binning size of 1 nm in 5 minutes should have a S/N~10, which is better than WUPPE (Figure 8, right panels).

Figure 2 .

 2 Figure 2. (1) & (2): Schematic of a planetary system in which the unpolarized stellar light becomes polarized through reflection by a planetary atmosphere. Panel 2 illustrates how the unpolarized stellar light coming from the top-left corner becomes linearly polarized following reflection by the planetary surface/atmosphere. Left (3): degree of polarization (in %) as a function of planetary orbital phase, labelled as in panel (1), at 300 nm (from Loic Rossi). The different lines indicate different atmospheric cloud coverages, where zero corresponds to the cloudless condition. POLLUX will measure the degree of polarization as a function of wavelength and planetary orbital phase.

Figure 3 :

 3 Figure 3: Average S/N in the near-UV obtained with 3 hours of POLLUX shutter time as a function of the distance of the target Sun-like star (line colour) and size of the telescope primary mirror. The red dashed horizontal line indicates the maximum near-UV polarization signal of a hot-Jupiter with a 2 days orbital period. The black dotted vertical line indicates the size of LUVOIR's primary mirror.

Figure 4 :

 4 Figure 4: Synthetic polarization (%) map of the simulated diffuse ISM (1pc 2 ). Contour color: percentage of polarization in the SII 1250 Å absorption line. Orientation of the bars: polarization direction. Mean magnetic field =3 μG, oriented along the X-axis. This figure shows that the direction of polarization correctly indicates the magnetic field direction and that the expected polarization is mostly above 5%, within POLLUX capabilities.

Figure 5 :

 5 Figure 5: Effective area of POLLUX (in black; assuming 135 m² for LUVOIR with a RC telescope), compared to those of previous space-based UV spectrographs, as indicated. Only 4 AGN were observed with WUPPE. A total of 117 were observed with different polarimetric instruments on board HST, out of which only 35 in the UV with HST/FOS (for which an effective area similar to that of HST/COS was adopted). POLLUX will enable observations of hundreds of AGN over a wide UV spectral domain, and with much larger spectral resolution than ever achieved.

Figure 8 .

 8 Figure 8. Left: auroral emissions observed on Jupiter by HST. The Lya emission spectrum in the polar auroral region was obtained with STIS. Its strong asymmetry can be reproduced with a wind shear reaching 4-8 km/s 16 . Right: Io observed by Galileo (©Galileo Project, JPL, NASA). The linear polarisation of the surface measured by WUPPE between 220-320 nm presents complex variations with wavelength and provides information on the volcanic activity of the surface of Io 16 .

ACKNOWLEDGEMENTS

The authors acknowledge contributions from the entire LUVOIR and POLLUX consortia. We would like to especially thank Kevin France (Univ. of Colorado -Boulder), Matthew Bolcar (NASA GSFC) and Aki Roberge (NASA GSFC). We also want to acknowledge the crucial contribution of Eduard Muslimov (LAM), Arturo Lopez Ariste (IRAP) and Maelle Le Gal to the optical design of POLLUX and its polarimeters.

HIGH-LEVEL REQUIREMENTS

The science goals described in the previous section lead to the technical requirements for POLLUX presented in the Table 1 below. Based on these requirements, a design concept for POLLUX instrument has been proposed [START_REF] Muslimov | POLLUX: a UV spectropolarimeter for the future LUVOIR space telescope[END_REF] . In short:

• The spectral range will be split between 3 channelsfar-UV (90-124.5 nm), medium-UV (118.5-195 nm) and near-UV (195-390 nm), essentially because it allows to use dedicated/optimised optical elements, coatings and detectors and polarimeter for each band, hence obtain a gain in efficiency. It also allows us to maintain a reasonable size for each channel and respective components, while maintaining the required resolution. • Each channel is equipped with its own dedicated polarimeter followed by a high-resolution spectrograph.

• MUV+NUV channels are recorded simultaneously, while the FUV is recorded separately (temporal separation).

• The FUV and MUV boundaries are set relative to the Lyman-α line, such that this line is always present on both channels. The MUV and NUV boundary is set such that a maximum of one full octave falls in each channel (here the NUV) but this may be changed in a later design. • The instrument will allow either spectropolarimetric, or pure spectroscopic modes.

• POLLUX can be fed by the light coming from the telescope or from sources in the calibration unit.

SUMMARY

In this paper, we presented an overview of the wide variety of science that could be addressed with POLLUX onboard LUVOIR. This instrument has unique capabilities, as it will provide high-resolution (R ≥ 120,000) spectropolarimetry across the UV bandpass (90-390nm). Such capabilities are essential for determining the physical, chemical and/or magnetic properties of astrophysical objects. They will help bring decisive clues to questions at the core of the program for the LUVOIR observatory, concerning the origin and evolution of the Universe and the life within it.


Edouard Leurent
email: edouard.leurent@inria.fr

Yann Blanco
email: yann.blanco@renault.com

Denis Efimov
email: denis.efimov@inria.fr

Odalric-Ambrym Maillard
email: odalric.maillard@inria.fr

Approximate Robust Control of Uncertain Dynamical Systems

This work studies the design of safe control policies for large-scale non-linear systems operating in uncertain environments. In such a case, the robust control framework is a principled approach to safety that aims to maximize the worst-case performance of a system. However, the resulting optimization problem is generally intractable for non-linear systems with continuous states. To overcome this issue, we introduce two tractable methods that are based either on sampling or on a conservative approximation of the robust objective. The proposed approaches are applied to the problem of autonomous driving.

Introduction

Reinforcement Learning is a general framework that allows the optimal control of a Markov Decision Process (S, A, T, r) with state space S, action space A, reward function r ∈ [0, 1] S×A and unknown transition dynamics T (s |s, a) ∈ M(S) S×A by searching for the policy π ∈ M(A) S with maximal expected value v T π of the total discounted reward R T π :

R T π (s) def = ∞ t=0 γ t r(s t , a t ), v T π (s) def = E(R T π (s)), (1)
where s 0 = s , a t ∼ π(s t ), s t+1 ∼ T (s t+1 |s t , a t ), γ ∈ [0, 1) is the discount factor and M(X) denotes the set of probability measures over X.

Unfortunately, its application to real-world tasks has so far been limited by its considerable need for experiences. It is generally recognized [START_REF] Sutton | Integrated Architectures for Learning, Planning, and Reacting Based on Approximating Dynamic Programming[END_REF][START_REF] Atkeson | A comparison of direct and model-based reinforcement learning[END_REF]) that the most sample-efficient approach is the family of model-based methods which learn a nominal model T of the environment dynamics that is leveraged for policy search:

max π v T π (2)
One drawback of such methods is that they suffer from model bias; that is, they ignore the error between the learned dynamics T and the real environment T . It has been shown that model bias can dramatically degrade the policy performances [START_REF] Schneider | Exploiting model uncertainty estimates for safe dynamic control learning[END_REF].

Model errors can instead be explicitly considered and expressed through an ambiguity set of all possible dynamics models. Such a set can be constructed from a history of observations by computing the confidence regions associated with the system identification process [START_REF] Garud | Robust Dynamic Programming[END_REF][START_REF] Nilim | Robust Control of Markov Decision Processes with Uncertain Transition Matrices[END_REF][START_REF] Dean | On the Sample Complexity of the Linear Quadratic Regulator[END_REF][START_REF] Maillard | Self-normalization techniques for streaming confident regression[END_REF]. In this work, we will consider ambiguity sets of parametrized deterministic dynamical systems s = T θ (s, a) whose unknown parameters θ lie in a compact set Θ of R p .

In the optimal control framework, model uncertainty is handled by maximizing the expected performances with respect to unknown dynamics. In stark contrast, in real-world applications where failures may turn out very costly, the decision maker often prefers to minimize the risk of the policy, which can be defined with several metrics characterizing the distribution of the policy outcome [START_REF] García | A Comprehensive Survey on Safe Reinforcement Learning[END_REF].

The robust control framework is a popular setting in which the risk of a policy is defined as the worst possible outcome realization among the ambiguity set, to guarantee a lower-bound performance of the robust policy when executed on the true model:

max π min T v T π (3)
Robust optimization has been studied in the context of finite Markov Decision Processes (MDP) with uncertain parameters by Iyengar ( 2005), [START_REF] Nilim | Robust Control of Markov Decision Processes with Uncertain Transition Matrices[END_REF] and [START_REF] Wiesemann | Robust Markov Decision Processes[END_REF].

They show that the main results of Dynamic Programming can be extended to their robust counterparts only when the dynamics ambiguity set verifies certain rectangularity properties. In the control theory community, the robust control problem is mainly restricted to the context of linear dynamical systems with bounded uncertainty in the time or frequency domain, where the objective is to guarantee stability (e.g. H ∞ -optimal control, see [START_REF] Basar | H infinity -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach[END_REF] or performance (e.g. LQ optimal control theory, see [START_REF] Petersen | Robust control of uncertain systems: Classical results and recent developments[END_REF]. The existing nonlinear robust control approaches such as sliding mode control [START_REF] Li | Advances in Variable Structure Systems and Sliding Mode Control -Theory and Applications[END_REF], feedback linearization, backstepping, passivation and input-to-states stabilization [START_REF] Hassan | Nonlinear Control[END_REF] are usually based on canonical representations of regulated dynamics and admit constructive numeric realizations for systems of rather low dimensions.

There have been few attempts of robust control of large-scale systems with both continuous states and non-linear dynamics, which is the focus of this paper. Our contribution is twofold. In section 2, we first consider a simpler case where the ambiguity set Θ and action space A are both finite and introduce a sampling-based planner that approximately maximizes the robust objective (3). In section 3, we move to continuous ambiguity sets and form a conservative relaxation of the robust policy evaluation problem using interval predictors. In section 4, we illustrate the benefits of both techniques (for discrete, versus continuous Θ) on a problem of tactical decision-making for autonomous driving.

Sampling-based planning

If the true dynamics model T θ were known and the action-space A finite, sampling-based algorithms could be used to perform approximate optimal planning. In order to generalize to the robust setting, we need to make the following assumption about the structure of the ambiguity set:

Assumption 1 (Structure). The ambiguity set Θ and the action space A are discrete and finite:

A = {a k } k∈[1,K] and Θ = {θ m } m∈[1,M ] (4)
We slightly abuse notation and denote T m = T θm . Such a structure of the ambiguity set typically stems directly from expert knowledge of the problem at hand. In general, it is nonrectangular, which implies that the Robust Bellman Equation does not hold [START_REF] Wiesemann | Robust Markov Decision Processes[END_REF]. This prevents us from building on planners that implicitly use this property and generate trajectories step-by-step by picking promising successor states, such as MCTS [START_REF] Coulom | Efficient Selectivity and Backup Operators in Monte-Carlo Tree Search[END_REF] or UCT [START_REF] Kocsis | Bandit Based Monte-Carlo Planning[END_REF]. Instead, we turn to algorithms that perform optimistic sampling of entire sequences of actions and work directly at the leaves of the expanded tree (see, e.g. [START_REF] Bubeck | Open Loop Optimistic Planning[END_REF]. More precisely, we build on the work of [START_REF] Hren | Optimistic planning of deterministic systems[END_REF] on optimistic planning for deterministic dynamics, which we extend to the robust setting. We use similar notations and consider the infinite look-ahead tree T composed of all reachable states. Each node corresponds to a joint state {s m,t } m∈[1,M ] associated with the different dynamics T m .

The root starts at the current state, and all nodes have K children, each corresponding to an action a k ∈ A and associated with the successor joint state {s m,t+1 = T m (s m,t , a k )} m∈[1,M ] . We use the standard notations over alphabets to refer to nodes in T as action sequences. Thus, a finite word i ∈ A * of length d represents the node obtained following the action sequence (i 0 , • • • , i d ) from the root. Sequences i ∈ A * and j ∈ A * can be concatenated as ij ∈ A * , the set of suffixes of i is iA ∞ = {j ∈ A ∞ : ∃h ∈ A ∞ such that j = ih}, and the empty sequence is ∅.

The sample complexity is expressed in terms of number n of expanded nodes. It is related to the number of calls to dynamics models: when a node i is expanded, all successor states are computed for all K actions and M dynamics. At an iteration n, we denote T n the tree of already expanded nodes, and L n the set of its leaves. Figure 2: A few trajectories are sampled from an initial state s 0 following a policy π with various dynamics parameters θ m (in black). The union of reachability sets is shown in green, and its interval hull in red.

Definition Fix a dynamics model m ∈ [1, M ]. [START_REF] Hren | Optimistic planning of deterministic systems[END_REF] define for any node i ∈ T of depth d the optimal value v m i , its lower bound u-value u m i and upper-bound b-value b m i . These variables depend on the dynamics m and will therefore be referred to with a superscript m notation.

We extend these dynamics-dependent variables to the robust setting, using superscript r in notations.

• The robust value v r i of a path i ∈ A * as the restriction of (3) to policies that start with the action sequence i:

v r i def = max π∈iA ∞ min m∈[1,M ] R Tm π
(5) By definition, the robust value of ( 3) is recovered at the root v r ∅ = v r . Moreover, for i ∈ T n \ L n we have

v r i = max π∈iA ∞ min m∈[1,M ] R Tm π = max a∈A max π∈iaA ∞ min m∈[1,M ] R Tm π = max a∈A v r ia (6)
• The robust u-value u r i of a leaf node i ∈ L n is the worst-case discounted sum of rewards r t = r(s m,t , i t ) from the root to i. It is then backed-up to the rest of the tree:

u r i (n) def = min m∈[1,M ] d-1 t=0 γ t r t if i ∈ L n ; max a∈A u r ia (n) if i ∈ T n \ L n (7)
• Likewise, the robust b-value b r i is defined at leaf nodes and backed-up to the rest of the tree:

b r i (n) def = u r i (n) + γ d 1-γ if i ∈ L n ; max a∈A b r ia (n) if i ∈ T n \ L n (8)
An illustration of the computation of the robust b-values is presented in Figure 1.

Remark 1 (On the ordering of min and max). In the definition of u r i (n) it is essential that the minimum among the models is only taken at the end of trajectories, in the same way as for the robust objective (3) in which the worst-case dynamics is only determined after the policy has been fully specified. Assume that u r i (n) is instead naively defined as:

u r i (n) = min m∈[1,M ]
u m i (n), This would not recover the robust policy, as we show in Figure 3 with a simple counter-example. From these definitions we introduce Algorithm 1, and analyse its sample-efficiency in Theorem 1. Lemma 1 (Robust values ordering). The robust values, u-values and b-values exhibit similar properties as the optimal values, u-values and b-values, that is: for all 0 < t < n and i ∈ T n ,

u r i (t) ≤ u r i (n) ≤ v r i ≤ b r i (n) ≤ b r i (t) (9)
./min-max-order-eps-converted-to.pdf

r i (n). Expand argmax i∈Ln b r i (n). n = n + 1 return argmax a∈A u r a (n)
Proof. This result stems directly from the definitions, see more details in Appendix A.1.

The simple regret of the action a returned by Algorithm 1 after n rounds is defined as:

R n = v r -v r a (10)
We will say that R n = O(ε) for some ε > 0 if there exist ρ > 0 and n 0 > 0 such that R n ≤ ρε for all n ≥ n 0 . A node i ∈ T is said to be -optimal, in a robust sense, if and only if v r i ≥ v rfor some > 0. The proportion of -optimal nodes at depth d is then defined as p d ( ) = |i ∈ A d s.t i is -optimal|/K d . Further we will assume that for the graph T the following hypothesis is satisfied:

Assumption 2 (Proportion of near-optimal nodes). There exist β ∈ [0, log K log 1/γ ], c > 0 and d 0 > 0 such that p d ( ) ≤ c β for all > 0 and d ≥ d 0 .

Theorem 1 (Regret bound). Let κ = Kγ β ∈ [1, K].
Then the simple regret of Algorithm 1 is:

If κ > 1, R n = O n -log 1/γ log κ (11) If κ = 1, R n = O γ (1-γ) β c n (12)
Proof. We use the properties shown in Lemma 1 and derive a robust counterpart of the proof of [START_REF] Hren | Optimistic planning of deterministic systems[END_REF], which we only modify slightly. See more details in Appendix A.2

Interval predictors

In this section, we assume that the ambiguity set Θ is continuous and bounded.

In the robust objective (3), the min operator only requires us to describe the set of states that can be reached with non-zero probability.

Definition The reachability set S at time t is the set of all states that can be reached by starting from initial state s 0 ∈ S and following a policy π ∈ A S along the transition dynamics T θ ∈ S S×A . S(t, s 0 , π)

def = {s t : ∃θ ∈ Θ s.t. s k+1 = T θ (s k , a k ), a k = π(s k ), k = 0, • • • , t -1} (13)
This set can still have a complex shape. We approximate it by an overset easier to represent and manipulate: its interval hull.

Definition

The interval hull of S, denoted S = [s, s] is the smallest interval containing it: s(t, s 0 , π) def = min S(t, s 0 , π) s(t, s 0 , π) def = max S(t, s 0 , π) (14) The max and min operators are applied element-wise. This set is illustrated in Figure 2.

State intervals S have been used to describe the evolution of uncertain systems and derive feedback laws that achieve closed-loop stability in the presence of bounded disturbances [START_REF] Stinga | Robust interval observer and nonlinear predictive control of an active sludge process[END_REF][START_REF] Efimov | Design of interval observers for uncertain dynamical systems[END_REF][START_REF] Thach | Decentralization of Interval Observers for Robust Controlling and Monitoring a Class of Nonlinear Systems[END_REF]. The main techniques of interval simulation have been listed and described in a survey by [START_REF] Puig | Simulation of Uncertain Dynamic Systems Described By Interval Models: a Survey[END_REF], in which they are sorted into two categories. Region-based methods use the estimate of S at previous timestep t -1 to bootstrap the current estimate at time t. They are based on application of the theory of positive systems, which are frequently computationally efficient. However, the positive inclusion dynamics of a system may lead to overestimations of the true S and even unstable behaviour. Trajectory-based methods estimate S by taking the max and min in ( 14) over sampled trajectories for θ ∈ Θ. These methods produce subset estimates of the true S, do not suffer from the wrapping effect, but are often more computationally costly. In this work, we leverage them to derive a proxy for the robust objective (3).

Definition Let us denote the robust objective of equation (3) as v r (π) def = min θ∈Θ v T θ π . We define the surrogate objective vr on a finite horizon H > 0 as: (2011). In this case, derivative-free methods such as evolutionary strategies (e.g. CMAES) would be more appropriate than policy gradient methods, since vr cannot be easily differentiated. Planning algorithms can also be used to exploit the dynamics and structure of the surrogate objective.

vr (π)

Experiments

Most autonomous driving architectures perform sequentially the prediction of other drivers' trajectories and the planning of a collision-free path for the ego-vehicle. As a consequence, they fail to account for interactions between the traffic participants and the ego-vehicle, leading to overly conservative decisions and a lack of negotiation abilities [START_REF] Trautman | Unfreezing the robot: Navigation in dense, interacting crowds[END_REF]. In this work, we perform both tasks jointly to anticipate the effect of our own decisions on the dynamics of the nearby traffic. But human decisions are not fully predictable and cannot be reduced to a single deterministic model. To avoid model bias, we provide a whole ambiguity set of reasonable closed-loop behavioural models for other vehicles, and plan robustly with respect to this ambiguity. We introduce a new environment for simulated highway driving and tactical decision-making.1 Vehicle motion is described by the Kinematic Bicycle Model (see, e.g. [START_REF] Polack | The Kinematic Bicycle Model : a Consistent Model for Planning Feasible Trajectories for Autonomous Vehicles[END_REF]. They follow a lane keeping lateral behaviour, and a longitudinal behaviour inspired by the Intelligent Driver Model [START_REF] Treiber | Congested traffic states in empirical observations and microscopic simulations[END_REF] which balances reaching a desired velocity and respecting a safe time [START_REF] Kesting | General Lane-Changing Model MOBIL for Car-Following Models[END_REF]: they must increase the vehicles accelerations while satisfying safe braking decelerations. The behaviour parameters θ of each traffic participant are sampled uniformly from a set Θ.

The ego-vehicle can be controlled with a finite set of tactical decisions A = {no-op, right-lane, left-lane, faster, slower} implemented by low-lever controllers. It is rewarded for driving fast along a planned route while avoiding collisions. More information on the environment modelling is provided in the appendices. We carry out two experiments2 : First, the behavioural parameters of traffic participants are fixed but their planned routes are unknown: we enumerate every direction they can take at their next intersection (see Figure 4a) and plan robustly with respect to this finite ambiguity set using Algorithm 1. Second, we assume on the contrary that the agents' planned routes are known but not their behavioral parameters (see Figure 4b). We plan robustly with respect to this continuous ambiguity set using Algorithm 2. Crucially, the state intervals prediction is conditioned on the planned policy π.

In both experiments, we compare the performance of the robust planner to an oracle model that has perfect knowledge of the systems dynamics, and to a nominal planner that plans optimistically with respect to a dynamics model sampled uniformly from the ambiguity set. Statistics are collected from 100 episodes with random environment initialization. Results are presented in Table 1.

Conclusion

This paper has presented two methods for approximately solving the robust control problem. In the simpler case of finite ambiguity set and action space, we use optimistic planning and provide an upper bound for the simple regret. A direct consequence is that we recover the robust policy as the computational budget increases. In the general case, we use interval prediction to efficiently solve a conservative approximation of the robust objective while providing a lower bound for the performance of a policy when applied to the unknown true model. However, this method is lossy and does not enjoy asymptotic consistency. Both algorithms are flexible, allowing to handle a variety of parametrized dynamical systems, and practical, with a focus on computational efficiency. The two methods are also orthogonal, which means they can be combined to deal with complex ambiguity sets that display both continuous and discrete features, such as disjoint unions of connected sets.

Supplementary material

A Detailed proofs A.1 Lemma 1

Proof. By definition, when starting with sequence i, the value u m i (n) represents the minimum admissible reward, while b m i (n) corresponds to the best admissible reward achievable with respect to the the possible continuations of i. Thus, for all i ∈ A * , u m i (n) and u r i (n) are non-decreasing functions of n and b m i (n) and b r i (n) are a non-increasing functions of n, while v m i and v r i do not depend on n. Moreover, since the reward function r is assumed to have values in [0, 1], the sum of discounted rewards from a node of depth d is at most

γ d + γ d+1 + • • • = γ d 1-γ .
As a consequence, for all n ≥ 0 , i ∈ Ln of depth d, and any sequence of rewards (rt) t∈N obtained from following a path in iA ∞ with any dynamics m ∈ [1, M ]:

d-1 t=0 γ t rt ≤ d-1 t=0 γ t rt + ∞ t=d γ t rt ≤ d-1 t=0 γ t rt + γ d 1 -γ
That is equivalent to:

u m i (n) ≤ ∞ t=0 γ t rt ≤ b m i (n) Hence, min m∈[1,M ] u m i (n) ≤ min m∈[1,M ] ∞ t=0 γ t rt ≤ min m∈[1,M ] b m i (n) (17)
And as the left-hand and right-hand sides of ( 17) are independent of the particular path that was followed in iA ∞ , it also holds for the robust path:

min m∈[1,M ] u m i (n) ≤ max π∈iA ∞ min m∈[1,M ] ∞ t=0 γ t rt ≤ min m∈[1,M ] b m i (n) that is, u r i (n) ≤ v r i ≤ b r i (n) (18)
Finally, ( 18) is extended to the rest of Tn by recursive application of ( 6), ( 7) and (8).

A.2 Theorem 1

Proof. [START_REF] Hren | Optimistic planning of deterministic systems[END_REF] first show in Theorem 2 that the simple regret of their optimistic planner is bounded by γ dn 1-γ where dn is the depth of Tn. This properties relies on the fact that the returned action belongs to the deepest explored branch, which we can show likewise by contradiction using Lemma 1. This yields directly that a = i0 where i is some node of maximal depth dn expanded at round t ≤ n, which by Algorithm 1 verifies b r a (t) = b r i (t) = maxx∈A b r x (t) and:

v r -v r a = v r a * -v r a ≤ b r a * (t) -v r a ≤ b r a (t) -u r a (t) = b r i (t) -u r i (t) = γ dn 1 -γ (19)
Secondly, they bound the depth dn of Tn with respect to n. To that end, they show that the expanded nodes always belong to the sub-tree T∞ of all the nodes of depth d that are γ d 1-γ -optimal. Indeed, if a node i of depth d is expanded at round n, then b r i (n) ≥ b r j (n) for all j ∈ Ln by Algorithm 1, thus the max-backups of (8) up to the root yield b r i

(n) = b r ∅ (n). Moreover, by Lemma 1 we have that b r ∅ (n) ≥ v r ∅ = v r and so v r i ≥ u r i (n) = b r i (n) -γ d 1-γ ≥ v r -γ d 1-γ , thus i ∈ T∞.
Then from Assumption 2 and the definition of β applied to nodes in T∞, there exists d0 and c such that the number n d of nodes of depth d ≥ d0 in T∞ is bounded by c Hence,

γ d 1-γ β K d . As a consequence, n = dn d=0 n d = n0 + dn d=d 0 +1 n d ≤ n0 + dn d=d 0 +1 c γ d 1-γ β K d = n0 + c dn d=d 0 +1 κ d where c = c (1-γ) β . • If κ > 1, then n ≤ n0 + c κ d 0 +1 κ dn -d 0 -1 κ-1 and thus dn ≥ d0 + log κ (n-n 0 )(κ-1) c κ d 0 +1 . We conclude from (19) that Rn ≤ γ dn 1-γ = 1 1-γ (n-n 0 )(κ-1) c κ d 0 +1 log γ log κ = O n - log 1/γ log κ . • If κ = 1, then n ≤ n0 + c (dn -d0),
R T θ π = ∞ t=0 γ t r(st, at) ≥ H t=0 γ t r(st, at) ≥ H t=0 min s∈ S(t,s 0 ,π) γ t r(s, π(s)) = vr (π)
And finally,

v r (π) = min θ∈Θ v T θ π = min θ∈Θ E(R T θ π ) ≥ vr (π) B Environment dynamics B.

Kinematics

The vehicles kinematics are represented by the Kinematic Bicycle Model:

ẋ = v cos(ψ), (20)
ẏ = v sin(ψ), (21) v = a, (22) ψ
= v l tan(β), (23)
where (x, y) is the vehicle position, v its forward velocity and ψ its heading, l is the vehicle half-length, a is the acceleration command and β is the slip angle at the center of gravity, used as a steering command.

Each vehicle i is represented by its kinematics Xi = [xi, yi, vi, ψi]. The joint state is represented by s = {X1, • • • , XN }

B.2 Longitudinal control

The acceleration control is assumed to be linearly parametrized:

a = θ T a φa(s, i), ( 24
)
where θa is an uncertain weight vector, and φa(s, i) is a feature vector that depends on the joint state s and considered vehicle i.

It is composed of:

• a target velocity seeking term,

• a braking term to adjust velocity w.r.t. the front vehicle ,

• a braking term to respect a safe distance w.r.t. the front vehicle.

Denoting fi the front vehicle preceding vehicle i, φa is defined by

φa(s, i) =   v0 -vi n(v f i -vi) n(x f i -xi -(d0 + viT ))   ( 25
)
where n is the negative part function n(x) = min(x, 0) and v0, d0 and T respectively denote the speed limit, jam distance and time gap given by traffic rules.

We observe that this model exhibits similar qualitative behaviours to the IDM's.

B.3 Lateral control

A non-linear lane-keeping controller is implemented as follows: a lane L with lateral position yL and heading ψL is tracked by performing

β = tan -1 ( l v ψ cmd ) (29)
Finally,

β = tan -1 ( l v Kp ψ (ψL + sin -1 Kp y yL -y v -ψ)) (30)
This non-linear controller presented in subsection can be linearised around its equilibrium (y, ψ) = (yL, ψL).

B.4 Discrete behaviour

The MOBIL model [START_REF] Kesting | General Lane-Changing Model MOBIL for Car-Following Models[END_REF], which stands for Minimizing Overall Braking Induced by Lane

Figure 1 :

 1 Figure 1: The computation of robust b-values in Algorithm 1. The simulation of trajectories for every dynamics model T m is represented as stacked versions of the expanded tree T n .

Figure 3 :

 3 Figure3: From left to right: two simple models and corresponding u-values with optimal sequences in blue; the naive version of the robust values returns sub-optimal paths in red; our robust u-value properly recovers the robust policy in green.

  Figure 4: The highway-env environment. The ego-vehicle (green) is approaching a roundabout with flowing traffic (yellow).The robust objective error v rvr stems from two terms: the interval approximation of the reachable set and the loss of time-dependency between the states within a single trajectory. If both these approximations are tight enough, maximizing the lower bound vr will increase the true objective v r , which is the idea behind Algorithm 2. It is classically structured as an alternation of a Policy Evaluation step , during which the surrogate objective vr (π) is evaluated for a set of policies Π, and a Policy Search step which aims to steer the set of policies Π towards regions where the surrogate objective is maximal. The main Policy Search algorithms are listed in a survey by Deisenroth (2011). In this case, derivative-free methods such as evolutionary strategies (e.g. CMAES) would be more appropriate than policy gradient methods, since vr cannot be easily differentiated. Planning algorithms can also be used to exploit the dynamics and structure of the surrogate objective.

  Kp ψ (ψL + sin -1 Kp y yL -

Table 1 :

 1 Performances of robust planners on two ambiguous environments.

	Ambiguity set	Agent	Worst-case return Mean return ± std
	True model	Oracle	9.83	10.84 ± 0.16
	Discrete	Nominal Algorithm 1	2.09 8.99	8.85 ± 3.53 10.78 ± 0.34
	Continuous	Nominal Algorithm 2	1.99 7.88	9.95 ± 2.38 10.73 ± 0.61
	gap. The lane-change decisions are determined by the MOBIL model

  hence from (19) we have Rn = O γ nc . Proof. For any θ ∈ Θ, t ∈ [0, H] and any trajectory (s0, • • • , st) sampled from π and T θ ,

	A.3 Property 1
	st ∈ S(t, s0, π) ⊂ S(t, s0, π)

  1. Position controlvy cmd = Kp y (yL -y) (26) 2. Lateral velocity to heading conversionψ ref = ψL + sin -1 vy cmd v(27)3. Heading controlψ cmd = Kp ψ (ψ ref -ψ)(28) 4. Heading rate to steering angle conversion

32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada.

Source code is available at https://github.com/eleurent/highway-env

Video and source code are available at https://eleurent.github.io/robust-control/

It enables the vehicle and (with a politeness factor p) its following vehicles on both current and target lanes to increase their overall acceleration:∆ v + p(∆ vrear, current + ∆ vrear, target) ≥ amin(37)This model describes changes in the target lane L.

Acknowledgments

This work has been supported by CPER Nord-Pas de Calais/FEDER DATA Advanced data science and technologies 2015-2020, the French Ministry of Higher Education and Research, INRIA, and the French Agence Nationale de la Recherche (ANR).

C Interval Predictor

In this section, we design an interval predictor for our system.

C.1 Notations

For any real variable z, we denote an interval containing z as z = [z, z], such that z ≤ z ≤ z. As elements of R 2 , they can be scaled and offset by scalars. This definition is extended element-wise to vector variables.

where p(•) and n(•) are the projections onto R + and R -, respectively.

• The difference operator -

• The cosine and sine operators

• The inverse operator / over a positive interval z > 0

• Any other function f is assumed increasing on the interval z and is applied coefficient-wise

We start with an initial estimate of the intervals over state variables xI , yI , vI and ψI . Typically, we use zero-width intervals centred on the current state observation. Likewise, any variable z used in place of an interval corresponds to the zero-width interval [z, z].

C.2 Intervals for features

We use ( 25) and ( 35) respectively to derive intervals for the features φa and φ b from the intervals over the states.

We index the front vehicle intervals with the subscript

and

C.3 Intervals for controls

The controls intervals are derived from ( 24) and ( 33)

C.4 Intervals for velocity and heading

The velocity interval is derived from ( 22) and the heading interval from ( 23)

C.5 Intervals for positions

Likewise, the positions interval are derived from the kinematics ( 20) and ( 21)


Thomas Hélie

Fabrice Silva

Victor Wetzel

Port-Hamiltonian approach to self-sustained oscillations in the vocal apparatus

Phonation is a natural example of self-sustained oscillations in a nonlinear dynamical system. It results from the controlled nonlinear coupling between the deformable vocal folds and the airflow expired from the lungs through the glottis and the vocal tract. As a proof of concept, we explore the port-Hamiltonian approach to propose a minimal model of the full vocal apparatus. This port-Hamiltonian theory emphasises on the structure of a system, i.e., on the separation between the behaviour of the subsystems and their assembly, with two essential consequences : the modularity of the approach (enabling improvement in modelling of some subsystem independently of the others) and guaranteed power balance. We here present the minimal model of the vocal apparatus, and compare the time-domain simulations based on the numerical methods designed in the port-Hamiltonian approach, to results on bifurcations obtained by means of continuation methods.

Motivations

The physics of voice production has motivated a wide variety of models, from full-featured numerical ones (mainly based on the finite element methods for vocal folds and finite volume methods for the airflow) to reduced order models focusing on the essential phenomenon underlying the phonation [START_REF] Titze | The physics of small-amplitude oscillation of the vocal folds[END_REF]. A large body of work in the latter category relies on the description of the glottal aerodynamics provided by van den Berg in the late 1950 [START_REF] Van Den Berg | On the Air Resistance and the Bernoulli Effect of the Human Larynx[END_REF]. This work and its extensions are based on experiments made on synthetic or moulded human larynx, i.e. considering airflow in channel with rigid static boundaries. However phonation intrinsically implies the vibration of the vocal folds periodically closing the glottis. Consequently, there is a significant paradox in those simplified models: the vocal folds move as a consequence of the power exchanged with the glottal flow, but the description of the latter assumes that it does not receive or provide any power to the folds. This leads to an inconsistent power balance in the vocal apparatus model. The historical reasons for the simplified flow modelling are related to the source-filter separation theorised by Fant [START_REF] Fant | Acoustic Theory of Speech Production[END_REF]. Concisely, the vocal tract is supposed to act as a mere wave guide between the larynx and the outer space. In this representation, the larynx is then required to act as an acoustic source, the simplest one being a fluctuating flow rate through the glottis. Thus, simplified models of phonation mainly focused at a physically realistic way to produce the so called glottal flow signal. From a fluid-structure interaction perspective, the power imbalance introduces a bias in the analysis of the coupling occurring in the larynx and of the instability leading to voice production. The objective of the present communication is to present a minimal model of the vocal apparatus that is able to simulate oscillating regime similar to patterns observed in vivo while being energetically consistent. The port-Hamiltonian theory provides a valuable framework for this purpose and more generally to the modelling, analysis and control of complex nonlinear dynamical systems.

Port-Hamiltonian systems (PHS)

The port-Hamiltonian theory combines the views of the (geometric) Hamiltonian mechanics and of port-based modelling approach. It emphasises the separation between the behaviour of subsystems and their interconnection. The lingua franca of this theory is energy : port-Hamiltonian systems are open passive systems that can store, dissipate and exchange power with their neighbourhood. Energy-storage is defined by the Hamiltonian H(x) as a function of state variables x. This dependency, formulated as the Hamiltonian gradient ∇ x H, expresses the effort of the energy-storing component. Conversely, the evolution of this component is described by the flux variable ẋ. Effort and flux are power dual variables, i.e., they jointly define the energy flow dH/dt = ẋ • ∇ x H(x). Dissipating components are described by dissipation variable w and their constitutive laws z(w), such that the dissipated power is P diss = w • z(w) ≥ 0. Finally, external interactions are classically described in terms of inputs u (also called efforts) and output y (fluxes) that are dual with respect to the external power: P ext = y • u (≥ 0 when the system yields power). The geometric structure (Dirac structure, see [START_REF] Van Der Schaft | Port-Hamiltonian Systems Theory: an Introductory Overview[END_REF]) accounting for the interconnection of the subsystems is then described by a matrix S relating efforts to fluxes:

  ẋ w y   = S(x, w)   ∇ x H z(w) u   . (1)
When the interconnection is conservative, the power balance writes down as

0 = dH dt + P diss + P ext =   ẋ w y   •   ∇ x H z(w) u   =   ∇ x H z(w) u   • S(x, w) •   ∇ x H z(w) u   (2)
which is ensured when the matrix S is skew-symmetric (e • S • e = 0 for any effort e).

Minimal port-Hamiltonian model of the vocal apparatus

As depicted in Fig. 1, the vocal apparatus is modelled as the interconnection of vocal folds, glottal flow, vocal tract and a subglottal pressure supply. Every component is designed as minimal, but with an emphasis on the dual pairing on the interconnection ports. For instance, each vocal fold is considered to be a single-d.o.f. oscillator with an elastic cover (as in Ref. [START_REF] Awrejcewicz | Numerical Analysis of the Oscillations of Human Vocal Cords[END_REF]), and is submitted to pressure on the upstream and downstream faces, and interacts with the flow at the glottal face. Reciprocally, the glottal flow is modelled as the simplest kinematics of potential incompressible flow of inviscid air preserving the continuity of the normal velocity on the surface of the vocal folds (see details in Ref. [START_REF] Helie | Self-oscillations of a Vocal Apparatus: A Port-Hamiltonian Formulation[END_REF]). Downstream the glottis, the flow separates from the folds into a jet that spreads and dissipates its kinetic energy into heat. Finally, the vocal tract is represented as seen by the larynx, i.e. by means of its input impedance. We use an analytical modal formulation. When considering a single mode, the vocal tract acts as an Helmholtz resonator providing an acoustic feedback on the larynx. The full vocal apparatus is obtained by conservative interconnection of the previous subsystems. For the sake of conciseness, the resulting Hamiltonian and matrix S are not reported here but can be efficiently computed using the Python package PyPHS [START_REF] Falaize | PyPHS: Passive modeling and simulation in python[END_REF] designed for the symbolic manipulation of PHS systems.

P sub Q sub P sub r Q sub r P sub l Q sub l P - tot Q - F p l v l F p r vr P + Q + P sup l Q sup l P sup r Q sup r P ac Q ac

Numerical results

Time-domain simulations are performed using the numerical scheme designed after the principles of the port-Hamiltonian theory [START_REF] Lopes | Approche passive pour la modélisation, la simulation et l'étude d'un banc de test robotisé pour les instruments de type cuivre[END_REF]. It is shown that it not only preserves power balance, but it also more consistent than standard integrators (such as the Runge-Kutta schemes). The simulations evidence the ability of the model to produce oscillating regimes above some pressure threshold, and also quasi-periodic vibrations when vocal folds are detuned. The model are also been analysed from the point of view of the bifurcations by means of continuation methods such as the prediction-correction used in the AUTO software or the numerical asymptotic methods of the MANLAB software [START_REF] Karkar | A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities[END_REF]. They confirme qualitatively the results of the time-domain simulations, notably with the existence of two stable oscillating regimes for detuned folds.

Figure 1 :

 1 Figure 1: Components of the vocal apparatus. The interconnection takes place via pairs of effort (P ) and flux (Q) variables. The 0 connection expresses the equality of efforts and the division of flux. See Ref.[START_REF] Maschke | An intrinsic Hamiltonian formulation of network dynamics: non-standard Poisson structures and gyrators[END_REF] for an introduction to bond graphs.

Acknowledgements

We acknowledge the support of the ANR-DFG (French-German) project INFIDHEM ANR-16-CE92-0028.