Bérengère Argence

Bruno Chanteau

Olivier Lopez

Daniele Nicolodi

Michel Abgrall

Christian Chardonnet

Christophe Daussy

Benoît Darquié

Yann Le Coq

Anne Amy-Klein

Quantum cascade laser frequency stabilisation at the sub-Hz level

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Molecules are increasingly being used in precision tests of physics thanks to progress made in controlling molecular degrees of freedom [START_REF] Carr | Cold and ultracold molecules: science, technology and applications[END_REF][START_REF] Barry | Magnetooptical trapping of a diatomic molecule[END_REF] . They are now being used, for example, to test fundamental symmetries [START_REF] Tokunaga | Probing weak force-induced parity violation by high-resolution midinfrared molecular spectroscopy[END_REF][START_REF] Baron | Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron[END_REF] , and to measure fundamental constants [START_REF] Koelemeij | Vibrational spectroscopy of HD + with 2-ppb accuracy[END_REF][START_REF] Daussy | First direct determination of the Boltzmann constant by an optical method: towards a new definition of the Kelvin[END_REF][START_REF] Moretti | Determination of the Boltzmann Constant by Means of Precision Measurements of H 2 O Line Shapes at 1.39 µm[END_REF] and their possible variation in time [START_REF] Shelkovnikov | Stability of the proton-toelectron mass ratio[END_REF][START_REF] Hudson | Cold molecule spectroscopy for constaining the evolution of the fine structure constant[END_REF] . Most of these experiments are spectroscopic precision measurements, and are often in the mid-infrared (MIR) domain where the molecules exhibit intense and narrow rovibrational transitions. This creates a need for efficient laser sources in this spectral region, prompting efforts to develop ultra-stable (US) and accurate continuous wave (cw) laser sources as well as MIR frequency combs (see for instance [START_REF] Hugi | Mid-infrared frequency comb based on a quantum cascade laser[END_REF][START_REF] Schliesser | Mid-infrared frequency combs[END_REF] ). Quantum Cascade Lasers [START_REF] Faist | Quantum cascade laser[END_REF][START_REF] Yao | Mid-infrared quantum cascade lasers[END_REF] (QCL) are promising cw sources, as they are available anywhere in the 3-25 µm MIR range, and each QCL can be tuned over several hundreds of gigahertz. However they have a freerunning line width of tens to thousands of kilohertz, making their frequency stabilisation challenging [START_REF] Bielsa | HCOOH high resolution spectroscopy in the 9.18 µm region[END_REF][START_REF] Bartalini | Frequency-comb-referenced quantum-cascade laser at 4.4 μm[END_REF][START_REF] Bressel | 5 µm laser source for frequency metrology based on difference frequency generation[END_REF][START_REF] Borri | Direct link of a mid-infrared QCL to a frequency comb by optical injection[END_REF][START_REF] Cappelli | Subkilohertz linewidth room-temperature mid-infrared quantum cascade laser using a molecular sub-Doppler reference[END_REF][START_REF] Mills | Coherent phase lock of a 9 µm quantum cascade laser to a 2 µm thulium optical frequency comb[END_REF][START_REF] Galli | Comb-assisted subkilohertz linewidth quantum cascade laser for high-precision mid-infrared spectroscopy[END_REF][START_REF] Hansen | Robust, frequency-stable and accurate mid-IR laser spectrometer based on frequency comb metrology of quantum cascade lasers up-converted in orientationpatterned GaAs[END_REF][START_REF] Sow | A widely tunable 10-μm quantum cascade laser phase-locked to a state-ofthe-art mid-infrared reference for precision molecular spectroscopy[END_REF][START_REF] Tombez | Active linewidth-narrowing of a midinfrared quantum cascade laser without optical reference[END_REF] .

In general, frequency stabilisation of a laser requires one to first choose a frequency reference and then to transfer its spectral properties to the laser. The most common references used in the MIR spectral region are molecular rovibrational absorption lines [START_REF] Tokunaga | Probing weak force-induced parity violation by high-resolution midinfrared molecular spectroscopy[END_REF][START_REF] Acef | Metrological properties of CO 2 /OsO 4 optical frequency standard[END_REF] . However, molecular degrees of freedom cannot be controlled as efficiently as atomic ones, leading to limited frequency reproducibility and accuracy. Alternatively, several attempts to develop MIR US cavities have been made, but the performance is far from what is reported in the near-infrared (NIR) or visible spectral regions [START_REF] Bernard | Spectral purity and long-term stability of CO 2 lasers at the Hertz level[END_REF][START_REF] Fasci | Narrow-linewidth quantum cascade laser at 8.6 µm[END_REF] .

It is thus appealing to use as frequency references the best US lasers. Those being mostly in the NIR region, one has to bridge the gap between the NIR and MIR domains. This is possible using an optical frequency comb (OFC). The MIR frequency is locked to a high harmonic of the OFC repetition rate, using sum-or difference-frequency generation processes.

This not only provides the ultimate stabilities of lasers locked to state-of-the-art US cavities [START_REF] Kessler | A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity[END_REF][START_REF] Hagemann | Ultra-stable laser with average fractional frequency drift rate below[END_REF] , but also allows one to benefit from the direct link between such NIR sources and primary frequency standards [START_REF] Guéna | Progress in atomic fountains at LNE-SYRTE[END_REF] . A few groups have already demonstrated the stabilisation of a MIR laser to a primary standard traceable frequency reference using an OFC [START_REF] Bartalini | Frequency-comb-referenced quantum-cascade laser at 4.4 μm[END_REF][START_REF] Bressel | 5 µm laser source for frequency metrology based on difference frequency generation[END_REF][START_REF] Borri | Direct link of a mid-infrared QCL to a frequency comb by optical injection[END_REF][START_REF] Mills | Coherent phase lock of a 9 µm quantum cascade laser to a 2 µm thulium optical frequency comb[END_REF][START_REF] Galli | Comb-assisted subkilohertz linewidth quantum cascade laser for high-precision mid-infrared spectroscopy[END_REF][START_REF] Hansen | Robust, frequency-stable and accurate mid-IR laser spectrometer based on frequency comb metrology of quantum cascade lasers up-converted in orientationpatterned GaAs[END_REF][START_REF] Amy-Klein | Absolute frequency measurement of a SF 6 two-photon line by use of a femtosecond optical comb and sum-frequency generation[END_REF][START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF][START_REF] Vainio | Frequency-comb-referenced molecular spectroscopy in the mid-infrared region[END_REF] .

Moreover, with the rapid expansion of US frequency transfer over optical fibre links, signals from a NIR US laser referenced to a primary frequency standard can indeed be transferred a few hundreds of kilometers away without any degradation of its stability [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF][START_REF] Lopez | Ultra-stable long distance optical frequency distribution using the Internet fiber network[END_REF][START_REF] Newbury | Coherent transfer of an optical carrier over 251 km[END_REF][START_REF] Predehl | A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place[END_REF] . In the future, the development of a local reference laser may no longer be required.

In this paper we demonstrate the stabilisation of a QCL onto the signal from a remote NIR US laser. This signal is transferred using a 43-km fibre link and its frequency is monitored by primary standards. This leads to a two-order-of-magnitude improvement in frequency stability and traceability over previous work on QCLs [START_REF] Sow | A widely tunable 10-μm quantum cascade laser phase-locked to a state-ofthe-art mid-infrared reference for precision molecular spectroscopy[END_REF] . To show the potential for precise spectroscopic measurements, we further demonstrate continuous tuning of this stabilised QCL and measure OsO 4 molecular absorption frequencies with a 8x10 -13 uncertainty, well below what has ever been reported with QCLs.

The experimental set-up is displayed in Fig. 1. We use a room-temperature distributed feedback QCL emitting up to 40 mW at 10.3 µm, with a tuning range of 60 nm (see methods).

It is phase-locked to an OFC, consisting of an Erbium-doped fibre mode-locked laser emitting around 1.55 µm. The comb repetition rate f rep ~250 MHz is stabilised by phase-locking one tooth of the OFC onto a NIR frequency reference generated at LNE-SYRTE laboratory and transferred to LPL laboratory through an optical fibre link [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF] (see methods). This NIR reference exhibits a relative frequency stability lower than 2x10 -15 between 1 and 100 s. Its absolute frequency is known with an uncertainty of 10 -14 on this time scale when referenced to a H-Maser via a local OFC at LNE-SYRTE (referred to as SYRTE-OFC in Fig. 1 and methods) [START_REF] Guéna | Progress in atomic fountains at LNE-SYRTE[END_REF] . The 3×10 -16 Cs fountain's accuracy [START_REF] Guéna | Progress in atomic fountains at LNE-SYRTE[END_REF] can be reached for long enough averaging time. At LPL, part of the 1.55 µm comb is amplified and fed into a non-linear fibre to generate an additional output of the OFC, centred at 1.82 µm [START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF] . After adjusting their shape and polarization, the 1.82-µm comb and QCL beams are overlapped in an AgGaSe 2 crystal to perform sum-frequency generation (SFG), resulting in a shifted comb centred at 1.55 µm. This shifted comb is then combined with the 1.55 µm output of the LPL OFC. A fibre delay line is used to overlap the pulses in the time domain. On photodiode PD1 (see Fig. 1), a beat note of frequency Δ 1 = ± [n f repν QCL ] is obtained between the QCL frequency n QCL and a high-harmonic (of order n) of the repetition rate (see methods). Note that this beat note is independent of the comb carrier-envelope offset frequency f 0 . This signal at frequency Δ 1 is filtered and mixed with a radio-frequency (RF) oscillator to generate the error signal that drives the phase lock loop used to stabilise the QCL. A bandwidth of several hundreds of kilohertz is obtained. This stabilisation setup can be used on a wide 9-11 µm spectral range and could be extended to any frequency in the whole 3-20 µm range with an appropriate choice of crystal and comb spectrum.

The performance of this set-up is assessed by measuring the relative frequency stability and frequency noise power spectral density (PSD) of the QCL. We first compare the QCL's stability to that of the state-of-the-art secondary frequency standard around 10 µm, a CO 2 laser stabilised on an OsO 4 saturated absorption line [START_REF] Acef | Metrological properties of CO 2 /OsO 4 optical frequency standard[END_REF][START_REF] Bernard | CO 2 laser stabilization to 0.1-Hz level using external electrooptic modulation[END_REF][START_REF] Ducos | Progress toward absolute frequency measurements of the 127 I 2 -stabilized Nd:YAG laser at 563.2 THz/532 nm[END_REF] (hereafter referred to as the "MIR reference", see methods for details). The beat note of frequency ∆ 2 between the stabilised QCL and this MIR reference is detected on photodiode PD 2 (see Fig. 1) and sent to a frequency counter. The stability of the beat note is the quadratic mean of the MIR reference and the QCL stabilities, and is obtained by calculating the overlapping Allan deviation of the data (see red squares on Fig. 2). The fractional frequency stability equals 5x10 -14 for 1-s averaging time (τ) and decreases as τ -1/2 up to a few tens of seconds. The stability of the beat note of frequency ∆ 1 ' between the MIR reference and the OFC, detected on PD1, is also plotted for comparison (blue circles). The two stabilities are almost identical except for small deviations due to non-stationary effects, and reflect the noise level of the MIR reference.

Incidentally, this is the same noise level as was measured previously [START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF] . The MIR reference contribution being dominant, this measurement can only provide an upper limit on the QCL stability.

The QCL frequency stability is expected to be given by the LPL OFC stability, with only a negligible contribution from the phase-lock loop residual frequency noise. In this section, we evaluate both contributions, one after the other. The OFC stability is assessed by beating an optical mode of the OFC with a second remote US laser located at LNE-SYRTE (US laser #2, see Fig. 1 andmethods). The Allan deviation, shown as green triangles in Fig. 2, is lower than 2x10 -15 from 1 to 100 s and equals the US laser stability [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF] . To evaluate the frequency noise contribution from the QCL phase-lock loop, we modify the experimental setup to simultaneously phase-lock the QCL and the CO 2 laser onto the OFC. We then detect the beat note between the OFC-stabilised QCL and CO 2 laser. Its stability is a measure of the residual non-common frequency fluctuations between these MIR sources and thus gives an estimation of the frequency noise contribution of the two phase-lock loops. As shown in Fig. 2 (black stars), it is 2x10 -16 at 1 s averaging time, one order of magnitude below the OFC stability and reaches 10 -17 at 100 s. This demonstrates that the QCL and the CO 2 laser frequency fluctuations are identical at a level well below the OFC stability and that both copy the OFC spectral properties. This stability is 10 times better at 1 s than any other MIR laser to date [START_REF] Ducos | Progress toward absolute frequency measurements of the 127 I 2 -stabilized Nd:YAG laser at 563.2 THz/532 nm[END_REF] and constitutes an improvement of a factor at least 30 over the previous record with a QCL [START_REF] Sow | A widely tunable 10-μm quantum cascade laser phase-locked to a state-ofthe-art mid-infrared reference for precision molecular spectroscopy[END_REF] .

To further characterise the QCL stabilisation, the frequency noise PSD of the aforementioned OFC-US laser #2 beat note signal was also measured. The result is displayed in Fig. 3 (green curve c) after being scaled to 10.3 µm (see methods). The frequency noise PSD is 10 3 Hz 2 /Hz at 100 kHz Fourier frequency (limited by the approximately 500 kHz locking bandwidth) and exhibits a plateau at 10 -2 Hz 2 /Hz between 1 and 100 Hz. Fig. 3 also shows the frequency noise PSD of the 43-km noise compensated fibre link [START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF] (brown line d).

It is below or equal to the OFC PSD. As such, the link adds a minor contribution to the OFC noise. Next, Fig. 3 displays the frequency noise PSDs of the beat notes at frequencies ∆ 2 between the MIR reference and the OFC-stabilised QCL (red curve a) and ∆' 1 between the MIR reference and the OFC (blue curve b). These two curves overlap almost perfectly below 100 kHz Fourier frequency. They are at a level of 10 Hz 2 /Hz from 1 Hz to 1 kHz and increase to 10 3 Hz 2 /Hz at 100 kHz. The noise level of these PSDs is the sum of the noise contributions from each laser. At low Fourier frequencies (below 1 kHz), the MIR reference noise dominates over the OFC noise, as previously measured in [START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF] . The bump around 400 Hz comes from the MIR reference locking bandwidth. Above 1 kHz, PSDs (a) and (b) overlap with the OFC noise PSD. Finally, we note a bump around 500 kHz on PSD (a), which corresponds to the QCL locking bandwidth. These measurements confirm that the QCL noise is indeed copying the OFC noise. Fig. 3 also displays the free-running QCL PSD (black curve e) for comparison, highlighting a 12 orders of magnitude reduction in the QCL frequency noise at 1 Hz.

The relevant parameter for spectroscopic applications is the emission line width, which we evaluate from the measured OFC frequency noise PSD following the approach given in [START_REF] Elliott | Extracavity laser band-shape and bandwidth modification[END_REF] . The low frequency contribution needed for this calculation is deduced from the 1s gate time frequency data used for deriving the OFC stability of Fig. 2. The resulting QCL line shape shown in Fig. 4a exhibits a full width at half maximum (FWHM) of 0.2 Hz (for a 10-mHz resolution bandwidth) which is the narrowest line width reported for a QCL by far. It can be compared to the calculated free-running line width of 300 kHz (found using a resolution bandwidth of 1 kHz). Fig. 4b also shows the beat note signal ∆ 2 between the phaselocked QCL and the MIR reference, recorded with a resolution of 125 mHz with a Fast Fourier Transform analyser. A lorentzian fit gives a FWHM of 10 Hz. Although dominated by the MIR reference contribution, such a narrow beat note had never been recorded with a QCL.

The frequency accuracy, or more specifically the frequency traceability to the primary standards of LNE-SYRTE [START_REF] Guéna | Progress in atomic fountains at LNE-SYRTE[END_REF] , is ensured by the use of phase-lock loops for the QCL frequency stabilisation. The QCL absolute frequency is thus known with an uncertainty of about 10 -14 after 100 s averaging time, when the NIR laser frequency is referenced to the H-maser only.

By averaging for long enough, the 3×10 -16 Cs fountain's accuracy [START_REF] Guéna | Progress in atomic fountains at LNE-SYRTE[END_REF] can ultimately be transferred to the NIR reference and in turn to the QCL frequency.

Such an accurate and ultra-stable QCL is ideal for carrying out molecular spectroscopy at the highest resolutions in the MIR spectral region. As a first demonstration, we perform saturated absorption spectroscopy of the OsO 4 molecule at an unprecedented resolution and accuracy for QCL sources. OsO 4 is a test molecule for high-precision frequency measurements in the MIR region as many tens of its lines have been accurately measured to serve as secondary frequency standards [START_REF] Acef | Accurate determination of OsO 4 absolute frequency grid at 28/29 THz[END_REF][START_REF] Chardonnet | Hyperfine interactions in the n 3 band of osmium tetroxide: accurate determination of the spin-rotation constant by crossover resonance spectroscopy[END_REF] . The QCL beam is sent through an OsO 4 -filled Fabry-Perot (FP) optical cavity. Saturated absorption spectra are recorded in transmission of this cavity by scanning the RF oscillator used to phase lock the QCL onto the OFC (see methods). This allows the laser to keep its extremely high spectral purity, frequency stability and accuracy while being swept. Exhaustive studies of systematic effects affecting OsO 4 line shifts have already been carried out [START_REF] Acef | Metrological properties of CO 2 /OsO 4 optical frequency standard[END_REF] . In this work, care was taken to be in similar experimental conditions as those reported in the literature for the reference line to allow the comparison of absolute frequencies. We evaluate the frequency of a given line as the weighted mean frequency of independent measurements (see methods). The 190 OsO 4 reference line frequency is found to be n OsO4/R(14) -9 Hz , with a standard uncertainty of the mean of 22 Hz. The frequency of the unreported line shown in Fig. 5 is measured to be n OsO4/R(14) + 4 147 399 (23) Hz. The absolute frequencies of two other lines have been measured and are listed in Table 1. Our results, in very good agreement with those reported in the literature [START_REF] Acef | Accurate determination of OsO 4 absolute frequency grid at 28/29 THz[END_REF][START_REF] Chardonnet | Hyperfine interactions in the n 3 band of osmium tetroxide: accurate determination of the spin-rotation constant by crossover resonance spectroscopy[END_REF][START_REF] Chardonnet | Spectroscopie de saturation de hautes précision et sensibilité en champ laser fort[END_REF] , lead to fractional uncertainties on absolute frequencies as low as 8x10 -13 . This is one of the lowest uncertainties ever reported using saturated absorption spectroscopy in the MIR region [START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF][START_REF] Daussy | Long-distance frequency dissemination with a resolution of 10 -17[END_REF] . In particular, the uncertainty of the reference line is improved by a factor of 2, demonstrating the potential of our set-up for precision measurements devoted to metrological applications. Owing to the high spectral purity and direct traceability to the LNE-SYRTE frequency standards, this uncertainty is not limited by our ability to control the laser frequency, but by systematic effects such as the pressure shift. Note that thanks to the QCL's tuning range, we were able to record two unreported lines. Our stabilisation setup can straightforwardly be extended to the entire 9-11 µm spectral range without any loss of performance, by using QCLs of adjacent emission spectra.

We have demonstrated the frequency stabilisation of a 10.3-µm QCL on an OFC phase-locked to a remote US NIR frequency reference. We find the stability of the QCL to be at the level of the reference, below 0.06 Hz (2x10 -15 in relative value) from 1 to 100 s, which is a new record for such lasers. We derive a line width of the order of 0.2 Hz, which to our knowledge makes our QCL the narrowest to date. Furthermore, our set-up allows for frequency traceability to the primary standards, so the QCL absolute frequency is known to within an uncertainty of 10 -14 after 100 s (given by the LNE-SYRTE H-maser standard). The 3x10 -16 accuracy of the Cs fountains can be achieved by sufficiently increasing integration times [START_REF] Guéna | Progress in atomic fountains at LNE-SYRTE[END_REF] . Moreover, the ongoing work on dissemination of an optical reference on a continental scale using Internet fibre networks will eventually enable many laboratories to access an US optical reference. The level of stability and accuracy obtained using this method is better than any other reported in the MIR region to date, whether it be using QCLs or otherwise. It also frees us from having to lock the QCL to any particular reference (another laser or a molecular transition), leading to a considerable increase in the accessible spectral window.

QCLs with performances comparable to the most stable of oscillators pave the way for precise spectroscopic measurements on molecules in the MIR spectral region. Our work demonstrates frequency tuning over 100 MHz, allowing high-resolution saturated absorption spectroscopy of molecular transitions, including those in spectral regions which were previously inaccessible. A much broader range of a few gigahertz would be obtained by scanning the OFC repetition rate. This is achievable by generating a tunable sideband of the NIR frequency reference and locking the comb on it. Moreover, this set-up can be extended to the whole MIR spectral region up to approximately 20 µm simply by adapting the comb output wavelength to the frequency to be measured via sum or difference frequency generation in the appropriate crystal (using for instance orientation-pattern GaAs combined with SFG [START_REF] Eyres | All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion[END_REF] ).

At LPL, this QCL-based spectrometer is critical for our ongoing efforts to make the first observation of parity violation (PV) by Ramsey interferometry of a beam of chiral molecules [START_REF] Tokunaga | Probing weak force-induced parity violation by high-resolution midinfrared molecular spectroscopy[END_REF] . In this project, vibrational frequencies of two enantiomers are measured and compared. The expected frequency difference is small, and is predicted to be on the order of 10 -13 , at most. Furthermore, the project requires a certain flexibility to be able to switch from one molecular species to another in keeping with the on-going production of novel, promising exotic candidate species. Naturally, these molecules have resonances in different spectral regions, many of which only QCLs can attain. Thus, this particular combination of precision, stability and flexibility of tuning in the mid-IR requires a very particular laser setup, of which the first proof-of-principle is successfully demonstrated by results presented in this paper. correction of the phase noise accumulated by light propagation in the optical fibre is implemented [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF] . The residual phase noise instability added by the compensated link (detected with full bandwidth) lies below 10 -15 at 1 s averaging time (τ) and decreases with a τ -1 slope [START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF] .

It is below the stability of the US laser. The signal of the US frequency reference is thus transferred to LPL without degradation (up to Fourier frequencies of a few tens of hertz [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF] ). At LPL, it is used to phase-lock a local 1.54 µm laser diode for signal regeneration. The repetition rate f rep of the LPL OFC is in its turn phase-locked to this laser diode after removal of the comb carrier-envelope offset frequency f 0 [START_REF] Telle | mode-locked lasers as transfer oscillators for optical frequency measurements[END_REF] . Note that for this lock we use some small fraction of light at 1.55 µm present in the comb output at 1.82 µm (see next section) in order to take into account the possible phase fluctuations arising from the comb amplifier used to generate this 1.82 µm output [START_REF] Nicolodi | Spectral purity transfer between optical wavelengths at the 10 -18 level[END_REF] . Fast and slow corrections are respectively applied to an intracavity electro-optic modulator and a piezo-electric transducer (PZT) acting on the cavity length, resulting in a bandwidth of more than 500 kHz. The comb is then operating in the "narrow-line width regime" [START_REF] Zhang | Characterizing a fiber-based frequency comb with electro-optic modulator[END_REF] .

QCL stabilisation onto the OFC. The 1.55 µm LPL-OFC is used to transfer the spectral properties of the NIR reference to the QCL. An additional output of the comb centred at 1.82 µm (total power 25 mW, mode frequencies q f rep + f 0 where q is an integer, f rep is the OFC repetition rate, and f 0 is the carrier-envelope offset frequency) is overlapped with the QCL beam (5 mW, frequency ν QCL ) in an AgGaSe 2 crystal to perform the SFG. A set of lenses is used to focus the QCL beam in the crystal. An iris diaphragm and an optical isolator, composed of a wire-grid polarizer and a quarter-wave plate, minimize the optical feedback to the QCL mainly caused by back-reflection from the AgGaSe 2 crystal facets [START_REF] Hansen | Robust, frequency-stable and accurate mid-IR laser spectrometer based on frequency comb metrology of quantum cascade lasers up-converted in orientationpatterned GaAs[END_REF] . SFG between the 1.82 µm comb and the 10.3 µm cw beam results in a shifted comb (30 nW, mode frequencies q f rep + f 0 + ν QCL ), centred at 1.55 µm. This shifted comb is then combined with the 1.55 µm output of the LPL OFC (10 mW, mode frequencies p f rep + f 0 where p is an integer , )

resulting in a beat note of frequency Δ 1 = ± [(p-q) f repν QCL ] (with p-q = n an integer of the order of 120000), independent of the OFC carrier-envelope offset frequency f 0 . This beat note exhibits a signal to noise ratio (SNR) of about 25-30 dB in a 100 kHz bandwidth. This SNR is found to be independent of the QCL power since the same SNR is obtained with a higher power (50 mW) CO 2 laser [START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF] . It is limited by the broad frequency noise of the 1.82 µm output originating from the amplified spontaneous emission of the amplifier used to generate this 1.82 µm output. The beat note at frequency Δ 1 is then filtered and mixed with a RF oscillator to generate the error signal that drives a proportional-integrator phase-lock loop. The correction signal is applied to the QCL's current.

MIR reference.

The MIR reference is a CO 2 laser stabilised onto an OsO 4 saturated absorption line. This system constitutes the state-of-the-art MIR secondary frequency standard [START_REF] Acef | Metrological properties of CO 2 /OsO 4 optical frequency standard[END_REF][START_REF] Bernard | CO 2 laser stabilization to 0.1-Hz level using external electrooptic modulation[END_REF] in the 10 µm region. In this work, the CO 2 laser is stabilised on the 190 OsO 4 reference line in the vicinity of the 10 µm R( 14) CO 2 emission line [START_REF] Acef | Accurate determination of OsO 4 absolute frequency grid at 28/29 THz[END_REF] . The OsO 4 gas fills a 1.5-m long Fabry-Perot cavity which has a finesse of about 100 [START_REF] Bernard | CO 2 laser stabilization to 0.1-Hz level using external electrooptic modulation[END_REF] . For practical reasons, we use a second CO 2 laser phase-locked to the OsO 4 -stabilised CO 2 laser. As previously demonstrated [START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF] and expected from a properly working phase lock loop, no phase noise is added to the second CO 2 laser. Note that the experimental set-up allows the CO 2 laser to be sent to the SFG crystal (instead of the QCL, see Fig. 1) in order to evaluate the MIR reference frequency noise and stability.

OFC frequency noise and stability. In order to obtain the LPL OFC spectral characteristics, we evaluate the stability and the frequency noise of the beat note between an optical mode of this comb and another US laser (after removing the comb offset frequency f 0 [START_REF] Telle | mode-locked lasers as transfer oscillators for optical frequency measurements[END_REF] ). This US laser #2 consists of a 1.54-µm cw source frequency locked onto an US cavity similar to the one used to provide the NIR reference. The two US lasers thus present the same spectral properties. Since they are transferred from LNE-SYRTE to LPL through the same optical fibre link and since their frequencies are very close, only 375 MHz apart, the phase noise accumulated along the fibre link is approximately the same for both lasers. Thus any propagation noise is cancelled in the beat note between the OFC and the second US signal and compensating the link noise is not needed for this measurement. The beat note PSD (green curve c in Fig. 3) resulting from the sum of the OFC and the US laser #2 noises thus gives an upper limit on the OFC frequency noise. The residual noise PSD of the stabilised fibre link (brown curve d in Fig. 3) should be added to it to obtain the overall frequency noise of the OFC. Nonetheless, the link noise contribution is found to be negligible except for Fourier frequencies between 20 and 400 Hz, where it becomes comparable to that of the OFC, leading however to a minor overall contribution. Note that these PSDs have been scaled to 10.3 µm by the square of the wavelengths ratio for a direct comparison with PSDs recorded at 10.3 µm. From the beat note frequency measurement, we also derive the OFC stability by calculating the overlapping Allan deviation of the data and dividing it by 2 , assuming that the two systems are independent and identical. It is displayed as green triangles in Fig. 2, after removal of the residual linear drift of the cavities (0.3 Hz/s) from the raw frequency data. The compensated link stability is 9x10 -16 at 1s and decreases as τ -1 , thus hardly contributing to the OFC stability [START_REF] Chanteau | Mid-infrared laser phase-locking to a remote near-infrared frequency reference for high-precision molecular spectroscopy[END_REF] .

OsO 4 spectroscopy. Saturated absorption spectroscopy is performed inside a FP cavity to increase the SNR [START_REF] Acef | Metrological properties of CO 2 /OsO 4 optical frequency standard[END_REF][START_REF] Bernard | CO 2 laser stabilization to 0.1-Hz level using external electrooptic modulation[END_REF] . The molecular signal is detected in transmission of the cavity. For detection purpose, we apply two frequency modulations to the QCL. The first modulation at 18 kHz with a deviation of about 90 kHz allows the FP cavity to be locked onto the laser using first-harmonic detection. The correction is applied to a PZT on which is mounted one of the FP cavity mirror. The second modulation at 9.5 kHz with a deviation of 36 kHz is used to detect the third harmonic of the saturated absorption signal with a lock-in amplifier. The choice of modulation parameters results from a trade-off between an increase in the SNR and a broadening of the molecular absorption line. Typical experimental parameters are a pressure of 0.04 Pa (corresponding to an absorption of 50 % detected in transmission of the cavity), a power of 50 µW inside the cavity and a beam diameter of 6 mm. The QCL frequency is swept by scanning the frequency of the RF oscillator used to phase lock the QCL onto the OFC.

Each spectrum is recorded by scanning the frequency up and down in turns in order to avoid any line shift induced by the sweeping [START_REF] Rohart | Absorption line shape recovery beyond the detection bandwidth limit: application to the precision spectroscopic measurement of the Boltzmann constant[END_REF] . The spectrum in the inset of Fig. 5 is the result of 10 up and down scans of 200 steps of 1 kHz with a recording time constant of 5 ms per point and 100 ms between points. The total recording time is thus 400 s. Each line frequency is obtained as the weighted mean of the line centres determined from each measurement. The uncertainty on one measurement, given by the nonlinear regression, is a conservative value since the fitting profile does not perfectly match the experimental line shape (as shown by the small remaining residuals). The uncertainty of the mean is evaluated as the experimental standard uncertainty (calculated from the uncertainties of each single measurement). It gives a conservative value of the error since it is typically higher than the standard deviation of the mean. The present set-up allows scanning continuously over about 100 MHz with a limit given by f rep /2. To go beyond, it is possible to directly sweep the OFC repetition rate while keeping its spectral purity. Currently, the repetition rate is in fact locked to a local laser diode, itself phase-locked to the NIR reference (see above). By using a wideband electro-optic modulator (EOM), one can generate a sideband to this laser diode and phase-lock it onto the NIR reference. The laser diode frequency, and in turn the OFC repetition rate and the QCL

Figure 5

 5 Figure 5 shows two OsO 4 lines in a spectrum spanning 5 MHz in the vicinity of the

Figure 1 :

 1 Figure 1: Experimental setup. The NIR frequency reference ν ref generated at LNE-SYRTE

Figure 2 :

 2 Figure 2: Fractional frequency stability of (a) the beat note between the QCL phase-locked

Figure 3 .

 3 Figure 3. Frequency noise Power Spectral Density (PSD) of (a) the beat note between the

Figure 4 .

 4 Figure 4. QCL line shape and beat note with the MIR reference. a) Calculated QCL line

Acknowledgements

We are very grateful to Paul-Eric Pottie, Giorgio Santarelli, Rodolphe Le Targat and Won-Kyu Lee for fruitful discussions as well as helping us with the US NIR lasers or the active compensated optical link. We also thank Sean Tokunaga for carefully reading the manuscript and for helping us with line width calculations. The authors acknowledge financial support from CNRS, Agence Nationale de la Recherche (ANR BLANC LIOM 2011 BS04 009 01, ANR QUIGARDE 2012 ANR-12-ASTR-0028-03 and ANR NCPCHEM 2010 BLAN 724 3), Labex First-TF (ANR 10 LABX 48 01), AS GRAM and Université Paris 13.

Methods

The quantum cascade laser. We use a commercial device from Alpes Laser. It can be tuned from 967 to 973 cm -1 (corresponding to a tuning range of 60 nm or 180 GHz) by varying the temperature (between 283 K and 243 K) and the driving current. Cooling is ensured by a chilled water circuit and a Peltier module driven by a commercial temperature controller. The threshold current is 570 mA. The QCL can be operated at up to 870 mA and exhibits a maximum output power of 40 mW. It is driven by a homemade low-noise current source resulting in a negligible contribution of the current driver to the free-running frequency noise (below 40 Hz/√Hz down to 1 kHz) [START_REF] Sow | A widely tunable 10-μm quantum cascade laser phase-locked to a state-ofthe-art mid-infrared reference for precision molecular spectroscopy[END_REF] . The frequency noise of the free-running QCL is shown in Fig. 3 (black curve e). It is about 10 10 Hz 2 /Hz at 1 Hz and has approximately a 1/f slope up to 250 kHz, followed by a steeper slope for higher Fourier frequencies.

LPL OFC stabilisation to a NIR reference. The NIR frequency reference signal is generated at LNE-SYRTE laboratory. This signal is provided by a 1.54 µm cw laser whose frequency is locked onto an US cavity with the Pound-Drever-Hall method [START_REF] Drever | Laser phase and frequency stabilization using an optical resonator[END_REF] . The relative stability of this laser (US laser #1) is approximately 2x10 -15 at 1 s averaging time [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF] . This stability is transferred to the repetition rate of a local OFC (SYRTE-OFC), by phase-locking one tooth of the comb onto the US laser signal. This enables us to measure its absolute frequency by comparison against a primary frequency standard steered H-maser of LNE-SYRTE [START_REF] Guéna | Progress in atomic fountains at LNE-SYRTE[END_REF] . Furthermore, the cw US laser frequency is shifted by an acousto-optic modulator (whose frequency is controlled by the SYRTE-OFC measurement system) so as to remove the slow frequency drift of the US cavity. This leads to an US NIR frequency reference traceable to primary standards (H-maser) with a 10 -14 uncertainty after 100 s. This reference signal, of frequency ν ref , is then transmitted to LPL through a 43-km long optical fibre link. An active frequency can be swept by scanning the sideband frequency. This way scanning the QCL over 1 GHz is achievable.

Author contributions

BA, BC, OL, YLC, AAK conceived and designed the experiment. BA, BC, DN, BD, AAK performed the experiment. BA, OL, DN, MA, CD, BD, YLC, AAK analysed the data. BA, MA, BD, YLC, AAK contributed materials/analysis tools. All authors discussed the results. BA, OL, DN, MA, CC, CD, BD, YLC, AAK wrote the paper.

Competing Financial Interests statement

The authors declare no competing financial interests.

Additional information

Correspondence and requests for materials should be addressed to B.A. or A.A.K. [START_REF] Acef | Accurate determination of OsO 4 absolute frequency grid at 28/29 THz[END_REF] . In the second column we report the absolute frequencies calculated from references [START_REF] Acef | Accurate determination of OsO 4 absolute frequency grid at 28/29 THz[END_REF][START_REF] Chardonnet | Spectroscopie de saturation de hautes précision et sensibilité en champ laser fort[END_REF] with 1σ-uncertainty. The third column displays the results of this work, where the uncertainty is the experimental standard uncertainty of the mean. The R(46)A 1 3 (-) line has previously been recorded at lower pressure [START_REF] Chardonnet | Hyperfine interactions in the n 3 band of osmium tetroxide: accurate determination of the spin-rotation constant by crossover resonance spectroscopy[END_REF] . Our measurement is thus expected to be pressure-shifted by approximately +10

Hz [START_REF] Acef | Metrological properties of CO 2 /OsO 4 optical frequency standard[END_REF] .


John W Perram

Anton Shiriaev

Carlos Canudas De Wit

Frédéric Grognard

EXPLICIT FORMULA FOR A GENERAL INTEGRAL OF MOTION FOR A CLASS OF MECHANICAL SYSTEMS SUBJECT TO HOLONOMIC CONSTRAINT 1

Keywords: Mechanical systems, conserved quantities, holonomic constraints, zero dynamics

The paper suggests an explicit form of a general integral of motion for the dynamics of a mechanical system with n degrees of freedom, while the system is subject to n -1 holonomic constraints. The computation of this integral is given for the cart-pendulum with three different holonomic constraints.

INTRODUCTION AND MOTIVATING EXAMPLE

This paper suggests an explicit formula of a general integral for the dynamical system α(q)q + β(q) q2 + γ(q) = 0 (1)

where q ∈ R 1 ; α(q), β(q) and γ(q) are smooth scalar functions.

Exploring the properties of system (1) is mainly motivated by the fact that any n-degree of freedom mechanical system with (n -1) holonomic 1 The work has been supported by the European Commission via the Nonlinear and Adaptive Control network (NACO2) and the Danish Technical Research Council, the grant 26-01-0164 2 The corresponding author constraints, can be rewritten in the form (1).

From the point of view of controlling a mechanical system, such holonomic constraints could be seen as 'artificial'; they are forced by an appropriate feedback strategy.

To clarify the last point, consider the ubiquitous cart-pendulum system M (q)q + C(q, q) q + G(q) = τ, (

where q = [x, θ] T ∈ R 1 × S 1 , x is the horizontal displacement of the cart, θ is the angle between the pendulum rod and the vertical which is zero at the upright position;

M (q) = M + m ml • cos θ ml • cos θ ml 2 , C(q, q) = 0 -ml • sin θ • θ 0 0 , m, M are the masses of the pendulum and the cart respectively; l is the length of the rod;

G(q) = 0 -mgl • sin θ , τ = f 0
Fig. 1. The cart pendulum system Suppose that the external control variable f is chosen such that the center of mass of the pendulum is forced to belong to a given vertical line x * = a: the variables x and θ of the system (2) have to satisfy the constraint

x + l • sin θ = a (3)
Taking the first and second time derivatives of (3), one gets two new relations

ẋ + l • cos θ • θ = 0 (4) ẍ -l • sin θ • θ2 + l • cos θ • θ = 0 (5)
As one can expect, the dynamics of (2) subject to the constraint (3) result in a system with 1 degree of freedom. To get its expression, one can substitute the relations (3)-( 5) into the second equation of (2), which is

ml • cos θ • ẍ + ml 2 • θ -mgl • sin θ = 0 (6)
This leads to the following equation with respect to the remaining variable θ

ml 2 •sin 2 θ• θ+ml 2 •cos θ•sin θ• θ2 - -mgl • sin θ = 0 (7)
System (7) has the same structure as the general equation ( 1) with the parameters

α(θ) = ml 2 • sin 2 θ, β(θ) = ml 2 • cos θ • sin θ, γ(θ) = -mgl • sin θ
It will be shown in the next section that, for system (1), there exists a nontrivial function I, which preserves its value along any solution of (1). It will be written explicitly, and this is the main contribution of the paper. The orbital stabilization of cycles in the zero-dynamics and in the original system are then discussed as direct consequences of this development. Afterwards two illustrative examples are given.

MAIN RESULT

Theorem 1. Let q(t), q(t) be the solution of system (1) with given initial conditions q 0 , q0 . The function

I q, q, q 0 , q0 = q2 -exp    -2 q q 0 β(τ ) α(τ ) dτ    q2 0 + + exp    -2 q q0 β(τ ) α(τ ) dτ    × (8) × q q 0 exp    2 s q 0 β(τ ) α(τ ) dτ    2γ(s) α(s) ds
is finite and preserves its value along the solution q(t), q(t) .

Proof. Observe that for the solution q(t), q(t) the following identity

d 2 dt 2 q(t) = d dq 1 2 q2 (t) (9)
holds. Indeed,

d 2 dt 2 q(t) = d dt q(t) = ∂ ∂q q(t) • d dt q(t) = ∂ ∂q q(t) • q(t) = d dq 1 2 q2 (t)
Introducing the new variable

Y = q2 (t)
and taking (9) into account, one can rewrite the differential equation (1) in the equivalent form

α(q) 1 2 d dq Y + β(q) Y + γ(q) = 0 (10)
This is a linear equation with respect to function Y and with q (instead of t) as independent variable.

Let us first consider the case where, along the solution q(t), q(t) , α(q) is separated from zero.

Under this assumption, one can divide equation (10) by α(q) and consider the equation

d dq Y + 2β(q) α(q) Y + 2γ(q) α(q) = 0 (11)
Its general solution looks as follows

Y (q) = exp -2 q q0 β(τ ) α(τ ) dτ •Y (q 0 )- -exp    -2 q q 0 β(τ ) α(τ ) dτ    × (12) × q q 0 exp    2 s q 0 β(τ ) α(τ ) dτ    2γ(s) α(

s) ds

Introducing function I as

I(q, q, q 0 , q0 ) = Y (q) -exp    -2 q q0 β(τ ) α(τ ) dτ    • Y (q 0 ) + + exp    -2 q q 0 β(τ ) α(τ ) dτ    × × q q0 exp    2 s q0 β(τ ) α(τ ) dτ    2γ(s) α(s) ds
results in I = 0 along the solution q(t), q(t) .

This proves the case when α(q) is separated from zero along the solution q(t), q(t) .

Suppose now that there exists a finite time moment T * such that:

(a) the solution q(t), q(t) is well defined and remains continuous on the interval T * -ε, T * + ε for some ε > 0; (b) the function α(q) equals to zero at q = q(T * ), that is, α (q(T * )) = 0.

Under these assumptions one cannot write the descriptor equation ( 10) in the form (11) along the solution q(t), q(t) for the whole time interval

T * -ε, T + ε . Rather one should consider two equations α(q)q + β(q) q2 + γ(q) = 0, t ∈ T * -ε, T * (13) α(q)q + β(q) q2 + γ(q) = 0, t ∈ T * , T * + ε (14)
and one matching condition

lim t→T * - q(t), q(t) = lim t→T * + q(t), q(t) , ( 15
)
where t → T * -is the abbreviation for "approaching T * from the left", and t → T * + "from the right". The equality ( 15) is valid due to assumption (a) of continuity of the solution on the interval T * -ε, T + ε . This assumption also implies that q(t), q(t) remains bounded on the interval

T * -ε, T + ε .
For systems ( 13) and ( 14), one could write (12) for the subintervals T * -ε, T * and T * , T * + ε , while the matching condition (15) and boundedness of the solution provide that the function

q2 (t) = exp      -2 q(t) q0 β(τ ) α(τ ) dτ      • q2 0 - -exp      -2 q(t) q 0 β(τ ) α(τ ) dτ      × × q(t) q 0 exp    2 s q 0 β(τ ) α(τ ) dτ    2γ(s) α(s) ds
remains continuous for all t ∈ 0, T + ε . This finishes the proof.

Let us discuss this result:

(1) Function ( 8) is not the first integral of system (1). Indeed, following the definition, a first integral of a system

d dt y = f (t, y)
is a function U = U (t, y) that preserves its value along any solution y(t) of the system, see (Hartman, 1964). As shown in the Theorem, I preserves its value along the solutions, but it depends on the initial conditions [ q 0 , q0 ]. On the other hand, such a function is known as a general or full integral of the system. In turn, one can expect that, in many cases, this function leads to the explicit form of the first integral of the system. This point is illustrated in the next section, where the cart-pendulum system is presented.

(2) The knowledge of the explicit form of function (8) may be useful when the dynamics of the constraint system exhibits periodic motions for some initial conditions. Choosing 'interesting' initial conditions [ q 0d , q0d ], one can use a control Lyapunov function candidate of the form V (q, q) = 1 2 I 2 (q, q, q 0d , q0d )

for the orbital stabilization of a periodic solution corresponding to the choice [ q 0d , q0d ].

ORBITAL STABILIZATION OF CYCLES IN ZERO DYNAMICS

Suppose that, for some initial conditions [q d 0 , qd 0 ], the corresponding solution of system (1) is a cycle Γ. In order to orbitally stabilize this cycle in the controlled system

α(q)q + β(q) q2 + γ(q) = u (16)
we can proceed as follows: Choose a Lyapunov function candidate

V (q, q) = 1 2 I(q, q, q d 0 , qd 0 ) 2 (17)
Its time derivative along any solution [q(t), q(t)] of system ( 16) has the form

d dt V = I q(t), q(t), q d 0 , qd 0 ∂I ∂q q(t) + ∂I ∂ q q(t) = I × ∂I ∂q q(t)+ + ∂I ∂ q - β(q(t)) α(q(t)) q2 (t) - γ(q(t)) α(q(t)) + ∂I ∂ q u α(q(t)) = -k I q(t), q(t), q d 0 , qd 0 × q(t) 2 ≤ 0 ( 18
)
where the controller u = u good is defined to force the last equality:

u good = - k 2 I q(t), q(t), q d 0 , qd 0 q(t)α(q(t))+ (19) +β(q(t))      q2 (t) -exp   -2 q(t) q d 0 β(τ ) α(τ ) dτ    × ×   ( qd 0 ) 2 - q(t) q d 0 exp      2 s q d 0 β(τ ) α(τ ) dτ      2γ(s) α(s) ds        
with some positive parameter k.

Under appropriate conditions, the following statement is true: Consider system (1) and one of its cycles Γ. Choose any point [q d 0 , qd 0 ] ∈ Γ Then Γ is an orbitally asymptotically stable cycle of the closed loop system ( 16), ( 19).

ORBITAL STABILIZATION OF CYCLES FOR THE ORIGINAL SYSTEM

In the previous section a procedure for the stabilization of a cycle in the zero dynamics of the system has been proposed. In fact, controller ( 19) is not defined in the case where the system does not satisfy the holonomic constraint. Suppose that the dynamics of the original mechanical system can be rewritten as follows

q = - β(q) α(q) q2 - γ(q) α(q) + G(q, q, y, ẏ, v) α(q) (20) ÿ = -k 1 y -k 2 ẏ + v ( 21
)
where v is a control variable, and

G(q, q, y, ẏ, v) = 0, if y = ẏ = v = 0
where y designates the configuration variables not present in the state of the zero-dynamics (possibly after some feedback transformation).

Suppose again that we are given a cycle Γ of subsystem (20) (characterized by some [q d 0 , qd 0 ]), and that we are asked to orbitally stabilize this cycle. To proceed we can utilize the back-stepping idea: Consider a Lyapunov function candidate

W (q, q, y, ẏ) = 1 2 I(q, q, q d 0 , qd 0 ) 2 + (22) + ε 2 G(q, q, y, ẏ, v) -u good 2
where u good is defined in (19). Taking the time derivative of W along any solution of system ( 20)-( 21), we get

d dt W = ε G(q, q, y, ẏ, v) -u good × × d dt G(q, q, y, ẏ, v) -u good + I q, q, q d 0 , qd 0 × × ∂I ∂q q(t) - ∂I ∂ q β(q) α(q) q2 + γ(q) α(q) + ∂I ∂ q G α(q) = ε G -u good × d dt G -u good - -k I q, q, q d 0 , qd 0 × q 2 + +I q, q, q d 0 , qd 0 × 2 q × G -u good α(q(t)) = -k I q, q, q d 0 , qd 0 × q 2 + G -u good × (23) × ε d dt G -ε d dt u good + I q, q, q d 0 , qd 0 × 2 q α(q)
If we choose the control variable v as a solution of the equation

ε d dt G(q, q, y, ẏ, v)-ε d dt u good + +I q, q, q d 0 , qd 0 × 2 q α(q) = (24) = -ρ G(q, q, y, ẏ, v) -u good
with some ρ > 0, then the time derivative of W takes the form

d dt W = -k I q, q, q d 0 , qd 0 × q 2 - ( 25
) -ρ G(q, q, y, ẏ, v) -u good 2
The last equality (25) implies that the function W is not increasing along the solution of the closed lop system ( 20), ( 21), ( 24), ( 19). The function W may not be proper even locally around the cycle Γ. Therefore, convergence of solutions of the closed loop system to the cycle Γ remains unproven in general (while it could be proven in some examples).

EXAMPLE: THE CART-PENDULUM

In this section, we analyze the effect of different holonomic constraints on the zero-dynamics and the general integral of the cart-pendulum.

The Center of Mass of the Pendulum Belongs to a Given Vertical

This example has been discussed in the Introduction, and the constrained dynamics of the system is covered by equation ( 7). Straightforward calculations, based on formula (8), result in the following expression

I(θ, θ, θ 0 , θ0 ) = θ2 - sin θ 0 sin θ 2 θ2 0 + (26) + 2g l • sin 2 θ cos θ -cos θ 0
Theorem 1 implies that this function remains zero along the solution θ(t), θ(t) that starts at θ 0 , θ0 . Therefore, function

sin 2 θ•I(θ, θ, θ 0 , θ0 ) = sin 2 θ• θ2 - (27) -sin 2 θ 0 • θ2 0 + 2g l cos θ -cos θ 0
will also be zero along the solution. Introducing a function

U (θ, θ) = sin 2 θ • θ2 + 2g l cos θ one gets sin 2 θ • I(θ, θ, θ 0 , θ0 ) = U (θ, θ) -U (θ 0 , θ0 )
From the last relation one can conclude that U is a first integral of system (7).

A Fixed Point on the Pendulum's Rod Belongs to a Given Vertical

To extend the previous example, consider again the constraint

x + L u • sin θ = a (28)
where L u is a fixed parameter (possibly different from the distance l to the center of mass of the pendulum). Taking the first and second timederivatives of (28), one gets

ẋ + L u • cos θ • θ = 0 (29) ẍ -L u • sin θ • θ2 + L u • cos θ • θ = 0 (30)
The substitution of ẍ from the second relation into (6) results in the zero-dynamics

l -L u • cos 2 θ • θ+ (31) + L u • cos θ • sin θ • θ2 -g • sin θ = 0
The straightforward calculation of formula (8) results in the following expression

I(θ, θ, θ 0 , θ0 ) = θ2 - l -L u • cos 2 θ 0 l -L u • cos 2 θ θ2 0 - (32) - 2g l -L u cos 2 θ θ θ0 sign l -L u cos 2 s sin sds
If one assumes that along the solution θ(t), θ(t) the function

l -L u • cos 2 θ(t) > 0
then, along this solution, the formula (32) could be rewritten as

I(θ, θ, θ 0 , θ0 ) = θ2 - l -L u • cos 2 θ 0 l -L u • cos 2 θ θ2 0 + (33) + 2g cos θ 0 -cos θ l -L u • cos 2 θ
Theorem 1 indicates that this function remains equal to zero along the solution. Therefore, the function

l -L u • cos 2 θ •I(θ, θ, θ 0 , θ0 ) = = l -L u • cos 2 θ θ2 + 2g cos θ U (θ, θ) - -l -L u • cos 2 θ 0 θ2 0 + 2g cos θ 0 = U (θ, θ) -U (θ 0 , θ0 ) (34)
will also be zero along the solution, while U is the first integral. It is interesting to observe that, if the parameter L u is chosen as

L u > l
then the upright equilibrium of system ( 31) is neutrally stable, and the system possesses stable cycles around the upright equilibrium.

Constraints without Physical Interpretation

In the previous examples, the computation of general integrals of the system can easily be transfered to corresponding first integrals. This is not the case in general. To observe this consider the constraint

x + L u • sin θ = ε • sin(kθ), (35)
which is a modified version of (28) with new parameters ε and k. Taking the first and second time-derivatives of (28), one gets

ẋ + L u • cos θ • θ = ε • k • cos(kθ) • θ ẍ -L u • sin θ • θ2 + L u • cos θ • θ = ε • k • cos(kθ) • θ -ε • k 2 sin(kθ) • θ2
Substitution of ẍ from the second relation into (6) results in the dynamics

l + ε • k • cos(kθ) • cos θ -L u • cos 2 θ • θ+ + L u • cos θ • sin θ -ε • k 2 • sin(kθ) • cos θ • θ2 - -g • sin θ = 0 (36)
For this system the function I in (8) looks as follows

I θ, θ, θ 0 , θ0 = θ2 - -exp    -2 θ θ 0 L u cos τ sin τ -εk 2 sin(kτ ) cos τ l + εk cos(kτ ) cos τ -L u cos 2 τ dτ    θ2 0 + + exp    -2 θ θ 0 L u cos τ sin τ -εk 2 sin(kτ ) cos τ l + εk cos(kτ ) cos τ -L u cos 2 τ dτ    × × θ θ0 exp    2 s θ0
L u cos τ sin τ -εk 2 sin(kτ ) cos τ l + εk cos(kτ ) cos τ -L u cos 2 τ dτ    × × -2g • sin s l + εk cos(ks) cos s -L u cos 2 s ds

The direct integration inside the last formula may be difficult for different values of parameters L u , ε, k, and the form of the corresponding first integral, if it exists, is not straightforward, as it was in the previous examples. Nevertheless, this expression could be used for analysis and control of the system.

CONCLUSIONS

This paper is aimed at a development of a new method for construction and local orbital stabilization of cycles in mechanical systems. The main idea comes from the fact that we were able to explicitly build a a general integral of motion for any mechanical system with configuration space of dimension n with (n -1) holonomic constraints.

The dynamics of such a system could be reduced to a system with one degree of freedom. If such a system has a cycle, then the explicit form of its general integral of motion enables us to describe this cycle as a zero-level set of the square of this integral function, and use it as a Lyapunov function for deriving a controller and checking orbital asymptotic stability of this cycle in the closed loop system.

For the stabilization of this cycle in the original n-degree of freedom mechanical system, an algorithm based on the extension of the back-stepping idea, has been suggested.


Y Chitour
email: yacine.chitour@math.u-psud.fr

F Grognard

G Bastin
email: bastin@auto.ucl.ac.be

Stability analysis of a metabolic model with sequential feedback inhibition

This paper deals with the stability analysis of a simple metabolic system with feedback inhibition. The system is a sequence of monomolecular enzymatic reactions. The last metabolite acts as a feedback regulator for the rst enzyme of the pathway. The enzymatic reactions of the pathway satisfy Michaelis-Menten kinetics. The inhibition is described byanhyperbolic model. Without inhibition, it is clear that the system is cooperative and has a single globally asymptotically stable equilibrium. But, in the common situation where there is inhibition, the system is no longer cooperative and the stability analysis is more intricate. In this paper we exhibit su cient conditions on the kinetic parameters in order to guarantee that this simple metabolic system with inhibition still has a single globally asymptotically stable equilibrium.

Introduction

The huge set of biochemical reactions which occur inside living cells is called the Cellular Metabolism. It is usually represented byanintricate network connecting the involved biochemical species (called "metabolites"). The pathways of the network are called "metabolic pathways". In the metabolic engineering literature, it is widely accepted that "despite their immense complexity, metabolic systems are characterized by their ability to reach stable steady states" (quoted from 6], Chapter 4). It should however be fair torecognize that a mathematical analysis of this fundamental stability propertyisa di cult question whichwas not muchinvestigated. Our objective in this paper is to provide a modest contribution to this question. We shall limit ourselves to simple metabolic pathways which are made up of a sequence of mono-molecular enzyme-catalysed reactions as where X i (i =1 n) represent the successive metabolites of the pathway:

X 1 ! X 2 ! !X n :
Atypical situation is when such a simple pathway is located between two branchpoints of a complex metabolic network. We shall consider the case of a so-called sequential feedback inhibition (cf. 6])

where the last metabolite X n acts as an inhibitor of the rst reaction X 1 ;! X 2 . This inhibition is represented by the dotted feedbackarrow in Fig. 1. The velocityofeach enzymatic reaction X i ;! X i+1 is represented by a Michaelis-Menten kinetic function :

' i (x i )= a i x i k i + x i (1. 1)
where x i denotes the intracellular molar fraction of the metabolite X i , a i is the maximal velocity and k i the so-called half-saturation constant. It is assumed that the velocity of the rst reaction X 1 ;! X 2 is inhibited by the last metabolite with a multiplicativehyperbolic inhibition function of the form:

(x n )= 1 1+ x n (1. 2)
In addition, it is assumed that the cell metabolism is analysed during a period of exponential cell growth with a constant speci c growth rate . Under these assumptions and notations, a mass balance dynamical model is formulated as:

( ) 8 > > > > < > > > > : _ x 1 = ; a 1 x 1 (k 1 +x 1 )(1+ xn) ; x 1 + c _ x 2 = a 1 x 1 (k 1 +x 1 )(1+ xn) ; a 2 x 2 (k 2 +x 2 ) ; x 2 _ x i = a i;1 x i;1 (k i;1 +x i;1 ) ; a i x i (k i +x i ) ; x i 3 i n (1. 3)
where n 3 is a positivei n teger, x =(x 1

x n ) T 2 IR n , and all the a i k i 's, c are positive constants.

In this model c denotes the in ow rate of the rst metabolite X 1 of the sequence and is assumed to be constant. Without inhibition (i.e. = 0), the system is clearly compartmental and cooperative which implies that it has a single globally asymptotically stable equilibrium. But if there is inhibition, the system is no longer cooperative and the stability analysis is more di cult. Our contribution in this paper will be to exhibit su cient conditions on the kinetic parameters that guarantee that the simple metabolic system with feedback inhibition still has a single globally asymptotically stable equilibrium.

2 Notations and statement of the theorem 2.1 Notations Consider the metabolic system ( ) with feedback inhibition (1. 3). Up to a change of variable (the x i 's are multiplied by ) and a time reparameterization (the time is multiplied by ), wemay assume that = = 1 and the model is rewritten:

( ) 8 > > > > < > > > > : _ x 1 = ;' 1 (x 1 ) (x n ) ; x 1 + c _ x 2 = ' 1 (x 1 ) (x n ) ; ' 2 (x 2 ) ; x 2 _ x i = ' i;1 (x i;1 ) ; ' i (x i ) ; x i 3 i n (2. 1)
where ' i is de ned in (1. 1) and := 1 in (1. 2). In this model, c stands for c (with the original c in that last formula) and similarly, for 1 i n, a i stands for a i and k i for k i .

Weintroduce some notations: for 2 i n, f i (x)=x + ' i (x)andf 1 xn (x)=x + (x n )' 1 (x). It is clear that the ' i 's, 1 i n, are strictly monotone functions on IR + and realize bijections between IR + and 0 a i ). We use ' ;1 i to denote the inverse function. For 2 i n,thef i 's are strictly monotone functions on IR + and realize bijections from IR + to IR + f ;1 i denotes the inverse function of f i and g i = ' i f ;1 i . Let M : IR + ! IR + by M = f ;1 n g n;1 g 2 . For every x 0 and 2 i n,wehave

1 <f 0 i (x)=1+' 0 i (x) 1+ a i k i k i a i + k i (f ;1 i ) 0 (x)= 1 1+' 0 i (f ;1 i (x)) < 1: (2. 2)
Note that the f i 's, the ' i 's are concave functions on IR + (negative second derivative) for 2 i n. i (x)= ' 0

i (f ;1 i (x)) 1+' 0 i (f ;1 i (x)) g 0 i (0) = a i k i + a i (2. 3) and since M 0 (x)=(f ;1 i ) 0 (g n;1 g 2 (x)) " n;1 Y i=3 g 0 i (g i;1 g 2 (x)) # g 0 2 (x)
we can then conclude from (2. 2) and (2. 3) that, for every x 0 0 <M 0 (x) < n;1 Y i=2 a i k i + a i :

(2. 4)

As for f 1 xn , for every x n 2 IR + , it behaves likea n y f i ,2 i n. De ne z : IR + ! 0 c)b y z(b)=f ;1 1 b (c). Later we will study in more details that application. Let F be the vector eld on IR n simply de ned by the right-hand side of ( ). Let K = IR n + the non-negative orthantandK + the positive orthant. The positive cone K de nes a closed partial order relation on IR n de ned by x y if and only if y ; x 2 K. It means that x i y i holds for every 1 i n.W ewritex<yif x y and x 6 = y, and x y whenever y ; x 2 Int(K)=K + . This notation extends trivially to subsets of IR n . Moreover, if x y, then the set P x y = fz 2 IR n x z yg is a parallellepiped. Let v 2 K + de ned by v =(1 1).

If f : IR + ! IR,setlimf = lim sup t!1 f and limf = lim inf t!1 f. This notation is naturally extended to the vectorial case using the partial order de ned previously.W e will also consider sometimes the function V : IR n ! IR de ned by V (x)= P n i=1 x i .L e tT d , d>0, be the simplex of K de ned as the the set of x 2 K so that V (x) d.

If x y 2 X, then x y] denotes the segment with extremities x and y,i . e . the set of points tx +(1; t)y for t 2 0 1]. A set X IR n is said to be p-convex if for every x y 2 X with x y then x y] X. Let m be a positiveinteger. An m m matrix A =(a ij ) is said to be irreductible if for every nonempty, proper subset I f1 ng, there is an i 2 I and j 2f1 ng=I such that a ij 6 = 0. There is a graph-theoretic formulation of irreducibility(cf. 5 ]): consider the directed graph G whose set of vertices is f1 ng twovertices i j have a directed edge from i to j if a ij 6 = 0. Then A is irreducible if its directed graph G is connected.

A dynamical system (G)given by_ x = G(x), x 2 D with D open, G : D ! IR n of class C 1 is said to be cooperative (see 5]) if, for every x 2 D,1 i j n and i 6 = j, @G i @x j 0:

If in addition the jacobian matrix DG(x) is irreducible for every x 2 D, then (G) is said to be irreducible cooperative.

Remark 2. 1 It is worth noticing that ( ) is not cooperative with respect to the partial order de ned by K. It should be pointed out that the coe cient in the jacobian DF (x) that renders ( ) non cooperativeis @f 2 @xn , which is indeed negative. We will consider auxiliary systems ( ) b , b 0, given by

( ) b 8 > > > > < > > > > : _ x 1 = ;' 1 (x 1 ) (x n ) ; x 1 + c _ x 2 = ' 1 (x 1 ) (b) ; ' 2 (x 2 ) ; x 2 _ x k = ' k;1 (x k;1 ) ; ' k (x k ) ; x k 3 k n (2. 5)
where the di erence with ( ) lies in the equation de ning _

x 2 : the variable x n is frozen at the constant value b.W euseF b (x) to denote the right-hand side of ( ) b .N o w, the (2 n)-coe cientinDF b (x)is identically equal to zero. If x 2 K,w euse x , b

x respectively, to denote the trajectory of ( ), ( ) b respectively, which starts at x.

Preliminary results and statement of the theorem

Proposition 2. 2 The system ( ) has the following properties:

(1) ( ) has a unique equilibrium point x 2 K +

(2) For every x 2 K and every t>0, x (t) 2 K + i.e. K is a positively invariant set for ( ) (3) For every x 2 K, lim V ( x ) c i.e. T c is a global attractor of all the trajectories starting in K.

Remark 2. 3 Eventhough ( ) is not cooperative, it has some of the basic features that are required for the investigation of the !-limit sets of cooperative systems: an invariant cone with a repelling boundary, a bounded attractor and a unique equilibrium point.

Proof of Proposition 2. 2: we start with the argument of (1). Let x 2 K an equilibrium pointof ( ), i.e. F ( x) = 0. For 3 k n,w ehave x k = f ;1 k ' k;1 ( x k;1 ). In addition, we deduce from c ; x 1 = '( x 1 ) ( x n )=f 2 ( x 2 ) that x 2 = f ;1 2 (c ; x 1 ). Then x n = M (c ; x 1 ). Therefore, since M is strictly increasing, ( ) admits as many equilibrium points in K as solutions in 0 c] of the equation c = x + ' 1 (x) (M(c ; x)):

(2. 6) De ne J : 0 c] ! IR + by J(x)=x + ' 1 (x) (M(c ; x)). Since is strictly decreasing, so is J.

Moreover J(0) = 0 and J(c)=c + ' 1 (c) >c . Then J takes the value c exactly once, which implies that equation (2. 6) has exactly one solution in (0 c). Claim (1) is proved.

To establish (2), it is enough to show that @K, the boundary of K is repelling for ( ). Let y =(y 1 y n ) T 2 @K and x = y . Let i 0 =maxfi y i =0g.I fi 0 =1,then _

x 1 (0) > 0 and x 1 > 0 in a neighborhood of t =0andwe are done.

Assume now that i 0 > 1. If y i 0 ;1 > 0, then _

x i 0 (0) > 0 and the result follows. If not, then y i 0 ;1 =_ x i 0 (0) = 0. The next alternativeiswhetheri 0 ; 1 = 1 or not. If i 0 ; 1 = 1, then _

x i 0 ;1 (0) > 0 and x i 0 (0) > 0 again the conclusion holds in this case. If i 0 ; 1 > 1, weha ve a new alternative, whether y i 0 ;2 > 0 or not. If y i 0 ;2 > 0, then _ x i 0 ;1 (0) > 0, case already studied. If y i 0 ;2 =0,then y i 0 ;l = 0 for l =0 1 2. Proceeding in a similar manner, one can show that the last case to be treated corresponds to y i 0 ;l = 0 for l =0 i 0 . It implies that x (l) i 0 =0forl =0 i 0 ; 1. Since, anyhow _ x 1 (0) > 0, wemust have_ x (i 0 ) i 0 > 0 and we are done. Along trajectories of ( ), the derivativeofV veri es _ V = c ; V ; ' n (x n ):

(2. 7) Therefore, if x = x (0) 2 K and taking into account (2), then ' n ( x n (t)) 0. Then, _ V c ; V , which clearly implies (3).

Remark 2. 4 At the light of Proposition 2. 2, the relevance of the auxiliary systems ( ) b for understanding the dynamics of ( ) can be put forward. It is based on the twofollowing remarks:

(a) for every b 0, ( ) b is an irreducible cooperative system (use the graph-theoretic formulation of irreducibility). This easily implies that ( ) b veri es (2) (cf. Theorem 1:1 p.56 of 5]) and (3) (with possibly another positive constant instead of c). In fact ( ) b is a hypercycle for whicha Poincar e-Bendixon theory was developped for the compact !-limit sets of ( ) b , regardless of the dimension of the system(cf. 4]). Then, we expect taking advantage of the many deep results relative to that class of irreducible cooperative systems (for an excellent reference, cf 5]).

(b) for every x 2 K and 0 b 0 <b 1 ,wehave b 0 x n b 1 ) F b 1 (x) F (x) F b 0 (x)

(2. 8) and, if x 1 > 0, then can be replaced everywhere by < in the above equation. The monotonicity property expressed in (2. 8) translates to the trajectories of F and F b as explained next. Assume that wehaveshown the existence of 0 b 0 <b 1 such that for every x 2 K, there is some t x > 0 for which (2. 10)

Assume now that, according to Part (a),the!-limit sets of ( ) b 0 and ( ) b 1 are investigated in details and one is able to show e.g. that every trajectory of ( ) b 0 (( ) b 1 respectively) starting in K converges to x b 0 (x b 1 respectively). In addition, assume that b 0 x b 0 and x b 1 b 1 . Then the pair (x b 0 n x b 1 n ) can be used in (2. 9) instead of (b 0 b 1 ) in the bounding process for x (t) described above. If that procedure can be reproduced, one may hope to get more and more precise information on the !-limit sets of ( ).I ti se v en tempting to conjecture that every trajectory of ( ) starting in K converges to x.W e prove it but for a restricted set of the problem's parameters.

Theorem 2. 5 Under the following condition (C), (C)( a 1 + c) (2. 12)

It is not surprising that if =0(i.e. no inhibition) or if is large enough then the condition expressed in (2. 12) holds true.

3 Proof of Theorem 2. 5

Technical lemmas

Before starting the proof of the theorem, we establish a series of useful lemmas. We rst study the

application z : IR + ! 0 c)by z(b)=f ;1 1 b (c). Wehave Lemma 3. 1 (i)
The application z is strictly increasing from IR + to 0 c) (ii) The application ' 1 z is strictly increasing from IR + to 0 ' 1 (c)) and is concave.

Proof of Lemma 3. 1: The expression of z(y) comes from the equation _ x 1 = 0. Di erentiating this equation with respect to y results in an expression z 0 (y)=l(z(y) 1 1+y ) for some function l.W ethen isolate 1 1+y in the equation _ x 1 = 0 and substitute the resulting expression into l, so that z 0 (y)= (c ; z(y)) 2 (z(y)+k 1 ) 2 a 1 (z(y) 2 + ck 1 ) (3. 1) and z 0 (y) > 0forally 0 because z(y) <cfor all y 0. Item (i) is proved. The expressions of d' 1 (z(y)) dy and d 2 ' 1 (z(y))

dy 2 result from straightforward computations: Proof of Lemma 3. 2: if e 2 E b , then e n = M(' 1 (e 1 ) (b)). Then (3. 4) follows by de nition of z. When considering the solutions of (3. 4), we are looking for the zeros of the function h b : IR + ! IR, de ned by h b (y)=y ; M ' 1 (z(y)) (b) : Note that h b (0) < 0a n dh b tends to +1 if y tends to +1 (indeed, ' 1 (z(y)) (b) is bounded). Therefore, if h b is strictly increasing, then E b will have a unique equilibrium point. We show next that this holds true under condition (C).

d' 1 (z(y)) dy = ' 0 1 (z)z 0 (y)= k 1 (c ; z(y)) 2 z(y) 2 + ck 1 (3. 2)
The derivativeofh b is

h 0 b (y)=1; M 0 ' 1 (z(y)) (b) d' 1 (z(y)) dy (b) (3. 5)
Equation (2. 4) gives an upper bound on M 0 and concavityof' 1 (z(y)) implies that

d' 1 (z(y)) dy (y) (b) d' 1 (z(y)) dy (0) (b):
We also see from (

We then have a unique zero to h b if

1 ; n;1 Y i=2 a i k i + a i c>0
which is satis ed if condition (C) holds. The lemma is proved.

From now on, assume that condition (C) holds. Then, for every b 0, ( ) b has a unique equilibrium point e(b)inK + . All the assumptions of Theorem 3:1of 5] are satis ed. Therefore, ( ) b is globally (with respect to initial states in K) asymptotically stable with respect to e(b).

The next lemma studies the application e : 0 c] ! K + that associates to b 2 0 c], e(b). Set e n : 0 c] ! IR + for the application that associates to b the n-th coordinate of e(b). Note that x n is a xed pointofe n . Lemma 3. 4 Assume that condition (C) holds. Then with I 0 =0and S 0 so that e(0) S 0 and, for every x 2 T c , x S 0 .H e r ee(0) is the equilibrium pointof( ) 0 . Then, for every l 0,wehave I l x S l and lim l!1

I l =l i m l!1 S l = x (3. 8)
where x is the equilibrium pointof( ).

Proof of Lemma 3. 5: we rst prove that for every l 0, I l x S l ,(I l ) is increasing and (S l ) is decreasing with respect to the partial order . The argumentgoesby induction. More precisely, we showby induction that, for every l 0, the next proposition (P l ) holds, (P l ) I l x S l I l I l+1 S l+1 S l :

(3. 9)

Clearly (P 0 ) holds. Assume now that (P l ), l 0 is true. Then Because an upper bound of M 0 is given in equation (2. 4), ' 1 is bounded aboveby a 1 ,and d'(z(y)) dy by c, then (3. 11) is implied by condition (C). Since the latter already holds true, the proof of the lemma is nished. Lemma 3. 6 Let x 2 K. Then 0 lim x Proof of Lemma 3. 6: Without loss of generality,w etake0 x (cf. Proposition 2. 2, (2)) and argue by contradiction. Then there exists 1 i n so that P i holds:

(P i ): 8">0 8t>0 9t 0 >t x i (t 0 ) <" (3. 12) where x j := ( x ) j ,1 j n. Consider the largest interval I containing t 0 so that x i (s) " for s 2 I.B yt a king" small enough, the interval I is of the type t 0 with t 0 > 0. By continuity, x i (t 0 )=" and _

x i (t 0 ) 0. Clearly, i cannot be equal to 1, otherwise, from _

x i (t 0 ) 0, wewould deduce that c a 1 ", which is impossible for " small enough.

If (P 2 ) holds, then x 2 (t 0 )=" and _ x 2 (t 0 ) 0 imply that x 1 (t 0 ) C 1 ",w ithC 1 only depending on the positive parameters a 1 a 2 k 2 k 1 . This means that (P 1 ) holds and wehaveacontradiction.

Replacing 2 byanyindexi 3 in the previous sentence indicates that if (P i ) holds, then the same is true for (P i;1 ) and by a trivial induction we again have(P 1 ). Therefore, if any(P i ) holds, we get a contradiction. The proof of Lemma 3. 6 is complete.

3.2 Final part of the proof of Theorem 2. 5

We are now ready to establish Theorem 2. [START_REF] Mallet-Paret | \The Poincar e-Bendixson theorem for monotone cyclic feedback systems[END_REF]. From what precedes, the conclusion is the consequence of the next statement: for every x 2 K and for every l 0 (Q l ) I l lim x lim x S l :

(3. 13) Fix x 2 K. Proposition (Q l )isproved inductively.F or l = 0, this is a consequence of Proposition 2.

2, (3). Applying Lemma 3. 6, there exists ">0a ndt 0 (") > 0such that for every t t 0 ("), "v

x (t) S 0 ; "v:

(3. 14) Then, passing to the limit wehave e(S 0 n ; ") lim x lim x e("):

(3. 15)

Note that, in equation (3. 14), " may be replaced byany0< ". Since e is globally Lipschitz over IR + , equation (3. 15) implies (Q 1 ) but also the existence of t 1 (") > 0suchthatforevery t t 1 ("), I 1 + C 1 "v x (t) S 1 ; C 1 "v (3. 16) for some 0 <C 1 1 independantof". Notice that equation (3. 16) is of the same type as equation (3. 14) and then leads to equations similar to (3. 15) and again (3. 16). In that way,we obtain, for every l 2, e(S l n ; C l ") lim x lim x e(I l n + C l ") (3. 17) and the existence of t l (") such that, for every t t l ("), I l + C l+1 "v x (t) S l ; C l+1 "v with C l+1 C l 1 independantof". Letting " tend to zero in (3. 17), we get (Q l ).

3 (

 3 y)=; 2k 1 c(c ; z(y))3 (z(y)+k 1 ) z(y) <c.Next, we determine, for b 2 0 c], the equilibrium set E b of ( ) b , i.e. the set of the equilibrium points of ( ) b . Lemma 3. 2 A point e 2 K is an elementofE b if and only if its n-th coordinate e n is solution in IR + of the following equation in the unknown y y = M ' 1 (z(y)) (b) : (3. 4) Moreover the previous equation has always solutions and it has exactly one if condition (C) holds. Remark 3. 3 Wechose in this paper to investigate the sets E b 's in an elementary way rather than using the deep work of 4] and 3]. Doing so leads to obtain results on E b which are only valid under condition (C) eventhough they are more general. The more complete characterization of the E b 's will appear in the nal version of the paper, see 1].

  (1) e n is a strictly decreasing function and x n is its unique xed point (2) if b 1 <b 2 , then e(b 2 ) e(b 1 ) (3) if b< x n , then b< x n <e n (b) and x e(b) similarly if x n <b, then e n (b) < x n <band e(b) x. Proof of Lemma 3. 4: De ne H(w b):=h b (w) for (y b) 2 IR 2 + . The di erentiation of H(e n (b) b1 (z(e n (b))) (b) ' 1 (z(e n (b))) 0 (b) decreasing. In addition, recall that e 1 (b)=z(e n (b)) and e k (b)=' ;1 k (f k+1 (e k+1 (b)))for 2 k n ; 1. Then (2) holds since z and the ' ;1 k f k+1 's are strictly increasing functions. Part (3) is immediate by taking into account the facts that the function e n is strictly decreasing, x n is its unique xed point and (2). Lemma 3. 5 Consider (I l ) and (S l ) the sequences of points of K de ned inductively as follows 8

  x n <S l n . By Lemma 3. 4, (3) wehave I l+1 = e(S l n ) e( x n )= x: Similarly,wehave x S l+1 . Since I l I l+1 , then I l+1 : Similarly, I l+1 I l+2 .Since the sequences (I l ) and (S l ) are monotone (componentwise) they converge to I and S with I x S. By passing to the limit in (3. 7), wehave I = e(S n ) S = e(I n ) (3. 10) which implies that I n = e n (S n )a n dS n = e n (I n ), i.e. I n and S n are xed points of e n e n . The derivative of that function is den db (e n ) den db 0. If it is smaller than one then x n is the unique xed point of e n e n on IR + and (3. 8) is proved. Finally to get den db (e n ) den db 1, it is enough to have j den db j 1. Taking into account(3. 6),itis enough that, for every b 2 0 c) M 0 ' 1 (z(e n (b))) (b) ' 1 (z(e n (b)))j 0 (b)j h 0 b (e n (b)) which in turn follows from the next inequality M 0 ' 1 (z(e n (b))) (b) (' 1 (z(e n (b))) + d'(z(y)) dy (e n (b)) 1: (3. 11)

This implies that the g i 's are also concave. Therefore, wehavefor2 i n and every x 0


Dominique Lecomte
email: dominique.lecomte@upmc.fr

Miroslav Zelen
email: zeleny@karlin.mff.cuni.cz

December

Borel chromatic number of closed graphs

Keywords: 2010 Mathematics Subject Classification. Primary: 03E15, Secondary: 54H05 Borel chromatic number, Borel class, coloring

We construct, for each countable ordinal ξ, a closed graph with Borel chromatic number two and Baire class ξ chromatic number ℵ 0 .

Introduction

The study of the Borel chromatic number of analytic graphs on Polish spaces was initiated in [K-S-T]. In particular, the authors prove in this paper that the Borel chromatic number of the graph generated by a partial Borel function has to be in {1, 2, 3, ℵ 0 }. They also provide a minimum graph G 0 of uncountable Borel chromatic number. This last result had a lot of developments. For example, B. Miller gave in [Mi] some other versions of it, which helped him to generalize a number of known dichotomy theorems in descriptive set theory. The first author generalized in [L2] the G 0 -dichotomy to any dimension making sense in classical descriptive set theory, and also used versions of G 0 to study the non-potentially closed subsets of a product of two Polish spaces (see [L1]).

A study of the ∆ 0 ξ chromatic number of analytic graphs on Polish spaces was initiated in [L-Z1] and was motivated by the G 0 -dichotomy. More precisely, let B be a Borel binary relation, on a Polish space X, having a Borel countable coloring (i.e., a Borel map c :

X → ω such that c(x) = c(y) if (x, y) ∈ B).
Is there a relation between the Borel class of B and that of the coloring? In other words, is there a map k : ω 1 \ {0} → ω 1 \ {0} such that any Π 0 ξ binary relation having a Borel countable coloring has in fact a ∆ 0 k(ξ) -measurable countable coloring, for each ξ ∈ ω 1 \{0}?

In [L-Z2], the authors give a negative answer: for each countable ordinal ξ ≥ 1, there is a partial injection with disjoint domain and range i : ω ω → ω ω , whose graph -is D 2 (Π 0 1 ) (i.e., the difference of two closed sets), -has Borel chomatic number two, -has no ∆ 0 ξ -measurable countable coloring.

On the other hand, they note that an open binary relation having a finite coloring c has also a ∆ 0 2 -measurable finite coloring (consider the differences of the c -1 ({n})'s, for n in the range of the coloring). Note that an irreflexive closed binary relation on a zero-dimensional space has a continuous countable coloring (this coloring is ∆ 0 2 -measurable in non zero-dimensional spaces). So they wonder whether we can build, for each countable ordinal ξ ≥ 1, a closed binary relation with a Borel finite coloring but no ∆ 0 ξ -measurable finite coloring. This is indeed the case:

Theorem Let ξ ≥ 1 be a countable ordinal. Then there exists a partial injection with disjoint domain and range f : ω ω → ω ω whose graph is closed (and thus has Borel chromatic number two), and has no ∆ 0 ξ -measurable finite coloring (and thus has ∆ 0 ξ chromatic number ℵ 0 ).

The previous discussion shows that this result is optimal. Its proof uses, among other things, the method used in [L-Z2] improving Theorem 4 in [M]. This method relates topological complexity and Baire category.

Mátrai sets

Before proving our main result, we recall some material from [L-Z2].

Notation. The symbol τ denotes the usual product topology on the Baire space ω ω . Definition 2.1 We say that a partial map f :

ω ω → ω ω is nice if its graph Gr(f ) is a (τ ×τ )-closed subset of ω ω ×ω ω .
The construction of P ξ and τ ξ , and the verification of the properties (1)-(3) from the next lemma (a corollary of Lemma 2.6 in [L-Z2]), can be found in [M], up to minor modifications. Lemma 2.2 Let 1 ≤ ξ < ω 1 . Then there are P ξ ⊆ ω ω , and a topology τ ξ on ω ω such that (1) τ ξ is zero-dimensional perfect Polish and

τ ⊆ τ ξ ⊆ Σ 0 ξ (τ ), (2) P ξ is a nonempty τ ξ -closed nowhere dense set, (3) if S ∈ Σ 0 ξ (ω ω , τ ) is τ ξ -nonmeager in P ξ , then S is τ ξ -nonmeager in ω ω , (4) if V, W are nonempty τ ξ -open subsets of ω ω , then we can find a τ ξ -dense G δ subset H of V \P ξ , a τ ξ -dense G δ subset L of W \P ξ ,
and a nice (τ ξ , τ ξ )-homeomorphism from H onto L.

The following lemma (a corollary of Lemma 2.7 in [L-Z2]) is a consequence of the previous one. It provides, among other things, a topology T ξ that we will use in the sequel.

Lemma 2.3 Let 1 ≤ ξ < ω 1 .
Then there is a disjoint countable family G ξ of subsets of ω ω and a topology T ξ on ω ω such that (a) T ξ is zero-dimensional perfect Polish and

τ ⊆ T ξ ⊆ Σ 0 ξ (τ ), (b) for any nonempty T ξ -open sets V, V ′ , there are disjoint G, G ′ ∈ G ξ with G ⊆ V , G ′ ⊆ V ′ ,
and there is a nice (T ξ , T ξ )-homeomorphism from G onto G ′ , and, for every G ∈ G ξ , (c) G is nonempty, T ξ -nowhere dense, and in

Π 0 2 (T ξ ), (d) if S ∈ Σ 0 ξ (ω ω , τ ) is T ξ -nonmeager in G, then S is T ξ -nonmeager in ω ω . The construction of G ξ and T ξ ensures that T ξ is (τ ξ ) ω
, where τ ξ is as in Lemma 2.2. This topology is on (ω ω ) ω , identified with ω ω . We will need the following consequence of the construction of G ξ and T ξ .

Lemma 2.4 Let 1 ≤ ξ < ω 1 , and V be a nonempty T ξ -open set. Then V τ is not τ -compact.

Proof. The fact that T ξ is (τ ξ ) ω gives a finite sequence U 0 , ..., U n of nonempty open subsets of (ω ω , τ ξ ) with U 0 ×...×U n ×(ω ω ) ω ⊆ V . Thus V τ contains the τ -closed set U 0 τ ×...×U n τ ×(ω ω ) ω , and it is enough to see that this last set is not τ -compact. This comes from the fact that the Baire space (ω ω , τ ) is not compact.

Proof of the main result

Before proving our main result, we give an example giving the flavour of the sequel. In [Za], the author gives a Hurewicz-like test to see when two disjoint subsets A, B of a product Y ×Z of Polish spaces can be separated by an open rectangle. We set

A := {(n ∞ , n ∞ ) | n ∈ ω}, B 0 := 0 m+1 (n+1) ∞ , (m+1) n+1 0 ∞ | m, n ∈ ω and B 1 := (m + 1) n+1 0 ∞ , 0 m+1 (n + 1) ∞ | m, n ∈ ω . Then A is not separable from B
by an open rectangle exactly when there are ε ∈ 2 and continuous maps g :

ω ω → Y , h : ω ω → Z such that A ⊆ (g×h) -1 (A) and B ε ⊆ (g×h) -1 (B).
Example. Here we are looking for closed graphs with Borel chromatic number two and of arbitrarily high finite ∆ 0 ξ chromatic number n. There is an example with ξ = 1 and n = 3 where B 0 is involved. We set

C := (2m) ∞ , (2m+1) ∞ | m ∈ ω ∪ B 0 , D := {(2m) ∞ | m ∈ ω} ∪ {0 m+1 (n+1) ∞ | m, n ∈ ω}, R := {(2m+1) ∞ | m ∈ ω} ∪ {(m+1) n+1 0 ∞ | m, n ∈ ω}, f (2m) ∞ := (2m+1) ∞ and f 0 m+1 (n+1) ∞ := (m+1) n+1 0 ∞ .
This defines f : D → R whose graph is C. The first part of C is discrete, and thus closed. Assume that (α k , β k ) := 0 m k +1 (n k + 1) ∞ , (m k + 1) n k +1 0 ∞ ∈ B 0 and converges to (α, β) ∈ ω ω × ω ω as k goes to infinity. We may assume that (m k ) is constant, and (n k ) too, so that (α, β) ∈ B 0 , which is therefore closed. This shows that C is closed. Note that D, R are disjoint and Borel, so that C has Borel chromatic number two. Let ∆ be a clopen subset of ω ω . Let us prove that C ∩ ∆ 2 or C ∩ (¬∆) 2 is not empty. We argue by contradiction. Then ∆ or ¬∆ has to contain 0 ∞ . Assume that it is ∆, the other case being similar. Then

0 m+1 (n+1) ∞ ∈ ∆ if m is big enough. Thus (m+1) n+1 0 ∞ / ∈ ∆ if m is big enough. Therefore (m+1) ∞ / ∈ ∆ if m is big enough. Thus (2m) ∞ , (2m+1) ∞ ∈ C ∩ (¬∆) 2 if m is big enough, which is absurd.
We now turn to the general case. Our main lemma is as follows. We equip ω m with the discrete topology τ d , for each m > 0.

Lemma Let ξ ≥ 1 be a countable ordinal, n ≥ 1 be a natural number, and X := ω ×ω ω . Then we can find a partial injection f : X → X and a disjoint countable family F of subsets of X such that (a) f has disjoint domain and range, Proof. We argue by induction on n.

(b) Gr(f ) is (τ d ×τ )×(τ d ×τ ) -closed, (c) there is no sequence (∆ i ) i<n of ∆ 0 ξ subsets of (X, τ d ×τ ) such that (i) ∀i < n Gr(f ) ∩ ∆ 2 i = ∅, (ii) i<n ∆ i is (τ d ×T ξ )-comeager in X, (

The basic case n = 1

Let G ξ be the family given by Lemma 2.3. We split G ξ into two disjoint subfamilies G 0 ξ and G 1 ξ having the property (b) in Lemma 2.3. This is possible since the elements of G ξ are T ξ -nowhere dense. Let G 0 , G 1 ∈ G 0 ξ be disjoint, and ϕ be a nice (T ξ , T ξ )-homeomorphism from G 0 onto G 1 . We then set f (0, α) := 0, ϕ(α) if α ∈ G 0 , and

F := {n}× G | n ∈ ω ∧ G ∈ G 1
ξ . It remains to check that the property (c) is satisfied. We argue by contradiction, which gives ∆ 0 ∈ ∆ 0 ξ . By property (d) in Lemma 2.3,

∆ 0 ∩ ({0} × G ε ) is (τ d × T ξ )-comeager in {0} × G ε for each ε ∈ 2. As f is a (τ d ×T ξ , τ d ×T ξ )-homeomorphism, ∆ 0 ∩ ({0}×G 0 ) ∩ f -1 ∆ 0 ∩ ({0}×G 1
) is (τ d ×T ξ )-comeager in {0}×G 0 , which contradicts the fact that Gr(f ) ∩ ∆ 2 0 = ∅.

  d) F has the properties (b)-(d) in Lemma 2.3, where G ξ , ω ω , T ξ and τ are respectively replaced with F, X, τ d ×T ξ and τ d ×τ , (e) ( F) ∩ Domain(f ) ∪ Range(f ) = ∅.

thanks Charles University in Prague for the hospitality. The research was supported by the grant GA ČR P201/12/0436 for the second author.

The induction step from n to n+1

The induction assumption gives f and F. Here again, we split F into two disjoint subfamilies F 0 and F 1 having the property (b) in Lemma 2.3, where G ξ , ω ω , T ξ and τ are respectively replaced with F, X, τ d ×T ξ and τ d ×τ . Let (V p ) be a basis for the topology τ d ×T ξ made of nonempty sets. Fix p ∈ ω. By Lemma 2.4, there is a countable family (W p q ) q∈ω , with (τ d ×τ )-closed union, and made of pairwise disjoint (τ d ×τ )-clopen subsets of X intersecting V p .

• Let b : ω → ω 2 be a bijection. We construct, for v = (p, q) ∈ ω 2 and ε ∈ 2, and by induction on b

We want these objects to satisfy the following:

-

).

• We now define the desired partial map f : ω ×ω ×ω ω → ω ×ω ×ω ω , as well as F ⊆ 2 ω×ω×ω ω , as follows:

and

Identifying X with ω ×ω ×ω ω , we can consider f as a partial map from X into itself and F as a family of subsets of X (this identification is based on the identification of ω with ω×ω). (b) Assume that (l k , x k ), (m k , y k ) ∈ Gr( f ) tends to (l, x), (m, y) ∈ (ω×X) 2 as k goes to infinity. We may assume that (l k ) and (m k ) are constant.

If l = 0, then there is p such that p+1 = m and (x k , y k ) ∈ G p,q k 0 ×G p,q k 1 . As G p,q k 0 ⊆ W p q k , we may also assume that (q k ) is also constant and equals q. As ϕ p,q is nice, (l, x), (m, y) ∈ Gr( f ).

(c) We argue by contradiction, which gives (∆ i ) i≤n . We may assume, without loss of generality, that

1 , for each q.

1 , for each q.

Claim The set

Indeed, we argue by contradiction. This gives W ∈ (τ d ×T ξ )\{∅} such that

×ω ω , which contradicts the induction assumption.

In order to get our main result, it is enough to apply the main lemma to each n ≥ 1. This gives f n : ω × ω ω → ω × ω ω . It remains to define f : n≥1 ({n} × ω × ω ω ) → n≥1 ({n} × ω × ω ω ) by f (n, x) := f n (x) (we identify (ω\{0})×ω×ω ω with ω ω ).