Takuto Honda

Mayuka Kanaya

Isao T Tokuda

Anne Bouvet

Annemie Van Hirtum

Xavier Pelorson

Experimental study on quasi-steady approximation of glottal flows

Keywords: numbers: 43.70.Aj, 43.70.Bk, 43.70.-h, 43.70.Gr Intraglottal Pressure, Replical Model of Vocal Folds, Unsteady Flow, Boundary Layer Model

, the vocal fold model was constructed in an enlarged scale (7.5 times of the human hemilarynx). To realize unsteady flow conditions, the flow rate was sinusoidally modulated with a frequency of 2 Hz, which corresponds to 112.5 Hz in real life. Measurements of the intraglottal pressures under both steady and unsteady flows revealed that the pressure profiles of the unsteady flow conditions are non-distinguishable from those of the steady flow conditions as far as they have the same subglottal pressure as an input pressure. Our study therefore suggests that the unsteady glottal flow can be approximated as a series of steady flow states with a time-varying subglottal pressure in the range of normal vocalization frequencies.

I. INTRODUCTION

In voice production, the vocal folds play a vital role in generating the sound source.

As an air-flow expelled from the lungs passes through the glottis, the vocal folds vibrate.

For efficient transfer of energy from the glottal air-flow to the vibrating vocal fold tissues [START_REF] Titze | The physics of small-amplitude oscillation of the vocal folds[END_REF], switching between convergent and divergent shapes of the left and right vocal folds in one glottal cycle is known to be essential. Of particular note is the air-jet separation point, which changes its location so that the intraglottal pressure increases during the opening phase of the glottis, while it decreases during the closing phase of the glottis [START_REF] Pelorson | Theoretical and experimental study of quasisteady-flow separation within the glottis during phonation. application to a modified two-mass model[END_REF]. To validate such change in the intraglottal pressure, experimental studies have been conducted to measure detailed pressure distributions inside physical models of the vocal folds. Sherer et al. measured intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees and observed the flow separation point [START_REF] Scherer | Intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees[END_REF]. The effect of glottal shapes (e.g., left-right asymmetry, slanted glottis, and convergent, unifom, and divergent shapes) on the glottal pressure profiles were further studied (Li et al., 2006a;[START_REF] Scherer | Intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees[END_REF][START_REF] Scherer | Intraglottal pressure distributions for a symmetric and oblique glottis with a uniform duct (l)[END_REF][START_REF] Scherer | Intraglottal pressures in a three-dimensional model with a non-rectangular glottal shape[END_REF]. Numerical simulations (e.g., based on FLUENT) were performed to elucidate the experimental observations (Li et al., 2006a,b;[START_REF] Scherer | Intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees[END_REF][START_REF] Scherer | Intraglottal pressure distributions for a symmetric and oblique glottis with a uniform duct (l)[END_REF][START_REF] Scherer | Intraglottal pressures in a three-dimensional model with a non-rectangular glottal shape[END_REF]. It should be noted that these studies have been based on the steady-flow conditions. In real phonations, however, the flow is inevitably unsteady due to the oscillatory nature of the glottis. For comprehensive understanding of the intraglottal pressures that support the vocal fold oscillations, effect of the unsteady flow should be taken into account. From such viewpoint, Alipour et al. utilized a finite-volume computational model to solve the time-dependent glottal air-flow in a forcedoscillation model of the glottis and predicted that the flow separation is delayed during the vibratory cycle [START_REF] Alipour | Flow separation in a computational oscillating vocal fold model[END_REF]. Kucinschi et al. constructed a driven mechanical model of the vocal fold and experimentally measured the subglottal and supraglottal pressures by tracing detailed opening angles of the glottis [START_REF] Kucinschi | An experimental analysis of the pressures and flows within a driven mechanical model of phonation[END_REF]. Although their study clarified the effect of the moving boundary conditions, the glottal pressures have been monitored only at two points, which do not provide detailed pressure profile inside the glottis.

The aim of the present study is to measure a detailed glottal pressure distribution under unsteady flow condition. We examine whether the quasi-steady approximation of the glottal air-flow, widely assumed in a mathematical modeling of the vocal fold oscillations, is valid in reality. To tackle this problem, physical model of the vocal folds was constructed based on the standard M5 geometry [START_REF] Scherer | Intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees[END_REF]. The unsteady flow condition was realized by sinusoidally modulating the flow rate with a frequency of 2 Hz by a solenoid valve.

Measurements of the intraglottal pressures under both steady and unsteady flows revealed that the pressure profiles of the unsteady flow conditions are quite similar to those of the steady flow conditions as far as they have the same subglottal pressure as an input pressure.

The air-jet separation point is also found to be located in a very similar position between the unsteady and steady data. This supports the quasi-steady approximation of the glottal flow, in which unsteady flow is represented as a series of steady flow states. A theoretical model based on the boundary layer theory is further simulated to validate our experimental measurements.

The present paper is organized as follows. Section II describes the experimental methods.

Our experimental set-up as well as the vocal fold replica model is explained in detail. A theoretical model is also introduced here. Section III presents experimental as well as simulation results. The final Section is devoted to conclusions and discussions on the influence of unsteady flow on the intraglottal pressure profiles as well as on the vocal fold oscillations.

II. METHODS

A. M5 Vocal Fold Model

Based on the M5 geometry of Scherer et al. [START_REF] Scherer | Intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees[END_REF], acrylic replica model of the vocal fold has been fabricated and used for the experiment. As shown in Fig. 1 (a), the vocal fold replica represents an upscaled structure (7.5 times) of the human hemilarynx. The glottal entrance and exit are indicated on the lower and upper sides, respectively. As shown in Fig. 1 (b), the left and right vocal folds were set to a symmetric and oblique configuration with a divergence angle of 5 • and 5 • . The airstream goes along the x-axis (dorsoventral direction). The channel height h(x) is defined on the y-axis (mediolateral direction), where its minimum represents the aperture h 0 . After the set-up of the left and right vocal folds, the aperture was measured at 10 points uniformly located along the glottal length. At each point, a stack of feeler gages was placed through the glottis. As summarized in Fig. 1 (c), the averaged aperture was h 0 = 0.375±0.005 mm.

On the vocal fold surface, 14 pressure taps were placed 2 mm apart from each other.

They were staggered on either side of the centerline, which run along the airstream direction.

The pressure taps were made flush and perpendicular to the vocal fold surface. The inner diameter of each pressure tap was 2 mm, while the diameter was narrowed to 0.3 mm at the vocal fold surface. In addition to the pressure taps on the vocal folds, another pressure tap was located upstream inside of the rectangular pipe, so as to measure the subglottal pressure as a reference. The 14 pressure taps were led to 14 pressure sensors (Differential pressure transducer, DP15-28-N1S4A, Validyne Engineering, Northridge, CA; Pressure amplifier PA501, KRONE Corporation, Tokyo, Japan) via silicon tubing. Another pressure sensor (Differential pressure transducer, PDS-70GA, Kyowa; Signal conditioner, CDV-700A, Kyowa, Tokyo, Japan) was used to monitor the subglottal pressure.

Prior to the experiment, the pressure sensors were calibrated using two liquid column manometers (MG 80, Sauermann; TJ 300, Sauermann, Montpon-Ménestérol, France). The sensors were labelled from P 1 to P 15 towards the downstream direction. P 1 corresponds to the subglottal pressure (x = 0 [mm]), whereas P 15 indicates the pressure at supraglottal exit (x = 59.8[mm]). The location that corresponds to the aperture lies between P 7 and P 8.

B. Flow Experiment

Intraglottal pressures were measured by injecting an air-flow through the vocal fold model (see the experimental set-up of Fig. 1 (d)). To transfer an air-flow from air compressors (SilentAirCompressor SC820, Hitachi Koki, Tokyo, Japan; Rebicon 0.2LE-8SBA, Hitachi, Tokyo, Japan; UA 3810A, Hikoki, Tokyo, Japan) to the vocal fold model, a long rectangular pipe (inner cross-sectional rectangular area: 0.08 m × 0.09 m, length: 2.8 m) was inserted between them (Fig. 1 (f)).

The Reynolds number at the constriction is given by Re = Uh 0 /ν, where h 0 is the characteristic linear dimension of the flow channel (in this case, the aperture), U = V /A g is the average flow velocity in the minimum cross section area (V : volumetric flow rate, A g = h 0 l g : minimum area as a product of aperture and glottal width l g = 0.09 m), and ν = 15•10 -6 m 2 /s is the kinematic viscosity of air. In the present steady flow experiment, the volumetric flow rate was varied from V = 12 l/min to V = 53 l/min. The corresponding Reynolds number ranges from Re = 148 to 654, that is in the range of laminar flows.

According to [START_REF] Shah | Bifurcations in an asymmetric vocal-fold model[END_REF], the hydrodynamic entry length is given approximately as l h = 0.05•Re•D h and, in our experiment, it ranges from 0.62 m to 2.78 m (here, hydraulic diameter of the rectangular pipe is approximated as D h = 2•0.08•0.09/(0.08 + 0.09) = 0.085 m). The pipe length of 2.8 m, used in the present set-up, covered most of the entry length D h , which is required to reach to the fully developed flow.

For steady flow experiments, the flow rate from the air compressors was controlled by a pressure regulator (10202U, Fairchild, Winston-Salem, NC) and a digital mass flow controller (CMQ-V, Azbil, Santa Clara, CA). For unsteady flow experiments, it was controlled by a solenoid valve (Type 8605, Burkert; Type 2873, Burkert, Ingelfingen, Germany) combined with a function generator (33500B Series, Keysight, Santa Rosa, CA). The volume flow rate was also monitored by a flow meter (Thermal Mass Flowmeters Series 4000, TSI, Houston, TX). For steady flow experiment, the flow rate was varied from 12 l/min to 53 l/min with an increment of 1 l/min. For unsteady flow experiment, the flow rate was sinusoidally modulated with a frequency of 2 Hz by the solenoid valve and the function generator. All signals were stored into a digital recorder (Controller, PXIe-8840, National Instruments; Input/output card, BNC-2110, National Instruments; Software, Labview, National Instruments, Tokyo, Japan) with a sampling frequency of 20 kHz.

According to the experimental study of measuring the intraglottal pressures [START_REF] Scherer | Intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees[END_REF], one side of the vocal fold forms a "flow wall," while the other forms a "nonflow wall."

In the flow wall, the flow is attached to the glottal surface until it separates near the glottal exit, whereas, in the nonflow wall, the flow separation occurs further upstream in the glottis.

In our experiments, they were detected manually by placing a hand in front of the vocal fold model to sense the flow direction. The direction of the glottal flow was switched by inserting a thin paper inside of the glottis and pulling it to a desired direction.

C. Theoretical Model

To elucidate the steady-flow experiment, Thwaites boundary layer model was solved numerically [START_REF] Deverge | Influence of collision on the flow through in-vitro rigid models of the vocal folds[END_REF][START_REF] Hofmans | Vortex sound in confined flows[END_REF][START_REF] Schlichting | Boundary-layer theory[END_REF]. The theoretical model is based on the incompressible quasi-steady boundary layer theory. At moderate Reynolds numbers, viscous effects can indeed be assumed to be concentrated in a thin region, i.e., the boundary layer, near the walls of the channel [START_REF] Schlichting | Boundary-layer theory[END_REF]. Outside of the boundary layer, the flow is assumed to be inviscid and governed by Bernoulli's law together with the equation of mass conservation:

P (x) + 1 2 ρv(x) 2 = P 1, (1)
l g {h(x) -2δ(x)}v(x) = const, ( 2
)
where ρ is the air density, v(x) the flow velocity and δ(x) is the thickness of the boundary layer. Within the boundary layer, the two-dimensional viscous flow is described by the von Kármán equation, which is solved using the Thwaites method. Details about this resolution and validation against various glottal replicas can be found in [START_REF] Vilain | Experimental validation of a quasi-steady theory for the flow through the glottis[END_REF].

Solving the boundary layer equation provides δ(x) in equation 1 and predicts the position of the flow separation point, for which the shear stress at the walls equals zero. The pressure profile P (x) along the glottis can then be computed using equation 1. Figure 2 (e) shows dependence of the flow separation point on the subglottal pressure. The observed tendency is consistent with the earlier experimental and modelling studies on a round VF shape [START_REF] Cisonni | Experimental validation of quasi-one-dimensional and two-dimensional steady glottal flow models[END_REF][START_REF] Cisonni | Theoretical simulation and experimental validation of inverse quasi-one-dimensional steady and unsteady glottal flow models[END_REF][START_REF] Scherer | Intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees[END_REF][START_REF] Van Hirtum | On quasi-steady laminar flow separation in the upper airways[END_REF].

III. RESULTS

A. Steady Flow Experiment

B. Theoretical Model

To elucidate the steady-flow experiment, Thwaites boundary layer model was simulated.

F y = l g x 2
x 1 P (x)dx, respectively. Here, x 1 and x 2 determine the integration range, which were for the glottal forces F x and F y , respectively. For mathematical modeling of the vocal fold oscillations [START_REF] Ishizaka | Synthesis of voiced sounds from a two-mass model of the vocal cords[END_REF][START_REF] Pelorson | Theoretical and experimental study of quasisteady-flow separation within the glottis during phonation. application to a modified two-mass model[END_REF][START_REF] Shah | Bifurcations in an asymmetric vocal-fold model[END_REF][START_REF] Titze | Principles of voice production[END_REF][START_REF] Titze | The myoelastic aerodynamic theory of phonation (National Center for Voice and Speech)[END_REF], the 10 % error may produce a certain mismatch, e.g., in phonation onset pressures, but it is within a tolerance range for simulation of a normal oscillation mode. for the steady and unsteady conditions, respectively. Their difference in the the pressure forces were quite small (e x = 0.3 % and e y = 0.2 %).

Figure 4 (c) shows another snap shot of the pressure profile observed at t = 2.06 s.

The unsteady data (subglottal pressure: 153.0±2.4 Pa) is compared with the closest steady flow data (subglottal pressure: 153.0±3.3 Pa). Again, the two pressure profiles were not distinguished from each other (t-tests for P6: p = 0.94, P7: p = 0.90, P8: p = 0.88). The glottal pressure forces were (F x , F y ) = (-0.422N, 0.459N) and (-0.422N, 0.459N) for the steady and unsteady conditions, respectively, with the errors of e x = 0.1 % and e y = 0.1 %.

To confirm that these two points represent general property of the unsteady flow data, other data points were further examined. Namely, 200 time points were randomly selected from the unsteady data and the same comparisons were made. No difference was detected for 192 pairs of unsteady and steady data (significance level of α = 0.05), while a slight difference was detected only for 8 pairs. The averaged difference in the glottal pressure forces were less than 1 % (i.e., e x = 0.75±0.45 %, e y = 0.79±0.46 %). This implies that the pressure profiles of the unsteady flow data are non-distinguishable from the steady flow data as far as they have the same subglottal pressure as an input pressure. Thus, unsteady flow can be viewed as a series of steady flow states with a time-varying subglottal pressure.

For the flow wall data, similar results have been obtained. Among 200 pairs of steady and unsteady data, 186 pairs indicated no significant difference in their pressure profiles. Their difference in the pressure forces were also very low (i.e., e x = 0.66±0.45 %, e y = 0.70±0.51 %).

To demonstrate that detailed information on the steady flow is contained in a short time window of the unsteady flow, the air-jet separation points were detected from the snapshots of the unsteady flow data and drawn as a function of the subglottal pressure in Their differences in the glottal pressure forces were also very small (e x = 0.87 %, e y = 0.87 %). In (significance level of α = 0.05). The averaged differences in the glottal pressure forces were less than 1 % (i.e., e x = 0.68±0.67 %, e y = 0.73±0.73 %).

Similar results were obtained also for the flow wall data. Among 200 pairs of steady and unsteady data, 188 pairs indicated no significant difference in their pressure profiles.

Their differences in the pressure forces were again less than 1 % (i.e., e x = 0.69±0.63 %, e y = 0.75±0.74 %).

Finally, the air-jet separation points were detected from the snapshots of the unsteady flow data and drawn as a function of the subglottal pressure in Fig. 5 (d). The red plus signs and the blue crosses correspond to the results estimated from the nonflow wall and the flow wall data, respectively. They agree quite well with the steady flow data (black circles and black crosses). Compared to Fig. 4 (d), which utilized only a single time trace of the unsteady flow data, the range of the subglottal pressure has been largely extended. This implies that our view of approximating the unsteady flow as a series of steady flow states is valid for a wider range of unsteady flow rates.

IV. CONCLUSIONS AND DISCUSSIONS

In the present study, the effect of unsteady flow on the intraglottal pressure distribution was examined. Based on the M5 geometry of Scherer et al. [START_REF] Scherer | Intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees[END_REF], replica model of the vocal folds were constructed in an enlarged scale (7.5 times of the human hemilarynx). The left and right vocal folds were set to be symmetric with a divergence angle of 5 • and 5 • . The aperture was set to be small (i.e., 0.375±0.005 mm). Unsteady flow condition was realized by sinusoidally modulating the input air-flow to the vocal fold model with a frequency of 2 Hz. The 2 Hz in the enlarged model corresponds to 112.5 Hz in real life, that is within a typical frequency range of male vocalizations. By comparing with steady flow data measured with variable subglottal pressures, we have shown that the pressure profiles of the unsteady flow data can not be distinguished from those of the steady flow data as far as they have the same subglottal pressure as an input pressure. This has been confirmed for a wide range of subglottal pressures (from 51 Pa to 423 Pa) and for both flow wall and nonflow wall conditions. Our study therefore suggests that the unsteady glottal flow can be approximated as a series of steady flow states with a time-varying subglottal pressure. This supports quasi-steady approximation of the glottal flow widely employed in mathematical modeling of the vocal fold oscillations, where the air-jet separation point is determined by assuming a steady flow inside the glottis [START_REF] Ishizaka | Synthesis of voiced sounds from a two-mass model of the vocal cords[END_REF][START_REF] Pelorson | Theoretical and experimental study of quasisteady-flow separation within the glottis during phonation. application to a modified two-mass model[END_REF][START_REF] Shah | Bifurcations in an asymmetric vocal-fold model[END_REF][START_REF] Titze | Principles of voice production[END_REF][START_REF] Titze | The myoelastic aerodynamic theory of phonation (National Center for Voice and Speech)[END_REF].

Further studies are needed for a comprehensive understanding of the effect of unsteady flow on the glottal pressure. In the present study, oscillation frequency of the unsteady flow was limited up to 2 Hz. For unsteady flow having a frequency higher than 2 Hz, oscillation range of the subglottal pressure was strongly diminished due to the limited capability of the solenoid valve. Under such diminished oscillations, effect of the unsteadiness could not be examined in the present experimental set-up. A more efficient system is needed for the flow regulation, which enables large-amplitude oscillations in a frequency higher than 2 Hz. A simple way is to make the aperture smaller so that the subglottal pressure becomes more sensitive to the volume flow rate of the glottis. Such system may detect the frequency range, above which the effect of unsteady flow appears to influence the intraglottal pressures. Our future study may focus on the high frequency region that would introduce unsteady effect to the glottal flow. The influence of left-right asymmetry between the vocal folds as well as the glottal shape (e.g., convergent and divergent shapes) on the time-varying pressure profile should be also investigated. As another factor of unsteadiness, the moving boundary condition of the glottal surface should be also taken into account.

APPENDIX

In this Appendix, we show how the time scale of the enlarged vocal fold model is translated into that of the real-life. For the sake of simplicity, we consider here the equations of an incompressible fluid:

∇• v = 0 (A1) ρ 0 ∂ v ∂t + ρ 0 ( v• ∇) v = ∇p + µ 0 ∇ 2 v, (A2)
where p = p(t, x, y, z) and v = v(t, x, y, z) are respectively the flow pressure and velocity, µ 0 the dynamic viscosity coefficient, ρ 0 the air density, and ∇ the gradient operator. For dimensional analysis, we introduce the following dimensionless variables:

v * = v v 0 , p * = p ρ 0 v 2 0 , t * = t t 0 , x * = x l 0 , y * = y l 0 , z * = z l 0 ,
where v 0 is a flow characteristic (typical) value, l 0 a characteristic dimension of the flow channel, and t 0 a characteristic time scale. Equation A2 can be rewritten as:

Sr ∂ v * ∂t * + ( v * • ∇ * ) v * = ∇ * p * + 1 Re ∇ * 2 v * (A3)
with Sr = l 0 t 0 v 0 and Re = ρ 0 v 0 l 0 µ 0 , which represent the Strouhal and Reynolds numbers,

Re = ρ 0 v 0 h 0 µ 0 .
For the Strouhal number, the typical time scale should be the period of oscillation (1/f o , where f o is the fundamental frequency of the oscillation) and the characteristic dimension should be the thickness of the glottis (d 0 ). Indeed, the time needed by the flow to pass through the glottis can be estimated by d 0 /v 0 and must be compared with t 0 . The relevant

Figure 2

 2 Figure 2 shows the results of the steady-flow experiment. Panels (a) and (b) indicate the

Figure 3

 3 Figure3shows intraglottal pressure distributions of the boundary layer model, in which the

  set to x 1 = 10 mm and x 2 = 60 mm. The model errors can be quantified as e α = 100|F e α -F m α |/F e α %, where the subscript α indicates x or y and the superscript indicates experiment (e) or model (m). Figure 3 (e) shows the results. For 20 model simulations (the subglottal pressure ranges between 29.7 Pa and 442.9 Pa), the modeling error was below 2 % and 11 %

Finally, the model

  prediction of the flow separation point is compared with the experiment in Fig.3(f). The model reproduced qualitative feature of the experiment. Namely, in both model and experiments, the separation point moved towards an upstream region as the subglottal pressure was increased. Compared to the experimental curves, the model generally located the separation point in a more upstream region. The gap between the model and the nonflow wall experiment ranged between -2 mm and -0.38 mm. In the sense that they are less than the distance between the pressure sensors (i.e., 2 mm), the model predicted the separation point within our experimental precision. As the subglottal pressure is increased to about 200 Pa, the gap decreases and reaches to around 0.4 mm.Considering the simplified modeling assumptions, this level of agreement is considered good enough.C. Unsteady Flow ExperimentIn the next experiments, unsteady flow was injected into the vocal fold model. Figure4 (a)shows time trace of the subglottal pressure sinusoidally modulated with 2 Hz. The subglottal pressure oscillated in the range between 115 and 365 Pa. Because of the long and large pipe, which prevents instantaneous transfer of the modulated air pressure at the solenoid valve to the subglottal area, there was a limitation in the oscillation range. The oscillation range of 250 Pa was in the maximal level in the present setting. According to our manual detection of the flow direction at the glottal exit, the pressure profile measured by the sensors was on the side of the nonflow wall. By randomly selecting two points (red squares) in the time trace, the corresponding pressure profiles are drawn. Figure4(b) shows a snap shot of the pressure profile (dashed red line) observed at t = 1.19 s (average and standard deviation of 50 consecutive points are indicated). The subglottal pressure was 314.5±2.4 Pa. To compare this pressure profile with that under steady flow condition, a steady flow data was selected in such a way that its subglottal pressure is located closely to that of the unsteady data. The pressure profile of the selected steady data, which had the subglottal pressure of 314.5±2.0 Pa, is drawn simultaneously in Fig.4(b) (average and standard deviation of 50 consecutive points are indicated by solid black line). The two pressure profiles were quite similar to each other. The t-tests indicated no pressure difference between the steady and unsteady flow conditions at taps P6 (p = 0.29), P7 (p = 0.57), and P8 (p = 0.45), where the separation point is located. Following the formula introduced in Subsection III B, the glottal pressure forces were calculated as (F x , F y ) = (-0.867 N, 0.946 N) and(-0.865 N, 0.943 N)

Fig. 4

 4 Fig. 4 (d). The red plus signs and the blue crosses correspond to the results estimated from

  Fig. 5 (c), the same comparison was made for another pressure profile observed at t = 2.04 s in the upper time trace ofFig. 5 (a). Again, the pressure profile of the unsteady flow (subglottal pressure: 339.0±2.5 Pa) was not distinguished from that of the steady flow (subglottal pressure: 339.1±3.4 Pa) by the t-tests (P6: p = 0.54, P7: p = 0.72, P8: p = 0.93). Their differences in the glottal pressure forces were tiny (e x = 0.58 %, e y = 0.64 %). Next, for 200 time points randomly selected from the upper and lower time traces of the unsteady flow data, the corresponding steady flow data, which showed the closest subglottal pressure, were searched. Among such 200 pairs of unsteady and steady flow data, their pressure profiles were compared. For 185 pairs, no significant difference was detected

  respectively. Because of the normalization, all terms in equations A3 are now comparable with each others. The Strouhal number appears thus as an indicator of the importance of the inertial terms (ρ 0 ∂ v ∂t ) with respect to the convective ones (ρ 0 ( v• ∇) v). In the same way, the Reynolds number (or 1/Re) measures the importance of the convective terms with respect to the viscous terms (µ 0 ∇ 2 v). Now the relevance of using Sr and Re relies on a sensible choice of the characteristic values and thus on some knowledge about the flow. Concerning the flow through the glottis, viscous losses are mainly dependent on the aperture (h 0 ) of the glottis. The relevant Reynolds number is thus:

  FIG. 1. (a): Geometry and detailed dimensions [mm] of the M5 model. (b): Symmetric divergent configuration for the left and right vocal fold models. The left and right angles are set equally to 5 • and 5 • . The x-and y-axes are located towards dorsoventral and mediolateral directions. The airstream goes along the x-axis. The channel height h(x) is defined on the y-axis, where its minimum represents the aperture h 0 . Locations of the 14 pressure taps are indicated by the arrows. (c): Dependence of the aperture h 0 on the glottal location. (d): Experimental setup. Air-flow from the air compressors is regulated by the valve and the flow controller (or solenoid valve). It is then transferred to the vocal fold model through a long rectangular pipe. The subglottal pressure and the intraglottal pressures were measured simultaneously.

FIG. 2 .FIG. 3 .

 23 FIG. 2. Steady flow experiment. (a): Intraglottal pressure distribution (abscissa: x [mm], ordinate: pressure [Pa]) measured under nonflow wall condition. The flow rate was varied as 10, 20, 30, 40, 50 l/min. (b) Enlarged graph of (a). The air-jet separation points (-5 % level of the subglottal pressure) are indicated by circles. (c): Intraglottal pressure distribution (abscissa: x [mm], ordinate: pressure [Pa]) measured under flow wall condition. The flow rate was varied as 10, 20, 30, 40, 50 l/min. (d) Enlarged graph of (c). (e): Dependence of the air-jet separation point on the subglottal pressure. The dotted red line and the solid black line correspond to the nonflow wall and flow wall conditons, respectively. The flow-rate was varied from 12 l/min to 53 l/min with an increment of 1 l/min.

Strouhal number is thus:

Using an up-scaled (or down-scaled) replica makes sense as long as Re and Sr (i.e., the physical effects of viscosity and inertia) are kept constant. Thus, if the replica is upscaled by a factor of N , keeping the Reynolds number constant means that the velocity on the replica must be N times lower than that in real life (because h 0 is N time larger than that in real life). Concerning the Strouhal number, d 0 is N times larger than that in real life, v 0 is N times lower than that in real life, thus t 0 is a factor N 2 times larger than that in real life. This means that an oscillation of 1 Hz on the replica corresponds to N 2 Hz in real life.

Note that the same demonstration could be done considering compressibility, in which the Mach number (Ma = v 0 /c 0 , where c 0 is the speed of sound) would appear as an extra dimensionless term. This is because, for voicing, v 0 c 0 , the compressible effects are considered of second order.


Antoine Chaillet
email: antoine.chaillet@centralesupelec.fr

Iasson Karafyllis

Yuan Pierdomenico Pepe

Yuan Wang

Growth conditions for global exponential stability and exp-ISS of time-delay systems under point-wise dissipation

Keywords: non-linear time-delay systems, stability analysis, input-to-state stability

come

Growth conditions for global exponential stability and exp-ISS of time-delay systems under point-wise dissipation

Introduction

The input-to-state stability (ISS) framework has become a central and classical tool to study stability and robustness of nonlinear systems. Originally developped for systems described by ordinary differential equations [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF][START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF], it has progressively been extended to infinite-dimensional systems [START_REF] Mironchenko | Input-to-state stability of infinite-dimensional systems: recent results and open questions[END_REF][START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF].

A particular class of infinite-dimensional systems is that of time-delay systems. Due to the peculiarities of this subclass, it has been the subject of specific ISS developments initiated with the works [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF][START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF]. ISS for time-delay systems has now become a mature topic, but some fundamental questions remain open.

In particular, a Lyapounov-Krasovskii functional (LKF) characterization of ISS was provided in [START_REF] Karafyllis | Input-to-Output Stability for systems described by retarded functional differential equations[END_REF][START_REF] Kankanamalage | On Lyapunov-Krasovskii Characterizations of Input-to-Output Stability[END_REF]. This characterization requests that the dissipation of the LKF along the system's solutions is expressed in terms of the LKF itself (LKF-wise dissipation). This requirement turns out to be rather unhandy in practice. More crucially, it is not in line with the LKF characterization of global asymptotic stability, in which the LKF is allowed to dissipate merely in terms of the current value of the solution's norm (point-wise dissipation) [START_REF] Krasovskii | Problems of the theory of stability of motion[END_REF]. To date, it is not known whether ISS can be ensured through a point-wise dissipation [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems?[END_REF],

although this question has recently received a positive answer for the weaker notion of integral ISS [START_REF] Chaillet | Lyapunov-Krasovskii characterizations of integral input-to-state stability of delay systems with non-strict dissipation rates[END_REF].

Even in the absence of exogenous inputs, point-wise dissipation is not yet fully understood. In particular, the possibility to ensure global exponential stability with a point-wise dissipation still constitutes an open question. A partial answer was given in [START_REF] Chaillet | A relaxed Lyapunov-Krasovskii condition for global exponential stability of Lipschitz time-delay systems[END_REF], but only for systems ruled by a globally Lipschitz vector field. A positive answer to this question would complement the existing arsenal to establish GES, including the Razumikhin approach [START_REF] Wang | Exponential stability for impulsive delay differential equations by Razumikhin method[END_REF] and Halanay's inequality [14, p. 378].

In this paper, we significantly enlarge the class of systems for which ISS or GES can be established under a point-wise dissipation. To that aim, we focus on the exp-ISS property, which is a particular case of ISS in which the influence of the initial state is requested to decay exponentially. In particular, exp-ISS ensures GES for the corresponding input-free system. We provide growth conditions under which a point-wise dissipation is enough to conclude exp-ISS. These conditions may take two forms: either left or right, depending on whether the increase rate or the decay rate is restricted. Both these conditions turn out to be automatically satisfied when the vector field is globally Lipschitz, but we

show through examples that the proposed growth conditions encompass a much wider class of systems, thus significantly generalizing the results in [START_REF] Chaillet | A relaxed Lyapunov-Krasovskii condition for global exponential stability of Lipschitz time-delay systems[END_REF].

While making the analysis simpler, the use of a point-wise dissipation complicates the analysis of robustness to modeling errors or parameter uncertainties, as compared to an LKF-wise dissipation or a history-wise one (in which the LKF is requested to dissipate in terms of the whole state history norm).

To address this issue, we allow for an additional quadratic positive term in the LKF's derivative. This term involves the full state history. Since the dissipation is only in terms of the current solution's norm, the derivative of the considered LKF is thus no longer guaranteed to be non-positive, even in the absence of an input. Despite this severe issue, we show that exp-ISS does hold provided that this extra term is sufficiently small. Our proofs being constructive, we actually provide explicit estimates of the strength of this additional quadratic term. We show through an example how this result can indeed be useful for robustness analysis with respect to modeling errors.

On our way to establish these results, we provide several technical lemmas that may be of interest on their own. In particular, we propose a novel sufficient condition for a property known as robust forward completeness or bounded reachability property, which plays a central role in the stability analysis of timedelay systems [START_REF] Mironchenko | Input-to-state stability of time-delay systems: criteria and open problems[END_REF][START_REF] Karafyllis | Is global asymptotic stability necessarily uniform for time-delay systems?[END_REF]. We also show that exp-ISS can be reformulated as two specific inequalities on the system's solutions.

1 ≥ t 2 ≥ 0, u [t1,t2] denotes the restriction of u to [t 1 , t 2 ], namely u : [t 1 , t 2 ] → R m is defined as u [t1,t2] (t) := u(t) for all t ∈ [t 1 , t 2 ]. Given a continuously differentiable function V : R n → R, ∇V denotes its gradient. Given a ∈ R ≥0 , ⌈a⌉ denotes the smallest q ∈ N for which q ≥ a. A function α : R ≥0 → R ≥0 is said to be of class N if it is continuous, non-decreasing and zero at zero. It is said of class K if, in addi- tion, it is increasing. It is said to be of class K ∞ if it is of class K and satisfies lim s→+∞ α(s) = +∞. Given a functional V : X n → R ≥0 , its Driver derivative D + V : X n × R n → [-∞,
+∞] is defined, for all φ ∈ X n and all w ∈ R n , as

D + V (φ, w) := lim sup h→0 + V (φ h,w )-V (φ) h
where, for each h ∈ [0, ∆), φ h,w ∈ X n is given by

φ h,w (s) :=    φ(s + h), if s ∈ [-∆, -h), φ(0) + (h + s)w, if s ∈ [-h, 0].

Context

Global exponential stability

We start by considering input-free delay systems, namely:

ẋ(t) = f 0 (x t ), (1)
where x t ∈ X n and f 0 : X n → R n is a vector field which is Lipschitz on bounded sets and satisfies f 0 (0) = 0. For such class of systems, we recall the definition of global exponential stability.

Definition 1 (GES) The origin of (1) is said to be globally exponentially stable (GES) if there exist k, η > 0 such that, for all x 0 ∈ X n , the corresponding solution of (1) satisfies |x(t)| ≤ k x 0 e -ηt for all t ≥ 0.

The GES property therefore ensures that all solutions converge at the exponential rate η and that their transient overshoot is bounded by a linear function of the initial state norm. A powerful tool to establish GES is through a Lyapunov-Krasovskii functional (LKF). In particular, we recall the following from [START_REF] Krasovskii | Problems of the theory of stability of motion[END_REF][START_REF] Pepe | Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form[END_REF].

Theorem 1 (Existing LKF characterization for GES) The following statements are equivalent:

i) the origin of (1) is GES

ii) there exist a functional V : X n → R ≥0 , Lipschitz on bounded sets, and a, a, a, ρ > 0 such that, for all φ ∈ X n ,

a|φ(0)| ρ ≤ V (φ) ≤ a φ ρ D + V (φ, f 0 (φ)) ≤ -aV (φ)
iii) there exist a functional V : X n → R ≥0 , Lipschitz on bounded sets, and a, a, a > 0 such that, for all φ ∈ X n ,

a φ ≤ V (φ) ≤ a φ D + V (φ, f 0 (φ)) ≤ -a φ .
In this statement, D + V (φ, f (φ, v)) denotes Driver's derivative of V along the system's solutions, as recalled in the Notation paragraph. In particular, D + V (x t , f (x t , u(t)) coincides almost everywhere with the upper-right Dini derivative of t → V (x t ) on its maximal interval of existence: see [START_REF] Driver | Existence and stability of solutions of a delay-differential system[END_REF] and [START_REF] Pepe | On Liapunov-Krasovskii functionals under Caratheodory conditions[END_REF]Theorem 2].

Beyond the fact that ρ does not need to be equal to 1 in Item ii), there are two key differences between Items ii) and iii). The first one lies in the fact that the LKF V in Item iii) is coercive, in the sense that it can only vanish if the whole state history is identically zero. On the contrary, the LKF in Item ii) is lower-bounded only in terms of |φ(0)|, meaning that it may vanish even if φ = 0.

The possibility offered by Item ii) to consider non-coercive LKFs turns out to be very convenient in practice, whereas the fact that V is coercive in Item iii)

often proves useful to conduct further robustness analysis.

The second difference is in the way the LKF dissipates along solutions: in Item ii), this dissipation is in terms of the LKF itself (LKF-wise dissipation), whereas, in Item iii), the dissipation is requested to involve the whole history norm (history-wise dissipation). When the LKF is coercive, there is no qualitative difference between these two types of dissipation, but LKF-wise dissipation does not necessarily guarantee a history-wise dissipation when V is not coercive.

To sum up, while Item ii) is more convenient to establish GES, Item iii) constitutes a powerful converse result once GES is known to hold.

Exponential ISS

We may consider the impact of an exogenous input on the system, which then takes the form

ẋ(t) = f (x t , u(t)), (2)
where x t ∈ X n , u ∈ U m , and f : X n ×R m → R n is a functional which is Lipschitz on bounded sets and satisfies f (0, 0) = 0. A powerful way to assess robustness with respect to the input signal u is through the input-to-state stability (ISS) framework. The ISS property requests in particular that the steady-state error of solutions is small if the applied input is of sufficiently small amplitude [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF][START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF].

In this paper, we focus on the following particular declination of ISS.

Definition 2 (Exp-ISS, linear gain) The system (2) is said to be exponentially input-to-state stable (exp-ISS) if there exist k, η > 0 and µ ∈ N such that, for all x 0 ∈ X n and all u ∈ U m , the corresponding solution of (2) satisfies

|x(t)| ≤ k x 0 e -ηt + µ( u [0,t] ), ∀t ≥ 0.
It is said to be exp-ISS with linear gain if, in addition, there exists µ 0 ≥ 0 such that µ(s) = µ 0 s for all s ≥ 0.

This property has already been used in the literature of infinite-dimensional systems. In particular, it was studied in [START_REF] Mironchenko | Monotonicity methods for inputto-state stability of nonlinear parabolic PDEs with boundary disturbances[END_REF] in the context of parabolic partial differential equations. Before that, it was used for general infinite-dimensional systems under the name eISS [START_REF] Dashkovskiy | Input-to-state stability of infinitedimensional control systems[END_REF]. It should not be confused with the exponential ISS notion employed in [START_REF] Praly | Stabilization in spite of matched unmodelled dynamics and an equivalent definition of Input-to-State Stability[END_REF], which rather considers ISS with respect to a filtered version of the input.

The above exp-ISS property clearly ensures 0-GES, meaning that the origin of the input-free system ẋ(t) = f (x t , 0) is GES. But it also ensures that, in response to a bounded input u, any solution is attracted by a µ( u )-neighborhood of the origin. In particular, solutions are bounded in response to any bounded input, the steady-state error is small for sufficiently small inputs, and solutions converge to the origin in response to any vanishing input. In the particular case of exp-ISS with linear gain, the steady-state error is at most proportional to the amplitude of the applied input. This feature mimics what happens for linear systems and turns out particularly useful when invoking small-gain results for the stability analysis of interconnected systems [START_REF] Karafyllis | A small-gain theorem for a wide class of feedback systems with control applications[END_REF].

Exp-ISS can be established by existing LKF tools. In particular, the following result can be proved using classical manipulations.

Proposition 1 (LKF-wise dissipation for exp-ISS) Assume that there exist a functional V : X n → R ≥0 which is Lipschitz on bounded sets, constants a, a, a, ρ > 0 and a function γ ∈ N such that, for all φ ∈ X n and all v ∈ R m ,

a|φ(0)| ρ ≤ V (φ) ≤ a φ ρ (3)
D + V (φ, f (φ, v)) ≤ -aV (φ) + γ(|v|). (4)
Then (2) is exp-ISS. Moreover, if there exists γ 0 ≥ 0 such that γ(s) = γ 0 s ρ for all s ≥ 0, then (2) is exp-ISS with linear gain.

Just like Theorem 1, this result requests a LKF-wise dissipation. Interestingly, Proposition 1 can actually be strengthened by allowing a positive square term of the state history norm in the expression of the LKF's derivative, as stated next.

Theorem 2 (Alternative LKF-wise dissipation for exp-ISS) Assume that there exist a functional V : X n → R ≥0 which is Lipschitz on bounded sets, constants a, a, a, ρ > 0, c ≥ 0, and a function γ ∈ N such that, for all φ ∈ X n and

all v ∈ R m , a|φ(0)| ρ ≤ V (φ) ≤ a φ ρ (5)
D + V (φ, f (φ, v)) ≤ -aV (φ) + c φ ρ + γ(|v|). (6)
Then, provided that

c < aae -a∆ , (7)
the system (2) is exp-ISS . If, in addition, there exists γ 0 ≥ 0 such that γ(s) = γ 0 s ρ for all s ≥ 0, then (2) is exp-ISS with linear gain.

This result, proved in Section 5.2, suggests some robustness of the exp-ISS property: as compared to Proposition 1, the additional positive term c φ ρ does not compromise exp-ISS provided that c is small enough to satisfy [START_REF] Karafyllis | Input-to-Output Stability for systems described by retarded functional differential equations[END_REF]. This would come as no surprise if the considered LKF was coercive (meaning lower bounded by a term of the form a φ ρ ), which is however not requested here.

Note that, even in the input-free case, this result seems to be novel and provides a useful sufficient condition for GES (simply consider v = 0 in Theorem 2).

Growth conditions for exp-ISS

While Theorems 1 and 2 provide powerful conditions for GES and exp-ISS, the requirement of a LKF-wise (or history-wise) dissipation often significantly complicates the analysis. It also lacks homogeneity with the existing LKF condition for global asymptotic stability, in which a point-wise dissipation is enough [START_REF] Krasovskii | Problems of the theory of stability of motion[END_REF]. Here, we investigate whether such a point-wise dissipation would be sufficient to ensure GES or exp-ISS.

Partial answers in this direction were given in [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems?[END_REF], where growth conditions were provided to ensure that a point-wise dissipation ensures ISS. For the GES property, it was shown in [START_REF] Chaillet | A relaxed Lyapunov-Krasovskii condition for global exponential stability of Lipschitz time-delay systems[END_REF] that such a point-wise dissipation is indeed sufficient provided that the vector field is globally Lipschitz, which constitutes a conservative constraint in practice. To date, it is not known whether a point-wise dissipation can be used to establish GES or exp-ISS, but the results presented below enlarge the class of systems for which this holds true by imposing specific growth conditions. These conditions take two possible forms: either right or left, depending on whether we assume a sufficiently slow increase or decrease.

We detail these results in two separate sections.

Right growth condition

Our first result ensures exp-ISS under a point-wise dissipation provided an upper bound on a specific function involving the vector field. In line with

Theorem 2, we additionally allow for the presence of a positive term in the state history norm, but the result is new also when this term is taken as zero. Its proof is reported in Section 5.3.

Theorem 3 (Exp-ISS under an alternative right growth condition) Assume that there exist a functional V : X n → R ≥0 which is Lipschitz on bounded sets, a, a > 0, c ≥ 0, and γ ∈ N such that, for all φ ∈ X n and all v ∈ R m ,

0 ≤ V (φ) ≤ a φ 2 (8)
D + V (φ, f (φ, v)) ≤ -a|φ(0)| 2 + c φ 2 + γ(|v|). (9)
Assume further that there exists a symmetric positive definite matrix P ∈ R n×n and a constant σ > 0 such that, for all φ ∈ X n and all v ∈ R m , φ(0

) ⊤ P f (φ, v) ≤ σ( φ 2 + γ(|v|)). ( 10
)
Then there exists c > 0 such that, if c ∈ [0, c), the system (2) is exp-ISS. If, in addition, there exists γ 0 ≥ 0 such that γ(s) = γ 0 s 2 for all s ≥ 0, then (2) is exp-ISS with linear gain.

This result provides a right growth condition on the vector field f under which a point-wise dissipation is enough to guarantee exp-ISS. It is worth stressing that condition [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems?[END_REF] holds automatically for any arbitrary positive definite matrix P when f is globally Lipschitz in the state, uniformly in the input, namely if there exists ℓ ≥ 0 such that

|f (φ, v) -f (ψ, v)| ≤ ℓ φ -ψ , ∀φ, ψ ∈ X n , v ∈ R n .
In this case, [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems?[END_REF] Rather surprisingly, the above result does not impose any lower bound on the considered LKF, not even in terms of |φ(0)|: see [START_REF] Kankanamalage | On Lyapunov-Krasovskii Characterizations of Input-to-Output Stability[END_REF]. In particular, it allows for non-coercive LKFs, which is precious from an application viewpoint.

An immediate consequence of Theorem 3 is the following sufficient condition for GES.

Corollary 1 (GES under a right growth condition) Assume that there exist a functional V : X n → R ≥0 which is Lipschitz on bounded sets, a, a > 0 and c ≥ 0 such that, for all φ ∈ X n , (8) holds and

D + V (φ, f 0 (φ)) ≤ -a|φ(0)| 2 + c φ 2 .
Assume further that there exists a symmetric positive definite matrix P ∈ R n×n and a constant σ > 0 such that, for all φ ∈ X n , φ(0

) ⊤ P f 0 (φ) ≤ σ φ 2 .
Then there exists c > 0 such that, if c ∈ [0, c), the origin of (1) is GES.

Remark 1 (Estimate of c) The proof shows that c can be picked in Theorem 3 and Corollary 1 as

1 c = min p m σ e -2∆ , 1 ap m 2p M e -2∆ ,
where p m , p M > 0 denote the smallest and largest eigenvalues of P respectively.

This may however constitute a conservative estimate of the maximal positive quadratic term appearing in Driver's derivative of the considered LKF.

The proof of Theorem 3 consists in explicitly constructing a coercive LKF with history-wise dissipation in order to invoke Theorem 1. It crucially relies on the following technical lemma, which originally appeared in [START_REF] Karafyllis | Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization[END_REF]Lemma 5.3].

Lemma 1 (From point-wise to LKF-wise) Given any continuously differ-

entiable function W : R n → R ≥0 , the functional V 0 : X n → R ≥0 defined as V 0 (φ) := max τ ∈[-∆,0] e 2τ W (φ(τ )), ∀φ ∈ X n ,
is Lipschitz on bounded sets and satisfies

V 0 (φ) > W (φ(0)) ⇒ D + V 0 (φ, f (φ, v)) ≤ -2V 0 (φ) V 0 (φ) = W (φ(0)) ⇒ D + V 0 (φ, f (φ, v)) ≤ max {-2V 0 (φ), ∇W (φ(0))f (φ, v)} .
We will see in the proof of Theorem 3 that this result turns out to be particularly useful not only to construct a coercive LKF based on a non-coercive one, but also to impose a LKF-wise dissipation rate.

Left growth condition

The growth condition of Theorem 3 imposes that solutions do not grow too fast. The following result shows that a similar statement can be obtained by requiring that they do not decay too fast. Its proof is provided in Section 5.5.

1 See (49) and (47).

Theorem 4 (Exp-ISS under a left growth condition) Assume that there exist a functional V : X n → R ≥0 which is Lipschitz on bounded sets, a, a, a > 0, c ≥ 0 and γ ∈ N such that, for all φ ∈ X n and v ∈ R n ,

a |φ(0)| 2 ≤ V (φ) ≤ a φ 2 (11)
D + V (φ, f (φ, v)) ≤ -a |φ(0)| 2 + c φ 2 + γ (|v|) . ( 12
)
Assume further that there exists a symmetric positive definite matrix P ∈ R n×n and a constant σ > 0 such that, for all φ ∈ X n and all v ∈ R m , φ(0

) ⊤ P f (φ, v) ≥ -σ φ 2 + γ (|v|) . ( 13
)
Then there exists c > 0 such that, if c ∈ [0, c), the system (2) is exp-ISS. If, in addition, there exists γ 0 ≥ 0 such that γ(s) = γ 0 s 2 for all s ≥ 0, then (2) is exp-ISS with linear gain.

Unlike Theorem 3, this result does require a lower bound on the LKF in terms of |φ(0)|. Note however that V is not requested to be coercive.

Remark 2 (Estimate of c) Although Theorem 4 is an existence result, its proof provides an explicit estimate of the constant c: see (59) where p m , p M > 0 denote the minimal and maximal eigenvalues of P respectively. In particular, c is independent of the function γ. However, here also the reader should be warned that the above estimate is conservative and in practice larger values of c could be allowed.

The following result is an immediate consequence of Theorem 4 for input-free systems.

Corollary 2 (GES under a left growth condition) Assume that there exist a functional V : X n → R ≥0 which is Lipschitz on bounded sets, a, a, a > 0 and c ≥ 0 such that, for all φ ∈ X n , (11) holds and

D + V (φ, f 0 (φ)) ≤ -a |φ(0)| 2 + c φ 2 . ( 14
)
Assume further that there exists a symmetric positive definite matrix P ∈ R n×n and a constant σ > 0 such that, for all φ ∈ X n , φ(0

) ⊤ P f 0 (φ) ≥ -σ φ 2 . ( 15
)
Then there exists a constant c > 0 such that, if c ∈ [0, c), the origin of (1) is GES.

The proof of Theorem 4 relies on the following two lemmas, which may be of independent interest. The first one, proved in Section 5.1, is used to show that, under the assumptions of Theorem 4, the solutions of ( for all T, r > 0, all x 0 ∈ X n , and all u ∈ U m , the corresponding solution of (2)

satisfies the following implication:

x 0 ≤ r, u ≤ r ⇒ sup t∈[0,T ] |x(t)| < +∞.
In other words, starting from any bounded set of initial states and considering any input lying in any given bounded set, solutions can only reach a bounded set over any finite time interval.

Lemma 2 (Sufficient condition for RFC) Assume that there exist a func-

tional V : X n → R ≥0 which is Lipschitz on bounded sets, a, c, c ≥ 0, α, α ∈ K ∞ ,
and γ ∈ N such that, for all φ ∈ X n and all v ∈ R m ,

α(|φ(0)|) ≤ V (φ) ≤ α( φ ) + c ( 16
)
D + V (φ, f (φ, v)) ≤ aα( φ ) + γ(|v|) + c. ( 17
)
Then the system (2) is robustly forward complete.

From the regularity assumptions made on both V and f , the upper bound ( 17) is necessarily satisfied for some α ∈ K ∞ , γ ∈ N , and a, c ≥ 0. The key requirement in the above lemma therefore lies in the fact that α can also be picked as a lower bound on V , as imposed by [START_REF] Karafyllis | Is global asymptotic stability necessarily uniform for time-delay systems?[END_REF].

The second lemma allows to convert the proof of exp-ISS into specific inequalities on the solutions norm.

Lemma 3 (Exp-ISS characterization) The system (2) is exp-ISS if and only if there exist constants ℓ, T > 0, λ ∈ (0, 1) and a function µ ∈ N such that, for all x 0 ∈ W n and all u ∈ U m , its corresponding solution satisfies

x t ≤ ℓ x 0 + µ u [0,t] , ∀t ∈ [0, T ], (18)
x

T ≤ λ x 0 + µ u [0,T ] . (19)
Moreover, (2) is exp-ISS with linear gain if and only if the above holds with a linear function µ ∈ N .

Note that this result allows to consider only initial states x 0 that are absolutely continuous, which significantly increases the regularity of the resulting solutions. In particular it can be shown that, if V is Lipschitz on bounded sets then, given any x 0 ∈ W n and any u ∈ U m , the map t → V (x t ) is locally absolutely continuous (and consequently differentiable almost everywhere) on its domain of existence: see [27, Theorem 5, Remark 6] or [7, Lemma 2.5].

Examples

We illustrate our results through the following three academic examples.

Example 1 (Theorems 3 or 4) Consider the system

ẋ1 (t) = - 1 2 x 1 (t) + x 2 (t -∆) + x 2 (t) x 1 (t) 2 + x 2 (t -∆) 2 (20a) ẋ2 (t) = -2x 2 (t) -x 1 (t) x 1 (t) 2 + x 2 (t -∆) 2 + u(t). ( 20b
)
and let f (x t , u(t)) denote its right-hand side. Consider the LKF defined for all

φ = (φ 1 , φ 2 ) ⊤ ∈ X 2 as V (φ) := φ 1 (0) 2 + φ 2 (0) 2 + 2 0 -∆ φ 2 (τ ) 2 dτ. (21)
Then it holds that

|φ(0)| 2 ≤ V (φ) ≤ (1 + 2∆) φ 2 , ∀φ ∈ X 2 , (22)
which establishes (8) and (11) with a = 1 and a = 1 + 2∆. For all φ ∈ X 2 and all v ∈ R, its Driver's derivative along the system's solutions reads

D + V (φ, f (φ, v)) = -φ 1 (0) 2 + 2φ 1 (0)φ 2 (-∆) -4φ 2 (0) 2 + 2φ 2 (0)v + 2 φ 2 (0) 2 -φ 2 (-∆) 2 ≤ -φ 1 (0) 2 + 1 2 φ 1 (0) 2 + 2φ 2 (-∆) 2 -4φ 2 (0) 2 + φ 2 (0) 2 + v 2 + 2 φ 2 (0) 2 -φ 2 (-∆) 2 ≤ - 1 2 φ 1 (0) 2 -φ 2 (0) 2 + v 2 ≤ - 1 2 |φ(0)| 2 + v 2 , (23)
thus establishing (9) and ( 12) with a = 1/2 and γ(s) = s 2 for all s ≥ 0. Finally,

φ(0) ⊤ f (φ, v) = - 1 2 φ 1 (0) 2 + φ 1 (0)φ 2 (-∆) -2φ 2 (0) 2 + φ 2 (0)v ≤ - 1 4 φ 1 (0) 2 + φ 2 (-∆) 2 - 3 2 φ 2 (0) 2 + 1 2 v 2 ≤ φ 2 + γ(|v|), (24)
thus fulfilling (10) with σ = 1 and P = I. Hence, Theorem 3 applies and we conclude that, for any value of the delay ∆ ≥ 0, ( 20) is exp-ISS with linear gain.

Similarly,

φ(0) ⊤ f (φ, v) = - 1 2 φ 1 (0) 2 + φ 1 (0)φ 2 (-∆) -2φ 2 (0) 2 + φ 2 (0)v ≥ -φ 1 (0) 2 - 1 2 φ 2 (-∆) 2 - 5 2 φ 2 (0) 2 - 1 2 v 2 ≥ -3( φ 2 + γ(|v|)), (25)
thus making the left growth condition (13) fulfilled with σ = 3. So Theorem 4.

can also be invoked to conclude exp-ISS.

Note that the vector field in (20) is not globally Lipschitz, thus making [12, Corollary 1] inapplicable to conclude exp-ISS. Also, it is worth stressing that the considered LKF V does not admit an LKF-wise dissipation rate, so Theorem 2 cannot be invoked either. It is however fair to stress that such a LKF-wise dissipation could have been obtained by considering the following LKF:

W (φ) = φ 1 (0) 2 + φ 2 (0) 2 + 0 -∆ e -ρτ φ 2 (τ ) 2 dτ,
for some conveniently chosen ρ > 0. The addition of an exponential term under the integral is standard in the analysis of time-delay systems. While this trick does work for some relevant classes of systems [28, Lemma 1], it is not universal and the possibility offered by Theorems 3 and 4 to consider a point-wise dissipation significantly simplifies the analysis.

The second example illustrates how the c φ 2 term appearing in ( 9) and ( 12) can be used to guarantee some robustness to modeling errors.

Example 2 (Theorem 3 or 4 for robustness) Consider the system

ẋ1 (t) = - 1 2 x 1 (t) + x 1 (t -∆) + x 2 (t) x 1 (t) 2 + x 2 (t -∆) 2 + εd 1 (x t ) (26a) ẋ1 (t) = -2x 2 (t) -x 1 (t) x 1 (t) 2 + x 2 (t -∆) 2 + u(t) + εd 2 (x t ). ( 26b
)
The only difference with respect to [START_REF] Mironchenko | Monotonicity methods for inputto-state stability of nonlinear parabolic PDEs with boundary disturbances[END_REF] lies in the extra terms d 1 (x t ) and d 2 (x t )

that represent modeling uncertainties. ε ∈ R represents the intensity of these uncertainties. We assume that d 1 , d 2 : X 2 → R are Lipschitz on bounded sets and satisfy, for all φ ∈ X 2 ,

|d i (φ)| ≤ φ , ∀i ∈ {1, 2}. (27)
Considering the LKF V defined in [START_REF] Dashkovskiy | Input-to-state stability of infinitedimensional control systems[END_REF], it can be seen from ( 23) and (27) that

D + V (φ, f (φ, v)) ≤ - 1 2 |φ(0)| 2 + v 2 + 2ε (φ 1 (0)d 1 (φ) + φ 2 (0)d 2 (φ)) ≤ - 1 2 |φ(0)| 2 + v 2 + 4|ε| φ 2 ,
where f is such that the right-hand side of (26) reads f (x t , u(t)). We conclude, with either Theorem 3 or Theorem 4, that there exists ε > 0 so that exp-ISS is preserved under such model uncertainties provided that ε ∈ (-ε, ε). Recalling that, for this example, a = σ = p m = p M = 1 and a = 1/2, the estimate of ε given by Theorem 3 (see Remark 1) reads ε1 = e -4∆ /16, thus suggesting a decrease of robustness when the delay ∆ increases. Similarly, recalling that σ = 3 in the application of Theorem 4, we get from Remark 2 the following alternative estimate of this tolerable uncertainty:

ε2 = 1 8 ⌈2304(1 + 2∆) 4 ⌉ ∆ + 1 48(1+2∆) 2
, which can be checked to be greater than ε1 whenever ∆ ≥ 4.5. Combining Theorems 3 and 4 and letting ε := max{ε 1 , ε2 }, any ε ∈ (-ε, ε) preserves exp-ISS of (26) if the modeling uncertainty satisfies (27).

The following other variant of Example 1 shows that, on some occasions, condition [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems?[END_REF] can be fulfilled while condition ( 13) is violated.

Example 3 (Theorem 3, but not Theorem 4) Consider the system

ẋ1 (t) = - 1 2 x 1 (t) + x 1 (t -∆) + x 2 (t) x 1 (t) 2 + x 2 (t -∆) 2 (28a) ẋ1 (t) = -2x 2 (t) -x 2 (t) 3 -x 1 (t) x 1 (t) 2 + x 2 (t -∆) 2 + u(t). ( 28b
)
The only difference with respect to [START_REF] Mironchenko | Monotonicity methods for inputto-state stability of nonlinear parabolic PDEs with boundary disturbances[END_REF] is the extra term -x 2 (t) 3 appearing in (28b). This term brings only non-positive terms in Driver's derivative of the function V defined in [START_REF] Dashkovskiy | Input-to-state stability of infinitedimensional control systems[END_REF], meaning that (23) remains valid when f denotes the vector field in [START_REF] Or Lowski | Adaptive control of Lipschitz time-delay systems by sigma modification with application to neuronal population dynamics[END_REF]. For the same reason, (24) remains valid as well. Theorem 3 thus applies and the system (28) is exp-ISS with linear gain. Nevertheless,

φ(0) ⊤ f (φ, v) = - 1 2 φ 1 (0) 2 + φ 1 (0)φ 1 (-∆) -2φ 2 (0) 2 -φ 2 (0) 4 + φ 2 (0)v.
The term -φ 2 (0) 4 cannot be lower-bounded by a term proportional toφ 2 .

Reasoning similarly, we can see that (13) cannot be fulfilled for this system, no matter the choice of the positive definite matrix P ∈ R 2×2 , thus making Theorem 4 inapplicable.

Proofs

Proof of Lemma 2

Invoking the density of W n in X n and the continuity of solutions with respect to the initial state, it can be shown that robust forward completeness holds even if we restrict the analysis to initial states in W n . In other words, it is enough to show that, given any r, T > 0, there exist R > 0 such that, given any x 0 ∈ W n and any u ∈ U m , the following implication holds:

x 0 ≤ r , u ≤ r ⇒ sup t∈[0,T ] |x(t)| ≤ R. ( 29
)
Consider any x 0 ∈ W n and any u ∈ U m and let x(•) denote the maximal solution of ( 2 Integrating, it follows that, for all t ∈ [0, t max ),

V (x t ) ≤ V (x 0 ) + a t 0 α( x τ )dτ + t 0 (γ(|u(τ )|) + c) dτ.
We then get from ( 16) that, for all t ∈ [0, t max ),

α(|x(t)|) ≤ α( x 0 ) + c + a t 0 α( x τ )dτ + t 0 (γ(|u(τ )|) + c) dτ. ( 30
)
It follows in particular that2 , for all t ∈ [∆, t max ),

α( x t ) ≤ α( x 0 ) + c + a t 0 α( x τ )dτ + t 0 (γ(|u(τ )|) + c) dτ. (31)
Moreover, for all t ∈ [0, min{∆, t max }), it holds that

α( x t ) = max max τ ∈[t-∆,0] α(|x(τ )|) , max τ ∈[0,t] α(|x(τ )|) ≤ max α( x 0 ) , max τ ∈[0,t] α(|x(τ )|) .
Using again (30) and observing that [START_REF] Chaillet | Lyapunov-Krasovskii characterizations of integral input-to-state stability of delay systems with non-strict dissipation rates[END_REF] imposes that α(s) + c ≥ α(s) for all s ≥ 0, it follows that, for all t ∈ [0, min{∆, t max }),

α( x t ) ≤ max α( x 0 ) , α( x 0 ) + c + a t 0 α( x τ )dτ + t 0 (γ(|u(τ )|) + c) dτ ≤ α( x 0 ) + c + a t 0 α( x τ )dτ + t 0 (γ(|u(τ )|) + c) dτ. ( 32
)
In view of (31), we conclude that (32) holds for all t ∈ [0, t max ). Invoking

Gronwall-Bellman's inequality, it follows that, for all t ∈ [0, t max ),

α( x t ) ≤ (α( x 0 ) + c) e at + t 0 (γ(|u(τ )|) + c) e a(t-τ ) dτ,
meaning that

x t ≤ α -1 (α( x 0 ) + c) e at + t 0 (γ(|u(τ )|) + c) e a(t-τ ) dτ . (33)
If t max were finite, then we would have lim sup t→t - max x t = +∞ [29, Theorem 3.2, p. 43], which is incompatible with (33). Hence t max = +∞. In particular, given any r, T > 0, if x 0 ≤ r and u ≤ r, then it holds that sup

t∈[0,T ] |x(t)| ≤ α -1 (α(r) + c) e aT + (γ(r) + c) e aT a ,
thus fulfilling (29) and establishing robust forward completeness.

Proof of Theorem 2

First observe that Lemma 2 readily ensures that all solutions of (2) exist on R ≥0 . Using density of C 1 ([-∆, 0]; R n ) in X n and continuity of solutions with respect to the initial state, it is enough to show that there exist k, η > 0 and

µ ∈ N such that, for all x 0 ∈ C 1 ([-∆, 0]; R n ) and all u ∈ U m , |x(t)| ≤ k x 0 e -ηt + µ( u [0,t] ), ∀t ≥ 0. ( 34
)
Considering any x 0 ∈ C 1 ([-∆, 0]; R n ) and any locally bounded u ∈ U m , since

V is Lipschitz on bounded sets, the mapping t → V (x t ) is locally absolutely continuous on R ≥0 (see [27, Theorem 5, Remark 6] or [7, Lemma 2.5]) and we get from (6) that

d dt V (x t ) ≤ -aV (x t ) + c x t ρ + γ (|u(t)|) , ∀t ≥ 0 a.e.
It follows that

d dt e at V (x t ) ≤ ce at x t ρ + e at γ (|u(t)|) , ∀t ≥ 0 a.e.
Since t → V (x t ) is locally absolutely continuous, we get for all t ≥ 0 that

e at V (x t ) -V (x 0 ) = t 0 d dτ (e aτ V (x τ )) dτ ≤ c t 0 e aτ x τ ρ dτ + t 0 e aτ γ (|u(τ )|) dτ.
It follows from ( 5) that, for all t ≥ 0,

|x(t)| ρ ≤ e -at a a x 0 ρ + c t 0 e aτ x τ ρ dτ + t 0 e aτ γ (|u(τ )|) dτ . (35)
In particular, for all t ≥ ∆, we get that

x t ρ ≤ e -a(t-∆) a a x 0 ρ + c t 0 e aτ x τ ρ dτ + t 0 e aτ γ (|u(τ )|) dτ . (36)
Moreover, using again (35), it holds for all t ∈ [0, ∆] that

x t ρ = max max τ ∈[t-∆,0] |x(τ )| ρ , max τ ∈[0,t] |x(τ )| ρ ≤ max x 0 ρ , 1 a a x 0 ρ + c t 0 e aτ x τ ρ dτ + t 0 e aτ γ (|u(τ )|) dτ ≤ 1 a a x 0 ρ + c t 0 e aτ x τ ρ dτ + t 0 e aτ γ (|u(τ )|) dτ ,
where the last bound comes from the fact that a ≥ a as ensured by [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF]. We conclude that (36) actually holds for all t ≥ 0, which implies that, for all t ≥ 0,

x t ρ ≤ ae -a(t-∆) a x 0 ρ + e a∆ aa sup τ ∈[0,t] c x τ ρ + γ u [0,τ ] .
Invoking [4, Lemma 7.1], we conclude that for every ε > 0 there exists δ > 0 (independent of the solution) such that, for all t ≥ 0,

x t ρ ≤ ae a∆ a x 0 ρ e -δt + e a∆ aa (1 + ε) sup τ ∈[0,t] c x τ ρ + γ( u [0,τ ] ) e -δ(t-τ ) ,
which in turn implies that

sup τ ∈[0,t] x τ ρ e δτ ≤ ae a∆ a x 0 ρ + ce a∆ aa (1 + ε) sup τ ∈[0,t] x τ ρ e δτ + e a∆ aa (1 + ε)γ( u [0,t] )e δt . (37)
Pick ε > 0 small enough that ce a∆ (1 + ε)/aa < 1, the existence of which is ensured by [START_REF] Karafyllis | Input-to-Output Stability for systems described by retarded functional differential equations[END_REF]. Then the constant ξ := 1ce a∆ (1 + ε)/aa is positive and we get from (37) that, for all t ≥ 0,

x t ρ e δt ≤ sup τ ∈[0,t] x τ ρ e δτ ≤ ae a∆ aξ x 0 ρ + e a∆ aaξ (1 + ε)γ( u [0,t] )e δt .
The function s → s 1/ρ being increasing, it holds that (r + s) 1/ρ ≤ (2r) 1/ρ + (2s) 1/ρ for all r, s ≥ 0. We conclude that

x t ≤ 2ae a∆ aξ 1/ρ
x 0 e -δt/ρ + 2e a∆ aaξ (1 + ε)

1/ρ γ u [0,t] 1/ρ ,
which establishes (34) with η := δ and

k := 2ae a∆ aξ 1/ρ , µ(s) := 2e a∆ aaξ (1 + ε) 1/ρ γ (s) 1/ρ , ∀s ≥ 0.
We conclude that (2) is exp-ISS. Finally, if γ(s) = γ 0 s ρ for some γ 0 ≥ 0, then the obtained gain µ is linear and ( 2) is exp-ISS with linear gain.

Proof of Theorem 3

Consider the functional V 0 : X n → R ≥0 defined as

V 0 (φ) := max τ ∈[-∆,0] e 2τ φ(τ ) ⊤ P φ(τ ), ∀φ ∈ X n . (38)
Then it holds that

e -2∆ p m φ 2 ≤ V 0 (φ) ≤ p M φ 2 , (39)
where p m , p M > 0 denote the smallest and largest eigenvalues of P respectively.

In view of Lemma 1, V 0 is also Lipschitz on bounded sets and satisfies

V 0 (φ) > φ(0) ⊤ P φ(0) ⇒ D + V 0 (φ, f (φ, v)) ≤ -2V 0 (φ) (40) V 0 (φ) = φ(0) ⊤ P φ(0) ⇒ D + V 0 (φ, f (φ, v)) ≤ 2 max -V 0 (φ), φ(0) ⊤ P f (φ, v) . (41)
Now let W (φ) := V (φ) + εV 0 (φ) for some ε > 0 to be chosen later. Then W is Lipschitz on bounded sets and it holds that

εe -2∆ p m φ 2 ≤ W (φ) ≤ (a + εp M ) φ 2 . ( 42
)
Thus, W a coercive LKF. In view of ( 40)-( 41), we need to consider two cases.

Case 1: V 0 (φ) > φ(0) ⊤ P φ(0). Then we get from ( 9) and (40) that

D + W (φ, f (φ, v)) ≤ -2εV 0 (φ) -a|φ(0)| 2 + c φ 2 + γ(|v|) ≤ -2εp m e -2∆ -c φ 2 + γ(|v|), (43)
where the last bound comes from (39).

Case 2: V 0 (φ) = φ(0) ⊤ P φ(0). Then we have from (39) that

|φ(0)| 2 ≥ V 0 (φ) p M ≥ p m p M e -2∆ φ 2 . ( 44
)
It follows from ( 9) and (41) that

D + W (φ, f (φ, v)) ≤ -a|φ(0)| 2 + c φ 2 + γ(|v|) + 2ε max -V 0 (φ) , φ(0) ⊤ P f (φ, v) ≤ -a p m p M e -2∆ -c φ 2 + γ(|v|) + 2ε max -V 0 (φ) , φ(0) ⊤ P f (φ, v) . (45)
Here again, we need to consider two subcases.

Subcase 2.1: -V 0 (φ) ≥ φ(0) ⊤ P f (φ, v). Then we get from (45) that

D + W (φ, f (φ, v)) ≤ -a p m p M e -2∆ -c φ 2 + γ(|v|) -2εV 0 (φ) ≤ -a p m p M e -2∆ -c φ 2 + γ(|v|). ( 46
)
Subcase 2.2: -V 0 (φ) < φ(0) ⊤ P f (φ, v). Then we make use of condition [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems?[END_REF] to get from (45) that

D + W (φ, f (φ, v)) ≤ -a p m p M e -2∆ -c φ 2 + γ(|v|) + 2εφ(0) ⊤ P f (φ, v) ≤ -a p m p M e -2∆ -c φ 2 + γ(|v|) + 2εσ( φ 2 + γ(|v|)) ≤ -a p m p M e -2∆ -c -2εσ φ 2 + (1 + 2εσ)γ(|v|).
By picking

ε := a p m 4σp M e -2∆ , (47)
it follows in particular that

D + W (φ, f (φ, v)) ≤ -a p m 2p M e -2∆ -c φ 2 + (1 + 2εσ)γ(|v|). (48)
Combining ( 43), ( 46) and (48), we obtain the following inequality, valid for all φ ∈ X n and all v ∈ R m :

D + W (φ, f (φ, v)) ≤ -(c -c) φ 2 + γ(|v|)
where

c := min 2ε , a 2p M p m e -2∆ (49)
γ(s) := (1 + 2εσ)γ(s), ∀s ≥ 0.

Using (42), we obtain that

D + W (φ, f (φ, v)) ≤ - c -c a + εp M W (φ) + γ(|v|).
We conclude that W has a history-wise dissipation rate provided that c ∈ [0, c).

Exp-ISS of (2) then follows from classical LKF sufficient conditions for exp-ISS, such as the one recalled as Proposition 1. In the case when γ is a square function, we see from (50) that γ results in a square function. Exp-ISS with linear gain then also follows from Proposition 1.

Proof of Lemma 3

If ( 2) is exp-ISS then there exist constants k ≥ 1, η > 0 and a function µ ∈ N such that, for all x 0 ∈ W n and all u ∈ U m , its solution satisfies

|x(t)| ≤ k x 0 e -ηt + µ( u [0,t] ), ∀t ≥ 0, (51)
which implies in particular that

x t ≤ k x 0 e -ηt + µ( u [0,t] ), ∀t ≥ ∆. ( 52
)
In addition, we have from (51) that, for all t ∈ [0, ∆],

x t = max max τ ∈[t-∆,0] |x(τ )| , max τ ∈[0,t] |x(τ )| ≤ max x 0 , k x 0 e -ηt + µ( u [0,t] ) .
Since k ≥ 1, it follows that, for all t ∈ [0, ∆],

x t ≤ k x 0 e -η(t-∆) + µ( u [0,t] ), ∀t ∈ [0, ∆]. (53)
In view of (52), we conclude that (53) actually holds for all t ≥ 0. In particular, given any λ ∈ (0, 1), inequalities [START_REF] Driver | Existence and stability of solutions of a delay-differential system[END_REF] and [START_REF] Pepe | On Liapunov-Krasovskii functionals under Caratheodory conditions[END_REF] hold with ℓ := ke η∆ and T := ∆ + 1 η ln k λ . We now proceed to showing the converse. To that aim, suppose that there exist ℓ, T > 0, λ ∈ (0, 1) and µ ∈ N such that, for all x 0 ∈ W n and all u ∈ U m , the solution satisfies [START_REF] Driver | Existence and stability of solutions of a delay-differential system[END_REF] and [START_REF] Pepe | On Liapunov-Krasovskii functionals under Caratheodory conditions[END_REF]. Invoking the density of W n in X n and the continuity of solutions with respect to the initial state, it can be shown that [START_REF] Driver | Existence and stability of solutions of a delay-differential system[END_REF] and [START_REF] Pepe | On Liapunov-Krasovskii functionals under Caratheodory conditions[END_REF] actually hold for all x 0 ∈ X n and all u ∈ U m . Considering any x 0 ∈ X n and any u ∈ U m , let x(•) denote the corresponding solution of (2).

Clearly, ( 18)-( 19) and the semigroup property imply that x(•) is defined on R ≥0 .

Using [START_REF] Pepe | On Liapunov-Krasovskii functionals under Caratheodory conditions[END_REF] inductively in conjunction with the semigroup property, we conclude that, for every i ∈ N ≥1 ,

x iT ≤ λ i x 0 + µ u [0,iT ] i-1 j=0 λ j ≤ λ i x 0 + 1 1 -λ µ u [0,iT ] .
Defining η := 1 T ln 1 λ > 0, we get that

x iT ≤ e -iηT x 0 + 1 1 -λ µ u [0,iT ] .
Using [START_REF] Driver | Existence and stability of solutions of a delay-differential system[END_REF] and the semigroup property, it follows that, for every i ∈ N ≥1 and for every τ ∈ [0, T ],

x iT +τ ≤ ℓe -iηT x 0 + ℓ 1 -λ + 1 µ u [0,iT +τ ] .
Notice that, by [START_REF] Driver | Existence and stability of solutions of a delay-differential system[END_REF], the above estimate holds for i = 0 as well. Since for every t ≥ 0 there exists i ∈ N and τ ∈ [0, T ) such that t = iT + τ , we obtain that

x t ≤ ℓe -η(t-τ ) x 0 + ℓ 1 -λ + 1 µ u [0,t] ≤ ℓe ηT x 0 e -ηt + ℓ 1 -λ + 1 µ u [0,t] ,
and exp-ISS follows. If µ is linear then the obtained ISS gain is linear too.

Proof of Theorem 4

Let p m , p M > 0 denote respectively the smallest and largest eigenvalues of P .

We assume that (11), ( 12) and ( 13) hold for some c ∈ (0, c], where c is a positive constant to be defined later. The proof for the case c = 0 is completely similar (and far easier). First observe that, by Lemma 2, the system is robustly forward complete. In particular, its solutions exist on R ≥0 . The proof relies on Lemma 3. Given any x 0 ∈ W n and any u ∈ U m , let x(•) denote the corresponding solution of (2). Since V is Lipschitz on bounded sets, the mapping t → V (x t ) is then locally absolutely continuous (as ensured by [27, Theorem 5, Remark 6] or [7, Lemma 2.5]), and we get from (12) that

d d t V (x t ) ≤ -a |x(t)| 2 + c x t 2 + γ (|u(t)|) , ∀t ≥ 0 a.e.
Integrating, we get that, for all t ≥ 0,

V (x t ) + a t 0 |x(τ )| 2 dτ ≤ V (x 0 ) + ct max τ ∈[0,t] x τ 2 + tγ u [0,t] . (54)
Using [START_REF] Chaillet | Lyapunov-Krasovskii characterizations of integral input-to-state stability of delay systems with non-strict dissipation rates[END_REF], it follows in particular that, for all t ≥ 0,

|x(t)| 2 ≤ a a x 0 2 + ct a max τ ∈[-∆,t] |x(τ )| 2 + t a γ u [0,t] .
Consequently, for all t ≥ 0,

max τ ∈[-∆,t] |x(τ )| 2 = max max τ ∈[-∆,0] |x(τ )| 2 , max τ ∈[0,t] |x(τ )| 2 ≤ max x 0 2 , a a x 0 2 + ct a max τ ∈[-∆,t] |x(τ )| 2 + t a γ u [0,t] ≤ a a x 0 2 + ct a max τ ∈[-∆,t] |x(τ )| 2 + t a γ u [0,t] ,
where we used the fact that a ≥ a as ensured by [START_REF] Chaillet | Lyapunov-Krasovskii characterizations of integral input-to-state stability of delay systems with non-strict dissipation rates[END_REF]. In particular, for all

t ∈ [0, a 2c ], it holds that max τ ∈[-∆,t] |x(τ )| 2 ≤ a a -ct x 0 2 + t a -ct γ u [0,t] ≤ 2a a x 0 2 + 2t a γ u [0,t] , (55)
which in turn implies that, for all T ∈ [0, a 2c ] and all t ∈ [0, T ],

x t ≤ 2a a x 0 + 2T a γ u [0,t] . (56)
On the other hand, we have from ( 11) and (54) that, for all t ∈ [0, T ],

T 0 |x(τ )| 2 dτ ≤ a a x 0 2 + cT a max τ ∈[0,t] x τ 2 + T a γ u [0,T ] . (57)
Combining ( 55) and (57), we get for all T ∈ 0, a 2c :

T 0 |x(τ )| 2 dτ ≤ a a 1 + 2cT a x 0 2 + T a 1 + 2cT a γ( u [0,T ] ) ≤ 2a a x 0 2 + 2T a γ( u [0,T ] ). ( 58
) Pick ε := p m a 2 16a 2 σ q := p M σε 2 , T := q(∆ + ε), c := a 2T , (59)
and consider the function µ ∈ N defined for all s ≥ 0 as

µ(s) := 4 √ a max √ 2T , a p m p M T aqε + σε 1 + 2T a γ(s). (60)
Then it follows readily from the definition of c in (59) that, for all c ∈ (0, c],

T ∈ 0, a 2c , (61)
and it can be checked from the definitions of q and ε in (59) that

qε p M p m a 2a - 4εσa a > 2a a .
Consequently, there exists λ ∈ (0, 1) such that

qε p M p m a 2a λ 2 - 4εσa a > 2a a . (62)
Also, considering separately the two possible values that the max in (60) may take, it can be checked that the following two properties hold for all s ≥ 0:

µ(s) - 2T a γ(s) ≥ µ(s) 2 (63) qε p M ap m 8a µ(s) 2 -2σε 1 + 2T a γ(s) ≥ 2T a γ(s). (64)
We show next that the estimate

x T ≤ λ x 0 + µ u [0,T ] (65)
holds with the constant T given in (59), the function µ provided in (60), and any λ ∈ (0, 1) satisfying (62) with the constants provided in (59). The proof of (65) is made by contradiction. Suppose that it does not hold for some x 0 ∈ W n and some u ∈ U m , namely:

x T > λ x 0 + µ( u [0,T ] ).

(66)

We know from (56) and the semi-group property that

x T ≤ 2a a x t + 2T a γ u [0,T ] , ∀t ∈ [0, T ].
This, combined with (66), gives for all t ∈ [0, T ]:

x t > a 2a λ x 0 + µ( u [0,T ] ) - 2T a γ( u [0,T ] ) .
It follows that, for each i ∈ {0, . . . , q-1}, there exists

t i ∈ [i(∆ + ε), i(∆ + ε) + ∆] such that |x(t i )| > a 2a λ x 0 + µ( u [0,T ] ) - 2T a γ( u [0,T ] ) ≥ λ a 2a x 0 + 1 2 a 2a µ( u [0,T ] ), (67)
where we used (63) to obtain the last bound. On the other hand, inequality [START_REF] Wang | Exponential stability for impulsive delay differential equations by Razumikhin method[END_REF] Confronting this lower bound with the upper bound provided in (58), we obtain a contradiction. Thus, given any x 0 ∈ W n and any u ∈ U m , estimate (65) holds with the constant T given in (59), the function µ ∈ N proposed in (60), and any λ ∈ (0, 1) satisfying (62) for the constants introduced in (59). Exp-ISS then directly follows from Lemma 3, by invoking (56) and (65). Finally, in view of (60), the obtained gain µ is indeed linear if γ is a square function.

Conclusion

We have presented two growth conditions under which a point-wise dissipation is enough to establish exp-ISS for time-delay systems. These results are new even in the input-free case and significantly extend the class of systems that can be addressed by a point-wise dissipation. They also show that a positive quadratic term involving the whole state history function can be added to the LKF derivative without compromising exp-ISS. Although probably conservative, explicit estimates are provided on the strength of this additional term.

We have shown with an example how this extra term can be useful to conduct robustness analysis with respect to modeling uncertainties.

Notation.

  Given n ∈ N ≥1 and ∆ ≥ 0, X n denotes the set of all continuous functions from [-∆, 0] to R n , whereas W n denotes the Sobolev space of absolutely continuous functions mapping [-∆, 0] into R n with essentially bounded derivative, and C 1 ([-∆, 0]; R n ) denotes the set of all continuously differentiable functions from [-∆, 0] to R n . Given a continuous signal x : [-∆, T ) → R n with T ∈ (0, +∞] and any t ∈ [0, T ), x t ∈ X n denotes the history function defined as x t (τ ) := x(t + τ ) for all τ ∈ [-∆, 0]. Given a non-empty interval I ⊂ R and an essentially bounded Lebesgue measurable signal u : I → R m , u := ess sup t∈I |u(t)|, where | • | denotes the Euclidean norm. Given m ∈ N ≥1 , U m denotes the set of all signals u : R ≥0 → R m that are Lebesgue measurable and locally essentially bounded. Given u ∈ U m and t

  turns out to be satisfied with any γ ∈ N satisfying |f (0, v)| 2 ≤

γ(|v|) for all v ∈ R m . Nevertheless, as can be seen from Examples 1-3 below, the proposed growth condition is far from being restricted to such class of systems, thus significantly generalizing [12,

Corollary 1]

. The requirement

[START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems?[END_REF]

can be interpreted as a condition to guarantee that the current history norm x t is not too much greater than the current solution norm |x(t)|. More precisely, it imposes a restriction on the increase rate of the function t → x(t) ⊤ P x(t).

  implies that, for almost all t ≥ 0, Recalling that t i ≤ i(∆ + ε) + ∆, it holds from the definition of T in (59) thatt i + ε ≤ T for all i ∈ {0, . . . , q -1}. It follows from (55) that, for almost all t ∈ [t i , t i + ε],for all t ∈ [0, T ]. Noticing that the intervals (t i , t i + ε), i ∈ {0, ..., q -1}, are disjoint subsets of [0, T ], we conclude that Observe that x 0 = 0. Indeed, if x 0 was zero, then we would have from (56) and (63) that x T ≤ γ( u [0,T ] ) 2T /a ≤ µ( u [0,T ] )/2, which is incompatible with (66). Based on this, we finally get from (62), (64) and (68) that

	0	T	|x(τ )|	d dt 2 dτ ≥	x(t) ⊤ P x(t) ≥ -2σ x t q-1 i=0 ti+ε ti |x(τ )| 2 dτ	2 + γ u [0,t]	.
							≥	qε p M	p m λ 2 a 2a	-	4εσa a	x 0	2	(68)
									+	qε p M	p m a 8a	µ( u [0,T ] ) 2 -2σε 1 +	2T a	γ( u [0,T ] ) .
			d d t		x(t) ⊤ P x(t) ≥ -	4σa a	x 0	2 -2σ 1 +	2T a	γ( u [0,t] )
													≥ -	4σa a	x 0	2 -2σ 1 +	2T a	γ( u [0,T ] ).
									0	T	|x(τ )| 2 dτ >	4εσa a 2a a x 0 x 0 2 + 2 -2σε 1 + 2T a γ( u [0,T ] ). a 2T	γ( u [0,T ] ).
			|x(t)|	2 ≥	p m p M	|x(t i )|	2 -	4εσa p M a	x 0	2 -	2σε p M	1 +	2T a	γ( u [0,T ] ).
	Combining this with (67) yields
			|x(t)|	2 ≥	p m p M		λ	a 2a	x 0 +	1 2	2a a	µ( u [0,T ] )	2
							-	4εσa p M a	x 0	2 -	2σε p M	1 +	2T a	γ( u [0,T ] )
						≥	p m a p M a	λ 2 2	x 0	2 +	1 8	µ( u [0,T ] ) 2
							-	4εσa p M a	x 0	2 -	2σε p M	1 +	2T a	γ( u [0,T ] )
						≥	1 p M			p m λ 2 a 2a	-	4εσa a	x 0	2
							+	1 p M		p m a 8a	µ( u [0,T ] ) 2 -2σε 1 +	2T a	γ( u [0,T ] ) ,

By integration, we get for all i ∈ {0, . . . , q -1} and all t ∈ [t i , t i + ε] that

x(t) ⊤ P x(t) ≥ x(t i ) ⊤ P x(t i ) -

It follows that, for all i ∈ {0, . . . , q -1} and all t ∈ [t i , t i + ε],

Note that the interval [∆, tmax) may be empty.

Despite their apparent similarities, the proofs of Theorems 3 and 4 are radically different: the former relies on the explicit construction of a coercive LKF with history-wise dissipation, whereas the latter relies on a solutions-based contradiction argument. We have not been able to unify the two proofs which, we believe, could provide an interesting direction of research in order to gain insights on these conditions and envision possible ways to further improve them.

Another obvious line of research would be to investigate whether exp-ISS (or GES) can be established under a point-wise dissipation without any growth restrictions. This question remains open and is in line with the conjecture proposed in [START_REF] Chaillet | Is a point-wise dissipation rate enough to show ISS for time-delay systems?[END_REF] that ISS can be established through a point-wise dissipation.


Oxana Govokhina

Gérard Bailly

Gaspard Breton

Paul Bagshaw

A new trainable trajectory formation system for facial animation

A new trainable trajectory formation system for facial animation is here proposed that dissociates parametric spaces and methods for movement planning and execution. Movement planning is achieved by HMM-based trajectory formation. Movement execution is performed by concatenation of multi-represented diphones. Movement planning ensures that the essential visual characteristics of visemes are reached (lip closing for bilabials, rounding and opening for palatal fricatives, etc) and that appropriate coarticulation is planned. Movement execution grafts phonetic details and idiosyncratic articulatory strategies (dissymetries, importance of jaw movements, etc) to the planned gestural score.

Introduction

The modeling of coarticulation is in fact a difficult and largely unsolved problem [START_REF] Hardcastle | Coarticulation: Theory, Data, and Techniques[END_REF]. The variability of observed articulatory patterns is largely planned [START_REF] Whalen | Coarticulation is largely planned[END_REF]) and exploited by the interlocutor [START_REF] Munhall | Audiovisual gating and the time course of speech perception[END_REF]. Since the early work of Öhman on tongue movements (1967), several coarticulation models have been proposed and applied to facial animation. Bailly et al [START_REF] Bailly | Evaluation of movement generation systems using the point-light technique[END_REF] implemented some key proposals and confronted them to ground-truth data: the concatenation-based technique was shown to provide audiovisual integration close to natural movements. The HMM-based trajectory formation technique was further included [START_REF] Govokhina | Evaluation de systèmes de génération de mouvements faciaux[END_REF]. It outperforms both objectively and subjectively the other proposals. In this paper we further tune the various free parameters of the HMM-based trajectory formation technique using a large motion capture database [START_REF] Gibert | Analysis and synthesis of the 3D movements of the head, face and hand of a speaker using cued speech[END_REF]) and compare its performance with the winning system of Bailly et al study. We finally propose a system that aims at combining the most interesting features of both proposals.

Audiovisual data and articulatory modelling

The models are benchmarked using motion capture data. Our audiovisual database consists of 238 (228 for training and 10 for test) French utterances spoken by a female speaker. Acoustic and motion capture data are recorded synchronously using a Vicon© system with 12 cameras [START_REF] Gibert | Analysis and synthesis of the 3D movements of the head, face and hand of a speaker using cued speech[END_REF]. The system delivers the 3D positions of 63 infra-red reflexive markers glued on the speaker's face at 120 Hz (see Figure 1). The acoustic data is segmented semi-automatically into phonemes. An articulatory model is built using a statistical analysis of the 3D positions of 63 feature points. The cloning methodology developed at ICP [START_REF] Badin | Three-dimensional linear articulatory modeling of tongue, lips and face based on MRI and video images[END_REF][START_REF] Revéret | MOTHER: a new generation of talking heads providing a flexible articulatory control for video-realistic speech animation[END_REF] consists of an iterative Principal Component Analysis (PCA) performed on pertinent subsets of feature points. First, jaw rotation and protrusion (Jaw1 and Jaw2) are estimated from the points on jaw line and their effects subtracted from the data. Then the lip rounding/spreading gesture (Lips1), the proper vertical movements of upper and lower lips (Lips2 and Lips3), of the lip corners (Lips4) and of the throat (Lar1) are further subtracted from the residual data. These parameters explain 46.2, 4.6, 18.7, 3.8, 3.2, 1.6 and 1.3% of the movement variance. The analysis of geometric targets of the 5690 allophones produced by the speaker (see Figure 2) reveals confusion trees similar to previous findings [START_REF] Odisio | Tracking talking faces with shape and appearance models[END_REF]. Consequently 3 visemes are considered for vowels (grouping respectively rounded

HMM-based synthesis

The principle of HMM-based synthesis was first introduced by Donovan for acoustic speech synthesis [START_REF] Donovan | Trainable speech synthesis[END_REF]. This was extended to audiovisual speech by the HTS working group [START_REF] Tamura | Text-to-audio-visual speech synthesis based on parameter generation from HMM[END_REF].

Training. An HMM and a duration model for each state are first learned for each segment of the training set. The input data for the HMM training is a set of observation vectors. The observation vectors consist of static and dynamic parameters, i.e. the values of articulatory parameters and their derivatives. The HMM parameter estimation is based on ML (Maximum-Likelihood) criterion [START_REF] Tokuda | Speech parameter generation algorithms for HMM-based speech synthesis[END_REF]. Here, for each phoneme in context, a 3-state left-to-right model with single Gaussian diagonal output distributions and no skips is learned. Synthesis. The phonetic string to be synthesized is first chunked into segments and a sequence of HMM states is built by concatenating the corresponding segmental HMMs. State durations for the HMM sequence are determined so that the output probability of the state durations is maximized. From the HMM sequence with the proper state durations assigned, a sequence of observation parameters is generated using a specific ML-based generation algorithm [START_REF] Zen | An introduction of trajectory model into HMM-based speech synthesis[END_REF]). Note that HHM synthesis imposes some constraints on the distribution of observations for each state. The ML-based parameter generation algorithm requires Gaussian diagonal output distributions. It thus best operates on an observation space that has compact targets and characterizes targets with maximally independent parameters. We compared the dispersion of visemes obtained using different observation spaces: articulatory vs. geometric. Only lip geometry (aperture, width and protrusion) is considered. Despite its lower dimension, the geometric space provides less confusable visemes.

Detailed analysis

We compared phoneme-HHM with and without contextual information for selection. Table 1 summarizes our findings: anticipatory coarticulation is predominant, grouping context into visemes does not degrade performance. This contextual information enables the HMM system to progressively capture variability of allophonic realizations (see Figure 3). Syllable boundaries are known to influence coarticulation patterns. For this data, however, adding presence/absence of syllabic boundary does not improve the results (see bottom of Table 1). Sentence-internal (syntactic) pauses behave quite differently from initial and final pauses: initial pauses are characterized visually by prephonatory lips opening that reveals presence of initial bilabial occlusives if any; final pauses are characterized by a complete closure whereas the mouth often remains open during syntactic pauses especially when occurring between open sounds. We show that the viseme class immediately following the syntactic pause provides efficient contextual information for predicting lip geometry (see Table 2).

Consonantal visemes Vocalic visemes Articulation

Geometry Articulation Geometry Figure 3. Projecting the consonantal and vocalic visemes on the first discriminant plane (set using natural reference) for various systems and two different parametric spaces: articulatory versus geometric. From top to bottom: phoneme HMM, phoneme HMM with next segment information, TDA and natural reference. The proposed trajectory formation system TDA (Task Dynamics for Animation), the proposed trajectory formation system, combines the advantages of both HMM-and concatenation-based techniques. The proposed system (see Figure 4) is motivated by articulatory phonology and its first implementation by the task dynamics model [START_REF] Saltzman | A dynamical approach to gestural patterning in speech production[END_REF]. Articulatory phonology put forward underspecified gestures as primary objects of both speech production and perception. In the task dynamics model, context-independent underspecified gestures first give spatio-temporal gauges of vocal tract constrictions for each phoneme. Then a trajectory formation model executes this gestural score by moving articulatory parameters shaping the vocal tract. In this proposal, the gestural score specifying the lip geometry (lip opening, width and protrusion) is first computed by HMM models. Then execution of this score is performed by a concatenation model where the selection score penalizes segments according to their deviation from this planned geometry. The stored segments are thus characterized both by lip geometry for selection and by detailed articulation (jaw, separate control of upper and lower lips as well as rounding, etc) for the final generation.

Planning gestures by HMM synthesis. HMM-based synthesis outperforms both in objective and subjective terms concatenative synthesis and phoneme or diphone HMMs, when all these systems are trained to generate directly articulatory parameters. When trained on geometric parameters, these systems generate targets that are more discriminated and the correlation between original trajectories and those generated by all systems is substantially higher when considering geometry (see Table 3). This confirms previous studies that promote constrictions as the best characteristics for speech planning [START_REF] Bailly | Learning to speak. Sensori-motor control of speech movements[END_REF].

Executing gestures by concatenative synthesis. While diphone HMMs generate smooth trajectories while preserving visually relevant phonetic contrasts, concatenative synthesis has the intrinsic properties of capturing inter-articulatory phasing and idiosyncratic articulation. Concatenative synthesis also intrinsically preserves the variability of natural speech.

Performance analysis

Table 3 summarizes the comparative performance of the different systems implemented so far. Performance of the concatenation system is substantially increased when considering a selection cost using target parameters computed HMM trajectory planner. This is true whenever considering geometry or articulatory planning space. The performance of the current implementation of the TDA is however deceptive: the articulatory generation often degrades the quality of the planned geometric characteristics. If the TDA compensates well for the bad planning of movement during syntactic pauses, it often degrades the timing (see Figure 5). We are currently reconsidering the procedure that warps stored articulatory segments to planned gestures.

Conclusions and perspectives

The TDA system is a trajectory formation system for generating speechrelated facial movement. It combines a HMM-based trajectory formation system responsible for planning long-term coarticulation in a geometric space with a trajectory formation system that selects and concatenates segments that are best capable of realizing this gestural score. Contrary to most proposals, this system builds on motor control theory -that identifies distinct modules for planning and execution of movements -and implements a theory of control of speech movements that considers characteristics of vocal tract geometry as primary cues of speech planning. In the near future we will exploit in a more efficient way the information delivered by the HMM-based synthesis e.g. adding timing and spatial gauges to the gestural score in order to guide more precisely the segment selection.

  [uy ], mid-open [ieøoa ] and open vowels [aoeoe  ]) and 4 visemes for consonants (distinguishing respectively bilabials [pbm], labiodentals [fv], rounded fricatives [] from the others).

Figure

  Figure 1: Motion capture data and videorealistic clone mimicking recorded articulation.

Figure 2 .

 2 Figure 2. Grouping phonemes into viseme classes according to geometric confusability. Left: consonantal targets. Right: vocalic targets.

Figure 4 :

 4 Figure 4: The proposed trajectory formation system TDA. A geometric score is thus computed by HMM-based synthesis. Segments are then retrieved that best match this planned articulation. Articulatory trajectories also stored in the segment dictionary are then warped, concatenated and smoothed and drive the talking head. Since the speech signal is generated using the same warping functions, audiovisual coherence of synthetic animation is preserved.

Figure 5 .

 5 Figure 5. Comparing trajectory formation systems (blue: natural reference; red: concatenation/selection TDA; green: contextual phoneme-HMM) with a natural test stimulus (blue).From top to bottom: geometric parameters: lip aperture, width and protrusion; articulatory parameters: jaw aperture, lips rounding/spreading. Major discrepancies between TDA and contextual phoneme-HMM are enlighten.

Table 1 :

 1

	Phoneme HMM	Correlation	Coverage	Mean nb. of samples
	Without context	0.77±0.07	1.00	164±112
	Prev. phoneme	0.78±0.09	0.13	20±11

Adding contextual information to an initial context-independent phoneme HMM. Mean correlation (±standard deviation) between observed and predicted trajectories using different phoneme HMM systems for geometric space; coverage (nb. of segments which number of samples is superior to ten divided by total nb. of segments) and mean nb. of samples (±standard deviations) are computed.