N Carrasco

E Hébrard

M Banaszkiewicz

M Dobrijevic

Pascal Pernot
email: pascal.pernot@lcp.u-psud.fr

Influence of neutral transport on ion chemistry uncertainties in Titan ionosphere

Keywords: Atmosphere chemistry, Atmosphere composition, Atmosphere dynamics, Ionospheres, Titan

Models of Titan ionospheric chemistry have shown that ion densities depend strongly on the neutral composition. The turbulent diffusion transport conditions, as modeled by eddy coefficients, can spectacularly affect the uncertainty on predicted neutral densities. In order to evaluate the error budget on ion densities predicted by photochemical models, we perform uncertainty propagation of neutral densities by Monte Carlo sampling and assess their sensitivity to two turbulent diffusion profiles, corresponding to the extreme profiles at high altitudes described in the literature. A strong sensitivity of the ion density uncertainties to transport is observed, generally more important than to ion-molecule reaction parameters themselves. This highlights the necessity to constrain eddy diffusion profiles for Titan ionosphere, which should progressively be done thanks to the present and future measurements of the orbiter Cassini.

Introduction

Discrepancies between the outputs of different models and available data are difficult to assess in the absence of quantified uncertainties. In particular, modelling the chemistry of planetary ionospheres involves numerous physical and chemical parameters, which values are known from laboratory measurements with experimental uncertainty factors. These uncertainty sources should be accounted for in the modelling, in order to quantify the uncertainties on the model outputs and more generally to evaluate the model predictivity [START_REF] Wakelam | Estimation and reduction of the uncertainties in chemical models: application to hot core chemistry[END_REF]Zádor et al. 2006;[START_REF] Hébrard | Consequences of chemical kinetics uncertainties in modeling Titan's atmosphere[END_REF]).

In a previous work [START_REF] Carrasco | Uncertainty analysis of bimolecular reactions in Titan ionosphere chemistry model[END_REF], we evaluated the uncertainties on a Titan ionospheric chemistry model (based on the work of [START_REF] Banaszkiewicz | A coupled model of Titan's atmosphere and ionosphere[END_REF], due to the rate constants and branching ratios of ion-molecule reactions. Neutral densities were considered as fixed inputs, with the neutral density profiles calculated by [START_REF] Lara | Vertical distribution of titan's atmospheric neutral constituents[END_REF]. In parallel, [START_REF] Hébrard | Consequences of chemical kinetics uncertainties in modeling Titan's atmosphere[END_REF] studied the chemical kinetics uncertainties in a photochemical model for neutral species in Titan atmosphere.

Considering that ion densities closely depend on the neutral atmosphere composition [START_REF] Keller | Model of Titan's ionosphere with detailed hydrocarbon ion chemistry[END_REF], one can expect a direct impact of neutral uncertainties on ion uncertainties. In order to evaluate this influence, we built a semi-coupled Titan ionospheric model for neutral and ion species, using the chemistry model described in [START_REF] Carrasco | Uncertainty analysis of bimolecular reactions in Titan ionosphere chemistry model[END_REF] with neutral density profiles and their uncertainties as calculated by [START_REF] Hébrard | Consequences of chemical kinetics uncertainties in modeling Titan's atmosphere[END_REF].

Dynamics plays undoubtedly some role in the distribution of Titan's both neutral and ionic constituents but yet, the significance of eddy diffusion processes is not entirely known and still requires some attention. The eddy diffusion coefficient K(z) usually acts in photochemical modeling of planetary atmospheres as a free parameter that must be estimated to fit observations (see Fig. 1). In particular, the eventuality that the globally averaged distribution of Titan's constituents may be accurately and simultaneously described with a single eddy diffusion profile is still discussed

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A N U S C R I P T ACCEPTED MANUSCRIPT (Wilson and Atreya 2004). [START_REF] Hidayat | Millimeter and submillimeter heterodyne observations of Titan: Retrieval of the vertical profile of HCN and the 12C/13C solar ratio[END_REF] inferred a low homopause profile (840 km) from their millimeter observations of HCN vertical profile in much of the lower regions of the atmosphere; [START_REF] Strobel | Titan's upper atmosphere: structure and ultraviolet emissions[END_REF] inferred a higher homopause profile (1040 km) from their analysis of Voyager UVS solar occultation and airglow data; [START_REF] Toublanc | Photochemical modeling of Titan's atmosphere[END_REF] developed a very low homopause profile (680 km) from [START_REF] Toon | A physical model of Titan's aerosols[END_REF] profile and adapted it to fit [START_REF] Tanguy | Stratospheric profile of HCN on Titan from millimeter observations[END_REF] HCN distribution and Voyager UVS data for methane CH 4 . This diversity of profiles is in part due to the differences in the chemical scheme adopted by the authors. In fact, [START_REF] Hébrard | Consequences of chemical kinetics uncertainties in modeling Titan's atmosphere[END_REF] showed that the eddy diffusion profile may not currently be constrained as tightly as expected. Uncertainties attached to the computed abundances can be indeed so important that modifying the eddy diffusion coefficient K(z) does not change significantly their agreement with the different abundances inferred from the available observations. It appears moreover that the uncertainty factors of computed abundances are very sensitive to the choice of the eddy diffusion profile adopted, especially to the choice of a high-or low-homopause profile.

[Figure 1 about here.]

In order to assess their effect on ion densities, we considered two turbulent diffusion profiles for the neutral species, corresponding to the extreme profiles at high altitudes described in the literature [START_REF] Strobel | Titan's upper atmosphere: structure and ultraviolet emissions[END_REF][START_REF] Toublanc | Photochemical modeling of Titan's atmosphere[END_REF].

We first evaluated the uncertainties on the ion densities for both neutral turbulent transport cases, with a fixed ion-molecule chemistry. Then, we calculated the contributions of both uncertainty sources (neutral densities and ion-molecule reaction parameters), and identified the main sources of uncertainty for all the major ions predicted by our model of Titan ionosphere.

A C C E P T E D

M A N U S C R I P T ACCEPTED MANUSCRIPT 2 Methods
2.1 Ion and neutral chemistry semi-coupled models for Titan ionosphere Titan ionospheric chemistry model [START_REF] Banaszkiewicz | A coupled model of Titan's atmosphere and ionosphere[END_REF][START_REF] Carrasco | Uncertainty analysis of bimolecular reactions in Titan ionosphere chemistry model[END_REF] is semi-coupled with a photochemistry model of neutral species [START_REF] Hébrard | Consequences of chemical kinetics uncertainties in modeling Titan's atmosphere[END_REF]. Neutral density profiles are calculated with their uncertainties by the neutral photochemistry model. These profiles, with their uncertainties, are taken as inputs of the ionospheric chemistry model. Furthermore the correlations between neutral densities are taken into account through their covariance matrix. Density profiles are built with a 5 km scale in the 800-1300 km altitude range.

Uncertainty propagation

Because of non-linearities in the model and large uncertainties on numerous parameters, chances to be outside the validity range of linear uncertainty propagation are important. To avoid this bias, we used a Monte Carlo sampling method, which requires the definition of a probability density function for input parameters. As their are not correlated, we design separately the probability density functions for the kinetic parameters and for the neutral density profiles.

Uncertainties of kinetic parameters of ion-molecule reactions. The distributions are parametrized from the preferred values and the uncertainties reported in the review of [START_REF] Anicich | Ion-molecule chemistry in Titan's ionosphere[END_REF]. If an uncertainty value is not given for rate constants, the preferred value is considered as being inaccurate, with a relative uncertainty of 60% (highest uncertainty value reported in the review). The global rate k is depicted by a log-uniform distribution [START_REF] Carrasco | Uncertainty analysis of bimolecular reactions in Titan ionosphere chemistry model[END_REF]).

As uncertainty is not quantified for branching ratios in the reference review, uncertainty intervals have been defined according to the statistical deviations reported

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in [START_REF] Carrasco | Uncertainty analysis of bimolecular reactions in Titan ionosphere chemistry model[END_REF]: 10% for branching ratios larger than 0.5, 30% for branching ratios between 0.1 and 0.5 and 100% for the smaller values. For a few reactions, branching ratios are not reported. In such cases the pathways are considered as equiprobable with an uncertainty of 90%. Branching ratios were previously (Carrasco and Pernot 2007) modeled by Dirichlet distributions, which respect the sum rule for these parameters. In this work, we refine our elicitation of the branching ratios with Dirichlet uniform distributions (DIUD), which have the additional property not to favor any value within the given intervals (see the Appendix).

The sample for probability density function of ion-molecule reaction parameters can be produced with the following procedure. In order to preserve the intrinsic correlation due to the sum rule, partial reaction rates for a reaction with n pathways are produced following three steps:

1. a global rate k is sampled from a log-uniform distribution; 2. the branching ratios, b i , are sampled from the DIUD method (see Appendix), 3. the partial rate constants, k i , are products of two random numbers

(k i = kb i ) n 1 .
Uncertainties on neutral densities. The different chemical sources of uncertainties in photochemical models of Titan's atmosphere and their associated probability density functions were recently reviewed and evaluated at representative temperatures through a comprehensive cross-examination of extensive reaction rates database [START_REF] Hébrard | Photochemical kinetics uncertainties in modeling Titan's atmosphere: a review[END_REF] and implemented through a Monte-Carlo procedure into a 1D photochemical model of Titan's neutral atmosphere [START_REF] Hébrard | Consequences of chemical kinetics uncertainties in modeling Titan's atmosphere[END_REF].

A sample of N run = 500 density profiles for each neutral species is generated at all altitudes up to 1300 km and stored for analysis. The data are in a three dimensional table

{c ijk ; i = 1, N sp ; j = 1, N alt ; k = 1, N run } (1)
where N sp is the number of species, N alt is the size of the altitude grid.

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These profiles could be used directly in the 0D ionospheric model. However we preferred to build an intermediate probability density function (PDF) because it provides insight into the structure of neutral densities uncertainties. In addition this procedure enables a much larger number of runs to be used for uncertainty propagation and analysis in ionospheric chemistry. Typically, we found that cumulative density functions for ion densities were satisfyingly converged for 10 4 samples, a number presently out of reach of the 1D model.

To design the PDF for neutral densities, we first analyze their correlation structure. We consider two sources of correlation :

• ρ alt ij,il = c ijk , c ilk k : spatial correlations for a given species, resulting mainly from continuity laws of chemistry-transport processes, where ... k denotes the correlation coefficient calculated over the sample;

• ρ sp il,jl = c ilk , c jlk k : inter-species correlations, resulting from the chemical processes and mass conservation law.

To account for non-linearities, Rank Correlation Coefficients (RCC) have been used.

They convert nonlinear but monotonic relationship into a linear relationship by replacing the values of the sampled inputs/outputs by their respective ranks [START_REF] Hamby | A review of techniques for parameter sensitivity analysis of environmental models[END_REF][START_REF] Helton | Survey of sampling-based methods for uncertainty and sensitivity analysis[END_REF].

The spatial correlation of densities for a species is linked to the deformations caused to the density profile by chemistry fluctuations. If the density variations are similar at all altitudes (all curves in a sample remain parallel), the RCC should be equal to 1. A negative correlation would indicate opposite variations between two altitudes. We calculated the RCC for altitudes between 800 and 1300 km in the case of [START_REF] Strobel | Titan's upper atmosphere: structure and ultraviolet emissions[END_REF] eddy diffusion profile. As expected, spatial correlation is high: for all species, ρ alt ij,il is above 0.4, and for most species, the RCC distribution over altitudes is strongly peaked at the maximal value. Globally, when a species undergoes a density increase/decrease at the base of the ionosphere, there is a similar

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
increase/decrease in all the upper column. As we are performing 0D calculations for ionospheric chemistry, we can assume that it is safe to consider that all altitudes are maximally correlated, ρ alt ij,il =1 for all altitude-pairs and species.

To evaluate the correlation between neutral densities, we chose an altitude representative for ionospheric chemistry, i.e. 1200 km, and RCC's for all pairs of species have been evaluated at that altitude. We checked that the RCC's matrix was globally constant in the ionospheric altitude range. It appears that some neutral densities are significantly correlated. A good approximation of the correlation matrix was obtained by using the linear correlations between the log-densities log 10 c il . This representative correlation matrix was used at every altitudes. The probability density function is thus finally built as a multivariate normal density, parametrized by the average values log 10 c il and uncertainty factors F il , such as log 10 c il = log 10 c il ± log 10 F il , and the correlation matrix with elements ρ sp il,jl .

The sample for the full neutral densities probability density function, assuming unity correlation between altitudes and altitude-independent inter-species correlations, can be produced with the following procedure:

• Initializations:

estimate average values and uncertainty factors of log-densities for all species, at all altitudes;

evaluate inter-species linear correlation matrix of log-densities C at a representative altitude and calculate its Cholesky decomposition [START_REF] Gelman | Bayesian Data Analysis[END_REF].

• Monte Carlo loop:

generates N sp standard normal deviates {u i ∼ N(0, 1); i = 1, N sp };

combines these into N sp correlated numbers ε i by the Cholesky procedure;

loop over altitudes (j);

c ij = log 10 c ij + ε i × log 10 F ij ; i = 1, N sp .
3 Results and Discussion

Correlation of neutral densities

[Table 1 mass conservation. Full exploration of the other correlation is not relevant to the present study, but will be detailed in a future article. A consequence to be kept in mind for data fitting is that, within the framework of a consistent photochemical model, densities of species should not be adjusted independently of each other. For this study, we conclude that those strong correlations cannot be a priori neglected in the uncertainty propagation to ionospheric chemistry.

In order to check this point, we generated two samples in the Strobel case, one correlated and one uncorrelated (setting the correlation matrix to identity: C = I).

We compared the densities and their uncertainties obtained with the two samples.

The effect is illustrated on the 40 most abundant ions (Fig. 2). Except for very few ions, the impact is negligible. Predicted ion densities and uncertainties are practically insensitive to the correlation of neutral densities.

[Figure 2 about here.]

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A significant effect can however be observed on the correlation between ion densities, as can be seen from the compared cumulative density functions of correlation coefficients of all ions pairs on Fig. 3. For uncorrelated neutral densities, the RCC's are globally weak, massively located, with 90% probability, between -0.1 and 0.6, whereas for the correlated case the corresponding probability interval spreads between -0.3 and 0.8. The proportion of significant correlations increases thus notably when neutrals correlation is taken into account. We observe therefore a correlation transfer from neutrals to ions. This might be relevant for the computation of observables combining ion densities. As observed before, this correlation restricts considerably the degrees of freedom when, for instance, adjusting the ion densities to match an observable (as an ion mass spectrum).

[Figure 3 about here.]

In conclusion to this section, we note that correlation between neutral densities has certainly to be taken into account for data fitting or sensitivity analysis, but that it can safely be neglected for the sole purpose of uncertainty propagation.

Effect of the neutral transport processes on the ion concentrations

As shown in [START_REF] Hébrard | Consequences of chemical kinetics uncertainties in modeling Titan's atmosphere[END_REF], the turbulent macroscopic transport of the neutral species in Titan ionosphere is not yet tightly constrained by the existing observations and might vary between two extreme cases described in [START_REF] Strobel | Titan's upper atmosphere: structure and ultraviolet emissions[END_REF] and [START_REF] Toublanc | Photochemical modeling of Titan's atmosphere[END_REF]. For simplicity, these two cases will be called the "Strobel were used as inputs of our simulations, allowing us to evaluate the impact of the poorly known neutral transport processes on the ionic species.

The density profile of the major ion , HCNH + in Titan ionosphere, is represented on Fig. 4 for both neutral transport cases. The nominal profiles are significantly different for altitudes lower than 950 km: HCNH + density is up to 10 times larger at 800 km in the Strobel case. This confirms the sensitivity of the ionospheric chemistry model to the neutral density profiles previously noticed by [START_REF] Keller | Model of Titan's ionosphere with detailed hydrocarbon ion chemistry[END_REF]. Moreover, the uncertainties on HCNH + density differ significantly from one case to the other, with a larger uncertainty in the Toublanc case, at all altitudes.

[Figure 4 about here.]

The altitude 1200 km is of specific interest, being the average altitude of the first Cassini's flyby T5 providing data on ion densities. We compared the effect of both neutral diffusion cases on the ten major ions calculated by our model at this altitude:

HCNH + , C 2 H + 5 , CH + 5 , N + 2 , C 3 H + 5 , CH + 3 , N 2 H + , N( 3 P) + , C 2 H + 4 and C 2 H + 3 .
The densities with their uncertainties are reported on Fig. 5. The ions are ranked by decreasing density. The results are highly dependent on the eddy diffusion profile:

quite precise ion densities in the Strobel case, and relative uncertainties of one or two orders of magnitude for all ions in the Toublanc case.

[Figure 5 about here.]

One can conclude that the eddy coefficient chosen to describe the transport of the neutral species influences significantly the nominal density profiles of the ion species but also their uncertainty. The chemistry of the ions in Titan ionosphere cannot be decorrelated from transport considerations.

There is still to understand why the uncertainties in the Toublanc case are much larger than in the Strobel case. A reason lies probably in the uncertainty on the major neutral reactant, methane (CH 4 ). Indeed the uncertainty on the methane

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
profiles at the ionospheric altitudes is substantially larger in the Toublanc case (see Fig. 6). The uncertainty factor F CH 4 at 1200 km is equal to about 1.1 in the Strobel case and about 5.7 in the Toublanc case. High-homopause profiles seem to restrain to some extent the propagation of chemical uncertainties in the upper atmosphere contrary to low-homopause profiles.

[Figure 6 about here.]

The reason for such a discrepancy on methane uncertainty is not yet fully established. However, the vertical transport is more important in the case of a high homopause. This means that the vertical renewal of the species through transport, compensates more efficiently for the chemical losses. The transport can thus be considered as an attenuation factor of the chemical uncertainties on the neutral densities.

Comparison of two uncertainty sources: neutral densities and ion-molecule reactions parameters

We compared the respective contributions of the neutral densities and of the parameters of ion-molecule reactions to the ion density uncertainty by first performing uncertainty propagation on both uncertainty sources separately. As earlier, both

Strobel and Toublanc cases were considered to encompass extreme transport processes in Titan ionosphere.

The ten major ion densities are reported in Fig. 7, with their uncertainties for both Strobel and Toublanc cases. Each case is compared with the uncertainty contribution of the ion-molecule reaction parameters. In the Toublanc case, the uncertainty due to the ion-molecule reaction parameters is negligible in comparison to the uncertainty due to the neutral densities. This implies that in this case, the uncertainty on the ion densities is mainly controlled by the uncertainty on the neutral atmosphere rather than by the ion reactivity itself. In the Strobel case, [Figure 7 about here.]

A C C E P T E D M A N U S C
The Strobel case corresponds to a high homopause configuration. One can notice that the hypothesis of a high homopause seems to be consolidated by the recent INMS data (Yelle et al., 2006).

Conclusion

We presented the first evidence of the influence of the modeling of turbulent transport of neutral species on the uncertainty of ion chemistry in Titan ionosphere.

We found a strong sensitivity of ion chemistry to the description of turbulent neutral transport. The uncertainties on the ion densities were much higher with a low homopause hypothesis than with a high homopause level. This effect can be explained by a competition between vertical transport and chemistry: an efficient vertical transport attenuates the chemical uncertainties. In the case of a low homopause, the uncertainty of the ion densities due to the ion chemistry was even We showed that neutral species are strongly correlated by the photochemical model and transfer an important correlation between ion densities. This provides chemistry-based constraints that might be useful when trying to fit models to measured Mass Spectra, for instance by tuning neutral densities.

The present simulations provide a basis for a sensitivity analysis, from which to identify the key reactions and species strongly responsible for outputs uncertainty.

In a forthcoming paper, we will present results along this line, pertaining to both ion-molecule and neutral chemistry.

Appendix:

Dirichlet Uniform Distribution (DIUD) for branching ratios of ion-molecule reactions

An elicitation scheme is based on considerations about the nature of the uncertainties to be represented. Branching ratios are often specified by intervals [START_REF] Jenkinson | The elicitation of probabilities -A review of the statistical literature[END_REF][START_REF] Bates | Bayesian uncertainty assessment in multicompartment deterministic simulation models for environmental risk assessment[END_REF], and it is assumed that there is no reason to favor any value within a given interval. This case seems indeed to be the one favored by experts in the measurement of branching ratios. They consider that they report intervals accounting for systematic errors with enough latitude to encompass all acceptable values [START_REF] Carrasco | Uncertainty analysis of bimolecular reactions in Titan ionosphere chemistry model[END_REF]).

The distribution is defined uniformly over the (n -1)-simplex

(b 1 , . . . , b n ) ∼ Dirichlet(1, . . . , 1) (2) A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C

  * generates samples of correlated log-densities with altitude dependent uncertainties log 10

  case" and the "Toublanc case". The eddy coefficient profiles are given on Fig.1. The Strobel case corresponds to a high homopause level (1040 km) whereas the Toublanc case corresponds to a low homopause level (680 km). The two corresponding neutral density profiles obtained by[START_REF] Hébrard | Consequences of chemical kinetics uncertainties in modeling Titan's atmosphere[END_REF], for diurnally averaged chemistry,

  found negligible in comparison to the uncertainty due to the neutral densities. This highlights the necessity to constrain eddy diffusion profiles for Titan ionosphere, which should progressively be done thanks to the present and future measurements of the Cassini orbiter(Yelle et al. 2006).The present study focused on the sensitivity of a Titan ionospheric model to two particular sources of uncertainty: the description of the neutral density profiles, and the ion-molecule chemistry parameters. Additional sources of uncertainty are still to be studied, such as uncertainties on the photo-dissociation or recombination would allow a complete overview of the needs and limits of the actual model, in order to improve it.

Figure 1 :Fig. 2 -Figure 3 :Figure 5 :

 1235 Figure 1: Eddy diffusion profiles -Hidayat et al. (1997) profile (solid line), Strobel et al. (1992) profile (dashed line) and[START_REF] Toublanc | Photochemical modeling of Titan's atmosphere[END_REF] profile (dot dashed line). The methane molecular diffusion coefficient profile is also included.

Figure 7 :

 7 Figure 7: Densities of the ten major ions at altitude 1200 km for the two eddy coefficients : (a) Strobel case, (b) Toublanc case. Boxplots depict 50% and 90% confidence intervals. For each ion density, the upper boxplot corresponds to uncertainty propagation of the neutral density, whereas the lower boxplot depicts uncertainty propagation of ion chemistry parameters.

  about here.] We evaluated the correlation between the neutral density profiles in the Strobel case. Representative values for some important species are reported in Table 1. Strong correlations are observed between some neutrals, e.g. H 2 /CH 4 , N 2 /CH 4 or C 2 H 2 /C 2 H 4 , and result from the chemical network. A salient feature is the negative correlation of CH 4 with all other species in the table (increasing the density of CH 4

causes a decrease in the density of those species). The strong negative correlation of CH 4 with N 2 and H 2 has probably not a chemical origin, but a physical one, i.e.

  R I P T ACCEPTED MANUSCRIPT similar amplitudes for both uncertainty sources are observed. The conclusions are therefore depending on the ions: a prevalence of the neutral density uncertainty is observed for C 3 H + 5 and C 2 H + 3 , whereas a prevalence of the ion-molecule reaction parameters is observed for C 2 H + 4 . Similar contribution of both uncertainty sources affect HCNH + , C 2 H + 5 , CH + 5 , N + 2 , CH + 3 , N 2 H + , and N( 3 P) + .

Table 1 :

 1 Rank correlation coefficients between selected neutral densities at 1200 km. Coefficients with an absolute value above 0.5 are in bold face.

	R I P T

Acknowledgments

This work was partly supported by EuroPlaNet through travel grants to M.D. and M.B. We also acknowledge the support received from Centre National de Recherche Scientifique (CNRS) and from Centre National d'Etude Spatiale (CNES) through postdoctoral positions for N.C. We gratefully thank O. Dutuit, R. Thissen and C. Alcaraz for fruitful discussions.

(3)

The definition of the limits of the intervals depends on the available information. If one gets a set as (b i , b i ) n i=1 , one has simply b i,min = b i -Δb i and b i,max = b i + Δb i .

For the sake of convenience, we name this truncated uniform Dirichlet distribution DIUD in the following. It has 2n parameters

Samples corresponding to the same branching ratios (0.6, 0.3, 0.1) for 10% and 90%

relative uncertainty are displayed in Fig. 8. Note that, although the distribution is uniform over the restricted (n -1)-simplex, this is not the case for the marginal distributions.

The most thorough method to produce samples from this distribution is to draw samples from the uniform Dirichlet distribution (generated by the Gamma algorithm, [START_REF] Gelman | Bayesian Data Analysis[END_REF] and to reject those lying outside the prescribed intervals.

However for fairly accurate branching ratios, this method can spill a lot of random draws.

[Figure 8 about here.]

List of Figures


Samir Ladaci
email: samirladaci@yahoo.fr

Emmanuel Moulay
email: emmanuel.moulay@irccyn.ec-nantes.fr

L p -stability analysis of a class of nonlinear fractional differential equations

This paper investigates the L p -stability properties of fractional nonlinear differential equations. Systems defined on a finite time interval are considered. The principal contributions are summarized in a theorem which give sufficient conditions for bounded stability of fractional order systems. We show that the proposed results can not be extended to the case of systems defined on an infinite time interval.

I. INTRODUCTION

The fractional calculus and fractional order differential equations attracted a great attention these last decades (see [START_REF] Miller | An Introduction to the Fractional Calculus and Fractional Differential Equations[END_REF], [START_REF] Podlubny | Fractional Differential Equations[END_REF]). One of the most important reasons for this interest is their ability to model many natural systems and their seducing properties like robustness and dynamical behavior. Fractional order systems have found many applications in various domains such as heat transfer, viscoelasticity, electrical circuit, electro-chemistry, dynamics, economics, polymer physics and control. The study of stability for this kind of systems focuses a great interest in the research community. We can cite in this domain the works of Matignon [START_REF] Matignon | Stability result on fractional differential equations with applications to control processing[END_REF] and Bonnet and Partington [START_REF] Bonnet | Coprime factorizations and stability of fractional differential systems[END_REF] for the stability of linear fractional systems, those of Khusainov [START_REF] Khusainov | Stability Analysis of a Linear-Fractional Delay System[END_REF], Bonnet and Partington [START_REF] Bonnet | Analysis of fractional delay systems of retarded and neutral type[END_REF], Chen and Moore [START_REF] Chen | Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems[END_REF] and Deng et al. [START_REF] Deng | Stability analysis of linear fractional differential system with multiple time delays[END_REF] for fractional systems with time delay and Ladaci et al. [START_REF] Ladaci | Fractional Order Adaptive High-Gain Controllers for a Class of Linear Systems[END_REF] for fractional adaptive control systems. Ahn et al. have proposed robust stability test methods for fractional order systems [START_REF] Ahn | Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality[END_REF], [START_REF] Chen | Robust stability check of fractional order linear time invariant systems with interval uncertainties[END_REF]. Recently Lazarević [START_REF] Lazarević | Finite time stability analysis of PD α fractional control of robotic time-delay systems[END_REF] has studied the finite time stability of a fractional order controller for robotic time delay systems.

In this paper we are concerned by the stability analysis of fractional order systems represented by the following nonlinear differential equation:

D α x(t) = h (t, x(t)) , x ∈ R n , t ∈ R + (1)
where

0 < α < 1, h ∈ C (R + × R n , R n+ ) is a continuous positive function, with the initial condition D α-1 x(t 0 ) = x 0 . (2)
In the following we will use the notation h x (t) = h(t, x(t)).

This paper is organized as follows. In Section II, we present some useful theoretical background. In section III, the main result on L p -stability of nonlinear fractional order systems defined on a finite interval is given with an illustrative example. Section IV concludes the paper.

II. THEORETICAL BACKGROUND

The mathematical definition of fractional derivatives and integrals has been the subject of several different approaches [START_REF] Oldham | The Fractional Calculus[END_REF]. In this paper we consider the following Riemann-Liouville definition [START_REF] Miller | An Introduction to the Fractional Calculus and Fractional Differential Equations[END_REF],

Definition 1 (Fractional integral): Let ν ∈ C such that Re(ν) > 0 and let g be piecewise continuous on (0, ∞) and integrable on any finite subinterval of [0, ∞). Then for 0 ≤ t 0 < t we call:

t0 D -ν t g(t) = 1 Γ(α) t t0 (t -τ ) ν-1 g(τ )d(τ )
the Riemann-Liouville fractional integral of g of order ν where

Γ(x) = ∞ 0 y x-1 e -y dy
is the Gamma function.

For simplicity we will note D µ g(t) for t0 D µ t g(t).

The Riemann-Liouville definition of fractional order derivative of g is now recalled.

Definition 2 (Fractional derivative):

Let g be a continuous function and let µ > 0. Let m be the smallest integer that exceeds µ. The fractional derivative of g of order µ is defined as,

D µ g(t) = D m D -ν g(t) , µ > 0, (if it exists) where ν = m -µ > 0.
Lemma 3: The solution of System (1)-( 2) is given by the vector equality

x(t) = x 0 Γ(α) (t -t 0 ) α-1 + 1 Γ(α) t t0 (t -τ ) α-1 h x (τ )d(τ ).
For the proof see for instance [START_REF] Diethelm | Analysis of Fractional Differential Equations[END_REF], [START_REF] Hadid | Lyapunov stability of differential equations of non-integer order[END_REF].

With no loss of generality we can take x 0 = 0 and t 0 = 0. This reduces our solution to,

x(t) = 1 Γ(α) t 0 (t -τ ) α-1 h x (τ )d(τ ) (3)
Let us recall the definition of L p -stability.

Definition 4: Let 1 ≤ p ≤ ∞ and Ω ⊂ R + , the system (1) is L p (Ω)-stable if the solution x(t) T = (x 1 , . . . , x n ) defined by Equation (3) belongs to L p (Ω).
We now introduce the convolution product on C([t 0 , +∞), R) with t 0 ≥ 0. Definition 5: For all functions f, g ∈ C(R + , R) we define the operator f g as follows:

f g(t) = t 0 f (t -τ )g(τ )dτ, t ≥ 0. ( 4
)
Lemma 6: The product is a commutative internal composition rule on C(R + , R).

Proof: By using the change of variables τ = tu, we obtain

f g(t) = t 1 0 f (t(1 -u))g(tu)du.
Due to the theorem about continuity of an integral depending on a parameter, we deduce that t → f g(t) belongs to C(R + , R). The commutativity property of the operator can be easily proven by using the change of variables u = tτ in (4).

In the next we show interesting properties of this product operator.

Lemma 7: Let f, g ∈ C(R + , R + ) ∩ L 1 ([0, T ]) with T > 0, the following properties hold: (i) T 0 f g(t)dt = T 0 g(τ ) T -τ t0 f (t)dt dτ, (ii)
T 2 0 f (t)dt T 2 0 g(τ )dτ ≤ T 0 f g(t)dt ≤ T 0 f (t)dt T 0 g(τ )dτ ,
(iii) For any p ≥ 1 we have:

T 2 0 g(τ ) p dτ T 2 0 f (t)dt p ≤ T 0 (f g(t)) p dt ≤ T 0 g(τ ) p dτ T 0 f (t)dt p , (iv) Moreover, if f, g ∈ L 1 (R +
), the integral of f g converges and we have:

+∞ 0 f g(t)dt = +∞ 0 f (t)dt +∞ 0 g(τ )dτ , (v) Moreover, if f ∈ L 1 (R + ) and g ∈ L p (R + ), we have for any p ≥ 1: +∞ 0 (f g(t)) p dt ≤ +∞ 0 f (t)dt p +∞ 0 g(τ ) p dτ .
Proof: (i)-By using the theorem of Fubini for positive functions, we have:

T 0 f g(t)dt = T 0 t 0 f (t -τ )g(τ )dτ dt = T 0 T τ f (t -τ )g(τ )dt dτ
Then by using the change of variables u = tτ , we obtain:

T 0 f g(t)dt = T 0 g(τ ) T τ f (t -τ )dt dτ = T 0 g(τ ) T -τ 0 f (u)du dτ. (5) (ii)-Since f is positive, we have T -τ 0 f (t)dt ≤ T 0 f (t)dt
and the right inequality is immediate from [START_REF] Chen | Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems[END_REF]. For the left inequality we remark that

T 0 g(τ ) T -τ 0 f (t)dt dτ ≥ T 2 0 g(τ ) T -τ t0 f (t)dt dτ Now if 0 ≤ τ ≤ T 2 , then T -τ ≥ T 2 and, T 0 g(τ ) T -τ 0 f (t)dt dτ ≥ T 2 0 g(τ ) T 2 0 f (t)dt dτ ≥ T 2 0 f (t)dt T 2 0 g(τ )dτ.
(iii)-From (i) we have,

T 0 f g(t)dt = T 0 g(τ ) T -τ 0 f (t)dt dτ,
then for the right inequality,

T 0 (f g(t)) p dt = T 0 g(τ ) T -τ 0 f (t)dt p dτ ≤ T 0 g(τ ) p T 0 f (t)dt p dτ ≤ T 0 g(τ ) p dτ T 0 f (t)dt p
For the left inequality we use the same proof as in (ii). Since 0 ≤ τ ≤ T 2 , we have Tτ ≥ T 2 and:

T 0 g(τ ) T -τ 0 f (t)dt p dτ ≥ T 2 0 g(τ ) p dτ T 2 0 f (t)dt p .
(iv)-If T tends to infinity, then both of the right and the left sides of the inequality (5) converge to the same limit, that is:

+∞ 0 f (t)dt +∞ 0 g(τ )dτ .
(v)-By using the preceding reasoning for the double inequality (5) we get,

+∞ 0 (f g(t)) p dt ≤ +∞ 0 f (t)dt p +∞ 0 (g(τ )) p dτ .

III. MAIN RESULTS

The solution (3) of ( 1) can be rewritten using the product operator defined in (4) as follows:

x(t) = (K α h x ) (t) (6)
where K α with 0 < α < 1 is the so called convolution kernel defined by

K α (u) = u α-1 Γ(α) (7)
Lemma 8: Consider the convolution kernel function K α defined in [START_REF] Deng | Stability analysis of linear fractional differential system with multiple time delays[END_REF], let > 0 then K α ∈ L p ([0, ]) if and only if p-1 p < α < 1 and p ≥ 1.

Proof: We have

0 K α (u) p du = 0 u α-1 Γ(α) p du (8)
= 0 u (α-1)p Γ(α) p du = u (α-1)p+1 ((α -1)p + 1)Γ(α) p 0
The generalized Riemann integral (8) is convergent if and only p ≥ 1 and (α -1)p + 1 > 0 that is p-1 p < α < 1. Then we have the main result on the L p -stability of the solution of the system (1)-(2) defined on a finite time interval.

Theorem 9: Let p ≥ 1 and consider the system defined by the fractional order differential equation ( 1)-( 2) where the time t ∈ [0, t f ] then the system (1)-( 2) is L p ([0, t f ])-stable if and only if p-1 p < α < 1 and h x ∈ L 1 ([0, t f ]).

Proof: As the operator is commutative, we have:

t f 0 x(t) p = t f 0 (K α h x ) p 1/p dt = t f 0 (h x K α ) p 1/p
dt and again from Lemma 7-(iii) and Equality (6) we get:

t f 0 x(t) p ≤ t f 0 h x (τ )dτ t f 0 K α (t) p dt 1/p . (9)
From Lemma 8, we have that

t f 0 K α (t) p dt < +∞ if and only if p-1 p < α < 1. If in addition h x ∈ L 1 ([0, t f ]), Equation (9) implies that t f 0 x(t) p ≤ t (α-1)p+1 p f ((α -1)p + 1) 1 p Γ(α) t f 0 h x (τ )dτ < +∞.
Example 10: Let us consider the system

D 3 4 x(t) = B(x) (t + 1) 3 , 0 ≤ t ≤ 1
where B(x) is a positive real function bounded by a constant c > 0 as for instance the function x → |sin(x)|. Then, by using Theorem 9 with p = 2 and α = 3 4 , we deduce that

1 0 x(t) 2 ≤ 3 √ 2 c 8 Γ 3 4 .
Remark 11: The generalization to the infinite case where t ∈ R + is not possible because the kernel function u → K α (u) does not belong to L 1 (R + ). Thus we can not use points (iv) and (v) of lemma 7. Even if the system is defined since t 0 > 0, the generalization to the infinite case where t ∈ [t 0 , +∞) is not possible. Indeed the solution of System (1)-(2) given by

x(t) = t t0 K α (t -τ ) h x (τ )d(τ ), t ≥ t 0 > 0 (10)
can not be defined by using a convolution product which is commutative. So, Theorem 9 can not be extended to the case where t ∈ [t 0 , +∞) with t 0 > 0. A commutative convolution product with initial condition t 0 > 0 which can be used would be f g(t) = t t0 f (t + t 0τ )g(τ )dτ, t ≥ 0.

However, our kernel function K α (tτ ) as it appears in the definition of the solution of System (1)-( 2) given by Equation [START_REF] Khusainov | Stability Analysis of a Linear-Fractional Delay System[END_REF] does not depend on the initial time t 0 .

IV. CONCLUSION

In this paper the L p -stability properties of fractional nonlinear differential equations are considered. Sufficient conditions for L p -stability of the fractional order system are presented in the case of finite time window, with analytical proofs based on the properties of a special convolution product. This work opens a new method for the L p -stability analysis of a class of nonlinear fractional order systems.


Emmanuel Moulay
email: emmanuel.moulay@irccyn.ec-nantes.fr

W Perruquetti

Finite time stability conditions for non autonomous continuous systems

published or not. The documents may come

Finite time stability conditions for non autonomous continuous systems

Introduction

Since the end of the 19 th and the beginning of the 20 th century, various concepts dedicated to the qualitative behavior of the solutions for dynamical systems have been introduced (for example the seminal definitions and results from A.M. [START_REF] Lyapunov | Stability of Motion: General Problem[END_REF]). But rapidly, one had to face some more precise time specifications of the behavior of the state variables (or outputs) for real process. For example, some finite time stabilizing control design relies upon sliding mode theory (see [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF]) and another one upon optimal control theory (see [START_REF] Ryan | Singular Optimal Controls for Second-Order Saturating Systems[END_REF]). In both cases the controller leads to some state discontinuous feedback control laws. Over the years much work has been dedicated to this concept, getting some sufficient conditions and the application to the construction of finite-time stabilization control (see for example [START_REF] Bhat | Continuous Finite-Time Stabilization of the Translational and Rotational Double Integrator[END_REF], [START_REF] Hong | Finite-Time Stabilization and Stabilizability of a Class of Controllable Systems[END_REF], [START_REF] Hong | On an Output Feedback Finite-Time Stabilization Problem[END_REF][START_REF] Hong | Finite-Time Control for Robot Manipulators[END_REF], Moulay and Perruquetti (2006a)). The main idea lies in assigning infinite eigenvalue to the closed loop system at the origin. Let us mention the following illustrative example ẋ = -|x| a sgn(x), x ∈ 4, a ∈ ]0, 1[ , (1) for which the solutions starting at τ = 0 from x are:

φ x (τ ) =    sgn(x) |x| 1-a -τ (1 -a) 1 1-a ) if 0 ≤ τ ≤ |x| 1-a 1-a 0 if τ > |x| 1-a 1-a , (2)
and they reach the origin in finite time. In fact, there exists a function called settling time that increases the time for a solution to reach the equilibrium. Usually, this function depends on the initial condition of a solution. But, for non autonomous systems, it may also depend on the initial time. Lastly, notice from this example, that in order to obtain finite time stability, the right hand side of the ordinary differential equation cannot be locally Lipschitz at the origin.

The paper is organized as follows. After defining the notion of finite time stability for continuous non autonomous systems in section 2 , we recall the necessary and sufficient conditions for finite time stability of autonomous scalar systems (section 3) (result which appears in [START_REF] Haimo | Finite Time Controllers[END_REF], Moulay and Perruquetti (2006b) and whose proof is given here. Then in section 4, we give a generalization for continuous non autonomous systems of any dimension. For this we introduce a Lyapunov function whose derivative satisfies an increase to obtain sufficient conditions for finite time stability. In the subsection 4.1, one uses a smooth Lyapunov function and in the subsection 4.2, one uses a nonsmooth one. Nevertheless, as we will see the use of a smooth Lyapunov function is not a weaker result, only a different one. Then, a Lyapunov function whose derivative verifies a decrease is used in order to obtain necessary conditions for finite time stability (see subsection 4.3). Through the paper, the following notations will be used:

• for a > 0, the following function will be used ϕ a (x) = |x| a sgn(x),

• V denotes a neighborhood of the origin in 4 n ,

• B ǫ is the open ball centered at the origin of radius ǫ > 0.

The upper Dini derivative of a function f : 4 → 4 is the function D + f : 4 → 4 defined by:

D + f (x) = lim sup h→0 + f (x + h) -f (x) h .
2 What is finite time stability?

Consider the system ẋ = f (t, x), t ∈ 4 ≥0 , x ∈ 4 n , (3)
where f : 4 ≥0 × 4 n → 4 n is a continuous function. Then φ x t (τ ) denotes a solution of the system ( 3) starting from (t, x) ∈ 4 ≥0 × 4 n and S (t, x) represents the set of all solutions φ x t . The existence of solutions is given by the well known Cauchy-Peano Theorem given for example in [START_REF] Hale | Ordinary Differential Equations, 2nd Edition, Pure and applied mathematics XXI[END_REF]

. A continuous function α : [0, a] → [0, +∞[ belongs to class K if it is strictly increasing and α(0) = 0. It is said to belong to class K ∞ if a = +∞ and α(r) → +∞ as r → +∞. Moreover, a continuous function V : 4 ≥0 × V → 4 ≥0 such that L1) V is positive definite, L2) V (t, x) = D + V • φx t (0) is negative definite with φx t (τ ) = (τ, φ x t (τ )), is a Lyapunov function for (3). If V is smooth then V (t, x) = ∂V ∂t (t, x) + n i=1 ∂ i V ∂x i (t, x)f i (t, x). A continuous function v : 4 ≥0 × V → 4 is decrescent if there exists a K-function ψ such that |v(t, x)| ≤ ψ (||x||) ∀(t, x) ∈ 4 ≥0 × V. A continuous function v : 4 ≥0 ×4 n → 4 is radially unbounded if there exists a K ∞ -function ϕ such that v(t, y) ≥ ϕ(||y||), ∀t ∈ 4 ≥0 ∀y ∈ 4 n .
The definition of asymptotic stability is well known (see [START_REF] Hahn | Theory and Application of Liapunov's Direct Method[END_REF]). In this case, the solutions of the system (3) tend to the origin (but without information about the time transient). This information comes from the notion of "settling time" which, when finite and combined with the stability concept, leads to the following definition for finite time stability Definition 2.1 The origin is weakly finite time stable for the system (3) if:

A1) the origin is Lyapunov stable for the system (3), A2) for all t ∈ I, there exists δ(t) > 0, such that if x ∈ B δ(t) then for all Φ x t ∈ S (t, x): a) φ x t (τ ) is defined for τ ≥ t, b) there exists 0 ≤ T (φ x t ) < +∞ such that φ x t (τ ) = 0 for all τ ≥ t + T (φ x t ).

T 0 (φ x t ) = inf{T (φ x t ) ≥ 0 : φ x t (τ ) = 0 ∀τ ≥ t + T (φ x t )} is called the settling time of the solution φ x t . A3) Moreover, if T 0 (t, x) = sup φ x t ∈S(t,x)
T 0 (φ x t ) < +∞, then the origin is finite time stable for the system (3).

T 0 (t, x) is called the settling time with respect to the initial conditions of the system (3).

Remark 1 When the system is asymptotically stable, the settling time of a solution may be infinite.

If the system (3) is continuous on I × V and locally Lipschitz on I × V \ {0}, because of solution uniqueness, the settling time of a solution and the settling time with respect to the initial conditions of the system are the same:

T 0 (t, x) = T 0 (φ x t ).
Definition 2.2 Let the origin be an equilibrium point of the system (3). The origin is uniformly finite time stable for the system (3) if the origin is B1) uniformly asymptotically stable for the system (3), B2) finite time stable for the system (3), B3) there exists a positive definite continuous function α : 4 ≥0 → 4 ≥0 such that the settling time with respect initial condition of the system (3) satisfies:

T 0 (t, x) ≤ α ( x ) .
3 Autonomous scalar systems

Let us recall the result for finite time stability of autonomous scalar systems of the form

ẋ = f (x), x ∈ 4, (4)
where f : 4 → 4 given in [START_REF] Haimo | Finite Time Controllers[END_REF]. In this particular case there exists a necessary and sufficient condition for finite time stability:

Lemma 3.1 Let the origin be an equilibrium point of the system (4) where f is continuous. The origin is finite time stable for the system (4) if and only if there exists a neighborhood of the origin V such that for all

x ∈ V \ {0} xf (x) < 0, ( 5
) 0 x dz f (z) < +∞. (6) Proof (⇐) As xf (x) < 0, V (x) =
x 2 is a Lyapunov function for the system. So, the origin of the system (4) is asymptotically stable. Let φ x (τ ) be a solution of the system which tend to the origin with time T (φ x ). We have to show that T (φ x ) < +∞. With the asymptotic stability, if x is small enough, then τ → φ x (τ ) is strictly monotone for τ ≥ 0. E. Moulay and W. Perruquetti Moreover,

T (φ x ) = T (φ x ) 0 dτ. As xf (x) < 0 for all x ∈ V\{0}, 1 f is defined on V\{0}. The following change of variables, [0, T (φ x )[ → ]0, x], τ → φ x (τ ) leads to 0 x dz f (z) = T (φ x ) 0 φx (τ ) f (φ x (τ )) dτ = T (φ x ) < +∞. T (x) = sup φ x ∈S(x)
T (φ x ) does not depend on φ x . So, we conclude that the origin of the system (4) is finite time stable with the following settling time

T 0 (x) = T (φ x ) satisfying T 0 (x) = 0 x dz f (z) .
(⇒) Suppose that the origin of the system (4) is finite time stable. Let δ > 0 given by definition 2.1. Suppose that there exists

x ∈ ]-δ, δ[ \ {0} such that xf (x) ≥ 0. • If xf (x) = 0, then f (x) = 0 et φ x (τ ) ≡
x is a solution of the system (4) which does not tend to the origin. • If xf (x) > 0, with no loss of generality, we can suppose that x > 0 et f (x) > 0. The continuity of f and the fact that f (x) > 0 lead to f (z) > 0 for z in a neighborhood of x. Then, the function τ → φ x (τ ) increases in a neighborhood of the origin. With its continuity, this solution can not tend to the origin.

Let x ∈ ]-δ, δ[ \ {0}
and consider the solution φ x (τ ). By assumption, there exists 0 ≤ T 0 (φ x ) < +∞ such that φ x (τ ) = 0 for all τ ≥ t + T 0 (φ x ). With the asymptotic stability, x can be chosen small enough such that τ → φ x (τ ) decreases for τ ≥ t. The following change of variables [0, T 0 (φ

x )[ → ]0, x], τ → φ x (τ ) leads to 0 x dz f (z) = T0(φ x ) 0 dτ = T 0 (φ x ) < +∞.
For x in a neighborhood of the origin, the settling time of a solution of (4) and the settling time with respect to the initial conditions of the system (4) are equal and

T 0 (x) = 0 x dz f (z)
.

If the system (4) is globally defined and if conditions ( 5) and ( 6) hold globally, then the origin is globally finite time stable. Moreover, it is obvious that for autonomous systems, uniform finite time stability is finite time stability.

Example 3.2 Let a ∈ ]0, 1[ and consider system (1). Obviously -xϕ a (x) < 0 for x = 0, and let x ∈ 4

then 0 x dz -|z| a sgn(z) = |x| 1-a 1 -a < +∞.
The assumptions of Lemma 3.1 are satisfied. Thus the origin is uniformly finite time stable and the solutions φ x (τ ) tend to the origin with the settling time T 0 (x) = |x| 1-a 1-a . These conclusions were directly obtained in the introduction by explicit computation of the solutions (2).

General case

For the more general systems described by Equation (3), a natural extension will invoke the use of Lyapunov functions to give sufficient or necessary conditions for finite time stability.

Sufficient condition using smooth Lyapunov function

In this section, we extend a result coming from [START_REF] Haimo | Finite Time Controllers[END_REF] to non autonomous systems. In the following, one needs the existence of a Lyapunov function V : 4 ≥ × V → 4 ≥ and a continuous function r : 4 ≥ → 4 ≥ such that r (0) = 0 satisfying the following differential inequality

V (t, x) ≤ -r (V (t, x)) (7)
for all (t, x) ∈ I × V.

Since the use of a Lyapunov function will lead to some scalar differential inequality, the following proposition will give sufficient condition for finite time stability: the existence of a Lyapunov function satisfying the condition (7).

Proposition 4.1 Let the origin be an equilibrium point for the system (3) where f is continuous.

i) If there exists a continuously differentiable Lyapunov function satisfying condition (7) with a positive definite continuous function r : 4 ≥0 → 4 ≥0 such that for some ǫ > 0

ǫ 0 dz r(z) < +∞ (8)
then the origin (3) is finite time stable for the system (3). ii) If in addition to i), V is decrescent, then the origin of the system (3) is uniformly finite time stable for the system (3). iii) If in addition to i), the system (3) is globally defined and V is radially unbounded, then the origin of the system (3) is globally finite time stable for the system (3).

Proof i) Since V : I × V → 4 ≥0 is a Lyapunov function (thus satisfies L1 and L2), then the Lyapunov Theorem tells us that the origin is asymptotically stable. Let φ x t (τ ) be a solution of (3) which tends to the origin with the settling time T 0 (φ x t (τ )) (0 ≤ T 0 (φ x t (τ )) ≤ +∞ : from the attractivity of the origin). We have to prove that T 0 (t, x) < +∞. By using the asymptotic stability definition, x can be chosen small enough to ensure that φ x t (τ ) ∈ V for τ ≥ t and τ → V (τ, φ x t (τ )) strictly decreases for τ ≥ t. By using the change of variables: [t, t + T 0 (φ x t )] → [0, V (t, x)] given by z = V (τ, φ x t (τ )), one obtains

0 V (t,x) dz -r(z) = t+T0(φ x t ) t V (τ, φ x t (τ )) -r(V (τ, φ x t (τ ))) dτ.
By assumption, V (τ, φ x t (τ )) ≤ -r(V (τ, φ x t (τ ))) ≤ 0 for all τ ≥ t. This shows that

T 0 (φ x t ) = t+T0(φ x t ) t dτ ≤ V (t,x) 0 dz r(z) .
This implies that T 0 (φ x t ) < +∞. As

V (t,x) 0 dz r(z)
is independent of φ x t , T 0 (t, x) < +∞. Thus, the origin of the system (3) is finite time stable. ii) If V is decrescent, then the system is uniformly asymptotically stable. Moreover, there exists a E. Moulay and W. Perruquetti

K-function β such that V (t, x) ≤ β ( x ). So T 0 (t, x) ≤ β( x ) 0 dz r(z) = α ( x )
with α positive definite.

iii) If V is radially unbounded, the system is globally asymptotically stable. Then, for all x in 4 n all the functions τ → V (τ, φ x t (τ )) decrease. Thus the system is globally finite time stable. Remark 1 The settling time with respect to initial conditions of the system (3) satisfies the following inequality

T 0 (t, x) ≤ V (t,x) 0 dz r(z)
.

so it is continuous at the origin.

Example 4.2 Scalar non autonomous systemConsider the following system:

ẋ = -(1 + t)ϕ a (x), t ≥ 0, x ∈ 4. (9)
The function V (x) = x 2 is a decrescent Lyapunov function for (9) and

V (t, x) = -2(1 + t)xϕ a (x) ≤ -2ϕ a+1 2 (x 2 ) = -r(V (x)). Since ǫ 0 dz r(z)
< +∞, the origin is uniformly finite time stable with the settling time with respect to initial conditions of the system

T 0 (t, x) ≤ 4 |x| 1-a 1 -a .
Example 4.3 two dimensional systemConsider a ∈]0, 1[ and the system:

ẋ1 = -ϕ a (x 1 ) -x 3 1 + x 2 ẋ2 = -ϕ a (x 2 ) -x 3 2 -x 1 . Taking V (x) = x 2 2 , we obtain V (x 1 , x 2 ) = - 2 i=1 (x 4 i + |x i | a+1 ) ≤ 0.
V is a Lyapunov function for the system. Moreover, if

r (z) = ϕ a+1 2 (z), then V (x 1 , x 2 ) ≤ -r (V (x 1 , x 2 )) .
In fact

2 i=1 (x 4 i + |x i | a+1 ) ≥ x 2 1 + x 2 2 a+1 2 = x a+1 .
Thus the origin is uniformly finite time stable with

T 0 (t, x) ≤ 2 1+a 2
x 1-a 1a .

In order to test condition (8) and to conclude to the finite time stability, first one must have the existence of a pair (V, r) satisfying condition (7). For this, it is enough that there exists a decrescent Lyapunov function for the system (3). Indeed, if V is decrescent, then there exists a class K-function α such that

V (t, x) ≤ α ( x )
for all (t, x) ∈ 4 ≥0 × V. As V is negative definite, there exists a class K-function β such that

-V (t, x) ≥ β( x )
for all (t, x) ∈ 4 ≥0 × V. Combining both results, one obtains that

V (t, x) ≤ -β α -1 (V (t, x))
for all (t, x) ∈ 4 ≥0 × V. Note that this condition does not imply that the constructed pair (V, r) will be a good candidate for condition (8).

Sufficient condition using nonsmooth Lyapunov function

The question is to know if it is possible to use a continuous only Lyapunov function to show finite time stability. This is possible by adding the condition that r is locally Lipschitz. We shall use the comparison lemma which can be found in (Khalil 1996, Lemma 3.4):

D + g(t) ≤ f (g(t)) then g(t) ≤ Φ(t, g(a)) for all t ∈ [a, b).
Now, one may give a proposition which generalizes a result given in [START_REF] Bhat | Finite Time Stability of Continuous Autonomous Systems[END_REF] to continuous non autonomous systems.

Proposition 4.5 Let the origin be an equilibrium point for the system (3) where f is continuous. If there exists a continuous Lyapunov function for the system (3) satisfying condition (7) with a positive definite continuous function r : 4 ≥0 → 4 ≥0 locally Lipschitz outside the origin such that ǫ 0 dz r(z) < +∞ with ǫ > 0, then the origin is finite time stable. Moreover, the settling time with respect to initial conditions of the system (3) satisfies

T 0 (t, x) ≤ V (t,x) 0 dz r(z) .
Proof Since V : 4 ≥0 × V → 4 ≥0 is Lyapunov function of the system (3) satisfying condition (7), then from the Lyapunov's theorem, the origin is asymptotically stable. Let x 0 ∈ V and φ x t (τ ) be a solution E. Moulay and W. Perruquetti of (3) which tends to the origin with the settling time T 0 (φ x t ). It remains to prove that T 0 (t, x) < +∞. Because of asymptotic stability, one may suppose with no loss of generality that φ x t (τ ) ∈ V for t ≥ 0 and V (t, x) ∈ [0, ǫ]. Let us consider the system ż = -r(z), z ≥ 0, with the global semi flow Φ(τ, z) for z ≥ 0. Now, applying the comparison lemma (4.4), one deduces that

V (τ, φ x t (τ )) ≤ Φ(τ, V (t, x)), τ ≥ 0, x ∈ V \ {0} .
From Lemma 3.1, one knows that

Φ(τ, z) = 0 for τ ≥ V (t,x) 0 dz r(z) .
With the positive definiteness of V , one concludes that

φ x t (τ ) = 0 for τ ≥ V (t,x) 0 dz r(z) .
Moreover,

V (t,x) 0 dz r(z) is independent of φ x t
, this shows that T 0 (t, x) < +∞. Thus, the origin of the system (3) is finite time stable.

One wants to give an example using a continuous only Lyapunov function.

Example 4.6 Let 0 < a < 1, if we consider the simplest system ẋ = -x a sgn(x), x ∈ 4, and the continuous Lyapunov function V (x) = |x|, we have

V (x) = -|x| a = -V (x) a .
So the system is finite time stable.

Let us sum up the two previous results: in order to obtain the finite time stability, we have the choice between a pair (V 1 , r 1 ) with V 1 which is continuously differentiable and r 1 only continuous, or a pair (V 2 , r 2 ) with V 2 which is continuous and r 2 locally Lipschitz. Moreover, the two conditions are not equivalent because there is no converse theorem for general non autonomous continuous systems.

Necessary conditions

For the moment, there is no necessary and sufficient condition for finite time stability of general continuous (even autonomous) systems. The only converse theorem appears in [START_REF] Bhat | Finite Time Stability of Continuous Autonomous Systems[END_REF] for autonomous systems with uniqueness of solutions in forward time and when the settling time is continuous at the origin. So, one proposes a sufficient condition for non autonomous continuous systems. In the following, one needs the existence of a Lyapunov function V : 4 ≥ × V → 4 ≥ and a continuous function s : 4 ≥ → 4 ≥ such that s (0) = 0 satisfying the following differential inequality

V (t, x) ≥ -s(V (t, x)) (10)
for all (t, x) ∈ 4 ≥0 × V. Then, the following proposition gives necessary conditions for finite time stability:

the existence of such a pair (V, s) such that the differential inequality (10) holds.

Proposition 4.7 Let the origin be an equilibrium point for the system (3) where f is continuous. If the origin is weakly finite time stable for the system (3) then for all Lyapunov functions for the system (3) satisfying condition (10) with a continuous positive definite function s : 4 ≥0 → 4 ≥0 , there exists ǫ > 0

such that ǫ 0 dz s(z) < +∞. ( 11
)
Proof Suppose that V : I × V → 4 ≥0 is a Lyapunov function satisfying condition 10. Let φ x t (τ ) be a solution of (3) with the settling time 0 ≤ T 0 (φ x t ) < +∞. Because of the asymptotic stability, one may choose x small enough to ensure that φ x t (τ ) ∈ V for all τ ≥ t and τ → V (τ, φ x t (τ )) strictly decreases for τ ≥ t. By using the change of variables, [t, t + T 0 (φ

x t )] → [0, V (t, x)] given by z = V (τ, φ x t (τ )), one obtains 0 V (t,x) dz -s(z) = t+T0(φ x t ) t V (τ, φ x t (τ )) -s(V (τ, φ x t (τ )))
dτ.

Since V (τ, φ x t (τ )) ≥ -s(V (τ, φ x t (τ ))) and -s(V (τ, φ x t (τ ))) < 0 for all τ ≥ t one obtains

V (t,x) 0 dz s(z) ≤ t+T0(φ x t ) t dτ = T 0 (φ x t ) < +∞.
Remark 2 This condition may be used to conclude to the non weakly finite time stability and in particular to the non finite time stability for some systems.

Example 4.8 Consider the following system:

ẋ = -|x| 1 + g(t)
where t ∈ 4 and g is a positive function bounded below by c > 0. The function V (x) = x 2 2 is a Lyapunov function for the system and

-V (t, x) ≤ x 2 1 + c = s x 2 2 with s(z) = 2z 1+c . Since x 0 dz s(z)
= +∞ for all x > 0, the origin is not finite time stable.

To test condition (11) and to conclude to the non finite time stability, one must have the existence of a Lyapunov function satisfying condition (10). A sufficient condition to obtain condition (10) is that there exists a Lyapunov function for the system (3) such that -V is decrescent.

There exists a gap between the sufficient and the necessary conditions. If r : 4 ≥0 → 4 ≥0 is a positive definite function such that V = -r (V ) for the system (3) then there exists a necessary and sufficient condition for finite time stability:

• if ǫ 0 dz r(z)
< +∞ then the origin is finite time stable;

• if In general, one uses for r (z) the following function r (z) = ϕ a (z) with 0 < a < 1. In the following example, one has forced to change the form for r (z) in order to show the finite time stability. Let the function u be defined on ]0, 1] by its graph given in figure 1 and such that u 1 n = n 2 for n ≥ 1 and 0 < a < 1. Consider the continuous function

f (x) = 0 if x = 0 r(x 2 ) 2x if x = 0 As f 1 n ≤ -1 2 n u 1 n 2 ≤ 1 2n 3 ,
f is continuous at the origin and thus on 4. Consider the system ẋ = f (x) , x ∈ 4.

V (x) = x 2 is a Lyapunov function for the system and V (x) ≤ -r (V (x)) .

So, the system is finite time stable. Nevertheless, there is no function ϕ b with 0 < b < 1 such that r is bounded below by ϕ b .

Conclusion

A necessary and a sufficient condition for finite time stability of non autonomous continuous systems are given. As mentioned, there is still a gap to obtain necessary and sufficient conditions. The main difficulty comes from the non existence of the flow for only continuous systems and the non continuity of the settling time. Thus it is not an easy task to prove the existence of a Lyapunov function satisfying condition 7 under the hypothesis of finite time stability of the origin. However, with the given sufficient conditions, it is possible to investigate the problem of finite time stabilization for general continuous and non autonomous systems.

Lemma 4. 4

 4 Comparison lemmaIf the scalar differential equation ẋ = f (x), x ∈ 4, has a global semi-flow Φ : 4 ≥0 × 4 → 4, where f is continuous, and if g : [a, b) → 4 (b could be infinity) is a continuous function such that for all t ∈ [a, b),

  = +∞ then the origin is not finite time stable. E. Moulay and W. Perruquetti 5 Application

Figure

  Figure 1. graph of u(z)

Acknowledgment

The authors thank the referees for careful reading and helpful suggestions on the improvement of the manuscript.