Raoul Jauberthie

Franck Rendell

Damien Rangeard

Laurent Molez

Stabilisation of estuarine silt with lime and/or cement

The tidal barrage situated at the mouth of the river Rance, near Saint Malo (France) has led to a considerable silting upstream of the barrage. The deposited material is fine estuarine silt which is frequently dredged to maintain navigable channels; the use of this material is only possible after stabilisation. This study evaluates the possible use of this material as a sub-base for lightly trafficked local roads; therefore the stabilisation has to achieve the required geotechnical characteristics. Physico-chemical characterisation of the silt was carried out using SEM with XRD. The estuarine deposit comprises clay, silt and sand; the characteristics of this material result from its mineral composition and its high organic content. Stabilisation tests were carried out with various mixtures of quicklime, Portland cement and a mixture of both. The results of the tests show an increase in the unconfined compressive strength and the CBR value with both lime and cement treatments. The characteristics of the deposit with lime-cement mixes were superior to those obtained with lime only and cement only treatments. Finally it is concluded that the estuarine silt has the potential to be stabilised to a standard suitable for use as a road sub-base; however further testing is required before proceeding to field trials.

Introduction

The Rance is a river on the north coast of Brittany; the geology of the catchment consists of granites, dolerites, diorites and briovenian schist. The river system consists of a fresh water river feeding into a saline estuary. A tidal barrage was constructed in the 1960s across the mouth of the Rance estuary near St Malo. An area of approximately 22 km² is impounded behind the barrage. The tidal range on the sea side of the barrage is of the order of 8 m whereas behind the barrage the range is reduced to approximately 2.5 m. Fifteen kilometres south of the barrage there is a second barrage at Châtelier, the site considered in this study, which is the upstream limit of the impounded tidal estuary. The tidal barrage at the mouth of the Rance estuary is responsible for the progressive silting of the impounded estuary. Dredging is regularly carried out to maintain navigable channels in the estuary, approximately 20 000 m 3 of silt per year; the storage and disposal of dredged silt pose a considerable problem. This study considers the reuse of estuarine deposits from the area of Châteauneuf d'Ille et Vilaine, at the upstream end of the tidal estuary. Several applications have been considered for the reuse of similar materials, however they all require stabilisation prior to use (Keith and Murray, 1994;Harvey and Murray, 1997;Murray, 2000). The reuse of soils as a road sub-base is generally not possible without stabilisation as its characteristics fall below the minimum required (Kergoet, 2001). A number of parameters influence the soil properties: the time lapse after construction (Khattab et al., 2007); the temperature which controls the kinetics of stabilisation (Rao and Shivananda, 2005); the use of admixtures (Jha and Gill, 2006); mineral additions to the soil (Sivapullaiah et al., 2006). Some authors have demonstrated the possibility of stabilising organic mud from mangroves in the Niger delta (Konyeha and Ogbeide, 2007). Studies and field trials on the use of low grade materials as road sub-base have been carried out in the UK. A detailed study of the stabilisation of clayey silt was carried out by Chaddock and Atkinson (1997); they demonstrated that a soil could be stabilised with a lime-cement mix to achieve a material with characteristics comparable to a Type 2 sub-base.

This work was followed up by field trials such as that carried out at Polgate by-pass UK by Waste & Resources Action Programme (WRAP) (WRAP, 2002). Sétra, a technical department within the French Ministry of Transport and Infrastructure, recently published a technical guide concerning the treatment of soils with lime and/or hydraulic binders (SETRA, 2007). This guide covers the mix design, pave-ment design, implementation and quality control of stabilised road base. The objective of the study is to evaluate the feasibility of using stabilised estuarine silt for a road base. This paper firstly considers the physico-chemical character of the dredged material and secondly, it studies stabilisation using lime and/or cement. The performance of the stabilised soil is evaluated by unconfined compressive strength (UCS) and Californian Bearing Ratio (CBR) value.

Materials

Untreated soil properties

Estuarine silt is a sediment that is deposited in tidal areas of coastal mudflats and is generally composed of a sand fraction, with calcareous shell debris and a fraction of silts and clays.

Mineralogical analysis

The mineralogical content of the silt was determined by X-ray diffraction (XRD). The XRD patterns were obtained using a Philips PW 3710 X-ray diffractometer with Bragg-Brentano geometry using Ni-filtered Cu Kα radiation, operating with the voltage of 30 kV and emission current of 20 mA. The step-scan covered the angular range 2-60° (2θ) in steps of 2θ= 0.02°, 2 s by step. The analysed material was ground to a powder passing a 50 μm sieve.

The recorded XRD pattern for the estuarine silt is shown in Fig. 1. The crystal phases are: quartz, calcite and the clay minerals illite and kaolinite. Examination of X-ray diffraction intensity, although qualitative, indicates that the proportion of clay minerals is very low with regard to other phases. Further examination of the material was carried out using Scanning Electron Microscopy-Energy Dispersive Spectroscopy (SEM-EDS). This was carried out using a JEOL 6400 scanning electron microscope (accelerating voltage 9 kV and working distance 15 mm) coupled with an Oxford INCA microanalysis system. Observations with better definition and higher magnification were obtained with an electron microscope with field emission JEOL JSM 6301 FEG. The observation shown in Fig. 2 and the micro analysis show the fine plates of clay with Si and Al atoms which correspond to the kaolin detected by XRD analysis; similar to the Si, Al and K corresponding to the illite (confirming a mineral with 1 nm diffraction). The larger and rounded minerals are composed of Si (quartz). The organic fraction consists of spicules, coccoliths and diatoms; these particles are in some cases siliceous and others are calcareous.

2.1.2

Granulometric analysis Granulometric analysis was carried out using a Laser Diffraction CILAS particle size analyser 1180 based on Mie and Fraunhofer scattering theories. A second analysis was carried out after a sample was attacked by hydrochloric acid to remove the calcareous material of organic origin. Fig. 3 and Table 1 show that the calcareous particles are of a larger particle size; after acid treatment the curve is moved towards the left. Table 2 shows the geotechnical classification of the material. The liquid limit was determined by cone penetration NF P94-052-1 and by the Casagrande Cup method (XP CEN ISO/TS, 17892-12). The linear shrinkage was determined by ASTM D423-1966 and the methylene blue test carried out to the standard NF P 94-068. The values shown in Table 2 are the average of four tests, each being carried out on separate sediment samples. The values of liquid and plastic limit obtained are characteristic of fine dredged sediments; the results are comparable with those obtained by other authors (Zentar et al., 2009;Boubaker et al., 2009).

Geotechnical characterisation

The Proctor Test is used to determine the optimal moisture content for compaction, that is to say the moisture content at which maximum dry density is achieved. The results from two Proctor Tests (carried out on different samples), dry density as a function of moisture content, are reported in Fig. 4. From these tests it was concluded that the optimum moisture content was of the order of 32% at which the dry density of about 1350 Mg m -3 was obtained. The Californian Bearing Ratio (CBR) Test is carried out on untreated soil with moisture content close to the optimum i.e. 30%. Table 3 sets out the force at the prescribed penetrations; from these results it is noted that the CBR for the untreated soil is 5.5.

Summary of the characterisation of the estuarine silt



The estuarine silt has a 9.4 ratio silt/clay with 11.6% organic matter. From the granulometric analysis it can be seen that most of the organic matter removed by acid attack is between 50 μm and 100 μm. The clay fraction is inorganic which is confirmed by XRD.



The methylene blue value of 2.7 and a Plastic Index of 25% classify the material as a silt of moderate plasticity (class A3 defined by NF P 11 300) the Casagrande classification for the clay is MH.



The Plastic Index of 25%, the clay/silt ratio of 9.4 and linear shrinkage of 12.5% would, according to Burroughs ( 2008), be a soil favourable for stabilisation, i.e. 93% probability of achieving a UCS of 2 MPa.

Characterisation of lime and cement used for treatment

The lime used for stabilisation is industrial quicklime (PIGEON Saint Pierre Lacour 35). This product is currently used for the stabilisation of road sub-bases and capping layers in the Brittany region. The mineralogical composition was determined by XRD, see Fig. 5; this analysis shows that the quicklime contained a small quantity of calcium carbonate in the form of calcite and vaterite. A very small quantity of lime was not burnt.

The cement was a CEMI (without admixtures) (LAFARGE Saint Pierre Lacour). The mineralogical composition was determined by XRD and the XRD pattern is shown in Fig. 6. This cement contained principally alite C3S but it was strongly aluminous with significant quantities of C3A and C4AF.

3 Methods

Lime and/or cement stabilisation

3.1.1

Lime stabilisation A sample of the estuarine silt was dried at a temperature of 100 °C and ground for 1 min with a planetary crusher. Quicklime was added to the ground sample to achieve a specified CaO/soil ratio defined by dry weight (1.5%, 3.0%, 4.0%, 5.0% and 7.0%). Water was then added to the mixture to achieve the optimal moisture content previously determined by the Proctor Test.

Cement stabilisation

The mixing and storage methods were the same as those used in the lime stabilisation tests. Cement was added to the ground sample to achieve a specified cement:soil ratio defined by dry weight (1.5, 3.0, 4.0, 5.0 and 7.0).

3.1.3

Lime-cement stabilisation For these tests the ratio of cement:lime was fixed at unity by weight; the 28 day strength of samples stabilised with various cement+ lime content of the soil (0.75% cement+ 0.75% lime, 1.5%+ 1.5%, 2.0% + 2.0%, and 3.5% + 3.5%).

Preparation and curing of UCS specimens

After mixing, the treated soil was statically compacted in a 40 mm diameter cylinder. This compaction was carried out on one layer of soil by two moveable pistons at the ends of the rigid mould. The applied compaction force and moisture content were set to achieve the optimum dry density with the untreated material; these parameters were then used for all mixes. Lime or cement treatments are known to increase the optimum moisture content and decrease the optimum dry density (Consoli et al. (2009)). It is to be noted that the compaction protocol adopted in this study resulted in dry densities lower than optimal for the various treatments. After compaction the samples were extruded from the mould; the samples were 40 mm diameter and had an aspect ratio of 1.5. Samples were stored in air for 7 or 28 days (15 °C and 50% RH) prior to strength testing. These storage conditions were appropriate as they reflect conditions typically employed on local sites where quicklime carbonates are exposed to atmospheric carbon dioxide.

CBR tests

A CBR test was carried out on the estuarine silt stabilised with 4% quicklime and 4% cement (CEMI), with a water content of 30%, which is close to the optimal value. A sample was compacted in a CBR mould; to obtain a stable humidity the sample was sealed in a polythene bag for a period of eight days. This test protocol is commonly used for French highway studies (SETRA, 2007).

Results and discussion

Results of treatments

4.1.1

Lime stabilisation The evolution of unconfined compressive strength (UCS) for the treated silt is shown in Fig. 7. As can be seen from this figure the UCS values at 7 days ranged from 0.8 MPa to 1.0 MPa for lime additions of between 3% and 7%. Samples stored for 28 days showed a significant increase in strength; as can be seen in Fig. 7 the UCS increased by a factor of two for 7% lime content. This increase is evidence of the relatively slow lime reaction kinetics in the storage conditions (approximately 15 °C, 50% RH).

Cement stabilisation

The UCS for the samples stabilised with cement CEMI are shown in Fig. 8. These strengths are greater than those seen in the case of lime stabilisation; 7% cement addition achieving a UCS of 1.7 MPa at 7 days whereas 7% lime achieved 0.8 MPa. However the increase in strength from 7 to 28 days seen in the case of lime treatment is less evident in the case of cement treatment; for 7% cement the strength increased from 1.7 MPa to 2.3 MPa.

4.1.3

Stabilisation with cement-lime mix The UCS at 28 days are shown in Fig. 9. Previous 28 day results are superimposed on this figure and it can be seen that the UCS of lime+ cement mixes were greater than those for lime and cement alone.

CBR results

At the end of the period the CBR value of the sample was 31%; the penetration force data are shown on Table 4. It can be seen that the CBR value of the estuarine silt stabilised with a cement lime mix achieved a CBR six times greater than that found for the non-stabilised soil.

Summary of the stabilisation treatments

The difference in the kinetics of the lime and cement reactions can be seen in Figs. 7 and8. In the case of lime treatment the increase in strength with time was dependent on the concentration, whereas for the range of cement content tested, treatment showed an increase in strength which is independent of the cement concentration. Fig. 9 shows the treatment with lime and cement led to the highest strengths; additionally a pre-treatment with lime would offer the possibility of controlling the moisture content prior to placement. Similarly quicklime added to a dry soil will modify the clay; the cement will bind the modified peds with a degree of long term strength gain due to the clay-lime reactions. With the French geotechnical classification of soil strength the estuarine silt compacted at optimal moisture content is class P1 (CBR 3% to 6%), considered deformable and liable to rutting. After stabilisation with cementlime at 5% the silt was class P4, slightly deformable and suitable for a 13t axle load. Referring to the UK specification for sub-base materials, it is seen that the estuarine silt stabilised with the cement-lime achieved a CBR value of 30% which is comparable with that required of a Type 2 unbound granular sub-base. The linear shrinkage of the material is of interest as many stabilised clay sub-bases tend to suffer early age shrinkage cracking. The reduction in linear shrinkage (LS) achieved by stabilisation is significant-LS of untreated silt being 12.5% which is reduced to the order of 3% for all the treatments studied.

Conclusion

1. Mineralogical, chemical and geological analyses of the estuarine silts, dredged from upstream of the tidal barrage, have been carried out. The mineral composition was principally quartz and calcare-ous materials emanating from biological species as well as clay minerals exist in the form of illite and kaolinite. 2. The un-stabilised soil had a Plastic Index of 25%, linear shrinkage of 12.5% and a CBR value of 5 at optimal compaction (30% m/c). 3. Stabilisation with a mixture of 3.5% cement plus 3.5% lime achieved the highest strengths, after 28 days the UCS being 3 MPa compared with 2 for 7% lime and 2.3 MPa with 7% cement. The CBR at 7 days was 31%, six times that of the un-stabilised material. 4. The stabilisation has achieved its objective in that the stabilised material fulfils the criteria required of a road sub-base-Class P4 and comparable with a Type 2 granular sub-base. It is concluded that it is possible to stabilise the estuarine silts from the Rance Estuary-this resource of reclaimed material would offer significant financial and ecological benefits. 5. Before adoption of this silt for an application as a road sub-base a second test programme is required:

(i) Improvement to the stabilisation process; analysis of the influence of complementary sulphate admixtures; optimisation of compaction energy; strength development of the stabilised soil under various levels of humidity.

(ii) A characterisation of the final material; tests for swelling, frost resistance, module of deformation and tensile strength. Table 1 Grain size distribution of the estuarine mud.

Table 2 Geotechnical properties of the estuarine silt (MB methylene blue test).

Fig. 1 .

 1 Fig. 1. XRD patterns of the estuarine silt (Cu Kα radiation).

Fig. 2 .

 2 Fig. 2. SEM observations of the silt sample.

Fig. 3 .

 3 Fig. 3. Grain size distribution of the estuarine silt.

Fig. 4 .

 4 Fig. 4. Proctor test for the un-stabilised soil.

Fig. 5 .

 5 Fig. 5. XRD pattern of quicklime (Cu Kα radiation).

Fig. 6 .

 6 Fig. 6. XRD pattern of cement (Cu Kα radiation).

Fig. 7 .

 7 Fig. 7. UCS of lime stabilised estuarine silt at 7 and 28 days.

Fig. 8 .

 8 Fig. 8. UCS of cement stabilised estuarine silt at 7 and 28 days.

Fig. 9 .

 9 Fig. 9. UCS of estuarine silt at 28 days for all treatments.

  • SETRA, 2007. (English Translation 2008) Technical Guide, treatment of soils with lime and/or hydraulic binders. Ref 0718, Ministry of Transport and Infrastructure, France. • Sivapullaiah, P.V., Sridharan, A., Ramesh, H.N., 2006. Effect of sulphate on the shear strength of limetreated kaolinitic soil. Ground Improvement 10, 23-30. • WRAP (Waste & Resources Action Programme), 2002. Polegate Bypass Lime modified, cement stabilised weald clay for sub-base. http://www.wrap.org.uk/downloads/ TheA27PolegateBypass_ABSOLUTE_FINAL_VERSION.f41b9266.1677.pdf2002. • XP CEN ISO/TS 17892-12, 2005. Essais de laboratoire sur les sols. Partie 12: détermination des limites d'Atterberg. (Laboratory soil tests Part 12: determination of Atterberg limits). • Zentar, R., Abriak, N.-E., Dubois, V., 2009. Effects of salts and organic matter on Atterberg limits of dredged marine sediments. Applied Clay Science 42, 391-397.

Table 3

 3 CBR test on sample un-stabilised soil.

	Penetration (mm)	Force (N)	Resistance (MPa)	ICBR
	2.5	703	0.36	5.1
	5	1140	0.58	5.5
	Table 4
	CBR test on sample treated with cement+ lime.
	Penetration (mm)	Force (N)	Resistance (MPa)	ICBR
	2.5	74280	2.167	30.95
	5	6030	3.067	29.21


Jean Lévine
email: jean.levine@minesparis.psl.eu

Differential Flatness by Pure Prolongation Differential Flatness by Pure Prolongation: Necessary and Sufficient Conditions

Keywords: differential flatness, Lie-Bäcklund isomorphism, Lie brackets, distributions of vector fields, prolongation of vector fields, dynamic feedback linearization

In this article, we introduce the notion of differential flatness by pure prolongation: loosely speaking, a system admits this property if, and only if, there exists a pure prolongation of finite order such that the prolonged system is feedback linearizable. We obtain Lie-algebraic necessary and sufficient conditions for a general nonlinear multi-input system to satisfy this property. These conditions are comprised of the involutivity and relative invariance of a pair of filtrations of distributions of vector fields. An algorithm computing the minimal prolongation lengths of the input channels that achieve the system linearization, yielding the associated flat outputs, is deduced. Examples that show the efficiency and computational tractability of the approach are then presented.

Introduction

The notion of dynamic feedback linearization of a nonlinear system, whose preliminary results where reported in [START_REF] Charlet | On dynamic feedback linearization[END_REF][START_REF]Sufficient conditions for dynamic state feedback linearization[END_REF] (see also [START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF][START_REF] Sluis | A bound on the number of integrators needed to linearize a control system[END_REF][START_REF] Battilotti | A constructive condition for dynamic feedback linearization[END_REF][START_REF] Franch | Linearization by prolongations: New bound on the number of integrators[END_REF]), was at the origin of a long chain of studies on nonlinear system theory. In particular, during the last three decades, it gave birth to the concept of differential flatness, that plays a prominent role in motion planning and trajectory tracking problems and their applications (see [START_REF] Ph | Contribution à l'étude des systèmes diffèrentiellement plats[END_REF][START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF][START_REF]A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF] and [START_REF]Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF] for a thorough presentation).

Sufficient or necessary conditions in special cases, as well as general differential flatness characterizations, though unable to detect if a system is non flat in a finite number of steps, have been obtained (see e.g. [START_REF]Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF][START_REF]On necessary and sufficient conditions for differential flatness[END_REF] for a historical review). Nevertheless, the question of obtaining computationally tractable necessary and sufficient conditions for dynamic feedback linearization as well as for differential flatness, remains open. In this paper, we restrict our study to the class of differentially flat systems by pure prolongation, i.e. , roughly speaking, the class of n-dimensional systems with m inputs ẋ = f (x, u 1 , . . . , u m )

for which there exists a multi-index j ≜ (j 1 , . . . , j m ) such that the prolonged system, denoting by u

(j i ) i
the j i -th order time derivative of u i , ẋ = f (x, u 1 , . . . , u m ), u

(j i +1) i = v i , i = 1, . . . , m,
of dimension n + m i=1 j i is locally diffeomorphic to the trivial system y (α i +1) i = w i , i = 1, . . . , m, with y = ψ(x, u 1 , . . . , u

(j 1 )
1 , . . . , u m , . . . , u (jm) m ) w i = W i (x, u 1 , . . . , u

(j 1 )
1 , v 1 , . . . , u m , . . . , u (jm) m , v m ), i = 1, . . . , m, and m i=1 α i = n + m i=1 j i . This diffeomorphism indeed locally implies that x = φ(y 1 , . . . , y

(α 1 ) 1
, . . . , y m , . . . , y (αm) m ) u = U (y 1 , . . . , y 1. proposition 2, extending results implicitly present in [START_REF] Battilotti | A constructive condition for dynamic feedback linearization[END_REF][START_REF] Franch | Linearization by prolongations: New bound on the number of integrators[END_REF], where we prove that, whatever the prolongation j ≜ (j 1 , . . . , j m ), the filtration made of the successive Lie brackets of the prolonged drift with the prolonged control vector fields is decomposable into the direct sum of two filtrations of distributions. The first one, denoted by {∆

(j)
k } k , is included in the original tangent bundle (of dimension n + m) and the other one, denoted by {Γ (j) k } k , is only made of the decreasing orders of the prolonged control vector fields i=1,...,m {g (j i -r) i | r = 0, . . . , max(j i -1, k)}, 2. theorem 3 giving the necessary and sufficient conditions: ∆ (j) k must be involutive with locally constant dimension and invariant by Γ (j) k for all k, and dim ∆ (j) k must be equal to n + m for all k large enough (strong controllability), [START_REF] Brunovský | A classification of linear controllable systems[END_REF]. formula (4.8) and theorem 4 giving the i-th prolongation length, j i , knowing the i-1 first minimal prolongation lengths j 1 = 0 ≤ . . . , ≤ j i-1 , or the criterion of non flatness by pure prolongation, 4. algorithm 1 whose input is the system vector fields and whose output is either the list of minimal prolongation lengths or the claim that the system is not flat by pure prolongation, finishing in a finite number of steps.

The paper is organized as follows: In section 2, we present short recalls of basic results on differential flatness and feedback linearization. Then we introduce and study the pure prolongation distributions and the associated vertical ones in section 3. The necessary and sufficient conditions for flatness by pure prolongation are then presented in section 4, followed by the pure prolongation algorithm. A series of four examples then illustrate our results in section 5, one of them, the pendulum example, being proven to be non flat by pure prolongation, though known to be differentially flat [START_REF]A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF][START_REF]Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF]. The paper ends with concluding remarks.

Recalls on Flatness and Feedback Linearization

Consider a non-linear system over a smooth n-dimensional manifold X given by ẋ = f (x, u)

(2.1)

where x is the n-dimensional state vector, u ∈ R m the input or control vector, with m ≤ n, and f a C ∞ vector field in the tangent bundle TX of X for each u ∈ R m and whose dependence on u is of class C ∞ .

In the sequel, we systematically use boldface letters j, k, . . . , to denote multiintegers (j 1 , j 2 , . . . , j m ), (k 1 , k 2 , . . . , k m ), . . . and overlined symbols to denote a collection of successive time derivatives of a time-dependent function as follows: given a multi-integer k = (k 1 , . . . , k m ) ∈ N m and a locally defined

C ∞ function t → ξ(t) ∈ R m ,
• ξ (k) denotes the vector ξ 1 , ξ1 , . . . , ξ i ≜ d j ξ i dt j , j = 1, . . . , k i , i = 1, . . . , m;

• ξ denotes the infinite sequence (ξ, ξ, ξ, . . .) ≜ ξ (k) i

; i = 1, . . . , m ; k ≥ 0 ∈ R m ∞ , where R m ∞ ≜ R m × R m × • • • is the product of an infinite number of copies of R m .
We also denote the minimum of two arbitrary integers k and l by k ∨ l ≜ min{k, l}, and their maximum by k ∧ l ≜ max(k, l). Also, for every j ≜ (j 1 , . . . , j m ) ∈ N m and k ∈ N, we introduce the componentwise minimum notation j k ≜ (j 1 ∨ k, . . . , j m ∨ k) = (min(j 1 , k), . . . , min(j m , k))

(2.2) and, more generally, for every k ≜ (k 1 , . . . , k m ) ∈ N m , j k ≜ (j 1 ∨ k 1 , . . . , j m ∨ k m ) = (min(j 1 , k 1 ), . . . , min(j m , k m )) .

(2.3)

Accordingly, the componentwise maximum is denoted by j k ≜ (j 1 ∧ k 1 , . . . , j m ∧ k m ) = (max(j 1 , k 1 ), . . . , max(j m , k m )) (2.4) with the notation

j k if k 1 = • • • = k m = k.

Recalls on Lie-Bäcklund Isomorphisms

Roughly speaking 1 , system (2.1) is said to be differentially flat at (x 0 , u 0 ) [START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF][START_REF] Ph | Contribution à l'étude des systèmes diffèrentiellement plats[END_REF][START_REF]A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF][START_REF]Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF] if, and only if, there exists an m-dimensional vector y = (y 1 , . . . , y m ) such that the following statements hold:

• y and its successive derivatives ẏ, ÿ, . . . are locally independent,

• y is a function of x, u and successive derivatives of u up to a finite order s ∈ N m :

y = Ψ(x, u (s) ), (2.5)
Ψ being defined and C ∞ in a suitable neighborhood of (x 0 , u 0 ),

• x and u are functions of y and its derivatives up to a finite order r ∈ N m :

x = Φ -1 (y (r) ), u = Φ 0 (y (r+1) ), (2.6)
with r + 1 ≜ (r 1 + 1, . . . , r m + 1), and where 2 Φ ≜ (Φ -1 , Φ 0 , Φ 1 , . . . , Φ l , . . .) is defined and C ∞ in a suitable neighborhood of y 0 = Ψ(x 0 , u 0 )), again with the notation Ψ ≜ Ψ, Ψ, . . . , Ψ (l) , . . . , where Ψ (l) , l ∈ N, stands for the total l-th order derivative of Ψ, i.e. Ψ (l) (x, u (s+l) ) = y (l) , again with the notation s + l ≜ (s 1 + l, . . . , s m + l).

• Moreover, Φ is such that equation (2.1) is identically satisfied, i.e.

d dt Φ -1 (y (r) ) = f (Φ -1 (y (r) ), Φ 0 (y (r+1) ))
for every C ∞ function t → y(t) such that y(t) remains in the above mentioned neighborhood of y 0 .

Then the vector y is called a flat output. This definition can be made rigorous by viewing Φ and Ψ as functions defined on infinite order jet spaces (see e.g. [START_REF]A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF][START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF][START_REF]Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF][START_REF]On necessary and sufficient conditions for differential flatness[END_REF][START_REF]A necessary condition for dynamic equivalence[END_REF]). They are then called Lie-Bäcklund isomorphisms and are inverse of one another. Let us now recall the definitions of system prolongation and Lie-Bäcklund isomorphism for the sake of completeness. Definition 1. The infinite order jet space prolongation of system (2.1) is given by the pair

(X × R m ∞ , C f ), where X × R m
∞ is the product of X with an infinite number of copies of R m , with coordinates (x, u), endowed with the Cartan field

C f = f (x, u) ∂ ∂x + j≥0 m i=1 u (j+1) i ∂ ∂u (j) i , (2.7)
that lifts the Lie derivative operator along f , defined on TX, to the Lie derivative along

C f on TX × TR m ∞ , the tangent bundle of X × R m ∞ . Definition 2. Consider two systems: ẋ = g(x, u) and ẏ = γ(y, v) (2.8)
and their prolongations (X × R m ∞ , C g ), with coordinates (x, u) and Cartan field

C g = g(x, u) ∂ ∂x + j≥0 m i=1 u (j+1) i ∂ ∂u (j) i , (2.9)
and (Y × R µ ∞ , C γ ), with coordinates (y, v), and Cartan field

C γ = γ(y, v) ∂ ∂y + j≥0 µ i=1 v (j+1) i ∂ ∂v (j) i .
(2.10)

We say that they are Lie-Bäcklund equivalent at a pair of points (x 0 , u 0 ) and (y 0 , v 0 ) if, and only if, there exists neighborhoods

N x 0 ,u 0 ⊂ X × R m ∞ and N y 0 ,v 0 ⊂ Y × R µ ∞ and a C ∞ isomorphism 3 Φ : N y 0 ,v 0 → N x 0 ,u 0 satisfying Φ(y 0 , v 0 ) = (x 0 , u 0 ), with C ∞ inverse Ψ, such that the respective Cartan fields are Φ and Ψ related, i.e. Φ * C γ = C g in N x 0 ,u 0 and Ψ * C g = C γ in N y 0 ,v 0 .
In other words, the two systems are Lie-Bäcklund equivalent at the points (x 0 , u 0 ) and (y 0 , v 0 ) if there exist neighborhoods of these points where every integral curve of the first system is mapped to an integral curve of the second one and conversely, with the same time parameterization.

We recall, without proof, a most important result from [START_REF] Ph | Contribution à l'étude des systèmes diffèrentiellement plats[END_REF] (see also [START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF][START_REF]A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF][START_REF]Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF]) giving an interpretation of the Lie-Bäcklund equivalence in terms of diffeomorphism in finite dimension and endogenous dynamic feedback, that will be useful in the next sections.

Theorem 1 (Martin [START_REF] Ph | Contribution à l'étude des systèmes diffèrentiellement plats[END_REF]). If the systems ẋ = g(x, u), ẏ = γ(y, v) are Lie-Bäcklund equivalent at a given pair of points, then (i) and (ii) must be satisfied:

(i) m = µ, i.e. they must have the same number of independent inputs;

(ii) there exist an endogenous dynamic feedback 4 u = α(x, z, w), ż = β(x, z, w), (2.11) where z belongs to Z, a finite dimensional smooth manifold, a multi-integer 5 r ≜ (r 1 , . . . , r m ),

-and a local diffeomorphism χ : Y × R |r| → X × Z,
all defined in a neighborhood of the considered points, such that the extended system ẏ = γ(y, v), v (r) = w (2.12)

and the closed-loop one

ẋ = g(x, α(x, z, w)), ż = β(x, z, w) (2.13)
are χ-related for all w ∈ R m , i.e.

(x, z) = χ(y, v, v, . . . , v (r-1) ), (y, v, v, . . . , v (r-1) ) = χ -1 (x, z) (2.14)

and ĝ = χ * γ, γ = χ -1 * ĝ (2.15)
where we have denoted

ĝ(x, z, w) ≜ g(x, α(x, z, w)) ∂ ∂x + β(x, z, w) ∂ ∂z γ(y, v, v, . . . , v (r-1) , w) ≜ γ(y, v) ∂ ∂y + m i=1 r i -1 j=0 v (j+1) i ∂ ∂v (j) i + w i ∂ ∂v (r i ) i .

Recalls on Differential Flatness

We say that system (2.1) is differentially flat (or, more shortly, flat) at the pair of points (x 0 , u 0 ) and y 0 ∈ R m ∞ if and only if, it is Lie-Bäcklund equivalent to the trivial system of R m ∞ endowed with the trivial Cartan field

τ ≜ j≥0 m i=1 y (j+1) i ∂ ∂y (j) i (2.

16)

4 A dynamic feedback is said endogenous if, and only if, the closed-loop system and the original one are Lie-Bäcklund equivalent, i.e. if, and only if, the extended state z can be locally expressed as a smooth function of x, u and a finite number of time derivatives of u (see [START_REF] Ph | Contribution à l'étude des systèmes diffèrentiellement plats[END_REF][START_REF] Fliess | Flatness and defect of nonlinear systems: introductory theory and examples[END_REF][START_REF]A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF][START_REF]Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF]). 5 Recall that we denote by v (r) ≜ v

(r 1 ) 1 , . . . , v (rm) m ≜ d r 1 v 1 dt r 1 , . . . , d rm vm dt rm .
at the considered points.

Otherwise stated, the locally defined flat output y = Ψ(x, u) is such that (x, u) = Φ(y) = (Φ -1 (y), Φ 0 (y), Φ 1 (y), . . .), with d dt Φ -1 (y) ≡ f (Φ -1 (y), Φ 0 (y)) for all sufficiently differentiable function y :

t ∈ R → y(t) ∈ R m .
This definition immediately implies that a system is flat if, and only if, there exists a generalized output y = Ψ(x, u) of dimension m, depending at most on a finite number of derivatives of u, with independent derivatives of all orders, such that x and u can be expressed in terms of y and a finite number of its successive derivatives, i.e. (x, u) = Φ(y), and such that the system equation d dt Φ -1 (y) = f •Φ(y) is identically satisfied for all sufficiently differentiable y : R → R m . We thus have recovered, in a rigorous mathematical framework, the definition of section 2.1 (see e.g. [17, § 5.2] for a related discussion).

For a flat system, with the notations of theorem 1, the vector field γ, or γ indifferently, corresponds to the linear system in Brunovský canonical form

y (r i +1) i = w i , i = 1, . . . , m, (2.17)
and C γ , defined by (2.10), satisfies C γ = τ , with τ given by (2.16). Theorem 1 reads:

Corollary 1. If system (2.1
) is flat at a given point, there exists an endogenous dynamic feedback of the form (2.11) and a finite dimensional local diffeomorphism χ such that the systems (2.17) and (2.13), with f in place of g, are χ-related for all w ∈ R m .

Recalls on Feedback Linearization and Lie brackets

Feedback linearizable systems [START_REF] Jakubczyk | On linearization of control systems[END_REF][START_REF]Global transformations of nonlinear systems[END_REF] (see also [START_REF] Isidori | Nonlinear control systems[END_REF][START_REF] Nijmeijer | Nonlinear dynamical control systems[END_REF][START_REF] Lévine | Static and dynamic state feedback linearization, Nonlinear Systems[END_REF]) constitute a subclass of differentially flat systems. We recall their definition. • a local diffeomorphism φ from a neighborhood of an equilibrium point of X (which may be chosen, without loss of generality, as the origin 0 ∈ X) to a suitable neighborhood of 0 ∈ R n ,

• and a static feedback u = α(x, v), v ∈ R m being the new control input and α being invertible with respect to v for all x in the above mentioned neighborhood of the origin, i.e. rank ∂α ∂v (x, v) = m for all x and v as above, such that the image by φ of the closed-loop vector field, namely φ * (f • α), is equal to the vector field associated to the Brunovsky controllability canonical form

y (r i ) i = v i , i = 1, . . . , m, (2.18)
where the multi-integer r ≜ (r 1 , . . . , r m ), whose components r i are called the controllability indices, satisfies |r| ≜ m i=1 r i = n = dim X.

Indeed, since the n-dimensional vectors y (r-1) and x are diffeomorphic (again we have noted r -1 ≜ (r 1 -1, . . . , r m -1)), it is immediate to verify that (y 1 , . . . , y m ) is a flat output and that a feedback linearizable system is flat.

These systems have been first characterized by [START_REF] Jakubczyk | On linearization of control systems[END_REF][START_REF] Hunt | Design for multi-input nonlinear systems, Differential Geometric Control Theory[END_REF][START_REF]Global transformations of nonlinear systems[END_REF] for control-affine systems, i.e. systems given by f (x, u) = f 0 (x) + m i=1 u i f i (x). More generally, it can easily be proven that systems of the form (2.1) are feedback linearizable if, and only if, the control-affine prolongation

ẋ = f (x, u), ui = u (1) i , i = 1, . . . , m (2.19)
with state x (0) ≜ (x, u) = (x 1 , . . . , x n , u 1 , . . . , u m ) ∈ X (0) ≜ X × R m and control vector u (1) ≜ (u

(1) 1 , . . . , u (1)
m ) ∈ R m , is feedback linearizable (see e.g. [START_REF] Charlet | On dynamic feedback linearization[END_REF][START_REF] Van Der Schaft | Linearization and input-output decoupling for general nonlinear systems[END_REF][START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF][START_REF] Lévine | Static and dynamic state feedback linearization, Nonlinear Systems[END_REF]). Indeed, in the local coordinates6 x (0) of X (0) , denoting the associated vector fields by

g (0) 0 (x (0) ) ≜ n i=1 f i (x, u) ∂ ∂x i , g (0)
i (x (0) ) ≜

∂ ∂u i , i = 1, . . . , m, (2.20)
defined on the tangent bundle TX (0) = TX × TR m , system (2.19) reads

ẋ(0) = g (0) 0 (x (0) ) + m i=1 u (1) i g (0) i (x (0) ) (2.21)
with the usual abuse of notations identifying a vector field expressed in local coordinates with its associated (Lie derivative) first order partial differential operator. Until now, for simplicity's sake, a system (2.1) will always be considered in the form (2.21), even if the vector-field f is already given in control-affine form. For the sake of coherence, we set u = (u 1 , . . . , u m ) ≜ u (0) = (u

(0) 1 , . . . , u (0)
m ), so that u(0) = u (1) .

We recall that the Lie bracket [η, γ] of two arbitrary vector fields η and γ of TX (0) is given, in the

x (0) -coordinates, by [η, γ] ≜ n+m i=1 n+m j=1 η j ∂γ i ∂x (0) j -γ j ∂η i ∂x (0) j ∂ ∂x (0) i , with x (0) = (x, u (0) ) ≜ (x (0) 1 , . . . , x (0)
n+m ). For iterated Lie brackets, we use the notation ad η γ ≜ [η, γ] and ad k η γ ≜ [η, ad k-1 η γ] for k ≥ 1, with the convention that ad 0 η γ = γ. In addition, if Γ is an arbitrary distribution of vector fields on TX (0) , we note ad k η Γ ≜ {ad k η γ : γ ∈ Γ}. The distribution Γ is said involutive if, and only if, [η, γ] ∈ Γ for every pair of vector fields η, γ ∈ Γ, to which case we note [Γ, Γ] ⊂ Γ, or Γ = Γ, where Γ denotes the involutive closure of Γ, i.e. the smallest involutive distribution containing Γ.

If the distribution Γ is locally generated by p vector fields γ 1 , . . . , γ p , with p arbitrary, we write Γ ≜ {γ 1 , . . . , γ p }. We also denote by Γ(ξ) ≜ {γ 1 (ξ), . . . , γ p (ξ)} the vector space generated by the vectors γ 1 (ξ), . . . , γ p (ξ) at a point ξ ∈ X (0) .

Consider the (0th order or non prolonged) filtration of distributions built on the vector fields (2.20) 7 :

G (0) 0 ≜ {g (0) 1 , . . . , g (0) m }, G (0) k+1 ≜ G (0) k + ad g (0) 0 G (0) k , ∀k ≥ 0, (2.22) indeed satisfying G (0) 0 ⊂ • • • ⊂ G (0) k ⊂ G (0) k+1 ⊂ • • • ⊂ TX (0) .
Theorem 2 ( [START_REF] Jakubczyk | On linearization of control systems[END_REF][START_REF]Global transformations of nonlinear systems[END_REF]). System (2.1), or equivalently system (2.19), is feedback linearizable in a neighborhood of the origin of X (0) if, and only if, in this neighborhood:

(i) G (0)
k is involutive with constant dimension for all k ≥ 0, (ii) there exists an integer k

(0) ⋆ ≤ n such that G (0) k = G (0) k (0) ⋆ = TX (0) for all k ≥ k (0) ⋆ .
Note that, according to (2.20), G

0 is involutive with constant dimension, equal to m, by construction.

Theorem 2 provides a construction of flat outputs via Frobenius theorem (see e.g. [START_REF] Chevalley | Theory of Lie groups[END_REF]) and the list of so-called Brunovský's controllability indices [START_REF] Brunovský | A classification of linear controllable systems[END_REF] as follows: Definition 4. Consider the sequence of integers

ρ (0) k ≜ dim G (0) k /G (0) k-1 ∀k ≥ 1, ρ (0) 0 ≜ dim G (0) 0 = m.
The Brunovský controllability indices κ

k 's are defined by

κ (0) k ≜ #{l | ρ (0) l ≥ k}, k = 1, . . . , m,
where #A denotes the number of elements of an arbitrary set A.

It can be proven (see e.g. [START_REF] Jakubczyk | On linearization of control systems[END_REF][START_REF]Global transformations of nonlinear systems[END_REF][START_REF] Isidori | Nonlinear control systems[END_REF][START_REF] Nijmeijer | Nonlinear dynamical control systems[END_REF][START_REF] Lévine | Static and dynamic state feedback linearization, Nonlinear Systems[END_REF]) that, for a feedback linearizable nonlinear system (2.1), or (2.19), we have: 7 As before, the superscript (0) is used to indicate that the distributions G •

• ρ (0) k 's and κ (0) k 's are non increasing sequences, • ρ (0) k ≤ m for all k, ρ (0) k = 0, for all k ≥ k (0) ⋆ + 1, • κ (0) 1 = k (0) ⋆ + 1, κ (0) m ≥ 1,
k (0) ⋆ k=0 ρ (0) k = m k=1 κ (0) k = dim G (0) k (0) ⋆ = n + m. The list κ (0) 1 , . . . , κ (0)
m is uniquely defined up to input permutation, invariant by static state feedback and state diffeomorphism, and is indeed equal to the list of controllability indices of the associated linear system (2.18) with r i = κ

(0) i , i = 1, . . . , m.
Moreover, for all k and all i = 1, . . . , m, and possibly up to a suitable input reordering, we have

G (0) k = m j=1 {ad l g (0) 0 g (0) j | l = 0, . . . , k ∨ (κ (0) j -1)}, G (0) κ (0) 1 -1 = G (0) k (0) ⋆ = TX (0) .
Then, flat outputs (y 1 , . . . , y m ) are locally non trivial solutions of the system of PDE's

L ad k g (0) 0 g (0) j y i = 0, k = 0, . . . , κ (0)
i -2, j = 1, . . . , m, with L ad κ (0) i -1 g (0) 0 g (0) i y i ̸ = 0, (2.23
) for i = 1, . . . , m, where we have denoted by L η φ the Lie derivative of a vector function φ along the vector field η. These solutions are such that the mapping

x (0) → (y 1 , . . . , y (κ (0) 1 -1) 1 , . . . , y m , . . . , y (κ (0) m -1) m ) is a local diffeomorphism.
3 System Pure Prolongation

Purely Prolonged distributions

We now introduce higher order prolongations of the vector fields defined by (2.20), called pure prolongations8 after [START_REF]Sufficient conditions for dynamic state feedback linearization[END_REF] (see also [START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF][START_REF] Sluis | A bound on the number of integrators needed to linearize a control system[END_REF][START_REF] Battilotti | A constructive condition for dynamic feedback linearization[END_REF][START_REF] Franch | Linearization by prolongations: New bound on the number of integrators[END_REF]).

Given a multi-integer j ≜ (j 1 , . . . , j m ) ∈ N m , we note, as before, |j| ≜ m i=1 j i and the prolonged state:

x (j) ≜ (x, u (j) ) ≜ (x 1 , . . . , x n , u (0) 1 , . . . , u (j 1 ) 1 , . . . , u (0) m , . . . , u (jm) m ),
with the convention u

(0) i = u i , i = 1, . . . , m. Let X (j) ≜ X × R m+|j|
be the associated j-th order jet manifold of dimension n + m + |j|, i.e. with coordinates x (j) .

The pure prolongation of order j of system (2.19), or otherwise said, of the vector fields (2.20), in the tangent bundle TX (j) = TX × TR m+|j| , is defined by

g (j) 0 (x (j) ) = n i=1 f i (x, u) ∂ ∂x i + m i=1 j i -1 k=0 u (k+1) i ∂ ∂u (k) i g (j) i (x (j) ) ≜ g (j i ) i (x (j) ) ≜ ∂ ∂u (j i ) i , i = 1, . . . , m, (3.1)
with the convention that

j i -1 k=0 u (k+1) i ∂ ∂u (k) i ≜ 0 if j i = 0.
They are naturally associated to the adjunction of j pure integrators to u = u (0) in (2.19) (with the same usual abuse of notations as in (2.21)):

ẋ(j) = g (j) 0 (x (j) ) + m i=1 u (j i +1) i g (j i ) i (x (j) ) (3.2) or ẋ = f (x, u), u(k) i = u (k+1) i , k = 0, . . . , j i , i = 1, . . . , m, u (j+1) = (u (j 1 +1) 1 , . . . , u (jm+1) m
) being the new control vector of this purely prolonged system, whose state is (x, u (j) ) = x (j) .

Remark 1. Note that the state of the j-th prolonged system, x (j) , coincides with the image of x by the projection p j : x ∈ X × R m ∞ → p j (x) = x (j) ∈ X (j) for all j. In addition, the family of projections p i,j : x (i) ∈ X (i) → x (j) = p i,j (x (i) ) ∈ X (j) for all i, j such that i k ≥ j k for all k = 1, . . . , m, that we note i ⪰ j, indeed satisfies p i,j • p j,k = p i,k for all i ⪰ j ⪰ k and thus allows us to identify the manifold X × R m ∞ with the projective limit of the family (X (i) , p i,j ) for all i and all j such that i ⪰ j, i.e. X × R m ∞ ≃ lim ← X (i) (see e.g. [2, Chap. I, §10]). A similar identification trivially holds for the associated tangent bundles, i.e. TX × TR m ∞ ≃ lim ← TX (i) relatively to the family Tp i,j of tangent projections, hence the identification of the Cartan field C f , defined by (2.7), with lim ← g (j) 0 , the projective limit of the vector fields g (j) 0 . Nevertheless, this property does not hold for the control vector fields g

(j) i since Tp j,k (g (j) i ) is not equal to g (k)
i , for i = 1, . . . , m and j ⪰ k. Moreover, the Lie bracket of vector fields is not preserved by this family of projections. This is one of the reasons why prolongations may enlarge the system transformation range.

Remark 2. Given an arbitrary point x

0 ≜ (x 0 , u 0 ) around which system (2.19) is defined, it is convenient to consider the shift θ : (x, u) ∈ X × R m ∞ → θ(x, u) = (x - x 0 , (u -u 0 )) ≜ (z, v) ∈ X × R m
∞ such that x 0 is mapped to the origin of TX × TR m ∞ , denoted by 0, thus inducing the shift of vector fields:

θ ⋆ (g (j) i )(z, v) ≜ g (j) i (z + x 0 , (v + u 0 )), i = 0, . . . m, (3.3)
now defined in a neighborhood of 0. For the sake of simplicity, we will only consider such shifted vector fields in the sequel while keeping the same notation g (j)

i as before, though abusive, but yet unambiguous.

We now introduce the following filtration of j-th order purely prolonged distributions of TX (j) :

G (j) 0 ≜ {g (j 1 ) 1 , . . . , g (jm) m }, G (j) k+1 ≜ G (j) k + ad g (j) 0 G (j) k , ∀k ≥ 0 (3.4)
Indeed, for j = 0 = (0, . . . , 0), i.e. j 1 = • • • = j m = 0, this filtration coincides with the 0th order one given by (2.22). Similarly to the 0th order case, G (j) 0 is involutive with constant dimension, equal to m, by construction.

Moreover, since every G

(j) k ⊂ TX (j) , with dim TX (j) = n + m+ | j |,

we have

Proposition 1. There exists a finite integer k

(j) ⋆ such that G (j) k = G (j) k (j) ⋆ for all k ≥ k (j)
⋆ and k

(j) ⋆ ≤ n+ | j | . (3.5)
Proof.

Since

n + m+ | j |≥ dim G (j) k (j) ⋆ = k (j) ⋆ k=1 dim G (j) k /G (j) k-1 + dim G (j) 0 ≥ k (j) ⋆ + m,
we get (3.5).

Remark 3. In full generality, k

⋆ depends on the point where it is evaluated. However, if dim G (j) k is constant in an open dense subset of X (j) for all large enough k, so is k

(j) ⋆ .
Let us inductively define the n-dimensional vector functions γ k,j,i , for k ≥ 1, i = 1, . . . , m, and arbitrary j = (j 1 , . . . , j m ) as follows:

γ k+1,j,i ≜ L g (j) 0 γ k,j,i -γ k,j,i ∂f ∂x = L g (0) 0 γ k,j,i + m p=1 jp-1 l=0 u (l+1) p ∂γ k,j,i ∂u (l) p -γ k,j,i ∂f ∂x (3.6) with γ 1,j,i = (-1) (j i +1) ∂f ∂u (0) i . (3.7)
For an arbitrary j and given i = 1, . . . , m, thanks to (3.7), it is readily seen that γ 1,j,i depends at most of x (0) and thus, if k ≤ j i -1, thanks to (3.6), γ k+1,j,i depends at most of x (j k) .

Vertical distributions of purely prolonged ones

Before stating the next Lemma, we need to recall the definition of vertical bundle. Given an arbitrary r ∈ N m and the fiber bundle π r : X (r) → R m+|r| , with π r (x (r) ) = u (r) , its vertical space at x (r) , denoted by V x (r) X (r) , is the tangent space T x X. Its vertical bundle, denoted by VX (r) , is the vector bundle made of the vertical spaces at each x (r) , i.e. the set of linear combinations n i=1 α i (x (r) ) ∂ ∂x i whose coefficients α i are smooth functions that depend at most on x (r) and where (x 1 , . . . , x n ) are local coordinates of X.

The same definition indeed holds for the vertical bundle V(X ×R m ∞ ) associated to the fiber bundle π : We now establish comparison formulae between Lie brackets of the vector fieds of the finitely purely prolonged system and those of the original (non prolonged) one.

X ×R m ∞ → R m ∞ , i.e.
Lemma 1 (Comparison formulae). For all j = (j 1 , . . . , j m ) ∈ N m satisfying 0 ≤ j 1 ≤ . . . ≤ j m , with j m finite, for all k ≤ j i and i = 1, . . . , m, we have:

ad k g (j) 0 g (j i ) i = (-1) k ∂ ∂u (j i -k) i = (-1) k g (j i -k) i (3.8)
and for all k ≥ 1:

ad j i +k g (j) 0 g (j i ) i = (-1) j i ad k g (j) 0 g (0) i = γ k,j,i ∂ ∂x ∈ VX (j (k-1)) , (3.9)
Moreover, we have

[g (jp-k) p , ad l-jq g (j) 0 g (0)
q ] = 0, ∀k < j p , ∀l ≥ j q s.t. k + l < j p + j q + 1.

(3.10)

Proof. It is immediately seen that ad g (j) 0 g (j i ) i =   f ∂ ∂x + m k=1 l≥0 u (l+1) k ∂ ∂u (l) k , ∂ ∂u (j i ) i   = - ∂ ∂u (j i-1 ) i = -g (j i -1) i .
Iterating this computation up to k = j i yields (3.8):

ad j i g (j) 0 g (j i ) i = (-1) j i ∂ ∂u (0) i = (-1) j i g (0) i .
Then, for k = j i + 1, using the fact that [ ∂ ∂u

(l) k , ∂ ∂u (0)
i ] = 0 for all i, k and l ≥ 0, we have:

ad j i +1 g (j) 0 g (j i ) i = (-1) j i f ∂ ∂x + m k=1 j k -1 l=0 u (l+1) k ∂ ∂u (l) k , ∂ ∂u (0) i = (-1) j i ad g (0) 0 g (0)
i .

Differential Flatness by Pure Prolongation

Moreover, a direct calculation shows that (-1)

j i ad g (0) 0 g (0) i = (-1) j i f ∂ ∂x , ∂ ∂u (0) i = (-1) (j i +1) ∂f ∂u (0) i (x (0) ) ∂ ∂x , (3.11)
which proves that ad

j i +1 g (j) 0 g (j i ) i
∈ VX (0) . Thus (3.9) holds at the order k = 1.

Assuming that (3.9) holds up to k = ν, with γ ν,j,i depending at most on x (j∨(ν-1)) , we have

ad j i +ν+1 g (j) 0 g (j) i = (-1) j i g (j) 0 , ad ν g (j) 0 g (0) i = L g (j) 0 γ ν,j,i -γ ν,j,i ∂f ∂x ∂ ∂x = γ ν+1,j,i ∂ ∂x ,
and, according to (3.6)-(3.7), for all j p + 1 ≤ ν < j p+1 , p = 1, . . . , m -1, since we differentiate γ ν,j,i with respect to m k=1

j k -1 l=0 u (l+1) k ∂ ∂u (l) k
, it is immediate to verify that γ ν+1,j,i depends at most on

x (j ν) = x, u (0) 1 , . . . , u (j 1 ) 1 , . . . , u (0) p , . . . , u (jp) p , u (0) p+1 , . . . , u (ν)
p+1 , . . . , u (0) m , . . . , u (ν) m and on q = ±γ l-jq,j,q ∂ ∂x depends only on x (j (l-jq-1)) , the derivative of γ l-jq,j,q with respect to g

x (j) = x (j ν) if ν ≥ j m , hence (3.
(jp-k) p is indeed null if j p -k > l -j q -1, hence the result.
Let us assume, without loss of generality, that the control components have been reordered in such a way that j 1 ≤ j 2 ≤ • • • ≤ j m . Moreover, we may suppose that j 1 = 0, as shown to be sufficient in the next section.

We now introduce two new filtrations of TX (j) , noted Γ

k and ∆

(j) k , for k ≥ 0, as follows Γ (j) k ≜ m p=2 g ((jp-l)) p | l = 0, . . . , k ∨ (j p -1) , (3.12)
∆ (j) k ≜ m p=1 ad l-jp g (j) 0 g (0) p | l = j p , . . . , k . (3.13)
with the convention that ad

k-jp g (j) 0 g (0) p = 0 if k < j p , p = 1, . . . , m.
We indeed have Γ

(j) k = Γ (j) jm-1 = m p=2 {g (jp) p , . . . , g (1)
p } for all k ≥ j m -1. Thus, dim Γ

k = Γ (j) k , [Γ (j) k , Γ (j) l ] ⊂ Γ (j) k∧l , ∆ (j) l ⊂ V(X × R m ∞ ), [Γ (j) k , ∆ (j) l ] ⊂ V(X × R m ∞ ).
(3.14)

Remark 4. Note that, in contrast to the increasing filtration {G

(j) k } 0≤k≤k (j) ⋆ , the mapping k → dim ∆ (j)
k is not non-decreasing in general.

Remark 5. In our definition of ∆ p used in [START_REF] Battilotti | A constructive condition for dynamic feedback linearization[END_REF].

Proposition 2. For all j such that 0 = j 1 ≤ . . . ≤ j m-1 ≤ j m , with j m finite, if j = (j 1 , . . . , j i , j i+1 , . . . , j m ) and j ′ = (j 1 , . . . , j i , j ′ i+1 , . . . , j ′ m ) for some (j ′ i+1 , . . . , j ′ m ), we have ∆

(j) k = ∆ (j ′ ) k for all k = 0, . . . , (j i+1 ∨ j ′ i+1 ) -1. Moreover, for all k ≥ 0, dim Γ (j) k =      i p=1 j p + (k + 1)(m -i) if j i ≤ k < j i+1 , i = 1, . . . , m -1 | j | if k ≥ j m (3.15) dim ∆ (j) k ≤      ((k + 1)i - i p=1 j p ) ∨ (n + i) if j i ≤ k < j i+1 , i = 1, . . . , m -1 ((k + 1)m-| j |) ∨ (n + m) if k ≥ j m (3.16
) and we have G

(j) k = Γ (j) k ⊕ ∆ (j)
k , ∀k ≥ 0.

(3.17)

Furthermore, the finite integer k

(j) ⋆ , satisfying (3.5), is such that ∆ (j) k = ∆ (j) k (j) ⋆ and Γ (j) k = Γ (j) k (j) ⋆ for all k ≥ k (j) ⋆ .
If, in addition, dim ∆

(j) k (j) ⋆ = m + n, then n+ | j |≥ k (j) ⋆ ≥ j m ∧ n+ | j | m . (3.18)
Proof. By definition, the generators of Γ (j)

k are independent for all k and thus their number is equal to dim Γ (j) k , hence (3.15). The dimension of ∆ (j) k , in turn, is lesser than, or equal to, the number of its generators, in number (k +1)+. . .+(k

-j i +1) = (k+1)i-i p=1 j p , if j i ≤ k < j i+1 (respectively (k+1)+. . .+(k-j m +1) = (k+1)m-| j |, if k ≥ j m )
, and, since, according to (3.9) of Lemma 1, ∆

(j) k is contained in { ∂ ∂u (0) 1 , . . . , ∂ ∂u (0) i , ∂ ∂x 1 , . . . , ∂ ∂xn } if j i ≤ k < j i+1 , i = 1, . . . , m -1 (respectively in { ∂ ∂u (0) 1 , . . . , ∂ ∂u (0) m , ∂ ∂x 1 , . . . , ∂ ∂xn } if k ≥ j m ), its dimension is bounded above by i + n (resp. m + n), hence (3.16)
The proof of (3.17) is by induction. For k = 0, by (3.12)-(3.13), we indeed have

G (j) 0 = g (0) 1 , g (j 2 ) 2 , . . . , g (jm) m = g (0) 1 ⊕ g (j 2 ) 2 , . . . , g (jm) m = ∆ (j) 0 ⊕ Γ (j)
0 . Thus, (3.17) is valid at the order 0.

Assume now that (3.17) holds true up to the order ν > 0 with j r ≤ ν < j r+1 for some r ∈ {1, . . . , m}.

Note that if j r = j r+1 , the reader may immediately go to the case ν + 1 = j r+1 below.

At the order ν +1, two cases are possible: either j r ≤ ν +1 < j r+1 or ν +1 = j r+1 . In the first case, using Lemma 1, we get:

G (j) ν+1 = G (j) ν + ad g (j) 0 G (j) ν = Γ (j) ν ⊕ ∆ (j) ν + ad g (j) 0 Γ (j) ν + ad g (j) 0 ∆ (j) ν = Γ (j) ν ⊕ ∆ (j) ν + ad ν+1 g (j) 0 g (0)
1 , . . . , ad ν+1-jr g (j) 0

g (0)
r , g

(j r+1 -ν-1) r+1 , . . . , g (jm-ν-1) m = Γ (j) ν+1 ⊕ ∆ (j)
ν+1 .

In the second case, namely if

ν + 1 = j r+1 , G (j) ν+1 = Γ (j) ν ⊕ ∆ (j) ν + ad ν+1 g (j) 0 g (0)
1 , . . . , ad ν+1-jr g (j) 0

g (0) r , g (0)
r+1 , g

(j r+2 -ν-1) r+2 , . . . , g (jm-ν-1) m = Γ (j) ν+1 ⊕ ∆ (j)
ν+1 .

The case j m ≤ ν + 1 follows the same lines:

G (j) ν+1 = Γ (j) ν ⊕ ∆ (j) ν + ad ν+1 g (j) 0 g (0)
1 , . . . , ad ν+1-jm g (j) 0

g (0) m = Γ (j) ν+1 ⊕ ∆ (j) ν+1
hence (3.17) is proven and the property of the number of iterations k ⋆ and the average value |j| m of the prolongation lengths j 1 , j 2 , . . . , j m , provided that the prolonged system satisfies the strong accessibility rank condition dim ∆ We say that a system is flat by pure prolongation at a point x 0 ∈ X × R m ∞ if, and only if, there exists a pure prolongation of finite order j such that the prolonged system is feedback linearizable at x 0 .

The feedback linearizability of the prolonged system of order j indeed implies that this prolonged system is flat and, consequently, the original system too, which justifies qualifying this property as flatness by pure prolongation. The corresponding class of systems is thus clearly a strict subclass of differentially flat systems.

The following Proposition is a straightforward adaptation of Theorem 2 for an arbitrary order j. Note that, at this stage, nothing is said about a possible choice of j, a question that will be dealt with in subsection 4.2, theorem 4.

Proposition 3. The prolonged system of order j is feedback linearizable at 0 if, and only if, G

k is involutive with locally constant dimension for all k and such that G

(j) k (j) * = TX (j) .
Again, flat outputs can be computed via Frobenius theorem, once established the list of Brunovský's controllability indices of order j, as follows: Definition 6. Consider the sequence of integers

ρ (j) k ≜ dim G (j) k /G (j) k-1 ∀k ≥ 1, ρ (j) 0 ≜ dim G (j) 0 = m.
The Brunovský controllability indices of order j are defined by

κ (j) k ≜ #{l | ρ (j) l ≥ k}, k = 1, . . . , m,
As in the 0th order case, if the prolonged system of order j is feedback linearizable at 0, we have:

• ρ (j) k 's and κ (j) k 's are non increasing sequences, • ρ (j) k ≤ m for all k and ρ (j) k = 0 for all k ≥ k (j) * + 1, • κ (j) 1 = k (j) * + 1, κ (j) m ≥ 1, • k (j) * k=0 ρ (j) k = m k=1 κ (j) k = dim G (j) k (j) * = n + m+ | j |. The list κ (j) 1 , . . . , κ (j)
m is uniquely defined up to input permutation, invariant by prolonged state feedback and prolonged state diffeomorphism, and is thus equal to the list of controllability indices of the associated linear system (2.18) with r i = κ

(j) i , i = 1, . . . , m.
Moreover, for all k and all i = 1, . . . , m, and possibly up to a suitable input reordering, we have

G (j) k = m p=1 ad l g (j) 0 g (j) p | l = 0, . . . k ∨ (κ (j) p -1) , G (j)
κ (j) 1 -1 = G (j) k (j) * = TX (j)
Differential Flatness by Pure Prolongation and flat outputs (y 1 , . . . , y m ) are locally non trivial solutions of the system of PDE's

G (j)
k , dy i = 0, k = 0, . . . , κ

(j) i -2, with G (j) κ (j) i -1
, dy i ̸ = 0, i = 1, . . . , m.

(4.1) Finally, the mapping

x (j) → (y 1 , . . . , y (κ (j) 1 -1) 1 , . . . , y m , . . . , y (κ (j) m -1) m ) is a local diffeomorphism.
The next Lemma extends a well-know result (see e.g. [START_REF] Charlet | On dynamic feedback linearization[END_REF][START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF]) to the context of jets of infinite order (see also [START_REF] Franch | Linearization by prolongations: New bound on the number of integrators[END_REF]).

Lemma 2. We consider system (3.2), denoted by Σ j , with j = (j 1 , . . . , j m ), assuming, without loss of generality, that 0 ≤ j 1 ≤ . . . ≤ j m , possibly up to input renumbering. We denote by j ′ = jj 1 = (0, j 2 -j 1 . . . , j m -j 1 ), and by Σ j ′ the corresponding system. Then Σ j is flat at a given point (x 0 , u 0 ) if, and only if, Σ j ′ is also flat at this point. Moreover, every flat output of Σ j at (x 0 , u 0 ) is a flat output of Σ j ′ at the same point, and conversely.

Proof. It is immediately seen that Σ j and Σ j ′ are Lie-Bäcklund equivalent. Hence, if Σ j is Lie-Bäcklund equivalent to the trivial system, the same property holds for Σ j ′ and conversely. Moreover, Σ j 's flatness implies the existence of y such that (x, u (j-1) , v) = (x, u) = Φ(y), with v = u (j) and y = Ψ(x, u) = Ψ(x, u (j-1) , v), and thus, setting w = u (j ′ ) , we immediately get that (x, u (j ′ -1) , w) = Φ(y), with y = Ψ(x, u (j ′ -1) , w). The converse is obvious and left to the reader.

In virtue of Lemma 2, it suffices to restrict our analysis to prolongations of order j = (j 1 , . . . , j m ) such that 0 = j 1 ≤ . . . , ≤ j m . Theorem 3. A necessary and sufficient condition for flatness by pure prolongation at 0 is that there exists j = (j 1 , . . . , j m ) ∈ N m , 0 = j 1 ≤ . . . , ≤ j m < +∞, such that

(i) ∆ (j) k = ∆ (j) k with dim ∆ (j)
k locally constant for all k ≥ 0 (involutivity with locally constant dimension of all ∆ (j)

k 's), (ii) [Γ (j) k , ∆ (j) k ] ⊂ ∆ (j) k for all k ≥ 0 (invariance of ∆ (j) k by Γ (j) k ), (iii) The integer k (j) * is such that ∆ (j) k = TX ×TR m and Γ (j) k = TR |j| for all k ≥ k (j) * (strong controllability).
Proof. By (3.17) of Proposition 2, we have G

(j) k = Γ (j) k ⊕ ∆ (j) k for all k ≥ 0. Then, G (j) k = G (j) k implies that [Γ (j) k ⊕ ∆ (j) k , Γ (j) k ⊕ ∆ (j) k ] ⊂ Γ (j) k ⊕ ∆ (j) k . Since Γ (j) k = Γ (j)
k for all k, and since [Γ

(j) k , ∆ (j) k ] ∩ Γ (j) k = {0} by Lemma 1, we deduce that [Γ (j) k , ∆ (j) k ] + [∆ (j) k , ∆ (j) k ] ⊂ ∆ (j) k , hence [Γ (j) k , ∆ (j) k ] ⊂ ∆ (j) k and [∆ (j) k , ∆ (j) k ] ⊂ ∆ (j) k , i.e. ∆ (j) k = ∆ (j) k , for all k ≥ 0. Conversely, [Γ (j) k , ∆ (j) k ] ⊂ ∆ (j) k and ∆ (j) k = ∆ (j) k for all k ≥ 0 trivially implies that G (j) k = G (j) k for all k ≥ 0. Moreover, since dim Γ (j)
k is constant by construction, the fact that G (j) k has locally constant dimension is equivalent to the fact that ∆ (j) k has locally constant dimension too for all k ≥ 0, hence (i).

Finally, (iii) is an immediate consequence of the condition that G (j) k = TX × TR m+|j| for all k ≥ k (j) * , and the theorem is proven.

The Pure Prolongation Algorithm

Let us now assume that conditions (i)-(iii) of theorem 3 are satisfied for some j = (j 1 , . . . , j i-1 ), 0 = j 1 ≤ . . . , ≤ j i-1 , for some i = 2, . . . , m. We denote by (j, l) = (j 1 , . . . , j i-1 , l i , . . . , l m ) for some l ≜ (l i , . . . , l m ) ∈ N m-i+1 , with j i-1 ≤ l i ≤ . . . ≤ l m .

For l i -1 ≥ k ≥ j i-1 , (3.12) and (3.13) read:

Γ (j,l) k = {g (j 2 ) 2 , . . . , g (1)
2 , . . . , g

(j i-1 ) i-1 , . . . , g (1)
i-1 , g

(l i ) i , . . . , g (l i -k) i , . . . , g (lm) m , . . . , g (lm-k) m } ∆ (j,l) k = {g (0) 1 , . . . , ad k g (j,l) 0 g (0) 1 , . . . , g (0)
i-1 , . . . , ad

k-j i-1 g (j,l) 0 g (0) i-1 } (4.2) and for l i+1 -1 ≥ k ≥ l i ≥ j i-1 : Γ (j,l) k = {g (j 2 ) 2 , . . . , g (1)
2 , . . . , g

(l i ) i , . . . , g (1)
i , g

(l i+1 ) i+1 , . . . , g (l i+1 -k) i+1
, . . . , g (lm) m , . . . , g

(lm-k) m } ∆ (j,l) k = {g (0) 1 , . . . , ad k g (j,l) 0 g (0) 1 , . . . , g (0)
i-1 , . . . , ad

k-j i-1 g (j,l) 0 g (0) i-1 , g (0)
i , . . . , ad k-l i g (j,l) 0 g

i } (4.3) Note that in (4.2), according to (3.9), ∆ (j,l) k is contained in VX (k-1) and thus does not depend on l at all, and in ( 4

.3), if k ≥ l i ≥ j i-1 , ∆ (j,l)
k , depends at most on l i only.

We deduce that ∆ (j,l) k must be involutive for all k ≥ j i-1 and all l such that l i -1 ≥ k, therefore Proposition 4. If conditions (i)-(iii) of theorem 3 are satisfied for some j = (j 1 , . . . , j i-1 ), 0 = j 1 ≤ . . . ≤ j i-1 , and some i = 2, . . . , m, then, if there exists an l i > j i-1 such that ∆ (j,l) k is not involutive for at least one k such that l i -1 ≥ k ≥ j i-1 and such that dim ∆ (j,l) k < n + m, we cannot conclude on the system flatness by pure prolongation with non prolonged input u (0) 1 . Moreover, if this property holds for every choice of non prolonged input, then the system is not flat by pure prolongation.

In the case k + 1 ≥ l i ≥ j i-1 , let us introduce the number

σ ∆ (k) ≜ min{l i | k + 1 ≥ l i ≥ j i-1 , ∆ (j,l) k = ∆ (j,l) k }. (4.4)
We indeed have

j i-1 ≤ σ ∆ (k) ≤ k + 1 for k ≥ j i-1 (4.5)
and σ ∆ (k) is independent of (l i+1 , . . . , l m ). Note also, again since ∆ 1) , that [g

(j,l) k is contained in VX (k-
(lp-r) p , ∆ (j,l)
k ] = 0 for all r = 0, . . . , k and l such that l i -k > k -1 with k ≥ j i-1 ∧ 2. Thus, by assumption (ii) of theorem 3 for j = (j 1 , . . . , j i-1 ), 0 = j 1 ≤ . . . , ≤ j i-1 , we get that [Γ

(j,l) k , ∆ (j,l) k ] ⊂ ∆ (j,l) k
for all l i > 2k -1. Thus, introducing the number

σ Γ,∆ (k) ≜ min{l i | l i -1 ≥ j i-1 , [Γ (j,l) k , ∆ (j,l) k ] ⊂ ∆ (j,l) k } for k ≥ j i-1 , (4.6)
we have

j i-1 ≤ σ Γ,∆ (k) ≤ 2k -2 for all k ≥ j i-1 (4.7)
and σ Γ,∆ (k) is independent of (l i+1 , . . . , l m ). We thus have proven the following alternative Theorem 4. Assume that conditions (i)-(iii) of theorem 3 are satisfied for some j = (j 1 , . . . , j i-1 ), 0 = j 1 ≤ . . . ≤ j i-1 , and some i = 2, . . . , m. Then 1. if, for every choice of non prolonged input u

(0) 1 , there exists a k ≥ j i-1 such that ∆ (j,l) k is not involutive for all l i > j i-1 or if max k dim ∆ (j,l) k
< n + m for all l, then the system is not flat by pure prolongation; 2. otherwise, j i is given by

j i = max k≥j i-1 (σ Γ,∆ (k) ∧ σ ∆ (k)) (4.8)
and

j i ≤ k (j)
⋆ for all i = 1, . . . , m, i.e. every j i is such that dim ∆ (j,l)

j i < n + m.
Proof. In view of what precedes, it only remains to be proven that j i is such that dim ∆ (j,l)

j i < n + m, which indeed implies that j i ≤ k (j) ⋆ . By (4.8), we clearly have that [Γ (j,l) k , ∆ (j,l) k ] ⊂ ∆ (j,l) k and ∆ (j,l) k = ∆ (j,l) k
for all k ≥ j i-1 . Moreover, by (iii) and (3.18), we have

j i ≤ j m ≤ k (j)
⋆ , hence the boundedness of every j i , i = 1, . . . , m.

We thus immediately deduce the following

Associated Non Prolonged Distributions

Let us start this section by showing that system (5.1) is not static feedback linearizable. We denote the state coordinates by

X ≜ (x 1,1 , x 1,2 , x 1,3 , x 2,1 , x 2,2 , x 3 , u (0) 1 , u (0) 2 ) (n = 6 and m = 2), with x 1,j ≜ x (j) 1 , j = 1, 2, 3, x 2,j ≜ x (j) 2 , j = 1, 2 and u (0) i = u i , i = 1, 2.
The two input variables are u

(1) i ≜ ui , i = 1, 2.
The system vector fields are

g (0) 0 ≜ x 1,2 ∂ ∂x 1,1 + x 1,3 ∂ ∂x 1,2 + u (0) 1 ∂ ∂x 1,3 + x 2,2 ∂ ∂x 2,1 + u (0) 2 ∂ ∂x 2,2 + u (0) 1 u (0) 2 ∂ ∂x 3 g (0) 1 ≜ ∂ ∂u (0) 1 , g (0) 2 ≜ ∂ ∂u (0) 2 (5.3) One can verify that ad g (0) 0 g (0) 1 = - ∂ ∂x 1,3 -u (0) 2 ∂ ∂x 3 , ad 2 g (0) 0 g (0) 1 = ∂ ∂x 1,2 , ad 3 g (0) 0 g (0) 1 = - ∂ ∂x 1,1 , ad 4 g (0) 0 g (0) 1 = 0 and ad g (0) 0 g (0) 2 = - ∂ ∂x 2,2 -u (0) 1 ∂ ∂x 3 , ad 2 g (0) 0 g (0) 2 = ∂ ∂x 2,1
, ad 3

g (0) 0 g (0) 2 = 0. Therefore G (0) 0 = ∂ ∂u (0) 1 , ∂ ∂u (0) 2 = G (0) 0 , G (0) 1 = G (0) 0 + - ∂ ∂x 1,3 -u (0) 2 ∂ ∂x 3 , - ∂ ∂x 2,2 -u (0) 1 ∂ ∂x 3 ̸ = G (0) 1 ,
since, e.g. , [g

(0) 2 , ad g (0) 0 g (0) 1 ] = -∂ ∂x 3 ̸ ∈ G (0)
1 , and dim

G (0) 1 = 4 whereas dim G (0) 1 = 5, G (0) 2 = G (0) 1 + ∂ ∂x 1,2 , ∂ ∂x 2,1 ̸ = G (0) 2 , dim G (0) 2 = 7, G (0) 3 = G (0) 2 + ∂ ∂x 1,1 ̸ = G (0) 3 and G (0) k = G (0) 3 for all k ≥ 3. Moreover, dim G (0) 3 = 7 < dim G (0) 3 = n + m = 8
. We conclude that the system is not feedback linearizable.

Flatness by Pure Prolongation of the First Input

Let us now apply theorem 3 and algorithm 1 with j 2 = 0, i.e. g

(j 2 ) 2 ≜ g (0) 2 = ∂ ∂u (0) 2
to determine if this system is flat by pure prolongation and compute j 1 ≥ 1. Recall that we have set g

(l 1 ,0) 0 = g (0) 0 + l 1 -1 p=0 u (p+1) 1 ∂ ∂u (p) 1 . • k = 0. Setting l 1 ≥ 1, we have Γ (l 1 ,0) 0 = g (l 1 ) 1 = ∂ ∂u (l 1 ) 1 , ∆ (l 1 ,0) 0 = g (0) 2 = ∂ ∂u (0) 2 = ∆ (l 1 ,0) 0 and Γ (l 1 ,0) 0 , ∆ (l 1 ,0) 0 ⊂ ∆ (l 1 ,0) 0 for all l 1 ≥ 1. • k = 1. If l 1 ≥ 2, Γ (l 1 ,0) 1 = g (l 1 ) 1 , g (l 1 -1) 1 = ∂ ∂u (l 1 ) 1 , ∂ ∂u (l 1 -1) 1 and ∆ (l 1 ,0) 1 = g (0) 2 , ad g (l 1 ,0) 0 g (0) 2 = ∂ ∂u (0) 2 , ∂ ∂x 2,2 + u (0) 1 ∂ ∂x 3 = ∆ (l 1 ,0) 1 .
Moreover, it is readily seen that Γ

(l 1 ,0) 1 , ∆ (l 1 ,0) 1 = {0} ⊂ ∆ (l 1 ,0) 1 . Now, for l 1 = 1, we have Γ (1,0) 1 = g (1) 1 = ∂ ∂u (1) 1 and ∆
(1,0) 1 = g (0) 1 , g (0) 2 , ad g (1,0) 0 g (0) 2 = ∂ ∂u (0) 1 , ∂ ∂u (0) 2 , ∂ ∂x 2,2 + u (0) 1 ∂ ∂x 3
which is not involutive.

Thus σ ∆ (1) = 2 and σ Γ,∆ (1) = 0 which implies that

j 1 = max k≥0 σ ∆ (k) ∧ σ Γ,∆ (k) ≥ σ ∆ (1) ∧ σ Γ,∆ (1) = 2. • k = 2. If l 1 ≥ 3, we have Γ (l 1 ,0) 2 = g (l 1 ) 1 , g (l 1 -1) 1 , g (l 1 -2) 1 = ∂ ∂u (l 1 ) 1 , ∂ ∂u (l 1 -1) 1 , ∂ ∂u (l 1 -2) 1 and ∆ (l 1 ,0) 2 = g (0) 2 , ad g (l 1 ,0) 0 g (0) 2 , ad 2 g (l 1 ,0) 0 g (0) 2 = ∂ ∂u (0) 2 , ∂ ∂x 2,2 + u (0) 1 ∂ ∂x 3 , ∂ ∂x 2,1 -u (1) 1 ∂ ∂x 3 = ∆ (l 1 ,0) 2 .
Moreover, it is readily verified that Γ

(l 1 ,0) 2 , ∆ (l 1 ,0) 2 ⊂ ∆ (l 1 ,0) 2 only if l 1 ≥ 4, condition (ii) of theorem 3 being violated if l 1 = 3 and we have σ ∆ (2) = 0 and σ Γ,∆ (2) = 4 which implies that j 1 = max k≥0 σ ∆ (k) ∧ σ Γ,∆ (k) ≥ max r=1,2 σ ∆ (r) ∧ σ Γ,∆ (r) ≥ 4. • k = 3. Again, if l 1 ≥ 4, we have: Γ (l 1 ,0) 3 = g (l 1 ) 1 , g (l 1 -1) 1 , g (l 1 -2) 1 , g (l 1 -3) 1 = ∂ ∂u (l 1 ) 1 , ∂ ∂u (l 1 -1) 1 , ∂ ∂u (l 1 -2) 1 , ∂ ∂u (l 1 -3) 1 and ∆ (l 1 ,0) 3 = g (0) 2 , ad g (l 1 ,0) 0 g (0) 2 , ad 2 g (l 1 ,0) 0 g (0) 2 , ad 3 g (l 1 ,0) 0 g (0) 2 = ∂ ∂u (0) 2 , ∂ ∂x 2,2 + u (0) 1 ∂ ∂x 3 , ∂ ∂x 2,1 -u (1) 1 ∂ ∂x 3 , u (2) 1
∂ ∂x 3 = ∂ ∂u (0) 2 , ∂ ∂x 2,2 , ∂ ∂x 2,1 , ∂ ∂x 3 = ∆ (l 1 ,0) 3 provided that u (2)
1 ̸ = 0. We also indeed have Γ

(l 1 ,0) 3 , ∆ (l 1 ,0) 3 ⊂ ∆ (l 1 ,0) 3 for l 1 ≥ 4 hence σ Γ,∆ (3) ≤ 4 and j 1 ≥ max r=1,2,3 σ ∆ (r) ∧ σ Γ,∆ (r) ≥ 4. • k = 4 If l 1 ≥ 5, we have Γ (l 1 ,0) 4 = ∂ ∂u (l 1 ) 1 , . . . , ∂ ∂u (l 1 -4) 1 and ∆ (l 1 ,0) 4 = ∆ (l 1 ,0) 3 since ad 4 g (l 1 ,0) 0 g (0) 2 = -u (3) 1 ∂ ∂x 3 ∈ ∆ (l 1 ,0) 3 . If now l 1 = 4, Γ (4,0) 4 = ∂ ∂u (4) 1 , . . . , ∂ ∂u (1) 1 and ∆
(4,0) 4 = g (0) 1 , g (0) 2 , ad g (4,0) 0 g (0) 2 , . . . , ad 4 g (4,0) 0 g (0) 2 = ∂ ∂u (0) 1 , ∂ ∂u (0) 2 , ∂ ∂x 2,2 , ∂ ∂x 2,1 , ∂ ∂x 3 = ∆ (4,0) 4 .
We thus immediately get Γ

(l 1 ,0) 4 , ∆ (l 1 ,0) 4 ⊂ ∆ (l 1 ,0) 4
for all l 1 ≥ 4.

• k ≥ 5 Finally, the reader may easily check that Γ

+ ad g (4,0) 0 g (0) 1 = ∂ ∂u (0) 1 , ∂ ∂x 1,3 , ∂ ∂u (0) 2 , ∂ ∂x 2,2 , ∂ ∂x 2,1 , ∂ ∂x 3 = ∆ (4,0) 5 ∆ (4,0) 6 = ∆ (4,0) 5 + ad 2 g (4,0) 0 g (0) 1 = ∂ ∂u (0) 1 , ∂ ∂x 1,3 , ∂ ∂x 1,2 , ∂ ∂u (0) 2 , ∂ ∂x 2,2 , ∂ ∂x 2,1 , ∂ ∂x 3 = ∆ (4,0) 5 ∆ (4,0) 7 = ∆ (4,0) 6 + ad 3 g (4,0) 0 g (0) 1 = TR 8 .
We also indeed have Γ

(4,0)) k , ∆ (4,0)) k ⊂ ∆ (4,0) k
for all k ≥ 5. Hence, for all k ≥ 0, the minimal j is equal to (4, 0) and we conclude that system (5.1), with the first input channel prolonged up to j 1 = 4, i.e. controlled by u The prolonged system is now expressed in the state coordinates

x (4,0) ≜ (x 1,1 , x 1,2 , x 1,3 , x 2,1 , x 2,2 , x 3 , u (0) 1 , u (1) 1 , u (2) 1 , u (3) 1 , u (4) 1 , u (0) 2 )
still with x 1,j ≜ x

(j-1) 1 , j = 1, 2, 3, x 2,j ≜ x (j-1) 2 , j = 1, 2 and u (0) i = u i , i = 1, 2.
We indeed still have n = 6 but the prolonged state dimension is now equal to n + m + |j| = 12 with the two input variables u 2 .

The prolonged system vector fields are

g (4,0) 0 ≜ g (0) 0 + 3 j=0 u (j+1) 1 ∂ ∂u (j) 1 g (4,0) 1 ≜ ∂ ∂u (4) 1 g (4,0) 2 ≜ g (0) 2 = ∂ ∂u (0) 2 with g (0) 0 and g (0)
2 given by (5.3). We check the linearizability by computing the successive Lie brackets: = TR 12 .

ad k g (4,0) 0 g (4,0) 1 = (-1) k ∂ ∂u (4-k) 1 k = 0, . . . , 4, ad 5 g (4,0) 0 g (4,0) 1 = ∂ ∂x 1,3 + u (0) 2 ∂ ∂x 3 , ad 6 g (4,0) 0 g (4,0) 1 =
- ∂ ∂x 1,2 + u (1) 2 ∂ ∂x 3 , ad 7 g (4,0) 0 g (4,0) 1 = u (2) 2
∂ ∂x 3 + ∂ ∂x 1,1 , ad k g (4,0) 0 g (4,0) 1 = u
(k-1) 2 ∂ ∂x 3 ∀k ≥ 8 and ad g (4,0) 0 g (4,0) 2 = - ∂ ∂x 2,2 -u (0) 1 ∂ ∂x 3 , ad 2 g (4,0) 0 g (4,0) 2 = ∂ ∂x 2,1 -u (1) 1 ∂ ∂x 3 , ad k g (4,0) 0 g (4,0) 2 = -u (k-1) 1 ∂ ∂x 3 ∀k ≥ 3 Hence, G (4,0) 0 = ∂ ∂u (4) 1 , ∂ ∂u (0) 2 =
G (4,0) 0 , G (4,0) 1 =
G (4,0) 0 + ∂ ∂u (3) 1 , - ∂ ∂x 2,2 -u (0) 1 ∂ ∂x 3 = G (4,0) 1 , G (4,0) 2 =
G (4,0) 1 + ∂ ∂u (2) 1 , ∂ ∂x 2,1 -u (1) 1 ∂ ∂x 3 = G (4,0) 2 , G (4,0) 3 =
G (4,0) 2 + ∂ ∂u (1) 1 , -u (2) 1
∂ ∂x 3 = G (4,0) 3 , G (4,0) 4 =
G (4,0) 3 + ∂ ∂u (0) 1 = G (4,0) 4 , G (4,0) 5 = G (4,0) 4 +
∂ ∂x 1,3 = G (4,0) 5 , G (4,0) 6 = G (4,0) 5 +
∂ ∂x 1,2 = G (4,0) 6 ,
It results that the system (5.1) is flat by pure prolongation in any neighborhood excluding u whose solution is

y 1 = x 1 , y 2 = x 3 -x 2,2 u (0) 1 + x 2,1 u (1) 1 = x 3 -ẋ2 u 1 + x 2 u1 .
(5.5)

Remark 7. Another linearization by pure prolongation could have been obtained by interchanging the first and second input. However, we leave to the reader the verification that, by prolonging the second input, one obtains a prolongation of order 6, with flat outputs given by (5.2), thus larger than the one given by (5.5), associated to the first input, of the 4th order. They conjectured r to be equal to 1 in the present case (with their notations, α 1 = 3, α 2 = 2 and min(α 1 , α 2 ) -1 = 1). As the reader may easily verify, it is indeed minimal since the minimal j is (4, 0) and moreover since, by the equations of the first line of (5.4), y 1 neither can depend on u

1 but explicitly depends on u [START_REF] Ph | Flat systems, Plenary Lectures and Minicourses, Proc. ECC[END_REF][START_REF] Ph | Any (controllable) driftless system with 3 inputs and 5 states is flat[END_REF][START_REF]Any (controllable) driftless system with m inputs and m+2 states is flat[END_REF]

4-dimensional Driftless Bilinear System

ẋ1 = u 1 ẋ2 = x 3 u 1 ẋ3 = x 4 u 1 ẋ4 = u 2 (5.6)
It is immediate to verify that this system is flat with flat output

y 1 = x 1 , y 2 = x 2 (5.7)
(see [START_REF] Ph | Any (controllable) driftless system with 3 inputs and 5 states is flat[END_REF][START_REF]Any (controllable) driftless system with m inputs and m+2 states is flat[END_REF] and [20, theorems 4 and 5]) but not static feedback linearizable. According to our formalism, we consider the state (x 1 , x 2 , x 3 , x 4 , u

2 ) of dimension 6, with n = 4 and m = 2, and the new inputs (u

(1) 1 , u (1)
2 ). The associated vector fields are

g (0) 0 = u (0) 1 ∂ ∂x 1 + x 3 ∂ ∂x 2 + x 4 ∂ ∂x 3 + u (0) 2 ∂ ∂x 4 , g (0) 1
= ∂ ∂u (0) 1 , g (0) 2
= ∂ ∂u (0) 2 .
We compute the distributions Γ

k and ∆

(j)
k and the conditions of theorem 3 using algorithm 1 with j = (j 1 , 0), j 1 ≥ 1.

We thus have

g (l 1 ,0) 0 = g (0) 0 + l 1 -1 p=0 u (p+1) 1 ∂ ∂u (p) 1
, g

(l 1 ) 1 = ∂ ∂u (l 1 ) 1 , g (0) 2
= ∂ ∂u (0) 2 and Γ (l 1 ,0) 0 = ∂ ∂u (l 1 ) 1 , ∆ (l 1 ,0) 0 = ∂ ∂u (0) 2
for all l 1 ≥ 1.

• k = 1. If l 1 ≥ 2, we have Γ l 1 ,0) 1 = ∂ ∂u (l 1 ) 1 , ∂ ∂u (l 1 -1) 1 , ∆ (l 1 ,0) 1 = ∆ (l 1 ,0) 0 + {ad g (l 1 ,0) 0 g (0) 2 } = ∂ ∂u (0) 2 , ∂ ∂x 4
Differential Flatness by Pure Prolongation

and if l 1 = 1, Γ (1,0) 1 = ∂ ∂u (1) 1 , ∆ (1,0) 1 = ∆ (1,0) 0 + {g (0) 1 , ad g (1,0) 0 g (0) 2 } = ∂ ∂u (0) 1 , ∂ ∂u (0) 2 , ∂ ∂x 4 .
We thus have ∆

(l 1 ,0) 1 = ∆ (l 1 ,0) 1 and Γ (l 1 ,0) 1 , ∆ (l 1 ,0) 1 = {0} ⊂ ∆ (l 1 ,0) 1 for all l 1 ≥ 1, hence σ Γ,∆ (1) = σ ∆ (1) = 0. • k = 2. If l 1 ≥ 3, we have Γ (l 1 ,0) 2 = ∂ ∂u (l 1 ) 1 , ∂ ∂u (l 1 -1) 1 , ∂ ∂u (l 1 -2) 1 , ∆ (l 1 ,0) 2 = ∂ ∂u (0) 2 , ∂ ∂x 4 , ∂ ∂x 3 , Γ (l 1 ,0) 2 , ∆ (l 1 ,0) 2 = {0} and if l 1 = 2, Γ (2,0) 2 = ∂ ∂u (2) 1 , ∂ ∂u (1) 1
, ∆

(2,0) 2

= ∂ ∂u (0) 1 , ∂ ∂u (0) 2 , ∂ ∂x 4 , ∂ ∂x 3 , Γ (2,0) 2 , ∆ (2,0) 2 = {0}. Finally, if l 1 = 1, Γ (1,0) 2 = ∂ ∂u (1) 1
, ∆

∂ ∂u (0) 1 , - ∂ ∂x 1 -x 3 ∂ ∂x 2 -x 4 ∂ ∂x 3 , ∂ ∂u (0) 2 , ∂ ∂x 4 , ∂ ∂x 3 ̸ = ∆ (1,0) 2 =
We thus have ∆

(l 1 ,0) 2 = ∆ (l 1 ,0) 2 and Γ (l 1 ,0) 2 , ∆ (l 1 ,0) 2 ⊂ ∆ (l 1 ,0) 2 for all l 1 ≥ 2 since, if l 1 = 1, ∆ (1,0) 2
is not involutive. Therefore, σ Γ,∆ (2) ≤ σ ∆ (2) = 2 and our search may be restricted to j 1 ≥ 2.

• k = 3. If l 1 ≥ 4, we have Γ (l 1 ,0) 3 = ∂ ∂u (l 1 ) 1 , ∂ ∂u (l 1 -1) 1 , ∂ ∂u (l 1 -2) 1 , ∂ ∂u (l 1 -3) 1 , ∆ (l 1 ,0) 3 = ∂ ∂u (0) 2 , ∂ ∂x 4 , ∂ ∂x 3 , ∂ ∂x 2 .
If

l 1 = 3, Γ (3,0) 3 = ∂ ∂u (3) 1 , ∂ ∂u (2) 1 , ∂ ∂u (1) 1 , ∆ (3,0) 3 = ∂ ∂u (0) 1 , ∂ ∂u (0) 2 , ∂ ∂x 4 , ∂ ∂x 3 , ∂ ∂x 2 . If l 1 = 2, Γ (2,0) 3 = ∂ ∂u (2) 1 , ∂ ∂u (1) 1 , ∆ (2,0) 3 = ∂ ∂u (0) 1 , ∂ ∂u (0) 2 , ∂ ∂x 4 , ∂ ∂x 3 , ∂ ∂x 2 , ∂ ∂x 1 = TR 6
Therefore, ∆

(2,0) k = ∆ (2,0) k and Γ (2,0) k , ∆ (2,0) k ⊂ ∆ (2,0) k
for all k ≥ 0 and ∆

(2,0) 3 = TR 6 . We conclude that the conditions of theorem 3 hold true whenever j 1 ≥ 2, which proves that system (5.6) with the pure prolongation of order j = (2, 0) is feedback linearizable, this prolongation being minimal.

Verification. The prolonged state is now x (2,0)

≜ (x 1 , x 2 , x 3 , x 4 , u (0) 1 , u (1) 1 , u (2) 1 , u (0)
2 ) of dimension 8, and the new inputs are (u

(3) 1 , u (1)
2 ).

G (2,0) 0 = ∂ ∂u (2) 1 , ∂ ∂u (0) 2 = G (2,0) 0 , G (2,0) 1 = ∂ ∂u (2) 1 , ∂ ∂u (1) 1 , ∂ ∂u (0) 2 , ∂ ∂x 4 = G (2,0) 1 , G (2,0) 2 = ∂ ∂u (2) 1 , ∂ ∂u (1) 1 , ∂ ∂u (0) 1 , ∂ ∂u (0) 2 , ∂ ∂x 4 , ∂ ∂x 3 = G (2,0) 2 , G (2,0) 3 = ∂ ∂u (2) 1 , ∂ ∂u (1) 1 , ∂ ∂u (0) 1 , ∂ ∂x 1 + x 3 ∂ ∂x 2 + x 4 ∂ ∂x 3 , ∂ ∂u (0) 2 , ∂ ∂x 4 , ∂ ∂x 3 , ∂ ∂x 2 = G (2,0) 3 = TR 8
hence the feedback linearizability of the purely prolonged system with ρ

(2,0) 0 = ρ (2,0) 1 = ρ (2,0) 2 = ρ (2,0) 3 = 2 and κ (2,0) 1 = κ (2,0) 2 = 4 (see definition 6).
Flat outputs (y 1 , y 2 ) are locally non trivial solutions of the system (5.8), i.e. : G (2,0) k , dy i = 0, k = 0, 1, 2, with G

(2,0) 3

, dy i ̸ = 0, i = 1, 2.

(5.8)

It is immediate to verify that

y 1 = x 1 , y 2 = x 2
is a solution of (5.8) and that the mapping

x (2,0) → y 1 , . . . , y (3)
1 , y 2 , . . . , y

is a local diffeomorphism.

Remark 9. The reader may easily check that, since ∆ (0,l 2 ) 2

is not involutive for all l 2 ≥ 3, no pure prolongation on the second channel, u (0) 2 , leads to the linearizability conditions.

An Example from [6]

In our formalism, considering the inputs (u 1 , u 2 ) ≜ (u

(0) 1 , u (0)
2 ) as part of the state, with n = 4 and m = 2, this example from [6, Example 2] reads:

ẋ1 = x 2 + x 3 u (0) 2 ẋ2 = x 3 + x 1 u (0) 2 ẋ3 = u (0) 1 + x 2 u (0) 2 ẋ4 = u (0) 2 u(0) 1 = u (1) 1 u(0) 2 = u (1) 2 .
(5.9)

It is shown in [START_REF]Sufficient conditions for dynamic state feedback linearization[END_REF] that this sytem does not satisfy the sufficient, but not necessary, condition for dynamic linearization of Theorem 4.2 of this paper. Nevertheless, it satisfies the algorithm of [1, section 3.1], that constitutes a sufficient condition for flatness by pure prolongation, without proof of minimality of the obtained prolongation. We show here that it is linearizable by pure prolongation by application of our algorithm and thus give the minimal prolongation.

We prolong the second input at an arbitrary order j 2 ≥ 1 and set j = (0, j 2 ). For l 2 ≥ 1, we denote, as before,

g (0,l 2 ) 0 = (x 2 + x 3 u (0) 2 ) ∂ ∂x 1 + (x 3 + x 1 u (0) 2 ) ∂ ∂x 2 + (u (0) 1 + x 2 u (0) 2 ) ∂ ∂x 3 + u (0) 2 ∂ ∂x 4 + l 2 -1 p=0 u (p+1) 2 ∂ ∂u (p) 2 g (0) 1 = ∂ ∂u (0) 1 , g (l 2 ) 2 = ∂ ∂u (l 2 ) 2
.

We have

Γ (0,l 2 ) 0 = ∂ ∂u (l 2 ) 2 , ∆ (0,l 2 ) 0 = ∂ ∂u (0) 1 = ∆ (0,l 2 ) 0 , Γ (0,l 2 ) 0 , ∆ (0,l 2 ) 0 ⊂ ∆ (0,l 2 ) 0 , ∀l 2 ≥ 1. • k = 1. For all l 2 ≥ 2, ad g (0,l 2 ) 0 g (0) 1 = - ∂ ∂x 3 and Γ (0,l 2 ) 1 = ∂ ∂u (l 2 ) 2 , ∂ ∂u (l 2 -1) 2 , ∆ (0,l 2 ) 1 = ∂ ∂u ((0)) 1 , - ∂ ∂x 3 = ∆ (0,l 2 ) 1 , Γ (0,l 2 ) 1 , ∆ (0,l 2 ) 1 ⊂ ∆ (0,l 2 ) 1
.

Differential Flatness by Pure Prolongation

For l 2 = 1:

Γ (0,1) 1 = ∂ ∂u (1) 2 , ∆ (0,1) 1 = ∂ ∂u (0) 2 , ∂ ∂u (0) 1 , - ∂ ∂x 3 = ∆ (0,1) 1 , Γ (0,1) 1 , ∆ (0,1) 1 ⊂ ∆ (0,1) 1 . • k = 2. For all l 2 ≥ 3, ad 2 g (0,l 2 ) 0 g (0) 1 = u (0) 2 ∂ ∂x 1 + ∂ ∂x 2 and Γ (0,l 2 ) 2 = ∂ ∂u (l 2 ) 2 , ∂ ∂u (l 2 -1) 2 , ∂ ∂u (l 2 -2) 2 , ∆ (0,l 2 ) 2 = ∂ ∂u (0) 1 , -∂ ∂x 3 , u (0) 2 ∂ ∂x 1 + ∂ ∂x 2 = ∆ (0,l 2 ) 2 , Γ (0,l 2 ) 2 , ∆ (0,l 2 ) 2 ⊂ ∆ (0,l 2 ) 2
.

But for l 2 = 2:

Γ (0,2) 2 = ∂ ∂u (2) 2 , ∂ ∂u (1) 2 , ∆ (0,2) 2 =
∂ ∂u (0) 2 , ∂ ∂u (0) 1 , - ∂ ∂x 3 , u (0) 2
∂ ∂x 1 + ∂ ∂x 2 ̸ = ∆ (0,2) 2
therefore, we must exclude j 2 = 2.

• k = 3. For all l 2 ≥ 4, ad 3

g (0,l 2 ) 0 g (0) 1 = (u (1)
2 -1)

∂ ∂x 1 -u (0) 2 2 ∂ ∂x 2 -u (
= 1, Γ (0,l 2 ) 3 = ∂ ∂u (l 2 ) 2 , ∂ ∂u (l 2 -1) 2 , ∂ ∂u (l 2 -2) 2 , ∂ ∂u (l 2 -3) 2 , ∆ (0,l 2 ) 3 = ∂ ∂u (0) 1 , - ∂ ∂x 3 , u (0) 2 ∂ ∂x 1 + ∂ ∂x 2 , (u (1) 2
2 -1)

∂ ∂x 1 + u (0) 2 2 ∂ ∂x 2 -u (0) 2 ∂ ∂x 3 = ∂ ∂u ((0)) 1 , ∂ ∂x 3 , ∂ ∂x 2 , ∂ ∂x 1 = ∆ (0,l 2 ) 3 , Γ (0,l 2 ) 3 , ∆ (0,l 2 ) 3 ⊂ ∆ (0,l 2 ) 3
.

The reader may then easily check that the same holds for l 2 = 3:

Γ (0,3) 3 = ∂ ∂u (3) 2 , ∂ ∂u (2) 2
, ∂ ∂u

,

∆ (0,3) 3 = ∂ ∂u (0) 2 , ∂ ∂u (0) 1 , ∂ ∂x 3 , ∂ ∂x 2 , ∂ ∂x 1 = ∆ (0, 3) 3 .
• For all k ≥ 4. Since ad k g (0,l 2 ) 0 g (0)

1 and ad k-3 g (0,l 2 ) 0 g (0)

2 are linear combinations of ∂ ∂x 1 , ∂ ∂x 2 and ∂ ∂x 3 only, we have, for all l 2 ≥ 4:

∆ (0,l 2 ) k = ∆ (0,l 2 ) k , Γ (0,l 2 ) k , ∆ (0,l 2 ) k ⊂ ∆ (0,l 2 ) k
, and, using the fact that ad g (0,l 2 )

0 g (0) 2 = -x 3 ∂ ∂x 1 -x 1 ∂ ∂x 2 -x 2 ∂ ∂x 3 -∂ ∂x 4 , we have Γ (0,3) 4 = ∂ ∂u (3) 2 , ∂ ∂u (2) 2
, ∂ ∂u

,

∆ (0,3) 4 = ∂ ∂u (0) 2 , -x 3 ∂ ∂x 1 -x 1 ∂ ∂x 2 -x 2 ∂ ∂x 3 - ∂ ∂x 4 , ∂ ∂u (0) 1 , ∂ ∂x 3 , ∂ ∂x 2 , ∂ ∂x 1 = ∂ ∂u (0) 2 , ∂ ∂u (0) 1 , ∂ ∂x 4 , ∂ ∂x 3 , ∂ ∂x 2 , ∂ ∂x 1 = ∆ (0,3) 4 = TR 6
and ∆ (0,3) k = TR 6 ∀k ≥ 4, hence k (0,3) * = 4. We conclude that the conditions of theorem 3 are satisfied for all k provided that j 2 ≥ 3 and thus that the minimal prolongation of the second input channel for which the system is feedback linearizable is equal to 3.

Let us finally give the construction of the prolonged state diffeomorphism. The prolonged state is (x 1 , x 2 , x 3 , x 4 , u , dy 2 ̸ = 0.

Its solution is given by y 1 = x 4 , y 2 = x 1 -u (0) 2 x 2 . Note, on the contrary, that no pure prolongation of the first channel u 1 can linearize the system. Indeed, if we consider j = (j 1 , 0), we may easily check that, for all l 1 ≥ 3,

∆ (l 1 ,0) 2 = ∂ ∂u (0) 2 , -x 3 ∂ ∂x 1 -x 1 ∂ ∂x 2 -x 2 ∂ ∂x 3 - ∂ ∂x 4 , (x 1 -u (0) 1 ) ∂ ∂x 1 -x 3 ∂ ∂x 3
is not involutive, thus contradicting condition (i) of Theorem 3.

The Pendulum Example [9, section II. C]

This model of pendulum in the vertical plane has been studied in [9, section II. C], [17, section 6.2.3], [18, section 5.3] where it is shown to be flat. We prove here that it is not flat by pure prolongation. Though naturally control-affine, it is presented here in its prolonged form (2. (5.10)

The state is (x 1 , x 2 , y 1 , y 2 , θ 1 , θ 2 , u

1 , u

2 ), of dimension n + m = 6 + 2 = 8. The associated non prolonged vector fields are 2 play a symmetric role and one can choose indifferently one of them as the non prolonged input. This assertion will be clarified later on. We choose u (0) 1 as non prolonged input. Thus, the vector fields associated to a prolongation of length l 2 on the second input read: .

g (0) 0 = x 2 ∂ ∂x 1 + y 2 ∂ ∂y 1 + θ 2 ∂ ∂θ 1 + 1 ε sin θ 1 ∂ ∂θ 2 + u (0) 1 ∂ ∂x 2 - 1 ε cos θ 1 ∂ ∂θ 2 + u (
g (0,l 2 ) 0 = x 2 ∂
(5.12)

• k = 1. For all l 2 ≥ 2,

Γ (0,l 2 ) 1 = { ∂ ∂u (l 2 ) 2
, ∂ ∂u

(l 2 -1) 2 }, ∆ (0,l 2 ) 1 = { ∂ ∂u (0) 1 , ∂ ∂x 2 - 1 ε cos θ 1 ∂ ∂θ 2 },
and if l 2 = 1, Γ (0,1) 1

= { ∂ ∂u (1) 2 }, ∆ (0,l 2 ) 1 = { ∂ ∂u (0) 1 , ∂ ∂x 2 - 1 ε cos θ 1 ∂ ∂θ 2 , ∂ ∂u (0) 2
}.

Thus, we have ∆ for all l 2 ≥ 1.

• k = 2. For all l 2 ≥ 3, Γ ] ⊂ ∆ (0,l 2 ) 2 for all l 2 ≥ 1.

, ∂ ∂x 2 - 1 ε cos θ 1 ∂ ∂θ 2 , - ∂ ∂x 1 + 1 ε sin θ 1 ∂ ∂θ 2 , - 1 ε θ 2 sin θ 1 ∂ ∂θ 1 + 1 ε θ 2 2 - 1 ε u (0) 1 sin θ 1 cos θ 1 + 1 ε 1 + u (0) 2 sin 2 θ 1 ∂ ∂θ 2 }.
The reader may easily verify that the Lie bracket of the last element of ∆ (0,l 2 ) 3 with the second or third one is a combination of ∂ ∂θ 1 and ∂ ∂θ 2 not in ∆ (0,l 2 ) 3

. Thus, ∆ (0,l 2 ) 3

is not involutive for all l 2 ≥ 4. Moreover, exchanging the non prolonged input u

(0) 1 in u (0)
2 , a similar calculation, left to the reader, shows that ∆ (l 1 ,0) 3

is not involutive for all l 1 ≥ 4, which proves, according to the first item of theorem 4, that system (5.10) is not flat by pure prolongation.

Concluding Remarks

We have established necessary and sufficient conditions for a system to be flat by pure prolongation, extending preliminary results of [START_REF] Charlet | On dynamic feedback linearization[END_REF][START_REF]Sufficient conditions for dynamic state feedback linearization[END_REF][START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF][START_REF] Sluis | A bound on the number of integrators needed to linearize a control system[END_REF][START_REF] Battilotti | A constructive condition for dynamic feedback linearization[END_REF][START_REF] Franch | Linearization by prolongations: New bound on the number of integrators[END_REF], and drawn up a precise picture on the role played by pure prolongations of vector fields.

These conditions allow the construction of a computationally tractable algorithm giving the minimal prolongation in a finite number of steps using only Lie brackets and linear algebra.

Possible extensions of this work towards general flatness necessary and sufficient conditions are under study.

(k 1 ) 1 ,

 11 . . . , ξ m , ξm , . . . , ξ (km) m of dimension m + |k|, with |k| ≜ m i=1 k i and ξ (j)

  k are built on the non prolonged vector fields(2.20) and to distinguish them from the prolonged distributions G (j) k of arbitrary jth order, j ∈ N m , and related indices, ρ (j) k and κ (j) k , introduced in sections 3 and 4.

  (j) k ≤ |j| for all k. The definitions (3.12)-(3.13) and Lemma 1 readily yield Γ (j)

}Remark 6 .

 6 ⋆ = m + n, we must have j m ≤ k (j) ⋆ since otherwise, using definition (3.13) for k (j) ⋆ < j m , ∆ and its dimension would not exceed m -1 + n. Consequently, applying once more (3.16) with dim ∆ (j) k (j) ⋆ = m + n, we get (k (j) ⋆ + 1)m-| j |≥ m + n, which, combined with (3.5), immediately yields (3.18). The inequality (3.18) reads k (j) ⋆ -|j| m ≥ n m and may thus be interpreted as an estimate of the gap between k (j)

1

 1 Necessary and Sufficient ConditionsDefinition 5.

Algorithm 1 .

 1 flatness by pure prolongation Input: the vector fields g the minimal lengths 0 = j 1 ≤ . . . ≤ j m and k (j) ⋆ or fail if the system is not flat by pure prolongation. Procedure: Initialization. Choose l = (l 1 , . . . , l m ), associated to the prolongations u (l i ) i , with l 1 = 0 and l i ≥ 0, i = 2, . . . , m arbitrary, and compute Γ (l) 0 and ∆ (l) 0 (at this step, ∆ (l) 0 depends only on the choice of u (0) 1). Note that for every choice of l 1 , we have ∆

  Note that the bounds (3.5) and (3.18) are indeed satisfied. They read k

G ( 4

 4

= 7 ., dy 2

 72 The reader may easily verify that ρ (4,0) k = 2 for k = 0, . . . , 3 and ρ (4,0) k = 1 for k = 4, . . . , 7, which yields κ The corresponding flat outputs are thus obtained by solving the set of P.D.E.'s G (4,0) k, dy 1 = 0, k = 0, . . . , = 0, k = 0, . . . ,

Remark 8 .

 8 In [20, section 3.1.1], the authors consider a dual notion of minimality, called r-flatness, where r is the minimal number over all possible flat outputs of the maximal number of derivatives of the inputs that appear in the flat outputs, i.e. r = min Y : flat output max i=1,...,m {s i | s = (s 1 , . . . , s m ), y = Y (x, u (s) )}.

2 = 4

 24 and the system of PDE's that the flat outputs must satisfy is :G (0,3) k , dy 1 = 0, k = 0, . . . , 3, G (0,3) 4 , dy 1 ̸ = 0 G (0,3) k, dy 2 = 0, k = 0, . . . , 2, G (0,3) 3

• k = 3 . 4

 34 For all l 2 ≥

  the set of linear combinations of ∂ ∂x 1 , . . . , ∂

∂xn whose coefficients are smooth functions of x.

See section

2.2 of this article for a precise definition.[START_REF] Bourbaki | Algèbre, Éléments de Mathématiques[END_REF] the components of Φ are indexed from -1 to ∞, to let the component u (k) of u be Φ k for all k ≥ 0.

Recall that a continuous function and, a fortiori, differentiable, resp. C ∞ , depends, by definition of the source and target product topologies, on a finite number of components of its variables, namely Φ (resp. Ψ) depends an a finite number of components of (y, v) (resp. (x, u)) (see e.g.[START_REF] Krasil'shchik | Geometry of jet spaces and nonlinear partial differential equations[END_REF][START_REF] Zharinov | Geometrical aspect of partial differential equations[END_REF][START_REF]Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF]).

We introduce the superscript (0) from now on to get ready to work with higher order prolongations (see section 3).

They are called pure dynamic extensions in[START_REF] Sluis | A necessary condition for dynamic feedback linearization[END_REF][START_REF] Sluis | A bound on the number of integrators needed to linearize a control system[END_REF], a terminology used for feedback design, whereas prolongation here refers to the jet space coordinates in the spirit of É. Cartan[START_REF] Cartan | Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes[END_REF] who used the French word "prolongement".

∧ j 1 = 5 and k (4,0) ⋆ = 7 ≤ n + j 1 = 10, but they are not tight. Accordingly, evaluating the bound on the number of integrators needed to linearize the system, proposed by[START_REF] Sluis | A bound on the number of integrators needed to linearize a control system[END_REF] for m = 2, we find 2n -3 = 9, and the one proposed by[START_REF] Franch | Linearization by prolongations: New bound on the number of integrators[END_REF] gives 2n -1 6 (8 + 24 -14) = 2n -3 = 9 as well.

Acknowledgements-The author wishes to express his warm thanks to Ph. Martin for many fruitful discussions.

Step 1. For k ≥ 1, compute ∆ (l) k for all 1 ≤ l 2 ≤ k + 1, σ ∆ (k) and σ Γ,∆ (k). Continue up to the first k 1 such that σ Γ,∆ (k 1 ) ∧ σ ∆ (k 1 ) is maximal. Then set

k is non involutive for some k ≤ k 1 , permute the inputs to change u (0) 1 and restart.

Step i, 1 ≤ i ≤ m. Same as step 1 with k ≥ j i-1 to obtain k i-1 and

k is non involutive for some k ≤ k i-1 , permute the inputs to change u (0)

k is non involutive for all such permutations, then fail.

Step i ≥ m. Determine k

Otherwise, if for every permutation of non prolonged input, there is an i = 2, . . . , m, a k ≥ j i-1 and if for all

Examples

All the examples of this paper concern two input systems, i.e. m = 2, with prolongation index j = (j 1 , j 2 ). Such prolongations are supposed to be such that 0 = j 1 ≤ j 2 , up to a suitable input permutation. However, for the sake of readability, we will keep the original input numbering unchanged and thus consider that j = (0, j 2 ) or (j 1 , 0) depending on the context. At the exception of this modification, we strictly apply algorithm 1 in all the examples.

The first example gives a detailed presentation of the application of algorithm 1, in particular the role played by σ Γ,∆ (k) and σ ∆ (k). The second one shows the importance of the number σ ∆ (k) alone to determine the prolongation length, and the third one, borrowed from [START_REF]Sufficient conditions for dynamic state feedback linearization[END_REF], and taken up again in [1, Section 3.1], is reported here to compare our approach with the one of [START_REF] Battilotti | A constructive condition for dynamic feedback linearization[END_REF]. Finally, the pendulum example is presented to show that non flat systems by pure prolongation can be detected in a finite number of steps. [START_REF] Ph | Flat systems, Plenary Lectures and Minicourses, Proc. ECC[END_REF] x

Chained System

This system has been proven to be flat in [20, section 3.1.1] with the following flat output y 1 = x 3 -ẍ1 u 2 + ẋ1 u2 -x 1 ü2 , y 2 = x 2 .

(5.2)