Mabrouk Boughdiri

Fabrice Cordey
email: fabrice.cordey@univ-lyon1.fr

Houaïda Sallouhi

Kamel Maâlaoui

Amara Masrouhi

Mohamed Soussi

Jurassic radiolarian-bearing series of Tunisia: biostratigraphy and significance to western Tethys correlations

Keywords: Jurassic, Tunisia, radiolaria, radiolarites, Maghrebides, biostratigraphy, paleogeography

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Introduction

Jurassic radiolarian-bearing siliceous successions has been, over the last 50 years, well studied and documented within the circum-Mediterranean ranges (see for instance synthesis by [START_REF] Bill | Radiolarite ages in Alpine-Mediterranean ophiolites: Constraints on the oceanic spreading and the Tethys-Atlantic connection[END_REF]. In northern Tunisia, the Jurassic series of the Jédidi formation (Fm) considered as "radiolarites" or "pseudoradiolarites" by the authors (e.g. [START_REF] Alouani | Découverte de radiolarites du Jurassique supérieur dans le « Sillon Tunisien »: facies et structures d'une marge SE de la Téthys[END_REF], Soussi 2002, respectively) have undergone various stratigraphic attributions, including controversial interpretations concerning the polarity of the Jurassic series to which they belong (see synthesis in Peybernès et al. 1994). Originally interpreted as lower Cretaceous in age [START_REF] Solignac | Etude géologique de la Tunisie septentrionale[END_REF][START_REF] Castany | Etude géologique de l'Atlas tunisien oriental[END_REF][START_REF] Castany | Les extrusions jurassiques en Tunisie[END_REF], they carried various age attributions (Callovian-Oxfordian, Jebels Jédidi, Chaâbane and Tamrhoura, [START_REF] Pini | Découverte de Jurassique à l'Ouest de Medjez el-Bab (Tunisie septentrionale)[END_REF]Kimmeridgian-Tithonian, Ghardimaou area, Rouvier 1985; late Oxfordian-lower Kimmeridgian, J. Jédidi, [START_REF] Alouani | Découverte de radiolarites du Jurassique supérieur dans le « Sillon Tunisien »: facies et structures d'une marge SE de la Téthys[END_REF]; late Kimmeridgian-late Tithonian, J. Amar, Peybernès et al. 1996; late Bajocian-late Oxfordian, [START_REF] Soussi | Faciès à affinité ammonitico rosso d'âge Toarcien supérieur de la Tunisie atlasique nord-occidentale. Conséquences pour les corrélations et la paléogéographie[END_REF][START_REF] Soussi | Le Jurassique de la Tunisie atlasique. Stratigraphie, dynamique sédimentaire, paléogéographie et intérêt pétrolier[END_REF]Soussi , 2004)).

The first preliminary direct age determination of the Jedidi Fm was recently established by [START_REF] Cordey | First direct age determination from the Jurassic radiolarian-bearing siliceous series (Jédidi Formation) of northwestern Tunisia[END_REF] who also questioned previous interpretations that these series are true radiolarites. We present here more complete results obtained at the type locality of the Jédidi Fm and other sections from the area (J. Chaabane, Oued Tazega). Direct radiolarian age determinations are established at the base, the middle part and the top of the siliceous series. In relation to these new results, we propose a paleogeographic interpretation for their onset and an attempt to focus on the main deposit controlling factors. Finally, regional correlations and comparisons to the equivalent successions within the Maghrebides and NE Sicily ranges place these deposits within their West-Tethyan context.

Stratigraphic and paleogeographic setting

In this stratigraphic and paleogeographical overview, we refer to the recent synthesis proposed by [START_REF] Soussi | Le Jurassique de la Tunisie atlasique. Stratigraphie, dynamique sédimentaire, paléogéographie et intérêt pétrolier[END_REF][START_REF] Soussi | Nouvelle nomenclature lithostratigraphique «événementielle» pour le Jurassique de la Tunisie atlasique[END_REF]. The Middle and Upper Jurassic successions of northern Tunisia are characterized by heterogeneous facies underlain by major discontinuities more or less marked from the Tunisian Dorsale northwestward (Tunisian Trough) (Fig. 2).

The nodular limestones with chert nodules and Zoophycos of Kef El Orma

Location, structural setting and lithology of studied sections

The sections of J. Chaâbane and J. Jédidi are located 10 km to the west of the town of Mejez el Beb, near highway 6 linking Tunis to Ghardimaou (Fig. 1A,B). Between Mejez el Beb and Oued Zarga, a gravel road heads south to Sidi Ahmed Jédidi then to the southeastern flank of jebels Jédidi and Chaâbane where sections are exposed (Fig. 1B). Fifteen km to the NE of these sections is located the studied area of Oued Tazega that can be reached by gravel roads that bifurcate from highway 64 linking Tébourba to Mejez el Beb.

Within the Jédidi-Chaâbane range, Jurassic units are bounded to the south by the Téboursouk fault (Fig 1A), cropping out along a NE-SW belt parallel to large exposures of Triassic salt formations. Lower Jurassic limestones, presently worked in a gravel quarry, bound the Jurassic southernmost exposures. Jurassic units extend to the north as the base of a wide syncline where are identified sub-vertical to slightly reverse Middle and Late Jurassic strata. These strata grade into Cretaceous and Eocene successions of Toukabeur and Chaouache that extend northward and northeastward within the Toukaber-Hidous syncline. This syncline constitutes a back folding of the Oued Zarga monocline generated by a strike slip faulting of J. Boulahouadjed [START_REF] Fournet | Notice de la carte géologique au 1/50000 de Tébourba[END_REF]. On the southeastern flank of J. Boulahouadjed, the studied Jurassic beds are subhorizontal to northwestward dipping, within a slice of terrane uplifted by the J. Berd diapir. As for the Jédidi and Chaâbane southern boundary of Jurassic units, the J. Berd diapir belongs to a wide NE-SW structure that crops out along the Téboursouk fault which separates in Tunisia the overthrust structural zone, to the west, from the "Tunisois Domes" zone, to the east [START_REF] Jauzein | Contribution à l'étude géologique des confins de la Dorsale tunisienne[END_REF]).

At J. Jédidi and J. Chaabane, the sections are dominantly composed of thick red pelites alternating with thinly laminated siliceous limestone, dark brown to brick red with tints of yellow and green. Within their middle part, these sections exhibit a lesser amount of pelites and siliceous facies. Calcareous beds are dominant and are harder, thicker and darker than those of the base and the top of the series. At O. Tazega, the exposure is essentially composed of red to dark brown pelites with thin siliceous limestone beds at the base and the middle part (Figs. 4,5). It is worth noting that through all studied successions, the siliceous facies varies along sections. Contrary to previous interpretations, [START_REF] Cordey | First direct age determination from the Jurassic radiolarian-bearing siliceous series (Jédidi Formation) of northwestern Tunisia[END_REF] established that these successions are actually not "true" radiolarites, as radiolarian-rich layers alternate with carbonates (locally silicified). Towards the top of the formation, carbonate proportions within siliceous limestone beds increase, grading into marl-limestone alternations of the Bene Kleb formation.

Radiolarian assemblages and age correlations

Previous attempts for extracting radiolaria from siliceous series of Tunisia were undertaken in the 1980ies and 1990ies by French and Italian teams, without success (R. Enay, pers. com., 2004). As a matter of fact, the quality of radiolarian fauna varying from bed to bed as well as within single beds, we based our field study on thorough sample selection [START_REF] Cordey | A field technique for identifying and dating radiolaria applied to British Columbia and Yukon[END_REF]) and adapted laboratory processing (low concentration acids). For age correlations, we used the unitary association-based biozonation from [START_REF] Baumgartner | Middle Jurassic to Early Cretaceous radiolarian biochronology of Tethys based on Unitary Associations; in Middle Jurassic to Lower Cretaceous Radiolaria of Tethys : occurrences, systematics, biochronology[END_REF]. The three sections described herein released radiolarian fauna of fairly good preservation (Figs 3,4, Plate 1). Nine beds and their faunal content are presented in this paper.

At the Jebel Jédidi type-section (Fig. 3), preliminary data reported the occurrence of two diagnostic associations within the lower and middle part of the succession [START_REF] Cordey | First direct age determination from the Jurassic radiolarian-bearing siliceous series (Jédidi Formation) of northwestern Tunisia[END_REF]. Sample MB4 (Fig. 3) contains an association correlative with UAZ 7 of late Bathonian or early Callovian age. Sample MB18 (Fig. 3) is correlative with UA7-8 of late Bathonian-early Oxfordian age. New and complementary data documented here provide age determinations near the base and the top of the Jedidi formation. Sample JD4b from sequence 2 at 5,5 m from the base bears an association corresponding to the UAZ 5 of latest Bajocian or early Bathonian age.

Sample JDI-1 collected at 2 m from the top of the type-section released an assemblage correlative to UAZ 9 of middle or late Oxfordian age.

The J. Chaâbane section (Figs. 1,3) was sampled as a complement to the study of the J. Jedidi type section. So far, two samples from sequences 2 and 4 released diagnostic radiolarians. Sample CHA2 corresponds to UAZ 5 of latest Bajocian-early Bathonian age. One meter above, sample CHA4 also released an association attributed to UAZ 5. These ages are consistent with those obtained in the J. Jedidi section. No radiolaria were yet successfully obtained from the middle part and the top of the J.

Chaâbane section.

The O. Tazega section (Figs. 1,4) provided three diagnostic samples. Sample OTA1 (sequence 1, Fig. 4) has an assemblage correlative with UAZ 7 of late Bathonian or early Callovian age. Sample OTA5 (sequence 5, Fig. 4) released an association attributed to UAZ 8 of mid-late Callovian or early Oxfordian age. In the middle part of the section, sample OTA9b (sequence 9b) contains an assemblage attributed to UAZ 9 of middle or late Oxfordian in age.

In summary, the youngest age obtained within the Jedidi formation is to be correlated with UAZ 5 of latest Bajocian or early Bathonian age (J. Jedidi and Chaâbane). At O. Tazega, the youngest age obtained is late Bathonian or early Callovian age (UAZ 7). These results show that the O. Tazega section probably does not expose the very base of the formation, as suggested by the presence of a fault (Fig. 4). The presence of UAZ 5 at about 6 m from the base of the Jedidi formation type section indicates that siliceous deposition probably started during the late or latest Bajocian rather than within the Bathonian stage. This is fairly consistent with the occurrence of late Bajocian ammonite data from the underlying Kel el Orma formation [START_REF] Soussi | Faciès à affinité ammonitico rosso d'âge Toarcien supérieur de la Tunisie atlasique nord-occidentale. Conséquences pour les corrélations et la paléogéographie[END_REF]. The youngest age obtained so far on these siliceous series of the Tunisian trough is middle or late Oxfordian in age, as shown by both the J. Jedidi and O. Tazega sections. Although the corresponding radiolarian association UAZ 9 is found only 1,6 m from the top of the formation in J. Jedidi, it is found about 18 m below the overlying Bene Kleb fm in O Tazega. This apparent discrepancy can be explained by either tectonic duplication affecting part of sequence 10 at O. Tazega, or significant differences in sedimentation rates, or else the occurrence of diachronism between the two areas. We cannot, at this time, differentiate between these hypotheses as we have not yet found any radiolarian associations younger than UAZ 9. However, a mid or late Oxfordian age for the top of the Jedidi formation seems consistent with the late Oxfordian data obtained on the Bene Kleb formation by [START_REF] Boughdiri | Calpionellid zonation of the Jurassic-Cretaceous transition in North-Atlasic Tunisia. Updated stratigraphy of the 'Tunisian trough' and Upper Jurassic regional correlations[END_REF].

Biostratigraphic data obtained on the J. Jedidi type section can be applied to assess sedimentation rates for Tunisian siliceous series and compare it with true radiolarite successions. Latest Bajocian to mid-late Oxfordian interval represent a time span of approximately 10 m.y. (Jurassic time scale, [START_REF] Gradstein | A Geologic Time Scale[END_REF]). Given the 40 m overall average thickness of the formation, the average sedimentation rate of these siliceous series would be of ~ 4 m per m.y.. This broad calculation is to be compared to average rates for 'true' Tethyan radiolarites s.s. of the same time interval, whose average rate is usually around 1 or 2 m per m.y. (see for instance De Wever & Cordey 1995).

This confirms the preliminary assessment by [START_REF] Cordey | First direct age determination from the Jurassic radiolarian-bearing siliceous series (Jédidi Formation) of northwestern Tunisia[END_REF] that the Jurassic siliceous series of Tunisia are indeed a mixture of siliceous and carbonate sedimentation whose sedimentation rates are commonly higher than those of true radiolarites.

The onset of siliceous deposition in northwest Tunisia and controlling factors

Rapid lateral changes of Upper Bajocian-Lower Callovian facies in northwest Tunisia (fig. 2) indicate an increase in Tethyan-rift-related individualization of the "Tunisian Trough" initiated in the lower Jurassic time period. To the relatively silica-free deposits of Bent Saïdane Fm within the relatively resistant but subsiding domain of the "Tunisian Dorsale" correspond the earliest radiolarian-bearing successions (lower part of the Jédidi Fm) of the deeper "Tunisian Trough", both domains being separated by coeval transitional slope facies. On the same tilted-block topography occurred the Ammonitico Rosso (A.R.) deposits of the Tunisian Dorsale during Middle Callovian-Oxfordian time, the onset of which coincides with the first siliceous argillites and limestones of Fahs Fm and the upper part of the Jédidi Fm in the "Tunisian Trough".

Probably resulting from this regional paleotopographic gradient, we note a tendency to progressive lime-free conditions either upsection or from the southeast (Dorsale) to the northwest (Tunisian Trough). Although the environmental conditions of radiolarian-rich deposition is still a matter of debate, we preliminarily consider that the relative unfavourable conditions for carbonate productivity (i.e. the onset of A.R. condensed units and the A.R./siliceous series heteropy) could be related to paleoceanographic conditions. As formulated by [START_REF] Baumgartner | Tethyan Jurassic Radiolarites[END_REF] Since the Kimmeridgian, the last occurrences of the A.R. facies of the Dorsale and coeval biosiliceous successions within the adjacent Tunisian Trough and nearby exposures are overlain by the carbonate series of Béni Kleb Fm [START_REF] Enay | La limite Kimméridgien-Tithonien et l'âge des formations du Jurassique supérieur de la « Dorsale » tunisienne, comparaisons avec l'Algérie et la Sicile[END_REF], Boughdiri et al. 2005). This dramatic facies change was interpreted by Peybernès et al. (1996) as linked to a drastic sea level fall. As formulated by [START_REF] Cecca | The disappearance of the 'Ammonitico Rosso[END_REF] for the disappearance of the Ammonitico Rosso, an eustatic cause of the onset or disappearance of radiolarian-rich successions is not likely because 1) the stratigraphy and geographic distribution of these successions do not depend on sea level fluctuation and 2) the broadly homogenous and constant radiolarian-bearing deposits within the Jédidi Fm encompasses different stratigraphic intervals that correspond to various sea level fluctuations as illustrated in charts (e.g. [START_REF] Haq | Chronology of fluctuating sea levels since the Triassic (250 million years ago to present)[END_REF][START_REF] Hardenbol | Mesozoic and Cenozoic sequence chronostratigraphic framework of European basins. Jurassic Sequence Chronostratigraphy[END_REF]. Our point of view favours the interpretation of a return to higher rates of carbonate sedimentation due to a marked change in currents paleocirculation regime [START_REF] Cecca | The middle Jurassic-Early Cretaceous Ammonitico rosso succession at Monte Inici (Trapanese domain): sedimentology, biostratigraphy and isotope stratigraphy[END_REF], Savary et al. 2003, among others). In fact, by the end of Jurassic, the change in Mediterranean Tethys location exposes this realm to weak surface circulation controlled by subtropical gyres rather than to prevailing equatorial current systems [START_REF] Hsü | Palaeoceanography of the Mesozoic Alpine Tethys[END_REF][START_REF] Weissert | Die Paläozeanographie der Südwestlichen Tethys in der Unterkreide[END_REF][START_REF] Baumgartner | Tethyan Jurassic Radiolarites[END_REF][START_REF] Savary | Etude stratigraphique du Rosso Ammonitico du Monte Inici (domaine Trapanais, Sicile occidentale) : événements biosédimentaires au Jurassique moyen-Crétacé inférieur[END_REF].

Comparison with radiolarian-rich deposits of the Maghrebian-Sicilian Range and geodynamic context

To our knowledge, only the Jurassic biosiliceous successions described in the Babors [START_REF] Ehrmann | Les radiolarites du Jurassique de la Kabylie des Babors[END_REF][START_REF] Leikine | Etude géologique des Babors occidentaux (Algérie)[END_REF][START_REF] Obert | Etude géologique des Babors orientaux (Domaine tellien, Algérie)[END_REF]) and the northwest Constantinois region of northern Algeria [START_REF] Durand-Delga | Les rapports de la chaîne numidique avec les Babors[END_REF], 1955) can be compared with the Jédidi Fm of Tunisia. Although direct age determination is still lacking, a Middle Jurassic age was previously assigned to these similar series. Middle Jurassic facies from the Mésorif (Morocco) may also be associated to this first group of radiolarian-bearing successions of the external segments of the Maghrebides (Atrops, pers. comm. 2006).

In the Maghrebian flysch area, Jurassic sections include more recent "true" radiolarites that clearly differ from Tunisian deposits. For example, within the Ouareg Fm of Targuist area (Rif, Morocco) are exposed red radiolarites with embedded breccias and oolitic limestones (Bajocian-Tithonian) [START_REF] Olivier | Le substratum jurassique des flyschs maurétaniens de l'ouest des Maghrébides : l'unité de Ouareg (région de Targuist, Rif, Maroc)[END_REF]. They overlay calcareous turbidites and cherty limestone of Upper Aalenian-Upper Bajocian age. Homologous successions can be followed eastward over long distances as they are known in the Lesser Kabylia (northern Algeria), where the Achaiches series also yield radiolarites supported by limestone microbreccias with basement debris [START_REF] Bouillin | La transversale de Collo et d'El Milia (Petite Kabylie): une régionclef pour l'interprétation de la tectonique alpine de la chaîne littorale d'Algérie[END_REF][START_REF] Durand-Delga | Les unités à Mésozoïque métamorphique d'El Milia à Texenna (Algérie) et leur cadre structural[END_REF]. In the Sicily-Calabrian Arc, Tithonian radiolarites of the Contarada Lanzeri Fm [START_REF] Bouillin | Un témoin des series jurassiques du Bassin Maghrébin en Sicile: la formation de la Contrada Lanzeri, à la base du flysch maurétanien du Monte Soro[END_REF] and the Caloveto type series (Santontonio & Teale 1986) are considered to be lateral equivalents of the flyschs substratum remnants.

In an inner structural setting, the Middle Jurassic successions of the Maghrebian "Dorsale calcaire" include thin radiolarite beds devoid of detrital intercalations. Among these series can be mentioned those of the Internal Rif (J.

Moussa, Haouz Chain) where red-to-green radiolarite layers and argillites were dated as Middle Jurassic in age. Radiolaritic argillites and radiolarites (Callovian-Oxfordian) are known in the Medium Dorsale of Great Kabylia [START_REF] El Hatimi | Les séries jurassiques et éocrétacées du groupe du Moussa (Rif septentrional, Maroc) témoins d'une tectonique distensive en blocs basculés[END_REF][START_REF] El Hatimi | La Dorsale calcaire du Haouz ancienne marge continentale passive téthysienne (Rif, Maroc)[END_REF][START_REF] Durand-Delga | Le domaine Tariquide (arc de Gibraltar, Espagne et Maroc). Succession et hiatus de la sédimentation du Jurassique supérieur au Paléocène[END_REF]. Lateral equivalent series described in the Peloritan dorsale of northeast Sicily (Toarmina-Monte san Pietro area) include filament-rich and cherty limestones, Upper Callovian radiolarites and calpionellid-bearing limestones [START_REF] Bouillin | Organisation structurale et sédimentaire de la paléomarge dans les monts Péloritains (Sicile, Italie)[END_REF]. This review suggests that three categories of Jurassic radiolarian-bearing successions can be distinguished in this region, characterizing three nappe complexes originating from three main paleogeographic domains: Internal units, Flysch basin and External units [START_REF] Durand-Delga | La Méditerranée occidentale, étapes de sa genèse et problèmes structuraux liés à celle-ci[END_REF]Fontboté 1980). To the latter domain belong the Tunisian series analysed here, deposited on the North Africa Margin of western Tethys. Replaced in their broader geodynamic setting, we consider as other authors that some elements of the Maghrebian inner zones were part of the Alkapeca microcontinent [START_REF] Bouillin | Betic-Rifian and Tyrrhenian Arcs: distinctive features, genesis and development stages[END_REF]) that also included present-day Calabrese and

Conclusion

Several radiolarian assemblages extracted from the siliceous series of the Jedidi Fm (Tunisian Trough) led to direct age determinations of the succession (uppermost Bajocian-Oxfordian). Our study also establishes an age reference for the top of the Kef El Orma Fm and the base of Beni Kleb Fm where biostratigraphic components are scarce or absent. These new data offer good arguments to seal the debate concerning the polarity of the Jurassic sequence of the Tunisian Trough that lasted for several decades.

As a perspective, radiolarian dating may still offers some potential for contributing to future paleoecological and paleogeographic reconstructions in the region, as some radiolarian-bearing carbonate and siliceous beds either within other localities of the Jedidi Fm or the Fahs Fm are still undated. These may also be needed for long distance correlations within the Maghrebian external segments, the Algerian northern Aures Mountains, the Babors and Numidian chains as well Mesorif (Morocco), as part of a broader south-Tethyan margin Jurassic reconstruction.

Captions

  Formation (Fm) (Aalenian-Lower Bajocian) are overlain by filament-rich marl/limestone alternations of Bent Saïdane Fm (Upper Bajocian-Lower Callovian) of the Dorsale. These facies are relayed laterally by black limestones bearing thin-shelled bivalves ("filaments") within the eastern flank of J. Zaghouan, by breccias-bearing marl/limestone alternations of J. Bou Kornine Hammam Lif, and then by the studied Jedidi Fm (Tunisian Trough). During Middle Callovian-Oxfordian time interval, to red nodular limestones and marls of Ammonitico Rosso facies (Zaress Fm) of the Dorsale correspond radiolarian-rich pelitic and carbonated units of Fahs Fm ("faciès ocelés" of Bonnefous 1972) within the Dorsale western nearby exposures and the upper part of the Jedidi Fm. Sealing all these successions, the Kimmeridgian-Berriasian carbonate series are mainly characterized by reefal and sub-reefal facies on risen blocks of the Dorsale (Ressas Fm) that grade into marls and pseudonodular limestones with Saccocoma and then calpionellids of Béni Kleb Fm which is enriched in radiolarians and chert nodules within the Tunisian Trough. The heterogeneity of Middle-Late Jurassic facies in northern Tunisia is the result of two main geodynamic and paleogeographical events. The first corresponds to the fragmentation of the initial Early Jurassic platform linked to Tethyan rifting that led to a mosaic of raised and drowned blocks. The second event, coeval to radiolarian-bearing series of the Tunisian Trough, is expressed by a clear deepening in northwest Tunisia region and, as a result, the individualization of two major paleogeographic realms: 1) a relatively resistant composite domain including the Tunisian Dorsale and 2) a deeper subsiding domain that differentiated from the Dorsale since the Upper Bajocian where the biosiliceous successions were deposited. The transition between these domains consists of a NE-SW intermediate area including the present range of jebels Bou Kornine of Hammam Lif, Oust, Aziz, Bou Kornine of Pont du Fahs and Raouas-Béni Kleb.

  for other Tethyan basins in different paleogeographic settings, bottom currents may have caused high surface fertility in the Tunisian Trough basin and winnowed light radiolarian tests from swells (Tunisian "Dorsale") by resuspension and then transported them into the adjacent current-protected basin (Tunisian Trough).

  Peloritan overthrusts. The western part of this microcontinent consisted of the Alboran terrane. During Upper Eocene-Oligocene times, the collision Alkapeca-Iberia generated back-thrust tectonics that evolved in a northwest-dipping subduction of the African Margin beneath Alkapeca. As recently formulated by Chalouan et al. (2001), Tertiary Alpine s.l. general convergence led to docking of the extending Alkapeca segments (present-day thrust sheets of north-African inner margin) along with closure of the Maghrebian Flysch oceanic trough, and took place onto the African ancient margins that were formed by the present day Tello-Rifan external zones.

Fig. 1 .

 1 Fig. 1. Location of studied sections.

Fig. 2 .

 2 Fig. 2. Overview of Tunisian stratigraphy.

Fig. 3 .

 3 Fig. 3. Lithological sections from jebels Jedidi and Chaabane with location of

Fig. 4 .

 4 Fig. 4. Lithological section from Oued Tazega with location of radiolarian samples,

Fig. 5 .

 5 Fig. 5. Exposures from Oued Tazega section. A: general view, from bottom to top. Two

Fig. 6 .

 6 Fig. 6. Middle Callovian paleogeography of North Africa region (after Thierry et al.

  Figure 5

Acknowledgements

MB thanks the staff of the laboratory Paléoenvironnements et Paléobiosphère of Université Lyon 1 (A. Armand, F. Atrops, J.-J. Cornée, P. Desvignes, P. Hantzpergue, C. Lécuyer, and P. Moissette). FC benefited from logistics and funding provided by University of Bizerte and the Tunisian "Institut National de la Recherche Scientifique et Technologique" (INRST). Elizabeth Carter (University of Oregon) and Luis O'Dogherty

(Universidad de Cádiz) are acknowledged for their thorough reviews. The authors wish also to thank Arlette Armand and Dominique Barbe (Lyon) for their help with SEM and reprography, respectively. Contribution UMR5125-07.029.


Marc Gouttefarde

Ghina Hassan

Ahmed Chemori

Ahmed Chemori Proof

Proof of an Algorithm to Compute Time-Optimal Third Order Polynomial S-curve Trajectories

Proof of an Algorithm to Compute Time-Optimal

Third Order Polynomial S-curve Trajectories Marc Gouttefarde, Ghina Hassan, Ahmed Chemori 1

I. INTRODUCTION

This research report deals with time-optimal third order polynomial S-curve trajectories. Methods to generate timeoptimal S-curve trajectories have been previously proposed. For instance, an algorithm to optimize an objective function composed of two terms, the execution time and the jerk, has been developed in [START_REF] Gasparetto | A technique for time-jerk optimal planning of robot trajectories[END_REF]. A binary search method to optimize the S-curve velocity profile has been introduced in [START_REF] Li | Optimizing s-curve velocity for motion control[END_REF].

In the present work, a new algorithm is introduced and proved. This algorithm computes the maximum velocity and acceleration of a 3rd order polynomial S-curve trajectory such the total time of the S-curve trajectory is minimized for a given desired displacement. Upper bound constraints on the S-curve maximum velocity and acceleration are taken into account. The resulting optimization problem is solved by analyzing in detail the Karush-Kuhn-Tucker (KKT) conditions. This mathematical analysis allows to figure out and prove the validity of the proposed algorithm. While the analysis is slightly tedious, the obtained algorithm is very efficient and, to the best of our knowledge, it has never been proposed.

II. MINIMUM-TIME S-CURVE TRAJECTORY

A. Polynomial S-Curve Motion Profile

The third order polynomial S-curve is considered in this report. As shown in Figure 1, the motion profile of the third order polynomial S-curve consists of seven segments, among which the first three and the last three constitute the acceleration and deceleration phases, respectively, and the fourth segment constitutes the constant velocity stage. The jerk is defined as the following function of time:

j(t) =                J, t 0 ≤ t ≤ t 1 ,t 6 ≤ t ≤ t 7 0, t 1 ≤ t ≤ t 2 ,t 3 ≤ t ≤ t 4 ,t 5 ≤ t ≤ t 6 -J, t 2 ≤ t ≤ t 3 ,t 4 ≤ t ≤ t 5 ( 1
)
where J is the maximum jerk value and the time instants t i are shown in Figure 1. The acceleration, velocity and displacement as functions of time can be deduced by integration of (1) with appropriate initial and final conditions. Referring to Figure 1, d j = t 1t 0 is the time needed to increase the acceleration from zero to the maximum acceleration A and to decrease the acceleration from the maximum 1 Marc Gouttefarde, Ghina Hassan and Ahmed Chemori are with LIRMM, Univ Montpellier, CNRS, Montpellier, France, {marc.gouttefarde}@lirmm.fr value A to zero, i.e., it is the time during which jerk stays constant at the maximum jerk value J. Besides, d a = t 2t 1 corresponds to the time during which the acceleration profile remains constant and equal to the maximum acceleration A and d v = t 4t 3 corresponds to the time during which the velocity profile remains constant and equal to V . Note that the symmetrical S-curve has been adopted in the present study (i.e. the acceleration and deceleration phases are symmetrical) and that d j , d a and d v are all greater than or equal to zero. The three time intervals can be written as follows:

d j = A J d a = V A - A J d v = P V - V A - A J (2)
A and V are the maximum velocity and acceleration achieved for a given displacement P. Both A and V should be less than or equal to the corresponding maximum actuator capabilities. The total time to travel a distance P is expressed as follows:

T = 4d j + 2d a + d v (3)

B. Minimization Problem Formulation

In this report, the proof of a new algorithm to optimize the values of the acceleration A and velocity V is provided. For a given displacement P and taking into account upper bound constraints on maximum acceleration and velocity, i.e., A ≤ A max and V ≤ V max , this algorithm efficiently computes the values of A and V of a 3rd order polynomial S-curve trajectory with minimum total time T . By substitution of the three time intervals (2) in the total time (3), the function T to be minimized is expressed as follows:

T = P V + V A + A J (4)
Let us define x = (x 1 , x 2 ), where x 1 = V , x 2 = A, as well as x 1max = V max and x 2max = A max . Hence, the objective function is:

f (x) = T = P x 1 + x 1 x 2 + x 2 J (5)
The minimization problem of finding the minimum-time Scurve trajectory is then formulated as follows:

min x f (x) subject to                          C 1 (x) = d a = x 1 x 2 - x 2 J ≥ 0 C 2 (x) = d v = P x 1 - x 1 x 2 - x 2 J ≥ 0 C 3 (x) = x 1max -x 1 ≥ 0 C 4 (x) = x 2max -x 2 ≥ 0 C 5 (x) = x 1 > 0 C 6 (x) = x 2 > 0 (6)

C. Algorithm to Compute the Minimum-Time S-Curve Trajectory

The KKT first-order necessary optimality conditions of the optimization problem (6) yield 11 cases: One case of no active constraint, four cases of two active constraints (C 1 , C 2 , C 3 or C 4 ) and six cases of two active constrains (C 1 = C 2 = 0, C 1 = C 3 = 0, etc.), being given that the constraints C 5 and C 6 are always inactive since V = x 1 > 0 and A = x 2 > 0. A detailed analysis of these 11 cases (cf. Section III) shows that the following four are impossible at a (local) optimal solution:

• No active constraint

• One active constraint C 1 = 0 • One active constraint C 4 = 0 • Two active constraints C 1 = C 4 = 0.
It also shows that, at a local optimal solution:

• One active constraint C 2 = 0 ⇐⇒ two active constraints

C 1 = C 2 = 0 • One active constraint C 3 = 0 ⇐⇒ two active constraints C 1 = C 3 = 0 • The case of two active constraints C 2 = C 3 = 0 turns out
to be a particular case of the case one active constraint C 3 = 0. Hence, out of the 11 cases, only four of them needs to be considered to determine the optimal solutions of the optimization problem (4):

• One active constraint

C 2 = 0 • One active constraint C 3 = 0 • Two active constraints C 2 = C 4 = 0 • Two active constraints C 3 = C 4 = 0.
As detailed in the Section III, further analyzing these four cases lead to Algorithm 1 which allows to efficiently determine the values of V and A yielding the minimum-time polynomial S-curve trajectory.

Algorithm 1 Minimum-Time S-Curve Trajectory Input: P, J, V max , A max Output: A and V yielding the minimum total time T

1: if (J 2 P ≤ 2A 3 max ) then 2: if (JP 2 ≤ 4V 3 max ) then 3: V = 3 JP 2 4 and A = 3 J 2 P 2 4: else 5: V = V max and A = √ JV max 6: end if 7: else 8: if ( √ JV max ≤ A max ) then 9: V = V max and A = √ JV max 10:
else 11:

A = A max 12: V = -A 2 max + √ A 4 max +4J 2 PA max 2J 13: if (V > V max ) then 14: V = V max 15: end if 16:
end if 17: end if

III. PROOF OF ALGORITHM 1

The KKT first-order necessary conditions [START_REF] Nocedal | Numerical Optimization[END_REF] for

x * = (x * 1 , x * 2 )
to be a local solution to the optimization problem (6) can be stated as follows.

First-order necessary conditions: There is a Lagrange multiplier vector λ * , with components λ * i , 1 ≤ i ≤ 6, such that the following conditions are satisfied:

∇ x L (x * , λ * ) = 0 (7) C i (x * ) ≥ 0, 1 ≤ i ≤ 6 (8) λ * i ≥ 0, 1 ≤ i ≤ 6 (9) λ * i C i (x * ) = 0, 1 ≤ i ≤ 6 ( 10
)
where the constraints C i (x) are defined in (6) and ∇ x L (x, λ ) is the gradient with respect to x of the Lagrangian function L (x, λ ) defined as:

L (x, λ ) = f (x) - 6 ∑ i=1 λ i C i (x) (11)
The first-order necessary conditions stated above in (7) to (10) are valid if the functions f (x) and C i (x) are continuously differentiable and the so-called Linear Independence Constraint Qualification (LICQ) holds at x * [START_REF] Nocedal | Numerical Optimization[END_REF]. According to the definitions of f (x) and C i (x), the continuous differentiability condition is true for problem (6). The fact that the LICQ holds is proved in Section III-P.

The well-known method to use the first-order necessary conditions to find local optimal solutions x * consists in distinguishing all the possible cases of sets of active constraints which allows the determination of x * and of the Lagrange multiplier vector λ * .

A constraint C i (x) is defined as being active if C i (x) = 0. According to (10), for a given i, either

C i (x * ) is active, i.e. C i (x) = 0, or λ * i = 0. Since C 5 (x) = x 1 = V > 0 and C 6 (x) =
x 2 = A > 0, C 5 (x) and C 6 (x) are never active, and according to (10), λ 5 = 0 and λ 6 = 0 at a local optimal solution. Then, only four constraints can be active, C 1 (x) to C 4 (x), and the problem (6) having two variables, x 1 and x 2 , the following 11 cases of possible active constraint sets must be studied:

• No active constraint (one case)

• Four cases of one active constraint • Six cases of two active constraints These relatively small number of possible active constraint sets and the rather simple expressions of the functions f (x) and C i (x) make the analysis of each of these eleven cases possible, as detailed in the following subsections.

A. Preliminaries: Expressions of the gradients

In (7), the gradient with respect to x of the Lagrangian function L (x, λ ) in ( 11) is given by:

∇ x L (x, λ ) = ∇ x f (x) - 4 ∑ i=1 λ i ∇ x C i (x) (12)
where the summation is taken for i = 1 to 4 since, as pointed above, λ 5 = 0 and λ 6 = 0 at a local optimal solution. According to (5) and ( 6), the gradients in (12) are as follows:

∇ x f (x) =       ∂ f ∂ x 1 ∂ f ∂ x 2       =       1 x 2 - P x 2 1 1 J - x 1 x 2 2       =       x 2 1 -Px 2 x 2 1 x 2 x 2 2 -Jx 1 Jx 2 2       (13)
∇ x C 1 (x) =       ∂C 1 ∂ x 1 ∂C 1 ∂ x 2       =       1 x 2 - 1 J - x 1 x 2 2       =       1 x 2 -x 2 2 -Jx 1 Jx 2 2       (14) ∇ x C 2 (x) =       -P x 2 1 - 1 x 2 x 1 x 2 2 - 1 J       =       -Px 2 -x 2 1 x 2 1 x 2 Jx 1 -x 2 2 Jx 2 2       (15) ∇ x C 3 (x) = -1 0 (16) ∇ x C 4 (x) = 0 -1 (17)

B. Case 1: No active constraint

According to (10), λ * i = 0 for i = 1 . . . 4 if there is no active constraint at a local optimal solution x * . Then, (7) yields:

∇ x L (x * , λ * ) = ∇ x f (x * ) = 0 ⇐⇒    x * 2 1 -Px * 2 = 0 x * 2 2 -Jx * 1 = 0 ⇐⇒    x * 3 1 = JP 2 x * 3 2 = J 2 P (18)
However, the constraint C 2 (x) is:

C 2 (x) = P x 1 - x 1 x 2 - x 2 J = J Px 2 -x 2 1 -x 1 x 2 2 Jx 1 x 2 (19)
so that, with (18), the fact that C 2 (x) is not active leads to:

C 2 (x * ) > 0 ⇐⇒ J Px * 2 -x * 2 1 -x * 1 x * 2 2 > 0 ⇐⇒ -x * 1 x * 2 2 > 0 ( 20
)
where the second equivalence comes from Px * 2 -x * 2 1 = 0 which is a consequence of (18). The inequality (20) is impossible since x * 1 > 0 and x * 2 > 0 according to constraints C 5 and C 6 . Consequently, this first case of no active constraint is not feasible and is discarded.

C. Case 2: One active constraint C 1 (x) = 0
According to (10), we have λ * i = 0 for i = 2, . . . , 4. Hence, x * 1 , x * 2 and λ * 1 can be determined from the following equation system obtained from (7) and C 1 (x * ) = 0:

     ∇ x L (x * , λ * ) = ∇ x f (x * ) -λ * 1 ∇ x C 1 (x * ) = 0 C 1 (x * ) = x * 1 x * 2 - x * 2 J = 0 (21)
From ( 13) and (14), the system (21) is equivalent to:

         x * 2 1 -Px * 2 -λ * 1 x * 2 1 = 0 x * 2 2 -Jx * 1 + λ * 1 x * 2 2 + Jx * 1 = 0 Jx * 1 -x * 2 2 = 0 (22)
This second equation system yields λ * 1 = 0, x * 3 1 = JP 2 and x * 3 2 = J 2 P. From these values of x * 1 and x * 2 , it follows that, similarly to Case 1 detailed in the previous subsection, the constraint C 2 is not feasible which discards Case 2.

D. Case 3: One active constraint C 2 (x) = 0 When the only active constraint is C 2 , (10) gives λ * i = 0 for all i = 2, and x * 1 , x * 2 and λ * 2 can then be determined from:

     ∇ x L (x * , λ * ) = ∇ x f (x * ) -λ * 2 ∇ x C 2 (x * ) = 0 C 2 (x * ) = J Px * 2 -x * 2 1 -x * 1 x * 2 2 Jx * 1 x * 2 = 0 (23)
which, from ( 13) and ( 15), is equivalent to:

         x * 2 1 -Px * 2 + λ * 2 Px * 2 + x * 2 1 = 0 x * 2 2 -Jx * 1 -λ * 2 Jx * 1 -x * 2 2 = 0 J Px * 2 -x * 2 1 -x * 1 x * 2 2 = 0 (24)
The second equation of this system can be written

x * 2 2 -Jx * 1 (1 + λ * 2 ) = 0 which implies x * 2 2 = Jx * 1 since λ * 2
≥ 0 according to (9). Then, the third equation of (24) gives x * 3 2 = J 2 P /2, which in turn leads to x * 3 1 = JP 2 /4 and λ * 2 = 1/2. Since λ * 2 = 1/2 > 0 and λ * i = 0 for all i = 2, (9) is true. For the first order necessary conditions to be all true, it remains to verify that (8) is satisfied for i = 1, 2 and 3, i.e.,

C 1 (x * ) ≥ 0, C 3 (x * ) ≥ 0 and C 4 (x * ) ≥ 0. C 1 (x * ) ≥ 0 is equivalent to Jx * 1 -x * 2 2 ≥ 0 which is true since x * 2 2 = Jx * 1 .
Finally, since x * 3 1 = JP 2 /4 and x * 3 2 = J 2 P /2, C 3 (x * ) ≥ 0 and C 4 (x * ) ≥ 0 yield the two following conditions on J, P, x 1max = V max and x 2max = A max :

x * 1 ≤ x 1max ⇐⇒ JP 2 ≤ 4V 3 max (25) x * 2 ≤ x 2max ⇐⇒ J 2 P ≤ 2A 3 max (26)
To conclude this third case, if the two conditions (25) and ( 26) are verified, the following vector is a local minimum candidate:

x * =       3 JP 2 4 3 J 2 P 2       (27) E. Case 4: One active constraint C 3 (x) = 0
When the only active constraint is C 3 , (10) gives λ * i = 0 for all i = 3 and

C 3 (x * ) = x 1max -x * 1 = 0 gives x * 1 = x 1max = V max . Moreover, (7) is ∇ x f (x * ) -λ * 3 ∇ x C 3 (x * ) = 0, i.e.: x * 2 1 -Px * 2 + λ * 3 x * 2 1 x 2 = 0 x * 2 2 = Jx * 1 (28)
With x * 1 = V max and the second equation of (28), after some calculations, the first equation of (28) leads to:

λ * 3 = P √ J - √ V max 3 V 2 max √ J (29)
Then, since λ * 3 ≥ 0 according to (9), the following condition must hold:

√ V max 3 ≤ P √ J ⇐⇒ V 3 max ≤ JP 2 (30)
For the first order necessary conditions to be true, it remains to verify that (8) is satisfied for i = 1, 2 and 4, i.e.,

C 1 (x * ) ≥ 0, C 2 (x * ) ≥ 0 and C 4 (x * ) ≥ 0. First, since x * 2 2 = Jx * 1 , we have C 1 (x * ) = 0 so that C 1 (x * ) ≥ 0, and C 2 (x * ) ≥ 0 is equivalent to x * 3
1 ≤ JP 2 /4, i.e., from x * 1 = V max , the following condition is obtained:

V 3 max ≤ JP 2 4 ( 31
)
Note that condition (31) implies condition (30) so that only (31) is to be retained. Finally,

C 4 (x * ) ≥ 0, x * 2 2 = Jx * 1 and x * 1 = V max yields the following condition: √ JV max ≤ A max (32)
To summarize this fourth case, if the two conditions (31) and (32) are verified, the following vector is a local minimum candidate:

x * = V max √ JV max (33)
F. Case 5: One active constraint C 4 (x) = 0 When the only active constraint is C 4 , (10) gives λ * i = 0 for all i = 4 and

C 4 (x * ) = x 2max -x * 2 = 0 gives x * 2 = x 2max = A max . Moreover, (7) is ∇ x f (x * ) -λ * 4 ∇ x C 4 (x * ) = 0 which is a system of two equations. The first equation of this system yields x * 2 1 = Px * 2 . Then, according to (19), C 2 (x * ) ≥ 0 is equivalent to -x * 1 x * 2 2 ≥ 0 which is impossible since x * 1 > 0 and x * 2 > 0.
In conclusion, this fifth case is not feasible and is thus discarded. shall be done to verify that (9) is satisfied. First, the x * 1 and x * 2 are calculated from the following system of two equations obtained from

C 1 (x * ) = C 2 (x * ) = 0: Jx * 1 -x * 2 2 = 0 J Px * 2 -x * 2 1 -x * 1 x * 2 2 = 0 ( 34
)
whose solution is:

x * =       3 JP 2 4 3 J 2 P 2       (35)
Then, Eq. ( 7) with λ * 3 = 0 and λ * 4 = 0 is used to determine λ * 1 and λ * 2 :

∇ x f (x * ) -λ * 1 ∇ x C 1 (x * ) -λ * 2 ∇ x C 2 (x * ) = 0 ⇐⇒ x * 2 1 -Px * 2 -λ * 1 x * 2 1 + λ * 2 Px * 2 + x * 2 1 = 0 x * 2 2 -Jx * 1 + λ * 1 x * 2 2 + Jx * 1 -λ * 2 Jx * 1 -x * 2 2 = 0 ⇐⇒ -x * 2 1 Px * 2 + x * 2 1 x * 2 2 + Jx * 1 x * 2 2 -Jx * 1 λ * 1 λ * 2 = Px * 2 -x * 2 1 Jx * 1 -x * 2 2
With the expressions of x * in (35), the determinant of the matrix in the last equation is equal to -3J 2 P 2 /2. and the equation system possesses a unique solution. Once solved, e.g. with Cramer's rule, it yields λ * 1 = 0 and λ * 2 = 1/3 which verify (9).

It remains to verify (8), i.e., C 3 (x * ) ≥ 0 and C 4 (x * ) ≥ 0 which leads to the two following conditions:

JP 2 ≤ 4V 3 max and J 2 P ≤ 2A 3 max (36)
To conclude on this sixth case, if conditions (36) are true, x * in (35) is a local minimum candidate. Moreover, it appears that Case 6 is equivalent to Case 3 since x * in (35) is the same as x * in (27) and conditions (36) are the same as those in (25) and (26) (which is a consequence of x * being the same).

H. Case 7: Two active constraints C

1 (x) = C 3 (x) = 0 C 1 (x * ) = 0 and C 3 (x * ) = 0 give: x * = V max √ JV max (37)
Moreover, C 2 (x * ) > 0 and C 4 (x * ) > 0 imply with (10) that λ * 2 = 0 and λ * 4 = 0, and (7) yields the following equation system in λ * 1 and λ * 3 37), the second equation yields λ * 1 = 0 and then the first equation gives λ

∇ x f (x * ) -λ * 1 ∇ x C 1 (x * ) -λ * 3 ∇ x C 3 (x * ) = 0 ⇐⇒ x * 2 1 -Px * 2 -λ * 1 x * 2 1 + λ * 3 x * 2 1 x * 2 = 0 x * 2 2 -Jx * 1 + λ * 1 x * 2 2 + Jx * 1 = 0 With (
* 3 = Px * 2 -x * 2 1 / x * 2 1 x * 2 .
With (37), for (9) to be true, i.e. λ * 3 ≥ 0, the following condition must hold:

V 3 max ≤ JP 2 (38)
It remains to verify that (8) is satisfied for i = 2 and i = 4, i.e., C 2 (x * ) ≥ 0 and C 4 (x * ) ≥ 0, which leads to the two following conditions:

V 3 max ≤ JP 2 4 and √ JV max ≤ A max ( 39
)
where the first condition in (39) is stronger than (38), i.e., (39) implies (38).

To conclude on case 7, if conditions (39) are true, x * in (37) is a local minimum candidate. Moreover, it turns out that Case 7 is equivalent to Case 4 since x * in (37) is the same as x * in (33) and conditions (39) are the same as those in (31) and (32) (which is a consequence of x * being the same).

I. Case 8: Two active constraints C 1 (x) = C 4 (x) = 0 C 1 (x * ) = 0 and C 4 (x * ) = 0 give: x * =   A 2 max J A max   (40)
Moreover, C 2 (x * ) > 0 and C 3 (x * ) > 0 imply with (10) that λ * 2 = 0 and λ * 3 = 0, and (7) yields the following equation system in λ * 1 and λ * 4 :

∇ x f (x * ) -λ * 1 ∇ x C 1 (x * ) -λ * 4 ∇ x C 4 (x * ) = 0 ⇐⇒ x * 2 1 -Px * 2 -λ * 1 x * 2 1 = 0 x * 2 2 -Jx * 1 + λ * 1 x * 2 2 + Jx * 1 + λ * 4 Jx * 2
2 = 0 From the first equation and (40), we have:

λ * 1 = x * 2 1 -Px * 2 x * 2 1 = 1 - J 2 P A 3 max (41)
and, from the second equation:

λ * 4 = Jx * 1 -x * 2 2 -λ * 1 x * 2 2 + Jx * 1 Jx * 2 2 = - 2 J λ * 1 (42) since, according to (40), Jx * 1 -x * 2 2 = 0 and x * 2 2 + Jx * 1 = 2x * 2 2 .
Eq. ( 42) and (9) imply that λ * 1 = λ * 4 = 0 so that (41) gives:

J 2 P = A 3 max (43)
However, C 2 (x * ) ≥ 0, which must hold true according to (8), leads to:

J Px * 2 -x * 2 1 -x * 1 x * 2 2 ≥ 0 =⇒ JPA max -2 A 4 max J ≥ 0 =⇒ J 2 P 2 ≥ A 3 max
which is impossible in view of (43). In conclusion, Case 8 is impossible and thus discarded.

J. Case 9: Two active constraints C 2 (x) = C 3 (x) = 0 C 3 (x * ) = 0 means that x * 1 = V max while C 2 (x * ) = 0 yields a second-order polynomial equation in x * 2 .
The latter equation can be solved to obtain two solutions for x * 2 , but it is better to first consider (7). Since, with (10), C 1 (x * ) > 0 and C 4 (x * ) > 0 imply that λ * 1 = 0 and λ * 4 = 0, (7) yields:

∇ x f (x * ) -λ * 2 ∇ x C 2 (x * ) -λ * 3 ∇ x C 3 (x * ) = 0 ⇐⇒ x * 2 1 -Px * 2 + λ * 2 x * 2 1 + Px * 2 + λ * 3 x * 2 1 x * 2 = 0 x * 2 2 -Jx * 1 + λ * 2 x * 2 2 -Jx * 1 = 0 (44)
The second equation implies that either

x * 2 2 = Jx * 1 or λ * 2 = -1. Since λ * 2 ≥ 0 according to (9), λ * 2 = -1 is not possible so that x * 2 2 = Jx * 1 , i.e., x * 2 = √ JV max . Then, with x * 1 = V max and x * 2 = √ JV max , C 2 (x * ) = 0 is equivalent to: V 3 max = JP 2 4 ( 45
)
Going back to the equation system (44), note that in the second equation, since x * 2 2 = Jx * 1 , λ * 2 is undetermined so that any λ * 2 ≥ 0 satisfies this equation and (9) as well. Now, let us verify if there exist λ * 2 ≥ 0 and λ * 3 ≥ 0 such that the first equation of (44) is verified. With x * 1 = V max and x * 2 = √ JV max , this first equation is equivalent to:

λ * 3 = P √ JV max -V 2 max -λ * 2 P √ JV max +V 2 max V 2 max √ JV max (46)
so that λ * 3 ≥ 0 if and only if:

λ * 2 ≤ P √ JV max -V 2 max P √ JV max +V 2 max ( 47
)
Since λ * 2 must be non-negative, (47) is possible if and only if:

P √ JV max -V 2 max ≥ 0 ⇐⇒ JP 2 ≥ V 3 max (48)
which, according to (45), is true. Hence, taking λ * 2 equal to the right-hand side of the inequality (47), we have λ * 2 ≥ 0 and λ * 3 in ( 46) is also non-negative which proves that there exist λ * 2 ≥ 0 and λ * 3 ≥ 0 such that (44) is verified. Finally, the inequalities C 1 (x * ) ≥ 0 and C 4 (x * ) ≥ 0 should be verified for (8) to be true. First, we have

C 1 (x * ) = 0 since x * 2 2 = Jx * 1 . Second, C 4 (x * ) ≥ 0 is x * 2 ≤ A max which yields: √ JV max ≤ A max (49)
To conclude Case 9, if conditions (45) and (49) are verified, the following vector is a local minimum candidate:

x * = V max √ JV max (50)
Comparing ( 45) with (31), ( 49) with ( 32) and ( 50) with (33), Case 9 appears to be a particular case of Case 4 where (31) is verified as an equality.

K. Case 10: Two active constraints

C 2 (x) = C 4 (x) = 0 C 4 (x * ) = x 2max -x * 2 = 0 gives x * 2 = x 2max = A max while C 2 (x * ) = 0 yields the following quadratic equation in x * 1 : Jx * 2 1 + x * 2 2 x * 1 -JPx * 2 = 0 (51)
The discriminant of this equation is:

∆ = x * 2 x * 3 2 + 4J 2 P > 0 (52)
and (51) possesses the following two solutions:

x * 1 = -x * 2 2 ± √ ∆ 2J (53)
Since x * 1 > 0 according to (6), the only possible solution is:

x * 1 = -x * 2 2 + √ ∆ 2J = -A 2 max + A 4 max + 4J 2 PA max 2J ( 54
)
since the other one is negative. Note that x * 1 in ( 54) is positive because -x * 2 2 + √ ∆ > 0, i.e., x * 4 2 < ∆ which can be deduced from (52) and 4J 2 P > 0.

Now, let us determine the Lagrange multipliers λ * 2 and λ * 4 and establish the conditions for λ * 2 and λ * 4 to be nonnegative and thus to satisfy (9). Since, with (10), C 1 (x * ) > 0 and C 3 (x * ) > 0 imply that λ * 1 = 0 and λ * 3 = 0, (7) yields:

∇ x f (x * ) -λ * 2 ∇ x C 2 (x * ) -λ * 4 ∇ x C 4 (x * ) = 0 ⇐⇒ x * 2 1 -Px * 2 + λ * 2 x * 2 1 + Px * 2 = 0 Jx * 1 -x * 2 2 + λ * 2 Jx * 1 -x * 2 2 = λ * 4 ( 55
)
Since x * 2 1 + Px * 2 > 0, we have from the first equation:

λ * 2 = Px * 2 -x * 2 1 Px * 2 + x * 2 1 ( 56
)
With the expression of x * 1 in (54), one can verify that

Px * 2 -x * 2 1 ≥ 0 is equivalent to ∆ ≥ x * 4
2 which is true according to (52) and 4J 2 P > 0. Consequently, λ * 2 in (56) is non-negative. Besides, from the second equation of (55), we have: 54) and (52), the latter inequality is equivalent to x * 3 2 ≤ J 2 P 2 . Hence, λ * 4 ≥ 0 leads to the following condition:

λ * 4 = Jx * 1 -x * 2 2 (1 + λ * 2 ) (57) Since λ * 2 ≥ 0, λ * 4 ≥ 0 is equivalent to Jx * 1 -x * 2 2 ≥ 0. With (
A 3 max ≤ J 2 P 2 (58)
Finally, the inequalities C 1 (x * ) ≥ 0 and C 3 (x * ) ≥ 0 should be verified for (8) to be true.

C 1 (x * ) ≥ 0 is equivalent to Jx * 1 -x * 2 2 ≥ 0 i.e. to λ * 4 ≥ 0 which is verified if (58) is true. C 3 (x * ) ≥ 0 is: x * 1 = -A 2 max + A 4 max + 4J 2 PA max 2J ≤ V max (59)
To conclude Case 10, if conditions (58) and (59) are verified, the following vector is a local minimum candidate:

x * =   -A 2 max + A 4 max + 4J 2 PA max 2J A max   (60)
L. Case 11: Two active constraints C 3 (x) = C 4 (x) = 0

In this last case, the two active constraints C 3 (x) = C 4 (x) = 0 yield directly the following local minimum candidate:

x * = V max A max (61)
Since, with (10), C 1 (x * ) > 0 and C 2 (x * ) > 0 imply that λ * 1 = 0 and λ * 2 = 0, (7) gives:

∇ x f (x * ) -λ * 3 ∇ x C 3 (x * ) -λ * 4 ∇ x C 4 (x * ) = 0 ⇐⇒            λ * 3 = Px * 2 -x * 2 1 x * 2 1 x * 2 λ * 4 = Jx * 1 -x * 2 2 Jx * 2 2 ( 62
)
so that, with (61), the following equality should hold for (9) to be true:

λ * 3 ≥ 0 ⇐⇒ Px * 2 -x * 2 1 ≥ 0 ⇐⇒ PA max ≥ V 2 max ( 63
)
and also:

λ * 4 ≥ 0 ⇐⇒ Jx * 1 -x * 2 2 ≥ 0 ⇐⇒ JV max ≥ A 2 max (64)
Finally, the inequalities C 1 (x * ) ≥ 0 and C 2 (x * ) ≥ 0 should be verified for (8) to hold.

C 1 (x * ) ≥ 0 is equivalent to Jx * 1 -x * 2 2 ≥ 0, i.e., to (64). C 2 (x * ) ≥ 0 is equivalent to: J Px * 2 -x * 2 1 -x * 1 x * 2 2 ≥ 0 ⇐⇒ JPA max -JV 2 max ≥ V max A 2 max
(65) In summary, if the inequalities (63), (64) and (65) are satisfied, x * in (61) is a local minimum candidate.

M. Synthesis of the 11 cases and Algorithm 2

Let us now summarize all the 11 cases:

• Cases 1, 2, 5 and 8 are impossible in the sense that they do not verify the KKT first-order necessary conditions. • Case 6 is equivalent to Case 3 and Case 7 is equivalent to Case 4. • Case 9 is a particular case of Case 4. Hence, only four cases, namely Cases 3, 4, 10 and 11, need to be considered to find (local) solutions x * = (x * 1 , x * 2 ) to the optimization problem (6).

In order to ease the analysis of the relationships between these cases, the conditions to be fulfilled for each one of these four cases are summarized below.

Conditions for Case 3:

JP 2 ≤ 4V 3 max (66) J 2 P ≤ 2A 3 max (67)
Conditions for Case 4:

4V 3 max ≤ JP 2 (68) JV max ≤ A 2 max ( 69
)
Conditions for Case 10:

2A 3 max ≤ J 2 P (70) JPA max -JV 2 max ≤ V max A 2 max (71)
where (71) is equivalent (59) as can be shown with some elementary calculations. Moreover, again after some calculations, it turns out that (70) and (71) imply that:

A 2 max ≤ JV max (72)
Conditions for Case 11:

V 2 max ≤ PA max (73)
A 2 max ≤ JV max (74)

V max A 2 max ≤ JPA max -JV 2 max ( 75
)
Note that, from (74) and (75), we have:

2A 3 max ≤ J 2 P (76)
Indeed, the sum of ( 76) and (74) multiplied by V max gives 2V max A max ≤ JP. Using the latter inequality and multiplying (74) by 2A max lead to (76). Carefully analyzing all these conditions leads to the following relationships between the four remaining cases (Cases 3, 4, 10 and 11) and in turn to Algorithm 1.

Let us first assume that condition (67) of Case 3 is satisfied as a strict inequality. Then, Cases 10 and 11 are not possible because of ( 70) and ( 76), respectively, and there are two possibles cases:

• If (66) of Case 3 is verified (as a strict inequality), Case 4 is not possible because of (68). The only possible case is Case 3 which means that the sole local minimum candidate is x * in (27) which corresponds to line 3 of Algorithm 1. • If (66) of Case 3 is not verified, Case 3 is not possible and condition (68) of Case 4 is verified. Case 4 is then the only possible case provided that condition (69) is true. It turns out that (67) and (68) imply that condition (69) is verified. Indeed, (67) is equivalent to:

3 J 4 P 2 4 ≤ A 2 max (77) (68) is equivalent to: 3 4 JP 2 ≤ 1 V max (78)
and ( 77) and (78) imply that:

3 √ J 3 ≤ A 2 max V max ⇐⇒ JV max ≤ A 2 max ( 79
)
which is (69). Hence, Case 4 is the only possible case meaning that the sole local minimum candidate is x * in (33) which corresponds to line 5 of Algorithm 1. Let us now assume that condition (67) of Case 3 is not satisfied. Case 3 is then not possible and:

• If (69) is satisfied (as a strict inequality), Cases 10 and 11 are not possible because of ( 72) and (74), respectively. Moreover, since (67) is not satisfied, we have:

2A 3 max < J 2 P ⇐⇒ A max < 3 J 2 P 2 (80)
and using the latter in (69) (satisfied as a strict inequality) yields:

J < A 2 max V max < 1 V max 3 J 4 P 2 4 (81)
which implies that: 73) is true since it turns out to be implied by (75). The latter result comes from the fact that ( 75) is equivalent to:

V max < 3 JP 2 4 ⇐⇒ V 3 max < JP 2 4 ( 82
V max A 2 max + JV 2 max ≤ JPA max ( 83
)
which implies that:

JV 2 max ≤ JPA max ⇐⇒ V 2 max ≤ PA max ( 84
)
the latter inequality being (73). Hence, Case 11 is the only possible case and the unique local minimum candidate is x * in (61) which corresponds to lines 11 and 14 of Algorithm 1. Algorithm 2 summarizes the above analysis. It is exactly the same as Algorithm 1 which proves that the latter is correct. Note that line 12 of Algorithm 2 corresponds to lines 11 and 12 of Algorithm 1 as can be seen from the expression of x * in (60). Moreover, line 14 of Algorithm 2 corresponds to lines 11 and 14 of Algorithm 1 since, with V =

-A 2 max + √ A 4 max +4J 2 PA max 2J
as calculated at line 12 of Algorithm 1, the condition V > V max at line 13 of Algorithm 1 means that (59) is not satisfied. Hence, since (59) and (71) are equivalent (as can be shown with some elementary calculations), (71) is also not satisfied which corresponds to the condition for line 14 of Algorithm 2.

It is important to note that, according to the above analysis, Algorithm 1 and Algorithm 2 computes in fact the global minimum of the minimization problem (6).

N. Particular cases

For the proof to be complete, it remains to be shown that Algorithm 2, and hence Algorithm 1, is correct in a number of Algorithm 2 Minimum-Time S-Curve Trajectory Rewritten end if 17: end if particular cases. The latter were overlooked at various places in the proof presented in Section III-M where conditions (66), (67), ( 69) and (71) were supposed to be verified as strict inequalities. In fact, the particular cases to be considered to complete the proof are the following ones:

• (67) and (66) satisfied as equalities;

• (67) satisfied as an equality and (66) satisfied as a strict inequality; • (67) satisfied as an equality and (66) not satisfied; • (67) not satisfied and (69) satisfied as an equality; • (67) and (69) not satisfied and (71) satisfied as an equality.

First particular case: (67) and (66) satisfied as equalities

J 2 P = 2A 3 max (85) JP 2 = 4V 3 max (86)
These two equalities imply that (27) (line 3 of Algorithm 2) is:

x * =       3 JP 2 4 3 J 2 P 2       =       V max A max       (87)
Now, let us consider x * in (33) (line 5 of Algorithm 2):

x * = V max √ JV max = V max A max (88)
To show that the second equality in the previous equation holds, consider P = 2A 3 max J 2 from (85) and P 2 = 4V 3 max J from (86). Then, we have:

4A 6 max J 4 = 4V 3 max J ⇐⇒ A max = √ JV max (89)
Next, (60) (line 12 of Algorithm 2) is:

x * =   -A 2 max + A 4 max + 4J 2 PA max 2J A max   = V max A max (90)
since, with J 2 P = 2A 3 max in (85):

-A 2 max + A 4 max + 4J 2 PA max 2J = -A 2 max + A 4 max + 8A 4 max 2J = -A 2 max + 9A 4 max 2J = A 2 max J = V max (91)
where the last equality is obtained from (89). Hence, in this first particular case, it turns out that x * in (27), ( 33) and ( 60) are all equal to x * in (61), i.e, all possible local minima are equal. Then, Algorithm 2 necessarily computes the optimal value x * = [V max , A max ] T at line 3.

Second particular case: (67) satisfied as an equality and (66) satisfied as a strict inequality

J 2 P = 2A 3 max (92) JP 2 < 4V 3 max (93)
First, note that Case 3 is possible since both (67) and (66) are satisfied and Case 4 is not possible since (68) is not satisfied. Moreover, Algorithm 2 computes x * at line 3 according to (27):

x * =     3 JP 2 4 3 J 2 P 2     =     3 JP 2 4 A max     (94)
where 3 JP 2 4 < V max since JP 2 < 4V 3 max . Now, let us check whether or not Cases 10 and 11 are possible and, if they are, verify that x * in (94) also corresponds to their minimum.

The first condition for Case 10 to be possible is (70). This condition is verified since J 2 P = 2A 3 max . The second condition for Case 10 is (71) which can equivalently be written as:

V max A 2 max + JV 2 max ≥ JPA max (95)
Since V max > 3 JP 2 4 and J 2 P = 2A 3 max , one can write:

V max A 2 max + JV 2 max > A 2 max 3 JP 2 4 + J 3 J 2 P 4 16 = 3 J 4 P 2 4 3 JP 2 4 + J 3 J 2 P 4 16 = 3 J 5 P 4 16 + 3 J 5 P 4 16 = 2 3 J 5 P 4 16 = 3 J 5 P 4 2 = JP 3 J 2 P 2 Hence, V max A 2 max + JV 2 max > JP 3 J 2 P 2 = JPA max , i.e., (95)
is verified as a strict inequality so that (71) is also verified as a strict inequality and Case 10 is possible. Furthermore, the minimum x * in Case 10 is given in (60):

x * =   -A 2 max + A 4 max + 4J 2 PA max 2J A max   =   A 2 max J A max   (96)
where the second equality is from (91). Since J 2 P = 2A 3 max , we have:

3 JP 2 4 = 3 J 4 4A 6 max J 4 = A 2 max J (97)
so that the minimum x * of Case 10 is equal to the minimum x * of Case 3 given in (94). Finally, Case 11 is not possible because (71) is verified as a strict inequality and, hence, (75) is not verified.

Summarizing, only Cases 3 and 10 are possible, the minimum x * of these two cases are equal and Algorithm 2 indeed computes this x * at line 3.

Third particular case: (67) satisfied as an equality and (66) not satisfied

J 2 P = 2A 3 max (98) JP 2 > 4V 3 max (99)
In this particular case, Case 3 is not possible since (66) is not satisfied and Algorithm 2 computes x * at line 5 according to (33):

x * = V max √ JV max (100)
First, note that Case 4 is possible since (68) is true since J 2 P = 2A 3 max . Moreover, (69) is also verified since:

J 2 P = 2A 3 max ⇐⇒ A 2 max = 3 J 4 P 2 4 = J 3 JP 2 4 ( 101
)
and:

JP 2 > 4V 3 max ⇐⇒ V max < 3 JP 2 4 ( 102
)
imply that:

JV max < J 3 JP 2 4 = A 2 max (103)
Furthermore, let us examine Cases 10 and 11. Case 10 is not feasible since (71) is not verified. Indeed, in the previous paricular case, it was shown that J 2 P = 2A 3 max and JP 2 < 4V 3 max leads to (71) being verified as a strict inequality, i.e., JPA max -JV 2 max < V max A 2 max . Then, the same reasoning allows to conclude that J 2 P = 2A 3 max and JP 2 > 4V 3 max imply JPA max -JV 2 max > V max A 2 max i.e. ( 71) is not verified in the present particular case. Besides, Case 11 is also not feasible since A 2 max > JV max from (103) so that (74) is not true. Hence, only Case 4 is possible and the corresponding minimum x * is indeed computed at line 5 of Algorithm 2.

Fourth particular case: (67) not satisfied and (69) satisfied as an equality J 2 P > 2A 3 max (104)

JV max = A 2 max (105)
First, note that Case 3 is not possible since (67) is not satisfied and that Algorithm 2 computes x * at line 9 according to (33):

x * = V max √ JV max = V max A max (106)
which is also the minimum x * of Case 11 as given in (61) and where the second equality in (106) comes from (105). Then, let us check whether or not Cases 4, 10 and 11 are possible. Case 4 is possible since (69) is truce according to (105). Moreover, (104) and (105) imply that:

4V 3 max = 4 A 6 max J 3 < 4 J 3 J 4 P 2 4 = JP 2 (107)
so that (68) is satisfied. Case 10 is not possible since (71) is not satisfied. Indeed, with (105), we have:

V max A 2 max + JV 2 max = 2A 4 max J (108)
and (104) then implies that:

2A 4 max J < J 2 P A max J = JPA max (109)
so that:

V max A 2 max + JV 2 max < JPA max ⇐⇒ JPA max -JV 2 max > V max A 2 max
(110) Case 11 turns out to be possible since (110) shows that (75) is true and (74) is satisfied according to (105). Moreover, (104) and (105) imply that:

V 2 max = A 4 max J 2 < A max J 2 J 2 P 2 = A max P 2 < PA max (111)
i.e. (73) is satisfied. In conclusion of this fourth particular case, only Cases 4 and 11 are possible, their minimum x * are equal and indeed computed by Algorithm 2 at line 9.

Fifth particular case: (67) and (69) not satisfied and (71) satisfied as an equality

J 2 P > 2A 3 max (112) JV max > A 2 max (113) JPA max -JV 2 max = V max A 2 max ( 114
)
Cases 3 and 4 are then not possible since (67) and (69) are not satisfied, respectively.

Case 10 is possible since (112) implies ( 70) and ( 71) is true according to (114).

Then, about the feasibility of Case 11, it is not straightforward to verify whether or not (73) is true from (112), ( 113) and (114). Fortunately, this verification is in fact not required. Indeed, in the present particular case, Algorithm 2 computes

x * of Case 10 at line 12 and, as proved below, this x * is equal to the one of Case 11, x * = [V max , A max ] T . Hence, if Case 11 is not possible, Algorithm 2 computes x * of Case 10 which is the only possible minimum and, if Case 11 is possible, x * of Cases 10 and 11 are equal and Algorithm 2 indeed computes this minimum at line 12.

It remains to prove that the x * of Case 10, which according to (60) is:

x * =   -A 2 max + A 4 max + 4J 2 PA max 2J A max   (115)
is equal to the one of Case 11 which is x * = [V max , A max ] T , i.e., to prove that

-A 2 max + A 4 max + 4J 2 PA max 2J = V max (116)
To this end, let us consider (114) as a quadratic equation in

V max : JV 2 max + A 2 max V max -JPA max = 0 (117)
whose two solutions are:

V max = -A 2 max ± A 4 max + 4J 2 PA max 2J (118)
The only possible solution for V max among those two is the following one since the other one is negative:

V max = -A 2 max + A 4 max + 4J 2 PA max 2J (119)
which shows that x * of Case 10 in (115) is equal to x * of Case 11.

O. Second-Order Sufficient Conditions

Algorithm 2 (and thus Algorithm 1) is based on the KKT first-order necessary conditions. Hence, the vectors x * computed in the various cases in Algorithm 2 are local minima candidates and it remains to be verified that these vectors x * are indeed local minima. To this end, the following secondorder sufficient conditions can be used [START_REF] Nocedal | Numerical Optimization[END_REF].

Second-Order Sufficient Conditions: Suppose that at some feasible vector x * , there exists a Lagrange multiplier vector λ * satisfying the KKT conditions and that the Lagrangian Hessian ∇ 2

xx L (x * , λ * ) is positive definite. Then, x * is a strict local minimum to the optimization problem (6).

From (12), the Lagrangian Hessian is:

∇ 2 xx L (x, λ ) = ∇ 2 xx f (x) - 4 ∑ i=1 λ i ∇ 2 xx C i (x) (120)
The individual Hessian matrices appearing in the above expression of ∇ 2 xx L are obtained from (13) to (17) as follows.

∇ 2 xx f (x) =       ∂ 2 f ∂ x 2 1 ∂ 2 f ∂ x 1 ∂ x 2 ∂ 2 f ∂ x 2 ∂ x 1 ∂ 2 f ∂ x 2 2       =       2P x 3 1 - 1 x 2 2 - 1 x 2 2 2x 1 x 3 2       (121) ∇ 2 xx C 1 (x) =       0 - 1 x 2 2 - 1 x 2 2 2x 1 x 3 2       (122) ∇ 2 xx C 2 (x) =       2P x 3 1 1 x 2 2 1 x 2 2 - 2x 1 x 3 2       (123) ∇ 2 xx C 3 (x) = 0 (124) ∇ 2 xx C 4 (x) = 0 (125)
Therefore, the Lagrangian Hessian is: In Case 3, we have λ * 1 = 0, λ * 2 = 1/2, x * 2 2 = Jx * 1 and:

∇ 2 xx L (x, λ ) =       (1 -λ 2 ) 2P x 3 1 (λ 1 -λ 2 -1) 1 x 2 2 (λ 1 -λ 2 -1) 1 x 2 2 (1 -λ 1 + λ 2 ) 2x 1 x 3 2       (
x * =       x * 1 x * 2       =       3 JP 2 4 3 J 2 P 2       (127)
so that:

∇ 2 xx L (x * , λ * ) =       P x * 3 1 - 3 2x * 2 2 - 3 2x * 2 2 3x * 1 x * 3 2       (128)
According to Sylvester's criterion, since

P x * 3 1 > 0, the Hes- sian ∇ 2 xx L (x * , λ *
) is positive definite if and only if its determinant is strictly positive. Being given that x * 2 2 = Jx * 1 and with (127), the determinant is:

det ∇ 2 xx L (x * , λ * ) = 3P x * 2 1 x * 3 2 - 9 4x * 4 2 (129) = 3 x * 4 2 J 2 P x * 3 2 - 3 4 (130) = 3 x * 4 2 2J 2 P J 2 P - 3 4 (131) = 3 x * 4 2 5 4 > 0 (132) which is positive since x * 2 = 3 J 2 P 2 > 0.
Hence, the secondorder sufficient conditions are satisfied and x * of Case 3, given in ( 27) and computed at line 3 of Algorithm 2, is indeed a strict local minimum.

In Case 4, we have λ * 1 = λ * 2 = 0, x * 2 2 = Jx * 1 and:

x * = x * 1 x * 2 = V max √ JV max (133)
so that:

∇ 2 xx L (x * , λ * ) =       2P x * 3 1 - 1 x * 2 2 - 1 x * 2 2 2x * 1 x * 3 2       (134)
whose determinant is:

det ∇ 2 xx L (x * , λ * ) = 4P x * 2 1 x * 3 2 - 1 x * 4 2 (135) = 4J 2 P x * 4 2 x * 3 2 - 1 x * 4 2 (136) = 1 x * 4 2 4J 2 P x * 3 2 -1 (137) = 1 x * 4 2 4J 2 P J 3 V 3 max -1 (138)
Since x * 2 > 0, this determinant is positive if 4J 2 P > J 3 V 3 max and hence if 16JP 2 > V 3 max . The latter inequality is true since it is a consequence of (68): In Case 10, we have λ * 1 = 0 and:

λ * 2 = Px * 2 -x * 2 1 Px * 2 + x * 2 1 (140) x * = x * 1 x * 2 =   -A 2 max + A 4 max + 4J 2 PA max 2J A max   (141)
With λ * 1 = 0, the Hessian in (126) becomes:

∇ 2 xx L (x * , λ * ) =       (1 -λ * 2 ) 2P x * 3 1 -(1 + λ * 2 ) 1 x * 2 2 -(1 + λ * 2 ) 1 x * 2 2 (1 + λ * 2 ) 2x * 1 x * 3 2       (142)
and with the expression of λ * 2 in (140):

∇ 2 xx L (x * , λ * ) = 1 Px * 2 + x * 2 1       4P x * 1 - 2P x * 2 - 2P x * 2 4Px * 1 x * 2 2       (143)
Referring to Section III-K, x Hence, the second-order sufficient conditions are satisfied and x * of Case 10, given in (60) and computed at line 12 of Algorithm 2, is a strict local minimum. Finally, in Case 11, we have λ * 1 = 0 and λ * 2 = 0 and:

x * = x * 1 x * 2 = V max A max (146)
so that: (149)

∇ 2 xx L (x * , λ * ) =       2P x * 3 1 - 1 x * 2 2 - 1 x * 2
= 1 A 3 max 4P V 2 max - 1 A max (150) = 1 A 3 max 4PA max -V 2 max V 2 max A max (151)
This determinant is positive since PA max ≥ V 2 max according to (73). Hence, the second-order sufficient conditions are satisfied and x * of Case 11, given in (61) and computed at line 14 of Algorithm 2, is a strict local minimum.

P. LICQ

In the KKT first-order necessary conditions stated at the beginning of Section III, the LICQ should hold. As defined in Definition 12.4 of [START_REF] Nocedal | Numerical Optimization[END_REF], the LICQ holds at a given x * if the set of active constraint gradients are linearly independent at x * .

For completeness of the proof of Algorithm 2 (and thus of Algorithm 1), let us verify that the LICQ holds at the local minimum x * of Case 3 in (27), Case 4 in (33), Case 10 in (60) and Case 11 in (61).

For Cases 3 and 4, only one constraint is active, C 2 (x) = 0 and C 3 (x) = 0, respectively. Hence, the LICQ holds at x * of Case 3 if ∇ x C 2 (x * ) in ( 15) is nonzero which is necessarily the case since -Px 2x 2 1 is strictly negative for any x. The LICQ holds at x * of Case 4 if ∇ x C 3 (x * ) in ( 16) is nonzero which is always the case. In Case 10, there are two active constraints, C 2 (x) = C 4 (x) = 0. From (15) and (17), ∇ x C 2 (x) and ∇ x C 4 (x) are easily seen to be linearly independent whatever x so that the LICQ holds.

Finally, in Case 11, there are two active constraints, C 3 (x) = C 4 (x) = 0, and, according to ( 16) and (17), ∇ x C 3 (x) and ∇ x C 4 (x) are trivially linearly independent whatever x so that the LICQ holds.

Fig. 1 .

 1 Fig. 1. Third order polynomial S-Curve model.

G. Case 6 :

 6 Two active constraints C 1 (x) = C 2 (x) = 0 When only C 1 and C 2 are active, C 3 (x * ) > 0 and C 4 (x * ) > 0 imply with (10) that λ * 3 = 0 and λ * 4 = 0. Then, x * 1 , x * 2 , λ * 1 and λ * 2 have to be determined. The determination of λ * 1 and λ *

2

 2

  126) Let us now verify the second-order sufficient conditions at the local minimum candidate x * of Case 3 in (27), Case 4 in (33), Case 10 in (60) and Case 11 in (61).

JP 2 ≥* 3 1> 0 ,

 230 4V 3 max ⇐⇒ 16JP 2 ≥ 64V 3 max (139) Hence, det ∇ 2 xx L (x * , λ * ) > 0 and since 2P x Sylvester's criterion implies that ∇ 2 xx L (x * , λ * ) is positive definite. The second-order sufficient conditions are thus satisfied and x * of Case 4, given in (33) and computed at lines 5 and 9 of Algorithm 2, is a strict local minimum.

  > 0 hold. Then, according to Sylvester's criterion, ∇ 2 xx L (x * , λ * ) is positive definite if and only if the determinant of the matrix in (143) is positive. This determinant is positive since it is calculated as follows:

	Px * 2 + x * 2 1 > 0 and	4P x * 1		* 1 > 0 and x * 2 > 0 so that
	4P x * 1 x * 2 -2P	2P x * 2 2 x * 2 -4Px * 1	=	16P 2 x * 2 2	-	4P 2 2 x * 2	(144)
				=	12P 2 2 x * 2			(145)