Rank-2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8

Abstract

Abstract. We study the rank-2 distributions satisfying so-called Goursat condition (GC); that is to say, codimension-2 differential systems forming with their derived systems a flag. Firstly, we restate in a clear way the main result of [7] giving preliminary local forms of such systems. Secondly -and this is the main part of the paper -in dimension 7 and 8 we explain which constants in those local forms can be made 0, normalizing the remaining ones to 1. All constructed equivalences are explicit. The complete list of local models in dimension 7 contains 13 items, and not 14, as written in [7], while the list in dimension 8 consists of 34 models (and not 41, as could be concluded from some statements in [7]). In these dimensions (and in lower dimensions, too) the models are eventually discerned just by their small growth vector at the origin.Résumé. Nousétudions les distributions de rang 2 vérifiant la condition de Goursat ; c'est-à-dire, les systèmes différentiels de co-rang 2 formant, avec leurs systèmes dérivés, un drapeau. Nous donnons d'abord unénoncé clair du résultat principal de Kumpera et Ruiz sur des formes locales préliminaires de ces systèmes. Puis, dans la partie principale de l'article, en dimension 7 et 8, nous expliquons quelles constantes dans les formes préliminaires de Kumpera et Ruiz peuvent passerà 0, en normalisant simultanément les constantes restantesà 1. Toutes leséquivalences proposées sont explicites. La liste complète des modèles locaux en dimension 7 contient 13 objets (et non 14énoncés dans [7]), tandis que celle en dimension 8 comporte 34 modèles (et non 41 comme on pouvait le déduire de [7]). En ces dimensions (et en dimensions inférieures aussi) on n'utilise pour distinguer les modèles que leur petit vecteur de croissanceà l'origine.

Flat outputs of two-input driftless control systems

“…Its proof will be based on Corollary 4.2 and on the result that assures that any Goursat structure can be brought into the following polynomial normal form, called Kumpera-Ruiz normal form, as proved by Pasillas-Lépine and Respondek [26] (see also Cheaito and Mormul [4], Mormul [22] and Kumpera and Ruiz [15]): 14) where the coordinates …”

“…By the result of Sections 2.2 and 2.3, the functionφ 2 must satisfy the equations 4 . Solving the above system of equations, we get…”

Flag systems and ordinary differential equations

“…The basic theory as well as an initial attempt towards a local classification of flag systems can be found in [15]. Further classification can also be found in [5], [6], [8] and [21]. in one independent variable x and two dependent variables y and z, by means of parametrized expressions:…”

Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus

“…Since then many important results have been obtained, including solutions of the motion planning and stabilization problemsÐ see the papers (Rouchon et al 1993, Sù rdalen 1993, Fliess et al 1995, Samson 1995, Teel et al 1995, Tilbury et al 1995, Lizarraga et al 1998, Jiang and Nijmeijer 1999, the book (Laumond 1997), and references given thereÐ with the main emphasis put on regular con® gurations. For singular con® gurations much less results are available (see, however, Jean 1996, Vendittelli et al 1998, Cheaito and Mormul 1999, although it is clear that for some steering situations, like turning and going back in a narrow street, it is necessary to cross the singular locus. The aim of this paper is to show that theoretical results of Kumpera and Ruiz (1982), Cheaito and Mormul (1999) and Pasillas-Le pine and Respondek (2001) can be applied to this problem.…”