Contrôlabilité des systèmes non linéaires en basse dimension : structure et classification

Abstract : This thesis has three independant parts : 1-study of the controllability, in dimension3, of a pair of linear vector fields, one of these being a nilpotent one of order 2 ; 2-classification, from the controllability point of view, of affine systems in the plane with unbounded controls ; 3-classification of quadratic systems in dimension 3. The first deal with a difficult problem : controllability of pairs of vectors fields in dimension 3. In fact this is equivalent to achieve controllability for a system with bounded controls. It is knon that such a problem is difficult. For the pairs being considered necessary and sufficient conditions are given. It seems that something deep is happening with nilpotent element in a Lie group. This point desserves certainly to be investigated further. The second part gives necessary and sufficient conditions for an affine system in the plane with unbounded controls to be controllable : x=ax+u(dx+b). These conditions are entirely of an algebric kind, so computable. This is due to low dimension. A topological study of the set of controllable systemscan be carried out. It turns out that the controllable systems are essentially those for wich the asociated linear system is also controllable. We conjectured that it is true for all the dimensions. The last part deal with the classification of quadratic systems in dimension 3. x=Q(x)+ub. The classification is from the controllability point of view and allows to check if a system is controllable or not
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Jean-Claude Vivalda. Contrôlabilité des systèmes non linéaires en basse dimension : structure et classification. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1986. Français. ⟨NNT : 1986METZ027S⟩. ⟨tel-01775714⟩

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