Local study of sub-Finslerian control systems

Abstract : In this thesis I study the local geometry of Finslerian and sub-Finslerian structures associated to the maximum norm in dimension 2 and 3 : short generalized geodesics, cut locus, generalized conjugate locus, switching locus, small spheres.To define such a structure in the neighborhood of a point $p$ of $mathbb{R}^n$, we fix a familly of vector fields $(F_1,dots,F_k)$ and consider the norm defined on the distribution $Delta=mbox{vect}{F_1,dots,F_k}$ by $|G|=inf{max{|u_i|} ; | ; G=sum_i u_i F_i} $.In dimension 2, for $k=2$, if $F_1$ and $F_2$ are not proportionnal at $p$ then we obtain a Finslerian structure. If not, the structure is sub-Finslerian on a distribution with non constant rank. We describe the geometric objects for the set of all generic couples $(F_1,F_2)$.In dimension 3, we studied the local geometry for contact distributions.
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Entisar Abdul-Latif Ali. Local study of sub-Finslerian control systems. Optimization and Control [math.OC]. Université Grenoble Alpes, 2017. English. ⟨NNT : 2017GREAM005⟩. ⟨tel-01681328v2⟩

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