Control Sets of Linear Systems and Classification of Almost-Riemannian Structures on Lie Groups

Abstract : This Thesis analyzes the control sets of linear control systems and the isometries of almost-Riemannian structures on Lie groups. The main goal for the first topic is to characterize the properties of control sets such as existence, uniqueness, boundedness and invariance. We study such properties for Lie groups decomposable by eigenvalues of the linear vector field and extend some results to non-compact semi-simple Lie groups with finite center. The second topic main objective is to characterize isometry properties of almost-Riemannian structures. We search for invariants under isometries such as the singular locus and the set of the linear vector field singularities. For nilpotent Lie groups, we prove that all isometries are affine, that is, a composition of a translation with a Lie group automorphisms. To finish this topic we use the obtained results to classify the almost-Riemannian structures on low dimensional Lie groups.
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Guilherme Zsigmond Machado. Control Sets of Linear Systems and Classification of Almost-Riemannian Structures on Lie Groups. Differential Geometry [math.DG]. Normandie Université, 2017. English. ⟨NNT : 2017NORMR058⟩. ⟨tel-01692456⟩

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