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Distance and geometry of the set of curves and approximation of optimal trajectories

Abstract : Optimization problems on the set of curves appear in many fields of applications such as industry, robotic, path-planning and aerospace. This thesis is devoted to study the set of curves and propose a general method for trajectory optimization problems, autonomous ODEs and control of autonomous ODEs. In the first part, we provide a normalization of parametrized curves up to increasing diffeomorphism and use it to define a distance between curves. Then, we study topologies and differential structures on the set of curves. The second part defines a norm on spaces of piecewise cubic Bézier curves and estimates equivalence constants for this norm and some classical norms. The last part proposes a general method to approximate optimal trajectories using piecewise cubic Bézier curves. This idea is applied to autonomous ODEs and control of autonomous ODEs.
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Submitted on : Monday, April 6, 2020 - 4:20:09 PM
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van Duc Hoang. Distance and geometry of the set of curves and approximation of optimal trajectories. Differential Geometry [math.DG]. Université de Limoges, 2020. English. ⟨NNT : 2020LIMO0013⟩. ⟨tel-02533808⟩

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