Contrôle optimal et applications au transfert d'orbite et à la géométrie presque-riemannienne

Abstract : In this thesis we focus on optimal control techniques as well as geometric control techniques applied to the orbital transfer problem and to almost-Riemannian geometry. In these cases, Pontryagin’s Maximum Principle allows to analyse the extremal flow of affine control systems.In the case of a satellite with low-thrust propulsion, averaging techniques give an approximated system. Averaging is explicit in the energy minimization case and is directly related to almost-riemannian problems. The geometric analysis of such problems is generalized by the study of metrics on the two-sphere of revolution. In this way it is possible to classify the situations considering the transcendance of the solutions and to discuss the optimality problem considering the cut locus and the conjugate locus.The analysis of the averaged problem for the orbital transfer and of generic situations on the two-dimensional sphere of revolution is motivated by the homotopic approach to solve numerically the orbital transfer problem.The homotopy method using simple shooting techniques is applied to solve a transfer minimizing the fuel consumption.The geometric tools are also useful in the local analysis of tangency points in two-dimensional almost-Riemannian geometry. In this framework, we compute wavefronts, sphere of small radius and cut and conjugate loci.
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Gabriel Janin. Contrôle optimal et applications au transfert d'orbite et à la géométrie presque-riemannienne. Mathématiques générales [math.GM]. Université de Bourgogne, 2010. Français. ⟨NNT : 2010DIJOS051⟩. ⟨tel-00633197v2⟩

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