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000064687 001__ 64687 000064687 005__ 20181205220240.0 000064687 0247_ $$2doi$$a10.5075/epfl-thesis-3481 000064687 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis3481-8 000064687 02471 $$2nebis$$a5102081 000064687 037__ $$aTHESIS 000064687 041__ $$aeng 000064687 088__ $$a3481 000064687 245__ $$aOptimal control of geodesics in Riemannian manifolds 000064687 269__ $$a2007 000064687 260__ $$aLausanne$$bEPFL$$c2007 000064687 300__ $$a135 000064687 336__ $$aTheses 000064687 502__ $$aBruno Colbois, Julius Natterer, Jacques Rappaz, Jacques Thévenaz, Rachid Touzani 000064687 520__ $$aThe aim of this dissertation is to solve numerically the following problem, denoted by P : given a Riemannian manifold and two points a and b belonging to that manifold, find a tangent vector T at a, such that expa(T) = b, assuming that T exists. This problem is set under an optimal control formulation, which requires the definition of an objective function and a space of control, the choice of a method for the calculation of the descent direction of that function in the space of control and the use of an optimization algorithm to find its minimum, which corresponds to the solution of the original problem by construction. Several techniques are necessary to be put together, coming from the fields of geometry, numerical analysis and optimization. The first part will concern a recalling of the mathematical context in which this formulation takes place. The general principles of optimal control will also be given. In the second part, we will present an intrinsic formulation of the optimal control problem associated to P, based on Jacobi fields, which will play the role of the so called adjoint state. This derivation leads to necessary optimality conditions. We will illustrate explicitly that formulation by treating the specific case of Riemannian manifolds with constant sectional curvature. Then, we will derive the optimal control problem in coordinates, not only to check the intrinsic formulation but also to reveal how it is hidden behind the expressions in coordinates. Their use reveals some quantities whose interpretation may be given this way. Moreover, we will show that more possibilities exist to chose the cost function and the control space in coordinates. In a second step, an alternative approach will consider the Hamiltonian formulation of geodesics. This is an incursion into symplectic geometry. We will then reformulate the Riemannian optimal control problem in its Hamiltonian version. In the third part, the numerical methods used for solving P will be presented. The discretization imposes the definition of new discrete optimal control problems. The technique shows that the discrete adjoint state equation strongly depends on the numerical scheme used to solve the direct problem. We will give a collection of numerical computations in the specific case of parametric piece of surfaces, where the surface can be defined by one or several Bézier patches, each one corresponding to a chart, which is representative of a Riemannian manifold. We will compare the different numerical approaches. The last but one part will be devoted to the interesting application of wooden roof building, where the structure is made of wooden boards, with geodesic trajectories on the designed piece of surface. The Geos (Geodesic solver) software has been developed for that purpose. After having introduced some specific numerical methods used in the code, we present the Geos application interface (AI) developed as a tool for the conception of such a roof. We then show an existing wooden structure built according to that mean. Finally, we will summarize the results of our research and discuss future possible prospects. 000064687 6531_ $$aRiemaniann Manifolds 000064687 6531_ $$aHamiltonian Manifolds 000064687 6531_ $$aGeodesics 000064687 6531_ $$aOptimization 000064687 6531_ $$aOptimal Control 000064687 6531_ $$aNumerical Methods 000064687 6531_ $$aBézier Surfaces 000064687 6531_ $$aWooden Roofs Design 000064687 6531_ $$aVariétés Riemanniennes 000064687 6531_ $$aVariétés Hamiltoniennes 000064687 6531_ $$aGéodésiques 000064687 6531_ $$aOptimisation 000064687 6531_ $$aContrôle Optimal 000064687 6531_ $$aMéthodes Numériques 000064687 6531_ $$aSurfaces de Bézier 000064687 6531_ $$aConception de Toits en Bois 000064687 700__ $$0241280$$aRozsnyo, Roland$$g135290 000064687 720_2 $$0244696$$aBuser, Jürg Peter$$edir.$$g104683 000064687 720_2 $$0244952$$aSemmler, Klaus-Dieter$$edir.$$g106409 000064687 8564_ $$s6395004$$uhttps://infoscience.epfl.ch/record/64687/files/EPFL_TH3481.pdf$$yTexte intégral / Full text$$zTexte intégral / Full text 000064687 909C0 $$0252345$$pGEOM$$xU10122 000064687 909CO $$ooai:infoscience.tind.io:64687$$pDOI$$pSB$$pthesis$$pthesis-bn2018$$qDOI2 000064687 918__ $$aSB$$bSB-SMA 000064687 919__ $$aGEOM 000064687 920__ $$a2006-3-24$$b2006 000064687 970__ $$a3481/THESES 000064687 973__ $$aEPFL$$sPUBLISHED 000064687 980__ $$aTHESIS