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Contrôle optimal géométrique : méthodes homotopiques et applications

Abstract : This work is about geometric optimal control applied to celestial and quantum mechanics. We first dealt with the minimum fuel consumption problem of transfering a satellite around the Earth. This brought to the creation of the code HamPath which permits first of all to solve optimal control problem for which the command law is smooth. It is based on the Pontryagin Maximum Principle (PMP) and on the notion of conjugate point. This program combines shooting method, differential homotopic methods and tools to compute second order optimality conditions. Then we are interested in quantum control. We study first a system which consists in two different particles of spin 1/2 having two different relaxation time. Both sub-systems are driven by the same control. The problem consists in bringing to zero the magnetization of one of the two system while maximizing the magnetization of the second one. This problem comes from constrast imaging in Nuclear Magnetic Resonance and consists in maximising the contrast between two areas of the image. The use of geometrical and numerical tools has given a very precise sub-optimal synthesis for two particular cases (deoxygenated/oxygenated blood and cerebrospinal fluid/water cases). The last contribution of this thesis is about the Lindblad equations in the two-level case. The model is based upon the minimisation of the transfer energy. We restrict the study to a particular case for which the Hamiltonian given by the PMP is Liouville integrable.We describe the conjugate and cut loci for this Riemannian with drift problem
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Submitted on : Thursday, February 21, 2013 - 1:30:14 PM
Last modification on : Monday, October 11, 2021 - 10:04:09 AM
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  • HAL Id : tel-00742927, version 4


Olivier Cots. Contrôle optimal géométrique : méthodes homotopiques et applications. Mathématiques générales [math.GM]. Université de Bourgogne, 2012. Français. ⟨NNT : 2012DIJOS008⟩. ⟨tel-00742927v4⟩



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