Couples de générateurs de certaines sous-algèbres de Lie de l'algèbre de Lie symplectique affine, et applications
Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet
Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal.
Une approche géométrique du contrôle optimal de l’arc atmosphérique de la navette spatiale
L’objectif de ce travail est de faire quelques remarques géométriques et des calculs préliminaires pour construire l’arc atmosphérique optimal d’une navette spatiale (problème de rentrée sur Terre ou programme d’exploration de Mars). Le système décrivant les trajectoires est de dimension 6, le contrôle est l’angle de gîte cinématique et le coût est l’intégrale du flux thermique. Par ailleurs il y a des contraintes sur l’état (flux thermique, accélération normale et pression dynamique). Notre étude...
Geometric optimal control and two-level dissipative quantum systems
Une approche géométrique du contrôle optimal de l'arc atmosphérique de la navette spatiale
The aim of this article is to make some geometric remarks and some preliminary calculations in order to construct the optimal atmospheric arc of a spatial shuttle (problem of reentry on Earth or Mars Sample Return project). The system describing the trajectories is in dimension 6, the control is the bank angle and the cost is the total thermal flux. Moreover there are state constraints (thermal flux, normal acceleration and dynamic pressure). Our study is mainly geometric and is founded on the...
Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet
Consider a where is a neighborhood of 0 in , is a identified to ker , being the 1-form: , and is a on which can be taken in the : , , , . In a previous article we analyze : ; we describe the , the and the . The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three of the flat case which clarify the role of the three parameters in the where: , ....
The smooth continuation method in optimal control with an application to quantum systems
The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system...
Riemannian metric of the averaged energy minimization problem in orbital transfer with low thrust
The smooth continuation method in optimal control with an application to quantum systems
The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system...
Conjugate-cut loci and injectivity domains on two-spheres of revolution
In a recent article [B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, 26 (2009) 1081–1098], we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is = d + ()d to the period mapping of the -variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties...
Riemannian metrics on 2D-manifolds related to the Euler−Poinsot rigid body motion
The Euler−Poinsot rigid body motion is a standard mechanical system and it is a model for left-invariant Riemannian metrics on (3). In this article using the Serret−Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S associated to the dynamics...
Conjugate and cut loci of a two-sphere of revolution with application to optimal control
Second order optimality conditions in the smooth case and applications in optimal control
The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control, and the article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. We derive an algorithm called COTCOT (Conditions...
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