Search results for "Cut locus"

showing 3 items of 3 documents

Optimal control and Clairaut-Liouville metrics with applications

2014

The work of this thesis is about the study of the conjugate and cut loci of 2D riemannian or almost-riemannian metrics. We take the point of view of optimal control to apply the Pontryagin Maximum Principle in the purpose of characterize the extremals of the problem considered.We use geometric, numerical and integrability methods to study some Liouville and Clairaut-Liouville metrics on the sphere. In the degenerate case of revolution, the study of the ellipsoid uses geometric methods to fix the cut locus and the nature of the conjugate locus in the oblate and prolate cases. In the general case, extremals will have two distinct type of comportment which correspond to those observed in the r…

Ising chains of spinsLiouville metricsCut LocusContrôle optimal géométrique[ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]Almost-Riemannian geometryChaînes de spins de type IsingGeometric optimal control[ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG]Conjugate Locus[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]Métriques de LiouvilleMétrique pseudo-riemannienneLieu conjugué[MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG]Lieu de coupure
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Geodesic flow of the averaged controlled Kepler equation

2008

A normal form of the Riemannian metric arising when averaging the coplanar controlled Kepler equation is given. This metric is parameterized by two scalar invariants which encode its main properties. The restriction of the metric to $\SS^2$ is shown to be conformal to the flat metric on an oblate ellipsoid of revolution, and the associated conjugate locus is observed to be a deformation of the standard astroid. Though not complete because of a singularity in the space of ellipses, the metric has convexity properties that are expressed in terms of the aforementioned invariants, and related to surjectivity of the exponential mapping. Optimality properties of geodesics of the averaged controll…

[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC]0209 industrial biotechnologyGeodesicGeneral MathematicsCut locusConformal map02 engineering and technologyKepler's equationFundamental theorem of Riemannian geometry01 natural sciencesConvexityIntrinsic metricsymbols.namesake020901 industrial engineering & automationSingularity0101 mathematicsorbit transferMathematicsApplied Mathematics010102 general mathematicsMathematical analysis[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]cut and conjugate lociRiemannian metrics49K15 70Q05symbols[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
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Conjugate and cut loci of a two-sphere of revolution with application to optimal control

2008

Abstract The objective of this article is to present a sharp result to determine when the cut locus for a class of metrics on a two-sphere of revolution is reduced to a single branch. This work is motivated by optimal control problems in space and quantum dynamics and gives global optimal results in orbital transfer and for Lindblad equations in quantum control.

[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC]0209 industrial biotechnologyWork (thermodynamics)Class (set theory)Quantum dynamicsCut locus02 engineering and technologySpace (mathematics)01 natural sciencesspace and quantum mechanicsoptimal control020901 industrial engineering & automationconjugate and cut loci0101 mathematics2-spheres of revolutionMathematical PhysicsMathematicsApplied Mathematics010102 general mathematicsMathematical analysis[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]53C20; 53C21; 49K15; 70Q05Optimal controlMetric (mathematics)[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]Orbital maneuverAnalysis
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