Search results for "Mathematica"
showing 10 items of 7971 documents
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…
Riemannian metric of the averaged energy minimization problem in orbital transfer with low thrust
2007
Abstract This article deals with the optimal transfer of a satellite between Keplerian orbits using low propulsion and is based on preliminary results of Epenoy et al. (1997) where the optimal trajectories of the energy minimization problem are approximated using averaging techniques. The averaged Hamiltonian system is explicitly computed. It is related to a Riemannian problem whose distance is an approximation of the value function. The extremal curves are analyzed, proving that the system remains integrable in the coplanar case. It is also checked that the metric associated with coplanar transfers towards a circular orbit is flat. Smoothness of small Riemannian spheres ensures global opti…
Optimality results in orbit transfer
2007
Abstract The objective of this Note is to present optimality results in orbital transfer. Averaging of the energy minimization problem is considered, and properties of the associated Riemannian metric are discussed. To cite this article: B. Bonnard, J.-B. Caillau, C. R. Acad. Sci. Paris, Ser. I 345 (2007).
Second order optimality conditions in the smooth case and applications in optimal control
2007
International audience; 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 of Order Two and COnjugate times), available on the web, and apply it to the minimal time problem of orbit transfer, and to the attitude control problem of a rigid spacecraft. …
On local optima in minimum time control of the restricted three-body problem
2016
International audience; The structure of local minima for time minimization in the controlled three-body problem is studied. Several homotopies are systematically used to unfold the structure of these local minimizers, and the resulting singularity of the path associated with the value function is analyzed numerically.
Energy minimization of single input orbit transfer by averaging and continuation
2006
AbstractThis article deals with the transfer between Keplerian coplanar orbits using low propulsion. We focus on the energy minimization problem and compute the averaged system, proving integrability and relating the corresponding trajectories to a three-dimensional Riemannian problem that is analyzed in details. The geodesics provide approximations of the extremals of the energy minimization problem and can be used in order to evaluate the optimal trajectories of the time optimal and the minimization of the consumption problems with continuation methods. In particular, minimizing trajectories for transfer towards the geostationary orbit can be approximated in suitable coordinates by straig…
Conjugate times for smooth singular trajectories and bang-bang extremals
2003
Abstract In this paper we discuss the problem of estimating conjugate times along smooth singular or bang-bang extremals. For smooth extremals conjugate times can be defined in the generic case by using the intrinsic second order derivative or the exponential mapping. An algorithm is given which was implemented in the SR-case to compute the caustic [1] or in recent applied problems [5],[9]. We investigate briefly the problem of using this algorithm in the bang-bang case by smoothing the corners of extremals
Non subanalyticity of sub-Riemannian Martinet spheres
2001
Abstract Consider the sub-Riemannian Martinet structure (M,Δ,g) where M= R 3 , Δ= Ker ( d z− y 2 2 d x) and g is the general gradated metric of order 0 : g=(1+αy) 2 d x 2 +(1+βx+γy) 2 d y 2 . We prove that if α≠0 then the sub-Riemannian spheres S(0,r) with small radii are not subanalytic.
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.
Time Versus Energy in the Averaged Optimal Coplanar Kepler Transfer towards Circular Orbits
2015
International audience; The aim of this note is to compare the averaged optimal coplanar transfer towards circular orbits when the costs are the transfer time transfer and the energy consumption. While the energy case leads to analyze a 2D Riemannian metric using the standard tools of Riemannian geometry (curvature computations, geodesic convexity), the time minimal case is associated to a Finsler metric which is not smooth. Nevertheless a qualitative analysis of the geodesic flow is given in this article to describe the optimal transfers. In particular we prove geodesic convexity of the elliptic domain.