Search results for "optimization"
showing 10 items of 2824 documents
Stratification du secteur anormal dans la sphère de Martinet de petit rayon
2007
L’objectif de cet article est de fournir le cadre geometrique pour faire une analyse de la singularite de l’application exponentielle le long d’une direction anormale en geometrie sous-Riemannienne. Il utilise les calculs de [9], [12], et conduit dans le cas Martinet a une stratification de la singularite en secteurs Lagrangiens.
Continuous optimal control sensitivity analysis with AD
2000
In order to apply a parametric method to a minimum time control problem in celestial mechanics, a sensitivity analysis is performed. The analysis is continuous in the sense that it is done in the infinite dimensional control setting. The resulting sufficient second order condition is evaluated by means of automatic differentiation, while the associated sensitivity derivative is computed by continuous reverse differentiation. The numerical results are given for several examples of orbit transfer, also illustrating the advantages of automatic differentiation over finite differences for the computation of gradients on the discretized problem.
Optimal control of the atmospheric arc of a space shuttle and numerical simulations with multiple-shooting method
2005
This article, continuation of previous works, presents the applications of geometric optimal control theory to the analysis of the Earth re-entry problem for a space shuttle where the control is the angle of bank, the cost is the total amount of thermal flux, and the system is subject to state constraints on the thermal flux, the normal acceleration and the dynamic pressure. Our analysis is based on the evaluation of the reachable set using the maximum principle and direct computations with the boundary conditions according to the CNES research project\footnote{The project is partially supported by the Centre National d'Etude Spatiales.}. The optimal solution is approximated by a concatenat…
Une approche géométrique du contrôle optimal de l'arc atmosphérique de la navette spatiale
2002
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 est essentiellement géométrique et fondée sur une évaluation de l'ensemble des états accessibles en tenant compte des contraintes sur l'état. On esquisse une analyse des extrémales du Principe du …
Sensitivity analysis for time optimal orbit transfer
2001
The minimum time transfer of a satellite around the Earth is studied. In order to deal numerically with low thrusts, a new method is introduced: Based on a so-called noncontrollability function, the technique treats the ha1 time as a parameter. The properties of the method arc studied by means of an infinite dimensional sensitivity analysis. The numerical results obtained by this approach for very low thrusts are given
3D Geosynchronous Transfer of a Satellite: Continuation on the Thrust
2003
The minimum-time transfer of a satellite from a low and eccentric initial orbit toward a high geostationary orbit is considered. This study is preliminary to the analysis of similar transfer cases with more complicated performance indexes (maximization of payload, for instance). The orbital inclination of the spacecraft is taken into account (3D model), and the thrust available is assumed to be very small (e.g. 0.3 Newton for an initial mass of 1500 kg). For this reason, many revolutions are required to achieve the transfer and the problem becomes very oscillatory. In order to solve it numerically, an optimal control model is investigated and a homotopic procedure is introduced, namely cont…
Coplanar control of a satellite around the Earth
2001
We investigate the minimum time transfer of a satellite around the Earth. Using an optimal control model, we study the controllability of the system and propose a geometrical analysis of the optimal command structure. Furthermore, in order to solve the problem numerically, a new parametric technique is introduced for which convergence properties are established.
Asymptotics of accessibility sets along an abnormal trajectory
2001
We describe precisely, under generic conditions, the contact of the accessibility set at time $T$ with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer $\gamma$ into two sectors, bordered by the first Pontryagin's cone along $\gamma$, called the $\xLinfty$-sector and the $\xLtwo$-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.
Averaging and optimal control of elliptic Keplerian orbits with low propulsion
2006
This article deals with the optimal transfer of a satellite between Keplerian orbits using low propulsion. It is based on preliminary results of Geffroy [Generalisation des techniques de moyennation en controle optimal, application aux problemes de rendez-vous orbitaux a poussee faible, Ph.D. Thesis, Institut National Polytechnique de Toulouse, France, Octobre 1997] where the optimal trajectories are approximated using averaging techniques. The objective is to introduce the appropriate geometric framework and to complete the analysis of the averaged optimal trajectories for energy minimization, showing in particular the connection with Riemannian problems having integrable geodesics.
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…